Commun. Math. Phys. 254, 1–22 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1240-2
Communications in
Mathematical Physics
Phase Transition in Dependent Percolation Christopher Hoffman Department of Mathematics, University of Washington, Seattle, WA 98195, USA. E-mail:
[email protected] Received: 28 November 2001 / Accepted: 19 August 2004 Published online: 25 November 2004 – © Springer-Verlag 2004
Abstract: In this paper we discuss two different models of dependent percolation on the graph Z2 . These models can be thought of as percolation in a random environment. They were inspired by the work of McCoy and Wu [7, 8] on the Ising model in a random environment as well as other models of particle systems in a random environment [9, 5, 6, 3]. We show that both models of dependent percolation exhibit phase transitions. This proves a version of stability for percolation on Z2 and proves a conjecture of Jonasson, Mossel and Peres [4], who proved a similar result on Z3 . 1. Introduction Since mathematical models only approximate physical systems the stability of the model is important. A small change in the model should ideally result in a small change in the qualitative behavior of the model. In this paper we consider the stability of phase transitions in percolation. In Bernoulli percolation the existence of a phase transition is both one of the most important results and one of the most elementary. However, in the models of dependent percolation that have been studied, establishing a phase transition has been often quite difficult. In this paper we show that two models of dependent percolation exhibit phase transitions. The pioneering work in studying the stability of statistical mechanics models was done by McCoy and Wu [7, 8]. They studied the Ising model in a random environment. Their models had interactions which depend only on a row or column. This produced models which were both tractable and had physically realistic results. In addition to the Ising model, people have studied other particle systems, such as the contact process and the voter model, in random environments [9, 5, 6, 3]. The models that we study in this paper are natural examples for percolation of this family of statistical mechanics models in a random environment.
Research supported in part by an NSF postdoctoral fellowship
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Let us also note that in addition to the stability of statistical mechanics models, phase transitions in dependent percolation are also related to problems in distributive computing. In 1990 Winkler made a conjecture about colliding random walks on graphs involving ‘clairvoyant demons’. Noga Alon reformulated this conjecture in terms of the existence of a phase transition in a simple model of oriented percolation. Before this conjecture was resolved progress was made on a number of related topics. Winkler [10] and Balister, Bollobas, and Stacey [1] independently proved that there exists a phase transition for unoriented percolation in this model. The first of our models was introduced by Jonasson, Mossel and Peres [4]. This model is on the graph with vertices Z2 and with edges between nearest neighbors (i.e., there is an edge between (x, y) and (x , y ) iff |x − x | + |y − y | = 1.) To define the probability that an edge is included in the graph (or open) we define two sequences of random variables. Let {xi }i∈Z
and
{yi }i∈Z
be i.i.d. random variables. They have geometric distribution with parameter 2−1000 . Thus P (xi > λ) = C(2−1000 )λ . Let p be a number 0 < p < 1. Then set Pp (e((i, j ), (i + 1, j )) is open) = pxi and Pp (e((i, j ), (i, j + 1)) is open) = pyi . Each edge is either open or closed independently of all other edges conditioned on the values given above. We refer to this model as percolation on the randomly stretched lattice in Z2 . It is given this name for the following reason. Consider a graph generated by the coordinate axis and lines parallel to the coordinate axis as follows. The distance between the i th line to the right of the y axis and the i +1st line is xi . The distance between the j th line to the left of the y axis and the j − 1st line is x−j . In a similar manner the {yi } determine the distance between the horizontal lines. Then the probability that the edge between two adjacent vertices is open is pd , where d is the distance between the vertices. All edges are open independently of each other. This generates the same percolation process as was described above. 2 For each p this defines a measure Pp on = {0, 1}E(Z ) . Let ω ∈ be a realization of percolation. An open cluster C for ω is a maximal connected subset of edges such that all edges e ∈ C are open (ω(e) = 1 for all e ∈ C). Jonasson, Mossel and Peres were able to show that a similar process on Z3 exhibits a phase transition [4]. That is they proved the following theorem. Theorem 1.1 [4]. For percolation on the randomly stretched lattice in Z3 there exists a value pc , 0 < pc < 1 such that if p < pc then Pp (∃ an infinite open cluster) = 0. If p > pc then Pp (∃ an infinite open cluster) = 1.
Phase Transition in Dependent Percolation
3
In this paper we establish that percolation on a randomly stretched lattice on Z2 also exhibits a phase transition. The proof of Jonasson, Mossel and Peres works to show that for the randomly stretched lattice in Z2 with p close to zero that there is no infinite cluster a.s. Thus we need only to show that for p close to one that there is an infinite cluster a.s. Before we do that we introduce a different model of dependent percolation which we call percolation on a regularly stretched lattice. This model will also exhibit a phase transition. We introduce this model because the proof of a phase transition is much simpler than on the randomly stretched lattice, yet it contains the essential ideas of the proof of a phase transition on the randomly stretched lattice. The only difference between these two models is the choice of the {xi }i∈Z and {yi }i∈Z . Instead of random variables as they were for the randomly stretched lattice they will be deterministic. Let x0 = y0 = 1. Given i let xi = yi = q if 4q−1 | i but 4q i. Then the edges are open or closed independently with the probabilities listed above. 2. Phase Transition on the Regularly Stretched Lattice In this section we show that the regularly stretched lattice exhibits a phase transition. First we show that for small p there is no infinite connected cluster. The proof is identical to the proof in [4] that percolation on a randomly stretched lattice does not have an infinite cluster when p is small. Theorem 2.1. On the regularly stretched lattice if p <
1 4
then
Pp (∃ an infinite connected cluster) = 0. Proof. For any j let δj =
(±4j , c), (±4j + 1, c) , |c| ≤ 4j (d, ±4j ), (d, ±4j + 1) , |d| ≤ 4j .
This group of edges separates a square containing 0 from infinity. Each edge in δj has length j . Then Pp (∃ an open edge in δj ) ≤ 4(2 · 4j + 1)p j ≤ 16(4p)j . Now suppose the vertex (a, b) is in an infinite cluster. Then if 4j > |a|, |b| there exists an open edge in δj . Thus by Borel Cantelli if p < 1/4 then Pp (∃ an infinite connected cluster) = 0.
To show that when p is close to 1 there is an infinite cluster we use an argument reminiscent of Chayes, Chayes, and Durrett’s proof that there exists crossings of a square in certain examples of fractal percolation [2]. First we define Bn (a, b) to be the set of vertices (i, j ) such that a < i ≤ a + 4n−1 and b < j ≤ b + 4n−1 . An n box is a graph that has vertices Bn (a, b), where 4n |a and 4n |b. It consists of all edges between two vertices in Bn (a, b). If a = 0 and b = 0 and Bn (a, b) is an n box then all the edges emanating from Bn (a, b) have length n in the regular stretch lattice.
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Now we will inductively define what it means for an n box to be good. Let ω be a realization of percolation. A 1 box is good if every edge in the 1 box is open. Assume we have defined a good n − 1 box in such a way that if an n − 1 box is good then there exists a unique cluster which intersects all four sides of the n − 1 box. This is called the crossing cluster. An n box is good if 1. at least 15 of the 16 n − 1 boxes inside the n box are good and 2. between each pair of adjacent n − 1 boxes which are good there exists an open edge which connects the crossing clusters of the good n − 1 boxes. It is easy to check that a good n box also contains a crossing cluster. Lemma 2.1. Every good n box has a crossing cluster. Proof. This follows easily by induction. The crossing cluster of the good n box contains the union of the crossing clusters of the good n − 1 boxes. The key idea in the proof of the phase transition for the regularly stretched lattice is the following lemma. Lemma 2.2. Let Bn (a, b) and Bn (a + 4n , b) be good n boxes. There are at least 2n edges which intersect the crossing clusters of both Bn (a, b) and Bn (a + 4n , b). Proof. Each pair of adjacent n boxes, Bn (a, b) and Bn (a + 4n , b), contains four pairs of adjacent n − 1 boxes, Bn (a + 3 · 4n−1 , b + i4n−1 ) and Bn (a + 4n , b + i4n−1 ) for i = 0, 1, 2, 3. If both Bn (a, b) and Bn (a + 4n , b) are good then at least six of the above n − 1 boxes are good. Thus for at least two of the above pairs both of the adjacent n − 1 boxes are good. By the induction hypothesis each of these pairs has at least 2n−1 edges which connect the crossing clusters of the n − 1 boxes and thus connect the crossing clusters of the n boxes. Thus there are at least 2n edges which connect the crossing clusters of Bn (a, b) and Bn (a + 4n , b). Lemma 2.3. If p > 1 − 2−1000 then for any n box Bn (a, b), 1 −n e . 1000 Proof. The proof is by induction. The lemma is clearly true when n = 10. The probability that the first condition for any n box to be not good is less than 2 1 1 −n 16 2 −2n+2 . e ≤ .5 e Pp (Bn−1 (a, b) is not good) ≤ 120 1000 1000 2 Pp (Bn (a, b) is good) ≥ 1 −
If Bn−1 (a, b) and Bn−1 (c, d) are neighboring n − 1 boxes which are both good then there must be at least 2n pairs of vertices, one in each of the crossing clusters, which are separated by one edge. This is true by Lemma 2.2. Each of these edges is open with probability pn . The probability that all 2n are closed is (1 − p n )2 ≤ 2−(2 n
n pn )
≤ 2−1.5 . n
There are 16 in an n box. Thus the probability that 2 pairs of adjacent n − 1 boxes −1.5n Condition 2 is not satisfied is less than 16 . Thus 2 2 16 −1.5n 1 −n 1 −n P (Bn (a, b) is good) ≥ 1 − .5 e e . − 2 ≥1− 1000 1000 2 From this it is easy to conclude that there is an infinite open cluster.
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Theorem 2.2. If p > 1 − 2−1000 then Pp (∃ an infinite open cluster) = 1. Proof. There exists some N such that Bn (0, 0) is good for all n ≥ N a.s. Thus there is some vertex (i, j ) which is contained in a good n box for all n. Thus it is in the crossing cluster for all n and is connected to an infinite number of other vertices. 3. Creating a Regular Lattice from a Random Lattice In trying to adapt the proof in the previous section to the case of the randomly stretched lattice we have two main obstacles to overcome. First we must figure out how to fit a regular lattice onto the random {xi } and {yi }. That is the subject of this section. Then in the next section we check to see that the proof still applies to the lattice structure that we imposed. Each integer y defines a row which is the set of all edges that connect (x, y) to (x, y + 1) for some x. Each integer x defines a column which is the set of all edges that connect (x, y) to (x + 1, y) for some y. In the regularly stretched lattice there were rows and columns which it was hard for the percolation cluster to get through. On the randomly stretched lattice we will group neighboring rows and columns into bands. In the regular lattice the row generated by i was assigned the number yi . In this model each band will be assigned a number. A higher number indicates that that there are fewer paths across the band. We will assign the rows (and columns) into bands in such a way that if two bands are both assigned a high number then they must be exponentially far apart. This will allow us to mimic the proof in the previous section. Starting with {xi } we will construct a sequence of functions fl : Z → N. An l band is a collection of consecutive integers [i, j ]. A necessary condition for [i, j ] to be an l band is that fl (i) = fl (i + 1) = · · · = fl (j ). A band is an interval of integers which is an l band for all l greater than some L. Each i ∈ Z is a 1 band. We assign the 1 bands labels by f1 (i) = xi , the smallest integer greater than or equal to xi . We also use the notation xi for the largest integer less than or equal to xi . We now begin the inductive procedure in which we combine multiple n bands into a single n + 1 band and assign this band a higher number. Find i such that |i + .1| is the smallest for which there exists j , with the following properties: 1. j = i, 2. |j | ≤ |i|, and 3. min(f1 (i), f1 (j )) −
1 6
log2 |i − j | > 1.
The 2 bands then consist of the interval [j, i] (or [i, j ] if i < j ) and every integer k ∈ [j, i]. We set
k ∈ [j, i] f1 (k) f2 (k) = 1 log2 |i − j | k ∈ [j, i]. f1 (i) + f1 (j ) − 18 Now assume that n bands and fn have been defined. Let i be the integer such that |i + .1| is the smallest for which there exists a j with the following properties:
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1. j is not in the same n band as i, 2. |j | ≤ |i|, and 3. min(fn (i), fn (j )) − 16 log2 (1 + # of n bands between i and j ) > 1. Then the n + 1 bands are the union of the n bands intersecting the interval [i, j ] and every n band not intersecting the interval [i, j ]. We set
fn+1 (k) =
fn (k) n band of k ∩ [i, j ] = ∅ 1 log (1 + # of n bands in (i, j )) n band of k ∩ [i, j ] = ∅ . fn (i) + fn (j ) − 18 2
Let f (k) = lim fn (k) and let a band be an interval of integers for which there is an N n→∞ such that the interval was an n band for all n > N . We will now work towards proving that f (k) exists and is finite for all k a.s. Lemma 3.1. If [j, k] is an n band and fn (j ) = m then |k − j + 1| ≤ 32m−1 . Proof. The proof is by induction on m. The statement is true for m = 1 and m = 2 because for these we must have j = k. Now suppose [j, k] is an n band, j = k, and fn (j ) = m > 2. Then there exists some n such that [j, k] is an n band but [j, k] is not an n − 1 band. Let j , k , p, and q be defined so that [j, j ] and [k , k] are n − 1 bands, p = fn −1 (j ) and q = fn −1 (k). Then p, q < m so the induction hypothesis holds for all n − 1 bands in [j, k]. For any l < min(p, q) there are at least 64l−1 − 1 n − 1 bands in between any two n − 1 bands in [j, k] each with labels ≥ l. Let N = the number of n − 1 bands between [j, j ] and [k , k]. Then there are at most N/64l−1 n − 1 bands with labels l between [j, j ] and [k , k]. Thus |k − j + 1| = |k − k + (k − j ) + (j − j ) + 1| |b − b + 1| ≤ |k − k + 1| + |j − j + 1| + n −1 bands in (j ,k )
≤ |k − k + 1| + |j − j + 1| + ≤ 32q−1 + 32p−1 +
l
|b − b + 1|
n −1 bands in (j ,k ) with label l
(N/64l−1 )32l−1
l
≤ 32q−1 + 32p−1 + 2N ≤ 2 · 32max(p,q)−1 + 26 min(p,q)−5 1 m−1 ≤ 32 + 26 min(p,q)−5 16 1 m−1 1 (15/2) min(p,q)−5 + 2 ≤ 32 16 2 1 m−1 1 (3/2) min(p,q)−1 ≤ 32 + 32 16 2 1 m−1 1 m−1 32 ≤ + 32 16 2 ≤ 32m−1 .
(1) (2) (3)
(4)
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Line 1 is true because min(fn −1 (j ), fn −1 (k)) − 16 log2 (1 + N ) > 1. Line 2 is true because m ≥ max(p, q) + 1. Line 3 is true because min(p, q) ≥ 2. Line 4 is true because m ≥ 23 min(p, q). Lemma 3.2. If [j, j ] and [k , k] are two n bands which are combined to form an n + 1 band [j, k] then N = the number of n bands in (j , k ) ≥ (k − j − 1)/2.
Proof. Let Nl be the number of n bands in (j , k ) with labels l. Then N = Nl . Also Nl ≤ N/64l−1 because if not then there would have been two other n bands which would have combined (b − b + 1) k − j − 1 = [b,b ] a band in (j ,k )
≤ N1 + N 2 +
(N/64l−1 )32l−1
l≥3
≤ N1 + N2 + N/2 ≤ 2N so N ≥ (k − j − 1)/2.
Before we proceed we must introduce some more notation. If the interval [j, k] is an n band then we say j and k are the n generators of the n band. The n − 1 generators of the n band [j, k] are the n − 1 generators of the n − 1 bands containing j and k. Suppose for all n, all n bands and some i the n − i generators of an n band have been defined. Then for any n band define the n − (i + 1) generators of the n band as the union of the n − (i + 1) generators of the n − (i + 1) bands which contain an (n − i) generator of the n band. We call the 1 generators of an n band the generators of an n band. We say that a generator g of a band [i, j] is a maximal generator of [i, j] if the following condition is satisfied. For each pair of k bands [i1 , j1 ] and [i2 , j2 ] which form the k + 1 band [i1 , j2 ] such that g ∈ [i1 , j2 ], the label of the k band that contains g is at least as big as the label of the k band that doesn’t contain g. x = B We define B0,k 0,k to be the k band containing 0 for x. For a fixed k we inductively define Bi,k . If Bi−1,k = [j, j ] then we let Bi,k be the k band containing j + 1. In an analogous manner we define Bi,k for i < 0. We also define Bix in the analogous manner using the bands of x. Lemma 3.3. Let i1 < i2 < · · · < ik be the generators of an n band with
k
f1 (ij ) = m.
1
Then k
log2 (ij − ij −1 + 1) ≤ 37m.
2
Proof. For each j , 1 < j ≤ k, let nj be the value such that there exists a and b so nj bands [ia , ij −1 ] and [ij , ib ] are merged into nj +1 band [ia , ib ]. Let qj be the number of nj bands between [ia , ij −1 ] and [ij , ib ]. By the previous lemma, 1 (ij − ij −1 + 1) ≤ 1 + qj . 2
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C. Hoffman
By the assumption that
k
f1 (ij ) = m we have that
1 k 1 log2 (1 + qj ) ≤ m. 18 j =2
Thus k 1 1 log2 ( (ij − ij −1 + 1)) ≤ m, 18 2 k 2
2
1 1 log2 ( (ij − ij −1 + 1)) − 1 ≤ m, 18 2 k 2
1 log2 ( (ij − ij −1 + 1)) ≤ 36m, 2 k
log2 (ij − ij −1 + 1) ≤ 37m.
2
Lemma 3.4. For any j and l, P (∃n such that j is contained in an n band with label ≥ l) ≤ 2−399l . Proof. If there exists an n and l then ∃ generators i1 , . . . , ik and m such that f1 (ik ) = m ≥ l. k
By Lemma 3.1 we have that i1 ≤ j ≤ i1 + 32l−1 − 1. By Lemma 3.3 there are at most 237m choices of log2 (i2 − i1 + 1) , . . . , log2 (ik − ik−1 + 1) . Given a choice of log2 (i2 − i1 + 1) , . . . , log2 (ik − ik−1 + 1) there are at most 237m choices of (i2 − i1 ), . . . , (ik − ik−1 ). Thus there are at most 32l−1 (274m ) ≤ 279m choices of i1 , . . . , ik . For each choice of i1 , . . . , ik there are at most 2m choices for f1 (i1 ), . . . , f1 (ik ). Thus there are at most 280m choices of i1 , . . . , ik and f1 (i1 ), . . . , f1 (ik ). Each choice has probability ≤ 2−1000(m−k) ≤ 2−500m . Thus P (∃n such that j is contained in an n band with labels ≥ l) ≤ 2−400m m≥l −399l
≤2
.
Lemma 3.5. For each j ∈ Z there exists N such that for all n > N the n band containing j is the same a.s. Proof. This follows easily from Lemma 3.4 and Borel-Cantelli.
We say that two (n) bands [i1 , j1 ] and [i2 , j2 ] with labels ≥ l are neighboring (n) bands with labels ≥ l if there exists no (n) band [i3 , j3 ] with j1 < i3 ≤ j3 < i2 with label ≥ l.
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Lemma 3.6. If Bm and Bm are neighboring bands with labels ≥ l then |m−m | ≥ 64l−1 . Proof. If not then the bands would have been combined at some level.
We call a sequence {xi } regular if the procedure above generates bands such that for all l and all neighboring bands, Bi and Bi , with labels greater than or equal to l, [i1 , j1 ] and [i2 , j2 ], we have |i − i | ∈ [64l−1 , 6 · 64l−1 ). It is easy to check that if {xi } is regular then for any l and between any two bands neighboring bands with labels ≥ l there exists between 10 and 383 bands with labels l − 1. Now we want to show that for almost every sequence {xi } there exists a sequence {x˜i } such that {x˜i } is regular and x˜i ≥ x˜i for all i. This is Lemma 3.9. To do this we will use the following lemma repeatedly. Lemma 3.7. If for some sequence {xi }, i and some n, 1. i is a maximal generator of Bn , 2. the label of Bn is l, 3. for all m, n such that the label of Bm and Bn are greater than l, we have |m−n| ≥ 64l , then the sequence
x˜i =
i = i
x˜i
x˜i + 1 i = i
has the following properties: x = B x˜ for all k and m, 1. Bm,k m,k x˜ equals the label of B x , and 2. for all m, k such that i ∈ Bm,k the label of Bm,k m,k x ˜ x . 3. for all m, k such that i ∈ Bm,k the label of Bm,k is one greater than the label of Bm,k
Proof. This follows easily by induction.
Lemma 3.8. For almost every sequence {xi } there exists sequences {xiL } such that 1) xiL ≥ xi for all i, x L = B x for all n, k and L, and 2) Bn,k n,k L
L
3) for all l ≤ L and n, m such that Bnx and Bnx are neighboring bands with labels ≥ l, |n − m| ∈ [64l−1 − 1, 3 · 64l−1 ). Furthermore we can do this construction such that {xiL } is nondecreasing for all i and is unbounded for at most one i. Proof. We prove this by induction on L. It is true for L = 1 with xi1 = xi for all i. Suppose the lemma is true for L. L L Let S(L) = {s : the label of Bsx ≥ L} and S (L) = {s : the label of Bsx ≥ L + 1}. ˜ Next we show that there exists a set S(L) such that ˜ ⊂ S(L), 1. S (L) ⊂ S(L) ˜ 2. |s − t| ≥ 64L for any s, t ∈ S(L), ˜ ˜ 3. for any s ∈ S(L) there exists s ∈ S(L) with s − s ≤ 3 · 64L .
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˜ For any s, t ∈ S (L) with S (L) ∩ (s, t) = ∅ we construct S(L) ∩ (s, t). We do this inde˜ pendently for each such pair s and t. If t − s ≤ 3 · 64L then we set S(L) ∩ (s, t) = ∅. If not we let s1 be the smallest element of S(L) ∩ [s + 64L , t). By the induction hypothesis we have that s1 ∈ [s + 64L , s + 64L + 3 · 64L−1 ). Thus t − s1 ≥ s + 3 · 64L − (s + 64L + 3 · 64L−1 ) ≥ 2 · 64l − 3 · 64L−1 > 64L . ˜ ˜ we still have that all elements of S(L) are separated by at least If we include s1 in S(L) 64L . If t − s1 > 3 · 64L then we find s2 ∈ S(L) ∩ [s1 + 64L , s1 + 64L + 3 · 64L−1 ). We repeat this procedure until we get sk such that t − sk ≤ 3 · 64L . Then we set ˜ ˜ S(L) ∩ (s, t) = {s1 , . . . , sk }. For each s˜ ∈ S(L) \ S (L) we pick as˜ a maximal generator L x of Bs˜ . Then we set
xiL+1
=
xiL + 1 xi
˜ if i = as˜ for some s˜ ∈ S(L) \ S (L) else.
By Lemma 3.7 the induction hypothesis is satisfied for L + 1. For any i the sequence {xiL } is clearly nondecreasing in L. As the bands for x L are independent of L there is at most one band [j, k] such that where the sequences may be unbounded. If we make a consistent choice of the maximal generator of this band then the sequence {xiL } is unbounded for at most one i. Lemma 3.9. For almost every sequence {xi } there exists a sequence { xi } such that 1) xi ≥ xi for all i, 2) the k bands for { xi } are the same as the k bands for {xi } for all k, and 3) { xi } is regular. Proof. By Lemma 3.8 we get a nondecreasing (in L) sequence of sequences {xiL }. We also have that limL xiL exists for all i except possibly one value of i. Call that value i . Set
limL xiL i = i xi = xi i = i. Then this sequence has the property that between any two neighboring bands with labels ≥ l the number of bands in between is in the interval [64l−1 , 6 · 64l−1 ). Such a sequence is regular and the lemma is true. When we defined what it means for a sequence to be regular we were concerned with neighboring bands with labels ≥ l being too far apart. We now define what it means for a sequence to be very regular. In addition to the bands being regular we demand a regularity condition of the k bands that are combined in order to form a k + 1 band. A vertical band is a band that is generated from the {xi }. A horizontal band is a band that is generated from the {yi }. Suppose [i1 , j1 ] and [i2 , j2 ] are neighboring horizontal bands with label ≥ n and [i3 , j3 ] and [i4 , j4 ] are neighboring vertical bands with label ≥ n. Then the graph with vertices V = {(x, y) : j3 + 1 ≤ x ≤ i4 , j1 + 1 ≤ y ≤ i2 }
Phase Transition in Dependent Percolation
11
and edges E = {edges between two vertices in V with at most one edge in δV } is called an n box. We write [j3 + 1, i4 ] × [j1 + 1, i2 ] is an n box. We will also call any subgraph [j3 + 1, i4 ] × [j1 + 1, i2 ] an n box if there exists {x¯i } and {y¯i } which are regular such that x¯m = xm
for all m such that j3 < m ≤ i4 ,
y¯m = ym
for all m such that j1 < m ≤ i2 ,
and [j3 + 1, i4 ] × [j1 + 1, i2 ] is an n box (for {x¯i } and {y¯i }). Suppose [i5 , j5 ] is a horizontal (vertical) band with label n and [j3 +1, i4 ]×[j1 +1, i2 ] is an m box. Then we say (j3 , i4 ] × [i5 , j5 + 1] ([i5 , j5 + 1] × [j1 + 1, i2 ]) is a horizontal (vertical) (m, n) strip. This graph has vertices V = {(x, y) : j3 + 1 ≤ x ≤ i4 , i5 ≤ y ≤ j5 + 1} and edges E = {edges between two vertices in V with at most one edge in δV }. We will also call any subgraph [j3 + 1, i4 ] × [i5 , j5 + 1] an (m, n) strip if there exists {x¯i } and {y¯i } which are regular such that x¯m = xm
for all m such that j3 < m ≤ i4 ,
y¯m = ym
for all m such that i5 < m ≤ j5 ,
and [j3 + 1, i4 ] × [i5 , j5 + 1] is an (m, n) strip (for {x¯i } and {y¯i }). Any k band of the form [a, a] is very regular. Any 1 box is very regular. Any 2 box [a, b] × [c, d] is very regular if 11 ≤ b − a ≤ 384 and 11 ≤ d − c ≤ 384. We now inductively define what it means for a k band with label l to be very regular and for an n box to be very regular. Let [a, b] be a k band with label l which was formed by combining the k − j band [a, a ] and [b , b] into the k − j + 1 band [a, b]. We say that [a, b] is very regular if there exists m ≤ 384, c1 = a , c2 , c3 , . . . , cm , d1 , d2 , . . . , dm = b , and q such that 1) all k − j bands in [a, b] are very regular k − j bands, 2) [ct , dt ] × [ct , dt ] in {xi } × {xi } is a very regular q box for all t, and 3) [dt , ct+1 − 1] is a very regular k − j band with label q. An l box is very regular if 1) {xi } and {yi } are regular and 2) all bands with labels l − 1 and l − 1 boxes inside the l box are very regular. We say a sequence {xi } is very regular if all the bands generated by {xi } are very regular. Lemma 3.10. If 1) Ba,k is a k band with label l, 2) a is a maximal generator of Ba,k , 3) the label of the j band containing Ba,k is lj , 4) for all j > k such that a is a generator of the j band containing Ba,k the 64lj j bands on either side of the band containing Ba,k have label < lj + 1,
12
C. Hoffman
and we set
xi =
xi xa + 1
if i = a , if i = a
then we have i) the j bands of {xi } and the j bands of {x i } are the same for all j , ii) all j bands of {x i } that have a as a generator have label one greater than they did for {xi }, and iii) all j bands of {x i } that do not have a as a generator have the same label as they did for {xi }. Proof. We prove the lemma by induction on j . It is clearly true for j = 1. We assume that it is proven for j . By Condition 4 and ii) and iii) for j we have that i) holds for j + 1. As the j bands are the same, ii) and iii) for j imply that ii) and iii) hold for j + 1. Lemma 3.11. If in addition to the hypotheses in Lemma 3.10 we add 5) Ba,k is very regular (for {xi }), then we get the additional conclusion iv) Ba,k is very regular (for {x i }). Proof. We can prove the lemma by proving that for any j all j bands in Ba,k are very regular. We do this by induction. It is clearly true for j = 1. Assume it is proven for j . All the j bands are very regular since Ba,k is very regular (for {xi }). All the other conditions are also satisfied since Ba,k is very regular (for {xi }). Thus the j + 1 bands are regular (for {x i }). xi } such that Lemma 3.12. For almost every sequence {xi } there exists a sequence { 1) xi ≥ xi for all i and 2) { xi } is very regular. j
Proof. By Lemma 3.9 we may assume that {xi } is regular. We construct sequences {xi } j j such that {xi } is nondecreasing as j increases. The k bands of {xi } are the same as the k j bands of {xi } for all j and k. Also the k bands of {xi } are very regular for all k ≤ j . We do this inductively. It can be done for j = 1. Suppose it can be done for j . Then there is at most one j + 1 band which is not a j band. We define a, b, and c so that the j bands between Ba,j and Bb,j are combined to form the j + 1 band Bc,j +1 . Hence Bc,j +1 is j the only j + 1 band which might not be very regular for {xi }. We now employ the method of Lemma 3.8 (with Lemma 3.7 replaced by Lemma j +1 j +1 3.11) to construct {xi } so that Bc,j +1 is very regular for {xi }. To do this we conj,L struct a (finite) sequence {xi } which is nondecreasing in L. We also want to show inductively that x x 1. Bs,j = Bs,j for all s, j and L, 2. all j bands are still very regular for {x j,L } and, 3. for all l ≤ L, m, m ∈ [a, b], such that Bm,j and Bm ,j are neighboring j bands with labels ≥ l we have that |m − m | ∈ [64l−1 , 3 · 64l−1 ). j,L
Phase Transition in Dependent Percolation
Then we set
j +1
xi
13
j,L
= max xi L
.
This will allow us to conclude that Bc,j +1 is a very good j + 1 band. j,1 j The induction hypothesis is true for L = 1 with xi = xi for all i. Suppose it is true for L. First we define the sets j,L
x S(L) = {s ∈ [a, b] : the label of Bs,j ≥ L}
and
x S (L) = {s ∈ [a, b] : the label of Bs,j ≥ L + 1}. j,L
˜ ˜ ⊂ S(L). For any As in Lemma 3.8 we will define a set S(L) such that S (L) ⊂ S(L) ˜ s, t ∈ S (L) with S (L)∩(s, t) = ∅ we construct S(L)∩(s, t). We do this independently ˜ for each such pair s and t. If t − s ≤ 3 · 64L then we set S(L) ∩ (s, t) = ∅. If not we let s1 be the smallest element of S(L) ∩ [s + 64L , t). By the induction hypothesis we have that s1 ∈ [s + 64L , s + 64L + 3 · 64L−1 ). Thus t − s1 ≥ s + 3 · 64L − (s + 64L + 3 · 64L−1 ) ≥ 2 · 64L − 3 · 64L−1 > 64L . ˜ ˜ If we include s1 in S(L) we still have that all elements of S(L) are separated by at least 64L . If t − s1 > 3 · 64L then we find s2 ∈ S(L) ∩ [s1 + 64L , s1 + 64L + 3 · 64L−1 ). We repeat this procedure until we get sk such that t − sk ≤ 3 · 64L . Then we set ˜ ˜ S(L) ∩ (s, t) = {s1 , . . . , sk }. For each s˜ ∈ S(L) \ S (L) we pick as˜ , a maximal generator j,L x of Bn . Then we set
j,L ˜ xi + 1 if i = as˜ for some s˜ ∈ S(L) \ S (L) j,L+1 xi = j,L xi else. By Lemma 3.11 we have that all j bands in Bc,j +1 are very good for {x j,L+1 } and are the same as the j bands for {x}. We also have that for all l ≤ L + 1 and m, m ∈ [a, b] such that Bm,j and Bm ,j are neighboring j bands with labels ≥ l we have that |m − m | ∈ [64l−1 , 3 · 64l−1 ). Thus the inductive hypothesis is true for L + 1. We repeat this procedure until we have a value q such that |S(q)| ≤ 384 and |S (q)| = 2. It is easy to check that |S(L)| is decreasing. Also if |S(L)| > 384 and |S (L)| = 2 then |b − a| > 384 · 64L−1 > 3 · 64L ˜ and |S(L)| = |S(L + 1)| > 2. Thus this process terminates with a q such that 2 < |S(q)| ≤ 384. We set j +1 j,q xi = xi . Now we check that the j + 1 band Bc,j +1 is very regular for x j +1 . By construction the x j +1 } j bands in [a, b] are very regular. The j bands {Bs,j s∈S(q)\[a,b] satisfy the second and third conditions in the definition of very regular. Thus Bc,j +1 is very regular for x j +1 . Finally set j xi = lim xi . j
j
The sequence { xi } is very regular as all the j bands in {xi } are very regular. It dominates j {xi } since each of the {xi } dominates {xi }. Thus the lemma is true.
14
C. Hoffman
4. Phase Transition on the Randomly Stretched Lattice We show that if {xi } and {yi } are very regular then the stretched lattice determined by {xi } and {yi } has an infinite cluster for p close to 1 a.s. We will do this by mimicking the proof that percolation on the regularly stretched lattice has an infinite cluster for p close to 1 a.s. This is enough to establish a phase transition as Jonasson, Mossel, and Peres have proven that if p is sufficiently close to one then there is no infinite cluster a.s. [4]. 2 Fix {xi } and {yi } which are very regular. Let ω ∈ {0, 1}E(Z ) be the realization of percolation. We will inductively define what it means for an n box to be good for ω. A cluster C in a subgraph S is a maximal connected subgraph of S such that ω(c) = 1 for all c ∈ C. For an n box a crossing cluster is a cluster in the n box which contains vertices on all four edges of the n box. Inductively it will be clear that every good n box contains a crossing cluster. A crossing of a horizontal (m, n) strip [a, b] × [c, d] is a cluster in the (m, n) strip which contains at least one vertex in [a, b] × [c] and at least one vertex in [a, b] × [d]. An n ≤ 200 box is good if all the edges in the n box are included in ω. Given an n box, n > 200, suppose we have defined what it means for an n − 1 box to be good and it being good implies that the n − 1 box contains a crossing cluster. Let a1 = the number of n − 1 boxes in the n box which are not good, and a2 = the number of pairs of neighboring good n−1 boxes such that the (n − 1, n − 1) strip between them doesn’t have a crossing that intersects the crossing clusters of the good n − 1 boxes. Then the n box is good if a1 + a2 is at most one. We define a (9, m) tree in a horizontal (m, n) strip inductively in m. A (9, m) tree is a set of vertices in the boundary of the (m, n) strip. For any n and any (2, n) strip [a, b] × [c, d] and any set I of 9 elements in [a, b] we define two (9, 2) trees T and T in a (2, n) strip by T = {(i, j ) : i ∈ I, j = c} and T = {(i, j ) : i ∈ I, j = d}. Each (m, n) strip contains at least (9, m) disjoint (m − 1, n) strips. A 9 tree in an (m, n) strip is the union of nine (9, m − 1) trees in disjoint (m − 1, n) strips within the (m, n) strip. Thus each (9, m) tree in an (m, n) strip consists of 9m−1 vertices. For any m < m the 9m−1 vertices are in 9m−m (9, m ) trees in disjoint (m , n) strips. Lemma 4.1. Every pair of good n boxes separated by an (n, n) strip defines at least one (9, n) tree on each side of the (n, n) strip. Proof. The proof is by induction. Every pair of good 2 boxes separated by a (2, 2) strip has at least 9 pairs of vertices, one in each of the 2 boxes, such that every pair is separated by one edge. This forms a (9, 2) tree in the (2, 2) strip. Every pair of good n boxes separated by an (n, n) strip has at least 11 pairs of n − 1 boxes, one in each of the n boxes, such that every pair is separated by an (n − 1, n − 1) strip. In each of the good n boxes at least 10 of the 11 n − 1 boxes are good. Thus every pair of good n boxes separated by an (n, n) strip has at least 9 pairs of n − 1 boxes, one in each of the boxes, such that every pair is separated by an (n − 1, n − 1) strip and both the n − 1 boxes are good. With the induction hypothesis this forms a (9, n) tree in the (n, n) strip. Many of our calculations will require the following lemma.
Phase Transition in Dependent Percolation
15
Lemma 4.2. For any c, p1 , . . . , pn , 0 < pi < 1, and a = 1−
n 1
Proof. We have that
n 1
n 1
pi ,
a (1 − pi ) ≥ min 1 − e−c , (1 − e−c ) . c
−pi = −a and n
n 1
ln(1 − pi ) ≤ −a. Thus
(1 − pi ) ≤ e−a
1
and
n 1 − (1 − pi ) ≥ 1 − e−a . 1
The inequality 1 − zd ≥ d(1 − z) holds for 0 ≤ z, d ≤ 1. If a ≤ c then applying this with z = e−c and d = a/c gives 1−
n
(1 − pi ) ≥ 1 − e−a = 1 − (e−c )a/c ≥
1
a (1 − e−c ). c
For any set T ∈ Z2 define R(T ) = {(x, y) ∈ Z2 : there exists x ∈ Z such that (x , y) ∈ T }. For any set V ∈ Z define R(V ) = {(x, y) ∈ Z2 : y ∈ V }. We can define C(T ) and C(V ) in an analogous manner. For the rest of the paper we fix the following. Let B be any n box. Let B1 = [a1 , b1 ]× [c1 , d1 ] and B2 = [a1 , b1 ] × [c2 , d2 ] be good n boxes separated by an (n, n) strip. Let S¯ = [a1 , b1 ] × [d1 , c2 ] be the (n, n) strip between B1 and B2 . Let R1 ⊂ R(d1 ) and R2 ⊂ R(c2 ) be two (9, n) trees defined by the crossing clusters of B1 and B2 . Let ˜ × [c, ˜ be a ( 2n/3 , n) strip, T ⊂ R(d) ˜ be a (9, 2n/3 ) tree in S and S = [a, ˜ b] ˜ d] T ⊂ C(T ) ∩ R(c) ˜ be a collection of (9, k) trees in S with k ≤ 2n/3 . Lemma 4.3. There exists pc < 1 such that if p > pc then 1. Pp (B is good) > 1 − 4−n , and |T | 2. Pp (∃ a crossing of S that intersects T and T ) > 9 2n/3 , −n ¯ 3. P (∃ a crossing of S intersecting R1 and R2 ) ≥ 1 − 4 . The proof is by induction with base case n = 200. Statement 1 is similar to the induction hypothesis for the regularly stretched lattice. Statement 3 is also similar to a step for the regularly stretched lattice. It does not work as an inductive step. Statement 2 is introduced because it is possible to induct on this Statement. Statement 3 then follows ˜ the proof of 2 is a simple easily from Statement 2. When the height of S is one (c˜ = d) calculation. The proof of Statement 2 when the height of S is greater than one is the most
16
C. Hoffman
complicated part of the proof of Lemma 4.3. Before we prove Lemma 4.3 we require a few additional lemmas. Now we consider the case that the height of S is greater than one. Because {xi } and {yi } are very regular S has the following structure. We can break S up into 3 parts. On ˜ × [c, the bottom we have a ( 2n/3 , m) strip S1 = [a, ˜ b] ˜ c˜ ]. On the top we have a ˜ ˜ ˜ ˜ b] × [d , d]. In the middle are up to 384 rows of q boxes ( 2n/3 , r) band S2 = [a, separated by l bands with labels q. By the definition of very regular and the way the labels were assigned to bands m, r > q and m + r − q/3 − n = 0 or 1.
(5)
2n/3 > 2m/3 + 2r/3 − q + 30.
(6)
In particular if q > 100 then
The outline of our argument is as follows. If 1. there are “enough” crossings of S1 which intersect T , 2. there is at least one crossing of S2 which intersects T , 3. there exists v contained in a crossing of S1 which intersects T and w contained in a crossing of S2 which intersects T such that v and w are contained in a column of q boxes, and 4. v and w are connected, then there exists a crossing of S intersecting T and T . In Lemma 4.4 we bound from below the probability that there is at least one crossing of S1 intersecting T . We define what “enough” is in line 7. In Lemma 4.5 we give a lower bound on the probability of 1. Then we use Lemma 4.4 to bound the probability of 2 and 3 conditioned on 1 occurring. Then we bound the probability of 4 in Lemma 4.6. Then the proof of Lemma 4.3 part 2 is done by combining the above bounds. Let S = [a , b ] × [c , d ] be a (J, j ) strip with j ≤ n and J ≥ 2j/3 . Let Sˆ = ∪Sˆi be a union of (l, j ) strips in S . We write Sˆi = [fi , gi ] × [c , d ]. Let T¯ ⊂ R(d ) be a (9, J ) tree in S¯ which intersects each Sˆi in a (9, 2j/3 ) tree. Let Tˆ ⊂ C(T¯ ) ∩ R(c ) be a union of (9, l) trees in disjoint (l, j ) strips in Sˆ where l ≤ 2j/3 . Lemma 4.4. Suppose the conclusions of Lemma 4.3 are satisfied for j ≤ n − 1. Then ˆ| | T P (∃ a crossing of Sˆ intersecting Tˆ and T¯ ) ≥ min .9, . 3 · 9 2j/3 Proof. Tˆ is the union of (9, l) trees so Tˆ = Ti , where each Ti is a union of (9, l) trees in ( 2j/3 , j ) strip Si . By the induction hypothesis we have P (∃ a crossing of Sˆi intersecting Tˆi and T¯ ) ≥ By Lemma 4.2 with c = 3,
|Tˆi | 9 2j/3
.
|Tˆi | P (∃ a crossing of Sˆ intersecting Tˆ and T¯ ) ≥ min 1 − e−3 , 3 · 9 2j/3 |Tˆ | ≥ min .9, . 3 · 9 2j/3
Phase Transition in Dependent Percolation
17
We have defined a ( 2n/3 , n) strip S, a (9, 2n/3 ) tree T , a union of (9, k) trees T , and constants q, m and k. Let q ∗ = max(100, q) and M = max( 2m/3 , 100, q). Also let k = min(k, 2m/3 ). Given T define T¯ = R(c˜ ) ∩ C(T ). Define T to be the union of the (9, q ∗ ) trees in T¯ such that for each Ti ⊂ T there exists v˜i ∈ Ti , vi ∈ T ∩ C(Ti ) and a crossing of S1 containing v˜i and vi . Define ∗ to be the event that |9q ∗ −1 |T ∗ −1 q |T| ≥ max . (7) ,9 1000 · 9M Lemma 4.5. If Lemma 4.3 is satisfied for j ≤ n − 1 then |T | P (∗) > min .9, . 100 · 9 2m/3 Proof. We break the proof up into cases based on the size of |T | and whether M = q ∗ , M = 100, or M = 2m/3 . If |T | ≤ 1000 · 9 2m/3 and M = 2m/3 then by Lemma 4.4 |T | P (∗) ≥ min .9, . 3 · 9 2m/3 If |T | ≤ 1000 · 9M and M = q ∗ then we write T = N i=1 Ti where each Ti is a union of (9, k ) trees in a ( 2m/3 , m) strip. Then for all i by Lemma 4.4, |Ti | ¯ P (∃ a crossing intersecting Ti and T ) ≥ min .9, . 3 · 9 2m/3 If for one i the minimum is .9 then we are done. Otherwise by Lemma 4.2 we have |T | P (∃ a crossing intersecting T and T¯ ) > min .9, . 9 · 9 2m/3 If |T | > 1000 · 9M , then write T =
N i=1
Ti , where each Ti is a union of (9, k ) trees in
a union of (M, m) strips. Do this in such a way that for each i, 3 · 9M ≤ |Ti | = 4 · 9M , and if i = j the unions of (M, m) strips for i and j are disjoint. Thus N is at least |T | ≥ 100. 4 · 9M By the fact that |Ti | ≥ 3 · 9M we have that P (∃ a crossing of Si intersecting Ti and T ∗ ) ≥ .9. Thus we have N ≥ 100 independent events with probability greater than or equal to .9. The probability that at least N/10 of these events happen is greater than the probability that 11 events happen with N = 100. This probability is at least .9. Finally if M = 100 then q ≤ 100 and m ≤ 150. Thus P (∗) ≥ 1 − 4−200 .
18
C. Hoffman
Let G be a graph which is composed of the union of a column of q boxes, [e, f ] × [g1 , h1 ] through [e, f ] × [gl , hl ], where q ≥ 200 and l ≤ 384. Also in G are horizontal (q, q) strips [e, f ] × [h1 , g2 ] through [e, f ] × [hl−1 , lk ]. Thus G = [e, f ] × [g1 , hl ]. We call such a graph a column of q boxes. Let v ⊂ R(g1 ) and w ⊂ R(hl ) be vertices on the bottom and top of G. We say G is normal for v and w if there exists a cluster in G connecting v and w. Lemma 4.6. Suppose the conclusions of Lemma 4.3 are satisfied for q ≤ n. Then P (G is normal for v and w) ≥ .99. Proof. G is normal for v and w if 1. all of the q boxes in G are good, 2. v and w are in the crossing clusters of their respective q boxes, and 3. all of the (q, q) strips in G have a cluster which connects the crossing clusters of the good q boxes on the top or bottom of the (q, q) strip. Thus by the induction hypothesis P (condition 1 is satisfied) > (1 − 4−q )384 > 1 − 2−100 . If the j box containing v (or w) is good for all j, 200 ≤ j ≤ q, then v (or w) is in the crossing cluster of the q box. Thus 4−j ) > 1 − 4−199 . P (condition 2 is satisfied) > (1 − 2 j ≥200
By Lemma 4.3 P (condition 3 is satisfied) > (1 − 4−q )384 > 1 − 2−100 . Thus P (G is normal for v and w) > 1 − 3 · 2−100 > .99.
Proof of Lemma 4.3. The proof is by induction. Choose pc such that for p > pc , the lemma is true for all n ≤ 200. Assume that the lemma is true for all j < n. Since {xi } and {yi } are regular there are at most 3842 n − 1 boxes in an n box and we have that Pp (a1 = 1) ≤ 3842 (4−n+1 ) ≤ 410 4−n and Pp (a1 ≥ 2) ≤ (3842 )2 (4−n+1 )2 ≤ 430 4−2n ≤ 4−50 4−n . There are at most 2(384)2 (n − 1, n − 1) strips in an n box. If such a strip is between two good n − 1 boxes then the probability that there exists a crossing which connects to both the crossing clusters of the good n boxes is calculated as follows. By the induction hypothesis Pp (a2 ≥ 2) ≤ (2(384)2 )2 (4−n+1 )2 ≤ 420 4−2n ≤ 4−50 4−n
Phase Transition in Dependent Percolation
19
and Pp (a2 ≥ 1|a1 = 1) ≤ 2(384)2 (4−n+1 ) ≤ 420 4−n . Thus Pp (a1 + a2 ≥ 2) ≤ Pp (a1 ≥ 2) + Pp (a2 = 1|a1 = 1) · Pp (a1 = 1) + Pp (a2 ≥ 2) ≤ 2(4−50 )4−n + 430 4−2n ≤ 4−n . This completes the proof of the first statement. Now we consider the second statement. First we must consider the case that the height of S is one. In this case the probability is easy to calculate. There are |T | edges which would form an appropriate crossing if they were open. Thus Pp (∃ a cluster in S which intersects T and T ) ≥ 1 − (1 − p n )|T
|
≥ min(1 − e−1 , |T |p n (1 − e−1 )) |T | ≥ 2n/3 9 by Lemma 4.2. Next it is easy to see that if either m < 200 or r < 200 then q < 200 and that the induction hypothesis is easily proven. Now we consider the case when m, r ≥ 200 and the height of S is greater than one. If q ≥ 100 and 1. the event ∗ happens on S1 . This forms T ⊂ C(T ) ∩ R(c˜ ). 2. Set T ∗ = C(T) ∩ R(d˜ ). There exists a crossing of S2 intersecting T ∗ and T . This gives us v ∈ T and w ∈ T ∗ . They are separated by a column of q boxes. And 3. the column of q boxes separating v and w is normal for v and w, then there exists a crossing of S that intersects T and T . By Lemma 4.5 |T | . P (1) = P (∗) ≥ min .9, 100 · 9 2m/3 If ∗ occurs then we get
q ∗ −1
|T˜ | ≥ max 9
If |T | ≤ 1000 · 9M then |T ∗ | = |T˜ | ≥ 9q
∗ −1
∗
9q −1 |T | , 1000 · 9M
.
and by Lemmas 4.5 and 4.6, ∗ P (2 and 3 | 1) ≥ P (∃ a crossing of S2 intersecting T ∗ and T |T ∗ | = 9q −1 )(.99) ∗ 9q −1 ≥ (.99) min .9, 3 · 9 2r/3 ∗ 9q −1 ≥ min .8, . 4 · 9 2r/3
20
C. Hoffman
If |T | < 100 · 9 2m/3 then P (∃ a crossing of S intersecting T and T ) ≥ P (1) · P (2 and 3|1) ∗ |T | 9q −1 ≥ min .8, 100 · 9 2m/3 4 · 9 2r/3 ∗ |T | 9q −1 |T | ≥ min , 200 · 9 2m/3 400 · 9 2m/3 9 2r/3 ≥
|T | 9 2n/3
.
If 100 · 9 2m/3 ≤ |T | ≤ 1000 · 9M then P (∃ a crossing of S intersecting T and T ) ≥ P (1) · P (2 and 3 | 1) ∗ 9q −1 ≥ (.9) min .8, 4 · 9 2r/3 ∗ 9q /3 1000 · 9 2m/3 ≥ min .5, 2r/3 + 2m/3 −q ∗ /2 9 1000 · 9M ≥ min .5, 2n/3 9 |T | ≥ min .5, 2n/3 9 |T | ≥ 2n/3 . 9 If |T | ≥ 1000 · 9M and ∗ occurs then ∗
|T |9q −1 |T| = |T ∗ | ≥ , 1000 · 9M and by Lemma 4.4 P (2 | 1) ≥ P
∃ a crossing of S2 intersecting
T∗
|T | ≥ min .9, ∗ 3000 · 9M+ 2r/3 −q +1 2|T | ≥ min .9, 2n/3 9 2|T | ≥ 2n/3 . 9 By Lemma 4.6, P (3 | 1 and 2) ≥ .99.
∗ |T |9q −1 and T |T ∗ | = 1000 · 9M
Phase Transition in Dependent Percolation
21
If |T | ≥ 1000 · 9M then P (∃ a crossing of S intersecting T and T ) ≥ P (1) · P (2| 1) · P (3 | 1 and 2) 2|T | ≥ (.9) 2n/3 (.99) 9 |T | ≥ 2n/3 . 9 For any two points (x1 , y1 ), (x2 , y2 ) ∈ Z2 we say the rectangle formed by v and w is the graph of all edges where both of vertices have x coordinate between x1 and x2 and have y coordinate between y1 and y2 . If q < 100 we substitute 3 with 3 ) all edges in the rectangle formed by v and w are open. Then everything goes through as above. This is because the rectangle between v and w is a portion of a 200 box. This completes the proof of Statement 2. Finally we prove Statement 3. The (9, n) trees R and R defines a set of 9n− 2n/3 ( 2n/3 , n) strips in S¯ . Let S˜ be one. By the induction hypothesis the probability that there is a crossing of S˜ which intersects R and R is at least 19 . As there are 9n− 2n/3 of these ( 2n/3 , n) strips and the events that there does not exist crossings that intersect both R and R are independent. The probability that there exists an appropriate crossing is at least 1−
9n− 2n/3 8 9n− 2n/3 −1 ≥ min 1 − e−2n , (1 − e−2n ) 9 2n ≥ 1 − e−2n ≥ 1 − 4−n .
Theorem 4.1. There exists pc < 1 such that for almost every {xi } and {yi } and all p > pc , Pp (∃ an infinite cluster) = 1. Proof. By Lemma 3.10 there exists {x¯i } and {y¯i } which are very regular and x¯i ≥ xi and y¯i ≥ yi for all i. Thus the estimates in Lemma 4.3 apply. Let pc be as in Lemma 4.3. It is easy to check that if v and w are two vertices in the same n box, v is in a good j box for all j ≤ n and w is in a good j box for all j ≤ n, then v and w are the same cluster. Thus if v is in a good n box for all n then v is in an infinite cluster. Thus Pp (∃ an infinite cluster) ≥ Pp ( 0 is in a good n box for all n) ∞ ≥ 1− 4−n n=200 −199
≥ 1−4
.
As Pp (∃ an infinite cluster) equals 0 or 1, Pp (∃ an infinite cluster) = 1.
22
C. Hoffman
Theorem 4.2. For percolation on the randomly stretched lattice in Z2 there exists a value pc , 0 < pc < 1 such that if p < pc then Pp (∃ an infinite open cluster) = 0. If p > pc then Pp (∃ an infinite open cluster) = 1. Proof. It follows from [4] that if p is sufficiently close to 0 then there is no infinite cluster a.s. Thus Theorem 4.1 completes the proof. Acknowledgement. I would like to thank David Levin andYuval Peres for introducing me to the problem. I would also like to thank Yuval Peres and Eric Babson for helpful conversations.
References 1. Balister, P., Bollobs, B., Stacey A.: Dependent percolation in two dimensions. Probab. Theory Related Fields 117(4), 495–513 (2000) 2. Chayes, J., Chayes, L., Durrett, R.: Connectivity properties of Mandelbrot’s percolation process. Probab. Theory Related Fields 77(3), 307–324 (1988) 3. Ferreira I.: The probability of survival for the biased voter model in a random environment. Stochastic Process. Appl. 34(1), 25–38 (1990) 4. Jonasson, J., Mossel, E., Peres Y.: Percolation in a dependent random environment. Random Struct. Alg. 16(4), 333–343 (2000) 5. Klein, A.: Extinction of contact and percolation processes in a random environment. Ann. Probab. 22(3), 1227–1251 (1994) 6. Liggett, T.: The survival of one-dimensional contact processes in random environments. Ann. Probab. 20(2), 696–723 (1992) 7. McCoy, B., Wu, T.: Theory of a two-dimensional Ising model with random impurities. I. Thermodynamics. Phys. Rev. (2) 176, 631–643 (1968) 8. McCoy, B., Wu, T.: Theory of a two-dimensional Ising model with random impurities. II. Spin correlation functions. Phys. Rev. (2) 188, 982–1013 (1969) 9. Newman, C., Volchan, S.: Persistent survival of one-dimensional contact processes in random environments. Ann. Probab. 24(1), 411–421 (1996) 10. Winkler, P.: Dependent percolation and colliding random walks. Random Structures Algorithms 16(1), 58–84 (2000) Communicated by M. Aizenman
Commun. Math. Phys. 254, 23–44 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1241-1
Communications in
Mathematical Physics
Compact Scalar-flat Indefinite K¨ahler Surfaces with Hamiltonian S 1 -Symmetry Hiroyuki Kamada, Numazu College of Technology, 3600 Ooka, Numazu, Shizuoka 410-8501, Japan Received: 1 December 2002 / Accepted: 30 March 2004 Published online: 25 Novemebr 2004 – © Springer-Verlag 2004
Abstract: The existence of scalar-flat indefinite K¨ahler metrics on compact complex surfaces is discussed. In particular, a compact scalar-flat indefinite K¨ahler surface admitting a Hamiltonian S 1 -symmetry is proved to be biholomorphic to the product of two complex projective lines, with the help of a generalization of the Bando-Calabi-Futaki character. In fact, it is shown that none of such metrics exist on Hirzebruch surfaces of positive degree. On the other hand, by employing an analogue of LeBrun’s hyperbolic ansatz, we construct a wealth of explicit scalar-flat indefinite K¨ahler metrics on the product of complex projective lines, and also prove that these explicit metrics provide infinitely many different isometry classes, by examining a necessary and sufficient condition for these metrics to be isometric to each other. 1. Introduction Since the work of Ooguri-Vafa [19], Ricci-flat indefinite K¨ahler metrics on complex surfaces have drawn considerable attention in mathematical physics. Recently Petean [20] studied the existence of indefinite K¨ahler Einstein metrics, including Ricci-flat ones, on compact complex surfaces. As another important branch of generalizations of the Ricci-flat case, we are interested in the scalar-flat case. The main theme of this paper is to study the existence of scalar-flat indefinite K¨ahler metrics on compact complex surfaces with certain symmetry. Let g be an indefinite Hermitian metric on a complex surface M = (M, J ), that is, g satisfies g(J ·, J ·) = g(·, ·). Then g is of type (2,2) (we often call an indefinite metric of neutral signature such as g simply a neutral metric). A triplet (M, J, g) is called an indefinite K¨ahler surface if the associated fundamental form := g(J ·, ·) is closed. For a given (indefinite) K¨ahler surface (M, J, g), the fundamental form and
Supported by JSPS–MEXT. Grant-in-Aid for Young Scientists (No. 13740053). Present address: The National University Corporation, Miyagi University of Education, 149 Aramaki Aoba, Aoba-ku, Sendai 980-0845, Japan.
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its cohomology class [] are called the K¨ahler form and the K¨ahler class of (M, J, g), respectively. Since g is indefinite, its K¨ahler form is a real symplectic (1,1)-form on M compatible with the nonstandard (anti-complex) orientation. In this paper, we equip an indefinite K¨ahler surface (M, J, g) with the complex orientation given by J . Let γ := Ric(J ·, ·) be the Ricci form of an indefinite K¨ahler surface (M, g) = (M, J, g). In terms of local holomorphic coordinates (z1 , z2 ) of M, the K¨ahler form √ and the Ricci form γ are expressed as = ( −1/2) 2α,β=1 gα β¯ dzα ∧ dzβ and √ γ = − −1∂∂ log | det(gα β¯ )|, respectively. Thus γ is also a closed real (1, 1)-form on M. Note that, since J is parallel with respect to the Levi-Civita connection ∇ of (M, g), the cohomology class [γ ] of γ determines the real first Chern class c1 (M)R of M by c1 (M)R = (1/2π)[γ ] ∈ H 1,1 (M; R). By definition, g is scalar-flat if and only if γ ∧ ≡ 0, which implies c1 (M)·[] = 0. A typical example of a compact scalar-flat indefinite K¨ahler surface is the product P1 × P1 of two complex projective lines with the indefinite product metric g0 = (−hP1 ) ⊕ hP1 , where hP1 denotes the standard unit round metric on P1 . Note that (P1 × P1 , g0 ) is not only scalar-flat but also conformally-flat, that is, the Weyl conformal tensor W of g0 identically vanishes. Furthermore, it is known that the conformal structure of a conformally-flat indefinite metric is unique on P1 × P1 . In fact, a compact simply-connected indefinite conformally-flat four-manifold is conformally equivalent to (P1 × P1 , g0 ) (see Kuiper [11]). As in the Riemannian case, it follows that an indefinite K¨ahler surface (M, g) is scalar-flat if and only if g is self-dual, that is, its Weyl conformal tensor W , regarded as an endomorphism on the vector bundle of two-forms 2 = 2 T ∗ M, satisfies ∗g W = W , where ∗g denotes the Hodge star operator of (M, g). It is known that the product P1 × P1 of two complex projective lines and the one-point blowing-up P2 # P2 of the complex projective plane P2 are typical examples of compact four-manifolds that admit no self-dual Riemannian metrics. To show these non-existence results for P1 × P1 and P2 # P2 , the Hirzebruch signature formula and Kuiper’s theorem are essentially used. Although the Hirzebruch signature formula is also known in the indefinite case (cf. Matsushita-Law [14]), we cannot conclude similar non-existence results at once, since the norms in the formula are indefinite in general. Related to complex structures, we then ask whether there exist non-conformally-flat scalar-flat indefinite K¨ahler metrics on P1 × P1 and P2 # P2 . Concerning this existence problem for P1 × P1 , we can show the following. Theorem 1. There exist scalar-flat indefinite K¨ahler metrics on P1 × P1 , which provide infinitely many different isometry classes. In §4, we construct explicit scalar-flat indefinite K¨ahler metrics on P1 ×P1 by employing an indefinite analogue of LeBrun’s hyperbolic ansatz, which is a method of constructing self-dual Riemannian metrics (cf. LeBrun [12]). By construction, each metric mentioned above has a nontrivial isometric S 1 -action preserving the K¨ahler form. Since P1 × P1 is simply-connected, there exists a moment map of the action with respect to the K¨ahler form. We call such an S 1 -action a Hamiltonian S 1 -symmetry. Comparing their Killing vector fields, we obtain a necessary and sufficient condition for these metrics to be isometric, and show that these provide infinitely many different isometry classes on P1 × P1 in §5 (cf. [5, 6]). In regard to S 1 -symmetry, we can show the following result. Theorem 2. Let (M, g) be a compact indefinite K¨ahler surface admitting a Hamiltonian S 1 -symmetry. Suppose that the K¨ahler class [] is orthogonal to the first Chern class
Compact Scalar-flat Indefinite K¨ahler Surfaces with Hamiltonian S 1 -Symmetry
25
c1 (M) with respect to the cup product. Then M must be biholomorphic to a Hirzebruch surface Fd (d = 0, 1, 2, . . . ). Theorem 2 is an immediate consequence of Theorem 5, which will be proved in §2. As in Theorem 1, there are many scalar-flat indefinite K¨ahler metrics on F0 = P1 × P1 . Then we also ask whether there exists such a metric on Fd (d ≥ 1), which includes the previous problem for F1 ∼ = P2 # P2 as a special case. Regarding this problem, we have the following. Theorem 3. If d ≥ 1, no scalar-flat indefinite K¨ahler metric exists on Fd . This non-existence result is proved by using a generalization of the Bando-CalabiFutaki character due to Futaki-Mabuchi [4] (see §3). Summarizing Theorems 2 and 3, we obtain the following characterization. Theorem 4. Let (M, g) be as in Theorem 2. Then, there exists a scalar-flat indefinite K¨ahler metric on M whose K¨ahler class coincides with that of g if and only if M is biholomorphic to P1 × P1 . 2. S 1 -Symmetry on Indefinite K¨ahler Surfaces In this section, we prove the following result. Theorem 5. Let (M, g) be a compact indefinite K¨ahler surface with K¨ahler form admitting an S 1 -symmetry and Fix(S 1 ) the fixed point set of the symmetry. If c1 (M)·[] = 0 and Fix(S 1 ) = ∅, then M must be biholomorphic to a Hirzebruch surface Fd (d = 0, 1, 2, . . . ). To begin with, we first show the following Proposition 1. Any Killing vector field ξ on a compact indefinite K¨ahler surface (M, g) is a real holomorphic vector field on M. Proof. As in §1, we equip M with the complex orientation. Any orientation-preserving isometry on (M, g) maps the set of parallel anti-self-dual two-forms to itself. Thus the K¨ahler form of (M, g) and its Lie derivative Lξ = dιξ are parallel and anti-self-dual. Then there exists a constant c such that (Lξ )2 = c 2 , whose integration over M leads us to c = 0, that is, (Lξ )2 ≡ 0. Furthermore, Lξ must be the real part of a parallel (2, 0)-form, since 2− = R ⊕ ((2,0 ⊕ 0,2 ) ∩ 2 ) and Lξ ∧ = (1/2)Lξ (2 ) ≡ 0, where 2 , 2− , 2,0 and 0,2 denote the spaces of real two-forms, anti-self-dual two-forms and (2, 0)- and (0,2)-forms on (M, g), respectively. Then (Lξ )2 ≡ 0 implies Lξ ≡ 0. Thus ξ is real holomorphic. Proof of Theorem 5. By Proposition 1, the underlying S 1 -action of the symmetry is a symplectic action on (M, ), where M denotes the manifold M with the opposite orientation. Let ξ be the vector field on M generating the S 1 -action. Since Fix(S 1 ) = ∅, it follows from a result of McDuff [15] that the S 1 -symmetry is Hamiltonian, that is, a moment map z : M → R, satisfying ιξ = dz, exists. Then, by the Frankel type result for (M, ), the first Betti number b1 (M) = b1 (M) is even (cf. Frankel [3]). It is well-known that a compact complex surface with even first Betti number admits a positive-definite K¨ahler metric (see Barth et al. [1]), so M admits such a metric.
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By Proposition 1 again, ξ yields a nontrivial holomorphic vector field on M with zero. Then the Kodaira dimension κ(M) of M is negative (cf. Kobayashi [9]). Hence M is biholomorphic to either a rational surface or a ruled surface (see [1]). In both cases, M admits a Riemannian metric of positive scalar curvature, thus all the Seiberg-Witten invariants of M and those of M vanish (cf. LeBrun [13]). Since b2− (M) ≥ 1, we have b2− (M) = b2+ (M) = 1; otherwise, by a result of Taubes [23], the Seiberg-Witten invariant for (M, ) is nonzero, which is a contradiction. Similarly, we obtain b2+ (M) = 1, via the existence of a positive-definite K¨ahler metric on M. Given that b2+ (M) and b2− (M) are both known to be one, and since 2 < 0 everywhere, the assumption c1 (M)·[] = 0 implies that c12 (M) > 0 unless c1 (M) is torsion. However, the latter is impossible, since b1 (M) is even and κ(M) < 0. By a result of Qin [22], M is then biholomorphic to a Hirzebruch surface Fd (d = 0, 1, 2, . . . ). In the course of the proof of Theorem 5, we also obtain the following Corollary 1. Let (M, g) be a compact scalar-flat indefinite K¨ahler surface of negative Kodaira dimension with even first Betti number. Then M must be biholomorphic to a Hirzebruch surface Fd (d = 0, 1, 2, . . . ). Remark 1. Let M be a compact complex surface of negative Kodaira dimension with odd first Betti number. Then M is of class VII. It is natural to ask whether there exists an indefinite K¨ahler metric on M. If it exists, then b2− (M) = 1 − b1 (M) + b2+ (M) is positive and even, since the K¨ahler form is a symplectic form on M. Hence the Seiberg-Witten invariant for (M, ) is nonzero, so that M admits no Riemannian metric of positive scalar curvature. However, as pointed out in Petean [20], every known example of surfaces of class VII is diffeomorphic to a connected sum (S 1 × S 3 )#kCP2 (k = 0, 1, 2, . . . ), which admits a Riemannian metric of positive scalar curvature. At present, no example is known of a compact complex surface of class VII admitting an indefinite K¨ahler metric. 3. Bando-Calabi-Futaki Character In this section, we first review a generalization of the Bando-Calabi-Futaki character due to Futaki and Mabuchi [4]. Let M be a compact complex manifold of complex dimension n endowed with a symplectic (1,1)-form and H 0 (M; O(T 1,0 M)) the space of all holomorphic vector fields on M. Define a set S = S(), which consists of symplectic (1,1)-forms on M contained in the cohomology class [], by √ S = {ϕ := + −1∂∂ϕ | ϕ ∈ C ∞ (M)R , ϕ is nondegenerate}. A holomorphic vector field V on (M, ) is said to be Hamiltonian holomorphic if [ιV ] = 0 in H 1 (M; OM ). This condition is independent of the choice of in S. Denote the set of all Hamiltonian holomorphic vector fields on (M, S) by Ham C (M, S). If the irregularity qM := dimH 1 (M; OM ) of M vanishes, then every holomorphic vector field is Hamiltonian holomorphic. Let (M, S) be as above. For each ∈ S and V ∈ Ham C (M, S), there exists a Cvalued smooth function v = v(, V ) on M such that ιV = ∂v. Note that v is unique up to an additive constant. Then we set (, V ) := n F v (γ ∧ n−1 − µn ), (1) (2π)n M
Compact Scalar-flat Indefinite K¨ahler Surfaces with Hamiltonian S 1 -Symmetry
where µ =
γ ∧ n−1 M
27
(, V ) is n = 2πc1 (M)·[]n−1 /[]n . Note that F M
independent of the choice of v. Let and be symplectic (1, 1)-forms in S. Suppose that√there exists a smooth family {ϕt }0≤t≤1 of smooth functions on M such that t = 0 + −1∂∂ϕt ∈ S satisfies t = if t = 0 and t = if t = 1. Let γt denote the Ricci form of the corresponding (possibly indefinite) K¨ahler metric of t . We also define a C-valued smooth function vt √ by vt := v + −1V ϕt . Then, by direct computations, one can show that d d ∂vt = ιV t , vt nt = vt (γt ∧ n−1 − µnt ) ≡ 0. t dt M dt M In the positive-definite case, the line segment t = (1 − t) + t gives a smooth path (t , V ) is independent of t, and therefore F (, V ) = F ( , V ). We then in S. Thus F (, V ) for any choice of in S, which is regarded as the Bandodefine FS (V ) := F Calabi-Futaki character of S. However, in the indefinite case, t may be degenerate and then γt may diverge at the zero locus of t for some t (0 < t < 1). In general, no smooth path in S may exist connecting and . To define FS (V ) in the indefinite case, we recall some basic facts for holomorphic line bundles. Let M be a compact complex manifold of complex dimension n, V a holomorphic vector field on M and E a holomorphic line bundle over M. Lemma 1. Let h and h be Hermitian fiber metrics on E satisfying h = e−ϕ h for some smooth function ϕ on M. Denote by θ (h) and (h) (resp. θ(h ) and (h )) the corresponding connection form on (E, h) (resp. (E, h )) and its curvature form, respectively. onto E, and fix such a lifting V . Then we Suppose that V has a holomorphic lifting V obtain ) + (h ))n+1 = ) + (h))n+1 . (−θ (h )(V (−θ (h)(V M
M
) = θ (h)(V ) + V ϕ. Take a smooth family {ht = Proof. We first note that θ (h )(V −ϕ t e h}0≤t≤1 of Hermitian fiber metrics on E satisfying ht = h if t = 0 and ht = h if t = 1, and denote by θt the connection form corresponding to ht . Then we have ) = θ (h)(V ) + V ϕt . Hence we obtain (ht ) = (h) + ∂∂ϕt and θ (ht )(V d ) + (ht ))n+1 ≡ 0, (−θ (ht )(V dt M which implies the desired result.
n+1
c ) by By using the quantity in Lemma 1, we define FE1 (V √ n+1 −1 c1n+1 ) + (h))n+1 . FE (V ) := (−θ (h)(V 2π M n+1
c ) depends on the choice of the lifting V of V . However, once V is Note that FE1 (V n+1
c ) is independent of the choice of h. chosen, FE1 (V Now, we go back to our situation. Let M be a compact complex manifold of complex dimension n with a symplectic (1, 1)-form . We assume that M admits a positivedefinite K¨ahler metric and also that the cohomology class [] is realized as (1/2π )[] =
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c1 (L) ∈ H 1,1 (M; Z) for some holomorphic line bundle L over √M. Because of the ∂∂lemma, there exists a Hermitian fiber metric h on L with = −1(h) and moreover the set S = S() depends only on the cohomology class [] = 2π c1 (L). Let V be a Hamiltonian holomorphic vector field on (M, S). Suppose that V has a onto L. Then θ (h)(V ) satisfies holomorphic lifting V √ )) = ιV (h) = − −1ιV , ∂(θ (h)(V where θ (h) denotes the connection form on L associated with h. We also write −θV (h) ), when V is fixed. Note that induces a Hermitian fiber metric | det(g)| on for θ(h)(V −1 the anti-canonical bundle KM , where g is a (possibly indefinite) K¨ahler metric on M associated with . As in Nakagawa [17], we can show that −g θV (h) = θV (| det(g)|) ∗ ∗ −1 , where g := −∂ g ∂ and ∂ g is the with respect to the canonical lifting of V onto KM formal adjoint of ∂ with respect to g. In particular, ∂(−g θV (h)) = −ιV (| det(g)|) follows. For another element in S, we expect that −g θV (h ) = θV (| det(g )|) holds. Regarding this, we show the following Lemma 2. Let and be symplectic (1, 1)-forms on M satisfying [ ] = [] = 2π c1 (L) and ( )n = e−ψ n for a smooth function ψ on M. Then θV (| det(g )|) = θV (| det(g)|) − V ψ = −g θV (h ) holds. Proof. By a similar computation in [17], we can show that ∂(−g θV (h )) = −ιV (| det(g )|) = ∂(θV (| det(g )|)) = ∂(θV (| det(g)|) − V ψ). On the other hand, we have the following computation: (θV (| det(g)|) − V ψ)(g )n = (−g θV (h) − V ψ)e−ψ ng M M ∗ = (e−ψ ∂ g ∂θV (h) + de−ψ (V ))ng M = (g(∂e−ψ , ∂θV (h)) + ∂e−ψ (V ))ng M = (∂θV (h)((∂e−ψ )# ) + ∂e−ψ (V ))ng M √ = ( −1(V , (∂e−ψ )# ) + ∂e−ψ (V ))ng M = (−∂e−ψ (V ) + ∂e−ψ (V ))ng = 0, M
which implies that θV (| det(g)|) − V ψ ≡ −g θV (h ).
By similar computations in Tian [24] (cf. [17]), we can obtain the following be as above and µ denote a constant, defined by Lemma 3. Let (M, ), L, V and V n−1 n (, V ) is expressed as µ = c1 (M)·c1 (L) /c1 (L) . Then F n+1 (, V ) = 2πnµ F c1 (V ) F L n+1 n+1 n 2π c j n ). − n F 1−1 n−2j(V (−1) j KM ⊗L 2 (n + 1)!
j =0
Compact Scalar-flat Indefinite K¨ahler Surfaces with Hamiltonian S 1 -Symmetry
29
Note that the left-hand side is independent of the choice of a holomorphic lifting V of V onto L, and that the right hand side is independent of the choice of a Hermitian (, V ) by FS (V ) or F c1 (L) (V ). If fiber metric h on L and hence of in S. We denote F S contains the K¨ahler form of a (possibly indefinite) K¨ahler metric of constant scalar curvature (i.e.,γ ∧ n−1 ≡ µn ), then FS (V ) = F c1 (L) (V ) ≡ 0 by (1). Summarizing these, we obtain the following Proposition 2. Let M be a compact complex manifold of complex dimension n with a symplectic (1, 1)-form on M and V a Hamiltonian holomorphic vector field on (M, ). Suppose the following conditions are satisfied: (i) M admits a positive-definite K¨ahler metric, (ii) [] = 2π c1 (L) for a holomorphic line bundle L over M, (iii) V has a holomorphic lifting V˜ onto L. (, V ) is independent of the choice of the K¨ahler form in Then F c1 (L) (V ) = F 2πc1 (L). In particular, if 2πc1 (L) contains the K¨ahler form of a (possibly indefinite) K¨ahler metric of constant scalar curvature, then F c1 (L) (V ) = 0. If M is a Hodge manifold, that is, if M admits a positive-definite K¨ahler metric whose K¨ahler class is integral, then the conditions (i) and (iii) are automatically satisfied (see Kobayashi [9]). In what follows, we discuss the existence of scalar-flat indefinite K¨ahler metrics on a Hirzebruch surface Fd , which is a compact complex surface, realized as the total space of a P1 -bundle over P1 , with the following numerical characters: κ(Fd ) = −∞, h1,0 (Fd ) = qFd = pg (Fd ) = 0, b2+ (Fd ) = b2− (Fd ) = 1. Here κ(Fd ), h1,0 (Fd ), qFd and pg (Fd ) denote the Kodaira dimension, the Hodge number (Fd ), the irregularity and the geometric genus of Fd , respectively. In particular, dimH 1,0 ∂ Fd admits a positive-definite K¨ahler metric. Let L be a complex line bundle over Fd satisfying c1 (L)·c1 (Fd ) = 0. If there exists a scalar-flat indefinite K¨ahler metric g on Fd , then the K¨ahler form [] of g is proportional to c1 (L), since b2 (Fd ) = 2. Taking account of κ(Fd ) = −∞ and qFd = 0, we may regard L as a holomorphic line bundle over Fd . Furthermore, we see that every holomorphic vector field V on Fd is on L (see [9]). Thus, by Hamiltonian holomorphic, and has a holomorphic lifting V Proposition 2, F c1 (L) (V ) is well-defined for any holomorphic vector field V on Fd . Regarding F c1 (L) (V ) for a specific V , we obtain the following. Theorem 6. Let L be a holomorphic line bundle over Fd with c1 (L)·c1 (Fd ) = 0 and V the holomorphic vector field on Fd associated with the standard C∗ -action along the fibers of Fd → P1 . If d ≥ 1, then F c1 (L) (V ) = 0. From Proposition 2 and Theorem 6, we obtain the following, which is referred to as Theorem 3. Corollary 2. If d ≥ 1, no scalar-flat indefinite K¨ahler metric exists on Fd . Before proving Theorem 6, we note that the Hirzebruch surface Fd is a typical example of a toric manifold. For compact toric manifolds, Nakagawa [18] obtained a combinatorial formula for the Bando-Calabi-Futaki character, by making use of the following localization result due to Bott [2], and points out that his formula is also available in the indefinite case (cf. [24]).
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Proposition 3. Let M be a compact complex manifold of complex dimension n and (E, h) → M a holomorphic line bundle with Hermitian metric h and connection form θ(h). Let V be a nondegenerate holomorphic vector field on M, that is, a holomorphic vector field on M such that the set Zero(V ) of zeros of V is finite and the endomorphism (LV )p : Tp1,0 M → Tp1,0 M induced from the Lie derivative is nonsingular at any point cn+1
) is onto E. Then F 1 (V p ∈ Zero(V ). Suppose that V has a holomorphic lifting V E expressed as √ n+1 ))n+1 −1 (−θ (h)(V c1n+1 FE (V ) = . √ 2π det(( −1/2π )(LV )p ) p∈Zero(V ) We now recall Nakagawa’s combinatorial formula for the Bando-Calabi-Futaki character of compact toric manifolds. Let Tn := (C∗ )n be an n-dimensional algebraic torus, (t 1 , . . . , t n ) the standard holomorphic coordinates for Tn , and {τ1 , . . . , τn } a basis for the Lie algebra tn of Tn given by τi := t i ∂/∂t i (i = 1, . . . , n). Let X be an n-dimensional compact toric manifold associated with a complete nonsingular fan in N := Zn and (i) denote the set of i-dimensional cones in (i = 0, 1, . . . , n). Then Tn acts biholomorphically on X , which has an open dense Tn -orbit isomorphic to Tn . Note that tn is regarded as a complex Lie subalgebra of H 0 (X ; O(T 1,0 X )). Then, for any S ∈ GLn (C), we define V1 (S), . . . , Vn (S), which we also regard as holomorphic vector fields on X , by (V1 (S), . . . , Vn (S)) = (τ1 , . . . , τn )S. For each σ ∈ (n) and S ∈ GLn (C), we set A(σ ) := (a 1 (σ ), . . . , a n (σ )) ∈ GLn (Z) and Q(S; σ ) := A(σ )−1 S ∈ GLn (C), where a 1 (σ ), . . . , a n (σ ) ∈ N form a generator of σ . Let a i (σ ) ∈ (1) be the one-dimensional cone R≥0 a i (σ ) generated by a i (σ ) ∈ N (i = 1, 2, . . . , n). For a map α : (1) → Z, we set β(S; σ, α) := (β1 (S; σ, α), . . . , βn (S; σ, α)) = α(σ )Q(S; σ ), where α(σ ) := (α(a 1 (σ )), . . . , α(a n (σ ))). Let Dν denote the Tn -invariant divisor correspondingto ν ∈ (1). Then a map α : (1) → Z defines a Tn -invariant divisor D(α) := − ν∈(1) α(ν)Dν on X . By Lα = O(D(α)) we denote the holomorphic line bundle over X corresponding to D(α). For example, the canonical bundle KX of X corresponds to the map α : (1) → Z defined by α(ν) ≡ 1 for all ν ∈ (1). Then we can state the following formula (see [18]). Proposition 4. Let X be an n-dimensional compact toric manifold associated with a complete nonsingular fan , and let S ∈ GLn (C) be a nondegenerate matrix, that is, all the components q j i (S; σ ) of Q(S; σ ) = A(σ )−1 S never vanish (i, j = 1, 2, . . . , n). For a map α : (1) → Z, we have βi (S; σ, α)n nj=1 q j i (S; σ ) √ c1 (Lα ) −1F (Vi (S)) = nj=1 q j i (S; σ ) σ ∈(n)
−
nµα βi (S; σ, α)n+1 n+1 nj=1 q j i (S; σ )
(2)
σ ∈(n)
for i = 1, 2, . . . , n, where µα = c1 (X )·c1 (Lα )n−1 /c1 (Lα )n . In our case, µα is always zero, so we omit the explicit expression of µα in terms of βi (S; σ, α) and q j i (S; σ ) (cf. [18]). Since F c1 (Lα ) is C-linear, we have (F c1 (Lα ) (τ1 ), . . . , F c1 (Lα ) (τn )) = (F c1 (Lα ) (V1 (S)), . . . , F c1 (Lα ) (Vn (S)))S −1 . Note that F c1 (Lα ) (τi ) are independent of the choice of S.
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31
To apply Proposition 4 to our problem, we next realize Hirzebruch surfaces Fd (d = 0, 1, 2, . . . ) as compact toric surfaces. Let N := Z2 and NR := N ⊗Z R, and set e1 := t (1, 0), e2 := t (0, 1). Define ν0 , . . . , ν3 and σ0 , . . . , σ3 by ν0 ν1 ν2 ν3
:= R≥0 e1 , := R≥0 e2 , := R≥0 (−e1 + de2 ), := R≥0 (−e2 ),
σ0 σ1 σ2 σ3
:= := := :=
R≥0 e1 + R≥0 e2 , R≥0 (−e1 + de2 ) + R≥0 e2 , R≥0 (−e1 + de2 ) + R≥0 (−e2 ), R≥0 e1 + R≥0 (−e2 ),
and let A(σk ) = (a 1 (σk ), a 2 (σk )) = (a i j (σk )) be a 2 × 2-matrix defined by A(σ0 ) = (e1 , e2 ), A(σ1 ) = (−e1 + de2 , e2 ), A(σ2 ) = (−e1 + de2 , −e2 ), A(σ3 ) = (e1 , −e2 ). Then := {0, ν0 , . . . , ν3 , σ0 , . . . , σ3 } is a complete nonsingular fan, and the corresponding toric surface X is known to be a toric realization of Fd . Recall the standard realization of Fd as a complex submanifold in P2 × P1 : Fd = {(w0 : w1 : w2 , z0 : z1 ) ∈ P2 × P1 | z0d w2 = z1d w1 }. Then, via the projection Fd to the second factor P1 , we can regard Fd as a P1 -bundle P(O(d) ⊕ O) → P1 , where O(d) and O denote the holomorphic line bundle of degree d and the trivial line bundle over P1 , respectively. Define a T2 = (C∗ )2 -action on Fd by (λ, µ)·(w0 : w1 : w2 , z0 : z1 ) := (w0 : µw1 : λd µw2 , z0 : λz1 ). Then the holomorphic vector fields τ1 and τ2 are obtained respectively by τ1 = ∂/∂λ and τ2 = ∂/∂µ. In particular, τ2 is induced by the standard C∗ -action along the fibers of Fd → P1 . Let Dνi be the T2 -invariant divisor corresponding to νi , which are given explicitly by the following fashion: Dν0 = {(w0 : w1 : 0, 1 : 0) ∈ Fd }, Dν1 = {(1 : 0 : 0, z0 : z1 ) ∈ Fd }, Dν2 = {(w0 : 0 : w2 , 0 : 1) ∈ Fd }, Dν3 = {(0 : z0d : z1d , z0 : z1 ) ∈ Fd }. Then the intersection numbers Dνi ·Dνj (0 ≤ i ≤ j ≤ 3) are given by Dν21 = −d, Dν23 = +d, Dν0 ·Dν1 = Dν0 ·Dν3 = Dν1 ·Dν2 = Dν2 ·Dν3 = 1, Dνi ·Dνj = 0 (otherwise). corresponds to the divisor 3i=0 Dνi . Note that the anti-canonical bundle KF−1 d Let L → Fd be a holomorphic line bundle satisfying c1 (KF−1 )·c1 (L) = 0. Here we d may assume that L = Lα corresponds to the divisor Dα = − 3i=0 (−1)i Dνi defined by α : (1) → Z, α(νi ) = (−1)i . Then we see that
α(σi ) = (1, −1) (i = 0, 1, 2, 3).
1 1 , we have π π2 1 1 −1 −1 Q(S; σ0 ) = , Q(S; σ ) = 1 2 , π π 2 π d + π d + 1 1 1 1 Q(S; σ2 ) = − , Q(S; σ3 ) = , d + π d + π2 −π −π 2
Setting S :=
32
H. Kamada
β(S; σ0 , α) = α(σ0 )Q(S; σ0 ) = (1 − π, 1 − π 2 ), β(S; σ1 , α) = α(σ1 )Q(S; σ1 ) = (−1 − d − π, −1 − d − π 2 ), β(S; σ2 , α) = α(σ2 )Q(S; σ2 ) = (−1 + d + π, −1 + d + π 2 ), β(S; σ3 , α) = α(σ3 )Q(S; σ3 ) = (1 + π, 1 + π 2 ). Since c1 (KF−1 )·c1 (Lα ) = 0, we can rewrite the formula (2) as d βi (S; σ, α)2 2j =1 q j i (S; σ ) √ c1 (Lα ) −1F (Vi (S)) = 2j =1 q j i (S; σ ) σ ∈(2) for i = 1, 2. By direct computations, we obtain
3 √ β1 (S; σk , α)2 q 1 1 (S; σk ) + q 2 1 (S; σk ) c1 (Lα ) −1F (V1 (S)) = q 1 1 (S; σk )q 2 1 (S; σk ) k=0
(1 − π)2 (1 + π) (−1 − d − π )2 (−1 + d + π ) + π −(d + π ) (−1 + d + π)2 (−1 − d − π ) (1 + π )2 (1 − π ) + + d +π −π = −2d 2 − 4πd,
=
√
−1F
c1 (Lα )
3 β2 (S; σk , α)2 q 1 2 (S; σk ) + q 2 2 (S; σk ) (V2 (S)) = q 1 2 (S; σk )q 2 2 (S; σk ) k=0
(1 − π 2 )2 (1 + π 2 ) (−1 − d − π 2 )2 (−1 + d + π 2 ) + π2 −(d + π 2 ) (−1 + d + π 2 )2 (−1 − d − π 2 ) (1 + π 2 )2 (1 − π 2 ) + + d + π2 −π 2 = −2d 2 − 4π 2 d, =
which is equivalent to √ √ (F c1 (Lα ) ( −1τ1 ), F c1 (Lα ) ( −1τ2 )) = (−2d 2 , −4d). In particular, this proves Theorem 6.
4. Construction of Scalar-Flat K¨ahler Metrics As in the previous section, F0 = P1 × P1 is the only possibility of a compact scalar-flat indefinite K¨ahler surface admitting a Hamiltonian S 1 -symmetry. In this section, we construct explicit scalar-flat indefinite K¨ahler metrics on P1 × P1 in the context of geometry of self-dual neutral metrics.
Compact Scalar-flat Indefinite K¨ahler Surfaces with Hamiltonian S 1 -Symmetry
33
Let S13 be the de Sitter three-space, which is a Lorentzian space-form of constant curvature +1 realized as a hyperquadric in the Minkowski space-time R41 , equipped with Lorentzian metric gS 3 (cf. Wolf [25]): 1
S13 := {(x0 , x1 , x2 , x3 ) ∈ R41 | − x02 + x12 + x22 + x32 = 1}, gS 3 := (−dx02 + dx12 + dx22 + dx32 )|S 3 . 1
(3)
1
Then we introduce an indefinite analogue of LeBrun’s hyperbolic ansatz (see [5, 6], Kamada-Machida [7], cf. LeBrun [12]): Proposition 5. Let V be a smooth positive function on the de Sitter three-space S13 such that ∗ˇ dV /2π is a closed two-form on S13 determining an integral class in H 2 (S13 ; R), where ∗ˇ is the Hodge star operator of S13 . Let M → S13 denote an S 1 -bundle over S13 with connection one-form θ whose curvature is given by dθ = ∗ˇ dV .
(4)
Then gV ,θ := −V −1 θ ⊗ θ + V gS 3 is a self-dual neutral metric on M. 1
√
If θ
−1φ
= θ + dφ√ for e : → S 1 , then gV ,θ and gV ,θ are isometric via the gauge transformation e −1φ : M → M. We often write gV for gV ,θ . Let (V , θ ) be a solution of (4) such that V > 0 and [ˇ∗dV ]/2π = 0 in the image ImH 2 (S13 ; Z) = Im(H 2 (S13 ; Z) → H 2 (S13 ; R)). Then we obtain a self-dual neutral metric gV on the total space M(∼ =S 1 × S13 ) of a trivial S 1 -bundle over S13 . The constant function V ≡ 1 is clearly a solution of (4), and the metric gV is conformal to the standard neutral metric g0 on the product S 2 × S 2 of unit round two-spheres. To see this, identify S13 with R × S 2 via S13
S13 (x0 , x1 , x2 , x3 ) = (sinh ρ, (cosh ρ)v) → (ρ, v) ∈ R × S 2 . Then gS 3 is expressed as 1
gS 3 = −dρ 2 + cosh2 ρ hS 2 1
(−∞ < ρ < +∞),
where hS 2 denotes the unit round metric on S 2 = {ρ = 0}. Take a connection one-form θ = dt and set r = eρ . Then g¯ V := (cosh2 ρ)gV coincides with g0 = −hS 2 ⊕hS 2 restricted to M = S 1 ×S13 = S 1 ×R×S 2 . Indeed, we see that g¯ V = −
dρ 2 + dt 2 4(dr 2 + r 2 dt 2 ) + h + hS 2 . 2 = − S (1 + r 2 )2 cosh2 ρ
This observation is a prototype of the following result: Proposition 6. Let (V , θ ) be a smooth solution of (4) such that V > 0 on S13 = R × S 2 and ∗ˇ dV is an exact two-form on S13 , where ∗ˇ denotes the Hodge star operator of S13 . Then a metric g¯ V := −sech2ρ(V dρ 2 + V −1 θ 2 ) + V hS 2
(5)
34
H. Kamada
on M = S 1 × R × S 2 extends smoothly to M ∼ = S 2 × S 2 , via polar coordinates at ρ −ρ r := e = 0 and at q := e = 0, if and only if there exist smooth functions F+ and F− on R × S 2 in variables r 2 , q 2 and ζ such that V = 1 + r 2 F− (r 2 , ζ ) and V = 1 + q 2 F+ (q 2 , ζ ),
(6)
as r → +0 and as q → +0, respectively. Here ζ is a complex coordinate of S 2 = P1 . Proof. Since the argument around r = 0 and q = 0 are similar, we discuss only the case near r = 0. After a gauge transformation, we may assume that θ contains no dρcomponent near r√= 0. Let t be a fiber-coordinate of the trivial S 1 -bundle M, and set √ xˆ + −1yˆ := re −1t . Then the following holds: r 2 = xˆ 2 + yˆ 2 , rdr = xd ˆ xˆ + yd ˆ y, ˆ r 2 dt = −yd ˆ xˆ + xd ˆ y, ˆ 2 2 2 2 2 dr + r dt = d xˆ + d yˆ , rdr∧dt = d x∧d ˆ y. ˆ
(7)
We first verify that the condition (6) is necessary. Suppose that g¯ V extends smoothly √ √ −1t to M via polar coordinate xˆ + −1yˆ = re . The restriction of g¯ V to S 2 × {ζ } 2 dt 2 (ζ ∈ S 2 ) is also smooth in ( x, ˆ y). ˆ In general, a metric a(r)dr 2 +2b(r)rdrdt +c(r)r √ √ √ 1 −1t 2 −1t on R+ × S = {(r, e )} extends smoothly to R via xˆ + −1yˆ = re if and only if a(r), b(r) and c(r) are smooth even functions in r satisfying a(0) = c(0)( = 0) and b(0) = 0, which implies that V is a smooth even function in r satisfying V (0, ζ ) = V (0, ζ )−1 (= 1) in our case (cf. Kazdan-Warner [8]). Then V should satisfy (6). For sufficiency, we see that dV , and hence ∗ˇ dV are smooth near r 2 = 0 in variables x, ˆ yˆ and ζ . By assumption, there exists a connection one-form θ = dt + A such that ∗ˇ dV = dθ = dA for some (real) one-form A on S13 . Now, comparing both sides of ∗ˇ dV = dA, we see that A is also smooth near r 2 = 0. Then g¯ V near r 2 = 0 is expressed as follows: − (r 2 , ζ )(r 2 dt)2 + F− (r 2 , ζ )(rdr)2 } 4(dr 2 + r 2 dt 2 ) 4{F − (1 + r 2 )2 (1 + r 2 )2 2 2 2 4{1 + r F− (r , ζ )}(2(r dt)A + r 2 A2 ) + (1 + r 2 F− (r 2 , ζ ))hS 2 , − (1 + r 2 )2
g¯ V = −
− (r 2 , ζ ) := (1+r 2 F− (r 2 , ζ ))−1 near r 2 = 0. Since rdr, r 2 dt, dr 2 +r 2 dt 2 where 1+r 2 F √ √ 2 on R \{(0, 0)} extends smoothly to R2 via the coordinates xˆ + −1yˆ = re −1t , we can regard g¯ V as a smooth metric on R2 ×S 2 . Similarly, we also see that g¯ V extends smoothly to a neighborhood of q 2 = 0. Thus g¯ V is regarded as a smooth metric on M. Concerning complex structures, we next prove the following Proposition 7. For a self-dual metric g¯ V on M = S 2 × S 2 , define an almost complex structure JV on M = S 1 × S13 = S 1 × R × S 2 by JV dρ := −V −1 θ,
JV dζ :=
√ −1dζ.
Then (g¯ V , JV ) defines an indefinite K¨ahler structure on M = S 2 × S 2 .
(8)
Compact Scalar-flat Indefinite K¨ahler Surfaces with Hamiltonian S 1 -Symmetry
35
Proof. Let p,q denote the space of (p, q)-forms on M with respect to JV . Then, JV is √ integrable if and only if d1,0 ⊂2,0 ⊕1,1 . Since 1,0 is generated by dρ + −1V −1 θ and dζ , the integrability of JV is equivalent to √ √ (dρ + −1V −1 θ )∧dζ ∧d(dρ + −1V −1 θ)≡0, (9) ¯ V is given by which follows from (4). Note that the fundamental form ¯ V = g¯ V (JV ·, ·) = −d tanh ρ∧θ + V ωS 2 ,
(10)
where ωS 2 is the volume form of the unit round sphere S 2 = {ρ = 0}. By (4), one can ¯ V ≡ 0. From (7), ¯ V is smooth and nondegenerate near r 2 = 0, and as also verify d 2 well as near q = 0, since it is written as ¯V =−
4rdr∧θ 4qdq∧(−θ) + V ωS 2 = − + V ωS 2 . 2 2 (1 + r ) (1 + q 2 )2
¯ V is regarded as a symplectic form on the whole M. Thus (g¯ V , JV ) is an indefiThen ¯ V , thus JV , are smooth on the whole M. nite K¨ahler structure on M, since g¯ V and Remark 2. For any solution (V , θ ) of (4) satisfying the conditions in Proposition 6, g¯ V is a scalar-flat indefinite K¨ahler metric on (S 2 × S 2 , JV ) endowed with a natural S 1 -symmetry. By Theorem 3, (S 2 × S 2 , JV ) is then biholomorphic to P1 × P1 . If (V , θ ) is another solution of (4) satisfying the same conditions as in Proposition 6, then so is (Vλ , θλ ) for each λ (0 ≤ λ ≤ 1), where (Vλ , θλ ) is defined by Vλ := λV +(1−λ)V , θλ := λθ +(1−λ)θ . Hence g¯ Vλ is a scalar-flat indefinite K¨ahler metric on ¯ V +(1−λ) ¯ V . (S 2 ×S 2 , JVλ ) = P1 ×P1 , and its K¨ahler form Vλ is given by Vλ = λ Note that the scalar-flatness of g¯ Vλ implies c1 (P1 × P1 )·[Vλ ] = 0. Furthermore, it follows from (10) that [Vλ ]·[ωpt×S 2 ] = [0 ]2 , where 0 is the K¨ahler form of the standard product metric g0 = −hP1 ⊕ hP1 . Then the cohomology class [Vλ ] is independent of λ. ¯ V ) is symplectomorphic to (S 2 × S 2 , 0 ). By Moser’s theorem [16], we see that (M, Concerning the Weyl conformal tensor W of g¯ V , we first note the following proposition, which is verified by a direct computation. Proposition 8. Let g¯ V be a self-dual neutral metric on S 2 × S 2 defined by (5). Then its Weyl conformal tensor W is completely determined by ˇ QV := V DdV − 3dV ⊗dV + dV 2 gS 3 , 1
where Dˇ is the Levi-Civita connection of gS 3 and · 2 denotes the indefinite squared 1 norm with respect to gS 3 . In particular, g¯ V is conformally-flat if and only if QV vanishes 1 identically. be the Levi-Civita connection We next examine the conformal-flatness of g¯ V . Let D 2 −2 of gS 3 := V gS 3 . Then, since DdlogV = V QV , g¯ V is conformally-flat if and only 1 1 log V ≡0, that is, V ≡ 1, for V → 1 as ρ → ±∞. Summarizing these, we obtain if Dd the following
36
H. Kamada
Theorem 7. Let g¯ V be a self-dual neutral metric on M = S 2 ×S 2 defined by (5). Then g¯ V is conformally-flat if and only if V ≡1. For a non-constant solution V of (4) satisfying the conditions in Proposition 6, we obtain a non-conformally-flat, self-dual neutral K¨ahler metric g¯ V on S 2 × S 2 . In the remainder of this section, we construct explicit solutions (V , θ ) of (4) satisfying the assumptions in Proposition 6. Let G0 be a function on S13 defined by G0 := (1 − tanh ρ)/2, which satisfies 1 1 ωS 2 , [ˇ∗dG0 ] = 1 ∈ ImH 2 (S13 ; Z) = Z. 2 2π From Proposition 5, we thus obtain a self-dual neutral metric gG0 on the total space S 3 × R of the Hopf bundle S 3 × R → S13 = S 2 × R. We remark that gG0 is conformal to a restriction of the Fubini-Study metric on the indefinite complex projective space CP21 . For an orientation-preserving isometry ϕ of S13 , the function G0 ◦ϕ satisfies ∗ˇ dG0 =
1 [ˇ∗d(G0 ◦ϕ)] = ±1 ∈ ImH 2 (S13 ; Z) = Z. 2π Recall that the group Isom+ (S13 ) of orientation-preserving isometries of S13 is isomorphic to the Lorentzian orthogonal group SO(1, 3), which also acts transitively on the disjoint union H+3 H−3 of hyperboloids in R41 : d(ˇ∗d(G0 ◦ϕ)) = 0,
H±3 = {(y0 .y1 , y2 , y3 ) ∈ R41 | y0 = ±(y12 + y22 + y32 + 1)1/2 }. Then, for any y ∈ H+3 H−3 , there exists an element ϕy ∈ SO(1, 3) such that ϕy (y) = e0 = (1, 0, 0, 0). Define a function G(x, y) on S13 × (H+3 H−3 ) by G(x, y) = (G0 ◦ϕy )(x), which is rewritten as ϕy (x), e0 1 x, y 1 = , 1+ 1+ G(x, y) = 2 2 1 + x, y2 1 + ϕy (x), e0 2 where ·, · denotes the standard inner product of R41 . Note that G(·, y) is independent of the choice of ϕy . For y = (y0 , y) ∈ H+3 H−3 (⊂ R41 ), we can express G(x, y), by replacing x with (sinh ρ, cosh ρ v), as follows: −y0 sinh ρ + cosh ρy ∗ v 1 . 1+ G(x, y) = 2 ∗ 2 1 + (−y0 sinh ρ + cosh ρy v) It is easily verified that G(x, y) is smooth near r 2 = 0 and near q 2 = 0, where r 2 = e2ρ and q 2 = e−2ρ . If y ∈ H+3 (resp. y ∈ H−3 ), then we have G(x, y) → 1 (resp. G(x, y) → 0) as r 2 → 0, G(x, y) → 0 (resp. G(x, y) → 1) as q 2 → 0. Let µ+ and µ− be probability measures with compact support on H+3 and H−3 , respectively. Define a smooth positive function V on S13 by G(x, y)dµ+ (y) + G(x, y)dµ− (y). V = H+3
H−3
Compact Scalar-flat Indefinite K¨ahler Surfaces with Hamiltonian S 1 -Symmetry
37
Then we see that [ˇ∗dV ]/2π = 0 in ImH 2 (S13 ; Z), and that V satisfies the conditions in Proposition 6. Thus the corresponding metric g¯ V is a self-dual neutral metric on S 2 ×S 2 . ± ± 3 As for explicit examples, we take µ± = (1/N ) N j =1 δpj± for {p1 , . . . , pN } ⊂ H± , respectively, where δy denotes the Dirac measure with support at y ∈ H+3 H−3 . Then V is expressed as V =
N 1 (G(x, pj+ ) + G(x, pj− )). N
(11)
j =1
Thus g¯ V is a self-dual neutral metric on S 2 × S 2 , and possesses an obvious S 1 -symmetry induced by the S 1 -bundle structure. According to the configuration of {pj± }N j =1 , the met− + ric g¯ V may have other extra symmetries. For example, if pj = −pj (j = 1, . . . , N ), then V ≡1. Hence g¯ V is the standard metric g0 , which has a natural S(O(3)×O(3))-symmetry. If {pj± }N j =1 are simultaneously collinear, that is, if they lie on a two-dimensional subspace in R41 , then g¯ V has an (S 1 ×S 1 )-symmetry. Indeed, the extra S 1 -symmetry is given by the rotation around the intersection of the subspace and the totally geodesic sphere S 2 = {ρ = 0}. In particular, if N = 1, then g¯ V always has an (S 1 ×S 1 )-symmetry (cf. Poon [21]). We also introduce another construction of explicit solutions (V , θ ) of (4) satisfying the conditions in Proposition 6, by using the separation method. Fix an identification S13 = R × S 2 , and let ρ be the coordinate of the factor R and ζ the complex coordinate of S 2 . To find solutions of (4), let us suppose that V = 1 + R(ρ)Y (ζ ) is a solution of d(ˇ∗dV ) = 0,
(12)
where R(ρ) is a smooth function on R satisfying limρ→±∞ R(ρ) = 0 and Y (ζ ) is also a smooth function on S 2 . By changing variable z = tanh ρ and setting Z(z) = R(ρ), we then see that (12) is equivalent to the following (1 − z2 )Z = −( + 1)Z, Z(±1) = 0, S 2 Y = −( + 1)Y
(13) (14)
( = 0, 1, 2, . . . ), where := d/dz. Let P (z) be the Legendre polynomial of degree . Then Z (z) := (1 − z2 )P (z) is a solution of (13) ( = 0, 1, 2, . . . ). Let {cm }|m|≤; ≥1 be a finite collection of real numbers and set V =1+ cm Z (z)Ym (ζ ), (15) ≥1 |m|≤
{Ym (ζ )}|m|≤
where is a basis for the eigenspace of S 2 satisfying (14). Then V is a solution of (12), and V > 0 for sufficiently small {|cm |}. The condition [ˇ∗dV ]/2π = 0 in H 2 (S13 ; R) is verified by the following computation:
∗ˇ dV = ≥1 |m|≤ cm Ym Z (z)ˇ∗dz + (1 − z2 )P ∗ˇ dYm = =
≥1
=−
|m|≤ cm
≥1
|m|≤ cm
≥1
−Z Ym ωS 2 + dP ∧
√
−1(∂ − ∂)Ym
( + 1)P Ym ωS 2 √ √ +d(P −1(∂ − ∂)Ym ) + 2P −1∂∂Ym
|m|≤ cm d(P
√
−1(∂ − ∂)Ym ).
38
H. Kamada
Then, as in Remark 2 and the description after Theorem 7, these non-constant solutions (V , θ ) of (4) yield non-conformally-flat, self-dual neutral K¨ahler metrics g¯ V on (M, JV ) = P1 × P1 . Remark 3. The isometry class of g¯ V = (sech2 ρ)gV depends not only on V but also on the identification S13 = R×S 2 (cf. §5). We call the totally geodesic sphere S 2 = {ρ = 0} in S13 the neck sphere (or the equatorial sphere). 5. Isometry Classes We next investigate the isometry classes of our indefinite K¨ahler metrics on P1 × P1 . In this section, we assume that every solution (V , θ ) of (4) under consideration satisfies the conditions in Proposition 6, and often use the notation g¯ V ,θ = sech2 ρ(−V −1 θ 2 +V gS 3 ) 1 instead of g¯ V . For a smooth function φ on S13 , (V , θ + dφ) is also a solution of (4). Such a mod√ ification stems from a gauge transformation = e −1φ : M → M with ∗ g¯ V ,θ = g¯ V ,θ+dφ . Let ϕ be an orientation-preserving isometry of S13 . Then (V ◦ϕ, ϕ ∗ θ) is also a solution of (4), and g¯ V ◦ϕ,ϕ ∗ θ and g¯ V ,θ are related by g¯ V ◦ϕ,ϕ ∗ θ =
cosh2 (ρ◦ϕ) ∗ ϕ g¯ V ,θ . cosh2 ρ
If ϕ also preserves the neck sphere S 2 = {ρ = 0}, then ϕ ∗ g¯ V ,θ = g¯ V ◦ϕ,ϕ ∗ θ , that is, g¯ V ,θ and g¯ V ◦ϕ,ϕ ∗ θ are isometric. It is natural to ask, for solutions (V , θ ) and (V , θ ) of (4), when the corresponding metrics g¯ V ,θ and g¯ V ,θ are isometric. The main goal in this section is to prove the following Theorem 8. Let g¯ V ,θ and g¯ V ,θ be non-conformally-flat, scalar-flat indefinite K¨ahler metrics on (M, J ) = P1 × P1 corresponding to solutions (V , θ ) and (V , θ ) of (4), respectively. Let ϕ be an orientation-preserving diffeomorphism on M and suppose that ϕ ∗ g¯ V ,θ = g¯ V ,θ . Then ϕ is induced from an isometry of S13 preserving the neck sphere S 2 . In particular, V ◦ϕ = V holds. Before proving Theorem 8, we first recall basic properties of holomorphic vector coordinate charts for the fields on P1 × P1 . Let (U, z) and (U , z ) be local holomorphic 1 first P satisfying z = 1/z on U U , and (V , ζ ) and (V , ζ ) be those for the second P1 satisfying ζ = 1/ζ on V V . It is known that any holomorphic vector field on P1 × P1 can be expressed in terms of (z, ζ ) as α(z)∂z + β(ζ )∂ζ , where ∂z := ∂/∂z and ∂ζ := ∂/∂ζ . Here α(z) and β(ζ ) are polynomials in z and ζ of degree at most two, respectively. By construction, each metric g¯ V ,θ has an S 1 -symmetry induced from the S 1 -bundle structure and the Killing vector field ξ generating the symmetry is time-like (i.e., g¯ V ,θ (ξ, ξ ) < 0 outside the set Zero(ξ ) of zeros of ξ ). Regarding time-like Killing vector fields, we prove the following Proposition 9. Let g be an indefinite K¨ahler metric on P1 × P1 and ξ ≡ 0 a time-like Killing vector field on (P1√ × P1 , g). Then, under suitable holomorphic coordinates, ξ may be expressed as ξ = −1a(z∂z − z∂z ) for some a ∈ R. In particular, Zero(ξ ) is identified with {0, ∞} × P1 .
Compact Scalar-flat Indefinite K¨ahler Surfaces with Hamiltonian S 1 -Symmetry
39
√ Proof. Let ξ C := ξ − −1J ξ denote the holomorphic vector field on P1 × P1 associated with ξ . At a point p ∈ Zero(ξ ), taking suitable holomorphic coordinates (z, ζ ) with (z(p), ζ (p)) = (0, 0), we may assume that ξ C = α(z)∂z +β(ζ )∂ζ for α(z) = z(a1 +a2 z) and β(ζ ) = ζ (b1 + b2 ζ ), where a1 , a2 , b1 , b2 ∈ C. Since ξ is a time-like vector field, we have 2g(ξ, ξ ) = |α(z)|2 g11¯ (z, ζ ) + α(z)β(ζ )g12¯ (z, ζ ) +β(ζ )α(z)g21¯ (z, ζ ) + |β(ζ )|2 g22¯ (z, ζ ) < 0
(16)
for (z, ζ ) ∈ / Zero(ξ ). Here gj l¯ = 2g(∂zj , ∂zl ) (j, l = 1, 2) denote the components of g with respect to {∂z1 := ∂z , ∂z2 := ∂ζ }. Then, since g is a J -invariant indefinite metric, the determinant of the matrix (gj l¯) is negative, that is, g11¯ (z, ζ )g22¯ (z, ζ ) − |g12¯ (z, ζ )|2 < 0.
(17)
The assumption that ξ is Killing (i.e., Lξ g ≡ 0) is equivalent to 2ξg11¯ + (α (z) + α (z))g11¯ = 0, 2ξg12¯ + (α (z) + β (ζ ))g12¯ = 0, 2ξg22¯ + (β (ζ ) + β (ζ ))g22¯ = 0.
(18)
At (z, ζ ) = (0, 0), we obtain (a1 + a1 )g11¯ = (a1 + b1 )g12¯ = (b1 + b1 )g22¯ = 0.
(19)
We first prove |a1 |2 + |b1 |2 > 0. If a1 = b1 = 0, then (18) is rewritten as Re(a2 z2 ∂z + b2 ζ 2 ∂ζ )g11¯ = −(a2 z + a2 z)g11¯ , Re(a2 z2 ∂z + b2 ζ 2 ∂ζ )g12¯ = −(a2 z + b2 ζ )g12¯ , Re(a2 z2 ∂z + b2 ζ 2 ∂ζ )g22¯ = −(b2 ζ + b2 ζ )g22¯ . √ If (z, ζ ) → (0, 0) along z = ζ ∈ R and along z = ζ ∈ −1R, then we have (a2 + a2 )g11¯ = (a2 + b2 )g12¯ = (b2 + b2 )g22¯ = 0, (a2 − a2 )g11¯ = (a2 − b2 )g12¯ = (b2 − b2 )g22¯ = 0 at (z, ζ ) = (0, 0). It follows from (17) that a2 = b2 = 0, which contradicts ξ ≡ 0. Thus we obtain |a1 |2 + |b1 |2 > 0. Setting ζ = λz in (16) for an arbitrary λ ∈ C and z → 0, we have |a1 |2 g11¯ + λa1 b1 g12¯ + λa1 b1 g21¯ + |λ|2 |b1 |2 g22¯ ≤ 0
(20)
at (z, ζ ) = (0, 0). In particular, we obtain |a1 |2 g11¯ (0, 0) ≤ 0
and |b1 |2 g22¯ (0, 0) ≤ 0.
(21)
Furthermore, (20) also implies that |a1 b1 |4 |g12¯ (0, 0)|2 (|g12¯ (0, 0)|2 − g11¯ (0, 0)g22¯ (0, 0)) ≤ 0. Then it follows from (17) that |a1 b1 |4 |g12¯ (0, 0)|2 = 0. By (19), we obtain g12¯ (0, 0) = 0,
and hence g11¯ (0, 0)g22¯ (0, 0) < 0,
(22)
40
H. Kamada
since |a1 |2 + |b1 |2 > 0. Then |a1 |2 |b1 |2 = 0 follows from (21). Furthermore, (22) implies that ξ has no isolated zero. To show this, we would suppose the contrary, that is, (|a1 |2 + |a2 |2 )(|b1 |2 + |b2 |2 ) > 0. Setting either z = 0 or ζ = 0 in (16), we would have g22¯ (0, ζ ) ≤ 0 and g11¯ (z, 0) ≤ 0, which contradicts (22). Therefore (z, ζ ) = (0, 0) is not an isolated zero of ξ . Thus we may assume that ξ C = z(a1 + a2 z)∂z for a1 =0. Setting z˜ := z/(a1 + a2 z), we have ξ C = a1 z˜ ∂z˜ . Here a1 + a1 = 0 follows from (19). Remark 4. On a Hirzebruch surface Fd of degree d ≥ 1, we can show the following result, by an argument similar to that in the proof of Proposition 9: Let g be an indefinite K¨ahler metric on Fd and ξ a time-like Killing vector field on (Fd , g) (d ≥ 1). Then ξ is the real part of a holomorphic vector field on Fd tangent to the fibers of Fd → P1 . Next, we study the case where two time-like Killing vector fields exist. Proposition 10. Let ξ1 and ξ2 be time-like Killing vector fields on an indefinite K¨ahler surface (P1 × P1 , g). Then ξ1 and ξ2 is linearly dependent over R if and only if [ξ1 , ξ2 ] ≡ 0. Furthermore, if ξ1 and ξ2 are linearly independent over R, then so are ξ1 , ξ2 and [ξ1 , ξ2 ]. √ Proof. We may assume that ξ1C = −1az∂z for some real number a =0. Since ξ2 is also a Killing vector field on (P1 × P1 , g), the holomorphic vector field ξ2C is expressed as either ξ2C = (a0 + a1 z + a2 z2 )∂z
or
ξ2C = (b0 + b1 ζ + b2 ζ 2 )∂ζ .
Note that [ξ1 , ξ2 ] ≡ 0 if and only if [ξ1C , ξ2C ] ≡ 0 in both cases. In the second case, ξ1 and ξ2 clearly commute, so we may also assume that ξ2C = √ −1bζ ∂ζ for a real constant b = 0. It follows from (19) and (16) that g12¯ (0, 0) = 0, g11¯ (0, 0) < 0 and g22¯ (0, 0) < 0. These imply that g11¯ (0, 0)g22¯ (0, 0) − |g12¯ (0, 0)|2 = g11¯ (0, 0)g22¯ (0, 0) > 0, which contradicts (17). In the case where ξ2C = (a0 + a1 z + a2 z2 )∂z , the Lie bracket [ξ1C , ξ2C ] is computed as √ [ξ1C , ξ2C ] = −1a(a2 z2 − a0 )∂z . If [ξ1 , ξ2 ] ≡ 0 everywhere on P1 × P1 , then, by (19), we see that ξ2C = a1 z∂z for a pure imaginary a1 , and hence that ξ1 and ξ2 are linearly dependent over R. If [ξ1 , ξ2 ] ≡ 0, then it can be verified that ξ1 , ξ2 , [ξ1 , ξ2 ] are linearly independent over R. Proof of Theorem 8. Let g¯ V := g¯ V ,θ and g¯ V := g¯ V ,θ be non-conformally-flat, selfdual neutral K¨ahler metrics on M = P1 × P1 given by g¯ V = sech2ρ(−V −1 θ 2 + V gS 3 ), 1
g¯ V = sech2ρ(−V
−1 2
θ
+ V gS 3 ), 1
ϕ ∗ g¯
respectively. Then the pull-back metric V is written as ϕ ∗ g¯ V = sech2(ρ◦ϕ) −(V ◦ϕ)−1 (ϕ ∗ θ )2 + (V ◦ϕ)ϕ ∗ gS 3 . 1
(23)
Suppose that there exists an orientation-preserving isometry ϕ : (M, g¯ V ) → (M, g¯ V ). Let ξ and ξ be the Killing vector fields tangent to the fibers on (M, g¯ V ) and (M, g¯ V ) satisfying θ (ξ ) = θ (ξ ) = 1, respectively. Since ϕ ∗ g¯ V = g¯ V , the pull-back vector field ϕ ∗ ξ of ξ is also a time-like Killing vector field on (M, g¯ V ). We have to consider the following two cases:
Compact Scalar-flat Indefinite K¨ahler Surfaces with Hamiltonian S 1 -Symmetry
(1) [ξ, ϕ ∗ ξ ] ≡ 0,
41
(2) [ξ, ϕ ∗ ξ ] ≡ 0.
In the case (1), we see from Proposition 10 that ξ and ϕ ∗ ξ are linearly dependent, that is, ξ = kϕ∗ ξ for some real constant k =0. It is clear that ϕ ∗ θ (ξ ) = k −1 θ (ξ ),
ϕ ∗ g¯ V (ϕ ∗ ξ , ·) = k g¯ V (ξ, ·).
(24)
Comparing the same quantity ϕ ∗ g¯ V (ϕ ∗ ξ , ϕ ∗ ξ ) = −
(V ◦ϕ)−1 cosh2 (ρ◦ϕ)
and
k 2 g¯ V (ξ, ξ ) = −
k 2 V −1 , cosh2 ρ
we then have k 2 (V ◦ϕ) cosh2 (ρ◦ϕ) = V cosh2 ρ.
(25)
Thus ϕ ∗ g¯ V is rewritten as ϕ ∗ g¯ V = −
V −1 V ◦ϕ ∗ 2 (kϕ ϕ ∗ gS 3 . θ ) + 1 cosh2 ρ cosh2 (ρ◦ϕ)
(26)
It follows from (24), (25) and (26) that ϕ ∗ θ = k −1 θ,
ϕ ∗ gS 3 = 1
k 2 cosh4 (ρ◦ϕ) gS 3 . 1 cosh4 ρ
(27)
In particular, ϕ determines a conformal transformation of S13 . It is well-known that the group Conf(S13 ) of (orientation-preserving) conformal transformations of S13 is isomorphic to SO(2, 3). Indeed, if we realize S13 as a hypersurface of the real projective space RP4 : −x 2 + x 2 + x 2 + x 2 − x 2 = 0, 3 0 1 2 3 4 , S1 = (x0 : x1 : x2 : x3 : x4 ) x4 = 1 then the action of Conf(S13 ) on S13 is induced from a linear transformation on R5 preserving the quadratic form −x02 + x12 + x22 + x32 − x42 . This action is also given by a linear fractional transformation as follows: ϕ(x) =
Px + q r ∗x + s
(x ∈ S13 ⊂ R41 ),
(28)
where P is a 4×4-matrix, q, r are column vectors of R4 and s ∈ R such that P ∗ P − rr ∗ = E, P ∗ q − rs = 0, −q ∗ q + s 2 = 1, P P ∗ − qq ∗ = E, −P r + qs = 0, −r ∗ r + s 2 = 1.
(29)
Here ∗ means the metric dual in R41 . Then we have ϕ ∗ gS 3 = (r ∗ x + s)−2 gS 3 . 1
1
In (28), we may express P , q, r, x respectively as q −r0 sinh ρ a b∗ , x= , P = , q = q0 , r = (cosh ρ)v c D r
(30)
42
H. Kamada
where a, q0 , r0 ∈ R, b, c, q, r ∈ R3 and D is a 3 × 3-matrix and x ∈ S13 . Then, from (28), we have cosh (ρ◦ϕ) = 1 + 2
a sinh ρ + cosh ρ b∗ v + q0 r0 sinh ρ + cosh ρ r ∗ v + s
2 .
(31)
Comparing (27) with (30), and using (31), we obtain 1+
a sinh ρ + cosh ρ b∗ v + q0 r0 sinh ρ + cosh ρ r ∗ v + s
2 =
cosh2 ρ . |k(r0 sinh ρ + cosh ρ r ∗ v + s)|
(32)
As ρ → ±∞, we can see that r = 0. Indeed, unless r = 0, the limit of the left-hand side is finite for some v ∈ S 2 , but that of the right is always infinite. This is a contradiction. Since r = 0, it follows from (29) that q = 0, s 2 = 1 and P ∈ O(1, 3). Then (32) is equivalent to k 2 (sech2 ρ + (a tanh ρ + b∗ v)2 )2 = 1.
(33)
Taking ρ → ±∞, and setting ρ = 0, we have k 2 (±a + b∗ v)4 = lim k 2 (sech2ρ + (a tanh ρ + b∗ v)2 )2 = 1,
(34)
k 2 (1 + (b∗ v)2 )2 = 1
(35)
ρ→±∞
for any v ∈ S 2 . Then (34) and (35) imply that k 2 = 1, b = 0 and a 2 = 1, and hence c = 0 and D ∈ O(3). By (25) and (28), it turns out that V ◦ϕ = V and that ϕ is induced from the linear action of O(1) × O(3). Therefore g¯ V and g¯ V are isometric if and only if ϕ belongs to S(O(1) × O(3)), since ϕ is orientation-preserving. Thus, we have verified Theorem 8 in the case (1). We next consider the case (2). In this case, since ξ1 := ξ and ξ2 := ϕ ∗ ξ do not commute, ξ1 , ξ2 and ξ3 := [ξ1 , ξ2 ]( ≡0) are linearly independent, and the corresponding holomorphic vector fields may be given by ξ1C =
√ √ −1az∂z , ξ2C = (a0 + a1 z + a2 z2 )∂z , ξ3C = −1a(a2 z2 − a0 )∂z .
The restrictions of ξ1 , ξ2 , ξ3 to the first factor S 2 ×{ζ }, the z-sphere, are linearly independent Killing vector fields on S 2 ×{ζ } with a negative definite metric g¯ V |S 2 ×{ζ } . It is wellknown that if a two-dimensional Riemannian manifold admits three linearly independent Killing vector fields, then it must be of constant curvature. Thus (S 2 × {ζ }, g¯ V |S 2 ×{ζ } ) is of constant curvature. ¯ V be the K¨ahler form of (g¯ V , JV ), and ωS 2 (z) and ωS 2 (ζ ) the volume forms Let of the unit round spheres S 2 × {ζ } and {z} × S 2 , respectively. Then the Lie derivatives ¯ V vanish identically (a = 1, 2, 3). In particular, we see Lξa ωS 2 (ζ ), Lξa ωS 2 (z) and Lξa ¯ that Lξa (ωS 2 (z) ∧ V ) = (ξa V )ωS 2 (z) ∧ ωS 2 (ζ ) ≡ 0 (a = 1, 2, 3). Thus V is independent of z, so that the equation d(ˇ∗dV ) ≡ 0 is reduced to the Laplace equation on {z}×S 2 . Then V is a constant function, namely, V ≡ 1. Hence g¯ V is the standard product metric on S 2 × S 2 , which however contradicts the assumption that g¯ V is non-conformally-flat.
Compact Scalar-flat Indefinite K¨ahler Surfaces with Hamiltonian S 1 -Symmetry
43
Remark 5. From Theorem 8, we see that indefinite K¨ahler metrics obtained in §4 give rise to infinitely many different isometry classes on P1 ×P1 . For example, let p1+ = e0 ∈ H+3 and p1− ∈ H−3 . Corresponding to a solution V of (4) defined by (11) for {p1+ , p1− } (N = 1), the isometry class of g¯ V is parameterized by the hyperbolic distance between p1+ and −p1− in H+3 . As other examples, let V = 1 + εZ Ym and V = 1 + ε Z Ym be solutions of (4) as special forms in (15) for sufficiently small positive constants ε and ε . Then the corresponding metrics g¯ V and g¯ V are not isometric, if either = , m = m and ε = ε or = . Acknowledgement. Several components of this work were carried out when the author was visiting the Mathematical Institute, Tˆohoku University in the academic year 2000. I would like to thank the participants of the Geometry Seminar at the institute for their friendship. Thanks also go to Professors Henrik Pedersen, Kazuo Akutagawa, Hiroyasu Izeki and Keisuke Ueno for a variety of discussion, suggestions and comments. I am particularly grateful to Professors Akito Futaki and Yasuhiro Nakagawa for many valuable comments on the Bando-Calabi-Futaki character, and to Professor Shin Nayatani for many helpful suggestions and constant encouragement since the early stage of this work. I would also like to thank the referee for useful suggestion and especially for correcting a flaw in the earlier proof of Proposition 1. Last but not least, I wish to express my sincere gratitude to Professor Seiki Nishikawa for his insightful advice and continuous help as well as for the warm welcome he extended to me while I visited to Tohoku University.
References 1. Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Berlin-Heidelberg-New YorkSpringer-Verlag, 1984 2. Bott, R.: A residue formula for holomorphic vector fields. J. Differ. Geom. 1, 311–330 (1967) 3. Frankel, T.: Fixed points and torsion on K¨ahler manifolds. Ann. of Math. II 70, 1–8 (1959) 4. Futaki, A., Mabuchi, T.: Moment maps and symmetric multilinear forms associated with integral symplectic classes. Asian J. Math. 6, 349–372 (2002) 5. Kamada, H.: Indefinite analogue of hyperbolic ansatz and its application. Proceedings of The Fifth Pacific Rim Geometry Conference (Sendai, 2000), Tohoku Mathematical Publications 20, Sendai: Tohoku University Mathematical Institute, 2001, pp. 69–73 6. Kamada, H.: Self-dual K¨ahler metrics of neutral signature on complex surfaces. Tohoku Mathematical Publications 24, Sendai: Tohoku University, 2002 7. Kamada, H., Machida, Y.: Self-duality of metrics of type (2, 2) on four-dimensional manifolds. Tˆohoku Math. J. II, 49, 259–275 (1997) 8. Kazdan, J. L., Warner, F. W.: Curvature function for open 2-manifolds.Ann. of Math. II, 99, 203–219 (1974) 9. Kobayashi, S.: Transformation groups in differential geometry. Berlin-Heidelberg-New York: Springer-Verlag, 1972 10. Kobayashi, S., Nomizu, K.: Foundations of differential geometry, I, II. New York-London: Interscience Publishers, 1963, 1969 11. Kuiper, N. H.: On conformally-flat spaces in the large. Ann. of Math. II, 50, 916–924 (1949) 12. LeBrun, C.: Explicit self-dual metrics on CP2 # · · · #CP2 . J. Differ. Geom. 34, 223–253 (1991) 13. LeBrun, C.: Kodaira dimension and the Yamabe problem. Comm. Anal. Geom. 7, 133–156 (1999) 14. Matsushita, Y., Law, P.: Hitchin-Thorpe type inequalities for pseudo-Riemannian 4-manifolds of metric signature (+ + −−). Geom. Dedicata 87, 65–89 (2001) 15. McDuff, D.: The moment map for circle actions on symplectic manifolds. J. Geom. Phys. 5, 149– 160 (1988) 16. Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965) 17. Nakagawa, Y.: Bando-Calabi-Futaki characters of K¨ahler orbifolds. Math. Ann. 314, 369–380 (1999) 18. Nakagawa, Y.: Bando-Calabi-Futaki character of compact toric manifolds. Tohoku Math. J. 53, 479–490 (2001) 19. Ooguri, H., Vafa, C.: Geometry of N = 2 strings. Nuclear Phys. B 361, 469–518 (1991) 20. Petean, J.: Indefinite K¨ahler-Einstein metrics on compact complex surfaces. Commun. Math. Phys. 189, 227–235 (1997)
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21. Poon,Y. S.: Compact self-dual manifolds with positive scalar curvature. J. Differ. Geom. 24, 97–132 (1986) 22. Qin, Z.: Complex structures on certain differentiable 4-manifolds. Topology 32, 551–566 (1993) 23. Taubes, C. H.: The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett. 1, 809–822 (1994) 24. Tian, G.: K¨ahler-Einstein metrics on algebraic manifolds. In: Transcendental methods in algebraic geometry (Cetraro, 1994), Lecture Notes in Math. 1646, Berlin: Springer, 1996, pp. 143–185 25. Wolf, J. A.: Spaces of constant curvature. 5th ed., Houston, TX: Publish or Perish, Inc., 1984 Communicated by M.R. Douglas
Commun. Math. Phys. 254, 45–89 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1235-z
Communications in
Mathematical Physics
Anti-Self-Dual Instantons with Lagrangian Boundary Conditions I: Elliptic Theory Katrin Wehrheim Department of Mathematics, Princeton University, Fine Hall, Princeton, NJ 08544-1000, USA. E-mail:
[email protected] Received: 6 February 2003 / Accepted: 20 July 2004 Published online: 11 November 2004 – © Springer-Verlag 2004
Abstract: We study nonlocal Lagrangian boundary conditions for anti-self-dual instantons on 4-manifolds with a space-time splitting of the boundary. We establish the basic regularity and compactness properties (assuming Lp -bounds on the curvature for p > 2) as well as the Fredholm theory in a compact model case. The motivation for studying this boundary value problem lies in the construction of an instanton Floer homology for 3-manifolds with boundary. The present paper is part of a program proposed by Salamon for the proof of the Atiyah-Floer conjecture for homology-3-spheres. 1. Introduction Let X be a manifold with boundary, let G be a compact Lie group, and consider a principal G-bundle P → X. The natural boundary condition for the Yang-Mills equation ∗ F = 0 on P is ∗F | dA A A ∂X = 0. (These are the Euler-Lagrange equations for the energy functional |FA |2 .) For this boundary value problem there are regularity and compactness results, see for example [U1, U2, W1]. Every solution is gauge equivalent to a smooth solution, and Uhlenbeck compactness holds: Every sequence of solutions with Lp -bounded curvature (where p > 21 dim X) contains a subsequence that is C ∞ -convergent up to a sequence of gauge transformations. On an oriented 4-manifold, the anti-self-dual instantons, i.e. connections satisfying FA + ∗FA = 0, are special first order solutions of the Yang-Mills equation. An important application of Uhlenbeck’s theorem is the compactification of the moduli space of anti-self-dual instantons over a manifold without boundary, leading to the Donaldson invariants of smooth 4-manifolds [D1] and to the instanton Floer homology groups of 3-manifolds [Fl]. On a 4-manifold with boundary the boundary condition ∗FA |∂X = 0 for anti-selfdual instantons implies that the curvature vanishes altogether at the boundary. This is an overdetermined boundary value problem comparable to Dirichlet boundary conditions for holomorphic maps. As in the latter case it is natural to consider weaker Lagrangian
46
K. Wehrheim
boundary conditions. The Cauchy-Riemann equation becomes elliptic when augmented with Lagrangian or more generally totally real boundary conditions. We consider a version of such Lagrangian boundary conditions for anti-self-dual instantons on a 4-manifold with a space-time splitting of the boundary, and prove that they suffice to obtain the analogue of the above mentioned regularity and compactness results for Yang-Mills connections. More precisely, we consider oriented 4-manifolds X such that each connected component of the boundary ∂X is diffeomorphic to S × , where S is a 1-manifold and is a closed Riemann surface. We shall study a boundary value problem associated to a gauge invariant Lagrangian submanifold L of the space of connections on : The restriction of the anti-self-dual instanton to each time-slice of the boundary is required to belong to L. This boundary condition arises naturally from examining the ChernSimons functional on a 3-manifold Y with boundary . Namely, the Lagrangian boundary condition renders the Chern-Simons 1-form on the space of connections closed, see [S]. The resulting gradient flow equation leads to the boundary value problem studied in this paper (for the case X = R × Y ). Besides the regularity and compactness properties on noncompact manifolds we also establish the Fredholm theory for the compact model case X = S 1 × Y . One motivation for studying the present boundary value problem lies in the AtiyahFloer conjecture for Heegaard splittings of a homology-3-sphere: A Heegaard splitting Y = Y0 ∪ Y1 of a homology 3-sphere Y into two handlebodies Y0 and Y1 with common boundary gives rise to two Floer homologies (i.e. generalized Morse homologies) as follows: Firstly, the moduli space M of gauge equivalence classes of flat connections on the trivial SU(2)-bundle over is a symplectic manifold (with singularities) and the moduli spaces LYi of flat connections over that extend to a flat connection over Yi are (singular) Lagrangian submanifolds of M as explained in [W2]. The symplectic Floer symp homology HF∗ (M , LY0 , LY1 ) is now generated by the intersection points of the Lagrangian submanifolds, and the generalized connecting orbits (that define the boundary operator) are pseudoholomorphic strips with boundary values in the two Lagrangian submanifolds. (In view of the singularities of M , an appropriate generalization of the concept of pseudoholomorphic strips will be required to give a strict definition of this Floer homology.) It was conjectured by Atiyah [A2] and Floer that this should be isomorphic to the instanton Floer homology HFinst ∗ (Y ). For the latter, the critical points are the flat SU(2)-connections over Y . These are the actual critical points of the ChernSimons functional, and the connecting orbits are given by its generalized flow lines, i.e. anti-self-dual instantons on R × Y . The program by Salamon [S] for the proof of this conjecture is to define the instanton Floer homology HFinst ∗ (Y, L) for 3-manifolds with boundary ∂Y = using boundary conditions associated to a Lagrangian submanifold L ⊂ M . Then the conjectured isomorphism might be established in two steps via the intermediate Floer homology HFinst ∗ ([0, 1] × , LY0 × LY1 ), as described in the outlook below. Boundary value problems for (Hermitian) Yang-Mills connections were also used by Donaldson [D2], who considered connections induced by Hermitian holomorphic bundles with a Dirichlet boundary condition on the metric. Fukaya [Fu] was the first to suggest the use of Lagrangian boundary conditions for anti-self-dual instantons in order to define a Floer homology for 3-manifolds with boundary. He studies a slightly different equation, involving a degeneration of the metric in the anti-self-duality equation, and uses SO(3)-bundles that are nontrivial over the boundary , so the moduli space M becomes smooth. However, when working on handlebodies
Anti-Self-Dual Instantons with Lagrangian Boundary Conditions I: Elliptic Theory
47
as 3-manifolds, or when considering the Lagrangian submanifold LY as in the AtiyahFloer conjecture, then one necessarily deals with the trivial bundle (or a non-connected Lie group). The present paper sets up the basic analysis for a construction of HFinst ∗ (Y, L) as outlined in [S], using trivial SU(2)-bundles. We will only consider trivial G-bundles for general compact Lie groups G. However, our main Theorems A, B, and C below generalize directly to nontrivial bundles – just the notation becomes more cumbersome. The main theorems are described below; they are proven in Sect. 2 and 3. The Appendix reviews the regularity theory for the Neumann and Dirichlet problem in the weak formulation that will be needed throughout this paper. Here we moreover introduce a technical tool for extracting regularity results for single components of a 1-form from weak equations that are related to a combination of Neumann and Dirichlet problems.
1.1. Notation and main results. Throughout this paper, we consider the trivial G-bundle over a 4-manifold X. Here G is a compact Lie group with Lie algebra g. We denote the Lie bracket on g by [·, ·], and we equip g with a G-invariant inner product ·, · . A connection on the trivial bundle G × X is a g-valued 1-form A ∈ 1 (X; g). We denote the space of smooth connections by A(X) := 1 (X; g). Associated to a connection A ∈ A(X) one has the exterior derivative dA on g-valued differential forms given by dA η = dη + [A ∧ η] ∀η ∈ k (X; g). Here the Lie bracket indicates how the values of the differential forms are paired. Now dA ◦ dA does not necessarily vanish, but it is a zeroeth order operator, dA dA η = [FA ∧ η] given by the curvature FA = dA + 21 [A ∧ A] ∈ 2 (X; g). So dA ◦ dA = 0 if and only if the connection is flat, that is its curvature vanishes. The gauge group G(X) := C ∞ (X, G) represents the smooth bundle isomorphisms. So a gauge transformation u ∈ G(X) acts on A ∈ A(X) by pullback, u∗ A = u−1 Au + u−1 du. On a compact base manifold M and for k ∈ N0 and 1 ≤ p ≤ ∞ we denote the Sobolev spaces of connections and gauge transformations by Ak,p (M) := W k,p (M, T∗ M ⊗ g), G k,p (M) := W k,p (M, G). For kp > dim M the latter is well-defined via an embedding G ⊂ R , and it forms a group G k,p (M) that acts smoothly on Ak−1,p (M), see e.g. [W1, Appendix B]. For nonk,p k,p compact base manifolds X we denote by Aloc (X) and Gloc (X) the spaces of sections and maps for which the regularity holds on all compact subsets of X. Next, we describe the class of 4-manifolds that we will be considering. Here and throughout all Riemann surfaces are closed oriented 2-dimensional manifolds. Moreover, unless otherwise mentioned, all manifolds are allowed to have a smooth boundary. Then the interior of a submanifold X ⊂ X is to be understood with respect to the relative topology, i.e. int X := X \ cl(X \ X ) might intersect ∂X.
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K. Wehrheim
Definition 1.1. A 4-manifold with a boundary space-time splitting is a pair (X, τ ) with the following properties: (i) X is an oriented 4-manifold (with boundary) which can be exhausted by a nested sequence X= Xk , k∈N
where all Xk are compact submanifolds and deformation retracts of X such that Xk ⊂ int Xk+1 for all k ∈ N. (ii) τ = (τ1 , . . . , τn ) is an n-tuple of embeddings τi : Si × i → X with disjoint images, where i is a Riemann surface and Si is either an open interval in R or is equal to S 1 = R/Z. (iii) The boundary ∂X is the union ∂X =
n
τi (Si × i ).
i=1
Definition 1.2. Let (X, τ ) be a 4-manifold with a boundary space-time splitting. A Riemannian metric g on X is called compatible with τ if for each i = 1, . . . n there exists a neighbourhood Ui ⊂ Si × [0, ∞) of Si × {0} and an extension of τi to an embedding τ¯i : Ui × i → X such that τ¯i∗ g = ds 2 + dt 2 + gs,t . Here gs,t is a smooth family of metrics on i and we denote by s the coordinate on Si and by t the coordinate on [0, ∞). We call a triple (X, τ, g) with these properties a Riemannian 4-manifold with a boundary space-time splitting. Remark 1.3. In Definition 1.2 the extended embeddings τ¯i are uniquely determined by the metric as follows. The restriction τ¯i |t=0 = τi to the boundary is prescribed, and the paths t → τ¯i (s, t, z) are normal geodesics. Example 1.4. Let X := R×Y , where Y is a compact oriented 3-manifold with boundary ∂Y = , and let τ : R× → X be the obvious inclusion. Given any two metrics g− and g+ on Y there exists a metric g on X such that g = ds 2 + g− for s ≤ −1, g = ds 2 + g+ for s ≥ 1, and (X, τ, g) satisfies the conditions of Definition 1.2. However, the metric g cannot necessarily be chosen in the form ds 2 + gs (one has to homotope the embeddings and the metrics). Now let (X, τ, g) be a Riemannian 4-manifold with a boundary space-time splitting and consider a trivial G-bundle over X for a compact Lie group G. Let p > 2, then for each i = 1, . . . ,n the Banach space of connections A0,p (i ) carries the symplectic form ω(α, β) = i α ∧ β . Note that the Hodge ∗ operator for any metric on i is an ω-compatible complex structure on A0,p (i ), since ω(·, ∗·) defines a positive definite inner product – the L2 -metric. We call a Banach submanifold L ⊂ A0,p (i ) Lagrangian if for all A ∈ L its tangent space is isotropic, ω|TA L ≡ 0, and coisotropic in the following sense: If α ∈ A0,p (i ) satisfies ω(α, TA L) = {0}, then α ∈ TA L.
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We fix an n-tuple L = (L1 , . . . , Ln ) of Lagrangian submanifolds Li ⊂ A0,p (i ) that are contained in the space of flat connections and that are gauge invariant, 0,p
Li ⊂ Aflat (i )
and
u∗ Li = Li
∀u ∈ G 1,p (i ).
(1)
Here Aflat (i ) is the space of weakly flat Lp -connections on i as introduced in [W2, Sect. 3]. For our purposes it is enough to know that this space coincides with G 1,p (i )∗ Aflat (i ), the set of connections that is W 1,p -gauge equivalent to a smooth flat connection, A ∈ A(i ) with FA = 0. Moreover, we recall from [W2, Lemma 4.2] the fact that the above assumptions on the Li imply that they are totally real with respect to the Hodge ∗ operator for any metric on i , i.e. for all A ∈ Li one has the topological sum 0,p
A0,p (i ) = TA Li ⊕ ∗TA Li . 1,p
We consider the following boundary value problem for connections A ∈ Aloc (X)
∗FA + FA = 0, τi∗ A|{s}×i ∈ Li
(2)
∀s ∈ Si , i = 1, . . . , n.
Observe that the above boundary condition is meaningful since for every neighbourhood U × ⊂ S × [0, ∞) × of a boundary slice {s} × {0} × one has the continuous embedding W 1,p (U × ) ⊂ W 1,p (U, Lp ()) → C 0 (U, Lp ()). The first nontrivial observation is that every connection in Li is gauge equivalent to a smooth connection on i and hence Li ∩ A() is dense in Li , as shown in [W2, Theorem 3.1]. Moreover, the 1,p Li are modelled on Lp -spaces, and every Wloc -connection on X satisfying the boundary condition in (2) can be locally approximated by smooth connections satisfying the same boundary condition, see [W2, Corollaries 4.4, 4.5]. Note that the present boundary value problem is a first order equation with first order boundary conditions (flatness in each time-slice). Moreover, the boundary conditions contain some crucial nonlocal Lagrangian information. We moreover emphasize that while Li is a smooth Banach submanifold of A0,p (i ), the quotient Li /G 1,p (i ) is not 0,p required to be a smooth submanifold of the moduli space Mi := Aflat (i )/G 1,p (i ), which itself might be singular. An example for these Lagrangians is Li = LY , the space of flat connections on i that extend to flat connections on a handlebody Y with ∂Y = i . The nonlocal Lagrangian information in this case is the extensibility condition, which is equivalent to the vanishing of the holonomies along those paths in i that are contractible in Y . See [W2, Lemma 4.6] for a detailed discussion of this example. To overcome the difficulties arising from the singularities in the quotient, we work with the (smooth) quotient by the based gauge group. The following two theorems are the regularity and compactness results for solutions of (2) generalizing the regularity theorem and the Uhlenbeck compactness forYang-Mills connections on 4-manifolds without boundary. They will be proven in Sect. 2. 1,p
Theorem A (Regularity). Let p > 2. Then every solution A ∈ Aloc (X) of the boundary value problem (2) is gauge equivalent to a smooth solution, that is there exists a gauge 2,p transformation u ∈ Gloc (X) such that u∗ A ∈ A(X) is smooth.
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Theorem B (Compactness). Let p > 2 and let g ν be a sequence of metrics compatible with τ that uniformly converges with all derivatives on every compact subset to a smooth 1,p metric. Suppose that Aν ∈ Aloc (X) is a sequence of solutions of (2) with respect to the ν metrics g such that for every compact subset K ⊂ X there is a uniform bound on the curvature FAν Lp (K) . Then there exists a subsequence (again denoted Aν ) and a sequence of gauge trans2,p formations uν ∈ Gloc (X) such that uν ∗ Aν converges uniformly with all derivatives on every compact subset to a smooth connection A ∈ A(X). The difficulty of these results lies in the global nature of the boundary condition. This makes it impossible to directly generalize the proof of the regularity and compactness theorems for Yang-Mills connections, where one chooses suitable local gauges, obtains the higher regularity and estimates from an elliptic boundary value problem, and then patches the gauges together. With the present global boundary condition one cannot obtain local regularity results. Thus we generalize a more global approach by Salamon to manifolds with boundary: Firstly, Uhlenbeck’s weak compactness theorem yields a weakly convergent subsequence. Its limit serves as a reference connection with respect to which one can achieve a global relative Coulomb gauge for a further subsequence. Then it remains to establish elliptic estimates and regularity results for the given boundary value problem together with the relative Coulomb gauge equations. The crucial point in this step is the regularity for the -component of the connections in a neighbourhood U × of a boundary component. Here one deals with a Cauchy-Riemann equation on U with values in the Banach space A0,p () and with Lagrangian boundary conditions. The regularity results for this boundary value problem are provided by [W2] in the general framework of a Cauchy-Riemann equation for functions with values in a complex Banach space and with totally real boundary conditions. The case 2 < p ≤ 4, when W 1,p -functions are not automatically continuous, poses some special difficulties in this last step. Firstly, in order to obtain regularity results from the Cauchy-Riemann equation, one has to straighten out the Lagrangian submanifold by going to suitable coordinates. This requires a C 0 -convergence of the connections, which in case p > 4 is given by a standard Sobolev embedding. In case p > 2 one still obtains a special compact embedding W 1,p (U × ) → C 0 (U, Lp ()) that suits our purposes. Secondly, the straightening of the Lagrangian introduces a nonlinearity in the Cauchy-Riemann equation that already poses some problems in case p > 4. In case p ≤ 4 this forces us to also deal with the Cauchy-Riemann equation with values in an L2 -Hilbert space and then use some interpolation inequalities for Sobolev norms. For the definition of the standard instanton Floer homology it suffices to prove a compactness result like Theorem B for p = ∞. In our case however the bubbling analysis [W3] requires the compactness result for some p < 3. This is why we have taken some care to deal with this case. Our third main result is the Fredholm theory in Sect. 3. It is a step towards proving that the moduli space of finite energy solutions of (2) is a manifold whose components have finite (but possibly different) dimensions. This also exemplifies our hope that the further analytical details of Floer theory will work out along the usual lines once the right analytic setup has been found in the proof of Theorems A and B. In the context of Floer homology and in Floer-Donaldson theory it is important to consider 4-manifolds with cylindrical ends. This requires an analysis of the asymptotic behaviour which will be carried out elsewhere. Here we shall restrict the discussion of the Fredholm theory to the compact case. The crucial point is the behaviour of the
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linearized operator near the boundary; in the interior we are dealing with the usual antiself-duality equation. Hence it suffices to consider the following model case. Let Y be a compact oriented 3-manifold with boundary ∂Y = and suppose that (gs )s∈S 1 is a smooth family of metrics on Y such that X = S 1 × Y,
τ : S 1 × → X,
g = ds 2 + gs
satisfy the assumptions of Definition 1.2. Here the space-time splitting τ of the boundary is the obvious inclusion τ : S 1 × → ∂X = S 1 ×, where = ni=1 i might be a disjoint union of an n-tuple of connected Riemann surfaces i . An n-tuple of Lagrangian submanifolds Li ⊂ A0,p (i ) as in (1) then constitutes a gauge invariant Lagrangian submanifold L := L1 × . . . × Ln of the symplectic Banach space 0,p A0,p () = A0,p (1 ) × . . . × A0,p (n ) such that L ⊂ Aflat (). In order to linearize the boundary value problem (2) together with the local slice condition, fix a smooth connection A + ds ∈ A(S 1 × Y ) such that As := A(s)|∂Y ∈ L for all s ∈ S 1 . Here ∈ C ∞ (S 1 × Y, g) and A ∈ C ∞ (S 1 × Y, T∗ Y ⊗ g) is an S 1 -fam1,p ily of 1-forms on Y (not a 1-form on X as previously). Now let EA be the space of S 1 -families of 1-forms α ∈ W 1,p (S 1 × Y, T∗ Y ⊗ g) that satisfy the boundary conditions ∗α(s)|∂Y = 0
and
α(s)|∂Y ∈ TAs L
for all s ∈ S 1 .
(3)
Then the linearized operator D(A, ) : EA × W 1,p (S 1 × Y, g) −→ Lp (S 1 × Y, T∗ Y ⊗ g) × Lp (S 1 × Y, g) 1,p
is given with ∇s = ∂s + [ , ·] by ∗ α . D(A, ) (α, ϕ) = ∇s α − dA ϕ + ∗dA α , ∇s ϕ − dA ∗ (α + ϕds), and the first boundThe second component of this operator is −dA+ ds ary condition is ∗(α + ϕds)|∂X = 0, corresponding to the choice of a local slice at A + ds. In the first component of D(A, ) we have used the global space-time splitting of the metric on S 1 × Y to identify the self-dual 2-forms ∗γs − γs ∧ ds with families γs of 1-forms on Y . The vanishing of this component is equivalent to the linearization + dA+ ds (α + ϕds) = 0 of the anti-self-duality equation. Furthermore, the boundary condition α(s)|∂Y ∈ TAs L is the linearization of the Lagrangian boundary condition in the boundary value problem (2).
Theorem C (Fredholm properties). Let Y be a compact oriented 3-manifold with boundary ∂Y = and let S 1 × Y be equipped with a product metric ds 2 + gs that is compatible with the embedding τ : S 1 × → S 1 × Y . Let A + ds ∈ A(S 1 × Y ) such that A(s)|∂Y ∈ L for all s ∈ S 1 . Then the following holds for all p > 2: (i) D(A, ) is Fredholm. 1,p (ii) There is a constant C such that for all α ∈ EA and ϕ ∈ W 1,p (S 1 × Y, g), (α, ϕ)W 1,p ≤ C D(A, ) (α, ϕ)Lp + (α, ϕ)Lp .
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(iii) Let q ∈ (1, 2) ∪ (2, ∞). There is a constant C such that the following holds: Suppose that β ∈ Lq (S 1 × Y, T∗ Y ⊗ g) and ζ ∈ Lq (S 1 × Y, g), and assume that there exists a constant c such that for all α ∈ C ∞ (S 1 × Y, T∗ Y ⊗ g) satisfying (3) and for all ϕ ∈ C ∞ (S 1 × Y, g), D(A, ) (α, ϕ) , (β, ζ ) ≤ c (α, ϕ)Lq ∗ . S 1 ×Y
Then β and ζ are of class W 1,q and (β, ξ )W 1,q ≤ Cc. Here and throughout we use the notation p1 + p1∗ = 1 for the conjugate exponent p ∗ of p. The above inner product ·, · is the pointwise inner product in (T∗ Y ⊗ g) × g. Theorem C (ii) actually extends to an L2 -estimate for W 1,p -regular (α, φ), that can be proven by more elementary methods than the general case, as will be shown in Sect. 3. In fact, this estimate was already stated in [S] as an indication for the well-posedness of the boundary value problem (2). The reason for our assumption q = 2 in Theorem C (iii) is a technical problem in dealing with the singularities of L/G 1,p (). We resolve them by dividing only by the based gauge group. This leads to coordinates of Lp (, T∗ ⊗ g) in a Banach space, part 1,p of which is a based Sobolev space Wz (, g) of functions vanishing at a fixed point z ∈ . So these coordinates that straighten out TL along A|S 1 ×∂Y are well-defined only for p > 2. We prove Theorem C (iii) by using such coordinates either for β or for the test 1-forms α, so we assume that either q > 2 or q ∗ > 2. This is completely sufficient for our purposes – proving the Fredholm property in Theorem C (i) for p > 2. The Fredholm property of a generalized operator D(A, ) for p = 2 follows from more general Hilbert space techniques, that will be carried out elsewhere. 1.2. Outlook. We give a brief sketch of Salamon’s program for the proof of the AtiyahFloer conjecture (for more details see [S]) in order to point out the significance of the present results for the whole program. The first step of the program is to define the instanton Floer homology HFinst ∗ (Y, L) of a 3-manifold Y with boundary ∂Y = and a (singular) Lagrangian submanifold L = L/G 1,p () ⊂ M in the moduli space of flat connections. The Floer complex will be generated by the gauge equivalence classes of irreducible flat connections A ∈ A(Y ) with Lagrangian boundary conditions A| ∈ L. 1 For any two such connections A+ , A− one then has to study the moduli space of Floer connecting orbits,
M(A− , A+ ) = A˜ ∈ A(R × Y ) A˜ satisfies (2), lim A˜ = A± /G(R × Y ). s→±∞
Theorem A shows that the boundary value problem (2) is well-posed. In particular, the spaces of smooth connections and gauge transformations in the definition of the above 1,p 2,p moduli space can be replaced by the Sobolev completions Aloc and Gloc . The next step in the construction of the Floer homology groups is the analysis of the asymptotic behaviour of the finite energy solutions of (2) on R × Y , which will be carried out elsewhere. 1 A connection A ∈ A (Y ) is called irreducible if its isotropy subgroup of G (Y ) (the group of gauge flat transformations that leave A fixed) is discrete, i.e. dA |0 is injective. There should be no reducible flat connections with Lagrangian boundary conditions other than the gauge orbit of the trivial connection. This will be guaranteed by certain conditions on Y and L, for example this is the case when L = LY ¯ such that Y ∪ Y is a homology-3-sphere. for a handlebody Y with ∂Y =
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Combining this with Theorem C one obtains an appropriate Fredholm theory and proves that for a suitably generic perturbation the spaces M(A− , A+ ) are smooth manifolds. In the monotone the connections in the k-dimensional part Mk (A− , A+ ) have a case 2 fixed energy |FA˜ | . Theorem B is a major step towards a compactification of the spaces Mk (A− , A+ ). It proves their compactness under the assumption of an Lp -bound on the curvature for p > 2, whereas a priori the L2 -norm is bounded due to the fixed energy. So the key remaining analytic task is an analysis of the possible bubbling phenomena. This is carried out in [W3] and draws upon the techniques developed in this paper. When this is understood, the construction of the Floer homology groups should be routine. In particular, for the metric independence note that one can interpolate between different metrics on Y as in Example 1.4, and Theorem B allows for the variation of metrics on X. So this paper sets up the basic analytic framework for the Floer theory of 3-manifolds with boundary. Moreover, the consideration of general 4-manifolds X (rather than just X = R × Y ) in Theorems A and B will allow for the definition of a product structure on this new Floer homology. The further steps in the program for the proof of the Atiyah-Floer conjecture are to consider a Heegaard splitting Y = Y0 ∪ Y1 of a homology 3-sphere, and identify symp inst HFinst (M , LY0 , LY1 ) respectively. ∗ ([0, 1] × , LY0 × LY1 ) with HF∗ (Y ) and HF∗ (These isomorphisms should also intertwine the ring structures on all three Floer homologies.) In both cases, the Floer complexes can be identified by elementary arguments, so the main task is to identify the connecting orbits. In the case of the two instanton Floer homologies, the idea is to choose an embedding (0, 1) × → Y starting from a tubular neighbourhood of ⊂ Y at t = 21 and shrinking {t} × to the 1-skeleton of Yt for t = 0, 1. Then the anti-self-dual instantons on R × Y pull back to anti-self-dual instantons on R × [0, 1] × with a degenerate metric for t = 0 and t = 1. On the other hand, one can consider anti-self-dual instantons on R × [ε, 1 − ε] × with boundary values in LY0 and LY1 . As ε → 0, one should be able to pass from this genuine boundary value problem to solutions on the closed manifold Y . This is a limit process for the boundary value problem studied in this paper. The identification of the instanton and symplectic Floer homologies requires an adaptation of the adiabatic limit argument in [DS] to boundary value problems for anti-selfdual instantons and pseudoholomorphic curves respectively. Here one again deals with the boundary value problem (2) studied in this paper. As the metric on is scaled to zero, the solutions, i.e. anti-self-dual instantons on R × [0, 1] × with Lagrangian boundary conditions in LY0 and LY1 should be in one-to-one correspondence with connections on R × [0, 1] × that descend to pseudoholomorphic strips in M with boundary values in LY0 and LY1 . The basic elliptic properties of the boundary value problem (2) that are established in this paper will also play an important role in this adiabatic limit analysis.
2. Regularity and Compactness The aim of this section is to prove the regularity Theorem A and the compactness Theorem B. Both theorems are dealing with a noncompact base manifold X that is exhausted by compact submanifolds Xk . We shall use an extension argument by Donaldson and Kronheimer [DK, Lemma 4.4.5] to reduce the problem to compact base manifolds. For the following special version of this argument a detailed proof can be found in
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K. Wehrheim
[W1, Propositions 7.6, 9.8]. At this point, the assumption that the exhausting compact submanifolds Xk are deformation retracts of X comes in crucially. It ensures that every gauge transformation on Xk can be extended to X, which is a central point in the argument due to Donaldson and Kronheimer. Proposition 2.1. Let the 4-manifold M˜ = k∈N Mk be exhausted by compact subman˜ and let p > 2. ifolds Mk ⊂ int Mk+1 that are deformation retracts of M, ˜ and suppose that for each k ∈ N there exists a gauge trans(i) Let A ∈ Aloc (M) formation uk ∈ G 2,p (Mk ) such that u∗k A|Mk is smooth. Then there exists a gauge 2,p ˜ transformation u ∈ Gloc (M) such that u∗ A is smooth. 1,p ˜ be given and suppose that the (ii) Let a sequence of connections (Aν )ν∈N ⊂ Aloc (M) following holds: For every k ∈ N and every subsequence of (Aν )ν∈N there exist a further subsequence (νk,i )i∈N and gauge transformations uk,i ∈ G 2,p (Mk ) such that sup uk,i ∗ Aνk,i W ,p (M ) < ∞ ∀ ∈ N. 1,p
k
i∈N
Then there exists a subsequence (νi )i∈N and a sequence of gauge transformations 2,p ˜ ui ∈ Gloc (M) such that sup ui ∗ Aνi W ,p (M ) < ∞ ∀k ∈ N, ∈ N. i∈N
k
So in order to prove Theorem A it suffices to find smoothing gauge transformations on the compact submanifolds Xk in view of Proposition 2.1 (i). For that purpose we shall use the so-called local slice theorem. The following version is proven e.g. in [W1, Theorem 8.1]. Note that we are dealing with trivial bundles, so we will be using the product connection as a reference connection in the definition of the Sobolev norms of connections. Proposition 2.2 (Local Slice Theorem). Let M be a compact 4-manifold, let p > 2, and let q > 4 be such that q1 > p1 − 41 (or q = ∞ in case p > 4). Fix Aˆ ∈ A1,p (M) and let a constant c0 > 0 be given. Then there exist constants ε > 0 and CCG such that the following holds. For every A ∈ A1,p (M) with ˆ q ≤ε A − A
and
ˆ W 1,p ≤ c0 A − A
there exists a gauge transformation u ∈ G 2,p (M) such that ∗ ∗ ˆ ˆ q ˆ q, dA ≤ CCG A − A u∗ A − A ˆ (u A − A) = 0, and ˆ W 1,p ≤ CCG A − A ˆ W 1,p . ˆ ∂M = 0, u∗ A − A ∗(u∗ A − A)| Remark 2.3. (i) If the boundary value problem in Proposition 2.2 is satisfied one says ˆ This is equivalent to v ∗ Aˆ being in that u∗ A is in Coulomb gauge relative to A. −1 Coulomb gauge relative to A for v = u , i.e. the boundary value problem can be replaced by ∗ ∗ ˆ dA (v A − A) = 0, ∗ ˆ ∗(v A − A)|∂M = 0.
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(ii) The assumptions in Proposition 2.2 on p and q guarantee that one has a compact Sobolev embedding W 1,p (M) → Lq (M). (iii) One can find uniform constants for varying metrics in the following sense. Fix a metric g on M. Then there exist constants ε, δ > 0, and CCG such that the assertion of Proposition 2.2 holds for all metrics g with g − g C 1 ≤ δ. 1,p
In the following we outline the proof of Theorem A. Given a solution A ∈ Aloc (X) of (2) one proves the assumption of Proposition 2.1 (i) for each of the exhausting submanifold Xk as follows. One finds a sufficiently large compact submanifold M ⊂ X with Xk ⊂ M. Then one chooses a smooth connection A0 ∈ A(M) sufficiently W 1,p close to A and applies the local slice theorem with the reference connection Aˆ = A to find a gauge transformation that puts A0 into relative Coulomb gauge with respect to A. This is equivalent to finding a gauge transformation that puts A into relative Coulomb gauge with respect to A0 . We denote this gauge transformed connection again by A ∈ A1,p (M). It satisfies the following boundary value problem: ∗ dA0 (A − A0 ) = 0, ∗FA + FA = 0, (4) ∗(A − A0 )|∂M = 0, τi∗ A|{s}×i ∈ Li ∀s ∈ Si , i = 1, . . . , n. More precisely, the Lagrangian boundary condition only holds for those s ∈ Si and i ∈ {1, . . . n} for which τi ({s} × i ) is entirely contained in ∂M. If M is chosen large enough (in particular, it has to contain the full boundary slice τi ({s} × i ) whenever this intersects Xk at all), then the regularity Theorem 2.6 below will assert the smoothness of A on Xk . The proof of Theorem B goes along similar lines. We will use Proposition 2.1 (ii) to reduce the problem to compact base manifolds. On these, we shall use the following weak Uhlenbeck compactness theorem (see [U1, W1, Theorem 7.1]) to find a subsequence of gauge equivalent connections that converges W 1,p -weakly. Proposition 2.4 (Weak Uhlenbeck Compactness). Let M be a compact 4-manifold and let p > 2. Suppose that the sequence of connections Aν ∈ A1,p (M) is such that FAν p is uniformly bounded. Then there exists a subsequence (again denoted (Aν )ν∈N ) and a sequence uν ∈ G 2,p (M) of gauge transformations such that uν ∗ Aν weakly converges in A1,p (M). The limit A0 of the convergent subsequence then serves as a reference connection Aˆ in the local slice theorem, Proposition 2.2, and this way one obtains a W 1,p -bounded sequence of connections A˜ ν that solve the boundary value problem (4). This makes crucial use of the compact Sobolev embedding W 1,p → Lq on compact 4-manifolds (with q from the local slice theorem). The estimates in the subsequent Theorem 2.6 then provide the higher W k,p -bounds on the connections that will imply the compactness. One difficulty in the proof of this regularity theorem is that due to the global nature of the boundary conditions one has to consider the -components of the connections near the boundary as maps into the Banach space A0,p () that solve a Cauchy-Riemann equation with Lagrangian boundary conditions. In order to prove a regularity result for such maps one has to straighten out the Lagrangian submanifold by using coordinates in A0,p ().
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(This is done in [W2].) Thus on domains U × at the boundary a crucial assumption is that the -components of the connections all lie in one such coordinate chart, that is one needs the connections to converge strongly in the L∞ (U, Lp ())-norm. In the case p > 4 this is ensured by the compact embedding W 1,p → L∞ on U × . To treat the case 2 < p ≤ 4 we shall make use of the following special Sobolev embedding: Lemma 2.5. Let M, N be compact manifolds and let p > max(dim M, dim N ). Then the following embedding is compact: W 1,p (M × N ) → L∞ (M, Lp (N )). Proof of Lemma 2.5. Since M is compact it suffices to prove the embedding in (finitely many) coordinate charts. These can be chosen as either balls B2 ⊂ Rm in the interior or half balls D2 = B2 ∩ Hm in the half space Hm = {x ∈ Rm x1 ≥ 0} at the boundary of M. We can choose both of radius 2 but cover M by balls and half balls of radius 1. So it suffices to consider a bounded set K ⊂ W 1,p (B2 × N ) and prove that it restricts to a precompact set in L∞ (B1 , Lp (N )), and similarly with the half balls. Here we use the Euclidean metric on Rm , which is equivalent to the metric induced from M. For a bounded subset K ⊂ W 1,p (D2 × N ) of functions over the half ball we define the subset K ⊂ W 1,p (B2 × N ) by extending every function u ∈ K to B2 \ Hm via u(x1 , x2 , . . . , xm ) := u(−x1 , x2 , . . . , xm ) for x1 < 0. The thus extended function is still W 1,p -regular with twice the norm of u. So K also is a bounded subset, and if this restricts to a precompact set in L∞ (B1 , Lp (N )), then also K ⊂ L∞ (D1 , Lp (N )) is precompact. Hence it suffices to consider the interior case of the full ball. The claimed embedding is continuous by the standard Sobolev estimates – check for example in the proof of [Ad, Theorem 5.4,] that the estimates generalize directly to functions with values in a Banach space. In fact, one obtains an embedding W 1,p (B2 × N ) ⊂ W 1,p (B2 , Lp (N )) → C 0,λ (B2 , Lp (N )) into some H¨older space with λ = 1 − m p > 0. One can also use this Sobolev estimate n 1,p for W (N ) with λ = 1 − p > 0 combined with the inclusion Lp → L1 on B2 to obtain a continuous embedding
W 1,p (B2 × N) ⊂ Lp (B2 , W 1,p (N )) → Lp (B2 , C 0,λ (N )) ⊂ L1 (B2 , C 0,λ (N )). Now consider a bounded subset K ⊂ W 1,p (B2 × N ). The first embedding ensures that the functions K u : B2 → Lp (N ) are equicontinuous. For some constant C, u(x) − u(y)Lp (N) ≤ C|x − y|λ
∀u ∈ K, x, y ∈ B2 .
The second embedding asserts that for some constant C , uC 0,λ (N) ≤ C ∀u ∈ K. B2
(5)
(6)
In order to prove that K ⊂ L∞ (B1 , Lp (N )) is precompact we now fix any ε > 0 and show that K can be covered by finitely many ε-balls. Pick J ∈ C ∞ (Rm , [0, ∞)) with supp J ⊂ B1 and Rm J = 1. Then the functions Jδ (x) := δ −m J (x/δ) are mollifiers for δ > 0 with supp Jδ ⊂ Bδ and Rm Jδ = 1.
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Let δ ≤ 1, then Jδ ∗ u|B1 ∈ C ∞ (B1 , Lp (N )) is well-defined. Moreover, choose δ > 0 sufficiently small such that for all u ∈ K, m Jδ ∗ u − u ∞ = sup J (y) (u(x − y) − u(x)) d y p δ p L (B1 ,L (N)) L (N) x∈B1 Bδ ≤ sup Jδ (y) C|y|λ dm y ≤ Cδ λ ≤ 41 ε. x∈B1 Bδ
Now it suffices to prove the precompactness of Kδ := {Jδ ∗ u u ∈ K}. If this holds then Kδ can be covered by 21 ε-balls around Jδ ∗ ui with ui ∈ K for i = 1, . . . , I 2 and the above estimate shows that K is covered by the ε-balls around the ui . Indeed, for each u ∈ K one has Jδ ∗ u − Jδ ∗ ui L∞ (B1 ,Lp (N)) ≤ 2ε for some i = 1, . . . , I and thus u − ui ≤ u − Jδ ∗ u + Jδ ∗ u − Jδ ∗ ui + Jδ ∗ ui − ui ≤ ε. The precompactness of Kδ ⊂ L∞ (B1 , Lp (N )) will follow from the Arz´ela-Ascoli theorem (see e.g. [L, IX §4]). Firstly, the smoothened functions Jδ ∗u are still equicontinuous on B1 . For all u ∈ K and x, y ∈ B1 use (5) to obtain (Jδ ∗ u)(x) − (Jδ ∗ u)(y)Lp (N) ≤ Bδ Jδ (z) u(x − z) − u(y − z)Lp (N) dm z ≤ Bδ Jδ (z) C|x − y|λ dm z = C|x − y|λ . Secondly, the L∞ -norm of the smoothened functions is bounded by the L1 -norm of the original ones, so for fixed δ > 0 one obtains a uniform bound from (6) : For all u ∈ K and x ∈ B1 , (Jδ ∗ u)(x)C 0,λ (N) ≤ Jδ (x − y) u(y)C 0,λ (N) dm y ≤ C Jδ ∞ . B2
Now the embedding C 0,λ (N ) → Lp (N ) is a standard compact Sobolev embedding, so this shows that the subset {(Jδ ∗ u)(x) u ∈ K} ⊂ Lp (N ) is precompact for all x ∈ B1 . Thus the Arz´ela-Ascoli theorem asserts that Kδ ⊂ L∞ (B1 , Lp (N )) is compact, and this finishes the proof of the lemma. In the proof of Theorem B, the weak Uhlenbeck compactness together with the local slice theorem and this lemma will put us in the position to apply the following main regularity theorem that also is the crucial point in the proof of Theorem A: Theorem 2.6. Let (X, τ, g0 ) be a Riemannian 4-manifold with a boundary space-time splitting. For every compact subset K ⊂ X there exists a compact submanifold M ⊂ X such that K ⊂ M and the following holds for all p > 2: (i) Suppose that A ∈ A1,p (M) solves (4). Then A|K ∈ A(K) is smooth.3 2 If a subset K ⊂ (X, d) of a metric space is precompact, then for fixed ε > 0 one firstly finds v1 , . . . , vI ∈ X such that for each x ∈ K one has d(x, vi ) ≤ ε for some vi . For each vi choose one such xi ∈ K, or simply drop vi if this does not exist. Then K is covered by 2ε-balls around the xi : For each x ∈ K one has d(x, xi ) ≤ d(x, vi ) + d(vi , xi ) for some i = 1, . . . , I . 3 More precisely, there is an open neighbourhood of K ⊂ X on which A is smooth.
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K. Wehrheim
∗ (ii) Fix a smooth connection A0 ∈ A(M) nsuch that τi A0 |{s}×i ∈ Li for all s ∈ Si and i = 1, . . . , n. Moreover, fix V = i=1 τ¯0,i (Ui × i ), a compact neighbourhood of K ∩ ∂X. (Here τ¯0,i denotes the extension of τi given by the geodesics of g0 .) Then for every given constant C1 there exist constants δ > 0, δk > 0, and Ck for all k ≥ 2 such that the following holds: Fix k ≥ 2 and let g be a metric on M that is compatible with τ and satisfies g − g0 C k+2 (M) ≤ δk . Suppose that A ∈ A1,p (M) solves the boundary value problem (4) with respect to the metric g and satisfies
A − A0 W 1,p (M) ∗ τ¯0,i (A − A0 )|i L∞ (Ui ,A0,p (i ))
≤ C1 , ≤ δ ∀i = 1, . . . , n.
Then A|K ∈ A(K) is smooth by (i) and A − A0 W k,p (K) ≤ Ck . We first give some preliminary results for the proof of Theorem 2.6. The interior regularity as well as the regularity of the Ui -components on a neighbourhood Ui × i of a boundary component Si × i will be a consequence of the following regularity result for Yang-Mills connections. The proof is similar to that of Lemma A.2 and can be found in full detail in [W1, Prop. 9.5]. Here M is a compact Riemannian manifold with boundary ∂M and outer unit normal ν. One then deals with two different spaces of test functions,
Cδ∞ (M, g) := φ ∈ C ∞ (M, g) φ|∂M = 0 ,
C ∞ (M, g) := φ ∈ C ∞ (M, g) ∂φ = 0 . ν
∂ν ∂M
Proposition 2.7. Let (M, g) be a compact Riemannian 4-manifold. Fix a smooth reference connection A0 ∈ A(M). Let X ∈ (TM) be a smooth vector field that is either perpendicular to the boundary, i.e. X|∂M = h · ν for some h ∈ C ∞ (∂M), or is tangential, i.e. X|∂M ∈ (T∂M). In the first case let T = Cδ∞ (M, g), in the latter case let T = Cν∞ (M, g). Moreover, let N ⊂ ∂M be an open subset such that X vanishes in a neighbourhood of ∂M \ N ⊂ M. Let 1 < p < ∞ and k ∈ N be such that either kp > 4 4p or k = 1 and 2 < p < 4. In the first case let q := p, in the latter case let q := 8−p . Then there exists a constant C such that the following holds. Let A = A0 + α ∈ Ak,p (M) be a connection. Suppose that it satisfies ∗ α dA = 0, 0 (7) ∗α|∂M = 0 on N ⊂ ∂M, and that for all 1-forms β = φ · ιX g with φ ∈ T , FA , dA β = 0. M
(8)
Then α(X) ∈ W k+1,q (M, g) and
α(X)W k+1,q ≤ C 1 + αW k,p + α3W k,p .
Moreover, the constant C can be chosen such that it depends continuously on the metric g and the vector field X with respect to the C k+1 -topology.
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Remark 2.8. In the case k = 1 and 2 < p < 4 the iteration of Proposition 2.7 also allows to obtain W 2,p -regularity and -estimates from initial W 1,p -regularity and -estimates.
To see this remark, first note that the Sobolev embedding W 2,q → W 1,p holds with 4q p = 4−q since q < 4. Now as long as p < 4 one can iterate the proposition and Sobolev embedding to obtain regularity and estimates in W 1,pi with p0 = p and pi+1 =
4qi 2pi = ≥ θpi > pi . 4 − qi 4 − pi
2 > 1 this sequence terminates after finitely many steps at some pN ≥ 4. Since θ := 4−p Now in case pN > 4 the proposition even yields W 2,pN -regularity and -estimates. In > 8 in order to conclude case pN = 4 one only uses W 1,pN for some smaller pN 3 W 2,pN +1 -regularity and -estimates for pN+1 > 4. Similarly, in case k = 1 and p = 4 one only needs two steps to reach W 2,p for some p > 4. The above proposition and remark can be used on all components of the connections in Theorem 2.6 except for the -components in small neighbourhoods U × of boundary components S × . For the regularity of their higher derivatives in the -direction we shall use the following lemma. The crucial regularity of the derivatives in the direction of U of the -components will then follow from the general regularity theory for Cauchy-Riemann equations in [W2].
Lemma 2.9. Let k ∈ N0 and 1 < p < ∞. Let be a compact manifold, let be a Riemann surface, and equip × with a product metric g ⊕ g, where g = (gx )x∈ is a smooth family of metrics on . Then there exists a constant C such that the following holds: ∗ α are of class W k,p Suppose that α ∈ W k,p ( × , T∗ ) such that both d α and d on × . Then ∇ α also is of class W k,p and one has the following estimate on × : ∗ ∇ αW k,p ≤ C d αW k,p + d αW k,p + αW k,p . Here ∇ denotes the family of Levi-Civita connections on that is given by the family of metrics g. Moreover, for every fixed family of metrics g one finds a C k -neighbourhood of metrics for which this estimate holds with a uniform constant C. Proof of Lemma 2.9. We first prove this for k = 0, i.e. suppose that α ∈ Lp ( × ∗ α (defined as weak derivatives) are also of class Lp . We , T∗ ) and that d α, d introduce the following functions: ∗ f := d α ∈ Lp ( × ),
g := − ∗ d α ∈ Lp ( × ),
and choose sequences f ν , g ν ∈ C ∞ ( × ), and α ν ∈ C ∞ (× , T∗ ) that converge to f, g, and α respectively in the Lp -norm. Note that f = g = 0 in Lp (), so the f ν and g ν can be chosen such that their mean value over also vanishes for all z ∈ . Then fix z ∈ and find ξ ν , ζ ν ∈ C ∞ ( × ) such that ξ ν = f ν , ζ ν = g ν , ξ ν (x, z) = 0 ∀x ∈ , ζ ν (x, z) = 0 ∀x ∈ .
60
K. Wehrheim j +2,p
j,p
These solutions are uniquely determined since : Wz () → Wm () is a bounded isomorphism for every j ∈ N0 depending smoothly on the metric, i.e. on j,p x ∈ . Here Wm () denotes the space of W j,p -functions with mean value zero and j +2,p Wz () consists of those functions that vanish at z ∈ . Furthermore, let πx : 1 () → h1 (, gx ) be the projection of the smooth 1-forms ∗ with respect to the metric g to the harmonic part h1 () = ker = ker d ∩ ker d x on . Then π is a family of bounded operators from Lp (, T∗ ) to W j,p (, T∗ ) for any j ∈ N0 , and it depends smoothly on x ∈ . So the harmonic part of α˜ ν is also smooth, π ◦ α˜ ν ∈ C ∞ ( × , T∗ ). Now consider α ν := d ξ ν + ∗ d ζ ν + π ◦ α˜ ν ∈ C ∞ ( × , T∗ ). We will show that the sequence α ν of 1-forms converges to α in the Lp -norm and that moreover ∇ α ν is an Lp -Cauchy sequence. For that purpose we will use the following estimate. For all 1-forms β ∈ W 1,p (, T∗ ) abbreviating d = d, βW 1,p () ≤ C d∗ βLp () + dβLp () + π(β)W 1,p () (9) ≤ C d∗ βLp () + dβLp () + βLp () . Here and in the following C denotes any finite constant that is uniform for all metrics gx on in a family of metrics that lies in a sufficiently small C k -neighbourhood of a fixed family of metrics. To prove (9) we use the Hodge decomposition β = dξ + ∗dζ + π(β). (See e.g. [Wa, Theorem 6.8] and recall that one can identify 2-forms on with func2,p tions via the Hodge ∗ operator.) Here one chooses ξ, ζ ∈ Wz () such that they solve ∗ ξ = d β and ζ = ∗dβ respectively and concludes from Proposition A.1 for some uniform constant C, dξ W 1,p () ≤ ξ W 2,p () ≤ Cd∗ βLp () , ∗dζ W 1,p () ≤ ζ W 2,p () ≤ CdβLp () . The second step in (9) moreover uses the fact that the projection to the harmonic part is bounded as map π : Lp (, T∗ ) → W 1,p (, T∗ ). Now consider the 1-forms α − α ν ∈ Lp ( × , T∗ ). For almost all x ∈ we have α(x, ·) − α ν (x, ·) ∈ Lp (, T∗ ) as well as ∗d (α(x, ·) − α ν (x, ·)) ∈ Lp () and ∗ (α(x, ·) − α ν (x, ·)) ∈ Lp (). Then for these x ∈ one concludes from the Hodge d decomposition that in fact α(x, ·) − α ν (x, ·) ∈ W 1,p (, T∗ ). So we can apply (9) and integrate over x ∈ to obtain for all ν ∈ N, p
α − α ν Lp (×) p ≤ α(x, ·) − α ν (x, ·)Lp (,gx ) ∗ p p p d (α − α ν )Lp () + d (α − α ν )Lp () + π(α − α˜ ν )W 1,p () ≤C p p p ≤ C f − f ν Lp (×) + g − g ν Lp (×) + α − α˜ ν Lp (×) . In the last step we again used the continuity of π. This proves the convergence α ν → α in the Lp -norm, and hence ∇ α ν → ∇ α in the distributional sense. Next, we use (9) to estimate for all ν ∈ N,
Anti-Self-Dual Instantons with Lagrangian Boundary Conditions I: Elliptic Theory p
∇ α ν Lp (×) =
61
p
∇ α ν (x, ·)Lp (,gx ) ∗ ν p ≤C d α Lp () + d α ν Lp () + α ν Lp () ∗ ν p p p ≤ C d α Lp (×) + d α ν Lp (×) + α ν Lp (×) .
∗ α ν = ξ ν = f ν → f = d∗ α, Here one deals with Lp -convergent sequences d ν ν ν − ∗ d α = ζ = g → g = − ∗ d α, and α ν → α. So (∇ α ν )ν∈N is uniformly bounded in Lp ( × ) and hence contains a weakly Lp -convergent subsequence. The limit is ∇ α since this already is the limit in the distributional sense. Thus we have proven the Lp -regularity of ∇ α on ×, and moreover the above estimate is preserved under the limit, which proves the lemma in the case k = 0,
∇ αLp (×) ≤ lim inf ∇ α ν Lp (×) ν→∞ ∗ ν ≤ lim inf C d α Lp (×) + d α ν Lp (×) + α ν Lp (×) ν→∞ ∗ = C d αLp (×) + d αLp (×) + αLp (×) . In the case k ≥ 1 one can now use the previous result to prove the lemma. Let ∗ α are of class W k,p . We denote by α ∈ W k,p ( × , T∗ ) and suppose that d α, d ∇ the covariant derivative on × . Then we have to show that ∇ k ∇ α is of class Lp . So let X1 , . . . , Xk be smooth vector fields on × and introduce α˜ := ∇X1 . . . ∇Xk α ∈ Lp ( × , T∗ ). ∗α Both d α˜ and d ˜ are of class Lp since
d α˜ = [d , ∇X1 . . . ∇Xk ]α + ∇X1 . . . ∇Xk d α, ∗ ∗ ∗ d α˜ = [d , ∇X1 . . . ∇Xk ]α + ∇X1 . . . ∇Xk d α. So the result for k = 0 implies that ∇ α˜ is of class Lp , hence ∇ k ∇ α also is of class Lp since for all smooth vector fields, ∇X1 . . . ∇Xk ∇ α = [∇ , ∇X1 . . . ∇Xk ]α + ∇ α. ˜ With the same argument – using coordinate vector fields Xi and cutting them off – one obtains the estimate ∇ k ∇ αLp (×) ∗ αLp (×) + ∇ k d αLp (×) + αW k,p (×) . ≤ C ∇ k d Now this proves the lemma, ∇ αW k,p (×) ≤ ∇ αW k−1,p (×) + ∇ k ∇ αLp (×) ∗ ≤ C d αW k,p (×) + d αW k,p (×) + αW k,p (×) .
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K. Wehrheim
Proof of Theorem 2.6. Recall that a neighbourhood of the boundary ∂X ⊂ X can be ∗ g = ds 2 + dt 2 + g covered by embeddings τ¯0,i : Ui × i → X such that τ¯0,i 0 0;s,t . (In the case (i) we put g0 := g.) SinceK ⊂ X is compact one can cover it by a compact subset Kint ⊂ int X and Kbdy := ni=1 τ¯0,i (I0,i × [0, δ0 ] × i ) for some δ0 > 0 and I0,i ⊂ Si that are either compact intervals in R or equal to S 1 . Moreover, one can ensure that Kbdy ⊂ int V lies in the interior of the fixed neighbourhood of K ∩ ∂X. Since X is exhausted by the compact submanifolds Xk one then finds M := Xk ⊂ X such that both Kbdy and Kint are contained in the interior of M (and thus also K ⊂ M). Now let A ∈ A1,p (M) be a solution of the boundary value problem (4) with respect to a metric g that is compatible with τ . Then we will prove its regularity and the corresponding estimates in the interior case on Kint and in the boundary case on Kbdy separately. Interior case. Firstly, since Kint ⊂ int M and Kint ⊂ int X = X \∂X we find a sequence of compact submanifolds Mk ⊂ int X such that Kint ⊂ Mk+1 ⊂ int Mk ⊂ M for all k ∈ N. We will prove inductively A|Mk ∈ Ak,p (Mk ) for all k ∈ N, in each step improving the differentiability of A at the expense of restriction to a smaller submanifold. This will imply that A|Kint ∈ A(Kint ) is smooth. Moreover, we inductively find constants Ck , δk > 0 such that the additional assumptions of (ii) in the theorem imply A − A0 W k,p (Mk ) ≤ Ck .
(10)
Here we use the fixed smooth metric g0 to define the Sobolev norms – for a sufficiently small C k -neighbourhood of metrics, the Sobolev norms are equivalent with a uniform constant independent of the metric. Moreover, recall that the reference connection A0 is smooth. To start the induction we observe that this regularity and estimate are satisfied for k = 1 by assumption. For the induction step assume this regularity and estimate to hold for some k ∈ N. Then we will use Proposition 2.7 on A|Mk ∈ Ak,p (Mk ) to deduce the regularity and estimate on Mk+1 . Every coordinate vector field on Mk+1 can be extended to a vector field X on Mk that vanishes near the boundary ∂Mk . So it suffices to consider such vector fields, i.e. use N = ∅ in the proposition. Then α := A − A0 satisfies the assumption (7). For the weak equation (8) we calculate for all β = φ · ιX g with φ ∈ T = Cδ∞ (Mk , g), − FA , dA β = dA (φ · ιX g) ∧ FA = φ · ιX g ∧ FA = 0. Mk
Mk
∂Mk
˜ for We have used Stokes’ theorem while approximating A by smooth connections A, which the Bianchi identity dA˜ FA˜ = 0 holds. Now Proposition 2.7 and Remark 2.8 imply that A|Mk+1 ∈ Ak+1,p (Mk+1 ). In the case (ii) of the theorem the proposition moreover provides δk+1 > 0 and a uniform constant C for all metrics g with g − g0 C k+1 (Mk ) ≤ δk+1 such that the following holds: If (10) holds for some constant Ck , then A − A0 W k+1,p (Mk+1 ) ≤ C 1 + A − A0 W k,p (Mk ) + A − A0 3W k,p (M ) k 3 ≤ C 1 + Ck + Ck =: Ck+1 . Here we have used the fact that the Sobolev norm of a 1-form is equivalent to an expression in terms of the Sobolev norms of its components in the coordinate charts.
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In case k = 1 and p ≤ 4, this uniform bound is not found directly but after finitely many iterations of Proposition 2.7 that give estimates on manifolds N1 = M1 and M2 ⊂ Ni+1 ⊂ int Ni . In each step one chooses a smaller δ2 > 0 and a bigger C2 . This iteration uses the same Sobolev embeddings as Remark 2.8. This proves the induction step on the interior part Kint . Boundary case. It remains to prove the regularity and estimates on the part Kbdy near the boundary. So consider a single boundary component K := τ¯0 (I0 × [0, δ0 ] × ). We identify I0 = S 1 ∼ = R/Z or shift the compact interval such that I0 = [−r0 , r0 ] and hence K = τ¯0 ([−r0 , r0 ] × [0, δ0 ] × ) for some r0 > 0. Since Kbdy (and thus also K ) lies in the interior of M as well as V, one then finds R0 > r0 and 0 > δ0 such that τ¯0 ([−R0 , R0 ] × [0, 0 ] × ) ⊂ M ∩ V. Here τ¯0 is the embedding that brings the metric g0 into the standard form ds 2 + dt 2 + g0;s,t . A different metric g compatible with τ defines a different embedding τ¯ such that τ¯ ∗ g = ds 2 + dt 2 + gs,t . However, if g is sufficiently C 1 -close to g0 , then the geodesics are C 0 -close and hence τ¯ is C 0 -close to τ¯0 . (These embeddings are fixed for t = 0, and for t > 0 given by the normal geodesics.) Thus for a sufficiently small choice of δ2 > 0 one finds R > r > 0 and > δ > 0 such that for all τ -compatible metrics g in the δ2 -ball around g0 , K ⊂ τ¯ ([−r, r] × [0, δ] × )
and
τ¯ ([−R, R] × [0, ] × ) ⊂ M ∩ V.
(In the case (i) this holds with r0 , δ0 , R0 , and 0 for the fixed metric g = g0 .) We will prove the regularity and estimates for τ¯ ∗ A on [−r, r] × [0, δ] × . This suffices because for C k+2 -close metrics the embedding τ¯ will be C k+1 -close to the fixed τ¯0 , so that one obtains uniform constants in the estimates between the W k,p -norms of A and τ¯ ∗ A. Furthermore, the families gs,t of metrics on will be C k -close to g0;s,t for (s, t) ∈ [−R, R] × [0, ] if δk is chosen sufficiently small. Now choose compact sub manifolds k ⊂ H := {(s, t) ∈ R2 t ≥ 0} such that for all k ∈ N, [−r, r] × [0, δ] ⊂ k+1 ⊂ int k ⊂ [−R, R] × [0, ]. We will prove the theorem by establishing the regularity and estimates for τ¯ ∗ A on the k × in Sobolev spaces of increasing differentiability. We distinguish the cases p > 4 and 4 ≥ p > 2. In case p > 4 one uses the following induction: I) Let p > 2 and suppose that A ∈ A1,2p (M) solves (4). Then we will prove inductively that τ¯ ∗ A|k × ∈ Ak,q (k × ) for all k ∈ N and with q = p or q = 2p according to whether k ≥ 2 or k = 1. Moreover, we will find a constant δ > 0 and constants Ck , δk > 0 for all k ≥ 2 such that the following holds: If in addition g − g0 C k+2 (M) ≤ δk and A − A0 W 1,2p (M) ∗ τ¯0 (A − A0 )| L∞ (U ,A0,p ()) then for all k ∈ N,
≤ C1 , ≤ δ,
τ¯ ∗ (A − A0 )W k,q (k ×) ≤ Ck .
This is sufficient to conclude the theorem in case p > 4 as follows. One uses I) with p replaced by 21 p to obtain regularity and estimates of A − A0 in A1,p (1 × ), p p A2, 2 (2 ×), and Ak, 2 (k ×) for all k ≥ 3. Recall that the component K of Kbdy is
64
K. Wehrheim p
contained in each τ¯ (k × ). In addition, one has the Sobolev embeddings W k+1, 2 → W k,p → C k−1 on the compact 4-manifolds k+1 × , cf. [Ad, Theorem 5.4]. So this proves the regularity and estimates on Kbdy . In the case 4 ≥ p > 2 a preliminary iteration is required to achieve the regularity and estimates that are assumed in I). In contrast to I) the iteration is in p instead of k. II) Let 4 ≥ p > 2 and suppose that A ∈ A1,p (M) solves (4). Then we will prove inductively that τ¯ ∗ A|j × ∈ A1,pj (j × ) for a sequence (pj ) with p1 = p and pj +1 = θ (pj ) · pj , where θ : (2, 4] → (1, 17 16 ] is monotonely increasing and thus the sequence terminates with pN > 4 for some N ∈ N. Moreover, we will find constants δ > 0 and C1,j , δ1,j > 0 for j = 2, . . . , N such that the following holds: If for some j = 1, . . . , N with g − g0 C 3 (M) ≤ δ1,j we have A − A0 W 1,p (M) ∗ τ¯0 (A − A0 )| L∞ (U ,A0,p ()) then
τ¯ ∗ (A − A0 )W 1,pj (
j ×)
≤ C1 , ≤ δ,
≤ C1,j .
Assuming I) and II) we first prove the theorem for the case 4 ≥ p > 2. After finitely many steps the iteration of II) gives regularity and estimates in A1,pN (N × ) with pN > 4 and under the assumption g − g0 C 3 (M) ≤ δ1,N on the metric. Now if necessary, decrease pN slightly such that 2p ≥ pN > 4, then one still has A1,pN -regularity and estimates on all components of Kbdy as well as on Kint (from the previous argument on the interior). Thus the assumptions of I) are satisfied with p replaced by 21 pN and C1 replaced by a combination of C1,N and a constant from the interior iteration (both of which only depend on C1 ). One just has to choose δ2 ≤ δ1,N and choose the δ > 0 in I) smaller than the δ > 0 from II). Then the iteration in I) gives regularity and estimates of 1 A − A0 in Ak, 2 pN (k × ) for all k ≥ 2. This proves the theorem in case 2 < p ≤ 4 1 due to the Sobolev embeddings W k+1, 2 pN → W k,p → C k−2 . So it remains to establish I) and II). Proof of I). The start of the induction k = 1 is true by assumption (after replacing C1 by a larger constant to make up for the effect of τ¯ ∗ ). For the induction step assume that the claimed regularity and estimates hold for some k ∈ N and consider the following decomposition of the connection A and its curvature: τ¯ ∗ A = ds + dt + B, τ¯ ∗ FA = FB + (dB − ∂s B) ∧ ds + (dB − ∂t B) ∧ dt +(∂s − ∂t + [ , ]) ds ∧ dt.
(11)
Here , ∈ W k,q (k × , g), and B ∈ W k,q (k × , T∗ ⊗ g) is a 2-parameter family of 1-forms on . Choose a further compact submanifold ⊂ int k such that k+1 ⊂ int . Now we shall use Proposition 2.7 to deduce the higher regularity of and on × . For this purpose one has to extend the vector fields ∂s and ∂t on × to different vector fields on k × , both denoted by X, and verify the assumptions (7) and
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(8) of Proposition 2.7. These extensions will be chosen such that they vanish in a neighbourhood of (∂k \ ∂H) × . Then α := τ¯ ∗ (A − A0 ) satisfies (7) on M = τ¯ (k × ) with N = τ¯ ((∂k ∩ ∂H) × ). Choose a cutoff function h ∈ C ∞ (k , [0, 1]) that equals 1 on and vanishes in a neighbourhood of ∂k \ ∂H. Then firstly, X := h∂t is a vector field as required that is perpendicular to the boundary ∂k × . For this type of vector field we have to check the assumption (8) for all β = φh · dt with φ ∈ Cδ∞ (k × , g). Note that τ¯∗ β = (φ · h) ◦ τ¯ −1 · ι(τ¯∗ ∂t ) g can be trivially extended to M and then vanishes when restricted to ∂M. So we can use partial integration as in the interior case to obtain Fτ¯ ∗ A , dτ¯ ∗ A β = FA , dA τ¯∗ β = − τ¯∗ β ∧ FA = 0. k ×
M
∂M
Secondly, X := h∂s also vanishes in a neighbourhood of (∂k \ ∂H) × and is tangential to the boundary ∂k × . So we have to verify (8) for all β = φh · ds with φ ∈ T = Cν∞ (k ×, g). Again, τ¯∗ β extends trivially to M. Then the partial integration yields Fτ¯ ∗ A , dτ¯ ∗ A β = − β ∧ τ¯ ∗ FA k × τ¯ −1 (∂M) =− φh · ds ∧ FB = 0. (k ∩∂ H)×
The last step uses the fact that B(s, 0) = τ ∗ A|{s}× ∈ L ⊂ Aflat (), and hence FB vanishes on ∂H × . However, we have to approximate A by smooth connections in order that Stokes’ theorem holds and FB is well-defined. So this calculation crucially uses the fact that a W 1,p -connection with boundary values in the Lagrangian submanifold L can be W 1,p -approximated by smooth connections with boundary values in L ∩ A(). This was proven in [W2, Cor. 4.5]. So we have verified the assumptions of Proposition 2.7 for both = τ¯ ∗ A(∂s ) and = τ¯ ∗ A(∂t ) and thus can deduce , ∈ W k+1,q ( × ). Moreover, under the additional assumptions of (ii) in the theorem we have the estimates s − 0 W k+1,q (×) ≤ Cs 1 + Ck + Ck3 =: Ck+1 , t − 0 W k+1,q (×) ≤ Ct 1 + Ck + Ck3 =: Ck+1 . (12) 0,p
The constants Cs and Ct are uniform for all metrics in some small C k+1 -neighbourhood of g0;s,t , so by a possibly smaller choice of δk+1 > 0 they become independent of gs,t . Note that in the above estimates we also have decomposed the reference connection in the tubular neighbourhood coordinates, τ¯ ∗ A0 = 0 ds + 0 dt + B0 . It remains to consider the -component B in the tubular neighbourhood. The boundary value problem (4) becomes in the coordinates (11) dB∗ 0 (B − B0 ) = ∇s ( − 0 ) + ∇t ( − 0 ), ∗FB = ∂t − ∂s + [, ], (13) ∂s B + ∗∂t B = dB + ∗dB , (s, 0) − 0 (s, 0) = 0 ∀(s, 0) ∈ ∂k , B(s, 0) ∈ L ∀(s, 0) ∈ ∂k .
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Here dB is the exterior derivative on that is associated with the connection B, similarly dB∗ 0 is the coderivative associated with B0 . Moreover, ∗ is the Hodge operator on with respect to the metric gs,t , and ∇s := ∂s +[ 0 , ], ∇t := ∂t +[0 , ]. We rewrite the first two equations in (13) as a system of differential equations for α := B − B0 on . For each (s, t) ∈ k , ∗ d α(s, t) = ξ(s, t),
d α(s, t) = ∗ζ (s, t).
(14)
Here we have abbreviated ξ = ∗[B0 ∧ ∗(B − B0 )] + ∇s ( − 0 ) + ∇t ( − 0 ), ζ = − ∗ d B0 − ∗ 21 [B ∧ B] + ∂t − ∂s + [, ]. These are both functions in W k,q ( × , g) due to the smoothness of A0 and the previously established regularity of and . (This uses the fact that W k,q · W k,q embeds into W k,q due to W k,q → L∞ .) So Lemma 2.9 asserts that ∇ (B − B0 ) is of class W k,q on × , and under the assumptions of (ii) in the theorem we obtain the estimate ∇ (B − B0 )W k,q (×) ≤ C ξ W k,q + ζ W k,q + B − B0 W k,q ≤ C 1 + B − B0 W k,q + − 0 W k+1,q + − 0 W k+1,q +B − B0 2W k,q + − 0 W k,q − 0 W k,q s t + Ck+1 + Ck2 =: Ck+1 . ≤ C 1 + Ck + Ck+1
(15)
Here C denotes any constant that is uniform for all metrics in a C k+1 -neighbourhood of the fixed g0;s,t , so this might again require a smaller choice of δk+1 > 0 in order that becomes independent of the metric g . the constant Ck+1 s,t Now we have established the regularity and estimate for all derivatives of B of order k + 1 containing at least one derivative in the -direction. (Note that in the case k = 1 we even have Lq -regularity with q = 2p, where only Lp -regularity was claimed. This additional regularity will be essential for the following argument.) It remains to consider the pure s- and t- derivatives of B and establish the Lp -regularity and -estimate k+1 B on k+1 × , where ∇H is the standard covariant derivative on H with for ∇H respect to the metric ds 2 + dt 2 . The reason for this regularity, as we shall show, is the fact that B ∈ W k,q (, A0,p ()) satisfies a Cauchy-Riemann equation with Lagrangian boundary conditions, ∂s B + ∗∂t B = G, (16) B(s, 0) ∈ L ∀(s, 0) ∈ ∂. The inhomogeneous term is G := dB + ∗dB ∈ W k,q (, A0,p ()). Here one uses the fact that W k,q ( × , T∗ ⊗ g) ⊂ W k,q (, A0,p ()) since the smooth 1-forms are dense in both spaces and the norm on the second space is weaker than the W k,q -norm on × , cf. [W2, Lemma 2.2]. Now one has to apply the regularity result [W2, Theorem 1.2] for the Cauchy-Riemann equation on the complex Banach space A0,p (). As reference complex structure
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J0 we use the Hodge ∗ operator on with respect to the fixed family of metrics g0;s,t on (that varies smoothly with (s, t) ∈ ). The smooth family J of complex structures in the equation is given by the Hodge operators with respect to the metrics gs,t . The Lagrangian submanifold L ⊂ A0,p () is totally real with respect to any Hodge operator, and it is modelled on a closed subspace of Lp (, Rn ) for some n ∈ N (see [W2, Lemma 4.2, Cor. 4.4] ) . In case (ii) of the theorem moreover a family of connections B0 ∈ C ∞ (, A()) is given such that B0 (s, 0) ∈ L for all (s, 0) ∈ ∂ and B satisfies B − B0 L∞ (,A0,p ()) = τ¯ ∗ (A − A0 )| L∞ (,A0,p ())
≤ Cτ¯0∗ (A − A0 )| L∞ (U ,A0,p ()) ≤ Cδ.
Here one uses the fact that τ¯ (×) ⊂ τ¯0 (U ×) lies in a component of the fixed neighbourhood V of K ∩ ∂X. The assumption of closeness to A0 in A0,p () was formulated for τ¯0∗ (A−A0 )| . However, for a metric g in a sufficiently small C 2 -neighbourhood of the fixed metric g0 the extensions τ¯ and τ¯0 are C 1 -close and one obtains the above estimate with a constant C independent of the metric. So the function B ∈ W k,q (, A0,p ()) satisfies the assumptions of [W2, Theorem 1.2] if δ > 0 is chosen sufficiently small. (Note that this choice is independent of k ∈ N.) Now [W2, Theorem 1.2] gives B ∈ W k+1,p (k+1 , A0,p ()). By [W2, Lemma 2.2] k+1 B ∈ Lp (k+1 , A0,p ()) = Lp (k+1 × , T∗ ⊗ g), and this this also proves ∇H finishes the induction step τ¯ ∗ A|k+1 × ∈ Ak+1,p (k+1 × ) for the regularity near the boundary. The induction step for the estimate in case (ii) of the theorem now follows from the estimate from [W2, Theorem 1.2], k+1 ∇H (B − B0 )Lp (k+1 ×) ≤ B − B0 W k+1,p (k+1 ,A0,p ()) ≤ C 1 + GW k,q (,A0,p ()) + B − B0 W k,q (,A0,p ()) s t H =: Ck+1 ≤ C 1 + Ck + Ck2 + Ck+1 + Ck+1 .
(17)
Here the constant from [W2, Theorem 1.2] is uniform for a sufficiently small C k+1 -neighbourhood of complex structures. In this case, these are the families of Hodge operators on that depend on the metric gs,t . Thus for sufficiently small δk+1 > 0 that constant (and also the further Sobolev constants that come into the estimate) becomes independent of the metric. The final constant Ck+1 then results from all the separate estimates, see the decomposition (11) and the estimates in (12), (15), and (17), s t H τ¯ ∗ (A − A0 )W k+1,p (k+1 ×) ≤ Ck + Ck+1 + Ck+1 + Ck+1 + Ck+1 .
Proof of II). Except for the higher differentiability of B in the direction of H this iteration works by the same decomposition and equations as in I. The start of the induction k = 1 is given by assumption. For the induction step we assume that the claimed W 1,pk regularity and -estimates hold for some k ∈ N with pk ≤ 4. Then Proposition 2.7 gives , ∈ W 2,qk ( × ) with corresponding estimates and 4p 8−pkk ; if pk < 4, qk = 3 ; if pk = 4.
(In the case pk = 4 one applies the proposition only assuming W 1,pk -regularity for 2,qk -regularity with q = 3.) Now the right-hand sides pk = 24 k 7 < 4, then one obtains W
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in (14) lie in W 1,qk ( × ), so Lemma 2.9 gives W 1,qk -regularity and -estimates for ∇ B on × . Next, B ∈ W 1,pk (, A0,p ()) satisfies the Cauchy-Riemann equation (16) with the inhomogeneous term G ∈ W 1,qk ( × , T∗ ( × ) ⊗ g). Now we shall use the Sobolev embedding W 1,qk ( × ) → Lrk ( × ) with 2p 4−pkk ; if pk < 4, 4qk = rk = 4 − qk 12 ; if pk = 4. Note that rk > pk ≥ p due to pk > 2, so that we have G ∈ Lrk (, A0,p ()). We cannot apply [W2, Theorem 1.2] directly because that would require the initial regularity B ∈ W 1,2p (, A0,p ()) for some p > 2. However, we still proceed as in its proof and introduce the coordinates from [W2, Lemma 4.3] that straighten out the Lagrangian submanifold, s,t : Ws,t → A0,p (). Here Ws,t ⊂ Y × Y is a neighbourhood of zero, Y is a closed subspace of Lp (, Rm ) for some m ∈ N, is in C k+1 -dependence on (s, t) in a neighbourhood U ⊂ of some (s0 , 0) ∈ ∩ ∂H and it maps diffeomorphically to a neighbourhood of B(s, t) or B0 (s, t) in case (ii). Thus one can write B(s, t) = s,t (v(s, t))
∀(s, t) ∈ U
with v = (v1 , v2 ) ∈ W 1,pk (U, Y × Y ). Moreover, we already have the W 1,qk -regularity of both B and ∇ B on U × , so B ∈ W 1,qk (U, A1,qk ()) ⊂ W 1,qk (U, A0,sk ()) with corresponding estimates. Here we use the Sobolev embedding [Ad, Theorem 5.4] W 1,qk () → Lsk () with
sk =
2qk 2−qk =
4pk 8−3pk
; if pk < 83 ,
44pk −80 8−pk
; if pk ≥ 83 ,
31 2
; if pk = 4.
(Here we have chosen suitable values of sk for later calculations in case pk ≥ 83 and thus qk ≥ 2.) The special structure of the coordinate map in [W2, Lemma 4.3] (it is a local diffeomorphism between A0,sk () and a closed subset of Lsk (, R2m ) since sk > pk > 2) implies that v ∈ W 1,qk (U, Lsk (, R2m )), which will be important later. The Cauchy-Riemann equation (16) now becomes ∂s v + I ∂t v = f, ∀s ∈ R. v2 (s, 0) = 0 Here I = (dv )−1 ∗ (dv ) ∈ W 1,pk (U, End(Y × Y )) and f = (dv )−1 (G − ∂s (v) − ∗∂t (v)) ∈ Lrk (U, Y × Y ). We now approximate f in Lrk (U, Y × Y ) by smooth functions that vanish on ∂U , then partial integration in [W2, (2.4) or (10)] yields for all φ ∈ C ∞ (U, Y ∗ ×Y ∗ ) and a smooth
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cutoff function h ∈ C ∞ (U, [0, 1]) with ∂t h|t=0 = 0 as in the proof of [W2, Theorem 1.2] hv , φ = f , ∂s (hφ) − ∂t (h · I ∗ φ) + F˜ , φ U U U + v1 , ∂t (hφ1 ) + ∂s (hφ2 ) . (18) ∂U ∩∂ H
Here F˜ = (h)v + 2(∂s h)∂s v + 2(∂t h)∂t v + h(∂t I )∂s v − h(∂s I )∂t v contains the crucial 1 terms (∂t I )(∂s v) and (∂s I )(∂t v) and thus lies in L 2 pk (U, Y × Y ). This is a weak Laplace equation with Dirichlet boundary conditions for hv2 , Neumann boundary conditions for hv1 , and with the inhomogeneous term in W −1,rk (U, Y × Y ). The latter is the dual space 1 of W 1,rk (U, Y ∗ × Y ∗ ) with r1k + r1 = 1. (The inclusion L 2 pk (U ) → W −1,rk (U ) is conk
tinuous as can be seen via the dual embedding that is due to 21 − r1 ≥ −1 + pk1/2 .) Recall k that Y ⊂ Lp (, Rm ) is a closed subspace. Since rk > p the special regularity result [W2, Lemma 2.1] for the Laplace equation with values in a Banach space cannot be applied to deduce hv ∈ W 1,rk (U, Y × Y ). However, the general regularity theory for the Laplace equation extends to functions with values in a Hilbert space. So we use the embedding Lp () → L2 (). Then (18) is a weak Laplace equation with the inhomogeneous term in W −1,rk (U, L2 (, R2m )) and enables us to deduce hv ∈ W 1,rk (U, L2 (, R2m )) and thus v ∈ W 1,rk (U˜ , L2 (, R2m )) with the corresponding estimates for some smaller domain U˜ (where h|U˜ ≡ 1; a finite union of such domains still covers a neighbourhood of ∩ ∂H). Furthermore, recall that v ∈ W 1,qk (U, Lsk (, R2m )). Now we claim that the following inclusion with the corresponding estimates holds for some suitable pk+1 : W 1,rk (U˜ , L2 ()) ∩ W 1,qk (U˜ , Lsk ()) ⊂ W 1,pk+1 (U˜ , Lpk+1 ()).
(19)
To show (19) it suffices to estimate the Lpk+1 (U˜ × )-norm of a smooth function by its Lrk (U˜ , L2 ())- and Lqk (U˜ , Lsk ())-norms. Let α > 2 and t ∈ [1, 2), then the H¨older inequality gives for all f ∈ C ∞ (U˜ × , R2m ), f αLα (U˜ ×) = |f |t |f |α−t ˜ U ≤ f tL2 () f α−t 2 α−t U˜
≤ ≤
L 2−t () t f Lr (U˜ ,L2 ()) f α−t α−t 2 α−t Lr r−t (U˜ ,L 2−t ()) f αLr (U˜ ,L2 ()) + f α r α−t . 2 α−t L r−t (U˜ ,L 2−t ())
Here we abbreviated r := rk > pk > 2. Now we want qk =
rk (α − t) rk − t
and
sk =
2(α − t) . 2−t
(20)
Indeed, in the case pk = 4 our choices qk = 3, rk = 12, and sk = 31 2 together with 17 solve these equations. So we obtain p = α = t := 53 and α := 17 k+1 4 16 pk . In case pk < 4 the first equation gives
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K. Wehrheim
α=
4+t pk . 8 − pk
(21)
If moreover pk ≥ 83 , then we choose t := also solves (20) with our choice sk = case
8 3
5 17 17 3 to obtain α = 24−3pk pk ≥ 16 pk . This 44pk −80 17 8−pk , so we obtain pk+1 = 16 pk . Finally, in
> pk > 2 one obtains from (20), t =
pk2 −pk2 + 7pk − 8
∈ [1, 2).
Inserting this in (21) yields α = θ (pk ) · pk with θ (pk ) =
3pk − 4 . −pk2 + 7pk − 8
One then checks that θ (2) = 1 and θ (p) > 0 for p > 2, thus θ(p) > 1 for p > 2. 8 Moreover, θ ( 83 ) = 98 , so θ (p ) = 17 16 for some p ∈ (2, 3 ). Now for p ≥ p we extend the function constantly to obtain a monotonely increasing function θ : (2, 4] → (1, 17 16 ]. With this modified function we finally choose pk+1 = θ(pk ) · pk for all 2 < pk ≤ 4. This finishes the proof of (19) and thus shows that v ∈ W 1,pk+1 (U˜ , Lpk+1 ()). In addition, note that our choice of pk+1 ≤ α will always satisfy pk+1 ≤ rk . In case pk = 4, see the actual numbers, in case pk < 4 this is due to (21), t ≤ 2, and pk > 2, α ≤
2 6 pk ≤ pk = rk . 8 − pk 4 − pk
Now we again use the special structure of the coordinates in [W2, Lemma 4.3] to deduce that B = ◦ v ∈ W 1,pk+1 (U˜ , A0,pk+1 ()) with the corresponding estimates. Above, we already established the W 1,rk - and thus W 1,pk+1 -regularity and -estimates for and as well as B ∈ Lpk+1 (U˜ , A1,pk+1 ()). (Recall the Sobolev embedding W 1,qk → Lrk , and that pk ≥ qk and rk ≥ pk+1 , so we have Lrk (U˜ , Lrk ())-regularity of B as well as ∇ B.) Putting all this together we have established the W 1,pk+1 -regularity and -estimates for τ¯ ∗ A over U˜ i × , where the U˜ i cover a neighbourhood of k+1 ∩ ∂H. The interior regularity again follows directly from Proposition 2.7. This iteration gives a sequence (pk ) with pk+1 = θ(pk ) · pk ≥ θ(p) · pk . So this sequence grows at a rate greater or equal to θ (p) > θ (2) = 1 and hence reaches pN > 4 after finitely many steps. This finishes the proof of (II) and the theorem. 1,p
Proof of Theorem A. Fix a solution A ∈ Aloc (X) of (2) with p > 2. We have to find 2,p ∗ a gauge transformation u ∈ Gloc (X) such that u A ∈ A(X) is smooth. Recall that the manifold X = k∈N Xk is exhausted by compact submanifolds Xk meeting the assumptions of Proposition 2.1. So it suffices to prove for every k ∈ N that there exists a gauge transformation u ∈ G 2,p (Xk ) such that u∗ A|Xk is smooth. For that purpose fix k ∈ N and choose a compact submanifold M ⊂ X that is large enough such that Theorem 2.6 applies to the compact subset K := Xk ⊂ M. Next, choose A0 ∈ A(M) such that A − A0 W 1,p (M) and A − A0 Lq (M) are sufficiently small for the local slice theorem, Proposition 2.2, to apply to A0 with the reference connection Aˆ = A. Here due to p > 2 one can choose q > 4 in the local slice
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theorem such that the Sobolev embedding W 1,p (M) → Lq (M) holds. Then by Proposition 2.2 and Remark 2.3 (i) one obtains a gauge transformation u ∈ G 2,p (M) such that u∗ A is in relative Coulomb gauge with respect to A0 . Moreover, u∗ A also solves (2) since both the anti-self-duality equation and the Lagrangian submanifolds Li are gauge invariant. The latter is due to the fact that u restricts to a gauge transformation in G 1,p (i ) on each boundary slice τi ({s} × i ) due to the Sobolev embedding G 2,p (Ui × ) ⊂ W 1,p (Ui , G 1,p (i )) → C 0 (Ui , G 1,p (i )). So u∗ A ∈ A1,p (M) is a solution of (4) and Theorem 2.6 (i) asserts that u∗ A|Xk ∈ A(Xk ) is indeed smooth. Such a gauge transformation u ∈ G 2,p (Xk ) can be found for every k ∈ N, hence 2,p Proposition 2.1 (i) asserts that there exists a gauge transformation u ∈ Gloc (X) on the ∗ full noncompact manifold such that u A ∈ A(X) is smooth as claimed. Proof of Theorem B. Fix a smoothly convergent sequence of metrics g ν → g that are 1,p compatible to τ and let Aν ∈ Aloc (X) be a sequence of solutions of (2) with respect to ν the metrics g . Recall that the manifold X = k∈N Xk is exhausted by compact submanifolds Xk meeting the assumptions of Proposition 2.1. We will find a subsequence 2,p (again denoted Aν ) and a sequence of gauge transformations uν ∈ Gloc (X) such that the sequence uν ∗ Aν is bounded in the W ,p -norm on Xk for all ∈ N and k ∈ N. Then due to the compact Sobolev embeddings W ,p (Xk ) → C −2 (Xk ) one finds a further diagonal subsequence that converges uniformly with all derivatives on every compact subset of X. By Proposition 2.1 (ii) it suffices to construct the gauge transformations and establish the claimed uniform bounds over Xk for all k ∈ N and for any subsequence of the connections (again denoted Aν ). So fix k ∈ N and choose a compact submanifold M ⊂ X such that Theorem 2.6 holds with K = Xk ⊂ M. Choose a further compact submanifold M ⊂ X such that Theorem 2.6 holds with K = M ⊂ M . Then by assumption of the theorem FAν Lp (M ) is uniformly bounded. So the weak Uhlenbeck compactness, Proposition 2.4, provides a subsequence (still denoted Aν ), a limit connection A0 ∈ A1,p (M ), and gauge transformations uν ∈ G 2,p (M ) such that uν ∗ Aν → A0 in the weak W 1,p -topology. The limit A0 then satisfies the boundary value problem (2) with respect to the limit metric g. (For the boundary conditions this follows from the compact embedding in Lemma 2.5 and the fact that every Li ⊂ A0,p (i ) is a Banach submanifold and hence Lp -closed.) So as in the proof of Theorem A one finds a gauge transformation u0 ∈ G 2,p (M) such that u∗0 A0 ∈ A(M) is smooth. (Note however that we do not have sufficient boundary conditions on ∂M \ ∂X to obtain smoothness on M . Thus we had to start out from the larger submanifold M = M.) Now replace A0 by u∗0 A0 and uν by uν u0 ∈ G 2,p (M), then still uν ∗ Aν → A0 (since ν∗ ν (uν u0 )∗ Aν − u∗0 A0 = u−1 0 (u A − A0 )u0 and conjugation by u0 is continuous in the weak W 1,p -topology). Thus one has a W 1,p -bound, uν ∗ Aν − A0 W 1,p (M) ≤ c0 for some constant c0 . Due to p > 2 one can now choose q > 4 in the local slice theorem such that the Sobolev embedding W 1,p (M) → Lq (M) is compact. Hence for a further subsequence of the connections uν ∗ Aν → A0 in the Lq -norm. Let ε > 0 be the constant from Proposition 2.2 for the reference connection Aˆ = A0 , then one finds a further subsequence such that uν ∗ Aν −A0 Lq (M) ≤ ε for all ν ∈ N. So the local slice theorem provides further gauge transformations u˜ ν ∈ G 2,p (M) such that the u˜ ν ∗ Aν are in relative Coulomb gauge with respect to A0 . The gauge transformed connections still solve (2), hence the u˜ ν ∗ Aν are solutions of (4). Moreover, we have u˜ ν ∗ Aν − A0 q ≤ CCG uν ∗ Aν − A0 q , hence u˜ ν ∗ Aν → A0 in the Lq -norm, and
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ν∗ ν u˜ A − A0 1,p ≤ CCG c0 . W (M) The higher W k,p -bounds will now follow from Theorem 2.6 after we verify its assumptions. Fix the metric g0 := g and a compact neighbourhood V = ni=1 τ¯0,i (Ui × i ) of ∗ (u K ∩ ∂X. Then the τ¯0,i ˜ ν ∗ Aν − A0 )|i are uniformly W 1,p -bounded and converge to q zero in the L -norm on Ui × Si as seen above. Now the embedding W 1,p (Ui × i , T∗ i ⊗ g) → L∞ (Ui , A0,p (i )) ∗ (u ˜ ν ∗ Aν )|i is compact by Lemma 2.5. Thus one finds a subsequence such that the τ¯0,i ∗ A | since this already is the converge in L∞ (Ui , A0,p (i )). The limit can only be τ¯0,i 0 i q L -limit. Now in Theorem 2.6 (ii) choose the constant C1 = CCG c0 and let δ > 0 be the constant determined from C1 . Then one can take a subsequence such that ∗ τ¯0,i (u˜ ν ∗ Aν − A0 )i L∞ (Ui ,A0,p (i )) ≤ δ
∀i = 1, . . . , n, ∀ν.
Now Theorem 2.6 (ii) provides the claimed uniform bounds as follows. Fix ∈ N, then g ν − gC +2 (M) ≤ δ for all ν ≥ ν with some ν ∈ N, and thus ν∗ ν u˜ A − A0 ,p ≤ C ∀ν ≥ ν . W (X ) k
This finally implies the uniform bound for this subsequence, sup u˜ ν ∗ Aν W ,p (X ) < ∞. ν∈N
k
Here the gauge transformations u˜ ν ∈ G 2,p (Xk ) still depend on k ∈ N and are only defined on Xk . But now Proposition 2.1 (ii) provides a subsequence of (Aν ) and gauge 2,p transformations uν ∈ Gloc (X) defined on the full noncompact manifold such that uν ∗ Aν is uniformly bounded in every W ,p -norm on every compact submanifold Xk . Now one can iteratively use the compact Sobolev embeddings W +2,p (X ) → C (X ) for each ∈ N to find a further subsequence of the connections that converges in C (X ). If in each step one fixes one further element of the sequence, then this iteration finally yields a sequence of connections that converges uniformly with all derivatives on every compact subset of X to a smooth connection A ∈ A(X). 3. Fredholm Theory This section concerns the linearization of the boundary value problem (2) in the special case of a compact 4-manifold of the form X = S 1 × Y , where Y is a compact orientable 3-manifold whose boundary ∂Y = is a disjoint union of connected Riemann surfaces. The aim of this section is to prove Theorem C. (The actual Fredholm property in part (i) will be proven last, building on (ii) and (iii), which are stated separately for future reference.) So we equip S 1 × Y with a product metric g˜ = ds 2 + gs (where gs is an S 1 -family of metrics on Y ) and assume that this is compatible with the natural space-time splitting of the boundary ∂X = S 1 × . This means that for some > 0 there exists an embedding τ : S 1 × [0, ) × → S 1 × Y
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preserving the boundary, τ (s, 0, z) = (s, z) for all s ∈ S 1 and z ∈ , such that τ ∗ g˜ = ds 2 + dt 2 + gs,t . Here gs,t is a smooth family of metrics on . This assumption on the metric implies that the normal geodesics are independent of s ∈ S 1 in a neighbourhood of the boundary. So in fact, the embedding is given by τ (s, t, z) = (s, γz (t)), where γ is the normal geodesic starting at z ∈ . This seems like a very restrictive assumption, but it suffices for our application to Riemannian 4-manifolds with a boundary space-time splitting. Indeed, the neighbourhoods of the compact boundary components are isometric to S 1 × Y with Y = [0, ] × and a metric ds 2 + dt 2 + gs,t . 0,p Now fix p > 2 and let L ⊂ Aflat () be a gauge invariant Lagrangian submanifold of A0,p () as in the introduction. Then for A˜ ∈ A1,p (S 1 × Y ) we consider the nonlinear boundary value problem ∗FA˜ + FA˜ = 0, (22) ˜ {s}×∂Y ∈ L ∀s ∈ S 1 . A| Fix a smooth connection A˜ ∈ A(S 1 × Y ) with Lagrangian boundary values (but not necessarily a solution of this boundary value problem). It can be decomposed as A˜ = A + ds with ∈ C ∞ (S 1 × Y, g) and with A ∈ C ∞ (S 1 × Y, T∗ Y ⊗ g) satisfying As := A(s)|∂Y ∈ L for all s ∈ S 1 . Similarly, a tangent vector α˜ to A1,p (S 1 × Y ) decomposes as α˜ = α + ϕds with ϕ ∈ W 1,p (S 1 × Y, g) and α ∈ W 1,p (S 1 × Y, T∗ Y ⊗ g). Now 1,p let EA ⊂ W 1,p (S 1 × Y, T∗ Y ⊗ g) be the subspace of S 1 -families of 1-forms α that satisfy the boundary conditions from the linearization of (22) and the Coulomb gauge, ∗α(s)|∂Y = 0
and
α(s)|∂Y ∈ TAs L
for all s ∈ S 1 .
Then the linearized operator for the study of the moduli space of gauge equivalence classes of solutions of (22) is as in the introduction D(A, ) : EA × W 1,p (S 1 × Y, g) −→ Lp (S 1 × Y, T∗ Y ⊗ g) × Lp (S 1 × Y, g) 1,p
given by
∗ α . D(A, ) (α, ϕ) = ∇s α − dA ϕ + ∗dA α , ∇s ϕ − dA Here dA denotes the exterior derivative corresponding to the connection A(s) on Y for all s ∈ S 1 , ∗ denotes the Hodge operator on Y with respect to the s-dependent metric gs on Y , and we use the notation ∇s α := ∂s α + [ , α]. Our main result, Theorem C (i), is the Fredholm property of D(A, ) . We now give an outline of its proof. The first crucial point is the estimate in Theorem C (ii), which ensures that D(A, ) has a closed image and a finite dimensional kernel. It can be rephrased as follows due to the identities d+˜ α˜ = 21 ∗ ∇s α − dA ϕ + ∗dA α − 21 ∇s α − dA ϕ + ∗dA α ∧ ds, A
∗ ∗ dA ˜ = −∇s ϕ + dA α. ˜α
(23)
Lemma 3.1. There is a constant C such that for all α˜ ∈ W 1,p (X, T∗ X ⊗ g) satisfying ∗α| ˜ ∂X = 0 one has the estimate
α| ˜ {s}×∂Y ∈ TAs L ∀s ∈ S 1
and
∗ ˜ p + dA ˜ p + α ˜ p . α ˜ W 1,p ≤ C d+˜ α ˜ α A
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K. Wehrheim
The second part of the Fredholm theory for D(A, ) is the identification of the cokernel with the kernel of a slightly modified linearized operator, which will be used to prove that the cokernel is finite dimensional. To be more precise let § : S 1 × Y → S 1 × Y denote the reflection given by §(s, y) := (−s, y), where S 1 ∼ = R/Z. Then we will establish the following duality: (β, ζ ) ∈ (im D(A, ) )⊥ ⇐⇒ (β ◦ §, ζ ◦ §) ∈ ker D§∗ (A, ) , where D§∗ (A, ) is the linearized operator at the connection §∗ A˜ = A ◦ § − ◦ §ds with respect to the metric §∗ g˜ on S 1 × Y . Once we know that im D(A, ) is closed, this gives an isomorphism between (coker D(A, ) )∗ ∼ = (im D(A, ) )⊥ and ker D§∗ (A, ) . Here Z ∗ denotes the dual space of a Banach space Z, and for a subspace Y ⊂ Z we denote by Y ⊥ ⊂ Z ∗ the space of linear functionals that vanish on Y . Now the estimate in Theorem C (ii) will also apply to D§∗ (A, ) , and this implies that its kernel – and hence the cokernel of D(A, ) – is of finite dimension. The main difficulty in establishing the above duality is the regularity result Theorem C (iii). This regularity as well as the estimate in Theorem C (ii) or Lemma 3.1 will be proven analogously to the nonlinear regularity and estimates in Sect. 2. Again, the interior regularity and estimate is standard elliptic theory, and one has to use a splitting near the boundary. We shall show that the S 1 - and the normal component both satisfy a Laplace equation with Neumann and Dirichlet boundary conditions respectively. The -component will again give rise to a (weak) Cauchy-Riemann equation in a Banach space, only this time the boundary values will lie in the tangent space of the Lagrangian. In contrast to the required Lp -estimates we shall first show that the L2 -estimate for Lp -regular 1-forms can be obtained by more elementary methods. These were already outlined in [S] as an indication for the Fredholm property of the boundary value problem (22). Let α˜ ∈ W 1,p (X, T∗ X ⊗ g) be as in Lemma 3.1 for some p > 2. From the first boundary condition ∗α| ˜ ∂X = 0 one obtains ∇ α ˜ 22 = dα ˜ 22 + d∗ α ˜ 22 − g(Y ˜ α˜ , ∇Yα˜ ν).
∂X
˜ so ∂X g(Y ˜ α˜ , ∇Yα˜ ν) ≥ −Cα ˜ 2L2 (∂X) . For Here the vector field Yα˜ is given by ιYα˜ g˜ = α, this last term one then uses the following special version of the Sobolev trace theorem for general 1 < q < ∞. Let τ : [0, ) × ∂X → X be a diffeomorphism to a tubular neighbourhood of ∂X in X. Then for all δ > 0 one finds a constant Cδ such that for all f ∈ W 1,q (X), q
f Lq (∂X) =
d (s − 1)|f (τ (s, z))|q ds d3 z ∂X 0 ds 1 1 q 3 |f (τ (s, z))| ds d z + q|f (τ (s, z))|q−1 |∂s f (τ (s, z))| ds d3 z ≤ ∂X
≤ ≤
1
∂X 0 q−1 C + f Lq (X) ∇f Lq (X) q δf W 1,q (X) + Cδ f Lq (X) .
0
q f Lq (X)
This uses the fact that for all x, y ≥ 0 and δ > 0, q δq y q ; if x ≤ δ q−1 y − 1 q q−1 x y ≤ ≤ δy + δ q−1 x . q q − q−1 δ x q ; if x ≥ δ q−1 y
(24)
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So we obtain ∗ α ˜ W 1,2 ≤ C dA˜ α ˜ 2 + dA ˜ 2 + α ˜ 2 . ˜ α
(25)
In fact, the analogous W 1,p -estimates hold true for general p, as is proven e.g. in [W1, Theorem 5.1]. However, in the case p = 2 one can calculate further for all δ > 0, dA˜ α ˜ 22 = dA˜ α˜ , 2d+˜ α˜ − dA˜ α˜ ∧ dA˜ α˜ A X X + 2 ˜ 2 − α˜ ∧ [FA˜ ∧ α] ˜ − α˜ ∧ dA˜ α˜ = 2d ˜ α A
X
∂X
≤ 2d+˜ α ˜ 22 + Cδ α ˜ 22 + δα ˜ 2W 1,2 . A
(26)
Here the boundary term above is estimated as follows. We use the universal covering of S 1 = R/Z to integrate over [0, 1] × ∂Y instead of ∂X = S 1 × ∂Y . Introduce A := (As )s∈S 1 , which is a smooth path in L. Then using the splitting α| ˜ ∂X = α + ϕds with α : S 1 × → T∗ ⊗ g and ϕ : S 1 × → g one obtains α˜ ∧ dA˜ α˜ − ∂X
=− 0
1
=
1
1
ϕ , dA α dvol ∧ ds − 0
α ∧ (dA ϕ − ∇s α) ∧ ds
α ∧ ∇s α ∧ ds
δα ˜ 2W 1,2 (X) 0
≤
+ Cδ α ˜ 2L2 (X) .
Firstly, we have used the fact that dA α| = 0 since α(s) ∈ TAs L ⊂ ker dAs for all s ∈ S 1 . Secondly, we have also used that both α and dA ϕ lie in TA L, hence the sym plectic form α ∧ dA ϕ vanishes for all s ∈ S 1 . This is not strictly true since α˜ only restricts to an Lp -regular 1-form on ∂X. However, as a 1-form on [0, 1] × Y it can be approximated as follows by smooth 1-forms that meet the Lagrangian boundary condition on [0, 1] × . We use the linearization of the coordinates in [W2, Lemma 4.3] at As for every s ∈ [0, 1]. Since the path s → As ∈ L ∩ A() is smooth, this gives a smooth path of diffeomorphisms s for any q > 2, s :
q ∗ Z × Z −→ m i L (, T ⊗ g)m (ξ, v, ζ, w) −→ dAs ξ + i=1 v γi (s) + ∗dAs ζ + i=1 w i ∗ γi (s),
with Z := Wz (, g) × Rm and where the γi ∈ C ∞ ([0, 1] × , T∗ ⊗ g) satisfy γi (s) ∈ TAs L for all s ∈ [0, 1]. We perform the above estimate on [0, 1] × Y since we cannot necessarily achieve γi (0) = γi (1). In these coordinates, we mollify to obtain the required smooth approximations of α˜ near the boundary. Furthermore, we use these coordinates for q = 3 to write the smooth approximations on the boundary as α(s) = i dAs ξ(s) + m i=1 v (s)γi (s) with ξ(s)W 1,3 () + |v(s)| ≤ Cα(s)L3 () . Then for all s ∈ [0, 1], 1,q
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K. Wehrheim
α(s) ∧ ∇s α(s) =
i α ∧ dAs ∂s ξ + m i=1 ∂s v · γi i + α ∧ [ , α] + [∂s A, ξ ] + m i=1 v · ∂s γi
≤ Cα(s)L2 () α(s)L3 () . Here the crucial point is that dA ∂s ξ and ∂s v i · γi are tangent to the Lagrangian, hence the first term vanishes. Now one uses (24) for q = 2 and the Sobolev trace theorem (the restriction W 1,2 (X) → L3 (∂X) is continuous by e.g. [Ad, Theorem 6.2] ) to obtain the estimate, 1 α ∧ ∇s α ∧ ds ≤ Cα ˜ L2 (∂X) α ˜ L3 (∂X) 0
≤ 2δ α ˜ 2W 1,2 (X) + Cδ α ˜ L2 (X) α ˜ W 1,2 (X) ≤ δα ˜ 2W 1,2 (X) + Cδ α ˜ 2L2 (X) . This proves (26). Now δ > 0 can be chosen arbitrarily small, so the term α ˜ W 1,2 can be absorbed into the left-hand side of (25), and thus one obtains the claimed estimate ∗ ˜ 2 + dA ˜ 2 + α ˜ 2 . α ˜ W 1,2 ≤ C d+˜ α ˜ α A
Proof of Theorem C (ii) or Lemma 3.1. We will use Lemma A.2 for the manifold M := S 1 × Y in several different cases to obtain the estimate for different components of α. ˜ The first weak equation in Lemma A.2 is the same in all cases. For all η ∈ C ∞ (M; g) α˜ , dη = d∗ α˜ , η + η , ∗α˜ M M ∂M ∗ ˜ ∧ ∗α] = dA α ˜ + ∗[ A ˜ , η = f , η . ˜ M
M
Here one uses the fact that ∗α| ˜ ∂M = 0 . Then f ∈
Lp (M, g)
and
∗ ˜ ∞ α ˜ p + 2A ˜ p. f p ≤ dA ˜ α
(27)
To obtain the second weak equation in Lemma A.2 we calculate for all λ ∈ 1 (M; g), α˜ , d∗ dλ = α˜ , d∗ dλ + d∗ ∗ dλ M M = γ , dλ − α˜ ∧ ∗dλ − α˜ ∧ dλ , (28) M
S 1 ×∂Y
S 1 ×∂Y
˜ + ∈ Lp (M, 2 T∗ M ⊗ g) and where γ = dα˜ + ∗dα˜ = 2d+˜ α˜ − 2[A˜ ∧ α] A
˜ ∞ α ˜ p + 4A ˜ p. γ p ≤ 2d+˜ α A
(29)
Now recall that there is an embedding τ : S 1 × [0, ) × → S 1 × Y to a tubular neighbourhood of S 1 × ∂Y such that τ ∗ g˜ = ds 2 + dt 2 + gs,t for a family gs,t of metrics on . One can then cover M = S 1 × Y with τ (S 1 × [0, 2 ] × ) and a compact subset V ⊂ M \ ∂M.
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For the claimed estimate of α˜ over V it suffices to use Lemma A.2 for vector fields X ∈ (TM) that are equal to coordinate vector fields on V and vanish on ∂M. So one has to consider (28) for λ = φ · ιX g˜ with φ ∈ Cδ∞ (M, g). Then both boundary terms vanish and hence Lemma A.2 directly asserts, with some constants C and CV , that ˜ Lp (M) α ˜ W 1,p (V ) ≤ C f Lp (M) + γ Lp (M) + α ∗ ˜ Lp (M) + dA ˜ Lp (M) + α ˜ Lp (M) . ≤ CV d+˜ α ˜ α A
So it remains to establish the estimate for α˜ near the boundary ∂M = S 1 × . For that purpose we introduce the decomposition τ ∗ α˜ = ϕds + ψdt + α, where ϕ, ψ ∈ W 1,p (S 1 × [0, ) × , g) and α ∈ W 1,p (S 1 × [0, ) × , T∗ ⊗ g). Let := S 1 × [0, 43 ] and let K := S 1 × [0, 2 ]. Then we will prove the estimate for ϕ and ψ on × and for α on K × . Firstly, note that ψ = α(τ ˜ ∗ ∂t ) ◦ τ , where −τ∗ ∂t |∂M = ν is the outer unit normal to ∂M. So one can cut off τ∗ ∂t outside of τ ( × ) to obtain a vector field X ∈ (TM) that satisfies the assumption of Lemma A.2, that is X|∂M = −ν is perpendicular to the boundary. Then one has to test (28) with λ = φ · ιX g˜ for all φ ∈ Cδ∞ (M, g). Again both boundary terms vanish. Indeed, on S 1 × ∂Y we have φ ≡ 0 and ιX g˜ = τ∗ dt, hence dλ|R×∂Y = 0 and ∗dλ|R×∂Y = − ∂φ ∂ν ∗ τ∗ (dt ∧ dt) = 0. Thus Lemma A.2 yields the following estimate: ˜ ψW 1,p (×) ≤ Cα(X) W 1,p (M) ≤ C f Lp (M) + γ Lp (M) + α ˜ Lp (M) ∗ ˜ Lp (M) + dA ˜ Lp (M) + α ˜ Lp (M) . ≤ Ct d+˜ α ˜ α A
Here C denotes any finite constant and the bounds on the derivatives of τ enter into the constant Ct . Next, for the regularity of ϕ = α(∂ ˜ s )◦τ one can apply Lemma A.2 with the tangential vector field X = ∂s . Recall that τ preserves the S 1 -coordinate. One has to verify the second weak equation for all φ ∈ Cν∞ (M, g), i.e. consider (28) for λ = φ · ιX g˜ = φ · ds. The first boundary term vanishes since one has ∗dλ|S 1 ×∂Y = − ∂φ ∂ν dvol∂Y = 0. For the second term one can choose any δ > 0 and then find a constant Cδ such that for all φ ∈ Cν∞ (M, g), α˜ ∧ dλ = α(s, 0) ∧ d (φ ◦ τ )(s, 0) ∧ ds 1 1 S ×∂Y S ˜ α˜ ∧ [A, φ] ∧ ds = S 1 ×∂Y
˜ ∞ φ p∗ ≤ α ˜ Lp (∂M) A L (∂M) ≤ δα ˜ W 1,p (M) + Cδ α ˜ Lp (M) φW 1,p∗ (M) . This uses the fact that α(s, 0) and dAs (φ ◦ τ )|(s,0)× both lie in the tangent space TAs L to the Lagrangian, on which the symplectic form vanishes. So we have the identity α ∧ dA (φ ◦ τ ) = 0. Moreover, we have used the trace theorem for Sobolev spaces, in particular (24) with q = p. Now Lemma A.2 and Remark A.3 yield with c1 = f p , c2 = γ Lp (M) + δα ˜ W 1,p (M) + Cδ α ˜ Lp (M) , and using (27), (29)
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K. Wehrheim
ϕW 1,p (×) ≤ C f Lp (M) + c2 + α ˜ Lp (M) ∗ ≤ δα ˜ W 1,p (M) + Cs (δ) d+˜ α ˜ Lp (M) + dA ˜ Lp (M) + α ˜ Lp (M) . ˜ α A
Here again δ > 0 can be chosen arbitrarily small and the constant Cs (δ) depends on this choice. It remains to establish the estimate for the -component α near the boundary. In the coordinates τ on × , the forms d∗˜ α˜ and d+˜ α˜ become A
A
∗ ∗ ˜ = −∂s ϕ − ∂t ψ + d α − τ ∗ (∗[A˜ ∧ ∗α]), ˜ τ ∗ dA ˜α ∗ + 1 τ d ˜ α˜ = 2 −(∂s α + ∗ ∂t α) ∧ ds + ∗ (∂s α + ∗ ∂t α) ∧ dt A ˜ + ). + 21 d α + (∗ d α)ds ∧ dt + τ ∗ ([A˜ ∧ α]
So one obtains the following bounds: The components in the mixed direction of and of the second equation yields for some constant C1 , ∂s α + ∗ ∂t αLp (×) ≤ τ ∗ d+˜ α˜ Lp (×) + τ ∗ ([A˜ ∧ α] ˜ + ) Lp (×) A ≤ C1 d+˜ α ˜ Lp (M) + α ˜ Lp (M) . A
Similarly, a combination of the first equation and the -component of the second equation can be used for every δ > 0 to find a constant C2 (δ) such that
∗ d αLp (×) + d αLp (×) + ∗ p ≤ C d ˜ α ˜ L (M) + dA ˜ Lp (M) + α ˜ Lp (M) + ϕW 1,p (×) + ψW 1,p ˜ α A ∗ ˜ Lp (M) + dA ˜ Lp (M) + α ˜ M) . ≤ δα ˜ W 1,p (M) + C2 (δ) d+˜ α ˜ α A
Now Lemma 2.9 provides an Lp -estimate for the derivatives of α in the -direction, ∇ αLp (×) ∗ ≤ C d αLp (×) + d αLp (×) + αLp (×) ∗ ≤ δα ˜ W 1,p (M) + C (δ) d+˜ α ˜ Lp (M) + dA ˜ Lp (M) + α ˜ Lp (M) , ˜ α A
where again C (δ) depends on the choice of δ > 0. For the derivatives in the s- and t-direction, we will apply [W2, Theorem 1.3] on the Banach space X = Lp (, T∗ ⊗g) with the complex structure ∗ determined by the metric gs,t on and hence depending smoothly on (s, t) ∈ . The Lagrangian submanifold L ⊂ X is totally real with respect to all Hodge operators and it is modelled on a closed subspace of Lp (, Rn ) as seen in [W2, Lemma 4.2, Cor. 4.4]. Now α ∈ W 1,p (, X) satisfies the Lagrangian boundary condition α(s, 0) ∈ TAs L for all s ∈ S 1 , where s → As is a smooth loop in L. Thus [W2, Cor. 1.4] yields a constant C such that the following estimate holds: ∇ αLp (K×) ≤ αW 1,p (K,X) ≤ C ∂s α + ∗ ∂t αLp (,X) + αLp (,X) ≤ CK d+˜ α ˜ Lp (M) + α ˜ Lp (M) . A
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Here CK also includes the above constant C1 . Now adding up all the estimates for the different components of α˜ gives for all δ > 0, ∗ α ˜ W 1,p ≤ (CV + Ct + Cs (δ) + C (δ) + CK ) d+˜ α ˜ p + dA ˜ p + α ˜ p ˜ α A
+2δα ˜ W 1,p . With δ = 41 the term α ˜ W 1,p can be absorbed into the left-hand side, which finishes the proof of the lemma. Proof of Theorem C (iii). Let β ∈ Lq (S 1 × Y, T∗ Y ⊗ g) and ζ ∈ Lq (S 1 × Y, g) be as supposed in Theorem C. Then there exists a constant c such that for all α ∈ C ∞ (S 1 × Y, T∗ Y ⊗ g) satisfying (3) and for all ϕ ∈ C ∞ (S 1 × Y, g), ∗ 1 ∇s α − dA ϕ + ∗dA α , β + 1 ∇s ϕ − dA α , ζ S Y S Y = D(A, ) (α, ϕ) , (β, ζ ) S 1 ×Y
≤ c(α, ϕ)q ∗ .
(30)
The higher regularity of ζ is most easily seen if we go back to the notation α˜ = α + ϕds. With this we can write D(A, ) (α, ϕ) = (2γ , −d∗˜ α), ˜ where d+˜ α˜ = ∗γ − γ ∧ ds . We A A ˜ ∂M = 0 abbreviate M := S 1 × Y , then we have for all α˜ ∈ C ∞ (M, T∗ M ⊗ g) with ∗α| and α| ˜ {s}×∂Y ∈ TAs L for all s ∈ S 1 , ∗ ≤ cα 2d+ α˜ , β ∧ ds + d α ˜ , ζ ˜ q∗ . ˜ ˜ A M
A
M
Now use the embedding τ : S 1 ×[0, )× → M to construct a connection Aˆ ∈ A(M) ˆ t, z) = As (z) near the boundary (this can be cut off and then extends such that τ ∗ A(s, trivially to all of M). Then α˜ := dAˆ φ satisfies the above boundary conditions for all ˆ φ ∈ Cν∞ (M, g) since dAˆ φ(ν) = ∂φ ∂ν + [A(ν), φ] = 0 and dAˆ φ|{s}×∂Y = dAs φ ∈ TAs L 1 ˆ φ]), for all s ∈ S . Thus we obtain for all φ ∈ Cν∞ (M, g) in view of φ = d∗ (α˜ − [A, φ , ζ M ∗ ∗ ˆ φ] , ζ = dA˜ α˜ + ∗[A˜ ∧ ∗α] ˜ − d [A, M + ∗ ˜ ˆ ≤ cα ˜ q ∗ + −2d ˜ dAˆ φ , β ∧ ds + ∗[A ∧ ∗dAˆ φ] − d [A, φ] , ζ A M M ≤ C c + βq + ζ q φW 1,q ∗ . ˜ A) ˆ that is inde(Here and in the following we denote by C any constant C = C(q, A, pendent of (β, ζ ).) The regularity theory for the Neumann problem, e.g. Proposition A.1, then asserts that ζ ∈ W 1,q (M) with ζ W 1,q ≤ C(c + (β, ζ )q ).
(31)
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K. Wehrheim
To deduce the higher regularity of β we will mainly use Lemma A.2. The first weak equation in the lemma is given by choosing α = 0 in (30). For all ϕ ∈ C ∞ (M, g), β , dϕ = β , d ϕ − [A, ϕ] A 1 M S Y ≤ cϕq ∗ + ∇s ζ , ϕ + [β ∧ ∗A] , ϕ S1 Y S1 Y ≤ c + C ζ W 1,q + βq ϕq ∗ . For the second weak equation let ϕ = 0 and α = ∗dλ − ∂s λ for λ = φ · ιX g˜ with φ in Cδ∞ (M, g) or Cν∞ (M, g) corresponding to the vector field X ∈ C ∞ (M, TY ). If the 1,p boundary conditions for α ∈ EA are satisfied, then we obtain with d = dY , β , d∗ dM (φ · ιX g) M M = β , ∗d ∗ dλ − ∂s2 λ − ∗(∂s ∗)∂s λ 1 S Y = β , ∗dA α − ∗[A ∧ ∗dλ] + ∗dA ∂s λ S1 Y +∇s α − [ , ∂s λ] − ∇s ∗ dλ − ∗(∂s ∗)∂s λ ∗ ≤ cαq ∗ + ζ , dA α + Cβq λW 1,q ∗ S1 Y ≤ C c + ζ W 1,q + βq φW 1,q ∗ . Here we have used the identity ∗dA ∂s λ − ∇s ∗ dλ = ∗[A ∧ ∂s λ] − [ , ∗dλ] − (∂s ∗)dλ. Moreover, partial integration with vanishing boundary term ∗α|∂Y = 0 gives ∗ ζ , dA α = dA ζ , ∗dλ − ∂s λ . S1
Y
S1
Y
Now let X ∈ C ∞ (M, TY ) be perpendicular to the boundary ∂M = S 1 × ∂Y , then α = ∗dλ − ∂s λ satisfies the boundary conditions (3) for every φ ∈ Cδ∞ (M). Indeed, on the boundary ∂M = S 1 × ∂Y the 1-form λ = φ · ιX g˜ vanishes, we have ιX g˜ = h · τ∗ dt for some smooth function h, and moreover dφ = − ∂φ ∂ν · τ∗ dt. Hence ∗α|∂Y = dλ|∂Y − ∗∂s λ|∂Y = 0, α|∂Y = ∗dλ|∂Y − ∂s λ|∂Y = − ∂φ ∂ν h ∗ (τ∗ dt ∧ τ∗ dt) = 0. Thus for all vector fields X ∈ C ∞ (M, TY ) that are perpendicular to the boundary, Lemma A.2 and Remark A.3 assert that β(X) ∈ W 1,q (M, g) and β(X)W 1,q ≤ C c + ζ W 1,q + βq ≤ C c + (ζ, β)q . Here we have also used the previously established regularity and estimate (31) for ζ . In particular, this implies W 1,q -regularity and -estimate for β on all compact subsets
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in the interior of S 1 × Y . So it remains to prove the regularity on the neighbourhood τ (S 1 × [0, 2 ] × ) of the boundary. We pull back (β, ζ ) from τ (S 1 × [0, ) × ) and write ˆ τ ∗ ζ = η. τ ∗ β = ξ dt + β, We have already established that η = ζ ◦ τ and ξ = β(τ∗ ∂t ) ◦ τ lie in W 1,q ( × , g) with the according estimate, where := S 1 × [0, 43 ). Here a vector field X on Y that is perpendicular to ∂Y is constructed by cutting off τ∗ ∂t outside of τ ( × ). So it remains to consider βˆ ∈ Lq ( × , T∗ ⊗ g) and establish its W 1,q -regularity and -estimate on S 1 × [0, 2 ] × . In order to derive a weak equation for βˆ on × from (30) we use the test 1-form α˜ = τ∗ (ϕds + ψdt + α) ˆ with ϕ, ψ ∈ C0∞ ( × , g) (supported in int() × )) and αˆ ∈ C ∞ ( × , T∗ ⊗ g) with compact support supp αˆ ⊂ S 1 × [0, 43 ) × and α(s, ˆ 0, ·) ∈ TAs L for all s ∈ S 1 . This α˜ satisfies the boundary conditions (3) and it can be extended trivially to a smooth 1-form on all of S 1 × Y . Thus we obtain ˆ ∇s αˆ + ∗∇t αˆ − dA ϕ − ∗dA ψ , β × ∗ −∇s ψ + ∇t ϕ − ∗dA αˆ , ξ + ∇s ϕ + ∇t ψ − dA αˆ , η ≤ ×
×
ˆ Lq ∗ (S 1 ×Y ) . +cτ∗ (ϕ ds + ψ dt + α) Here we have decomposed τ ∗ A˜ = ds +dt +A with A ∈ C ∞ ( × , T∗ ⊗ g) satisfying A(s, 0) = As ∈ L for all s ∈ S 1 . We also use the notation ∇t ϕ = ∂t ϕ + [, ϕ] and denote by dA and ∗ the differential and Hodge operator on . Now if we put αˆ = 0, then we obtain for all ϕ, ψ ∈ C0∞ ( × , g) by partial integration ≤ Cc + ∇t ξ − ∇s ηLq (×) ϕ q ∗ ˆ d ϕ , β A L (×) , × ≤ Cc + ∇s ξ + ∇t ηLq (×) ψ q ∗ ˆ ∗d ψ , β A L (×) . ×
∗ βˆ and d βˆ are of class Lq on × . Now This shows that the weak derivatives d Lemma 2.9 implies that ∇ βˆ is of class Lq on × with ˆ Lq (×) ≤ C c + ∇t ξ − ∇s ηq + ∇s ξ + ∇t ηq + β ˆ q ∇ β ≤ C c + (β, ζ )Lq (S 1 ×Y ) .
So it remains to deduce the Lq -regularity of ∂s βˆ and ∂t βˆ on S 1 × [0, 2 ] × from the above inequality for ϕ = ψ = 0, namely from ≤ Cc + dA η + ∗dA ξ Lq (×) α ˆ ˆ Lq ∗ (×) . (32) ∇ α ˆ + ∗∇ α ˆ , β s t ×
This holds for all αˆ ∈ C ∞ (×, T∗ ⊗g) with compact support and α(s, ˆ 0, ·) ∈ TAs L for all s ∈ S 1 . We now employ different arguments in the cases q > 2 and q < 2.
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Case q > 2. In this case the regularity of ∂s βˆ and ∂t βˆ will follow from [W2, Theorem 1.3] on the Banach space X = Lq (, T∗ ⊗ g) with the complex structure given by the Hodge operator on with respect to the metric gs,t . From (32) one obtains the following estimate for some constant C and all αˆ as above: ˆ β , ∂s αˆ + ∂t (∗α) ˆ ˆ Lq (×) α ≤ C c + ηW 1,q (×) + ξ W 1,q (×) + β ˆ Lq ∗ (×) ≤ C c + (β, ζ )Lq (S 1 ×Y ) α (33) ˆ Lq ∗ (,X∗ ) . ∗
∗
Note that this extends to the W 1,q (, Lq ())-closure of the admissible αˆ in (32). In particular the estimate above holds for all αˆ ∈ W 1,q (, X) that are compactly supported and satisfy α(s, ˆ 0, ·) ∈ TAs L for all s ∈ S 1 . To see that these can be approximated by smooth αˆ with Lagrangian boundary conditions one uses the Banach submanifold coordinates for L given by [W2, Lemma 4.3] as before. Here the Lagrangian submanifold L ⊂ X is totally real with respect to all Hodge operators as before, and it is the Lq restriction or -completion of the original submanifold in A0,p (), hence it is modelled 1,q on Wz (, g) × Rm , a closed subspace of Lq (, Rn ) (see [W2, Lemma 4.2, 4.3]). However, in order to be able to apply [W2, Theorem 1.3], we need to extend this estimate to all αˆ ∈ W 1,∞ (, X ∗ ) with compact support and α(s, ˆ 0) ∈ (∗TAs L)⊥ for all ∗ s ∈ S 1 . This is possible since any such αˆ can be approximated in W 1,q (, X∗ ) by ∞ αˆ i ∈ C (, X) that are compactly supported and satisfy the above stronger boundary condition αˆ i (s, 0) ∈ TAs L for all s ∈ S 1 . Indeed, [W2, Lemma 2.2] provides such an approximating sequence αi without the Lagrangian boundary conditions. From the proof via mollifiers one sees that the approximating sequence can be chosen with compact support in . Now for all s ∈ S 1 one has the topological splitting X = TAs L ⊕ ∗TAs L and thus X∗ = (TAs L)⊥ ⊕ (∗TAs L)⊥ . Since q > 2 the embedding X → X∗ is continuous. This identification uses the L2 inner product on X which equals the metric ω(·, ∗·) given by the symplectic form ω and the complex structure ∗. So due to the Lagrangian condition this embedding maps TAs L → (∗TAs L)⊥ and ∗TAs L → (TAs L)⊥ . We write αˆ = γ + δ and αi = γi + δi according to these splittings to obtain γ , δ ∈ C ∞ (, X∗ ) and γi , δi ∈ C ∞ (, X) such that ∗TA L γi → γ ∈ (TA L)⊥ and TA L δi → δ ∈ (∗TA L)⊥ with convergence ∗ in W 1,q (, X∗ ). The boundary condition on αˆ gives γ |t=0 ≡ 0. Moreover, ∂t γ is uniformly bounded in X∗ , so one can find a constant C such that γ (s, t)X∗ ≤ Ct for all t ∈ [0, 43 ) and hence for sufficiently small ε > 0, γ Lq ∗ (S 1 ×[0,ε],X∗ ) ≤
1
1+ q ∗ C . 1+q ∗ ε
Now let δ > 0 be given and choose 1 > ε > 0 such that γ Lq ∗ (S 1 ×[0,ε],X∗ ) ≤ εδ and γ W 1,q ∗ (S 1 ×[0,ε],X∗ ) ≤ δ. Next, choose a sufficiently large i ∈ N such that γi − γ W 1,q ∗ (,X∗ ) ≤ εδ, and let h ∈ C ∞ ([0, 43 ], [0, 1]) be a cutoff function with h(0) = 0, h|t≥ε ≡ 0, and |h | ≤ 2ε . Then αˆ i := hγi + δi ∈ C ∞ (, X) satisfies the Lagrangian boundary condition αˆ i (s, 0) ∈ TAs L and approximates αˆ in view of the following estimate:
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αˆ i − α ˆ W 1,q ∗ (,X∗ ) ≤ h(γi − γ )W 1,q ∗ (,X∗ ) + (1 − h)γ W 1,q ∗ (,X∗ ) ≤ γi − γ W 1,q ∗ (,X∗ ) + 2ε γi − γ Lq ∗ (,X∗ ) +γ W 1,q ∗ (S 1 ×[0,ε],X∗ ) + 2ε γ Lq ∗ (S 1 ×[0,ε],X∗ ) ≤ 6δ. This approximation shows that (33) holds indeed true for all αˆ ∈ W 1,∞ (, X ∗ ) with compact support and α(s, 0) ∈ (∗TAs L)⊥ for all s ∈ S 1 . Thus [W2, Theorem 1.3] asserts that βˆ ∈ W 1,q (S 1 × [0, 2 ], X), and hence ∂s βˆ and ∂t βˆ are of class Lq on K × wit K := S 1 × [0, 2 ] as claimed, with ˆ W 1,q (K,X) ≤ C c + (β, ζ )Lq (S 1 ×Y ) . ˆ Lq (K×) + ∂t β ˆ Lq (K×) ≤ β ∂s β Case q < 2. In this case we cover S 1 by two intervals, S 1 = I1 ∪ I2 such that there are isometric embeddings (0, 1) → S 1 identifying [ 41 , 43 ] with I1 and I2 respectively. Abbreviate K := [ 41 , 43 ] × [0, 2 ] and let ⊂ (0, 1) × [0, 43 ] be a compact submanifold of the half space H such that K ⊂ int . Then for each of the above identifications S 1 \ {pt} ∼ = (0, 1) one has Lq -regularity of βˆ on × by assumption and of ∗dA ξ + dA η from above. Now the task is to establish in both cases the Lq -regularity of ∂s βˆ and ∂t βˆ on K × using (32). For that purpose choose a cutoff function h ∈ C ∞ (H, [0, 1]) supported in such that h|K ≡ 1. Then it suffices to show that for all γ ∈ C0∞ ( × , T∗ ⊗ g) (compactly supported in int( ) × ) ≤ C c + (β, ζ )Lq (S 1 ×) γ q ∗ ˆ ∂ γ , h β s L ( ×) . ×
ˆ and This gives the required Lq -regularity and -estimate for the weak derivative ∂s (hβ) hence for ∂s βˆ on K × . The regularity and estimate for ∂t βˆ follows by the same argument with ∂s γ replaced by ∂t γ . We linearize the submanifold chart maps along (As )s∈(0,1) ∈ L ∩ A() given by ∗ ∗ [W2, Lemma 4.3] for the Lagrangian L ⊂ A0,q (). Note that this uses the Lq -com1,q ∗ pletion of the actual Lagrangian in A0,p (). Abbreviate Z := Wz (, g) × Rm and let ∗s,t denote the Hodge operator on with respect to the metric gs,t . Then one obtains a smooth family of bounded isomorphisms ∼
∗
s,t : Z × Z −→ Lq (, T∗ ⊗ g) =: X defined for all (s, t) ∈ by s,t (ξ, v, ζ, w) = dAs ξ +
m
i=1 v
iγ
i (s) + ∗s,t dAs ζ
+
m
i=1 w
i
∗s,t γi (s).
Here γi ∈ C ∞ ((0, 1) × , T∗ ⊗ g) with γi (s) ∈ TAs L for all s ∈ (0, 1). If we abbreviate Z ∞ := Cz∞ (, g) × Rm ⊂ Z, then s,t maps Z ∞ × Z ∞ into the set of smooth 1-forms 1 (, g). So given any γ ∈ C0∞ ( × , T∗ ⊗ g) we have f := −1 ◦ γ ∈ C0∞ ( , Z ∞ × Z ∞ ) and for some constant C, f Lq ∗ ( ,Z×Z) ≤ Cγ Lq ∗ ( ,X) = Cγ Lq ∗ ( ×) . Write f = (f1 , f2 ) with fi ∈ C0∞ ( , Z ∞ ) and note that ∂s f1 = 0 due to the compact support. So one can solve φ1 = ∂s f1 by φ1 ∈ Cν∞ ( , Z ∞ ) with φ1 = 0
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and φ2 = ∂s f2 by φ2 ∈ Cδ∞ ( , Z ∞ ). (For the Cz∞ (, g)-component of Z ∞ one has solutions of the Laplace equation on every × {x} that depend smoothly on x ∈ .) Now let := (φ1 , φ2 ) ∈ C ∞ ( , Z × Z) and consider the 1-form αˆ γ := h · (−∂s + J0 ∂t ) ∈ C ∞ ( , X). This extends to a 1-form on × that is admissible in (32). Indeed, αˆ γ vanishes for s close to 0 or 1 and thus trivially extends to s ∈ S 1 . The Lagrangian boundary condition is met since for all s ∈ S 1 , αˆ γ (s, 0) = h(s, 0) · s,0 (−∂s φ1 − ∂t φ2 , −∂s φ2 + ∂t φ1 ) ∈ s,0 (Z, 0) = TAs L. So from (32) we obtain for all αˆ γ of the above form βˆ , ∂s αˆ γ + ∂t (∗αˆ γ ) ≤ C c + (β, ζ )Lq (S 1 ×) αˆ γ Lq ∗ (,X) . ×
Moreover, one has for all γ ∈ C0∞ ( × , T∗ ⊗ g) and the associated f , and αˆ γ αˆ γ Lq ∗ (,X) ≤ C W 1,q ∗ ( ,Z×Z) ≤ Cf Lq ∗ ( ,Z×Z) ≤ Cγ Lq ∗ ( ×) . Here the second inequality follows from = ∂s f and [W2, Lemma 2.1] as follows. In the Rm -component of Z, it is the usual elliptic estimate for the Dirichlet or Neumann 1,q ∗ problem. For the components in the infinite dimensional part Y := Wz (, g) of Z (still denoted by φi and fi ) this uses the following estimate. For all ψ ∈ Cν∞ ( , Y ∗ ) in the case i = 1 and for all ψ ∈ Cδ∞ ( , Y ∗ ) in the case i = 2, φi , ψ = φi , ψ × × ∂s fi , ψ = × fi , ∂s ψ = ×
≤ fi Lq ∗ ( ,Y ) ψW 1,q ( ,Y ∗ ) . Now a calculation shows that ∂s αˆ γ + ∂t (∗αˆ γ ) = h · ( ) + ∂s (h · )(−∂s + J0 ∂t ) + ∂t (h · )(∂t − J0 ∂s ). We then use = ∂s f to obtain ˆ , ∂s γ h · β × βˆ , h · ( ) + h · ∂s (f ) = × βˆ , ∂s αˆ γ + ∂t (∗αˆ γ ) ≤ × ˆ Lq ( ,X∗ ) − ∂s + J0 ∂t ) q ∗ ∗ +Cβ L ( ,Z×Z) + f Lq ( ,Z×Z) ≤ C c + (β, ζ )Lq (S 1 ×) γ Lq ∗ ( ×) .
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This holds with uniform constants for all γ ∈ C0∞ ( × , T∗ ⊗ g) and thus implies the Lq -regularity of ∂s βˆ on K × together with the estimate ˆ Lq (K×) ≤ C c + (β, ζ )Lq (S 1 ×Y ) . ∂s β ˆ on This establishes the Lq -regularity and -estimate for ∂s βˆ (and analogously of ∂t β) S 1 × [0, 2 ] and thus finishes the proof of Theorem C (iii). Proof of Theorem C (i). Lemma 3.1 and the identities (23) imply that for some constant C and for all (α, ϕ) in the domain of D(A, ) , (α, ϕ)W 1,p ≤ C D(A, ) (α, ϕ)p + (α, ϕ)p . Note that the embedding W 1,p (X) → Lp (X) is compact, so this estimate already implies that ker D(A, ) is finite dimensional and im D(A, ) is closed (see e.g. [Z, 3.12]). So it remains to consider the cokernel of D(A, ) . For that purpose we abbreviate Z := Lp (S 1 × Y, T∗ Y ⊗ g) × Lp (S 1 × Y, g), then coker D(A, ) = Z/im D(A, ) is a Banach space since im D(A, ) is closed. So it has the same dimension as its dual space (Z/im D(A, ) )∗ ∼ = (im D(A, ) )⊥ . Now let § : S 1 × Y → S 1 × Y denote the reflection §(s, y) := (−s, y) on S 1 ∼ = R/Z, then we claim that there is an isomorphism ∼
(im D(A, ) )⊥ −→ ker D§∗ (A, ) (β, ζ ) −→ (β ◦ §, ζ ◦ §).
(34)
Here D§∗ (A, ) = D(A , ) is the linearized operator corresponding to the reflected connection §∗ A˜ = A + ds with respect to the metric §∗ g˜ on X. Note that ker D§∗ (A, ) has finite dimension since the estimate in Theorem C (ii) also holds for the operator D§∗ (A, ) . So this would indeed prove that coker DA˜ is of finite dimension and hence DA˜ is a Fredholm operator. To establish the above isomorphism consider any (β, ζ ) ∈ (im D(A, ) )⊥ , that is ∗ ∗ 1,p β ∈ Lp (S 1 × Y, T∗ Y ⊗ g) and ζ ∈ Lp (S 1 × Y, g) such that for all α ∈ EA and ϕ ∈ W 1,p (S 1 × Y, g), S 1 ×Y
D(A, ) (α, ϕ) , (β, ζ ) = 0.
Iteration of Theorem C (iii) implies that β and ζ are in fact W 1,p -regular: We start with ∗ q = p∗ < 2, then the lemma asserts W 1,p -regularity. Next, the Sobolev embedding theorem gives Lq1 -regularity for some q1 ∈ ( 43 , 2) with q1 > p ∗ . Indeed, the Sobolev 4p∗ 4p∗ 4p∗ 4 ∗ embedding holds for any q1 ≤ 4−p ∗ , and 3 < 4−p ∗ as well as p < 4−p ∗ holds due to p∗ > 1. So the lemma together with the Sobolev embeddings can be iterated to give 4qi Lqi+1 -regularity for qi+1 = 4−q as long as 4 > qi > 2 or 2 > qi ≥ p ∗ . This iteration i yields q2 ∈ (2, 4) and q3 > 4. Thus another iteration of the lemma gives W 1,q3 - and thus also Lp -regularity of β and ζ . Finally, since p > 2 the lemma applies again and asserts the claimed W 1,p -regularity of β and ζ . Now by partial integration
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0= =
S ×Y 1
S1
=
Y
D(A, ) (α, ϕ) , (β, ζ )
∇s α − dA ϕ + ∗dA α , β +
S1
α , −∇s β − dA ζ + ∗dA β + S1 Y S1 − α ∧ β − ϕ , ∗β . S1
S1
∗ ∇s ϕ − dA α, ζ ∗ ϕ , −∇s ζ − dA β
Y
Y
(35)
Testing this with all α ∈ C0∞ (S 1 × Y, T∗ Y ⊗ g) ⊂ EA and ϕ ∈ C0∞ (S 1 × Y, g) ∗ β = 0. Then furthermore we deduce implies −∇s β − dA ζ + ∗dA β = 0 and −∇s ζ − dA 1 ∞ 1 ∗β(s)|∂Y = 0 for all s ∈ S from testing with ϕ that run through all of C (S × , g) on the boundary. Finally, S 1 α ∧ β = 0 remains from (35). Since both α and β restricted to S 1 × are continuous paths in A0,p (), this implies that for all s ∈ S 1 and every α ∈ TAs L, α ∧ β(s) = ω(α, β(s)), 0 = 1,p
where ω is the symplectic structure on A0,p (). Since TAs L is a Lagrangian sub1,p space, this proves β(s)|∂Y ∈ TAs L for all s ∈ S 1 and thus β ∈ EA , or equivalently β ◦ § ∈ EA◦§ . So (β ◦ §, ζ ◦ §) lies in the domain of D§∗ (A, ) . Now note that §∗ A˜ = A ◦ § − ( ◦ §)ds, thus one obtains (β ◦ §, ζ ◦ §) ∈ ker D§∗ (A, ) since ∗ β) ◦ § = 0. D§∗ (A, ) (β ◦ §, ζ ◦ §) = (−∇s β − dA ζ + ∗dA β) ◦ § , (−∇s ζ − dA This proves that the map in (34) indeed maps into ker D§∗ (A, ) . To see the surjectivity of this map consider any (β, ζ ) ∈ ker D§∗ (A, ) . Then the same partial integration as in (35) shows that (β ◦ §, ζ ◦ §) ∈ (im D(A, ) )⊥ , and thus (β, ζ ) is the image of this element under the map (34). So this establishes the isomorphism (34) and thus shows that D(A, ) is Fredholm.
A. Dirichlet and Neumann Problem Throughout this paper we use various regularity results for the Laplace operator. For convenience these are summarized in this appendix. We deal with (homogeneous) Dirichlet boundary conditions and with possibly inhomogeneous Neumann boundary conditions. Often, the equations are formulated weakly with the help of the following test function spaces:
Cδ∞ (M) = φ ∈ C ∞ (M) φ|∂M = 0 ,
C ∞ (M) = φ ∈ C ∞ (M) ∂φ = 0 . ν
∂ν ∂M
Here and throughout this appendix let M be a manifold with boundary. We abbreviate := d∗ d, and denote by ∂φ ∂ν the Lie derivative in the direction of the outer unit normal. Moreover, we use the notation N = {1, 2, . . . } and N0 = {0, 1, . . . }. The regularity theory for the Dirichlet and Neumann problem that is used in this paper can be summarized as follows. References are for example [GT] and [W1, Theorems 2.3’, 3.2, D.2].
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Proposition A.1. Let 1 < p < ∞ and k ∈ N, then there exists a constant C such that the following holds. Let f ∈ W k−1,p (M) and G ∈ W k,p (M) and suppose that u ∈ W k,p (M) is a weak solution of the inhomogeneous Neumann problem (or the Dirichlet problem, in which case one can drop G), that is for all ψ ∈ Cν∞ (M) (or for all ψ ∈ Cδ∞ (M)) M
u · ψ =
M
f ·ψ +
∂M
G · ψ.
Then u ∈ W k+1,p (M) and
uW k+1,p ≤ C f W k−1,p + GW k,p + uW k,p .
In the special case k = 0 there exists a constant C such that the following holds: Suppose that u ∈ Lp (M) and that there exists a constant c such that for all ψ ∈ Cν∞ (M) (or for all ψ ∈ Cδ∞ (M)) M
u · ψ ≤ cψW 1,p∗ .
Then u ∈ W 1,p (M) and uW 1,p ≤ C c + uLp . We also frequently encounter Laplace equations for 1-forms, where the components satisfy different boundary conditions. In these cases the following lemma allows to obtain regularity results for the components separately. The proof relies on the above standard regularity theory for the Laplace operator. Lemma A.2. Let (M, g) be a compact Riemannian manifold (possibly with boundary), let k ∈ N0 and 1 < p < ∞. Let X ∈ (TM) be a smooth vector field that is either perpendicular to the boundary, i.e. X|∂M = h · ν for some h ∈ C ∞ (∂M), or tangential, i.e. X|∂M ∈ (T∂M). In the first case let T = Cδ∞ (M), in the latter case let T = Cν∞ (M). Then there exists a constant C such that the following holds: Fix a function f ∈ W k,p (M) and a 2-form γ ∈ W k,p (M, 2 T∗ M) and suppose that the 1-form α ∈ W k,p (M, T∗ M) satisfies α , dη = f ·η ∀η ∈ C ∞ (M), M M ∗ α, d ω = γ , ω ∀ω = d(φ · ιX g) , φ ∈ T . M
M
Then α(X) ∈ W k+1,p (M) and
α(X)W k+1,p ≤ C f W k,p + γ W k,p + αW k,p .
Remark A.3. In the case k = 0 let p1 + p1∗ = 1, then the weak equations for α can be replaced by the following: There exist constants c1 and c2 such that α , dη ≤ c1 ηp∗ ∀η ∈ C ∞ (M), M α , d∗ d(φ · ιX g) ≤ c2 φ 1,p∗ ∀φ ∈ T . W M
The estimate then becomes α(X)W 1,p ≤ C c1 + c2 + αp .
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Proof of Lemma A.2 and Remark A.3. Let α ν ∈ C ∞ (M, T∗ M) be an Lp -approximating sequence for α such that α ν ≡ 0 near ∂M. Then one obtains for all φ ∈ T ,
M
LX α ν , dφ − ιX dα ν , dφ ν→∞ M M ν = lim − α , LX dφ − α ν , divX · dφ ν→∞ M M ν − α , ιYdφ LX g − dα ν , ιX g ∧ dφ M M = α , d(−LX φ − divX · φ) − α , d∗ (ιX g ∧ dφ) M M + α , φ · d(divX) − ιYdφ LX g M = f , −LX φ − divX · φ + γ , d(φ · ιX g) M M − α , d∗ (φ · dιX g) + α , φ · d(divX) − ιYdφ LX g .
α(X) · φ = lim
M
M
Here the vector field Ydφ is given by ιYdφ g = dφ. In the case k ≥ 1 further partial integration yields for all φ ∈ T ,
M
α(X) · φ =
M
F ·φ+
∂M
G · φ,
where F ∈ W k−1,p (M), G ∈ W k,p (M), and for some constant C, F W k−1,p + GW k,p ≤ C f W k,p + γ W k,p + αW k,p . So the regularity Proposition A.1 for the weak Laplace equation with either Neumann (i.e. T = Cν∞ (M)) or Dirichlet (i.e. T = Cδ∞ (M)) boundary conditions proves that α(X) ∈ W k+1,p (M) with the according estimate. In the case k = 0 one works with the following inequality: Let p1∗ + p1 = 1, then there is a constant C such that for all φ ∈ T , α(X) · φ ≤ C f p + γ p + αp φ 1,p∗ . W M
(Under the assumptions of Remark A.3, one simply replaces f p and γ p by c1 and c2 respectively.) The regularity and estimate for α(X) then follow from Proposition A.1. Acknowledgement. I would like to thank Dietmar Salamon for his constant help and encouragement in pursuing this project. Part of this research was supported by the Swiss National Science Foundation.
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References [A1] Atiyah, M.F.: Instantons in two and four dimensions. Commun. Math. Phys. 93, 437–451 (1984) [A2] Atiyah, M.F.: New invariants of three and four dimensional manifolds. Proc. Symp. Pure Math. 48, 285–299 (1988) [Ad] Adams, R.A.: Sobolev Spaces. London NewYork: Academic Press, 1978 [D1] Donaldson, S.K.: Polynomial invariants of smooth four-manifolds. Topology 29, 257–315 (1990) [D2] Donaldson, S.K.: Boundary value problems for Yang-Mills fields. J. Geom. and Phys. 8, 89–122 (1992) [DK] Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford: Oxford Science Publications, 1990 [DS] Dostoglou, S., Salamon, D.A.: Self-dual instantons and holomorphic curves. Ann. Math. 139, 581–640 (1994) [Fl] Floer, A.: An instanton invariant for 3-manifolds. Commun. Math. Phys. 118, 215–240 (1988) [Fu] Fukaya, K.: Floer homology for 3-manifolds with boundary I. Preprint 1997. http://www.kusm.kyoto-u.ac.jp/ fukaya/fukaya.html [GT] Gilbarg D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin-Heidelberg New York: Springer, 1977 [L] Lang, S.: Analysis II. Reading MA: Addison-Wesley, 1969 [S] Salamon, D.A.: Lagrangian intersections, 3-manifolds with boundary, and the Atiyah–Floer conjecture, Proceedings of the ICM, Z¨urich, 1994, Basel: Birkh¨auser, Vol. 1, 1995, pp. 526–536 [U1] Uhlenbeck, K.K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys. 83, 11–29 (1982) [U2] Uhlenbeck, K.K.: Connections with Lp -bounds on curvature. Commun. Math. Phys. 83, 31–42 (1982) [Wa] Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Scott, Foresman and Company, 1971 [W1] Wehrheim, K.: Uhlenbeck Compactness. EMS Lectures in Mathematics, 2004 [W2] Wehrheim, K.: Banach space valued Cauchy-Riemann equations with totally real boundary conditions. Commun. Contemp. Math. 6, no. 4, 601–635 (2004) [W3] Wehrheim, K.: Anti-self-dual instantons with Lagrangian boundary conditions II: Bubbling. Preprint. http://www.math.princeton.edu/∼wehrheim [Z] Zeidler, E.: Applied Functional Analysis – Main Principles and their Applications. Berlin Heidelberg New York: Springer, 1995 Communicated by N.A. Nekrasov
Commun. Math. Phys. 254, 91–127 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1238-9
Communications in
Mathematical Physics
Power-Counting Theorem for Non-Local Matrix Models and Renormalisation Harald Grosse1 , Raimar Wulkenhaar2 1
Institut f¨ur Theoretische Physik, Universit¨at Wien, Boltzmanngasse 5, 1090 Wien, Austria. E-mail:
[email protected] 2 Max-Planck-Institut f¨ ur Mathematik in den Naturwissenschaften, Inselstraße 22-26, 04103 Leipzig, Germany. E-mail:
[email protected] Received: 3 July 2003 / Accepted: 25 May 2004 Published online: 2 December 2004 – © Springer-Verlag 2004
Abstract: Solving the exact renormalisation group equation a` la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties—typically arising from orthogonal polynomials—which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our powercounting theorem is the renormalisation of field theories on noncommutative RD in matrix formulation.
1. Introduction Noncommutative quantum field theories show in most cases a phenomenon called UV/IRmixing [1] which seems to prevent the perturbative renormalisation. There is an enormous number of articles on this problem, most of them performing one-loop calculations extrapolated to higher order.A systematic analysis of noncommutative (massive) field theories at any loop order was performed by Chepelev and Roiban [2, 3]. They calculated the integral of an arbitrary Feynman graph using the parametric integral representation and expressed the result in terms of determinants involving the incidence matrix and the intersection matrix. They succeeded to evaluate the leading contribution to the determinants in terms of topological properties of ribbon graphs wrapped around Riemann surfaces. In this way a power-counting theorem was established which led to the identification of two power-counting non-renormalisable classes of ribbon graphs. The Rings-type class consists of graphs with classically divergent subgraphs wrapped around the same handle of the Riemann surface. The Com-type class consists of planar graphs with external legs
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ending at several disconnected boundary components with the momentum flow into a boundary component being identically zero1 . Except for models with enough symmetry, noncommutative field theories are not renormalisable by standard techniques. One may speculate that the reason is the too na¨ıve way of performing the various limits. Namely, a field theory is a dynamical system with infinitely many degrees of freedom defined by a certain limiting procedure of a system with finitely many degrees of freedom. One may perform the limits formally to the path integral and evaluate it by Feynman graphs which are often meaningless. It is the art of renormalisation to give a meaning to these graphs. This approach works well in the commutative case, but in the noncommutative situation it seems to be not successful. A procedure which deals more carefully with the limits is the renormalisation group approach due to Wilson [5], which was adapted by Polchinski to a very efficient renormalisability proof of commutative φ 4 -theory in four dimensions [6]. There are already some attempts [7] to use Polchinski’s method to renormalise noncommutative field theories. We are, however, not convinced that the claimed results (UV-renormalisability) are so easy to obtain. The main argument in [7] is that the Polchinski equation is a oneloop equation so that the authors simply compute an integral having exactly one loop. It is, however, not true that nothing new happens at higher loop order. For instance, all one-loop graphs can be drawn on a genus-zero Riemann surface. The entire complexity of Riemann surfaces of higher genus as discussed by Chepelev and Roiban [2, 3] shows up at higher loop order and is completely ignored by the authors of [7]. As we show in this paper, the same discussion of Riemann surfaces is necessary in the renormalisation group approach, too. The mentioned complexity is due to the phase factors described by the intersection matrix which result in convergent but not absolutely convergent momentum integrals. As such it is very difficult to access this complexity by Polchinski’s procedure [6] which is based on taking norms of the contributions. Moreover, Chepelev and Roiban established the link between power-counting and the topology of the Riemann surface via the parametric integral representation, which is based on Gaußian integrations. These are not available for Polchinski’s method where we deal with cut-off integrals. In conclusion, we believe it is extremely difficult (if not impossible) to use the exact renormalisation group equation for noncommutative field theories in momentum space. The best one can hope is to restrict oneself to limiting cases where e.g. the non-planar graphs are suppressed [4, 8]. Even this restricted model has rich topological features. Fortunately, there exists a base fmn for the algebra under consideration,2 where the -product is reduced to an ordinary product of (infinite) matrices, fmn fkl = δnk fml , see [10]. The interaction d D x(φ φ φ φ) can then be written as tr(φ 4 ), where φ is now an infinite matrix (with entries of rapid decay). The price for the simplification of the interaction is that the kinetic matrix, or rather its inverse, the propagator, becomes very complicated. However, in Polchinski’s approach the propagator is anyway made complicated when multiplying it with the smooth cut-off function K[]. The parameter is an energy scale which varies between the renormalisation scale R and the initial scale 0 R . Introducing in the bilinear (kinetic) part of the action the cut-off function and replacing the φ 4 -interaction by a -dependent effective action L[φ, ], 1
These graphs were called “Swiss Cheese” in [4]. For another matrix realisation of the noncommutative RD and its treatment by renormalisation group methods, see [9]. 2
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the philosophy is to determine L[φ, ] such that the generating functional Z[J, ] is actually independent of . In this paper we provide the prerequisites to investigate the renormalisation of general non-local matrix models. We prove a power-counting theorem for the effective action L[φ, ] by solving (better: estimating) the Polchinski equation perturbatively. Our derivation and solution of the matrix Polchinski equation combines the original ideas of [6] with some of the improvements made in [11]. In particular, we follow [11] to obtain 0 -independent estimations for the interaction coefficients. The Polchinski equation for matrix models can be visualised by ribbon graphs. The power-counting degree of divergence of a ribbon graph depends on the topological data of the graph and on two scaling dimensions of the cut-off propagator. In this way, suitable scaling dimensions provide a simple criterion to decide whether a non-local matrix model has the chance of resulting in a renormalisable model or not. However, having the right scaling dimensions is not sufficient for the renormalisability of a model, because a divergent interaction is parametrised by an infinite number of matrix indices. Thus, a renormalisable model needs further structures3 which relate these infinitely many interaction coefficients to a finite number of base couplings. Nevertheless, the dimension criterion proven in this paper is of great value. For instance, it discards immediately the standard φ 4 -models on noncommutative RD , D = 2, 4, in the matrix base. These models have the wrong scaling dimension, which is nothing but the manifestation of the old UV/IR-mixing problem [1]. Looking closer at the origin of the wrong scaling dimensions it is not difficult to find a deformation of the free action which has the chance to be a renormalisable model. In the matrix base of the noncommutative RD , the Laplace operator becomes a tri-diagonal band matrix. The main diagonal behaves nicely, but the two adjacent diagonals are “too big” and compensate the desired behaviour of the main diagonal. Making the adjacent diagonals “smaller” one preserves the properties of the main diagonal and obtains the good scaling dimensions required for a renormalisable model. The deformation of the adjacent diagonals corresponds to the inclusion of a harmonic oscillator potential in the free field action. We treat in [13] the φ 4 -model on noncommutative R2 within the Wilson-Polchinski approach in more detail. We prove that this model is renormalisable when adding the harmonic oscillator potential. Remarkably, the model remains renormalisable when scaling the oscillator potential in a certain way to zero with the removal 0 → ∞ of the cut-off. We prove in [14] that the φ 4 -model on noncommutative R4 is renormalisable to all orders by imposing normalisation conditions for the physical mass, the field amplitude, the frequency of the harmonic oscillator potential and the coupling constant. In particular, the harmonic oscillator potential cannot be removed from the model. It gives the explicit solution of the UV/IR-duality which suggests that noncommutativity relevant at short distances goes hand in hand with a different physics at very large distances. The oscillator potential makes the φ 4 -action covariant with respect to a duality transformation [15] between positions and momenta. We stress that the power-counting theorem proven in this paper was indispensable to have from the start the right φ 4 -model for the renormalisation proof [13, 14]. Many noncommutative field theories have a matrix formulation, too. We think of fuzzy spaces 3 In the first version of this paper we had proposed a “reduction-of-couplings” mechanism [12] to get a finite number of relevant/marginal base couplings. Meanwhile it turned out [13, 14] that the models of interest provide automatically such structures in the form of orthogonal polynomials.
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and q-deformed models. Our general power-counting theorem can play an important rˆole in the renormalisation proof of these examples. 2. The Exact Renormalisation Group Equation We consider a φ 4 -matrix model with a general (non-diagonal) kinetic term, 1 λ S[φ] = VD Gmn;kl φmn φkl + φmn φnk φkl φlm , 2 4! m,n,k,l
(2.1)
m,n,k,l
where m, n, k, l ∈ Nq . For the noncommutative RD , D even, we have q = D2 . The factor VD is the volume of an elementary cell. The choice of φ 4 is no restriction but for us the most natural one because we are interested in four-dimensional models. Standard matrix models are given by q=1,
Gmn;kl =
1 δml δnk . µ20
(2.2)
For reviews on matrix models and their applications we refer to [16, 17]. The idea to apply renormalisation group techniques to matrix models is also not new [18]. The difference of our approach is that we will not demand that the action can be written as the trace of a polynomial in the field, that is, we allow for matrix-valued kinetic terms. The only restriction we are imposing is Gmn;kl = 0
unless m + k = n + l .
(2.3)
The restriction (2.3) is due to the fact that the action comes from a trace. It is verified D for the noncommutative RD due to the O(2) 2 -symmetry of both the interaction and the kinetic term. The kinetic matrix Gmn;kl contains the entire information about the differential calculus, including the underlying (Riemannian) geometry, and the masses of the model. More important than the kinetic matrix G will be its inverse, the propagator defined by Gmn;kl lk;sr = nm;lk Gkl;rs = δmr δns . (2.4) k,l
k,l
Due to (2.3) we have the same index restrictions for the propagator: nm;lk = 0
unless m + k = n + l .
(2.5)
Let us introduce a notion of locality: Definition 1. A matrix model is called local if nm;lk = (m, n)δml δnk for some function (m, n), otherwise non-local. We add sources J to the action (2.1) and define a (Euclidean) quantum field theory by the generating functional (partition function) Z[J ] = (2.6) dφab exp − S[φ] − VD φmn Jnm . a,b
m,n
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According to Polchinski’s derivation of the exact renormalisation group equation we now consider a (at first sight) different problem than (2.6). Via a cut-off function K[m, ], which is smooth in and satisfies K[m, ∞] = 1, we modify the weight of a matrix index m as a function of a certain scale : dφab exp − S[φ, J, ] , (2.7) Z[J, ] = a,b
1 S[φ, J, ] = VD φmn GK mn;kl () φkl + L[φ, ] 2 m,n,k,l
+
φmn Fmn;kl []Jkl +
m,n,k,l
GK mn;kl () :=
1 Jmn Emn;kl []Jkl +C[] , (2.8) 2
m,n,k,l
K[i, ]−1 Gmn;kl ,
(2.9)
i∈m,n,k,l
with L[0, ] = 0. Accordingly, we define K nm;lk () =
K[i, ] nm;lk .
(2.10)
i∈m,n,k,l D
D
For indices m = (m1 , . . . , m 2 ) ∈ N 2 we would write the cut-off function as a prod D2 mi uct K[m, ] = i=1 K , where K(x) is a smooth function on R+ with 2 (VD ) D 2
K(x) = 1 for 0 ≤ x ≤ 1 and K(x) = for x ≥ 2. In the limit → 0, the par2 i i 2 tition function (2.7) vanishes unless φmn = 0 for maxi (m , n ) ≥ 2(VD ) D , thus implementing a cut-off of the measure a,b dφab in (2.7). All other formulae involve mi positive powers of K which multiply through the cut-off propagator (2.10) 2 (VD ) D 2
the appearing matrix indices. In the limit → 0, K[m, ] has finite support in m so that all infinite-sized matrices are reduced to finite ones. The function C[] is the vacuum energy and the matrices E and F , which are not necessary in the commutative case, must be introduced because the propagator is non-local. It is, in general, not possible to separate the support of the sources J from the support of the -variation of K. Due to K[m, ∞] = 1 we formally obtain (2.6) for → ∞ in (2.9) if we set L[φ, ∞] =
λ φmn φnk φkl φlm , 4!
m,n,k,l
C[∞] = 0 ,
Emn;kl [∞] = 0 ,
Fmn;kl [∞] = δml δnk .
(2.11)
However, we shall expect divergences in the partition function which require a renormalisation, i.e. additional (divergent) counterterms in L[φ, ∞]. In the Feynman graph solution of the partition function one carefully adapts these counterterms so that all divergences disappear. If such an adaptation is possible with a finite number of local counterterms, the model is considered as perturbatively renormalisable. Following Polchinski [6] we proceed differently to prove renormalisability. We first ask ourselves how to choose L, C, E, F in order to make Z[J, ] independent of . After straightforward calculation one finds the answer
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∂ Z[J, ] = 0 iff (2.12) ∂
1 ∂K 1 ∂ 2 L[φ, ] ∂L[φ, ] nm;lk () ∂L[φ, ] ∂L[φ, ] , − = ∂ 2 ∂ ∂φmn ∂φkl VD ∂φmn ∂φkl φ m,n,k,l
(2.13)
∂Fmn;kl [] =− ∂
∂Emn;kl [] =− ∂
GK mn;m n ()
m ,n ,k ,l
T Fmn;m n []
m ,n ,k ,l
∂K n m ;l k () ∂
∂K n m ;l k () ∂
Fk l ;kl [] ,
(2.14)
Fk l ;kl [] ,
(2.15)
∂C[] 1 ∂ = ln K[m, ]K[n, ] ∂ VD m,n ∂ −
2 ∂K 1 nm;lk () ∂ L[φ, ] , 2VD ∂ ∂φmn ∂φkl φ=0
(2.16)
m,n,k,l
where f [φ] φ := f [φ] − f [0]. Na¨ıvely we would integrate (2.13)–(2.16) for the initial conditions (2.11). Technically, this would be achieved by imposing the conditions (2.11) not at = ∞ but at some finite scale = 0 , followed by taking the limit 0 → ∞. This is easily done for (2.14)–(2.16): K GK (2.17) Fmn;kl [] = mn;m n ()n m ;kl (0 ) , m ,n
Emn;kl [] =
m ,n ,k ,l
K K K ( ) G () − G ( ) K 0 0 mn;m n n m ;l k n m ;l k k l ;kl (0 ) ,
2 C[] = ln K[m, ]K −1 [m, 0 ] VD m 0 2 ∂K 1 nm;lk ( ) ∂ L[φ, ] d . + 2VD ∂ ∂φmn ∂φkl φ=0
(2.18)
(2.19)
m,n,k,l
At = 0 the functions F, E, C become independent of 0 and satisfy, in particular, (2.11) in the limit 0 → ∞. The partition function Z[J, ] is evaluated by Feynman graphs with vertices given by the Taylor expansion coefficients Lm1 n1 ;...;mN nN [] :=
∂ N L[φ, ] 1 N ! ∂φm1 n1 ∂φm2 n2 . . . ∂φmN nN φ=0
(2.20)
connected with each other by internal lines K () and to sources J by external lines K (0 ). As K[m, ] has finite support in m for finite , the summation variables in the above Feynman graphs are via the propagator K () restricted to a finite set. Thus, loop summations are finite, provided that the interaction coefficients Lm1 n1 ;...;mN nN [] are bounded. In other words, for the renormalisation of a non-local matrix model it is necessary to prove that the differential equation (2.13) admits a regular solution. As
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pointed out in the Introduction, to obtain a physically reasonable quantum field theory one has additionally to prove that there is a regular solution of (2.13) which depends on a finite number of initial conditions only. This requirement is difficult to fulfil because there is, a priori, an infinite number of degrees of freedom given by the Taylor expansion coefficients (2.20). This is the reason for the fact that renormalisable (four-dimensional) quantum field theories are rare. We are going to integrate (2.13) between a certain renormalisation scale R and the initial scale 0 . We assume that Lm1 n1 ;...;mN nN can be decomposed into parts (i) Lm1 n1 ;...;mN nN which for R ≤ ≤ 0 scale homogeneously: ∂L(i) m1 n1 ;...;mN nN [] . ≤ ri P qi ln ∂ R
(2.21)
Here, P q [X] ≥ 0 stands for some polynomial of degree q in X ≥ 0. Clearly, P q [X], for X ≥ 0, can be further bound by a polynomial with non-negative coefficients. As usual we define (i)
Definition 2. Homogeneous parts Lm1 n1 ;...;mN nN [] in (2.21) with ri > 0 are called relevant, with ri < 0 irrelevant and with ri = 0 marginal. There are two possibilities for the integration, either from 0 down to or from R up to , corresponding to the identities (i)
Lm1 n1 ;...;mN nN [] =
(i) Lm1 n1 ;...;mN nN [0 ] − (i)
= Lm1 n1 ;...;mN nN [R ] +
0
R
d ∂ (i) L [ ] (2.22a) ∂ m1 n1 ;...;mN nN d ∂ (i) L [ ] . (2.22b) ∂ m1 n1 ;...;mN nN
One has q x j q q! − r ln (−1) x r R + const x q r q+1 x j ! r−1 dx x = ln j =0 xR 1 x q+1 + const ln q+1 xR
for r = 0 , for r = 0 . (2.23)
At the end we are interested in the limit 0 → ∞. This requires that positive powers of 0 must be prevented in the estimations. For ri < 0 we can safely take the direction (2.22a) of integration and then, because all coefficients are positive, the limit 0 → ∞ in the integral of (2.22a). Thus, ∞ (i) (i) d ∂ (i) L L [ ] m1 n1 ;...;mN nN [] ≤ Lm1 n1 ;...;mN nN [0 ] + m n ;...;m n N N ∂ 1 1 (i) , for ri < 0. ≤ Lm1 n1 ;...;mN nN [0 ] + −|ri | P qi ln R (2.24)
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Here, P qi is a new polynomial of degree qi with non-negative coefficients. Now, the (i) limit 0 → ∞ carried out later requires that Lm1 n1 ;...;mN nN [0 ] in (2.24) is bounded, (i) [0 ] < Cs , with si > 0. As the resulting estimation (2.24) is further i.e. L m1 n1 ;...;mN nN
0i
iterated, si must be sufficiently large. We do not investigate this question in detail and simply note that it is safe to require < −|ri | P qi ln 0 , 0 R
(i) L
m1 n1 ;...;mN nN [0 ]
for ri < 0 ,
(2.25)
for the boundary condition. In the other case ri ≥ 0, the integration direction (2.22a) will produce divergences in 0 → ∞. Thus, we have to choose the other direction (2.22b). The integration (2.23) produces alternating signs, but these can be ignored in the maximisation. The only contribution from the lower bound R in the integral of (2.22b) is the term with j = 0 in (2.23). There, we can obviously ignore it in the difference r − rR . We thus obtain from (2.23) the estimation for ri > 0 (i) ri P qi ln R (i) L m1 n1 ;...;mN nN [] ≤ Lm1 n1 ;...;mN nN [R ] + qi +1 P ln R for ri = 0 . (2.26) 0 in Polchinski’s original work [6] to P ln R is due to The reduction from P ln R [11]. We can summarise these considerations as follows: ∂ (i) Definition/Lemma 3. Let ∂ Lm1 n1 ;...;mN nN [] be bounded by (2.21), ∂L(i) m1 n1 ;...;mN nN [] . ≤ ri P qi ln ∂ R
(2.27)
to The integration of (2.27) is for irrelevant interactions performed from−|ri |0 down (i) 0 q i starting from an initial condition bounded by Lm1 n1 ;...;mN nN [0 ] < 0 P ln R . For relevant and marginal interactions we have to integrate (2.27) from R up to , (i) starting from an initial condition Lm1 n1 ;...;mN nN [R ] < ∞. Under these conventions we have ≤ ri P qi +1 ln . [] m1 n1 ;...;mN nN R
(i) L
(2.28)
Let us give a few comments: – The stability (2.27) versus (2.28) of the estimation will be very useful in the iteration process. – Integrations according to the direction (2.22b), which entail an initial condition (i) Lm1 n1 ;...;mN nN [R ], are expensive for renormalisation, because each such condition (i)
(even the choice Lm1 n1 ;...;mN nN [R ] = 0) corresponds to a normalisation experiment. In order to have a meaningful theory, there has to be only a finite number of
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required normalisation experiments. Initial data at 0 do not correspond to normalisation conditions, because the interaction at 0 → ∞ is experimentally not accessible. Moreover, unless artificially kept alive4 , an irrelevant coupling scales away for (i) 0 → ∞ via its own dynamics. The property lim0 →∞ Lm1 n1 ;...;mN nN [0 ] = 0 for an irrelevant coupling is, therefore, a result and no condition. – There might be cases where the direction (2.22b) for ri < 0 gives convergence for 0 → ∞ nevertheless. This corresponds to the over-subtractions in the BPHZ renormalisation scheme. We shall not exploit this possibility. Unless there are further correlations between functions with different indices, specify(i) ing Lm1 n1 ;...;mN nN [R ] means to impose an infinite number of normalisation conditions (because of mi , ni ∈ ND/2 ). Hence, a non-local matrix model with relevant and/or marginal interactions can only be renormalisable if some additional structures exist which relate all divergent functions to a finite number of relevant/marginal base interactions. Such a distinguished property depends crucially on the model. Presumably, the class of models where such a reduction is possible is rather small. It cannot be the purpose of this paper to analyse these reductions. Instead, our strategy is to find the general power-counting behaviour of a non-local matrix model which limits the class of divergent functions among which the reduction has to be studied in detail. For example, we will find that under very general conditions on the propagator all non-planar graphs (as defined below) are irrelevant. Such a result is already an enormous gain5 for the detailed investigation of a model. Thus, our strategy is to integrate the Polchinski equation (2.13) perturbatively between two scales R and 0 for a self-determined choice of the boundary condition according to Definition/Lemma 3. The resulting normalisation condition for relevant and marginal interactions will not be the correct choice for a renormalisable model. Nevertheless, the resulting estimation (2.28) is compatible with a more careful treatment. Taking the example [14] of the φ 4 -model on noncommutative R4 , we would replace 2
P q [ln R ] in (2.28) by irrelevant – almost all of the relevant functions with bound µ2 2 µ2 q functions with bound max(m1 , n1 , . . . , mN , nN ) 2 P [ln R ], and – almost all marginal functions with bound P q [ln R ] in (2.28) by irrelevant functions 2
µ q with bound max(m1 , n1 , . . . , mN , nN ) 2 P [ln
R ],
for some reference scale µ. 3. Ribbon Graphs and Their Topologies We can symbolise the expansion coefficients Lm1 n1 ;...;mN nN as
(3.1) An example of an irrelevant coupling which remains present for 0 → in two-dimensional models [13]. 5 We recall [1] that non-planar graphs produce the trouble in noncommutative quantum field theories in momentum space. 4
∞ is the initial φ 4 -interaction
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The big circle stands for a possibly very complex interior and the outer (dotted) double lines stand for the valences produced by differentiation (2.20) with respect to the N fields φmi ni . The arrows are merely added for bookkeeping purposes in the proof of the power-counting theorem. Since we work with real fields, i.e. φmn = φnm , the expansion coefficients Lm1 n1 ;...;mN nN have to be unoriented. The situation is different for complex fields where φ = φ ∗ leads to an orientation of the lines. In this case we would draw both arrows at the double line either incoming or outgoing. The graphical interpretation of the Polchinski equation (2.13) is found when differentiating it with respect to the fields φmi ni :
(3.2) Combinatorical factors are not shown and a symmetrisation in all indices mi ni has to be performed. On the rhs of (3.2) the two valences mn and kl of the subgraphs are connected to the ends of a ribbon which symbolises the differentiated propagao ∂ tor n k / = ∂ K nm;lk . For local matrix models in the sense of Definition 1 we m l can regard the ribbon as a product of single lines with interaction given by (m, n). For non-local matrix models there is an exchange of indices within the entire ribbon. We can regard (2.13) construction scheme for L[φ, ] if we introduce as a formal V (V ) [φ, ] and additionally impose a cut-off in N for a grading L[φ, ] = ∞ V =1 λ L V = 1, i.e (1)
Lm1 n1 ;...;mN nN [] = 0
for N > N0 .
(3.3)
In order to obtain a φ 4 -model we choose N0 = 4 and the grading as the degree V in the (1) coupling constant λ. We conclude from (2.13) that Lm1 n1 ;...;m4 n4 is independent of so that it is identified with the original (λ/4!)φ 4 -interaction in (2.1): (1)
Lm1 n1 ;m2 n2 ;m3 n3 ;m4 n4 [] 1 = δn1 m2 δn2 m3 δn3 m4 δn4 m1 + δn1 m3 δn3 m4 δn4 m2 δn2 m1 4! 6 +δn1 m4 δn4 m2 δn2 m3 δn3 m1 + δn1 m4 δn4 m3 δn3 m2 δn2 m1 +δn1 m3 δn3 m2 δn2 m4 δn4 m1 + δn1 m2 δn2 m4 δn4 m3 δn3 m1 .
(3.4)
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To the first term on the rhs of (3.4) we associate the graph
(3.5) The graphs for the other five terms are obtained by permutation of indices. As mentioned before, a complex φ 4 -model would be given by oriented propagators −→ and examples for vertices are −→
(3.6) The consequence is that many graphs of the real φ 4 -model are now excluded. We can thus obtain the complex φ 4 -model from the real one by deleting the impossible graphs. The iteration of (3.2) with starting point (3.5) leads to ribbon graphs. The first examples of the iteration are
(3.7) We can obviously build very complicated ribbon graphs with crossings of lines which cannot be drawn any more in a plane. A general ribbon graph can, however, be drawn on a Riemann surface of some genus g. In fact, a ribbon graph defines the Riemann surfaces topologically through the Euler characteristic χ . We have to regard here the external lines of the ribbon graph as amputated (or closed), which means to directly connect the single lines mi with ni for each external leg mi ni . A few examples may help to understand this procedure:
(3.8)
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(3.9)
(3.10) The genus is computed from the number L˜ of single-line loops of the closed graph, the number I of internal (double) lines and the number V of vertices of the graph according to χ = 2 − 2g = L˜ − I + V .
(3.11)
˜ I, V There can be several possibilities to draw the graph and its Riemann surface, but L, and thus g remain unchanged. Indeed, the Polchinski equation (2.13) interpreted as in (3.2) tells us which external legs of the vertices are connected. It is completely irrelevant how the ribbons are drawn between these legs. In particular, there is no distinction between overcrossings and undercrossings. There are two types of loops in (amputated) ribbon graphs: – Some of them carry at least one external leg. They are called boundary components (or holes of the Riemann surface). Their number is B. – Some of them do not carry any external leg. They are called inner loops. Their number is L˜ 0 = L˜ − B. Boundary components consist of a concatenation of trajectories from an incoming index ni to an outgoing index mj . In the example (3.8) the inner boundary component → consists of the single trajectory − n− 1 m6 whereas the outer boundary component is made − − → − − → of two trajectories n3 m4 and n5 m2 . We let o[nj ] be the outgoing index to nj and i[mj ] be the incoming index to mj . We have to introduce a few additional notations for ribbon graphs. An external vertex is a vertex which has at least one external leg. We denote by V e the total number of external vertices. For the arrangement of external legs at an external vertex there are the following possibilities:
(3.12) We call the first three types of external vertices simple vertices. They provide one starting point and one end point of trajectories through a ribbon graph. The fourth vertex
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in (3.12) is called a composed vertex. It has two starting points and two end points of trajectories. A composed vertex can be decomposed by pulling the two propagators with attached external lines apart:
(3.13) In this way a given graph with composed vertices is decomposed into S segments. The external vertices of the segments are either true external vertices or the halves of a composed vertex. If composed vertices occur in loops, their decomposition does not always increase the number of segments. We need the following Definition 4. The segmentation index ι of a graph is the maximal number of decompositions of composed vertices which keep the graph connected. It follows immediately that if V c is the number of composed vertices of a graph and S the number of segments obtained by decomposing all composed vertices we have ι = Vc −S +1 .
(3.14)
In order to evaluate Lm1 n1 ;...;mN nN [] by connection and contraction of subgraphs according to (3.2) we need estimations for index summations of ribbon graphs. Namely, our strategy is to apply the summations in (3.2) either to the propagator or the subgraph only and to maximise the other object over the summation indices. We agree to fix all starting points of trajectories and sum over the end points of trajectories. However, due to (2.5) and (3.4) not all summations are independent: The sum of outgoing indices equals for each segment the sum of incoming indices. Since there are V e + V c (end points of) trajectories in a ribbon graph, there are s ≤ Ve +Vc −S = Ve +ι−1
(3.15)
independent index summations. The inequality (3.15) also holds for the restriction to each segment if V e includes the number of halves of composed vertices belonging to the segment. We let E s be the set of s end points of trajectories in a graph over which we are going to sum, keeping the starting points of these trajectories fixed. We define ≡ ··· if E s = {m1 , m2 , . . . , ms } . (3.16) Es
m1
ms
i[mj ]=const
Taking the example of the graph (3.8), we can due to V e +ι = 3 apply up to two index summations, i.e. a summation over at most two of the end points of trajectories m2 , m4 , m6 , where the corresponding incoming indices i[m2 ] = n5 , i[m4 ] = n3 and i[m6 ] = n1 are kept fixed. For the example of the graph (3.9) we can due to V e + ι = 2 apply at most one index summation, either over m1 for fixed i[m1 ] = n2 or over i[m2 ] = n1 . For E 1 = {m2 } we would consider
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(3.17) Note that for given n2 the other outgoing index is determined to m1 = n1 + n2 − m2 through index conservation at propagators (2.5) and vertices (3.5). It is part of the proof to show that the index summation (3.17) is bounded independently of the incoming indices n1 , n2 . 4. Formulation of the Power-Counting Theorem We first have to transform the Polchinski equation (2.13) into a dimensionless form. It is important here that in the class of models we consider there is always a dimensionful parameter, − 1 µ = VD D , (4.1) which instead of can be used to absorb the mass dimensions. The effective action L[φ, ] has total mass dimension D, a field φ has dimension D−2 2 and the dimension 4 of the coupling constant for the λφ interaction is 4 − D. We thus decompose L[φ, ] according to the number of fields and the order in the coupling constant: +2 ∞ 2V φ φ 1 λ V D (V ) m1 n1 mN nN L[φ, ] = µ A [] · · · . D−2 D−2 n ;...;m n m 1 1 N N N ! m ,n µ4−D µ 2 µ 2 V =1 N=2
i
i
(4.2) (V )
The functions Am1 n1 ;...;mN nN [] are assumed to be symmetric in their indices mi ni . Inserted into (2.13) we get
∂ (V ) [] A ∂ m1 n1 ;...;mN nN −1 N V 1 (V ) (V −V ) = Am11n1 ;...;mN −1 nN −1 ;mn []AmN nN1 ;...;mN nN ;kl [] Qnm;lk () 1 1 1 1 2 m,n,k,l N1 =2 V1 =1 N (V ) + (4.3) − 1 permutations − Am1 n1 ;...;mN nN ;mn;kl [] , N1 −1
where ∂ K () . (4.4) ∂ nm;lk The permutations refer to the possibilities to choose N1 − 1 of the pairs of indices m1 n1 , . . . , mN nN which label the external legs of the first A-function. The cut-off function K in (2.10) has to be chosen such that for finite there is a finite number of indices m, n, k, l with Qnm;lk () = 0. By suitable normalisation we can achieve that the volume of the support of Qnm;lk () with respect to a chosen index scales as D : Qnm;lk () := µ2
Power-Counting Theorem for Non-Local Matrix Models and Renormalisation
m
105
D signK[m, ] ≤ CD , µ
(4.5)
for some constant CD independent of . For such a normalisation we define two exponents δ0 , δ1 by µ δ0 δm+k,n+l , (4.6) max |Qnm;lk ()| ≤ C0 m,n,k,l µ δ1 max max |Qnm;lk ()| ≤ C1 . (4.7) n m,l k
In (4.7) the index n is kept constant for the summation over k. It is convenient to encode the dimension D in a further exponent δ2 which describes the product of (4.5) with (4.6): δ2 max |Qnm;lk ()| signK[m, ] ≤ C2 . (4.8) m,n,k,l µ m We have obviously C2 = CD C0 and δ2 = D − δ0 . Definition 5. A non-local matrix model defined by the cut-off propagator Qnm;kl given by (2.10) and (4.4) and the normalisation (4.5) of the cut-off function is called regular if δ0 = δ1 = 2, otherwise anomalous. The three exponents δ0 , δ1 , δ2 play an essential rˆole in the power-counting theorem (V ,V e ,B,g,ι) which yields the -scaling of a homogeneous part Am1 n1 ;...;mN nN [] of the interaction coefficients (V )
Am1 n1 ;...;mN nN [] =
1≤V e ≤V 1≤B≤N 0≤g≤1+ V − N − B 0≤ι≤B−1 2 4 2
(V ,V e ,B,g,ι) Am1 n1 ;...;mN nN []
2≤N≤2V +2
. (4.9)
It is important that the sums over the graphical (topological) data V e , B, g, ι in (4.9) are finite. We are going to prove (V ,V e ,B,g,ι)
Theorem 6. The homogeneous parts Am1 n1 ;...;mN nN [] of the coefficients of the effective action describing a φ 4 -matrix model with initial interaction (3.4) and cut-off propagator characterised by the three exponents δ0 , δ1 , δ2 are for 2 ≤ N ≤ 2V +2 and N i=1 (mi −ni ) = 0 bounded by (V ,V e ,B,g,ι) δ2 (V − N2 +2−2g−B) µ δ1 (V −V e −ι+2g+B−1+s) A m1 n1 ;...;mN nN [] ≤ µ s E µ δ0 (V e +ι−1−s) N × P 2V − 2 ln , (4.10) R provided that for all V
N N
posed at R (0 ), respectively, according to Definition/Lemma 3. The bound (4.10) is (V ,V e ,B,g,ι) / E s . We have Am1 n1 ;...;mN nN [] ≡ 0 for independent of the unsummed indices mi , ni ∈ N > 2V +2 or N i=1 (mi −ni ) = 0.
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The proof will be given in Sect. 5. We remark that L˜ 0 = V − N2 + 2 − 2g − B is the number of inner loops of a graph. The power-counting estimation (4.10) does not make any reference to the initial scale 0 [11] so that we can safely take the limit 0 → ∞. In this way we have constructed a regular solution of the Polchinski equation (2.13) associated with the non-local matrix model. However, this solution remains useless unless it can be achieved by a finite number of integrations from R to depending on a finite number of initial conditions at R . We refer to the remarks following Definition/Lemma 3. A first step would be to achieve regular scaling dimensions: Corollary 7. For regular matrix models according to Definition 5 we have independently of the segmentation index and the numbers of external vertices (V ,B,g) ω−D(2g+B−1) 2V − N A 2 P ln , (4.11) m1 n1 ;...;mN nN [] ≤ µ R s E
where ω = D +V (D −4)−N D−2 2 is the classical power-counting degree of divergence. We have derived the relation (4.11) with respect to the classical power-counting degree of divergence only for φ 4 -matrix models, but it is plausible that it also holds for more general interactions. 5. Proof of the Power-Counting Theorem We provide here the proof of Theorem 6, which is quite long and technical. The proof amounts to study all possible connections of two external legs of either different graphs or the same graph. It will be essential how the legs to connect are situated with respect to the remaining part of the graph. There are the following arrangements of the external legs at the distinguished vertex one (or two) of which we are going to connect:
(5.1) A big oval stands for other parts of the graph, the specification of which is not necessary for the proof. Dotted lines entering and leaving the oval stand for the set of all external legs different from the external legs of the distinguished vertex to contract. If two or three internal lines are connected to the oval this does not necessarily mean that these two lines are part of an inner loop.
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107
We are going to integrate the Polchinski equation (4.3) by induction upward in V and for constant V downward in N . Due to the grading (V , N ), the differential equation (4.3) is actually constructive. We consider in Sect. 5.1 the connection of two smaller graphs of (V1 , N1 ) and (V2 , N2 ) vertices and external legs and in Sects. 5.2 and 5.3 the self-contraction of a graph with (V1 = V , N1 = N + 2) vertices and external legs. These graphs are further characterised by Vie , Bi , gi , ιi external vertices, boundary components, genera and segmentation indices, respectively. Since the sums in (4.9) and the number of arrangements of legs in (5.1) are finite, it is sufficient to regard the contraction of subgraphs individually. That is, we consider individual subgraphs γ1 , γ2 the contraction of which produces an individual graph γ . We also ignore the problem of making the graphs symmetric in the indices mi ni of the external legs. At the very end we project the sum of graphs γ to homogeneous degree (V , V e , B, g, ι). To these homogeneous parts there contributes according to (4.9) a finite number of contractions of γi . We thus get the bound (4.10) if we can prove it for any individual contraction. The theorem is certainly correct for the initial φ 4 -interaction (3.4) which due to (4.2) (1,1,1,0,0) gives |Am1 n1 ;...;m4 n4 []| ≤ 1. 5.1. Tree-contractions of two subgraphs. We start with the first term on the rhs of (4.3) which describes the connection of two smaller subgraphs γ1 , γ2 of V1 , V2 vertices and N1 , N2 external legs via a propagator. The total graph γ for a tree-contraction has V = V1 +V2 vertices , I = I1 +I2 +1 propagators ,
N = N1 +N2 −2 external legs , L˜ = L˜ 1 +L˜ 2 −1 loops ,
(5.2)
because two loops of the subgraphs are merged to a new loop in the total graph. It follows from (3.11) that for tree-contractions we always have additivity of genus, g = g1 + g2 .
(5.3)
As an example for a contraction between graphs in the first line of (5.1) let us consider
(5.4) where σ m and σ n stand for the set of all other outgoing and incoming indices via external legs at the remaining part of the left subgraph γ1 and similarly for σ k and σ l for the right subgraph γ2 . The two boundary components to which the contracted vertices belong are joint in the total graph, i.e. B = B1 + B2 − 1. Moreover, we obviously have V e = V1e + V2e and ι = ι1 + ι2 . The graph (5.4) determines the -scaling
∂ (V ,V e ,B,g,ι)γ A [] ∂ m1 n1 ;m2 n2 ;σ m σ n;σ k σ l;k2 l2 ;k1 l1 1 (V1 ,V1e ,B1 ,g1 ,ι1 )γ1 (V2 ,V e ,B2 ,g2 ,ι2 )γ2 = Am1 n1 ;m2 n2 ;σ m σ n;mm1 [] Qm1 m;ll1 () Al1 l;σ k2σ l;k2 l2 ;k1 l1 [] . (5.5) 2 m,l
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Due to the conservation of the total amount of indices in γ1 and γ2 by induction hypothesis (4.10), both m = σ n − σ m + n2
and
l = σ k − σ l + k2
(5.6)
are completely fixed by the other external indices so that from the sum over m and l there survives a single term only. Then, because of the relation m1 + l = m + l1 from the propagator Qm1 m;ll1 (), see (4.6), it follows that the total amount of indices for (V ,V e ,B,g,ι)γ Am1 n1 ;m2 n2 ;σ m σ n;σ k σ l;k2 l2 ;k1 l1 is conserved as well. Let V¯ie and ι¯i be the numbers of external vertices and segmentation indices on the segments of the subgraphs γi on which the contracted vertices are situated. The induction hypothesis (4.10) gives us the bound if these segments carry s¯i ≤ V¯ie + ι¯i − 1 index summations. The new segment of the total graph γ created by connecting the boundary components of γi carries V¯1e + V¯2e external vertices and ι¯1 + ι¯2 segmentation indices and therefore admits up to s¯1 + s¯2 + 1 index summations. In (5.4) that additional index summation will be the m1 -summation. Due to (3.15) (for segments) there has to be an external leg on each segment the outgoing index of which is not allowed to be summed. If on the γ2 -part of the contracted segment there is an unsummed external leg, we can choose m as that particular index in γ1 . In this case we take in the propagator the maximum over m, l and sum the part γ2 for given l over those indices which belong to E s . The result is bounded independently of l and all other incoming indices. Next, we sum over the indices in E s which belong to γ1 , regarding m as an unsummed index. There is the possibility of an m1 -summation applied to the propagator in the last step, with l1 kept fixed, for which the bound is given by (4.7). In this case we therefore get ∂ (V ,V e ,B,g,ι)γ Am1 n1 ;m2 n2 ;σ m σ n;σ k σ l;k2 l2 ;k1 l1 [] ∂ e s s E , s¯2 ≤V¯2 +¯ι2 −1, m1 ∈E
≤
1 (V1 ,V1e ,B1 ,g1 ,ι1 )γ1 max Qm1 m;ll1 () Am1 n1 ;m2 n2 ;σ m σ n;mm1 [] max l1 m,l 2 s1 m E1
(V ,V e ,B ,g ,ι )γ A 2 2 2 2 2 2 [] × s
1
l1 l;σ k σ l;k2 l2 ;k1 l1
E22
≤
1 δ2 (V − N 2+2 +4−2g−(B+1)) µ δ1 (1+V −V e −ι+2g+(B+1)−2+(s−1)) C1 2 µ µ δ0 (V e +ι−2−(s−1)) N +2 × P 2V − 2 ln . (5.7) R
We have used the induction hypothesis (4.10) for the subgraphs as well as (4.7) for the propagator and have inserted N1 + N2 = N + 2, V1 + V2 = V , V1e + V2e = V e , ι1 + ι2 = ι, B1 + B2 = B + 1, g1 + g2 = g and s1 + s2 = s − 1, because there is an additional summation over m1 which belongs to E s but not to Eisi . If m1 ∈ E s we take instead the unsummed propagator and replace in (5.7) one factor (4.7) by (4.6) as well as (s − 1) by s. The total exponents of γ remain unchanged. Next, let there be no unsummed external leg on the contracted segment of γ2 viewed from γ . Now, we cannot directly use the induction hypothesis. On the other hand, for a given index configuration of γ2 and the propagator, the index k2 is not an independent summation index:
Power-Counting Theorem for Non-Local Matrix Models and Renormalisation
k2 = l + σ l − σ k = m − m 1 + l 1 + σ l − σ k .
109
(5.8)
See also (5.6). If m1 ∈ E s there must be an unsummed outgoing index on the contracted segment of γ1 . We can thus realise the k2 -summation as a summation over m in γ1 for fixed index configuration of γ2 and m1 , l1 . This m-summation is applied together with the summation over the γ1 -indices of E s to γ1 as the first step, taking again the maximum of the propagator over m, l. In the second step we sum over the restriction of E s to γ2 and the propagator. It is obvious that the estimation (5.7) remains unchanged, in particular, s1 +s2 = (s1 +1) + (s2 −1) = s−1. If m1 is the only unsummed index we realise the k2 -summation as a summation of the propagator over l. Here, one has to take into account that the subgraph γ2 is bounded independently of the incoming index l. Again we get the same exponents as in (5.7). We can summarise (5.7) and its discussed modification to ∂ (V ,V e ,B,g,ι)γ Am1 n1 ;m2 n2 ;σ m σ n;σ k σ l;k2 l2 ;k1 l1 [] ∂ s E
≤
δ2 (V − N +2−2g−B) µ δ1 (V −V e −ι+2g+B−1+s) µ δ0 (V e +ι−1−s) 2
µ ×P
2V − N2 −1
ln . R
(5.9)
For the choice of the boundary conditions according to Definition/Lemma 3, the -integration increases (again according to Definition/Lemma 3) the degree of the polynomial in ln R by 1. Hence, we have extended (4.10) to a bigger degree V for contractions of type (5.4). In particular, the bound is (by induction starting with (4.7), which represents the third graph in (3.7)) independent of the incoming indices ni , li . The verification of (4.10) for any contraction between graphs of the first line in (5.1) is performed in a similar manner. Taking the same subgraphs as in (5.4), but with a contraction of other legs, the discussion is in fact a little easier because there are no trajectories going through both subgraphs:
(5.10)
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The contractions
(5.11) are treated in the same way. The point is that the summation indices of the propagator (m, k for the upper graph and n, k for the lower graph in (5.11)) are fixed by index conservation for the subgraphs. In the same way one also discusses any contraction between the second and third graph in (5.1). Let us now contract the left graph in the second line of (5.1) with any graph of the first line of (5.1), e.g.
(5.12) The number of boundary components is reduced by 1, giving B1 + B2 = B + 1. We clearly have ι = ι1 + ι2 , but there is now one external vertex less on which we can apply an index summation, V e = V1e + V2e − 1. At the same time we need the index summation from the subgraph, because in the -scaling
∂ (V ,V e ,B,g,ι)γ [] A ∂ k1 l1 ;σ k σ l;σ m σ n 1 (V1 ,V1e ,B1 ,g1 ,ι1 )γ1 (V2 ,V e ,B2 ,g2 ,ι2 )γ2 = Amn;σ m [] Qnm;k1 k () Akk1 ;k12l1 ;σ k σ l [] σ n 2
(5.13)
m,n,k
there is now one undetermined summation index: k = l1 + σ l − σ k ,
m(n) = n + σ n − σ m .
(5.14)
First, let there be an additional unsummed external leg on the segment of m, n in γ1 . Then, the induction hypothesis (4.10) gives the bound for a summation over m. We thus fix n, k and all indices of γ2 in the first step and realise a possible k1 -summation due to k1 = m + k − n as an m-summation, which is applied together with the summation over the γ1 -indices of E s , after maximising the propagator over m, k1 . The result is independent of n. We thus restrict the n-summation to the propagator, see (4.7), and apply the remaining E s -summations to γ2 , where k remains unsummed. We have s1 + s2 = s and get the estimation
Power-Counting Theorem for Non-Local Matrix Models and Renormalisation
E s k1 , s¯1 ≤V¯1e +¯ι1 −2
≤
111
∂ (V ,V e ,B,g,ι)γ Ak1 l1 ;σ k σ l;σ m σ n [] ∂
1 (V1 ,V1e ,B1 ,g1 ,ι1 )γ1 [] max max Qnm;k1 k () Amn;σ m σ n k m,k1 2 s1 n m,E1
(V ,V e ,B ,g ,ι )γ A 2 2 2 2 2 2 [] × s
kk1 ;k1 l1 ;σ k σ l
E22 k1
≤
1 δ2 (V − N 2+2 +4−2g−(B+1)) µ δ1 (1+V −(V e +1)−ι+2g+(B+1)−2+s) C1 2 µ µ δ0 ((V e +1)+ι−2−s) 2V − N 2+2 × P ln . (5.15) R
If k1 ∈ E s we do not need the m-summation on γ1 . Again we have s = s1 + s2 and (5.15) remains unchanged. Here, we may allow for index summations at all other external legs on the segment of m, n in γ1 . If there is no unsummed external leg on the segment of m, n in γ1 , we must realise the k1 -summation as follows: We proceed as before up to the step where we sum the propagator for given k over n. For each term in this sum we have k1 = k + σ n − σ m. We thus achieve a different k1 if for given σ m, σ n we start from a different k. Since the result of the summations over γ1 and the propagator is independent of k, see (4.7), we realise the k1 -summation as a sum over k restricted to γ2 . We now get the same exponents as in (5.15) also for this case. According to Definition/Lemma 3, the -integration extends for contractions of type (5.12) the bound (4.10) to a bigger order V . The contraction of the other leg of the right vertex
(5.16) is easier to discuss because the k1 -summation is directly applied to γ2 . Taking the second vertex of the first line of (5.1) instead, we have two contractions which are identical to (5.12) and (5.16) and a third one with contractions as in the first and last graphs of (5.10) where γ1 and γ2 form different segments in γ . This case is much easier because there is no trajectory involving both subgraphs. Moreover, contracting the last instead of the first vertex of the second line of (5.1) gives the same estimates if the two propagators between the vertex and the oval belong to the same segment:
(5.17)
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The only modification to (5.15) and its variants is to replace (V e + 1) by V and ι by (ι + 1), because the total number of external vertices is unchanged whereas the total segmentation index is reduced by 1. If the contracted indices m, n belong to different segments of γ1 , e.g.
(5.18) they are actually determined by index conservation for the segments. The entire discussion of these examples is therefore similar to the graph (5.4) with bound (5.7) and its modifications. Note that we have V e = V1e + V2e and ι = ι1 + ι2 in (5.18). Accordingly, we can replace in all previous examples a vertex of the first line of (5.1) by the composed vertex under the condition that the two contracted trajectories at the composed vertex belong to different segments. It remains to study the contraction
(5.19) where two contraction indices (m or n and k or l) are undetermined. We have V e = V1e + V2e − 2 and ι = ι1 + ι2 . We first assume that at least one of the boundary components of γi to contract carries more than one external vertex. In this case we have B = B1 + B2 − 1. There has to be at least one unsummed external vertex on the segment, say on γ2 . We fix the indices of γ2 as well as n in the first step, take in the propagator the maximum over m, l and sum over the γ1 -indices of E s . Here, m can be regarded as an unsummed index. We take the maximum of γ1 over n so that the n-summation restricts to the propagator only. We take in the summed propagator the maximum over k so that the remaining k-summation is applied together with the summation over the γ2 -indices of E s . We thus need s1 +s2 = s +1 summations and the bound (4.7) for the propagator:
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113
∂ (V ,V e ,B,g,ι)γ Aσ k σ l;σ m σ n [] ∂ Es (V1 ,V e ,B1 ,g1 ,ι1 )γ1 1 1 A Qnm;lk () max [] max max ≤ k m,l 2 n s1 mn;σ m σ n n E1 (V ,V e ,B ,g ,ι )γ A 2 2 2 2 2 2 [] × kl;σ k σ l
s
k,E22
≤
1 δ2 (V − N 2+2 +4−2g−(B+1)) µ δ1 (1+V −(V e +2)−ι+2g+(B+1)−2+(s+1)) C1 2 µ µ δ0 ((V e +2)+ι−2−(s+1)) N +2 P 2V − 2 ln . (5.20) × R
Finally, we have to consider the case where the only external vertices of both boundary components of γi to contract are just the contracted vertices. In this case the contraction removes these two boundary components at the expense of a completely inner loop, giving B = B1 + B2 − 2. The differences n − m and k − l are fixed by the remaining indices of γi . For given m we may thus take the maximum of γ2 over l and realise the l-summation as a summation (4.7) over the propagator. We thus exhaust all differences m − l. In order to exhaust all values of m we take the maximum of γ1 over m, n and multiply the result by a volume factor (4.5). We thus replace in (5.20) (s + 1) → s and (B + 1) → (B + 2), and combine one factor (4.6) and a volume factor (4.5) to (4.8). We thus get the same total exponents as in (4.10) so that the -integration extends (4.10) to a bigger order V for all contractions represented by (5.19). The contractions
(5.21) are treated in the same way as (5.19), now with the two unknown summation indices taken into account by a reduction of V e + ι = (V1e + ι1 ) + (V2e + ι2 ) − 2. In particular, there is also the situation where m, n and k, l are the only external legs of their boundary components before the contraction. In this case the number of boundary components drops by 2, which requires a volume factor in order to realise the sum over the starting point of the inner loop. Thus, (4.10) is proven for any contractions produced by the first (bilinear) term on the rhs of (4.3). 5.2. Loop-contractions at the same vertex. It remains to verify the scaling formula (4.10) for the A-linear term on the rhs of the Polchinski equation (4.3), which describes
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self-contractions of graphs. The graphical data for the subgraph will obtain a subscript 1, such as the number of external vertices V1e , the segmentation index ι1 and the set E1s1 of summation indices. We always have V1 = V and N1 = N + 2. We first consider contractions of external lines at the same vertex, for which we have the possibilities shown in (5.1). The very first vertex leads to two different self-contractions:
(5.22)
(5.23) For the planar contraction (5.22) we estimate the l-summation by a volume factor so that we obtain (4.10) from (4.8). For the non-planar graph (5.23) we obtain (4.10) for s = 0 directly from (4.6). According to (3.15) we can apply one index summation which yields (4.10) via (4.7). For the second graph in the first line of (5.1) we first investigate the contraction
(5.24) The number of loops of the amputated graph is increased by 1, L˜ = L˜ 1 + 1, so that due to (3.11) and I = I1 + 1 we get g = g1 . The graph (5.24) determines the -variation
∂ (V ,V e ,B,g,ι)γ 1 (V1 ,V1e ,B1 ,g1 ,ι1 )γ1 Am1 n1 ;σ m σ n [] = − Qm1 k;ln1 () Am1 n1 ;n [] , (5.25) 1 l;σ m σ n;km1 ∂ 2 k,l
with one of the indices k, l being undetermined. First, let there be at least one further external leg on the same boundary component as l, k. In this case the number of boundary components is increased by 1, B = B1 + 1. If there is an unsummed index on the segment of k, l we can realise the k-summation in γ as a summation in γ1 after taking in the propagator the maximum over k, l. We thus have s1 = s + 1 and consequently
Power-Counting Theorem for Non-Local Matrix Models and Renormalisation
115
∂ (V ,V e ,B,g,ι)γ Am1 n1 ;σ m σ n [] ∂ E s , m1 ∈/ E s (V ,V1e ,B1 ,g1 ,ι1 )γ1 1 A [] max Qm1 k;ln1 () ≤ m1 n1 ;n1 l;σ m σ n;km1 2 k,l s
k,E1
≤
δ2 (V − N +2 +2−2g−(B−1)) µ δ1 (V −V e −ι+2g+(B−1)−1+(s+1))
1 2 C0 µ0 2 µ δ0 (1+V e +ι−1−(s+1)) N +2 P 2V − 2 ln . × R
(5.26)
We can sum the contracting propagator over m1 for fixed n1 , which amounts to replace one factor (4.6) by (4.7) compensated by s = s1 replacing s = s1 − 1. If k cannot be a summation index in γ1 then m1 must be unsummed in γ . We first apply the summation over o[l] for given l in γ1 . The result is independent of l so that, for given k, the l-summation can be restricted to the contracting propagator maximised over m1 , n1 . Finally, the remaining E s -summations are applied. We have to replace in (5.26) (s + 1) by s and one factor (4.6) by (4.7). Finally, let there be no further external leg on the same boundary component as l, k. Now the number of boundary components remains constant, B = B1 . Since k − l = n1 −m1 is a constant, the required summation over e.g. k has to be estimated by a volume factor (4.5). We thus replace in (5.26) (B − 1) → B and (s + 1) → s and combine one factor (4.6) and the volume factor to (4.8). In summary, we extend after -integration the scaling law (4.10) for the same degree V to a reduced number N of external lines. Next, we study the following contraction of the second graph in (5.1) which gives rise to an inner loop:
(5.27) It describes the -variation
(V ,V e ,B ,g ,ι )γ ∂ (V ,V e ,B,g,ι)γ 1 1 1 1 1 1 1 Qn1 l;ln1 () Am1 n1 ;n [] . (5.28) Am1 n1 ;σ m σ n [] = − 1 l;ln1 ;σ m σ n ∂ 2 l
The number of loops of the amputated graph is increased by 1 and the number of boundary (V ,V e ,B1 ,g1 ,ι1 )γ1
components remains constant, giving g = g1 and B = B1 . Note that Am1 n11;n1 l;ln1 ;σ m σ n is independent of l so that the l-summation acts on the propagator only. We estimate the l-summed propagator by (4.8) for the product of (4.6) with a volume factor (4.5). The factor (4.8) compensates the decrease N = N1 − 2; all other exponents remain unchanged when passing from γ1 to γ . Now the -integration extends the scaling law (4.10) to a reduced N.
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The third graph in the first line of (5.1) leads to the contracted graph
(5.29) There is one additional loop of the amputated graph, giving g = g1 . We have B = B1 if there are further external legs on the boundary component of n and B = B1 − 1 if no further external leg exists on the contracted boundary component. Very similar to (5.27), the l-summation is restricted to the propagator maximised over n, giving a factor (4.8) which compensates N = N1 − 2 in the first exponent of (4.10). For B = B1 the n-summation in (5.29) is provided by the subgraph γ1 , where the additional summation s1 = s + 1 compared with γ compensates the change V1e = V e + 1 of external vertices in the second and third exponent of (4.10). On the other hand, if B1 = B + 1 we have s = s1 and the summation over n has to come from a volume factor (4.5) combined with one factor (4.6) to (4.8). This verifies (4.10) for the contraction (5.29). The last case for which contractions of two external lines at the same vertex are to investigate is the last vertex in the second line of (5.1). As before in the proof for treecontractions, we have to distinguish whether the composed vertex under consideration appears inside a tree, in a loop but together with further composed vertices, or in a loop but as the single composed vertex. In the first case we have to analyse the graph
(5.30) Before the contraction, the indices m, n, k, l were all located on the same loop of the amputated graph and the same boundary component. After the contraction they are split into two loops, g = g1 . The number of boundary components is increased by 1 if both resulting boundary components of l, m and k, n carry further external legs, B = B1 + 1. We have B = B1 if only one of the resulting boundary components of l, m or k, n carries further external legs and B = B1 − 1 if there are no further external legs on these boundary components. We clearly have ι = ι1 and V e = V1e − 1. Due to index conservation for segments, either k or n is an unknown summation index, and either l or m. We first consider the case B = B1 + 1. In both segments of γ1 to contract there must be at least one unsummed outgoing index, which we can choose to be different from the vertex to contract. We thus take in the propagator the maximum (4.6) over all indices and restrict the required index summations over k, m to the segments of the subgraphs. This means that we have s1 = s + 2 summations, which compensate the change of the numbers of boundary components B1 = B − 1, external legs N1 = N + 2 and external vertices V1e = V e + 1:
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∂ (V ,V e ,B,g,ι)γ Aσ m σ n;σ m σ n [] ∂ E s ,B=B1 +1 (V1 ,V e ,B1 ,g1 ,ι1 )γ1 1 1 A [] max Qnm;lk () max ≤ σ m σ n;mn;σ m σ n;kl l,n 2 m,n,k,l s
k,m,E
≤
δ2 (V − N +2 +2−2g−(B−1)) µ δ1 (V −(V e +1)−ι+2g+(B−1)−1+(s+2))
1 2 C0 2 µ µ δ0 (1+(V e +1)+ι−1−(s+2)) N +2 × P 2V − 2 ln . R
(5.31)
We immediately confirm (4.10). Alternatively, instead of consuming a γ1 -summation to get the k-summation we can also sum the propagator for maximised l, m and given k over n. Compared with (5.31) we have to replace (s + 2) by (s + 1) and one factor (4.6) by (4.7), ending up in the same exponents. Next, we investigate the case B = B1 where, for example, the restriction of the boundary component to the left segment does not carry another external leg than m, l. The summation over m in γ1 is now provided by a volume factor, which means that in (5.31) we have to replace (s + 2) by (s + 1), (B − 1) by B and one factor (4.6) by (4.8). All exponents match again (4.10). Finally, let us look at the possibility B = B1 − 1, where the indices m, n, k, l to contract were the only external indices of the boundary component. We thus combine two volume factors (4.5) and two factors (4.6) to two factors (4.8), compensating (B − 1) → (B + 1) and (s + 2) → s. After -integration we extend (4.10) to a reduced N. The case that the two sides of the composed vertex to contract are connected but belong to different segments, e.g.
(5.32) is similar to treat concerning index summations, but for the interpretation of the genus there is a different situation possible. In the amputated subgraph γ1 the indices m, n and k, l may be situated on different loops and thus different boundary components. The contraction joins in this case the two loops, L˜ = L˜ 1 − 1, which results due to (3.11) in g = g1 + 1 and B = B1 − 1. There is at least one additional external leg on each of the boundary components of m, n and k, l before the contraction, because in order to close the loop we have to pass through the vertex m1 , n1 , m2 , n2 . Now we have to replace in (5.31) (B − 1) by (B + 1) and g by (g − 1), confirming (4.10) also in this case. If all indices m, n, k, l are on the same loop in γ1 , the contraction splits it into two and the entire discussion of (5.30) can be used without modification to the present example.
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It remains the case
(5.33) where the two halves of the composed vertex to contract belong to the same segment. Three of the indices m, n, k, l are now summation indices. We have ι = ι1 − 1 and V e = V1e − 1. Let first the indices m, n on one hand and k, l on the other hand be situated on different loops of the amputated graph γ1 . These are joint by the contraction, yielding g = g1 + 1. If there remain further external legs on the contracted loop we have B = B1 − 1, otherwise B = B1 − 2. We start with B = B1 − 1. Due to the segmentation index present in γ1 , the induction hypothesis for γ1 gives us the bound for two additional summations over m, k not present in γ . The third summation is provided by the propagator via (4.7). Assuming i[k] = l, n in γ1 we first take in the contracting propagator the maximum over m, l, then sum the m, n-boundary component over m and those indices of E s which belong to the m, n-boundary component, followed by the summation of the propagator over n for given k. Finally, we sum γ1 over the remaining indices of E s and over k: ∂ (V ,V e ,B,g,ι)γ Aσ m σ n [] ∂ Es (V1 ,V1e ,B1 ,g1 ,ι1 )γ1 1 A max Qmn;kl () [] max ≤ σ m σ n;mn;kl k l,m 2 s n k,m,E
≤
δ2 (V − N +2 +2−2(g−1)−(B+1))
1 2 C1 2 µ µ δ1 (1+V −(V e +1)−(ι+1)+2(g−1)+(B+1)−1+(s+2)) × µ δ0 ((V e +1)+(ι+1)−1−(s+2)) N +2 × P 2V − 2 ln . R
(5.34)
It is essential that the summation over k − i[k] is independent. If there are no external legs on the contracted loop, B = B1 − 2, then we have in γ1 either i[m] = n, i[k] = l or i[m] = l, i[k] = n. In the first case we would first fix n, k and maximise the propagator over m, l. Now the m-summation restricts to γ1 with bound independent of n. Thus, the n-summation for given k restricts to the propagator and delivers a factor (4.7), independent of k. However, since k − i[k] ≡ k − l = n − m is already exhausted in γ1 , the remaining k-summation has to come from a volume factor. We thus make in (5.34) the replacements (B+1) → (B+2), (s+2) → (s+1) and combine one factor (4.6) with a volume factor (4.5) to (4.8). The exponents match again (4.10). Next, we investigate the situation where all indices m, n, k, l are located on the same loop of the amputated subgraph γ1 . In this case the contraction to γ splits that loop into two so that we have g = g1 . As before we have B = B1 + 1 if both split loops contain further external legs, B = B1 if only one of the split loops contains further external legs, and B = B1 − 1 if the split loops do not contain further external legs. The discussion is
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similar as for (5.30), the difference is that three of m, n, k, l are now summations indices, which is taken into account by the replacement of ι in (5.31) by (ι + 1). We thus finish the verification of (4.10) for self-contractions of a vertex. 5.3. Loop-contractions at different vertices. It remains to check (4.10) for contractions of different vertices of the same graph. The external lines of the two vertices are arranged according to (5.1). We start with two vertices of the type shown as the second graph in (5.1). One possible contraction of their external lines is
(5.35) assuming that the vertices to contract are located on the same segment in γ1 . One of the indices m, l is a summation index. We first consider the case that the two vertices to contract are located on the same loop of the amputated graph γ1 . The contraction to γ splits that loop into two, giving g = g1 . We have B = B1 + 1 if the trajectory starting at l does not leave γ1 (and γ ) in m, whereas B = B1 if m, l are on the same trajectory in γ1 . In the case of B = B1 + 1 we keep i[m] in γ1 fixed, take in the propagator the maximum over m, l and restrict the m-summation to γ1 . Due to V1e = V e , ι1 = ι and B1 = B − 1 we have in the case that m1 remains unsummed ∂ (V ,V e ,B,g,ι)γ Ak2 l2 ;k1 l1 ;m1 n1 ;m2 n2 ;σ m σ n [] ∂ E s m1 ,B=B1 +1 (V1 ,V1e ,B1 ,g1 ,ι1 )γ1 1 A max Qm1 m;ll1 () [] ≤ k2 l2 ;k1 l1 ;l1 l;mm1 ;m1 n1 ;m2 n2 ;σ m σ n 2 m,l,m1 ,l1 s m,E
≤
δ2 (V − N +2 +2−2g−(B−1)) µ δ1 (V −V e −ι+2g+(B−1)+1+(s+1))
1 2 C0 2 µ µ δ0 (1+V e +ι−1−(s+1)) 2V − N 2+2 × P ln . R
(5.36)
Summing additionally over m1 we replace in (5.36) one factor (4.6) by (4.7). It is clear that this reproduces the exponents of (4.10) correctly. If l = i[m] in γ1 , we have to realise the m-summation by a volume factor. We thus replace in (5.36) (B − 1) → B, (s + 1) → s and combine (4.5) with one factor (4.6) to (4.8). Finally, the two vertices to contract in (5.35) may be located on different loops of the amputated graph γ1 . They are joint by the contraction to γ , giving g = g1 + 1, and because the newly created loop obviously has external legs, we have B = B1 − 1. As separated loops in γ1 , l cannot be the incoming index of the trajectory through m. Therefore, the m-summation gives the same bound as the rhs of (5.36), now with (B − 1) replaced by (B + 1) and g by (g − 1). We have thus extended (4.10) to a reduced N for all types of contractions (5.35).
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If the vertices to contract are located on different segments in γ1 , e.g.
(5.37) both indices m, l are determined by index conservation for segments. We can thus save an index summation compared with (5.36) and replace there and in its discussed modifications (s + 1) by s and ι1 = ι by ι1 = (ι − 1). Since the m-summation is not required, there is effectively an additional summation possible in agreement with (3.15). It is not possible that m and l are located on the same trajectory in γ1 so that either g = g1 , B = B1 + 1 or g = g1 + 1, B = B1 − 1. Let us make a few more comments on the segmentation index. It is essential that the contraction joins separated segments. For instance, the contraction
(5.38) does not increase the segmentation index, because in agreement with Definition 4 the number of segments remains constant. The graph on the left has ι = 1, and the internal indices m, l are determined by the external ones. The graph on the right has ι = 1 as well, and now one of the indices n, k becomes a summation index. Having several composed vertices in the middle link does not change the segmentation index:
(5.39)
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It makes, however, a difference if the two composed vertices are situated on different links:
(5.40) Here, the segmentation index increases from ι = 1 on the left to ι = 2 on the right, in agreement with Definition 4. The case
(5.41) is completely identical to (5.35). In the contraction
(5.42) the summation index n or l is provided by the propagator, replacing in (5.36) and its modifications (s+1) by s and one factor (4.6) by (4.7). It is not possible that n and l are located on the same trajectory in γ1 so that either g = g1 , B = B1 +1 or g = g1 +1, B = B1 −1. In order to treat the contraction
(5.43)
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one has to use that the summation over m1 can, due to m1 = k + m − k1 , be transferred as a k-summation of γ1 . The summation over the undetermined index m is applied in the last step. Finally,
(5.44) are similar to the ι-increased variant (5.37). The contraction
(5.45) is an example for a realisation of (3.14) where Vc is increased by 2 and S by 1, giving again a segmentation index increased by 1. It is obvious that the discussion of contractions involving the second and third or two of the third vertices of the first line in (5.1) is analogous. Let us now study loop contractions which involve the first graph in the second line of (5.1), assuming first that the vertices are situated on the same segment:
(5.46) = − 1. Two of the summation indices m, k, l are We thus have ι = ι1 and undetermined. Let first the two vertices to contract be located on the same loop of the amputated subgraph γ1 . The contraction splits that loop into two, giving g = g1 . The next question concerns the number of boundary components. We have B = B1 + 1 if there are further external legs on the loop through l, m and B = B1 if l = i[m] in γ1 . We start with B = B1 + 1. In general, the induction hypothesis provides us with bounds for summations over m and k, because l = i[m]. If m1 is an unsummed index we thus have Ve
V1e
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∂ (V ,V e ,B,g,ι)γ Am1 n1 ;σ m σ n [] ∂ E s m1 ,B=B1 +1 (V1 ,V1e ,B1 ,g1 ,ι1 )γ1 1 A [] max Qm1 m;lk () ≤ kl;nm ;m n ;σ m σ n 1 1 1 2 m,l,k,m1 s
m,k,E
δ2 (V − N +2 +2−2g−(B−1)) µ δ1 (V −(V e +1)−ι+2g+(B−1)+1+(s+2))
1 2 C0 2 µ µ δ0 (1+(V e +1)+ι−1−(s+2)) N +2 × P 2V − 2 ln . (5.47) R Now an additional summation over m1 can immediately be taken into account by replacing the maximised propagator (4.6) by the summed propagator (4.7), in agreement with (s + 2) replaced by (s + 1). The m1 -summation is applied before the k-summation is carried out. These considerations require an unsummed outgoing index on the contracted segment of γ1 . If this is not the case then m1 has to be the unsummed outgoing index. Now the l-summation for given m has to be restricted to the propagator and delivers a factor (4.7). The exponents match again (4.10). Next, for l = i[m] in γ1 we cannot use a summation over m in γ1 in order to account for the undetermined contraction index, because the incoming index l would change simultaneously. Instead we have to use a volume factor (4.5) combined with one factor (4.6) to (4.8). Additionally we have to replace in (5.47) (s + 1) by s and (B − 1) by B. Second, the two vertices to contract may be located on different loops of the amputated graph γ1 . They are joint by the contraction, giving g = g + 1. Because the loop carries at least the external leg m1 n1 , we necessarily have B = B1 − 1. Now, l = i[m] in γ1 so that we use summations over m, k in γ1 , giving the same balance (5.47) for the exponents, with (B − 1) → (B + 1) and g → (g − 1). The discussion is identical for the contraction ≤
(5.48) which in the case that k, l belong to the same segment in γ has two undetermined summation indices as well. We thus proceed as in (5.47) and its discussed modification and only have to replace (V e + 1) by V e and ι by (ι + 1). If k, l are situated on different segments in γ , e.g.
(5.49)
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there is only one undetermined summation index, which is reflected in the analogue of (5.47) by the fact that the segmentation index remains unchanged, ι = ι1 . Note that in the right graph of (5.49) we either have B = B1 − 1, g = g + 1 or B = B1 + 1, g = g. Of course we get the same estimations if the segment of γ1 with external lines σ k, σ l are connected by several composed vertices to the part of γ1 with external lines σ m, σ n. The contractions
(5.50) are a little easier because the contracting propagator does not have outgoing indices which for certain summations had to be transferred to the subgraph γ1 . If n = i[k] in γ1 , the k-summation for given n, n1 can be restricted to γ1 after maximising the propagator over all indices. Since the result for γ1 is independent of the starting point n, the k-summation can be regarded as a summation over all differences k − n. The final summation over all pairs k, n with fixed difference k − n is provided by a volume factor (4.5) combined with (4.6) to (4.8). The balance of exponents is identical to (5.47) and its discussed variants. It is clear that the analogue of (5.49) with the left vertex connected as in (5.50) is similar to treat. Next, we discuss the variant of (5.46) where the two vertices to contract belong to different segments in the subgraph γ1 :
(5.51) Now only one of the indices m, k, l is an undetermined summation index, with k being the most natural choice. We therefore get a bound for the -scaling analogous to (5.47) but with ι replaced by ι − 1, reflecting the increase of the segmentation index ι = ι1 + 1. There is now an additional index summation possible, here via (4.7) over the index m1 . Note that we have either B = B1 − 1, g = g + 1 or B = B1 + 1, g = g. The discussion of the variants of (5.51) with the right vertex taken as the second one in the last line of (5.1) and/or the left vertex arranged as in (5.50) is very similar. It remains to investigate contractions between two of the vertices in the second line of (5.1). We discuss in detail the contraction
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(5.52) All variants are similar as described between (5.46) and (5.51). Three of the four summation indices m, n, k, l in (5.52) are undetermined. We clearly have V1e = V e + 2 and ι = ι1 . We first consider the case where the four indices m, n, k, l are located on the same loop of the amputated subgraph γ1 . The contraction will split that loop into two, giving g = g1 . There are three possibilities for the change of the number of boundary components after the contraction. First, if on both paths of trajectories in γ1 from n to k and from l to m there are further external legs, we have B = B1 + 1. Second, if on one of these paths there is no further external leg, we have B = B1 . Third, if both paths contain no further external legs, i.e. m and k are the outgoing indices of the trajectories starting at l and n, respectively, we have B = B1 − 1. We start with B = B1 + 1. Then, i[k] and i[m] are fixed as external indices so that the induction hypothesis for γ1 provides the bounds for two summations over k, m. We first apply a possible summation to the outgoing index of the trajectory starting at l. The result is maximised independently from l so that we can restrict the l-summation to the propagator, maximised over k, n with m being fixed. Finally, we apply the summations over k, m and all remaining E s -summations to γ1 . We thus obtain ∂ (V ,V e ,B,g,ι)γ [] Aσ m σ n ∂ E s ,B=B1 +1 (V1 ,V1e ,B1 ,g1 ,ι1 )γ1 1 A max Qnm;lk () [] max ≤ mn;σ m σ n;kl n,k 2 m s l
≤
m,k,E
δ2 (V − N +2 +2−2g−(B−1)) µ δ1 (1+V −(V e +2)+ι+1+2g+(B−1)−1+(s+2))
1 2 C1 2 µ µ δ0 ((V e +2)+ι−1−(s+2)) N +2 × P 2V − 2 ln . R
(5.53)
The -integration verifies (4.10) in the topological situation under consideration. Next, we discuss the case B = B1 , assuming e.g. l = i[m] in γ1 . We maximise the propagator over k, n for given l so that the m-summation can be restricted to γ1 . Next, we apply the E s -summations and the k-summation to γ1 , still for given l. The final l-summation counts the number of graphs with different l, giving the bound (4.6) of the propagator times a volume factor. In any case the required modifications of (5.53), in particular (B − 1) → B, lead to the correct exponents of (4.10). If B = B1 − 1, i.e. l = i[m] and n = i[k], we take in the propagator the maximum over n, k so that for given l the m-summation can be restricted to γ1 . The result of that summation is bounded independently of l. Thus, each summand only fixes m−l = n−k, and the remaining freedom for the summation indices is exhausted by two volume factors and the bound (4.6) for the propagator. We thus replace in (5.53) (s + 2) → (s + 1), (B − 1) → (B + 1) and one factor (4.7) by (4.6). Then two factors (4.6) are merged with two volume factors (4.5) to give two factors (4.8).
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Finally, we have to consider the case where m, n are located on a different loop of the amputated subgraph γ1 than k, l. The contraction joins these loops, giving g = g1 + 1. If the resulting loop carries at least one external leg we have B = B1 − 1, whereas we get B = B1 − 2 if the resulting loop does not carry any external legs. We first consider the case that there is a further external leg on the n, m-loop in γ1 . We take in the propagator the maximum over k, n and sum the subgraph for given l, n over k and possibly the outgoing index of the n-trajectory. The result is independent of l, n. Next, we sum the propagator for given m over l and finally apply the remaining E s -summations and the summation over m to γ1 . We get the same estimates as in (5.53) with (B − 1) replaced by (B + 1) and g by (g − 1). If there are no further external legs on the contracted loop we would maximise the propagator over k, n, then sum γ1 over k for given l, next sum the propagator over l for given m. For each resulting pair k, l the remaining m-summation leaves m − n constant. We thus have to use a volume factor in order to exhaust the freedom of m − n, combining one factor (4.6) and the volume factor (4.5) to (4.8). We thus confirm (4.10) for any contraction of the form (5.52). It is obvious that all examples not discussed in detail are treated in the same manner. We conclude that (4.10) provides the correct bounds for the interaction coefficients of the φ 4 -matrix model with cut-off propagator described by the three exponents δ0 , δ1 , δ2 . 6. Discussion By solving the Polchinski equation perturbatively we have derived a power-counting theorem for non-local matrix models with arbitrary propagator. Our main motivation for the renormalisation group investigation of non-local matrix models was to tackle the renormalisation problem of field theories on noncommutative RD from a different perspective. The momentum integrals leading to the parametric integral representation are not absolutely convergent; nevertheless one exchanges the order of integration. In momentum space one can therefore not exclude the possibility that the UV/IR-mixing is due to the mathematically questionable exchange of the order of integration. The renormalisation group approach to noncommutative field theories in matrix formulation avoids these problems. We work with cut-off propagators leading to finite sums and take absolute values of the interaction coefficients throughout. Oscillating phases never appear; they are not required for convergence of certain graphs. Our power-counting theorem provides a necessary condition for renormalisability: The two scaling exponents δ0 , δ1 of the cut-off propagator have to be large enough relative to the dimension of the underlying space. In [13, 14] we determine these exponents for φ 4 -theory on noncommutative RD , D = 2, 4: Proposition 8. The propagator for the real scalar field on noncommutative RD , D = 2, 4, is characterised by the scaling exponents δ0 = 1 and δ1 = 0. Adding a harmonic oscillator potential to the action one achieves δ0 = δ1 = 2. We thus conclude that scalar models on noncommutative RD are anomalous unless one adds the regulating harmonic oscillator potential. The weak decay ∼ −1 of the propagator leads to divergences in ∼ 0 → ∞ of arbitrarily high degree. The appearance of unbounded degrees of divergences in field theories on noncommutative R4 is often related to the so-called UV/IR-mixing [1]. We learn from the power-counting theorem (Theorem 6) that similar effects will show up in
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any matrix model in which the propagator decays too slowly with . This means that the correlation between distant modes is too strong, i.e. the model is too non-local. Acknowledgement. We are grateful to Jos´e Gracia-Bond´ıa and Edwin Langmann for discussions concerning the integral representation of the -product and its matrix base. We would like to thank Thomas Krajewski for advertising the Polchinski equation to us and Volkmar Putz for the accompanying study of Polchinski’s original proof. We are grateful to Christoph Kopper for indicating to us a way to reduce 0 in our original power-counting estimation the polynomial in ln to a polynomial in ln , thus R R permitting immediately the limit 0 → ∞. We would also like to thank Manfred Schweda and his group for enjoyable collaboration. We are indebted to the Erwin Schr¨odinger Institute in Vienna, the Max-Planck-Institute for Mathematics in the Sciences in Leipzig and the Institute for Theoretical Physics of the University of Vienna for the generous support of our collaboration.
References 1. Minwalla, S., Van Raamsdonk, M., Seiberg, N.: Noncommutative perturbative dynamics. JHEP 0002, 020 (2000) 2. Chepelev, I., Roiban, R.: Renormalization of quantum field theories on noncommutative Rd . I: Scalars. JHEP 0005, 037 (2000) 3. Chepelev, I., Roiban, R.: Convergence theorem for non-commutative Feynman graphs and renormalization. JHEP 0103, 001 (2001) 4. Becchi, C., Giusto, S., Imbimbo, C.: The Wilson-Polchinski renormalization group equation in the planar limit. Nucl. Phys. B 633, 250 (2002) 5. Wilson, K.G., Kogut, J.B.: The Renormalization Group And The Epsilon Expansion. Phys. Rept. 12, 75 (1974) 6. Polchinski, J.: Renormalization and effective Lagrangians. Nucl. Phys. B 231, 269 (1984) 7. Griguolo, L., Pietroni, M.: Wilsonian renormalization group and the non-commutative IR/UV connection. JHEP 0105, 032 (2001) 8. Becchi, C., Giusto, S., Imbimbo, C.: The renormalization of non-commutative field theories in the limit of large non-commutativity. Nucl. Phys. B 664, 371 (2003) 9. Nicholson, E.: Perturbative Wilsonian formalism for noncommutative gauge theories in the matrix representation. Ph. D. thesis, http://arxiv.org/abs/hep-th/0305044, 2003 10. Gracia-Bond´ıa, J.M., V´arilly, J.C.: Algebras Of Distributions Suitable For Phase Space Quantum Mechanics. 1. J. Math. Phys. 29, 869 (1988) 11. Keller, G., Kopper, C., Salmhofer, M.: Perturbative renormalization and effective Lagrangians in φ44 . Helv. Phys. Acta 65, 32 (1992) 12. Zimmermann, W.: Reduction In The Number Of Coupling Parameters. Commun. Math. Phys. 97, 211 (1985) 13. Grosse, H., Wulkenhaar, R.: Renormalisation of φ 4 -theory on noncommutative R2 in the matrix base. JHEP 0312, 019 (2003) 14. Grosse, H., Wulkenhaar, R.: Renormalisation of φ 4 -theory on noncommutative R4 in the matrix base. To appear in Commun. Math. Phys., http://arxiv.org/abs/hep-th/0401128, 2004 15. Langmann, E., Szabo, R.J.: Duality in scalar field theory on noncommutative phase spaces. Phys. Lett. B 533, 168 (2002) 16. Dijkgraaf, R.: Intersection theory, integrable hierarchies and topological field theory. In: G. Maer, (ed.), New symmetry principles in quantum field theory, London: Plenum, 1993, pp. 95–158 17. Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2-D Gravity and random matrices. Phys. Rept. 254, 1 (1995) 18. Brezin, E., Zinn-Justin, J.: Renormalization group approach to matrix models. Phys. Lett. B 288, 54 (1992) Communicated by M.R. Douglas
Commun. Math. Phys. 254, 129–166 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1242-0
Communications in
Mathematical Physics
Quantum Hamiltonians and Stochastic Jumps Detlef Durr ¨ 1 , Sheldon Goldstein2 , Roderich Tumulka3 , Nino Zangh`ı3 1
Mathematisches Institut der Universit¨at M¨unchen, Theresienstraße 39, 80333 M¨unchen, Germany. E-mail:
[email protected] 2 Departments of Mathematics and Physics, Hill Center, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA. E-mail:
[email protected] 3 Dipartimento di Fisica and INFN sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy. E-mail:
[email protected];
[email protected] Received: 14 October 2003 / Accepted: 15 July 2004 Published online: 25 November 2004 – © Springer-Verlag 2004
Abstract: With many Hamiltonians one can naturally associate a ||2 -distributed Markov process. For nonrelativistic quantum mechanics, this process is in fact deterministic, and is known as Bohmian mechanics. For the Hamiltonian of a quantum field theory, it is typically a jump process on the configuration space of a variable number of particles. We define these processes for regularized quantum field theories, thereby generalizing previous work of John S. Bell [3] and of ourselves [11]. We introduce a formula expressing the jump rates in terms of the interaction Hamiltonian, and establish a condition for finiteness of the rates. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 2. The Jump Rate Formula . . . . . . . . . . . . . . . . . 2.1 Review of Bohmian mechanics and equivariance 2.2 Equivariant Markov processes . . . . . . . . . . 2.3 Equivariant jump processes . . . . . . . . . . . . 2.4 Integral operators correspond to jump processes . 2.5 Minimal jump rates . . . . . . . . . . . . . . . . 2.6 Bell-type QFT . . . . . . . . . . . . . . . . . . . 3. Examples . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A first example . . . . . . . . . . . . . . . . . . 3.2 Wave functions with spin . . . . . . . . . . . . . 3.3 Vector bundles . . . . . . . . . . . . . . . . . . . 3.4 Kernels of the measure type . . . . . . . . . . . . 3.5 Infinite rates . . . . . . . . . . . . . . . . . . . . 3.6 Discrete configuration space . . . . . . . . . . . 3.7 Bell’s process . . . . . . . . . . . . . . . . . . . 3.8 A case of POVM . . . . . . . . . . . . . . . . .
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3.9 Another case of POVM . . . . . . . . . 3.10 Identical particles . . . . . . . . . . . . 3.11 Another view of fermions . . . . . . . . 3.12 A simple QFT . . . . . . . . . . . . . . Existence Results . . . . . . . . . . . . . . . 4.1 Condition for finite rates . . . . . . . . 4.2 Integral operators . . . . . . . . . . . . 4.2.1 Hilbert–Schmidt operators. . . . 4.2.2 Complex-valued wave functions. 4.2.3 Vector-valued wave functions . 4.2.4 POVMs . . . . . . . . . . . . . 4.3 Global existence question . . . . . . . . 4.4 Extensions of bi-measures . . . . . . . Minimality . . . . . . . . . . . . . . . . . . Remarks . . . . . . . . . . . . . . . . . . . . 6.1 Symmetries . . . . . . . . . . . . . . . 6.2 Homogeneity of the rates . . . . . . . . 6.3 H + E . . . . . . . . . . . . . . . . . . 6.4 Nondegenerate eigenstates . . . . . . . 6.5 Left or right continuity . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . .
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1. Introduction The central formula of this paper is σ (dq|q ) =
[(2/) Im |P (dq)H P (dq )|]+ . |P (dq )|
(1)
It plays a role similar to that of Bohm’s equation of motion (2). Together, these two equations make possible a formulation of quantum field theory (QFT) that makes no reference to observers or measurements, while implying that observers, when making measurements, will arrive at precisely the results that QFT is known to predict. Special cases of formula (1) have been utilized before [3, 11, 31]. Part of what we explain in this paper is what this formula means, how to arrive at it, when it can be applied, and what its consequences are. Such a formulation of QFT takes up ideas from the seminal paper of John S. Bell [3], and we will often refer to theories similar to the model suggested by Bell in [3] as “Bell-type QFTs”. (What similar means here will be fleshed out in the course of this paper.) The aim of this paper is to define a canonical Bell-type model for more or less any regularized QFT. We assume a well-defined Hamiltonian as given; to achieve this, it is often necessary to introduce cut-offs. We shall assume this has been done where needed. In cases in which one has to choose between several possible position observables, for example because of issues related to the Newton–Wigner operator [26, 19], we shall also assume that a choice has been made. The primary variables of Bell-type QFTs are the positions of the particles. Bell suggested a dynamical law, governing the motion of the particles, in which the Hamiltonian H and the state vector determine the jump rates σ . We point out how Bell’s rates fit naturally into a more general scheme summarized by (1). Since these rates are in a sense
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the smallest choice possible (as explained in Sect. 5), we call them the minimal jump rates. By construction, they preserve the ||2 distribution. Most of this paper concerns the properties and mathematical foundations of minimal jump rates. In Bell-type QFTs, which can be regarded as extensions of Bohmian mechanics, the stochastic jumps often correspond to the creation and annihilation of particles. We will discuss further aspects of Bell-type QFTs and their construction in our forthcoming work [12]. The paper is organized as follows. In Sect. 2 we introduce all the main ideas and reasonings; a superficial reading should focus on this section. Some examples of processes defined by minimal jump rates are presented in Sect. 3. In Sect. 4 we provide conditions for the rigorous existence and finiteness of the minimal jump rates. In Sect. 5 we explain in what sense the rates (1) are minimal. Sect. 6 concerns further properties of processes defined by minimal jump rates. In Sect. 7 we conclude. 2. The Jump Rate Formula 2.1. Review of Bohmian mechanics and equivariance. Bohmian mechanics [4, 14, 16] is a non-relativistic theory about N point particles moving in 3-space, according to which the configuration Q = (Q1 , . . . , QN ) evolves according to1 dQ = v(Q) , dt
v = Im
∗ ∇ . ∗
(2)
= t (q) is the wave function, which evolves according to the Schr¨odinger equation i
∂ = H , ∂t
(3)
with H =−
2 +V 2
(4)
for spinless particles, with = div ∇. For particles with spin, takes values in the appropriate spin space Ck , V may be matrix valued, and the numerator and denominator of (2) have to be understood as involving inner products in spin space. The secret of the success of Bohmian mechanics in yielding the predictions of standard quantum mechanics is the fact that the configuration Qt is |t |2 -distributed in configuration space at all times t, provided that the initial configuration Q0 (part of the Cauchy data of the theory) is so distributed. This property, called equivariance in [14], suffices for empirical agreement between any quantum theory (such as a QFT) and any version thereof with additional (often called “hidden”) variables Q, provided the outcomes of all experiments are registered or recorded in these variables. That is why equivariance will be our guide for obtaining the dynamics of the particles. The equivariance of Bohmian mechanics follows immediately from comparing the continuity equation for a probability distribution ρ associated with (2), ∂ρ = − div (ρv) , ∂t
(5)
1 The masses m of the particles have been absorbed in the Riemann metric g µν on configuration k space R3N , gia,j b = mi δij δab , i, j = 1 . . . N, a, b = 1, 2, 3, and ∇ is the gradient associated with −1 gµν , i.e., ∇ = (m−1 1 ∇q 1 , . . . , mN ∇q N ).
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with the equation satisfied by ||2 which follows from (3), ∂||2 2 (q, t) = Im ∗ (q, t) (H )(q, t) . ∂t In fact, it follows from (4) that 2 Im ∗ (q, t) (H )(q, t) = − div Im ∗ (q, t)∇(q, t)
(6)
(7)
so, recalling (2), one obtains that ∂||2 = − div (||2 v) , ∂t
(8)
and hence that if ρt = |t |2 at some time t then ρt = |t |2 for all times. Equivariance is an expression of the compatibility between the Schr¨odinger evolution for the wave function and the law, such as (2), governing the motion of the actual configuration. In [14], in which we were concerned only with the Bohmian dynamics (2), we spoke of the distribution ||2 as being equivariant. Here we wish to find processes for which we have equivariance, and we shall therefore speak of equivariant processes and motions.
2.2. Equivariant Markov processes. The study of example QFTs like that of [11] has lead us to the consideration of Markov processes as candidates for the equivariant motion of the configuration Q for Hamiltonians H more general than those of the form (4). Consider a Markov process Qt on configuration space. The transition probabilities are characterized by the backward generator Lt , a (time-dependent) linear operator acting on functions f on configuration space: Lt f (q) =
d E(f (Qt+s )|Qt = q), ds
(9)
where d/ds means the right derivative at s = 0 and E( · | · ) denotes the conditional expectation. Equivalently, the transition probabilities are characterized by the forward generator Lt (or, as we shall simply say, generator), which is also a linear operator but acts on (signed) measures on the configuration space. Its defining property is that for every process Qt with the given transition probabilities, the distribution ρt of Qt evolves according to ∂ρt = L t ρt . ∂t Lt is the dual of Lt in the sense that f (q) Lt ρ(dq) = Lt f (q) ρ(dq) .
(10)
(11)
We will use both Lt and Lt , whichever is more convenient. We will encounter several examples of generators in the subsequent sections. We can easily extend the notion of equivariance from deterministic to Markov processes. Given the Markov transition probabilities, we say that the ||2 distribution is
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equivariant if and only if for all times t and t with t < t , a configuration Qt with distribution |t |2 evolves, according to the transition probabilities, into a configuration Qt with distribution |t |2 . In this case, we also simply say that the transition probabilities are equivariant, without explicitly mentioning ||2 . Equivariance is equivalent to Lt |t |2 =
∂|t |2 ∂t
(12)
for all t. When (12) holds (for a fixed t) we also say that Lt is an equivariant generator (with respect to t and H ). Note that this definition of equivariance agrees with the previous meaning for deterministic processes. We call a Markov process Q equivariant if and only if for every t the distribution ρt of Qt equals |t |2 . For this to be the case, equivariant transition probabilities are necessary but not sufficient. (While for a Markov process Q to have equivariant transition probabilities amounts to the property that if ρt = |t |2 for one time t, where ρt denotes the distribution of Qt , then ρt = |t |2 for every t > t, according to our definition of an equivariant Markov process, in fact ρt = |t |2 for all t.) However, for equivariant transition probabilities there exists a unique equivariant Markov process. The crucial idea for our construction of an equivariant Markov process is to note that (6) is completely general, and to find a generator Lt such that the right-hand side of (6) can be read as the action of L on ρ = ||2 , 2 Im ∗ H = L ||2 .
(13)
We shall implement this idea beginning in Sect. 2.4, after a review of jump processes and some general considerations. But first we shall illustrate the idea with the familiar case of Bohmian mechanics. For H of the form (4), we have (7) and hence that 2 Im ∗ H = − div Im ∗ ∇ = − div
∗ ∇ . ||2 Im ||2
(14)
Since the generator of the (deterministic) Markov process corresponding to the dynamical system dQ/dt = v(Q) given by a velocity vector field v is L ρ = − div (ρv) ,
(15)
we may recognize the last term of (14) as L ||2 with L the generator of the deterministic process defined by (2). Thus, as is well known, Bohmian mechanics arises as the natural equivariant process on configuration space associated with H and . To be sure, Bohmian mechanics is not the only solution of (13) for H given by (4). Among the alternatives are Nelson’s stochastic mechanics [25] and other velocity formulas [8]. However, Bohmian mechanics is the most natural choice, the one most likely to be relevant to physics. (It is, in fact, the canonical choice, in the sense of minimal process which we shall explain in [12, Sect. 5.2].) An important class of equivariant Markov processes are equivariant jump processes, which we discuss in the next three sections. They arise naturally in QFT, as we shall explain in Sect. 2.6.
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2.3. Equivariant jump processes. Let Q denote the configuration space of the process, whatever sort of space that may be (vector space, lattice, manifold, etc.); mathematically speaking, we need that Q be a measurable space. A (pure) jump process is a Markov process on Q for which the only motion that occurs is via jumps. Given that Qt = q, the probability for a jump to q , i.e., into the infinitesimal volume dq about q , by time t + dt is σt (dq |q) dt, where σ is called the jump rate. In this notation, σ is a finite measure in the first variable; σ (B|q) is the rate (the probability per unit time) of jumping to somewhere in the set B ⊆ Q, given that the present location is q. The overall jump rate is σ (Q|q). It is often the case that Q is equipped with a distinguished measure, which we shall denote by dq or dq , slightly abusing notation. For example, if Q = Rd , dq may be the Lebesgue measure, or if Q is a Riemannian manifold, dq may be the Riemannian volume element. When σ ( · |q) is absolutely continuous relative to the distinguished measure, we also write σ (q |q) dq instead of σ (dq |q). Similarly, we sometimes use the letter ρ for denoting a measure and sometimes the density of a measure, ρ(dq) = ρ(q) dq. A jump first occurs when a random waiting time T has elapsed, after the time t0 at which the process was started or at which the most recent previous jump has occurred. For purposes of simulating or constructing the process, the destination q can be chosen at the time of jumping, t0 + T , with probability distribution σt0 +T (Q|q)−1 σt0 +T ( · |q). In case the overall jump rate is time-independent, T is exponentially distributed with mean σ (Q|q)−1 . When the rates are time-dependent—as they will typically be in what follows—the waiting time remains such that t0 +T σt (Q|q) dt t0
is exponentially distributed with mean 1, i.e., T becomes exponential after a suitable (time-dependent) rescaling of time. For more details about jump processes, see [6]. The generator of a pure jump process can be expressed in terms of the rates: Lσ ρ(dq) = (16) σ (dq|q )ρ(dq ) − σ (dq |q)ρ(dq) , q ∈Q
a “balance” or “master” equation expressing ∂ρ/∂t as the gain due to jumps to dq minus the loss due to jumps away from q. We shall say that jump rates σ are equivariant if Lσ is an equivariant generator. It is one of our goals in this paper to describe a general scheme for obtaining equivariant jump rates. In Sect. 2.4 and 2.5 we will explain how this leads us to formula (1). 2.4. Integral operators correspond to jump processes. What characterizes jump processes versus continuous processes is that some amount of probability that vanishes at q ∈ Q can reappear in an entirely different region of configuration space, say at q ∈ Q. This is manifest in the equation for ∂ρ/∂t, (16): the first term in the integrand is the probability increase due to arriving jumps, the second the decrease due to departing jumps, and the integration over q reflects that q can be anywhere in Q. This suggests that Hamiltonians for which the expression (6) for ∂||2 /∂t is naturally an integral over dq correspond to pure jump processes. So when is the left-hand side of (13) an integral over dq ? When H is an integral operator, i.e., when q|H |q is not merely a formal symbol, but represents an integral kernel that exists as a function or a measure and satisfies
Quantum Hamiltonians and Stochastic Jumps
(H )(q) =
135
dq q|H |q (q ) .
(17)
(For the time being, think of Q as Rd and of wave functions as complex valued.) In this case, we should choose the jump rates in such a way that, when ρ = ||2 , 2 (18) Im ∗ (q) q|H |q (q ) , and this suggests, since jump rates must be nonnegative (and the right-hand side of (18) is anti-symmetric), that 2 + σ (q|q ) ρ(q ) = Im ∗ (q) q|H |q (q ) σ (q|q ) ρ(q ) − σ (q |q) ρ(q) =
(where x + denotes the positive part of x ∈ R, that is, x + is equal to x for x > 0 and is zero otherwise), or
+ (2/) Im ∗ (q) q|H |q (q ) σ (q|q ) = . (19) ∗ (q ) (q ) These rates are an instance of what we call the minimal jump rates associated with H (and ). They are also an instance of formula (1), as will become clear in the following section. The name comes from the fact that they are actually the minimal possible values given (18), as is expressed by the inequality (96) and will be explained in detail in Sect. 5. Minimality entails that at any time t, one of the transitions q1 → q2 or q2 → q1 is forbidden. We will call the process defined by the minimal jump rates the minimal jump process (associated with H ). In contrast to jump processes, continuous motion, as in Bohmian mechanics, corresponds to such Hamiltonians that the formal matrix elements q|H |q are nonzero only infinitesimally close to the diagonal, and in particular to differential operators like the Schr¨odinger Hamiltonian (4), which has matrix elements of the type δ (q − q ) + V (q) δ(q − q ). The minimal jump rates as given by (19) have some nice features. The possible jumps for this process correspond to the nonvanishing matrix elements of H (though, depending on the state , even some of the jump rates corresponding to nonvanishing matrix elements of H might happen to vanish). Moreover, in their dependence on the state , the jump rates σ depend only “locally” upon : the jump rate for a given jump q → q depends only on the values (q ) and (q) corresponding to the configurations linked by that jump. Discretizing R3 to a lattice εZ3 , one can obtain Bohmian mechanics as a limit ε → 0 of minimal jump processes [31, 32], whereas greater-than-minimal jump rates lead to Nelson’s stochastic mechanics [25] and similar diffusions; see [32, 17]. If the Schr¨odinger operator (4) is approximated in other ways by operators corresponding to jump processes, e.g., by Hε = e−εH H e−εH , the minimal jump processes presumably also converge to Bohmian mechanics. We have reason to believe that there are lots of self-adjoint operators which do not correspond to any stochastic process that can be regarded as defined, in any reasonable sense, by (19).2 But such operators seem never to occur in QFT. (The Klein–Gordon Consider, for example, H = p cos p, where p is the one-dimensional momentum operator −i ∂/∂q. Its formal kernel q|H |q is the distribution − 2i δ (q − q − 1) − 2i δ (q − q + 1), for which (19) would not have a meaning. From a sequence of smooth functions converging to this distribution, one can obtain a sequence of jump processes with rates (19): the jumps occur very frequently, and are by amounts of approximately ±1. A limiting process, however, does not exist. 2
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√ operator m2 c4 − 2 c2 does seem to have a process, but it requires a more detailed discussion which will be provided in a forthcoming work [13].) 2.5. Minimal jump rates. The reasoning of the previous section applies to a far more general setting than just considered: to arbitrary configuration spaces Q and “generalized observables”—POVMs—defining, for our purposes, what the “position representation” is. We now present this more general reasoning, which leads to formula (1). The process we construct relies on the following ingredients from QFT: 1. A Hilbert space H with scalar product | . 2. A unitary one-parameter group Ut in H with Hamiltonian H , Ut = e− tH , i
so that in the Schr¨odinger picture the state evolves according to i
dt = H t . dt
(20)
Ut could be part of a representation of the Poincar´e group. 3. A positive-operator-valued measure (POVM) P (dq) on Q acting on H , so that the probability that the system in the state is localized in dq at time t is Pt (dq) = t |P (dq)|t .
(21)
Mathematically, a POVM P on Q is a countably additive set function (“measure”), defined on measurable subsets of Q, with values in the positive (bounded self-adjoint) operators on (a Hilbert space) H , such that P (Q) is the identity operator.3 Physically, for our purposes, P ( · ) represents the (generalized) position observable, with values in Q. The notion of POVM generalizes the more familiar situation of observables given by a set of commuting self-adjoint operators, corresponding, by means of the spectral theorem, to a projection-valued measure (PVM): the case where the positive operators are projection operators. A typical example is the single Dirac particle: the position operators on L2 (R3 , C4 ) induce there a natural PVM P0 ( · ): for any Borel set B ⊆ R3 , P0 (B) is the projection to the subspace of functions that vanish outside B, or, equivalently, P0 (B)(q) = 1B (q) (q) with 1B the indicator function of the set B. Thus, |P0 (dq)| = |(q)|2 dq. When one considers as Hilbert space H only the subspace of positive energy states, however, the localization probability is given by P ( · ) = P+ P0 ( · )I with P+ : L2 (R3 , C4 ) → H the projection and I : H → L2 (R3 , C4 ) the inclusion mapping. Since P+ does not commute with most of the operators P0 (B), P ( · ) is no longer a PVM but a genuine POVM4 and consequently does not correspond to any position operator—although it remains true (for in the positive energy subspace) that |P (dq)| = |(q)|2 dq. That is why in QFT, the position observable is indeed more often a POVM than a PVM. POVMs are also relevant to photons [1, 22]. In one approach, the photon wave function : R3 → C3 3 The countable additivity is to be understood as in the sense of the weak operator topology. This in fact implies that countable additivity also holds in the strong topology. 4 This situation is indeed more general than it may seem. By a theorem of Naimark [7, p. 142], every POVM P ( · ) acting on H is of the form P ( · ) = P+ P0 ( · )I , where P0 is a PVM on a larger Hilbert space, P+ the projection to H and I the inclusion.
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is subject to the constraint condition ∇ · = ∂1 1 + ∂2 2 + ∂3 3 = 0. Thus, the physical Hilbert space H is the (closure of the) subspace of L2 (R3 , C3 ) defined by this constraint, and the natural PVM on L2 (R3 , C3 ) gives rise, by projection, to a POVM on H . So much for POVMs. Let us get back to the construction of a jump process. The goal is to specify equivariant jump rates σ = σ ,H,P , i.e., such rates that Lσ P =
dP . dt
(22)
To this end, one may take the following steps: 1. Note that 2 dPt (dq) = Im t |P (dq)H |t . dt 2. Insert the resolution of the identity I = P (dq ) and obtain
(23)
q ∈Q
dPt (dq) = dt
Jt (dq, dq ) ,
(24)
q ∈Q
where Jt (dq, dq ) =
2 Im t |P (dq)H P (dq )|t .
(25)
3. Observe that J is anti-symmetric, J(dq , dq) = −J(dq, dq ). Thus, since x = x + − (−x)+ ,
+
+ J(dq, dq ) = (2/) Im|P (dq)H P (dq )| − (2/) Im|P (dq )H P (dq)| . 4. Multiply and divide both terms by P( · ), obtaining that
J(dq, dq ) = q ∈Q
q ∈Q
[(2/) Im |P (dq)H P (dq )|]+ P(dq )− |P (dq )|
[(2/) Im |P (dq )H P (dq)|]+ − P(dq) |P (dq)|
.
5. By comparison with (16), recognize the right-hand side of the above equation as Lσ P, with Lσ the generator of a Markov jump process with jump rates (1), which we call the minimal jump rates. We repeat the formula for convenience: σ (dq|q ) =
[(2/) Im |P (dq)H P (dq )|]+ . |P (dq )|
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Mathematically, the right-hand side of this formula as a function of q must be understood as a density (Radon–Nikod´ym derivative) of one measure relative to another. The plus symbol denotes the positive part of a signed measure; it can also be understood as applying the plus function, x + = max(x, 0), to the density, if it exists, of the numerator. To sum up, we have argued that with H and is naturally associated a Markov jump process Qt whose marginal distributions coincide at all times by construction with the quantum probability measure, ρt ( · ) = Pt ( · ), so that Qt is an equivariant Markov process. In Sect. 4, we establish precise conditions on H, P , and under which the jump rates (1) are well-defined and finite P-almost everywhere, and prove that in this case the rates are equivariant, as suggested by Steps 1–5 above. It is perhaps worth remarking at this point that any H can be approximated by Hamiltonians Hn (namely Hilbert–Schmidt operators) for which the rates (1) are always (for all ) well-defined and equivariant, as we shall prove in Sect. 4.2.1. 2.6. Bell-type QFT. A Bell-type QFT is about particles moving in physical 3-space; their number and positions are represented by a point Qt in configuration space Q, with Q defined as follows. Let R3 denote the configuration space of a variable (but finite) number of identical particles in R3 , i.e., the union of (R3 )n modulo permutations,
R3 =
∞
(R3 )n /Sn .
(26)
n=0
Q is the Cartesian product of several copies of R3 , one for each species of particles. For a discussion of the space R3 , and indeed of S for any other measurable space S playing the role of physical space, see [12, Sect. 2.8]. A related space, for which we write = R3 , is the space of all finite subsets of R3 ; it is contained in R3 , after obvious identifications. In fact, = R3 = R3 \ , where is the set of coincidence configurations, i.e., those having two or more particles at the (n) (n) same position. = R3 is the union of the spaces Q = for n = 0, 1, 2, . . . , where Q = is the space of subsets of R3 with n elements, a manifold of dimension 3n (see [10] for a discussion of Bohmian mechanics on this manifold). The set of coincidence configurations has codimension 3 and thus can usually be ignored. We can thus replace
R3 by the somewhat simpler space = R3 . Qt follows a Markov process in Q, which is governed by a state vector in a suitable Hilbert space H . H is related to Q by means of a PVM or POVM P . The Hamiltonian of a QFT usually comes as a sum, such as H = H0 + H I
(27)
with H0 the free Hamiltonian and HI the interaction Hamiltonian. If several particle species are involved, H0 is itself a sum containing one free Hamiltonian for each species. The left-hand side of (13), which should govern our choice of the generator, is then also a sum, 2 2 Im ∗ H0 + Im ∗ HI = L ||2 .
(28)
This opens the possibility of finding a generator L by setting L = L0 + LI , provided we have generators L0 and LI corresponding to H0 and HI in the sense that
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2 Im ∗ H0 = L0 ||2 , 2 Im ∗ HI = LI ||2 .
(29a) (29b)
This feature of (13) we call process additivity; it is based on the fact that the left-hand side of (13) is linear in H . In a Bell-type QFT, the generator L is of the form L = L0 + LI , where L0 is usually the generator of a deterministic process, usually defined by the Bohmian or Bohm–Dirac law of motion, see below, and LI is the generator of a pure jump process, which is our main focus in this paper. The process generated by L is then given by deterministic motion determined by L0 , randomly interrupted by jumps at a rate determined by LI . We thus need to define two equivariant processes, one (the “free process”) associated with H0 and the other (the “interaction process”) with HI . The interaction process is the pure jump process with rates given by (1) with HI in place of H . We now give a description of the free process for the two most relevant free Hamiltonians: the second-quantized Schr¨odinger operator and the second-quantized Dirac operator. We give a more general and more detailed discussion of free processes in [12]; there we provide a formula, roughly analogous to (1), for L0 in terms of H0 , and an algorithm for obtaining the free process from a one-particle process that is roughly analogous to the “second quantization” procedure for obtaining H0 from a one-particle Hamiltonian. The free process associated with a second-quantized Schr¨odinger operator arises from Bohmian mechanics. Fock space H = F is a direct sum F =
∞
F
(n)
,
(30)
n=0
where F (n) is the n-particle Hilbert space. F (n) is the subspace of symmetric (for bosons) or anti-symmetric (for fermions) functions in L2 (R3n ,(C2s+1 )⊗n ) for spin-s parti cles. Thus, ∈ F can be decomposed into a sequence = (0) , (1) , . . . , (n) , . . . , the nth member (n) being an n-particle wave function, the wave function representing the n-particle sector of the quantum state vector. The obvious way to obtain a process on Q = R3 is to let the configuration Q(t), containing N = #Q(t) particles, move according to the N-particle version of Bohm’s law (2), guided by (N) .5 This is indeed an equivariant process since H0 has a block diagonal form with respect to the decomposition (30), H0 =
∞
(n)
H0 ,
n=0 (n)
and H0 is just a Schr¨odinger operator for n noninteracting particles, for which, as we already know, Bohmian mechanics is equivariant. We used a very similar process in [11] (the only difference being that particles were numbered in [11]). Similarly, if H0 is the second quantized Dirac operator, we let a configuration Q with N particles move according to the usual N -particle Bohm–Dirac law [5, p. 274] 5 As defined, configurations are unordered, whereas we have written Bohm’s law (2) for ordered configurations. Thanks to the (anti-)symmetry of the wave function, however, all orderings will lead to the same particle motion. For more about such considerations, see our forthcoming work [10].
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dQ ∗ (Q) αN (Q) =c , dt ∗ (Q) (Q)
(31)
where c denotes the speed of light and αN = (α (1) , . . . , α (N) ) with α (k) acting on the spin index of the k th particle. This completes the construction of the Bell-type QFT. An explicit example of a Belltype process for a simple QFT is described in [11], which we take up again in Sect. 3.12 below to point out how its jump rates fit into the scheme (1). Another such example, concerning electron–positron pair creation in an external electromagnetic field, is described in [12, Sect. 3.3.]. 3. Examples In this section, we present various special cases of the jump rate formula (1) and examples of its application. We also point out how the jump rates of the models in [11] and [3] are contained in (1). 3.1. A first example. To begin with, we consider Q = Rd , H = L2 (Rd , C), and P the natural PVM, which may be written P (dq) = |qq| dq. Then, P(dq) = |P (dq)| = |(q)|2 dq, and the jump rate formula (1) reads [(2/) Im ∗ (q) q|H |q (q )]+ ∗ (q ) (q ) 2 ∗ (q) q|H |q + = Im . ∗ (q )
σ (q|q ) =
(32a) (32b)
Note that (32a) is the same expression as (19). As a simple example of an operator HI with a kernel, consider a convolution operator, HI = V , where V may be complexvalued and V (−q) = V ∗ (q), (HI )(q) = V (q − q ) (q ) dq . The kernel is q|HI |q = V (q − q ). Together with H0 = − 2 , we obtain a baby example of a Hamiltonian H = H0 + HI that goes beyond the form (4) of Schr¨odinger operators, in particular in that it is no longer local in configuration space. Recall that H0 is associated with the Bohmian motion (2). Combining the two generators on the basis of process additivity, we obtain a process that is piecewise deterministic, with jump rates (19) and Bohmian trajectories between successive jumps. 2
3.2. Wave functions with spin. Let us next become a bit more general and consider wave functions with spin, i.e., with values in Ck . We have Q = Rd , H = L2 (Rd , Ck ) and P the natural PVM, which may be written P (dq) = ki=1 |q, iq, i| dq, where i indexes the standard basis of Ck . Another way of viewing P is to understand H as the tensor product L2 (Rd , C) ⊗ Ck , and P (dq) = P0 (dq) ⊗ ICk with P0 the natural PVM on L2 (Rd , C) and ICk the identity operator on Ck . Using the notation (q)|(q) for the
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scalar product in Ck , we can write P(dq) = |P (dq)| = (q)|(q) dq, and the jump rate formula (1) reads σ (q|q ) =
[(2/) Im (q)|K(q, q )|(q )]+ (q )|(q )
(33)
with K(q, q ), the kernel of H , a k×k matrix. If we write ∗ (q) (q) for (q)|(q), as we did in (2) and (31), and q|H |q for K(q, q ), (33) reads σ (q|q ) =
[(2/) Im ∗ (q) q|H |q (q )]+ , ∗ (q ) (q )
which is (19) again, interpreted in a different way. 3.3. Vector bundles. Next consider, instead of the fixed value space Ck , a vector bundle E over a Riemannian manifold Q, and cross-sections of E as wave functions. In order to have a scalar product of wave functions, we need every bundle fiber Eq to be equipped with a Hermitian inner product · | · q . We consider H = L2 (E) (the space of square-integrable cross-sections) and P the natural PVM. For any q and q , K(q, q ) then has to be a C-linear mapping Eq → Eq , so that the kernel of H is a cross-section of the bundle q,q Eq ⊗ Eq∗ over Q × Q. Equation (1) then reads σ (q|q ) =
[(2/) Im (q)|K(q, q ) (q )q ]+ . (q )|(q )q
(34)
In the following we will use the notation ∗ (q) (q) for (q)|(q)q and q|H |q for K(q, q ), so that σ (q|q ) =
[(2/) Im ∗ (q) q|H |q (q )]+ , ∗ (q ) (q )
which looks like (19) again. 3.4. Kernels of the measure type. The kernel q|H |q can be less regular than a function. Since the numerator of (1) is a measure in q and q , the formula still makes sense (for P the natural PVM) when the kernel q|H |q is a complex measure in q and q . The mathematical details will be discussed in Sect. 4.2. For instance, the kernel can have singularities like a Dirac δ, but it cannot have singularities worse than δ, such as derivatives of δ (as would arise from an operator whose position representation is a differential operator). It can happen that the kernel is not a function but a measure even for a very well-behaved (even bounded) operator. For example, this is the case for H a multiplication operator (i.e., a function V (q) ˆ of the position operator), q|H |q = V (q) δ(q −q ). Note, though, that multiplication operators correspond to zero jump rates. A nontrivial example of an operator with δ singularities in the kernel is H = 1 − cos(p/p0 ), where p = −i∂/∂q is the momentum operator in one dimension, H = L2 (R, C), and p0 is a constant. The dispersion relation E = 1−cos(p/p0 ) begins at p = 0 like 21 (p/p0 )2 but deviates from the parabola for large p. In the position representation, H is the convolution with ((2π)−1/2 times) the inverse Fourier transform of the function
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1 − cos(k/p0 ), and thus q|H |q = δ(q − q ) − 21 δ(q − q + p0 ) − 21 δ(q − q − p0 ). In this case, (1) leads to σ (q|q ) =
[(−1/) Im ∗ (q) (q )]+ + ) + δ(q − q − ) δ(q − q p0 p0 . ∗ (q ) (q )
(35)
(Note that nonnegative factors can be drawn out of the plus function.) This formula may be viewed as contained in (19) as well, in a formal sense. As a consequence of (35), only jumps by an amount of ± p0 can occur in this case.
3.5. Infinite rates. There also exist Markov processes that perform infinitely many jumps in every finite time interval (e.g., Glauber dynamics for infinitely many spins). These processes, which we do not count among the jump processes, may appear pathological, and we will not investigate them in this paper, but we note that some Hamiltonians may correspond to such processes. They could arise from jump rates σ ( · |q ) given by (1) that form not a finite but merely a σ -finite measure, so that σ (Q|q ) = ∞. Here is an (artificial) example of σ -finite (but not finite) rates, arising from an operator H that is even bounded. Let Q = R, H = L2 (R) with P ( · ) the √ position PVM, and let H , in Fourier representation, be multiplication by f (k) = π/2 sign(k). H is bounded since f is. f is the Fourier transform of i/x, understood as the distribution defined by the principal value integral. As a consequence, H has, in position representation, the kernel q|H |q = i/(q − q ). From (19) we obtain the jump rates σ (q|q ) =
Re ∗ (q) (q ) + 2 1 , ∗ (q ) (q ) q − q
(36)
which entails that σ (R|q ) = σ (q|q ) dq = ∞ at least whenever is continuous (and nonvanishing) at q . Nonetheless, since the rate for jumping anywhere outside the q +ε interval [q −ε, q +ε] is finite for every ε > 0 and since q −ε |q −q |σ (q|q ) dq < ∞, a process with these rates should exist: among the jumps that the process would have to make per unit time, the large ones would be few and the frequent ones would be tiny—too tiny to significantly contribute.
3.6. Discrete configuration space. Now consider a discrete configuration space Q. Mathematically, this means Q is a countable set. In this case, measures are determined by their values on singletons {q}, and we can specify all jump rates by specifying the rate σ (q|q ) for each transition q → q. Equation (1) then reads
+
(2/) Im |P {q}H P {q }| σ (q|q ) = |P {q }|
.
(37)
We begin with the particularly simple case that there is an orthonormal basis of H labeled by Q, {|q : q ∈ Q}, and P is the PVM corresponding to this basis,
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P {q} = |qq|. In this case, the notation q|H |q and the name “matrix element” can be taken literally. The rates (1) then simplify to [(2/) Im |qq|H |q q |]+ |q q | 2 |qq|H |q + = . Im |q
σ (q|q ) =
(38a) (38b)
Note that (38a) is the obvious discrete analogue of (19); in fact, one can regard (19) as another way of writing (38a) in this case. Consider now the more general case that a basis of Hilbert space is indexed by two “quantum numbers,” the configuration q and another index i. Then the POVM is given by the PVM P {q} = i |q, iq, i|, the projection onto the subspace associated with q (whose dimension might depend on q); such a PVM may be called “degenerate.” We have P(q) = |P {q}| = |q, iq, i|, and (1) becomes i
σ (q|q ) =
2
Im
i,i
+
|q, iq, i|H |q , i q , i | |q , i q , i |
.
(39)
i
We may also write (39) as (38a), understanding |q and q | as multi-component, q|H |q as a matrix, and products as inner products. In the case that the dimension of the subspace associated with q is always k, independent of q, (39) is a discrete analogue of the rate formula (33) for spinor-valued wave functions. Apart from serving as mathematical examples, discrete configuration spaces are relevant for several reasons: First, they provide particularly simple cases of jump processes with minimal rates that are easy to study. Second, any numerical computation is discrete by nature. Third, one may consider approximating or replacing the R3 that is supposed to model physical space by a lattice Z3 ; after all, lattice approaches have often been employed in QFT, for various reasons. Moreover, Bell-type QFTs will usually have as configurations the positions of a variable number of particles; so the configuration has a certain continuous aspect, the positions, and a certain discrete aspect, the number of particles. Sometimes one wishes to study simplified models, and in this vein it may be interesting to have only the particle number as a state variable, and thus the set of nonnegative integers as configuration space.
3.7. Bell’s process. The model Bell specified in [3] is a case of a minimal jump process on a discrete set. “For simplicity,” Bell considers a lattice instead of continuous 3-space, and a Hamiltonian of a lattice QFT. As a consequence, the configuration space Q = () is countable. (Bell even makes Q finite, but this is not relevant here. We also remark that according to Bell’s formulation, even distinguishable particles have configuration space ().) Bell chooses as the configuration the number of fermions at every lattice site, rather than the total particle number (i.e., in our terminology he takes P {q} to be the projection to the joint eigenspace of the fermion number operators for all lattice sites with eigenvalues the occupation numbers corresponding to q ∈ ()). He thus gives the fermionic
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degrees of freedom a status different from the bosonic ones. That is to say, boson particles do not exist in Bell’s model, despite the fact that H = Hfermions ⊗ Hbosons and the presence of bosonic terms in the Hamiltonian. Thus the PVM P {q} = Pfermions {q} ⊗ 1bosons is “doubly” degenerate: the fermionic occupation number operators do not form a complete set of commuting operators, because of both the spin and the bosonic degrees of freedom. Different spin states and different quantum states of the bosonic fields are compatible with the same fermion occupation numbers. So a further index i is necessary to label a basis {|q, i} of H . The jump rates Bell prescribes are then (39), and are thus a special case of (1). We emphasize that here the index i does not merely label different spin states, but states of the quantized radiation as well.
3.8. A case of POVM. Consider for H the space of Dirac wave functions of positive energy. The POVM P ( · ) we defined on it in Sect. 2.5 is, as we have already remarked, not a PVM but a genuine POVM and arises from the natural PVM P0 ( · ) on L2 (R3 , C4 ) by P ( · ) = P+ P0 ( · )I with P+ : L2 (R3 , C4 ) → H the projection and I : H → L2 (R3 , C4 ) the inclusion mapping. We can extend any given interaction Hamiltonian H on H to an operator on L2 (R3 , C4 ), Hext = I H P+ . If Hext possesses a kernel q|Hext |q , then H corresponds to a jump process, and the rates (1) can be expressed in terms of this kernel, since for ∈ H , |P (dq)H P (dq )| = |P+ P0 (dq)I H P+ P0 (dq )I | = |P0 (dq)Hext P0 (dq )| = ∗ (q) q|Hext |q (q ) dq dq . We thus obtain +
(2/) Im ∗ (q) q|Hext |q (q ) σ (q|q ) = ∗ (q ) (q )
.
(40)
This POVM is used in the pair creation model of [12, Sect. 3.3].
3.9. Another case of POVM. Let H = L2 (Rd ) and let P0 ( · ) be the natural PVM. We d obtain a POVM P by smearing out P0 with a profile function ϕ : R → [0, ∞) with ϕ(q) dq = 1 and ϕ(−q) = ϕ(q), e.g., a Gaussian:
P (B) =
dq q∈B
ϕ(q − q) P0 (dq ).
(41)
q ∈Rd
Whereas P0 (B) is multiplication by 1B , P (B) is multiplication by ϕ 1B . It leads to P(dq) = (ϕ ||2 )(q) dq. The jump rate formula (1) then yields
σ (q|q ) =
(2/) Im
dq
+
dq ϕ(q −q) ∗ (q ) q |H |q (q ) ϕ(q − q ) dq ϕ(q − q ) ∗ (q ) (q )
,
i.e., the denominator gets smeared out with ϕ, and the square bracket in the numerator gets smeared out with ϕ in each variable.
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3.10. Identical particles. The n-particle sector of the configuration space (without coincidence configurations) of identical particles = (R3 ) is the manifold of n-point subsets of R3 ; let Q be this manifold. The most common way of describing the quantum state of n fermions is by an anti-symmetric (square-integrable) wave function on Qˆ := R3n ; let H be the space of such functions. Whereas for bosons could be viewed as a function on Q, for fermions is not a function on Q. Nonetheless, the configuration observable still corresponds to a PVM P on Q: for B ⊆ Q, we set P (B)(q 1 , . . . , q n ) = (q 1 , . . . , q n ) if {q 1 , . . . , q n } ∈ B and zero otherwise. In other words, P (B) is multiplication by the indicator function of π −1 (B), where π is the obvious projection mapping Qˆ \ → Q, with the set of coincidence configurations. To obtain other useful expressions for this PVM, we introduce the formal kets |q ˆ ˆ the anti-symmetrization operator S for qˆ ∈ Qˆ (to be treated like elements of L2 (Q)), √ ˆ → H ), the normalized anti-symmetrizer6 s = n! S, and (i.e., the projection L2 (Q) the formal kets |s q ˆ := s|q ˆ (to be treated like elements of H ). The |q ˆ and |s q ˆ are normalized in the sense that
ˆ qˆ ) q| ˆ qˆ = δ(qˆ − qˆ ) and s q|s ˆ qˆ = (−1)(q, δ(q − q ),
where q = π(q), ˆ q = π(qˆ ), (q, ˆ qˆ ) is the permutation that carries qˆ into qˆ given that q = q , and (−1) is the sign of the permutation . Now we can write P (dq) = |q ˆ q| ˆ dq = n! S|q ˆ q| ˆ dq = |s qs ˆ q| ˆ dq, (42) q∈π ˆ −1 (q)
where the sum is over the n! ways of numbering the n points in q; the last two terms actually do not depend on the choice of qˆ ∈ π −1 (q), the numbering of q. The probability distribution arising from this PVM is 2 P(dq) = |(q)| ˆ 2 dq = n! |(q)| ˆ 2 dq = |s q|| ˆ dq (43) q∈π ˆ −1 (q)
with arbitrary qˆ ∈ π −1 (q). ˆ is permutation invariant, If an operator Hˆ on L2 (Q) U−1 Hˆ U = Hˆ for every permutation ,
(44)
ˆ performing the permutation , then Hˆ maps where U is the unitary operator on L2 (Q) anti-symmetric functions to anti-symmetric functions, and thus defines an operator H on H . If Hˆ has a kernel q| ˆ Hˆ |qˆ then the kernel is permutation invariant in the sense that (q)| ˆ Hˆ |(qˆ ) = q| ˆ Hˆ |qˆ
∀,
(45)
where (q 1 , . . . , q n ) := (q (1) , . . . , q (n) ), and H also possesses a kernel, ˆ Hˆ S|qˆ = s q|H ˆ |s qˆ = n! q|S
1 (q)| ˆ Hˆ | (qˆ ). n! ,
6
The name means this: since S is a projection, S is usually not a unit vector when is. Whenever ˆ ∈ L2 (Qˆ ) is supported by a fundamental domain of the permutation √ group, i.e., by a set ⊆ Q on which (the restriction of) π is a bijection to Q, the norm of S is 1/ n!, so that s is again a unit vector.
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In this case (1) yields σ (q|q ) =
Im
q, ˆ qˆ
+
∗ (q) ˆ q| ˆ Hˆ |qˆ (qˆ )
=
2
qˆ
(46a)
∗ (qˆ ) (qˆ ) +
ˆ q|H ˆ |s qˆ s qˆ | Im |s qs 2
|s qˆ s qˆ |
,
(46b)
where qˆ ∈ π −1 (q) and qˆ ∈ π −1 (q ), as running variables in (46a) and as arbitrary but fixed in (46b).
3.11. Another view of fermions. There is a way of viewing fermion wave functions as being defined on Q, rather than R3n , by regarding them as cross-sections of a particular 1-dimensional vector bundle over Q. To this end, define an n!-dimensional vector bundle E by C. (47) Eq := q∈π ˆ −1 (q)
Every function : R3n → C naturally gives rise to a cross-section of E, defined by (q) ˆ . (48) (q) := q∈π ˆ −1 (q)
The anti-symmetric functions form a 1-dimensional subbundle of E (see also [10] for a discussion of this bundle). The jump rate formula for vector bundles (34) can be applied to either the subbundle or E, depending on the way in which the kernel of H is given. The kernel q| ˆ Hˆ |qˆ above translates directly into a kernel on Q × Q with values in ∗ Eq ⊗ Eq , for which the rate formula for bundles (34) is the same as the rate formula for identical particles (46a) derived in the previous section. Another alternative view of a fermion wave function is to regard it as a complex differential form of full rank, a 3n-form, on Q. (See, e.g., [10]. This would not work if the dimension of physical space were even.) Of course, the complex 3n-forms are nothing but the sections of a certain 1-dimensional bundle, usually denoted C ⊗ 3n Q, which is equivalent to the subbundle of E considered in the previous paragraph, and which is contained in the bundle C ⊗ Q of Grassmann numbers over Q. 3.12. A simple QFT. We presented a simple example of a Bell-type QFT in [11], and we will now briefly point to the aspects of this model that are relevant here. The model is based on one of the simplest possible QFTs [30, p. 339]. The relevant configuration space Q for a QFT (with a single particle species) is the configuration space of a variable number of identical particles in R3 , which is the set (R3 ), or, ignoring the coincidence configurations (as they are exceptions), the set
= (R3 ) of all finite subsets of R3 . The n-particle sector of this is a manifold of dimension 3n; this configuration space is thus a union of (disjoint) manifolds of different
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dimensions. The relevant configuration space for a theory with several particle species is the Cartesian product of several copies of = (R3 ). In the model of [11], there are two particle species, a fermion and a boson, and thus the configuration space is Q = = (R3 ) × = (R3 ).
(49)
We will denote configurations by q = (x, y) with x the configuration of the fermions and y the configuration of the bosons. For simplicity, we replaced in [11] the sectors of = (R3 ) × = (R3 ), which are manifolds, by vector spaces of the same dimension (by artificially numbering the particles), and obtained the union Qˆ =
∞
n=0
(R3 )n ×
∞
(R3 )m ,
(50)
m=0
with n the number of fermions and m the number of bosons. Here, however, we will use (49) as the configuration space. In comparison with (50), this amounts to (merely) ignoring the numbering of the particles. H is the tensor product of a fermion Fock space and a boson Fock space, and thus the ˆ that are anti-symmetric in the fermion coordinates subspace of wave functions in L2 (Q) and symmetric in the boson coordinates. Let S denote the appropriate symmetrization ˆ → H , and s the normalized symmetrizer operator, i.e., the projection operator L2 (Q) √ s(x 1 , . . . , x n , y 1 , . . . , y m ) = n! m! S(x 1 , . . . , x n , y 1 , . . . , y m ), (51) √ i.e., s = N ! M! S with N and M the fermion and boson number operators, which commute with S and with each other. As in Sect. 3.10, we denote by π the projection mapping Qˆ \ → Q, π(x 1 , . . . , x n , y 1 , . . . , y m ) = ({x 1 , . . . , x n }, {y 1 , . . . , y m }). The configuration PVM P (B) on Q is multiplication by 1π −1 (B) , which can be underˆ since it is permutation invariant and stood as acting on H , though it is defined on L2 (Q), thus maps H to itself. We utilize again the formal kets |q, ˆ where qˆ ∈ Qˆ \ is a numbered configuration, for which we also write qˆ = (x, ˆ y) ˆ = (x 1 , . . . , x n , y 1 , . . . , y m ). We also use the symmetrized and normalized kets |s q ˆ = s|q. ˆ As in (42), we can write P (dq) = |q ˆ q| ˆ dq = n! m! S|q ˆ q| ˆ dq = |s qs ˆ q| ˆ dq (52) q∈π ˆ −1 (q)
with arbitrary qˆ ∈ π −1 (q). For the probability distribution, we thus have, as in (43), 2 P(dq) = |(q)| ˆ 2 dq = n! m! |(q)| ˆ 2 dq = |s q|| ˆ dq (53) q∈π ˆ −1 (q)
with arbitrary qˆ ∈ π −1 (q). The free Hamiltonian is the second quantized Schr¨odinger operator (with zero potential), associated with the free process described in Sect. 2.6. The interaction Hamiltonian is defined by HI = d 3 x ψ † (x) (aϕ† (x) + aϕ (x)) ψ(x) (54)
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with ψ † (x) the creation operators (in position representation), acting on the fermion † Fock space, and aϕ (x) the creation operators (in position representation), acting on the boson Fock space, regularized through convolution with an L2 function ϕ : R3 → R. HI has a kernel; we will now obtain a formula for it, see (60) below. The |s q ˆ are connected to the creation operators according to |s q ˆ = ψ † (x n ) · · · ψ † (x 1 )a † (y m ) · · · a † (y 1 )|0 ,
(55)
where |0 ∈ H denotes the vacuum state. A relevant fact is that the creation and annihilation operators ψ † , ψ, a † and a possess kernels. Using the canonical (anti-)commutation relations for ψ and a, one obtains from (55) the following formulas for the kernels of ψ(r) and a(r), r ∈ R3 :
ˆ xˆ ) 3m δ (y − y ), s q|ψ(r)|s ˆ qˆ = δn,n −1 δm,m δ 3n (x ∪ r − x ) (−1)((x,r),
s q|a(r)|s ˆ qˆ = δn,n δm,m −1 δ 3n (x − x ) (−1)
(x, ˆ xˆ )
δ
3m
(y ∪ r − y ),
(56) (57)
where (x, y) = q = π(q), ˆ and (x, ˆ xˆ ) denotes the permutation that carries xˆ to xˆ given that x = x . The corresponding formulas for ψ † and a † can be obtained by exchanging qˆ and qˆ on the right-hand sides of (56) and (57). For the smeared-out operator aϕ (r), we obtain ˆ xˆ ) δ 3m (y − y \ y ) ϕ(y − r). s q|a ˆ ϕ (r)|s qˆ = δn,n δm,m −1 δ 3n (x − x ) (−1)(x, y ∈y
(58) We make use of the resolution of the identity ˆ q| ˆ . I = dq |s qs
(59)
Q
Inserting (59) twice into (54) and exploiting (56) and (58), we find ˆ xˆ ) s q|H ˆ I |s qˆ = δn,n δm−1,m δ 3n (x − x ) (−1)(x,
y∈y
ˆ xˆ ) +δn,n δm −1,m δ 3n (x − x ) (−1)(x,
δ 3m (y \ y − y )
y ∈y
ϕ(y − x)
x∈x
δ 3m (y −y \ y )
ϕ(y −x) .
(60)
x∈x
This is another case of a kernel containing δ functions (see Sect. 3.4). By (52), the jump rates (1) are + 2 Im |s qs ˆ q|H ˆ I |s qˆ s qˆ | σ (q|q ) = . |s qˆ s qˆ |
(61)
More explicitly, we obtain from (60) the rates δ 3m (y \ y − y ) σcrea (q ∪ y|q ) σ (q|q ) = δnn δm−1,m δ 3n (x − x ) y∈y
+δnn δm,m −1 δ 3n (x − x )
y ∈y
δ 3m (y − y \ y ) σann (q \ y |q )
(62)
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with
√
σcrea (q ∪ y|q ) =
2 m + 1
2 σann (q \ y |q ) = √ m
ˆ xˆ ) Im ∗ (q) ˆ (−1)(x,
x ∈x
+
ϕ(y − x ) (qˆ )
,
∗ (qˆ ) (qˆ )
ˆ xˆ ) Im ∗ (q) ˆ (−1)(x,
x ∈x
(63a)
+
ϕ(y − x ) (qˆ )
∗ (qˆ ) (qˆ )
,
(63b)
for arbitrary qˆ ∈ π −1 (q ) and qˆ ∈ π −1 (q) with q = (x , y ∪ y) respectively q = (x , y \ y ). (Note that a sum sign can be drawn out of the plus function if the terms have disjoint supports.) Equation (62) is worth looking at closely: One can read off that the only possible jumps are (x , y ) → (x , y ∪ y), creation of a boson, and (x , y ) → (x , y \ y ), annihilation of a boson. In particular, while one particle is created or annihilated, the other particles do not move. The process that we considered in [11] consists of pieces of Bohmian trajectories interrupted by jumps with rates (62); the process is thus an example of the jump rate formula (1), and an example of combining jumps and Bohmian motion by means of process additivity. The example shows how, for other QFTs, the jump rates (1) can be applied to relevant interaction Hamiltonians: If HI is, in the position representation, a polynomial in the creation and annihilation operators, then it possesses a kernel on the relevant configuration space. A cut-off (implemented here by smearing out the creation and annihilation operators) needs to be introduced to make HI a well-defined operator on L2 . If, in some QFT, the particle number operator is not conserved, jumps between the sectors of configuration space are inevitable for an equivariant process. And, indeed, when HI does not commute with the particle number operator (as is usually the case), jumps can occur that change the number of particles. Often, HI contains only off-diagonal terms with respect to the particle number; then every jump will change the particle number. This is precisely what happens in the model of [11].
4. Existence Results The configuration space Q is assumed in this paper to be a measurable space, equipped with a σ -algebra A. Every set we consider is assumed to belong to the appropriate σ algebra: A on Q or the product σ -algebra A ⊗ A on Q × Q. If F is a quadratic form, we will usually use the notation |F | rather than F ( , ). If P (B) and P (C) lie in the form domain of H , we write |P (B)H P (C)| for P (B)|H |P (C).
4.1. Condition for finite rates. For the argument of Sect. 2.5 to work, it is necessary that (a) the bracket in the numerator of (1) exist as a finite signed measure on Q × Q, and (b) the Radon–Nikod´ym derivative of the numerator with respect to the denominator also be well defined. It turns out that, given (a), (b) is straightforward. However, contrary to what a superficial inspection might suggest, (a) is problematical even when H is bounded. To see this, consider the case H = L2 (R) with the natural PVM (corresponding to
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position) on Q = R, and with H the sum of the Fourier transform on H and its adjoint, given by the kernel 2 q|H |q = cos(qq ) . π Then, for real, the bracket in (1) would have to be understood as proportional to (q) cos(qq ) (q ) , and ∈ H could be so chosen that this does not define a signed measure on R × R because both its positive and negative part have infinite total weight. In fact, can be so chosen that the resulting σ ( · |q ) is an infinite measure, σ (Q|q ) = ∞, for all q , and thus does not define a jump process. Note, however, that for ∈ L1 (R) ∩ L2 (R), σ ( · |q ) is finite for this H . The following theorem provides a condition under which the argument sketched in Sect. 2.5 for the equivariance of the jump rates σ , Steps 1–5, can be made rigorous. Theorem 1. Let H be a Hilbert space, ∈ H with = 1, H a self-adjoint operator on H , Q a standard Borel space,7 and P a POVM on Q acting on H . Suppose that for all B ⊆ Q, P (B) lies in the form domain of H , and there exists a complex measure µ on Q × Q such that for all B, C ⊆ Q, µ(B × C) = |P (B)H P (C)| .
(64)
Then the jump rates (1) are well-defined and finite for P-almost every q , and they are equivariant if, in addition, ∈ domain(H ). Proof. We first show that under the hypotheses of the theorem, the jump rates (1) are well-defined and finite. Then we show that they are equivariant. To begin with, the measure µ whose existence was assumed in the theorem is conjugate symmetric under the transposition mapping (q, q )tr = (q , q) on Q × Q, i.e., µ(Atr ) = µ(A)∗ . To see this, note that a complex measure on Q × Q is uniquely determined by its values on product sets. µ( · tr ) and µ( · )∗ must thus be the same (64) measure since, by the self-adjointness of H , for A = B × C, µ(Atr ) = µ(C × B) = |P (C)H P (B)| = |P (B)H P (C)|∗ = µ(A)∗ . We define a signed measure J on Q × Q by J = 2 Im µ. Let J+ be the positive part of J (defined by its Hahn–Jordan decomposition, J = J+ − J− , see e.g. [21, p. 120]). Since µ is a complex measure (and thus assumes only finite values), J has finite positive and negative parts. Since µ is conjugate symmetric, J is anti-symmetric. We now show that for every B ⊆ Q, the measure J(B × · ) on Q is absolutely continuous with respect to P( · ), the “||2 ” measure defined in (21). If C is a P-null set, that is |P (C)| = 0, then P (C)| = 0: if P (C) is a projection, this is immediate, and if P (C) is just any positive operator, it follows from the spectral theorem—any component of orthogonal to the eigenspace of P (C) with eigenvalue zero would lie in the positive spectral subspace of P (C) and give a positive contribution to |P (C)|. 7 A standard Borel space is a measurable space isomorphic to a complete separable metric space with its Borel σ -algebra. Basically all spaces that arise in practice are in fact standard Borel spaces, and so are in particular all spaces that we have in mind for Q (which are countable unions of (separable) Riemannian manifolds). Thus, the condition of being a standard Borel space is not much of a restriction.
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From P (C) = 0 it follows that |P (B)H P (C)| = 0, so that J(B × C) = 0, which is what we wanted to show. Next we show that for every B ⊆ Q, the measure J+ (B × · ) is absolutely continuous with respect to P( · ). Suppose again that P(C) = 0. We have that J+ (B × C) ≤ J+ (B × C) + J− (B × C) = |J|(B × C) = sup
|J(Bi × Cj )|,
i,j
where the sup is taken over all finite partitions i Bi = B of B and j Cj = C of C. Now each J(Bi × Cj ) = 0 because J(Bi × · ) P( · ) and P(Cj ) ≤ P(C) = 0. Thus J+ (B × C) = 0. It follows from the Radon–Nikod´ym theorem that for every B, J+ (B × · ) possesses a density with respect to P( · ). The density is unique up to changes on P-null sets, and one version of this density is what we will take as σ (B|q ). We have to make sure, though, that σ is a measure in its dependence on B, and from the Radon–Nikod´ym theorem alone we do not obtain additivity in B. For this reason, we utilize a standard theorem [27, p. 147] on the existence of regular conditional probabilities, asserting that if Q (and thus also Q × Q) is a standard Borel space, then every probability measure ν on Q × Q possesses regular conditional probabilities, i.e., a function p( · |q ) on Q with values in the probability measures on Q×Q such that for almost every q , p( · |q ) is concentrated on the set Q × {q } ⊆ Q × Q, and for every A ⊆ Q × Q, p(A|q ) is a measurable function of q with p(A|q ) ν(Q × dq ) = ν(A). (65) q ∈Q
We set ν( · ) = J+ ( · )/J+ (Q × Q) and define σ as the corresponding regular conditional probability times a factor that takes into account that (1) involves the density of J+ relative to P (rather than to ν(Q × · ) or J+ (Q × · )): σ (B|q ) := p(B × Q|q )
dJ+ (Q × · ) (q ) . dP( · )
(66)
The last factor exists because we have shown above that J+ (Q × · ) P( · ). σ ( · |q ) is a (finite) measure because p( · |q ) is. For fixed B, σ (B|q ) as a function of q is a version of the Radon–Nikod´ym derivative dJ+ (B × · )/dP( · ) because dJ+ (Q × · ) (66) σ (B|q ) P(dq ) = p(B × Q|q ) (q ) P(dq ) dP( · ) q ∈C
q ∈C
+
= J (Q × Q) q ∈Q
p(B × C|q )
J+ (Q × dq ) (65) + = J (B × C). J+ (Q × Q)
According to the theorem on regular conditional probabilities that we used, σ is defined uniquely up to changes on a P-null set of q s.
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Now we check the equivariance of the jump rates σ : for any B ⊆ Q, (16) σ (B|q ) P(dq ) − σ (Q|q) P(dq) = J+ (B × Q) − J+ (Q × B) , Lσ P(B) = q ∈Q
q∈B
using that σ is a version of the Radon–Nikod´ym derivative of J+ relative to P. Since J is anti-symmetric with respect to the permutation mapping (q, q ) → (q , q) on Q × Q, we have that J+ (C × B) = J− (B × C), and therefore Lσ P(B) = J+ (B × Q) − J− (B × Q) = J(B × Q) (64)
= 2 Im µ(B × Q) = 2 Im |P (B)H |. It remains to be shown that Pt (B) = e−iH t/ |P (B) e−iH t/ is differentiable with respect to time at t = 0 and has derivative dPt (B) = 2 Im |P (B)H |. (67) t=0 dt If lies in the domain of H , t = e−iH t/ is differentiable with respect to t at t = 0 ˙ = − i H . Hence [28, p. 265] and has derivative 1 t |P (B)t − 0 |P (B)0 t = t |P (B)|(t − 0 )/ t + (t − 0 )/ t |P (B)0
converges, as t → 0, to ˙ + |P ˙ (B) = − i |P (B)H + i H |P (B) = 2 Im|P (B)H . |P (B)
It now follows that Lσ P = dP/dt, which completes the proof.
We remark that if, as supposed in Theorem 1, the measure µ exists, it is also unique. This follows from the fact, which we have already mentioned, that a (complex) measure on Q × Q is uniquely determined by its values on the product sets B × C. Another remark concerns how the (existence) assumption of Theorem 1 can be violated. Since the example Hamiltonian of Sect. 3.5 leads to infinite jump rates, it also provides an example for which the assumption of Theorem 1 is violated, in fact for every nonzero ∈ H . To see this directly, note that, while P (B) lies indeed in the form domain of H (which is H since H is bounded), ∗ (q) (q ) |P (B)H P (C)| = i dq P- dq , q − q B
C
where P- denotes a principal value integral. For B ∩ C = ∅, P- can be replaced by a Lebesgue integral. This, together with (64), would leave for µ only one possibility (up to addition of a complex measure concentrated on the diagonal {(q, q) : q ∈ Q}), namely µ(dq × dq ) = i
∗ (q) (q ) dq dq . q − q
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But this is not a complex measure for any since i ∗ (q) (q )/(q−q ) is not absolutely integrable. This example also nicely illustrates the difference between a complex bi-measure ν(B, C), i.e., a complex measure in each variable, and a complex measure µ( · ) on Q × Q: |P (B)H P (C)| is here a complex bi-measure and thus defines a finite-valued additive set function on the family of finite unions of product sets B × C ⊆ Q × Q, which, however, cannot be suitably extended to all sets A ⊆ Q×Q. The essential reason is that the positive and the negative singularity in 1/(q − q ) cancel (thanks to the use of principal value integrals) for every product set but do not for some nonproduct sets such as {(q, q ) : q > q }. In contrast, a (finite) non-negative bi-measure can always be extended to a (finite) measure on the product space; see Sect. 4.4. A related remark on the need for the existence assumption of Theorem 1. One might well have imagined that the complex measure µ on Q × Q, extending (64) from product sets, can always be constructed, at least when H is bounded, as the quantum expected value of the bounded-operator-valued measure (BOVM) P ×H P on Q × Q, the “H twisted product measure” P (dq)H P (dq ) of the POVM P with itself—or, equivalently, the product of the POVM P (dq) and the BOVM H P (dq ). Indeed, the nonexistence of µ for the Hamiltonian in the principal-value example that we have just discussed, as well as for the Hamiltonian in the Fourier-transform example at the beginning of this section, implies that P ×H P does not exist as a BOVM in these cases; if it did, so would µ, for all . The Fourier-transform example can also easily be adapted to show that the product P1 × P2 of two POVMs need not exist as a BOVM, and in fact does not exist when P1 and P2 are the most familiar PVMs for quantum mechanics, corresponding respectively to position and momentum. There is, however, an important special case for which the product P1 × P2 of two POVMs does exist, in fact as a POVM, namely when P1 and P2 mutually commute, i.e., when [P1 (B), P2 (C)] = 0 for all B and C. This will be discussed in Sect. 4.4.
4.2. Integral operators. In this section we make precise the statement that Hamiltonians with (sufficiently regular) kernels lead to finite jump rates. In particular, we specify a set of wave functions, depending on H, that lead to finite jump rates. 4.2.1. Hilbert–Schmidt operators. We begin with the simple case in which is a complex-valued wave function on Q, so that the natural configuration POVM P ( · ) is a “nondegenerate” PVM. What first comes to mind as a class of Hamiltonians possessing a kernel is the class of Hilbert–Schmidt operators; for these, the kernels are in fact square-integrable functions on Q × Q. Corollary 1. Let Q be a standard Borel space, H = L2 (Q, C, dq) with respect to a σ finite nonnegative measure on Q that we simply denote dq, let ∈ H with = 1, let H be a self-adjoint operator on H , and let P be the natural PVM on Q (multiplication by indicator functions) acting on L2 (Q, C, dq). Suppose that H is a Hilbert–Schmidt operator. Then, by virtue of Theorem 1, the jump rates given by (1) are well-defined and finite P-almost everywhere, and equivariant. In fact, the jump rates are given by (19) with q|H |q the kernel function of H . Proof. Since H is a Hilbert–Schmidt operator, it possesses an integral kernel K(q, q ) that is a square-integrable function [28, p. 210], i.e., there is a function K ∈ L2 (Q × Q, C, dq dq ) such that for all ∈ H ,
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H (q) =
K(q, q ) (q ) dq .
Q
Thus, for all
,
∈H, |H | =
dq
Q
dq ∗ (q) K(q, q ) (q )
Q
(by Fubini’s theorem, because the integrand is absolutely integrable) = dq dq ∗ (q) K(q, q ) (q ). Q×Q
It follows that
|P (B)H P (C)| =
dq dq 1B (q) ∗ (q) K(q, q ) 1C (q ) (q )
(68a)
dq dq ∗ (q) K(q, q ) (q ).
(68b)
Q×Q
= B×C
Note that since H is bounded, its form domain is H and thus contains all P (B). For A ⊆ Q × Q, define µ(A) = ∗ (q) K(q, q ) (q ) dq dq . A
Since
|(q)| |K(q, q )| |(q )| dq dq < ∞ ,
Q×Q
µ(A) is always finite, and thus a complex measure. Equation (68) entails that (64) is satisfied, so that Theorem 1 applies. We have already remarked that every Hamiltonian H can be approximated by Hilbert–Schmidt operators Hn . In this context, it is interesting to note that if H is itself a Hilbert–Schmidt operator, and if the Hn converge to H in the Hilbert–Schmidt norm, then the rates σ ,Hn converge to σ ,H in the sense that ,H n→∞ n (dq|q ) − σ ,H (dq|q ) |(q )|2 dq −→ 0. σ Q×Q
4.2.2. Complex-valued wave functions. In addition to the case of Hilbert–Schmidt operators, Theorem 1 applies in many other cases, in which the kernel K(q, q ) is not squareintegrable, nor even a function but instead a measure K(dq × dq ). More precisely, K(dq × dq ) should be a σ -finite complex measure, i.e., a product of a complex-valued measurable function Q × Q → C and a σ -finite nonnegative measure on Q × Q. (Note that this terminology involves a slight abuse of language since a σ -finite complex measure need not be a complex measure.) The complex measure µ assumed to exist in Theorem 1 is then
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µ(dq × dq ) = ∗ (q) K(dq × dq ) (q ) .
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(69)
This equation suggests that the minimal amount of regularity that we need to assume on the kernel of H is that it be a σ -finite complex measure. Otherwise, there would be no hope that (69) could be a complex measure for a generic wave function , that vanishes at most on a set of measure 0. The exact conditions that we need for applying Theorem 1 to a Hamiltonian H with kernel K(dq × dq ) are listed in the following statement: Corollary 2. Let Q be a standard Borel space, H = L2 (Q, C, dq) with respect to a σ finite nonnegative measure on Q that we simply denote dq, let ∈ H with = 1, let H be a self-adjoint operator on H , and let P be the natural PVM on Q acting on L2 (Q, C, dq). Suppose that H has a kernel K(dq × dq ) for ; i.e., suppose that K(dq ×dq ) is a σ -finite complex measure on Q×Q, and that some everywhere-defined version : Q → C of the almost-everywhere-defined function ∈ L2 (Q, C, dq) satisfies |(q)| |K(dq × dq )| |(q )| < ∞, (70a) Q×Q
P (B) ∈ form domain(H ) ∀B ⊆ Q, |P (B)H P (C)| = ∗ (q) K(dq × dq ) (q ) ∀B, C ⊆ Q.
(70b) (70c)
B×C
Then, by virtue of Theorem 1, the jump rates given by (1) are well-defined and finite P-almost everywhere, and they are equivariant if ∈ domain(H ). Proof. Set µ(A) =
∗ (q) K(dq × dq ) (q ) .
(71)
A
The integral exists because of (70a) and defines a complex measure µ, which satisfies (64) because of (70c). We remark that the choice of the everywhere-defined version : Q → C of the almost-everywhere-defined function ∈ L2 (Q, C, dq) does not affect the jump rates, since the measure µ is uniquely determined by its values on product sets, which are given in (64) in terms of the almost-everywhere-defined function ∈ H . The reader may be surprised that our notion of H having a kernel K seems to depend on , whereas one may expect that H either has a kernel or does not, independent of . The reason for our putting it this way is that domain questions are very delicate for such general kernels, and it is a tricky question for which ’s the expression |P (B)H P (C)| is actually given by the integral (70c). A discussion of domain questions would only obscure what is actually relevant for having a situation in which Theorem 1 applies, which is (70). Note, though, that if H has kernel K(dq × dq ) for , then it has kernel K also for every from the subspace spanned by P (B) for all B ⊆ Q. The conditions (70) become very transparent in the following case: Suppose H is a self-adjoint extension of the integral operator K arising from a kernel K(q, dq ) that is
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a σ -finite complex measure on Q for every q ∈ Q and is such that for every B ⊆ Q, K(q, B) is a measurable function of q. K is defined by K (q) = K(q, dq ) (q ) (72) q ∈Q
on the domain D containing the ’s satisfying |K(q, dq )| | (q )| < ∞ for almost every q
(73)
q ∈Q
and
K(q, dq ) (q ) is an L2 function of q.
(74)
Q
That H is an extension of K means that the domain of H contains D , and H = K for all ∈ D . Then, for a ∈ D satisfying K(q, dq ) (q ) ∈ L2 (Q, C, dq) ∀B ⊆ Q (75) q ∈B
and
|(q)| |K(q, dq )| |(q )| dq < ∞,
(76)
Q×Q
conditions (70) are satisfied with K(dq × dq ) = K(q, dq ) dq, and thus Corollary 2 applies. The jump rates (1) can still be written as in (19), understood as a measure in q. Corollary 2 defines a set of good ’s, for which the jump rates are finite, for the examples of Sects. 3.1, 3.4, and for (38a). 4.2.3. Vector-valued wave functions We now consider wave functions with spin, i.e., with values in Ck . In this case, let ∗ (q) denote, as before, the adjoint spinor, and ∗ (q) (q) the inner product in Ck . Corollary 2 remains true if we replace C by Ck everywhere and understand K(dq × dq ) as matrix-valued, i.e., as the product of a matrix-valued function and a σ -finite nonnegative measure. The proof goes through without changes. Let us now be a bit more general and allow the value space of the wave function to vary with q; we reformulate Corollary 2 for wave functions that are cross-sections of a vector bundle E over Q. The kernel is then matrix valued in the sense that q|H |q is a linear mapping Eq → Eq . Corollary 3. Let Q = n Q(n) be an (at most) countable union of (separable) Rie mannian manifolds, and E = n E (n) the union of vector bundles E (n) over Q(n) , where the fiber spaces Eq are endowed with Hermitian inner products, which we denote by ∗ (q) (q). Let H = L2 (E, dq) be the space of square-integrable (with respect to the Riemannian volume measure that we denote dq) cross-sections of E, let ∈ H
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with = 1, let H be a self-adjoint operator on H , and let P be the natural PVM on Q acting on H . Suppose that K(dq × dq ), the product of a σ -finite nonnegative measure on Q × Q and a section of the bundle q,q Eq ⊗ Eq∗ over Q × Q, is a kernel of H for ; i.e., suppose that some everywhere-defined version of the almost-everywhere-defined cross-section ∈ L2 (E, dq) satisfies (70a)–(70c) (where the integrand on the right hand side of (70c) should now be understood as involving the Hermitian inner product of Eq , and (70a) as involving the operator norm of K(dq × dq )). Then, by virtue of Theorem 1, the jump rates given by (1) are well-defined and finite P-almost everywhere, and they are equivariant if ∈ domain(H ). The proof of Corollary 2 applies here without changes. Equation (19) remains valid if suitably interpreted. Corollary 3 defines a set of good ’s, for which the jump rates are finite, for the examples of Sects. 3.2, 3.3, 3.10, 3.12, and for (39) in case the sum over i is always finite. 4.2.4. POVMs We now proceed to the fully general case of an arbitrary POVM. First, we provide two important mathematical tools for dealing with POVMs. • Any POVM corresponds to a PVM on a larger Hilbert space, according to the following theorem of Naimark [7, p. 142]: If P is a POVM on the standard Borel space Q acting on the Hilbert space H , then there is a Hilbert space Hext ⊇ H and a PVM Pext on Q acting on Hext such that P ( · ) = P+ Pext ( · )I with P+ : Hext → H the projection and I : H → Hext the inclusion, and Hext is the closed linear hull of , P is {Pext (B)H : B ⊆ Q}. The pair Hext , Pext is unique in the sense that if Hext ext fixing another such pair then there is a unitary isomorphism between Hext and Hext . H and carrying Pext to Pext We call Hext and Pext the Naimark extension of H and P . We recall that for the Hilbert space of positive energy solutions of the Dirac equation and the corresponding POVM introduced earlier, the Naimark extension is given by L2 (R3 , C4 ) and its natural PVM; this example indicates that the Naimark extension may be, in practice, something natural to consider. • In Corollaries 2 and 3, we were considering, for H and P , L2 spaces with their natural PVMs. But when we are given an arbitrary PVM on a Hilbert space, the situation is not genuinely more general, since it can be viewed as the natural PVM of an L2 space. We call this the naturalization of the PVM. It is based on the following version of the spectral theorem (which can be obtained from the representation theory of abelian operator algebras, see, e.g., [9]): If P is a PVM on the standard Borel space Q acting on the Hilbert space H , then there is a measurable field of Hilbert spaces8 Hq over Q, a σ -finite nonnegative measure dq on Q, and a unitary ⊕ Hq dq to the direct integral9 of Hq that carries P to the isomorphism U : H → ⊕ Hq dq. The naturalization is unique in the sense natural PVM on Q acting on that if {Hq }, (dq) , U is another such triple, then there is a measurable function 8 A measurable field of Hilbert spaces on Q is a family of Hilbert spaces H with scalar products q · | · q , endowed with a measurable structure that can be defined by specifying a family of cross-sections i (q) such that for all i, i the functions q → i (q)| i (q)q are measurable and for every q the family i (q) is total in Hq [18]. 9 This is the Hilbert space of square-integrable measurable cross-sections of the field {H }, i.e., crossq sections (q) such that all functions q → i (q)| (q)q are measurable and (q)| (q)q dq < ∞ [18].
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f : Q → (0, ∞) such that (dq) = f (q) dq and a measurable field of unitary isomorphisms Uq : Hq → Hq such that U (q) = f (q)−1/2 Uq U (q). A naturalized PVM is similar to a vector bundle in that with every q ∈ Q there is associated a value space Hq , which however may be infinite-dimensional, and ∈ H can be understood as a function on Q such that (q) ∈ Hq . Of course, instead of the differentiable structure of a vector bundle the naturalization of a PVM leads merely to a measurable structure. Thus, the situation with a general POVM is not much different from the situation with a vector bundle, as treated in Corollary 3. For Hilbert–Schmidt operators, the kernel is so well-behaved that no further conditions on are necessary: Corollary 4. Let H be a Hilbert space, ∈ H with = 1, H a self-adjoint operator on H , Q a standard Borel space, and let P be a POVM on Q acting on H . Suppose that H is a Hilbert–Schmidt operator. Then, by virtue of Theorem 1, the jump rates given by (1) are well-defined and finite P-almost everywhere, and they are equivariant. Proof. Let Pext be the Naimark extension PVM of P acting on Hext ⊇ H with P+ the ⊕ Hq dq be a naturalization of Pext . For projection Hext → H , and let U : Hext → every q ∈ Q, pick an orthonormal basis Iq = {|q, i} of Hq , with measurable dependence on q. When each set Iq is thought of as equipped with the counting measure, then ⊕ Hq dq is naturally identified from dq we obtain a measure on I = q Iq , and with L2 (I , C). Since H is a Hilbert–Schmidt operator, so is Hext = I H P+ , which thus possesses a kernel function K ∈ L2 (I × I , C) such that for all ∈ Hext , U Hext (q, i) =
dq
K(q, i, q , i ) U (q , i ).
i ∈Iq
Q
Since |P (B)H P (C)| = |Pext (B)Hext Pext (C)| , we have, for the same reasons as in the proof of Corollary 1, that |P (B)H P (C)| = dq dq U ∗ (q, i) K(q, i, q , i ) U (q , i ) . B×C
i∈Iq i ∈Iq
(77) For A ⊆ Q × Q, set µ(A) = dq dq U ∗ (q, i) K(q, i, q , i ) U (q , i ). A
i∈Iq i ∈Iq
Since U ∗ (q, i) K(q, i, q , i ) U (q , i ) is absolutely summable and integrable over q, i, q , and i , µ(A) is finite, and thus a complex measure. Equation (77) entails that Equation (64) is satisfied. Thus Theorem 1 applies.
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We now provide the most general version of our statement about jump rates for Hamiltonians with kernel measures. Let B (Hq , Hq ) denote the space of bounded linear operators Hq → Hq with the operator norm |O| =
O . ∈Hq , =0 sup
1/2
For the norm of (q) in Hq , (q)|(q)q , we also write |(q)|. Corollary 5. Let H be a Hilbert space, ∈ H with = 1, H a self-adjoint operator on H , Q a standard Borel space, and P a POVM on Q acting on H . Let Pext be ⊕ Hq dq the the Naimark extension PVM of P acting on Hext ⊇ H , and U : Hext → naturalization of Pext . Suppose that H has a kernel K(dq × dq ) for in the position representation defined by P ; i.e., suppose that K(dq × dq ) is the product of a σ -finite nonnegative measure on Q×Q and a measurable cross-section of the field B (Hq , Hq ) over Q × Q, that satisfies (70b), and that some everywhere-defined version (q) of ⊕ Hq dq satisfies (70a) and (70c) the almost-everywhere-defined cross-section U ∈ (where the integrand on the right hand side of (70c) is understood as involving the inner product of Hq ). Then, by virtue of Theorem 1, the jump rates given by (1) are well-defined and finite P-almost everywhere, and they are equivariant if ∈ domain(H ). The proof of Corollary 2 applies here without changes if one understands ∗ (q) K(dq × dq ) (q) as meaning (q)|K(dq × dq ) (q )q . Corollary 5 defines a set of good ’s, for which the jump rates are finite, for the examples of Sects. 3.7, 3.8, and 3.9. 4.3. Global existence question. The rates σt and velocities vt , together with Pt , define the process Qt associated with H, P , and , which can be constructed along the lines of Sect. 2.3. However, the rigorous existence of this process, like the global existence of solutions for an ordinary differential equation, is no trivial matter. In order to establish the global existence of the process (see [15] for an example), a variety of aspects must be controlled, including the following: (i) One has to show that for a sufficiently large set of initial state vectors, the relevant conditions for finiteness of the jump rates, see Sects. 4.1 and 4.2, are satisfied at all times. (ii) One has to show that there is probability zero that infinitely many jumps accumulate in finite time. (iii) One has to show that there is probability zero that the process runs into a configuration where σ is ill defined (e.g., where the denominator of (19) vanishes, if that equation is appropriate).
4.4. Extensions of bi-measures. We have pointed out in the next-to-last paragraph of Sect. 4.1 that a complex bi-measure need not possess an extension to a complex measure on the product space, a fact relevant to the conditions for finite rates. In this section we show, see Theorem 2 below, that nonnegative real bi-measures always possess such an extension. A useful corollary of Theorem 2, see Corollary 7 below, asserts that one can form the tensor product of any two POVMs. This is a special case of the more general statement, see Corollary 6 below, asserting that one can form the product of any two POVMs that commute with each other; this statement can be regarded as the generalization from
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PVMs to POVMs of the fact that two commuting observables can be measured simultaneously; it is also related to the discussion in the last paragraph of Sect. 4.1. Though we could not find the explicit statement of Corollary 6 in this form in the literature, it does follow from a part of a proof given by Halmos [20, p. 72]. Below, however, we give a somewhat different proof, using Theorem 2 instead of the lemma of von Neumann [33, p. 167] that Halmos uses. It is also presumably possible to derive Corollary 6 from Lemma 2.1 or Theorem 2.2 of [7]. Theorem 2. Let Q1 and Q2 be standard Borel spaces with σ -algebras A1 and A2 , and let ν( · , · ) be a finite nonnegative bi-measure, i.e., a mapping ν : A1 × A2 → [0, a], a > 0, that is a measure in each variable when the other variable is a fixed set. Then ν can be extended to a measure µ on Q1 × Q2 : there exists a unique finite nonnegative measure µ : A1 ⊗ A2 → [0, a] such that for all B1 ∈ A1 and B2 ∈ A2 , µ(B1 × B2 ) = ν(B1 , B2 ).
(78)
Proof. Suppose first that Q1 is finite or countably infinite. Then every set A ∈ A1 ⊗ A2 is an (at most) countable union of product sets,
A=
{q1 } × Bq1 ,
q1 ∈Q1
where every Bq1 ∈ A2 . Therefore, the unique way of extending ν is by setting µ(A) :=
ν({q1 }, Bq1 ).
(79)
q1 ∈Q1
One easily checks that (79) indeed defines a finite measure satisfying (78), noting first that the sum is always finite because ν({q1 }, Bq1 ) ≤ ν({q1 }, Q2 ) = ν(Q1 , Q2 ). The same argument can of course be applied if Q2 is finite or countably infinite. Suppose now that neither Q1 nor Q2 is finite or countable. Every uncountable standard Borel space Q is isomorphic, as a measurable space, to the space of binary sequences {0, 1}N (equipped with the σ -algebra generated by the family B of sets that depend on only finitely many terms of the sequence), i.e., there exists a bijection ϕ : Q → {0, 1}N that is measurable in both directions, see [24, p. 138] and [23, p. 358]. We may thus assume, without loss of generality, that Q1 = {0, 1}{−1,−2,−3,... } and Q2 = {0, 1}{0,1,2,... } , with Bi defined accordingly. Q1 × Q2 can then be canonically identified with {0, 1}Z . From the restriction of ν to sets B1 ∈ B1 and B2 ∈ B2 , one easily obtains a consistent family of finite-dimensional distributions, and hence, by the Kolmogorov extension theorem, e.g. [6, p. 24], a unique measure µ on Q1 × Q2 obeying (78) for all B1 ∈ B1 and B2 ∈ B2 . It remains to establish (78) for all B1 ∈ A1 and B2 ∈ A2 . First fix B2 in B2 . Then µ( · × B2 ) and ν( · , B2 ) are measures on A1 that agree on B1 . Hence they agree on A1 . Thus, fixing B1 in A1 , we have that µ(B1 × · ) and ν(B1 , · ) are measures on A2 that agree on B2 , and hence on all of A2 , completing the proof. In the following, we will again write B1 ⊆ Q1 instead of B1 ∈ A1 .
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Corollary 6. Let H be a Hilbert space, Q1 and Q2 standard Borel spaces, and P1 and P2 POVMs on Q1 and Q2 respectively, acting on H . If [P1 (B1 ), P2 (B2 )] = 0 for all B1 ⊆ Q1 and B2 ⊆ Q2 , then there exists a unique POVM P on Q1 × Q2 acting on H such that for all B1 ⊆ Q1 and B2 ⊆ Q2 , P (B1 × B2 ) = P1 (B1 )P2 (B2 ).
(80)
Proof. (We largely follow [20, p. 72].) For ∈ H we define a bi-measure ν by setting ν (B1 , B2 ) := |P1 (B1 )P2 (B2 )|. ν is obviously a complex bi-measure, and it takes values only in the nonnegative reals because P1 (B1 )P2 (B2 ) is a positive operator (since the two positive operators P1 (B1 ) and P2 (B2 ) can be simultaneously diagonalized). The values of ν are bounded by 2 . By Theorem 2, ν can be extended to a measure µ on Q1 × Q2 . We now define complex measures µ , on Q1 × Q2 by “polarization”: for every A ⊆ Q1 × Q2 and for every pair of vectors , we write µ , (A) := µ 1 + 1 (A) − µ 1 − 1 (A) + iµ 1 − i (A) − iµ 1 + i (A). 2
2
2
2
2
2
2
2
(81)
We assert that µ , (A) is, for each fixed set A, a symmetric bilinear functional. This assertion is proved by noting that (i) it is true if A = B1 × B2 , and (ii) the class of all sets for which it is true is closed under the formation of complements and countable ∞ unions. c) = µ To see (ii), note that µ (A (Q × Q ) − µ (A) and µ ( , , 1 2 , , k=1 Ak ) = limn→∞ nk=1 µ , (Ak ). Since µ, (A) = µ (A) ≤ 2 for every A ⊆ Q1 × Q2 , the bilinear functional µ , (A) is bounded and has, in fact, a norm ≤ 1. Therefore, there is a bounded operator P (A) such that µ , (A) = |P (A)|. P (A) is positive since µ, (A) ≥ 0 for every . P ( · ) is countably additive in the weak operator topology because µ , ( · ) is countably additive. P ( · ) satisfies (80), and thus P (Q1 × Q2 ) = I . Note that P1 need not be a commuting POVM, i.e., possibly [P1 (B1 ), P1 (C1 )] = 0, and correspondingly for P2 . An immediate consequence of Corollary 6, which we use in several places of [12], is Corollary 7. Let H1 and H2 be Hilbert spaces, Q1 and Q2 standard Borel spaces, and P1 and P2 POVMs on Q1 and Q2 respectively, acting on H1 and H2 respectively. Then there exists a unique POVM P on Q1 ×Q2 acting on H1 ⊗H2 such that for all B1 ⊆ Q1 and B2 ⊆ Q2 , P (B1 × B2 ) = P1 (B1 ) ⊗ P2 (B2 ).
(82)
5. Minimality In this section we explain in what sense the minimal jump rates (1)—or (19) or (38a)— are minimal. In so doing, we will also explain the significance of the quantity J defined in (25), and clarify the meaning of the steps taken in Sects. 2.4 and 2.5 to arrive at the jump rate formulas. Given a Markov process Qt on Q, we define the net probability current jt at time t between sets B and B by 1 jt (B, B ) = lim Prob Qt ∈ B , Qt+t ∈ B − (83) t 0 t −Prob Qt ∈ B, Qt+t ∈ B .
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This is the amount of probability that flows, per unit time, from B to B minus the amount from B to B . For a pure jump process, we have that σt (B|q ) ρt (dq ) − σt (B |q) ρt (dq) , (84) jt (B, B ) = q ∈B
q∈B
so that jt (B, B ) = jσ,ρ (B × B ),
(85)
where jσ,ρ is the signed measure, on Q × Q, given by the integrand of (16), jσ,ρ (dq × dq ) = σ (dq|q ) ρ(dq ) − σ (dq |q) ρ(dq) .
(86)
For minimal jump rates σ , defined by (1) or (19) or (38a) (and with the probabilities ρ given by (21), ρ = P), this agrees with (25), as was noted earlier, jσ,ρ = J,H,P ,
(87)
where we have made explicit the fact that J is defined in terms of the quantum entities , H , and P . Note that both J and the net current j are anti-symmetric, Jtr = −J and j tr = −j , the latter by construction and the former because H is Hermitian. (Here tr indicates the action on measures of the transposition (q, q ) → (q , q) on Q × Q.) The property (87) is stronger than the equivariance of the rates σ , Lσ Pt = dPt /dt: Since, by (16), (Lσ ρ)(dq) = jσ,ρ (dq × Q),
(88)
and, by (25), dP (dq) = J(dq × Q), (89) dt the equivariance of the jump rates σ amounts to the condition that the marginals of both sides of (87) agree, jσ,ρ (dq × Q) = J(dq × Q) .
(90)
In other words, what is special about processes with rates satisfying (87) is that not only the single-time distribution but also the current is given by a standard quantum theoretical expression in terms of H, , and P . That is why we call (87) the standard-current property—defining standard-current rates and standard-current processes. Though the standard-current property is stronger than equivariance, it alone does not determine the jump rates, as already remarked in [2, 29]. This can perhaps be best appreciated as follows: Note that (86) expresses jσ,ρ as twice the anti-symmetric part of the (nonnegative) measure C(dq × dq ) = σ (dq|q ) ρ(dq )
(91)
dq )
is absolutely continuous with respect to on Q × Q whose right marginal C(Q × ρ. Conversely, from any such measure C the jump rates σ can be recovered by forming the Radon–Nikod´ym derivative σ (dq|q ) =
C(dq × dq ) . ρ(dq )
Thus, given ρ, specifying σ is equivalent to specifying such a measure C.
(92)
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In terms of C, the standard-current property becomes (with ρ = P) 2 Anti C = J.
(93)
Since (recalling that J = J+ − J− is anti-symmetric) J = 2 Anti J+ ,
(94)
an obvious solution to (93) is C = J+ , corresponding to the minimal jump rates. However, (87) fixes only the anti-symmetric part of C. The general solution to (93) is of the form C = J+ + S,
(95)
where S(dq × dq ) is symmetric, since any two solutions to (93) have the same antisymmetric part, and S ≥ 0, since S = C ∧ C tr , because J+ ∧ (J+ )tr = 0. In particular, for any standard-current rates, we have that C ≥ J+ ,
or σ (dq|q ) ≥
J+ (dq × dq ) . P(dq )
(96)
Thus, among all jump rates consistent with the standard-current property, one choice, distinguished by equality in (96), has the least frequent jumps, or the smallest amount of stochasticity: the minimal rates (1). 6. Remarks 6.1. Symmetries. Quantum theories, and in particular QFTs, often have important symmetries. To name a few examples: space translations, rotations and inversion, time translations and reversal, Galilean or Lorentz boosts, global change of phase → eiθ , and gauge transformations. This gives rise to the question whether the process Qt of the corresponding Bell-type QFT respects these symmetries as well. Except for Lorentz invariance, which is difficult in that Lorentz boosts fail to map equal-time configurations into equal-time configurations, the answer is yes; a discussion is given in [12, Sec. 6.1]. An essential ingredient of this result is the manifest fact that the minimal jump rates (1) inherit the symmetries of the Hamiltonian (under which the POVM transforms covariantly). 6.2. Homogeneity of the rates. The minimal jump rates (1) define a homogeneous function of degree 0 in , i.e., σ λ = σ for every λ ∈ C \ {0}. This property is noteworthy since it forms the essential mathematical basis for a number of desirable properties of theories using such jump rates (such as that of [11]): (i) that (when P is a product PVM) unentangled and decoupled subsystems behave independently and follow the same laws as the entire system, (ii) that “collapsed-away,” i.e., sufficiently distant, parts of the wave function do not influence the future behaviour of the configuration Qt , (iii) invariance under a global change of phase → eiθ , (iv) invariance under the replacement → e−iEt/ for some constant E, which corresponds to adding E to the total Hamiltonian, (v) invariance under relabeling of the particles (which may cause a replacement → − due to the Pauli principle).
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6.3. H +E. Adding a constant E to the interaction Hamiltonian will not change the jump rates (1) provided P is a PVM. This is because |P (B)EP (C)| = E |P (B ∩ C)| has vanishing imaginary part. For a POVM, however, this need not be true.
6.4. Nondegenerate eigenstates. As mentioned earlier, after (19), it is a consequence of the minimal jump rate formula (1), in fact of the very minimality, that at each time t either σ (q|q ) or σ (q |q) is zero. It follows that for a time-reversible Hamiltonian H and POVM P , all jump rates vanish if is a nondegenerate eigenstate of H . This is because, in the simplest cases, q|H |q is real, and the coefficients q| can also be chosen real, or, more generally and more to the point, because in this case the process must coincide with its time reverse, which implies that the current from q to q is as large as the one from q to q, so that minimality requires both to vanish.
6.5. Left or right continuity. From what we have said so far, there remains an ambiguity as to whether Qt at the jump times should be the point of departure or the destination, in other words, whether the realization t → Qt should be chosen to be left or to be right continuous. Although we think there is not much physical content to this question, we should point out that demanding either left or right continuity will destroy time-reversal invariance (cf. Sect. 6.1). A prescription that preserves time-reversal invariance can, however, be devised provided the possible jumps can be divided into two classes, A and B, in such a way that the time reverse of a class-A jump necessarily belongs to class B and vice versa. Then class-A jumps can be chosen left continuous and class-B as right continuous. An example is provided by the model of [11]: since at every jump the number of particles either increases or decreases, the jumps naturally form two classes (“creation” and “annihilation”), and the time reverse of a creation is an annihilation. The prescription could be that if a particle is created (annihilated) at time t, then Qt already (still) contains the additional particle. But the opposite rule would be just as consistent with time-reversal symmetry, and we can see no compelling reason to prefer one rule over the other. 7. Conclusions We have investigated the possibility of understanding QFT as a theory about moving particles, an idea pioneered, in the realm of nonrelativistic quantum mechanics, by de Broglie and Bohm. The models proposed by Bell [3] and ourselves [11] turn out to be rather universal; that is, their construction can be transferred to a variety of situations, involving different Hamiltonians and configuration spaces, and invoking formulas of a canonical character. One ingredient of the construction is the use of stochastic jumps whose rates are determined by the quantum state vector (and the Hamiltonian). These rates can be specified through an explicit formula (1) that has a status similar to the velocity formula in Bohmian mechanics. We have provided a version of this jump rate formula that is more general than any previous one. Indeed, it seems to be the most general version possible: we need assume merely that the configuration space Q is a measurable space (the weakest notion of “space” available in mathematics), that the Hamiltonian is well-defined, and that Q and the Hilbert space are related through a generalized position observable (a positive-operator-valued measure, or POVM, the most general notion available in
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quantum theory of how a vector in Hilbert space may define a probability distribution). We have shown that these jump rates are well-defined and finite if the interaction Hamiltonian possesses a sufficiently regular kernel in the position representation defined by the POVM. We have also indicated that in a Bell-type QFT, the different contributions to the Hamiltonian correspond to different contributions to the motion of the configuration Qt . The relevant fact is process additivity, i.e., that the generator of the Markov process Qt is additive in the Hamiltonian. The free process usually consists of continuous trajectories, Bohmian or similar, an observation already made in [11] for the model considered there. Exploiting process additivity, we obtain that Qt is piecewise deterministic, the pieces being Bohm-type trajectories, interrupted by stochastic jumps. Given a Hamiltonian and POVM, our prescription determines the Markov process Qt . As an example, we have described the process explicitly for a simple QFT. The essential point of this paper is that there is a direct and natural way—a canonical way—of devising a Bell-type version of any QFT. Acknowledgement. We thank Ovidiu Costin, Avraham Soffer, and James Taylor of Rutgers University, Stefan Teufel of Technische Universit¨at M¨unchen, and Gianni Cassinelli and Alessandro Toigo of Universit`a di Genova for helpful discussions. R.T. gratefully acknowledges support by the German National Science Foundation (DFG). N.Z. gratefully acknowledges support by INFN and DFG. Finally, we appreciate the hospitality that some of us have enjoyed, on more than one occasion, at the Mathematisches Institut of Ludwig-Maximilians-Universit¨at M¨unchen, at the Dipartimento di Fisica of Universit`a di Genova, and at the Mathematics Department of Rutgers University.
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16. Goldstein, S.: Bohmian Mechanics (2001). In: Stanford Encyclopedia of Philosophy (Winter 2002 Edition), E.N. Zalta (ed.), http://plato.stanford.edu/archives/win2002/entries/qm-bohm/ 17. Guerra, F., Marra, R.: Discrete stochastic variational principles and quantum mechanics. Phys. Rev. D 29, 1647–1655 (1984) 18. Guichardet, A.: Special topics in topological algebras. NewYork, London, Paris: Gordon and Breach Science Publishers 1968 19. Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Berlin: Springer-Verlag, 1992 20. Halmos, P.R.: Introduction to Hilbert Space and the Theory of Multiplicity. Second Edition. New York: Chelsea, 1957 21. Halmos, P.R.: Measure Theory. Princeton: Van Nostrand, 1965 22. Kraus, K.: Position Observables of the Photon. In: W.C. Price, S.S. Chissick (eds.), The Uncertainty Principle and Foundations of Quantum Mechanics. New York: Wiley, 1977, pp. 293–320 23. Kuratowski, C.: Topologie Vol. I. Warsaw: Panstwowe Wydawnictwo Naukowe, 1948 24. Mackey, G.W.: Borel Structure in Groups and Their Duals. Trans. Amer. Math. Soc. 85, 134–165 (1957) 25. Nelson, E.: Quantum Fluctuations. Princeton: Princeton University Press, 1985 26. Newton, T.D., Wigner, E.P.: Localized States for Elementary Systems. Rev. Mod. Phys. 21, 400–406 (1949) 27. Parthasarathy, K.R.: Probability Measures on Metric Spaces. New York and London: Academic Press, 1967 28. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis. New York and London: Academic Press, 1972 29. Roy, S.M., Singh, V.: Generalized beable quantum field theory. Phys. Lett. B 234, 117–120 (1990) 30. Schweber, S.S.: An Introduction to Relativistic Quantum Field Theory. New York: Harper and Row, 1961 31. Sudbery, A.: Objective interpretations of quantum mechanics and the possibility of a deterministic limit. J. Phys. A: Math. Gen. 20, 1743–1750 (1987) 32. Vink, J.C.: Quantum mechanics in terms of discrete beables. Phys. Rev. A 48, 1808–1818 (1993) 33. von Neumann, J.: Functional Operators, Vol. I: Measures and Integrals. Princeton: Princeton University Press, (1950) Communicated by A. Connes
Commun. Math. Phys. 254, 167–178 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1248-7
Communications in
Mathematical Physics
Non-Abelianizable First Class Constraints Farhang Loran Department of Physics, Isfahan University of Technology (IUT), Isfahan, Iran. E-mail:
[email protected] Received: 12 January 2004 / Accepted: 20 July 2004 Published online: 2 December 2004 – © Springer-Verlag 2004
Abstract: We study the necessary and sufficient conditions on Abelianizable first class constraints. The necessary condition is derived from topological considerations on the structure of the gauge group. The sufficient condition is obtained by applying the implicit function theorem in calculus and studying the local structure of gauge orbits. Since the sufficient condition is necessary for the existence of proper gauge fixing conditions, we conclude that in the case of a finite set of non-Abelianizable first class constraints, the Faddeev-Popov determinant is vanishing for any choice of subsidiary constraints. This result is explicitly examined for the SO(3) gauge invariant model. 1. Introduction A gauge theory, in general, possesses a set of first class constraints, φi , i = 1, · · · , N , satisfying the algebra {φi , φj } =
N
Uij k φk ,
i, j = 1, · · · , N,
(1)
k=1
in which the structure functions Uij k are generally some functions of phase space coordinates. One of the most interesting questions in constraint systems is the possibility of converting a given set of first class constraints to an equivalent Abelian set. By definition, Abelian constraints commute with each other, i.e. their Poisson brackets are vanishing identically with each other. There are various motivations for examining such a possibility. For example, first class constraints are generators of the gauge transformation: φ δi F (z) = {F (z), φi } [1]. Since Uij k are functions of phase space coordinates, the full generator of the gauge transformation is a nontrivial combination of first class constraints [2]. This combination is the simplest if the first class constraints are Abelian, i.e. when the Poisson brackets of these constraints vanish identically with each other.
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Abelianization of first class constraints can also result in two more important simplifications. First, following Dirac’s arguments, quantization of a set of first class constraints satisfying the algebra (1), where Uij k ’s are not c-numbers, requires a definite operator ordering [1]. That is because in Dirac quantization, physical states are defined as null eigenstates of the operators φˆ i ’s, φˆ i |phys = 0,
(2)
in which the operator φˆ i ’s are defined corresponding to the constraints φi ’s. Definition (2) and the algebra (1) are consistent if the operators Uˆ ij k , defined corresponding to the structure functions Uij k , sit on the left of the operators φˆ i similar to Eq.(1). The existence of such an operator ordering is not evident generally. Apparently, when first class constraints are Abelian, no such operator ordering should be considered. Second, in BRST formalism, the algebra (1), in general, leads to a complicated expansion of the BRST charge in terms of the ghosts. When first class constraints are Abelian, the generator of the BRST transformation can be recognized in the most simple way [3]. Different methods for Abelianization of first class constraints are studied. Examples are, Abelianization via constraint resolution [3–5] or via generalized canonical transformation for general non-Abelian constraints (that satisfy a closed algebra) [6]. In reference [7] the authors study Abelianization via Dirac’s transformation. In this method, one assumes that linear combinations of non-Abelian first class constraints (satisfying a closed algebra) exist that converts the given set of non-Abelian constraints to an equivalent set of Abelian constraints. In this way the problem of Abelianization has led to that of solving a certain system of first order linear differential equations for the coefficients of these linear combinations. In [8], it is shown that mapping each first class constraint to the surface of the other constraints results in Abelian first class constraints. In [9] it is shown that the maximal Abelian subset of second class constraints can be obtained in the same way. The domain of validity and/or applicability of the above methods can be determined by studying the necessary conditions on Abelianizable first class constraints. Topology of the gauge group at each point p of the phase space, which is uniquely determined by the structure coefficients Uij k (p), provides the necessary tools for this purpose. In fact, if at some point p, a non-Abelian set of first class constraints can be made Abelian, the corresponding gauge group should be topologically equivalent to the Abelian gauge groups, i.e. the group of Euclidean translations. In [4] a method for Abelianization of first class constraints is proposed, which is based on the theorem of implicit differentiation (or the implicit function theorem) and gives a sufficient condition on Abelianizable first class constraints. According to that theorem, if at some point p, dφ is maximal (see Sect. 3), one can in principle, solve the equations φi (z1 , · · · , zN ; za ) = 0, i = 1, · · · , N as zi = zi (za ), i = 1, · · · , N . It is shown in [4] that the constraints ψi = zi − zi (za ), are Abelian. Therefore, maximality is the sufficient condition for constraints to be Abelianizable. Using these results, we conclude that Uij k (p)’s determine whether the maximality condition is satisfied at p or not. Violation of maximality causes serious problems. For example, the norm of the constraint surface is not well defined in the neighborhood of maximality-violated regions. Furthermore, the necessary condition on subsidiary constraints (gauge fixing conditions) ωi , i.e. det({φi , ωj })p = 0, can not be satisfied if maximality is violated at p. This means that the Faddeev-Popov determinant is vanishing regardless of our choice of subsidiary
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constraints. The equivalence of the Lagrangian and Hamiltonian formalism for constraint systems is also proven under the assumption that primary constraints satisfy the maximality condition over all the phase space [10]. Of course it should be noted that here we study only constraint systems with a finite set of first class constraints and it is not straightforward to generalize the results of this paper to systems with an infinite set of constraints, e.g. the SU(N) Yang-Mills theory (See Appendix 2). The organization of paper is as follows. In Sect. 2, we study the topological conditions on Abelianizable first class constraints. In Sect. 3, we discuss the maximality condition as the sufficient condition in two instructive ways: in Subsect. 3.1 by reviewing the Abelianization method of ref.[4] and in Subsect. 3.2 by studying the local structure of gauge orbits. In Sect. 4, we examine our results explicitly by considering two simple examples. Section 5 is devoted to the summary and conclusion. There are also two short appendices. Appendix 1 is a review of the method of obtaining orthogonal bases of a given vector space. The second appendix is a review of the constraint algebra of the SU (N ) Yang-Mills model. 2. Topological Considerations, A Necessary Condition Assume a finite set of irreducible first class constraints φi , i = 1, · · · , N . By definition, {φi , φj } = Uij k φk ,
(3)
where Uij k (zµ ) are some functions of phase space coordinates zµ . {φi , φj } stands for the Poisson bracket of φi and φj defined as follows: {φi , φj } =
∂φi µν ∂φj J , ∂zµ ∂zν
(4)
where J µν = {zµ , zν } is a full rank antisymmetric tensor, e.g. the symplectic two form, 01 J = . (5) 10 φ
The gauge transformation of any function of phase space, F (z), is given by δi F (zµ ) = {F, φi }| , where is the constraint surface corresponding to the constraints φi = 0, i = 1, · · · , N [1]. Using Eq. (3), and the Jaccobi identity for Poisson brackets, one can show that for an arbitrary analytic function of phase space coordinates F (z), φ φ [δi , δj ]F (z) = {F (z), φj }, φi | − {F (z), φi }, φj | = {φi , φj }, F (z) | = Uij k φk , F (z) | = −Uij k {F (z), φk }| φ
= −Uij k δk F (z).
(6)
φ
Consequently, δi are elements of a Lie algebra with structure coefficients −Uij k : φ
φ
φ
[δi (p), δj (p)] = −Uij k (p)δk (p),
p ∈ .
(7)
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The corresponding Lie group is called the gauge group and the gauge orbits are the φ integral curves of δi (p)’s. The main concept in our study is the concept of equivalence of two sets of constraints. Two sets of constraints are said to be equivalent at some point p of the phase space if 1) the corresponding constraint surfaces are similar at some neighborhood of p and 2) the resulting gauge transformations are equivalent [4, 11]. These conditions are fulfilled using the following definition of equivalence: Definiton. The set of constraints ψi , i = 1, · · · , N are equivalent to φi ’s, i = 1, · · · , N ψ φ at p if 1) p ∈ and p ∈ , 2) Tp is homeomorphic to Tp and 3) Gp = Gp . () is the constraint surface corresponding to the constraints φi ’s(ψi ’s) and Tp ψ φ (Tp ) is the tangent space of () at p. Gp and Gp are the gauge groups generated ψ φ by δi (p)’s and δi (p)’s respectively. Two topological spaces are said to be homeoφ morphic if there exists an invertible continuous map between them [12]. Since δi (p)’s ψ (δi (p)) expand some subspace of the tangent space Tp (Tp ) (see Subsect. 3.2) and ψ φ the equivalence of two Lie groups Gp and Gp requires that these subspaces be homeo1 morphic, the second and third conditions in the definition given above are consistent with each other. If two topological spaces are homeomorphic, their topological invariants should be the same. It is important to note that this is a necessary and not a sufficient condition. Examples of topological invariants are connectedness and compactness [12]. Abelianization of a set of non-Abelian first class constraints amounts to obtaining an equivalent set of Abelian constraints. The gauge group of Abelian constraints is homeomorphic to the group of Euclidean translations, i.e. RN , where N is the number of first class constraints.2 These are simply-connected and non-compact spaces. Consequently, the necessary (not sufficient) condition on constraints φi to be Abelianizable at some point p is that the corresponding gauge group determined by Uij k (p) should be simplyconnected and non-compact. For example, if Uij k (p) are the structure coefficients of some compact group, e.g. SO(N ), then the corresponding constraints can not be made Abelian at p. We call such sets of first class constraints, non-Abelianizable constraints. As an example consider the SO(3) gauge invariant model [13] where first class constraints are Li = ij k xj pk , i = 1, 2, 3 satisfying the algebra {Li , Lj } = ij k Lk , in which ij k is the Levi-Civita tensor. Consequently the gauge group is compact (homeomorphic to S 2 ) and Li ’s are non-Abelianizable. 3. Maximality, A Sufficient Condition In this section we obtain the sufficient condition, which we call the maximality condition, on constraints to be Abelianizable. Obviously the sufficient condition is not satisfied in the case of non-Abelianizable constraints. As we will show this means that the FaddeevPopov determinant in such systems is vanishing for any choice of subsidiary constraints. Equality, =, is a trivial homeomorphism given by the identity map. Some directions of RN can be compactified. A compactified direction corresponds to a U (1) gauge symmetry. The simplest example of such systems is the Friedberg model [14]. Although we do not consider such cases here, generalization of the final result to these systems is straightforward. 1 2
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In the following two subsections, we follow two different methods to study the maximality condition. The first one is based on the Abelianization via constraint resolution which is a well known method [4]. In the second method we study the local structure of gauge orbits. Although the following methods are basically equivalent, but they clarify different aspects of the maximality condition. 3.1. Resolution of constraints. Here we review the Abelianization method introduced in [4]. Assume a set of first class constraints φi , i = 1, · · · , N and the corresponding constraint surface . Consider a point p ∈ , where dφ is maximal. Maximality here means that there exist a subset of phase space coordinates, zi , i = 1, · · · , N , such that ∂φi = 0. (8) det ∂zj p In this case according to the theorem of implicit differentiation (or theorem of implicit function), one can in principle, solve equations φi (zi ; za ) = 0, i = 1, · · · , N to obtain zi = zi (za ), i = 1, · · · , N. One can show that the set of constraints ψi = zi − zi (za ) which by construction are equivalent to φi ’s, are Abelian. This can be verified noting that {ψi , ψj } = {zi , zj } − {zi , zj (za )} − {zi (za ), zj } + {zi (za ), zj (za )},
(9)
is independent of zi ’s because {zµ , zν } = 0, ±1 (see Eq. (5)). Since the left-hand side of Eq. (9) vanishes on the constraint surface (where zi = zi (za )), one concludes that it vanishes identically and consequently ψi ’s are Abelian [4]. Using the chain rule of partial differentiation, one can determine explicitly the gradient of constraints ψi in terms of the gradient of φi ’s, though ψi ’s are implicitly known. This has two consequences. Firstly, one can explicitly verify the equivalence of gauge transformations generated by ψi ’s and φi ’s. Secondly, it determines the homeomorphism mentioned in Sect. 2 between the tangent spaces (or gauge groups). The violation of the maximality condition has various geometrical consequences. For example, at any point p where maximality is not satisfied, the dimensionality and the norm of the constraint surface is not well-defined. Furthermore, the tangent space of the constraint surface at p is not homeomorphic to the tangent space at the regular points (where maximality is satisfied), though tangent spaces at regular points are all homeomorphic to each other. In maximality-violated regions, the condition det({φi , ωj }) = 0 on the subsidiary constraints (gauge fixing conditions) ωi , can not be satisfied for any choice of analytic functions ωi , because ∂φi µν ∂ωj , i, j = 1, · · · , N, (10) J det({φi , ωj }) = det ∂zµ ∂zν is vanishing if
∂φi rank ∂zµ
< N.
(11)
In other words, if a set of constraints do not satisfy the maximality condition, the Faddeev-Popov determinant (10) is vanishing for any choice of subsidiary constraints ωi .
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Although maximality is a sufficient condition for Abelianization of first class constraints, it is not a necessary condition. For example consider the constraints of the Friedberg model [14] φ1 = pz and φ2 = xpy − ypx . These constraints form a set of Abelian constraint though they do not satisfy the maximality condition at the origin. To emphasize the topological considerations discussed in Sect. 2, let us consider again the SO(3) gauge model. One can easily verify that the constraints Li ’s do not satisfy the maximality condition at any point of phase space. But, since maximality is not a necessary condition, using merely this result, one can not conclude that Li are not Abelianizable. 3.2. The space of gauge orbits. The method of resolution of constraints studied above considers only the equivalence of constraints in the sense that they should define the same constraint surface. But it does not verify the equivalence of the corresponding gauge groups explicitly. In the following, focusing on the local structure of gauge orbits, we obtain a complete description of the equivalence of constraints and the concept of Abelianizablity. Consider a point p on the constraint surface. The gauge transformation generated by the first class constraints φi , i = 1, · · · , N at p can be given as follows: µ
δi F = Xi (F ) = Xi where
µ Xi
=
∂φi νµ J ∂zν
∂ F, ∂zµ
(12)
.
(13)
p
The vectors Xi span a subspace of the tangent space Tp and determine the direction of gauge transformation on the constraint surface at p. It is obvious that the maximality condition, ∂φi rank = N, (14) ∂zµ p is the necessary condition to obtain exactly N independent vectors Xi . For further use, µ note that definition (13) implies that Xi J µν Xjν = {φi , φj }|p is vanishing identically. The gauge fixing conditions ωi can be defined as functions which gradients are proportional to Xi ’s, i.e. ∂ωi µ = fi Xi , ∂zµ
(no sum over i),
where fi is some function that can be determined by solving the condition, µ ∂ fi Xiν ∂ fi Xi = , ∂zν ∂zµ
(15)
(16)
which simply means that ∂ 2 ωi ∂ 2 ωi = ν µ. µ ν ∂z ∂z ∂z ∂z
(17)
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The gauge fixing conditions ωi defined by Eq. (15) satisfy the following relations: δi ωj = fi Xi .Xj ,
(18)
where Xi .Xj denotes the inner product of two vectors Xi and Xj given by the relation µ µ Xi .Xj = Xi Xj . µ The space of gauge orbits passing through p spanned by the vectors Xi , can be equivalently spanned by a set of orthonormal vectors X˜ i , (X˜ i .X˜ j = δij ). In Appendix 1, we review a well known method to obtain X˜ i ’s in terms of Xi ’s. Equations (13) and (15) can be used to define a new set of constraints φ˜ i and gauge fixing conditions ω˜ i in terms of X˜ i ’s. By construction, {φ˜ i , ω˜ j } = gi δij ,
(19)
where gi is some function of phase space. Since we are studying the system in an arbitrary small neighborhood of p, the function gi can be estimated as a constant that can be absorbed in ω˜ i . Thus Eq. (19) can be rewritten in a more interesting form {φ˜ i , ω˜ j } = δij . By construction, φ˜ i ’s are first class constraints equivalent to φi ’s.3 Now using the following theorem one can show that the set of constraints φ˜ i are Abelian. Theorem. Given a set of first class constraints φi , if there exists a set of gauge fixing conditions ωi such that {φi , ωj } = δij , then φi ’s are Abelian. Proof. Assume that the algebra of constraints is given by the relation {φi , φj } = Uij k φk . Consider the constraints φi and φj and one of gauge fixing conditions, say ωk . Using the Jaccobi identity, one can show that ωk , {φi , φj } = − φj , {ωk , φi } − φi , {φj , ωk } = {φj , δki } − {φi , δj k }, (20) which is vanishing identically. Therefore, 0 = ωk , {φi , φj }
= {ωk , Uij k φk }
= Uij k − Uij kk φk ,
(21)
where Uij kk = −{ωk , Uij k }. Consequently the algebra of constraints is given as follows:
{φi , φj } = Uij kk φk φk .
(22)
By repeating the above calculation, an arbitrary number of φi can appear on the right hand side of the above relation. To obtain a meaningful algebra, this chain of multiplying constraints should terminate somewhere which means that the right-hand side of the above equation is vanishing identically. In other words, φi ’s are Abelian. From Appendix 1 one verifies that X˜ i = Mij Xj , where M is some invertible matrix. Consequently one can show that firstly, X˜ i J X˜j is vanishing on the constraint surface which means that φ˜ i ’s are first class and secondly, the corresponding gauge groups are equivalent 3
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Summarizing our results, if at some point p on the surface of N first class constraints, the maximality condition is satisfied, then the space of gauge orbits at that point looks like RN (spanned by N orthogonal vectors X˜ i ). Furthermore there exists an equivalent set of Abelian constraints which can be recognized as generators of translation along orthogonal directions of RN . Consequently, if we are given a finite set of first class constraints, such that the corresponding space of gauge orbits looks like, say, a sphere, they satisfy the maximality condition nowhere on the constraint surface. Therefore for any choice of subsidiary constraints, the Faddeev-Popov determinant is vanishing. This result can also be verified noting that the Faddeev-Popov determinant is, in general, proportional to det(Xi .Xj )N×N which can be non-vanishing only if the maximality condition is satisfied and Xi ’s are exactly N independent vectors (see Eq. (14)). Finally let us briefly review the Abelianization method given in ref. [8]. There, using the Cauchy-Kowalevski theorem [15], it is concluded that for any analytic constraint φ, the partial differential equation {ω, φ} = 1 has at least one solution for ω that can be uniquely determined by the boundary conditions. ω is used to define a projection map to the constraint surface φ = 0. It is shown that after projecting the remaining constraints to the surface φ = 0, one finds a new set of constraints equivalent to the original set with an interesting property: the Poisson brackets of all mapped constraints with φ is vanishing identically. It is shown that using similar projection operators one can map all constraints to the surface of each other consistently, which results in an equivalent Abelian set of constraints. In addition it is shown that one can obtain a set of subsidiary constraints such that {φi , ωj } = δij .4 Obviously, the domain of validity of the this method is determined by the domain of validity of the Cauchy-Kowalevski theorem. As one anticipates, this theorem is valid as far as the maximality condition is satisfied. See ref. [15] for details. The result of this section can be used to prove the following theorem on second class constraints: Theorem. If a given set of second class constraints can be considered as the union of first class constraints and the corresponding gauge fixing conditions, then the subset of first class constraints is Abelianizable. To prove this theorem note that by definition a set of second class constraints ψI satisfies the relation det({ψI , ψJ })|ψ = 0. If ψI ’s are a combination of some first class constraints φi (namely {φi , φj }|ψ = 0) and gauge fixing conditions ωi , the definition of second class constraints gives, 0 A det = 0, (23) −A {ωi , ωj } where Aij = {φi , ωj }. Thus det A = 0 and the maximality condition is satisfied. For more details see ref. [9] and references therein. 4. Examples In this section we consider the simplest examples of Abelianizable and non-Abelianizable constraints. The non-Abelianizable constraints that we study here are the non-Abelian constraints of the SO(3) gauge model. We explicitly show that the Faddeev-Popov determinant is vanishing for any choice of gauge fixing conditions. 4 It is straightforward to show that obtaining the projected constraints is equivalent to calculating the orthogonal vectors X˜ i in terms of the original Xi ’s.
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1. Abelianizable Constraints. Consider a system given by the following constraints: φ1 = px ,
φ2 = py − ex py ,
(24)
which satisfy the algebra, {φ1 , φ2 } = ex φ1 . Obviously, this set of constraints is equivalent to the Abelian set φ˜ 1 = px and φ˜ 2 = py . It is interesting to obtain these Abelian constraints using the method of Subsect. 3.2. First note that x 1 −e 0 1 X1 = , X2 = , (25) 0 0 0 0 where, for example, X1 = J.∇φ1 |φ1 ,φ2 , in which ∂ ∂x
∂ ∂y ∇φ1 = ∂ ∂px
φ1 .
(26)
∂ ∂py
Using the method of Appendix 1, one obtains 1 0 0 1 ˜ ˜ X1 = , X2 = , 0 0 0 0
(27)
and consequently 0 0 ∇ φ˜ 1 = −J.X˜ 1 = , 1 0
0 0 ∇ φ˜ 2 = −J.X˜ 2 = , 0 1
(28)
which gives, φ˜ 1 = px and φ˜ 2 = py . 2. Non-Abelianizable Constraints. Consider the constraints of the SO(3) gauge model, Li = ij k xj pk , i = 1, 2, 3. Assume three arbitrary subsidiary constraints ωi . Since Li ’s are non-Abelianizable, the Faddeev-Popov determinant det({ωi , Lj }) is vanishing as can be verified as follows. Using the equality a11 a12 a13 (29) det a21 a22 a23 = ij k a1i a2j a3k , a31 a32 a33 one finds that det({ωi , Lj }) = ij k {ω1 , Li }{ω2 , Lj }{ω3 , Lk } 3 ∂ωc ∂ωc = − ij k ia1 b1 j a2 b2 ka3 b3 xbc + pbc . ∂xac ∂pac c=1
(30)
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Two generic terms in the above sum can be distinguished: ∂ω1 ∂ω2 ∂ω3 xa xb xc , P = ij k iaa j bb kcc ∂xa ∂xb ∂xc and
Q = ij k iaa j bb kcc
∂ω1 xa ∂xa
∂ω2 xb ∂xb
∂ω3 pc . ∂pc
To calculate P , one realizes three generic terms: ∂ω1 ∂ω2 ∂ω3 ij k i1a j 1b k1c (xa xb xc ), P1 = ∂x ∂x ∂x ∂ω1 ∂ω2 ∂ω3 P2 = ij k i1a j 1b k2c (xa xb xc ), ∂x ∂x ∂y ∂ω2 ∂ω3 ∂ω1 P3 = ij k i1a j 2b k3c (xa xb xc ). ∂x ∂y ∂z
(31)
(32)
(33)
P1 = 0 because here, (i, j, k) ∈ {2, 3} and consequently ij k is vanishing. P2 = 0 because i1a j 2b xa xb is symmetric under a ↔ b and consequently under i ↔ j , though ij k = − j ik . In addition, P3 = −yzx + zxy = 0. (The first term corresponds to a = 2 and the second term corresponds to a = 3 in Eq. (33).) Q is the sum of four generic terms: ∂ω1 ∂ω2 ∂ω3 ij k i1a j 1b k1c xa xb pc , ∂x ∂x ∂px ∂ω1 ∂ω2 ∂ω3 Q2 = ij k i1a j 1b k2c xa xb pc , ∂x ∂x ∂py ∂ω1 ∂ω2 ∂ω3 Q3 = ij k i1a j 2b k1c xa xb pc , ∂x ∂y ∂px ∂ω1 ∂ω2 ∂ω3 Q4 = ij k i1a j 2b k3c xa xb pc . ∂x ∂y ∂pz Q1 =
(34)
Q1 and Q2 are vanishing because under i ↔ j , i1a j 1b xa xb is symmetric but ij k is antisymmetric. Using the identity, ij k j 2b = −δi2 δkb + δib δk2 , one can show that Q3 and Q4 are some combinations of Li ’s. Therefore Q is also vanishing on the constraint surface. 5. Conclusion We found that first class constraints can be classified as Abelianizable and non-Abelianizable constraints. These classes are identified by topological invariants (e.g. compactness) of the corresponding gauge groups. The topology of a gauge group is uniquely determined by the structure coefficients of the gauge generators’ algebra which are simply the structure functions of the constraint algebra calculated at some particular point of phase space. Since maximality is the necessary condition on a given set of first class constraints for existence of a proper set of gauge fixing conditions, i.e. a set of subsidiary constraints such that the Faddeev-Popov determinant is non-vanishing, we concluded that
Non-Abelianizable First Class Constraints
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1. These constraints are Abelianizable if there exist a set of subsidiary constraints such that the Faddeev-Popov determinant is non-vanishing. 2. If these constraints are non-Abelianizable then the Faddeev-Popov determinant is vanishing for any choice of gauge fixing conditions. We studied the SO(3) gauge invariant model as an example. Using the first result mentioned above, we found that if a set of second class constraints is considered as the union of first class constraints and the corresponding gauge fixing conditions, then the subset of first class constraints is Abelianizable. Appendix 1 Here we briefly review the method of obtaining N orthogonal vectors u i in terms of a given set of N linearly independent vectors v i . This is a well known method that can be found in elementary textbooks in mathematics. Assuming that v i . vi = 1, the set of orthogonal vectors u i ’s, up to some normalization constants can be obtained as follows: u 1 = v 1 , u 2 = v 2 − ( u1 . v2 ) u1 , n u n+1 = vn − (ui . vn )ui ,
n = 2, · · · , N.
(35)
i=1
Appendix 2 In this appendix we study the algebra of SU (N ) Yang-Mills theory. It is well known that this model possesses an infinite set of non-Abelian constraints, φa ( x ) = ∂i ai − gfabc Abi ci ,
(36)
where fabc ’s are the structure coefficients of SU (N ) algebra and ai is the momentum x ), bj ( y )} = δ ab δij δ D ( x − y ). These field conjugate to the gauge field Aai , i.e. {Aai ( constraints satisfy the following algebra: {φa g1 , φb g2 } = gfabc φc g1 g2 ,
(37)
x ) and g2 ( x ) are some smooth functions and φa g = d 3 xg(x)φa (x) [3, in which g1 ( 4]. Assuming gi = e−ipi x , where the momentum space is a compactified lattice, which corresponds to lattice gauge theory on tori, we obtain a finite set of constraints satisfying the following closed algebra: p
{φam , φbn } = gfabc δm+n−p φc .
(38)
φam ’s are non-Abelianizable and consequently the corresponding Faddeev-Popov determinant is vanishing. It is important to examine any possible relation between this result and the appearance of Gribov copies in ordinary SU (N ) Yang-Mills theory. Acknowledgement. The financial support of Isfahan University of Technology (IUT) is acknowledged.
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References 1. Dirac, P.A.M.: Can. J. Math. 2, 129 (1950); Proc. R. Soc. London Ser. A 246, 326 (1958); Lectures on Quantum Mechanics. New York: Yeshiva University Press, 1964 2. Batlle, C., Gomis, J., Gracia, X., Pons, J.M.: J. Math. Phys. 30 (6), 1345 (1989); Pons, J.M., Garcia, J.A.: Int. J. Mod. Phys. A 15, 4681 (2000) 3. Henneaux, M.: Phys. Rep. 126, 1 (1985) 4. Henneaux, M., Teitelboim, C.: Quantization of Gauge System. Princeton, NJ: Princeton University Press, 1992 5. Goldberg, J., Newman, E.T., Rovelli, C.: J. Math. Phys. 32, 2739 (1991) 6. Bergman, P.G., Goldberg, I.: Phys. Rev. 98, 531 (1955); Bergman, P.G.: ibid. 98, 544 (1955) 7. Gogilidze, S.A., Khvedelidze, A.M., Pervushin, V.N.: J. Math. Phys. 37, 1760 (1996) 8. Loran, F.: Phys. Lett. B547, 63 (2002) 9. Loran, F.: Phys. Lett. B554, 207 (2003) 10. Battle, C., Gomis, J., Pons, J.M., Roman-Roy, N.: J. Math. Phys. 27 (12), 2953 (1986) 11. Govaerts, J.: Hamiltonian Quantisation and Constrained Dynamics. Leuven Notes in Theoretical and Mathematical Physics, Leuven: Leuven University Press, 1991 12. Munkres, J.R.: Topology, A First Course. Englewood Cliffs, NJ: Prentice-Hall, Inc, 1975 13. Govaerts, J., Klauder, J.R.: Ann. Phys. 274, 251 (1999) 14. Friedberg, R., Lee, T.D., Pang, Y., Ren, H.C.: Ann. Phys. 246, 381 (1996) 15. John, F.: Partial Differential Equations. Vol. 1, Fourth Edition, NewYork: Springer-Verlag NewYork Inc., 1981 Communicated by G.W. Gibbons
Commun. Math. Phys. 254, 179–189 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1197-1
Communications in
Mathematical Physics
Superpotentials for Gauge and Conformal Supergravity Backgrounds B.M. Gripaios Department of Physics – Theoretical Physics, University of Oxford, 1, Keble Road, Oxford, OX1 3NP, UK. E-mail:
[email protected] Received: 20 January 2004 / Accepted: 20 April 2004 Published online: 14 October 2004 – © Springer-Verlag 2004
Abstract: Effective superpotentials obtained by integrating out matter in super YangMills and conformal supergravity backgrounds in N = 1 SUSY theories are considered. The pure gauge and supergravity contributions (generalizing Veneziano-Yankielowicz terms) are derived by considering the case with matter fields in the fundamental representation of the gauge group. These contributions represent quantum corrections to the tree-level Yang-Mills and conformal supergravity actions. The classical equations of motion following from the conformal supergravity action require the background to be (super)conformally flat. This condition is unchanged by quantum corrections to the effective superpotential, irrespective of the matter content of the theory. 1. Introduction The effective action of a quantum field theory amounts to a solution of the field theory. Recently, following work by Dijkgraaf and Vafa [1], a step has been made in this direction for four-dimensional gauge theories with N = 1 supersymmetry. Specifically, if one considers the effective action defined by the path integral for an arbitrary gauge theory coupled to matter,1 then it has been shown how to compute the F -terms in the effective action (i. e. the effective superpotential) for a background gauge superfield Wα . This was shown perturbatively, by computing Feynman diagrams in superspace, in [2] and non-perturbatively, by solving the set of Ward identities following from a generalized Konishi anomaly, in [3]. These arguments determined the effective superpotential up to a pure gauge term (independent of the matter sector couplings), corresponding to the Veneziano-Yankielowicz term [4]. It was subsequently shown [5] that this term could in fact be derived by considering the special case with matter in the fundamental representation and a quartic tree-level matter superpotential. It was realized concurrently that these methods could be extended to computing the superpotential in a curved supergravity background [1], specified by an N = 1 Weyl 1
The theory is not quite arbitrary: the matter must be massive if one is to integrate it out.
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superfield Wαβγ . This was further developed in refs. [6–13]. Yet again, the arguments only determine the superpotential up to pure gauge and supergravity terms (independent of the matter sector couplings). This paper supplies a direct derivation of the pure gauge and supergravity superpotential using the generalized Konishi anomaly,2 extending the argument of [5]. In the next section, the effective superpotential is computed for matter consisting of F flavours of quarks (and their squark superpartners), transforming in the fundamental representation of the gauge group SU (N ),3 with a quartic tree-level matter superpotential. This matter superpotential allows gauge symmetry breaking via the Higgs mechanism. The computation was originally performed in [12]. (There is a discrepancy between the effective superpotenital obtained here and that in [12], explained in Sect. 2). One proceeds by solving the Ward identities following from the generalized Konishi anomaly. In a flat Minkowski background, correlation functions factorize and all the Ward identities are equivalent. In a supergravity background, the connected parts of two-point correlation functions are non-vanishing [12] and there are two independent identities. One can show [10] that terms of O((Wαβγ W αβγ )2 ) vanish in the chiral ring and so functions have a Taylor expansion in the Weyl superfield which terminates at O(Wαβγ W αβγ ). Having solved the Ward identities, one can use supersymmetry and holomorphy arguments to determine the effective superpotential up to a ‘constant’ term independent of the matter sector couplings, but depending on the gauge and supergravity background fields and their couplings. In Sect. 3, the superpotential for fundamental matter is used to derive the superpotential for the pure gauge and supergravity theory. The tree-level matter superpotential allows the gauge group SU (N ) to be broken to anything from SU (N ) to SU (N − F ) at low energy via the Higgs mechanism. One considers two vacua in which the unbroken gauge groups are SU (N − F1,2 ) respectively. Varying the matter couplings, one can take a limit in which the masses of both the quarks and the massive gauge bosons (and their superpartners) become large. In this limit, the massive matter decouples from the unbroken gauge group and its contribution to the effective superpotential can be discarded. What is left must represent the contribution from the unbroken gauge group and the supergravity background. If one subtracts the superpotentials for the two vacua, then the unknown ‘constants’ cancel, resulting in a difference equation whose solution yields the pure gauge and supergravity superpotential. In a suitable renormalization group scheme, the superpotential is S 1 S Weff = N −S log 3 + S + (N 2 − 1)G log 3 , (1.1) 6 1 1 α αβγ and , are dimensional transwhere S = − 32π 2 trWα W , G = 32π 2 trWαβγ W mutation scales corresponding to the gauge and supergravity couplings. This superpotential is discussed in Sect. 4. The first term (theVeneziano-Yankielowicz superpotential) comes from the tree-level superYang-Mills action, having accounted for quantum corrections. It is shown by analogy that the second term (depending on the Weyl superfield) comes from the (quantum-corrected) tree-level action for conformal supergravity. This is the unique supergravity action which preserves local superconformal symmetry (for a review see [14]). It is higher derivative and (in contrast to 2 Like the Veneziano-Yankielowicz term, the pure gauge and supergravity superpotential can also be determined from an extended U (1)R symmetry [12] or from the measure of a matrix model [13]. 3 The extension to other classical Lie groups is straightforward.
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Einstein-Hilbert gravity) has a dimensionless coupling constant, which has been replaced in (1.1) by the dimensional transmutation scale . Classically, the equation of motion following from the tree-level conformal supergravity action is Wαβγ = 0. This implies superconformal flatness, which is not quite trivial (for example, super-anti-de Sitter spacetimes are superconformally flat). Since terms in the effective superpotential can be at most quadratic in Wαβγ , it appears that the background Weyl superfield continues to satisfy the superconformal flatness condition irrespective of quantum effects or the actual gauge and matter content of the theory. This may seem paradoxical given that both the gauge and supergravity theories suffer from the conformal anomaly and that the matter sector violates the local superscale invariance even at tree-level if it has dimensionful couplings. However, the claim that conformal flatness persists is only at the level of F -terms. It is likely that radiative corrections will generate D-terms (e. g. super Einstein-Hilbert terms) which do not respect the local superscale invariance and lead to departures from superconformal flatness.
2. The Effective Superpotential with Fundamental Quarks Consider an N = 1 supersymmetric theory in four dimensions with matter chiral superfields coupled to a glueball superfield S and a supergravity superfield given by G. The matter superfields I transform in some representation of the gauge group denoted by the gauge group index I . The quantum theory has the Konishi anomaly [15, 16]; for the generalized chiral rotation δ = (), one has
I
∂Wtree + ∂I
1 G W K W αJ + δIK 32π 2 αJ I 3
∂K = 0, ∂I
(2.1)
where the tree-level matter superpotential Wtree = gk k is some (gauge- and flavourinvariant) polynomial in the matter superfields. The vacuum expectation values are to be evaluated in the presence of a background consisting of the light degrees of freedom. The matter is assumed to be massive and is integrated out. Furthermore, in a generic vacuum of the theory, the gauge symmetry will be spontaneously broken via the Higgs mechanism, with gauge superfields corresponding to broken generators of the gauge group becoming massive. The background should only contain the superfields corresponding to the broken generators (as well as the supergravity superfields). Let the corresponding background glueball superfield be denoted S . The first term in the parentheses in (2.1) splits into two parts. The part tracing over the unbroken gauge group is by definition S . The other part traces over the broken part of the gauge group. The associated superfields are massive, and their potential is (classically) quadratic and centred at the origin, such that their vevs are zero.4 The effective superpotential for the background superfields can then be determined by solving the partial differential equations ∂Weff = k , ∂gk
(2.2)
4 This is certainly true at the classical level, but it is possible that quantum corrections will modify this. However, the limit will be taken later later on in which the masses of the gauge bosons go to infinity and they decouple. In this limit, their vevs certainly are zero, and so the possibility of quantum corrections will not affect the argument below.
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which follow by holomorphy and supersymmetry. These determine Weff up to a constant term independent of the matter couplings gk . Now consider the particular case where the matter sector consists of F flavours of ‘quarks’, viz. F chiral superfields QiI transforming in the fundamental representation ˜ J in the anti-fundamental representation, of the gauge group and F chiral superfields Q j where i and j are flavour indices. The tree-level matter superpotential (for F < N ) is ˜ I as written in terms of the F × F gauge-invariant meson matrix Mji = QiI Q j Wtree = mtrM − λtrM 2 .
(2.3)
The classical equations of motion for the matter fields are j
j
mMi − 2λMik Mk = 0.
(2.4)
The meson matrix M can be brought to diagonal form via a global SU (F ) flavour transformation; the classical vacua then have F− eigenvalues at Mii = 0 and F+ = F − F− eigenvalues at Mii = m/2λ (no sum on i), with the low energy gauge group broken down to SU (N − F+ ). Turning to the generalized Konishi anomaly (2.1), the variations f
f
f
f
δQI = QI ,
δQI = QI Mhh , .. .
(2.5)
yield anomalous Ward identities for the meson matrix N G), 3 N 1 f f f f mMf Mhh − 2λ(M 2 )f Mhh = δf (S − G)Mhh − Gδfh Mh , 3 3 .. . (2.6) f
f
f
mMf − 2λ(M 2 )f = δf (S −
The aim is to solve the complete set of such Ward identities for the matter expectation values. Since terms of O(G2 ) or higher vanish in the chiral ring (and ergo in vacuum expectation values) it is convenient to Taylor expand everywhere in powers of G and to perform the analysis term by term. First recall the analysis at O(G0 ) performed previously [17, 5], corresponding to a flat background. At this order, correlation functions factorize, because connected n-point functions are of O(Gn ) [12]. The Ward identities (2.6) all reduce to the single identity mMi − 2λMik Mk = δi S . j
j
j
(2.7)
There is only one independent vev: the expectation value of the meson matrix. This can be diagonalized by a global SU (F ) rotation in flavour space. In such a basis, the Ward identities have the solution 8λS j j m (2.8) 1± 1− 2 . Mi = δi 4λ m
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The quantum vacua have F± eigenvalues at each of these two values corresponding to Higgsed or un-Higgsed quarks. At O(G1 ), things become more complicated, because the connected two-point correlation function is not zero. One must consider matrix vevs which are the vevs of products of the meson matrix. The global flavour symmetry can be used to diagonalize one such j matrix vev, chosen to be Mi as before. Since there is still a residual SU (F+ )×SU (F− ) global flavour symmetry, the vev of the meson matrix must take the form f
f
M±f = M± δf .
(2.9)
(The subscript ± labels the Higgsed and un-Higgsed spaces, to conform with the notation below.) It is claimed furthermore that all matrix vevs are block-diagonal in this basis. That is, they act reducibly on the Higgsed and un-Higgsed flavour subspaces of dimensions F+ and F− respectively.5 ,6 The Konishi anomaly equations (2.6) are valid verbatim in each subspace separately, with the flavour indices running from 1 to F+ in the Higgsed subspace (labelled by a plus sign as in (2.9) and from 1 to F− in the un-Higgsed subspace (labelled by a minus). The two-point function no longer factorises, but has a connected part of O(G), f
g
f
g
f g
2 M±f M±g = M±f M±g c + M± δf δg
(2.10)
and one makes the further ansatz that f
g
g f
M±f M±g c = δf δg B± .
(2.11)
The connected part of the three-point function vanishes to O(G), leaving f
g
f g
f
f
g
3 M±f M±g M±hh = M± δf δg δhh + M±f M±g c M± δhh + permutations, (2.12)
such that f
f
2 3 (M± )f M±hh = δf δhh (F± B± M± + M± ) + δfh δh 2B± M± .
(2.13)
Taylor expanding in G, there are no contributions beyond O(G); from (2.11), B± is already of O(G) and one can write M± = M0± + M1± G, B± = B1± G.
(2.14)
The set of Ward identities (2.6) reduce to three equations in three unknowns, viz. 2 S = mM0± − 2λM0± ,
0 = mM1± − 2λ(2M0± M1± + F± B1± ) + 0 = mB1± − 4λB1± M0± +
N , 3
M0± , 3
(2.15)
5 This claim is not an assumption, but an ansatz: one is trying to solve the set of Ward identities (2.6); it will be seen that the block-diagonalization does this. 6
f
f
The work [12] does not make this ansatz, but rather starts from the ans¨atze Mf = Mδf and
f
g
g f
Mf Mg c = δf δg B; this still yields two values for M and B, but the solution is only valid if all the f eigenvalues of Mf are the same. This is true for the cases F = F+ and F = F− , but not in the general case.
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with the solutions
M0±
m 8λS = 1± 1− 2 , 4λ m (2N − F± ) F± − , 8λS 6m(1 − 8λS ) 2 6m 1 − m2 m 1 1 . = 1± 12λ 8λS 1−
M1± = ±
B1±
(2.16)
m2
The partial differential equations (2.2) for the effective superpotential with respect to the matter sector couplings are ∂Weff = trM, ∂m ∂Weff = −trM 2 . ∂λ
(2.17)
Again it is convenient to Taylor expand Weff in G, Weff = W0eff + W1eff G.
(2.18)
At O(G0 ), one has [5, 17]
W0eff
m2 m2 8λS =F 1 − 2 + F S log m + (F+ − F− ) 8λ 8λ m F− F+ 1 1 8λS 8λS 1 1 1− 2 1− 2 + − +S log 2 2 m 2 2 m +c(S ),
(2.19)
where c(S ) is independent of m and λ. At O(G1 ), one has, from (2.14) and (2.17), the partial differential equations ∂W1eff = F+ M1+ + F− M1− , ∂m ∂W1eff = −F+ (F+ B+ + 2M0+ M1+ ) − F− (F− B− + 2M0− M1− ). ∂λ
(2.20)
Superpotentials for Gauge and Conformal Supergravity Backgrounds
Substituting from (2.16), one obtains the solution N 2N − F+ 1 1 8λS W1eff = −F+ log m + log − 1− 2 3 6 2 2 m F+ 8λS + log 1 − 2 m 12 N 8λS 2N − F− 1 1 −F− 1− 2 log m + log + 3 6 2 2 m F− 8λS + log 1 − 2 12 m + d(S ),
185
(2.21)
where d(S ) is independent of m and λ and is determined in the next section. 3. The Pure Gauge and Supergravity Superpotential The pure gauge and supergravity superpotential can be determined as follows. First √ take the limit in which both the quark masses, m, and the gauge boson masses, m/2λ, become large. The effective superpotential (2.21) becomes W1eff
m2 /λ→∞
→
−
2N − F+ NF 2λS log m − F+ log 2 . 3 6 m
(3.1)
What is the meaning of this expression? In this limit, the massive degrees of freedom decouple from the pure SU (N − F+ ) gauge theory in the supergravity background. The effective superpotential should consist of the superpotential for the low energy pure gauge and supergravity theory plus terms representing the contribution of the decoupled matter. The decoupled matter consists of the quarks and the massive gauge bosons corresponding to the broken generators of SU (N ) (and their superpartners). The quark superfields have been integrated out, and one can calculate their contribution to the effective superpotential as follows. The non-renormalization theorem applies and the contribution is found by replacing the quark fields in the tree-level superpotential (2.3) by their classical vacuum expectation values. This gives a contribution to W0eff and was discussed in [5]. The contribution of the massive gauge bosons was earlier seen to be zero. Thus one sees that the terms in (3.1) do indeed correspond to the contribution from the low energyunbroken pure gauge theory and supergravity. This contains the unknown constant d, which can be removed by considering the superpotentials for two distinct vacua in which the number of Higgsed quarks, F+ , takes the values F1 and F2 , but the argument S takes the same value, T say, in both.7 If one then subtracts the two effective superpotential functions, the unknown constant d cancels, giving W1eff = −F1
2N − F1 2N − F2 2λT 2λT log 2 + F2 log 2 . 6 m 6 m
(3.2)
7 Of course the physical interpretation of T is different in the two vacua. Here however one simply wants to determine the functional form of W1eff . W1eff is an unconstrained function of its arguments and so the arguments may be chosen arbitrarily.
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This expression ought to involve only the low energy degrees of freedom and the gauge and gravitational couplings, yet it appears to depend on the tree-level matter couplings m and λ. The same phenomenon was observed with the pure gauge theory superpotential in [5] and was shown to be due to the requirement of scale matching: the running gauge couplings of the Higgsed and un-Higgsed theories (which have different beta functions) must match at the Higgs scale, and this leads to a relation between the strong coupling scales of the high and low energy theories. In the gravitational case, the same thing happens: loop diagrams result in logarithmic corrections to the supergravity coupling which depend on the fields running in the loops and so there is again a scale-matching constraint for the Higgsed and un-Higgsed theories. The beta function can be found from the trace or conformal anomaly [18, 19]. The one-loop coefficient is given by b = −3Nv + Nχ ,
(3.3)
where Nv and Nχ are the number of vector and chiral SUSY multiplets respectively. The scale matching must be performed at two √ intermediate scales, corresponding to the quark mass, m, and the gauge boson mass, m/2λ. At high energies, one has N 2 − 1 massless vector supermultiplets and 2N F chiral supermultiplets. Below the Higgs scale, there are (N − F+ )2 − 1 vector supermultiplets and 2N F− + F+2 chiral supermultiplets,8 such that one has
N,F √ m/2λ
−3(N 2 −1)+2NF
=
N −F+ ,F− √ m/2λ
−3[(N−F+ )2 −1]+2NF− +F+2 (3.4)
.
Here, N,F is the dimensional transmutation scale corresponding to the supergravity coupling of the theory with gauge group SU (N ) coupled to F quark flavours. Below the quark mass scale one has only (N − F+ )2 − 1 vector supermultiplets, such that
N−F+ ,F− m
−3[(N−F+ )2 −1]+2NF− +F+2
=
N−F+ ,0 m
−3[(N−F+ )2 −1] .
(3.5)
The scale matching relation is therefore 3((N −F )2 −1)
N −F+ ,0+
m2 2λ
2NF+ −F+2
3(N 2 −1)−2NF
= N,F
m2NF .
(3.6)
Using this relation, one can replace the matter sector couplings m and λ in (3.1) by the appropriate gravitational scales to obtain W1,eff (N − F1 , T , N −F1 ,0 ) − W1,eff (N − F2 , T , N−F2 ,0 ) 1 T = ((N − F1 )2 − 1) log 3 6 N −F ,0 1
1 T − ((N − F2 )2 − 1) log 3 6 N −F
,
(3.7)
2 ,0
To see this, note that each of the 2NF+ −F+2 massless vector supermultiplets corresponding to broken generators eats a chiral supermultiplet to become a massive vector supermultiplet, leaving 2NF− + F+2 of the original 2NF chiral supermultiplets. 8
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where the functional dependence of W1eff has been indicated explicitly. This difference equation has the solution W1,eff (N, S, N,0 ) =
1 2 S (N − 1) log 3 + g(S), 6 N,0
(3.8)
where S has been re-instated and g(S) is an arbitrary function of S alone: it cannot depend on any of the other parameters. Furthermore, gG (which appears in the superpotential) must be of dimension three and g must therefore be of dimension zero, i.e. a pure 2 number. This ambiguity in W1eff corresponds to a re-scaling of N −1 , or equivalently to the freedom to choose a renormalisation group scheme. In a scheme in which g vanishes, the complete effective superpotential for the pure SU (N ) gauge and supergravity theory is S 1 S (3.9) Weff (N, S, N,0 ) = N −S log 3 + S + (N 2 − 1)G log 3 , 6 which is the form obtained by extended U (1)R symmetry considerations [12]. There it was noted that there is no need for the scales and to be the same to satisfy the U (1)R symmetry. The above derivation shows that these scales are indeed distinct: they represent the gauge and gravitational couplings. Now that the effective superpotential for the low energy gauge theory has been derived, one can determine d(S) in (2.21) as in [17] by demanding that W1eff in (2.21) reproduces the correct limit as m2 /λ → ∞ for the vacuum with F+ Higgsed quarks and low energy gauge group SU (N − F+ ). One obtains 2N − F+ F+ 8λS 1 1 8λS 1− 2 + log − log 1 − 2 W2 = −F+ + 6 2 2 m 12 m 2N − F− F− 8λS 1 1 8λS −F− + 1− 2 + log + log 1 − 2 6 2 2 m 12 m 1 S N −1 . + log 2 6 3(N −1)−2NF m2NF 2
(3.10)
The constant d is seen to depend only on the glueball and supergravity backgrounds and their couplings, justifying a posteriori the assumption that it cancels in the vacuum subtraction. 4. Discussion Having derived the form of the effective superpotential for the pure gauge and supergravity theory (3.9), it is of interest to study the vacuum structure obtained upon minimisation. Before doing so, it is important to clarify what is meant by ‘the supergravity theory’, since there exists more than one. To do this, one can proceed by analogy with the pure gauge theory case, with a flat background and no matter. There one starts with the tree-level Yang-Mills superpotential 2πiτ S (plus Hermitian conjugate) which is renormalized to N log 3 S. The non-perturbative dynamics modifies this further to S N −S log 3 + S . (4.1)
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The effective potential is thus generated in a natural way from the tree-level Yang-Mills superpotential. In a supergravity background, a term 1 2 S (N − 1) log 3 G 6
(4.2)
is added to the superpotential. Running the previous argument backwards, it seems reasonable that this comes from the renormalized superpotential 1 3 − (N 2 − 1) log G, 6
(4.3)
which in turn comes from the bare, tree-level superpotential 2π iτ G, where τ is some dimensionless coupling. Is this a bona fide supergravity action? Indeed it is, being (a gauge-fixed version of) the unique action preserving the group of local superscale transformations and known as conformal supergravity (for a review see [14]).9 It is higherderivative and has a logarithmically-renormalized dimensionless coupling constant. This suggests that the effective superpotentials (3.9) and (3.10) encode quantum corrections to conformal supergravity; if this is the case, what are the implications for the vacuum structure? Classically, the equation of motion for the Weyl superfield following from the conformal supergravity action is Wαβγ = 0. This is not quite trivial: it does not imply Minkowski flatness, but rather (super)conformal flatness. In particular, super anti-de Sitter spacetimes with a cosmological constant are permitted.10 When gauge and matter supermultiplets are added, along with their quantum effects, the effective superpotential contains terms of O(G0 ) and O(G). The former must be present even in a Minkowski background and so the latter represent the effects of a curved background in their entirety. Since G is quadratic in the Weyl superfield, the classical equation of motion Wαβγ = 0 is unchanged. Thus superconformal flatness is preserved irrespective of the gauge and matter content of the theory. If this were the whole story, it would be rather remarkable, since the addition of gauge fields, matter and quantum effects leads to the breaking of the local superscale invariance via the superconformal anomaly, even if the matter sector is superscale-invariant, which it need not be. There is, however, an important caveat. The above arguments only allow one to determine the effective superpotential, that is the F -terms in the effective action. Just as for the theory with global supersymmetry, one is unable to make general statements about the D-terms in the effective action. Since the superscale invariance is broken (either explicitly through dimensionful couplings in the tree-level matter superpotential, or via the superconformal anomaly), there is nothing to prevent the generation of nonscale-invariant gravitational corrections (such as super Einstein-Hilbert terms) via quantum effects, even if one starts with classical conformal supergravity. These corrections appear in the effective action as D-terms, and would in general affect the vacuum structure. Acknowledgement. The author would like to thank J. March-Russell for discussions and is supported by a PPARC studentship. 9 Conventional supergravities extending the Einstein-Hilbert action only preserve the subgroup of local super Poincar´e transformations. 10 But not super-de Sitter: there are no SUSY extensions of the de Sitter algebra.
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References 1. Dijkgraaf, R., Vafa, C.: A perturbative window into non-perturbative physics. http://xxx.lanl.gov/abs/hep-th/0208048 2. Dijkgraaf, R., Grisaru, M.T., Lam, C.S., Vafa, C., Zanon, D.: Perturbative computation of glueball superpotentials. http://xxx.lanl.gov/abs/hep-th/0211017 3. Cachazo, F., Douglas, M.R., Seiberg, N., Witten, E.: Chiral rings and anomalies in supersymmetric gauge theory. JHEP 12, 071 (2002) 4. Veneziano, G., Yankielowicz, S.: An effective Lagrangian for the pure N=1 supersymmetric YangMills theory. Phys. Lett. B113, 231 (1982) 5. Gripaios, B.M., Wheater, J.F.: Veneziano-Yankielowicz superpotential terms in N = 1 SUSY gauge theories. http://xxx.lanl.gov/abs/hep-th/0307176 6. Klemm, A., Marino, M., Theisen, S.: Gravitational corrections in supersymmetric gauge theory and matrix models. JHEP 03, 051 (2003) 7. Dijkgraaf, R., Sinkovics, A., Temurhan, M.: Matrix models and gravitational corrections. http://xxx.lanl.gov/abs/hep-th/0211241 8. Ooguri, H., Vafa, C.: The C-deformation of gluino and non-planar diagrams. http://xxx.lanl.gov/abs/hep-th/0302109 9. Ooguri, H., Vafa, C.: Gravity induced C-deformation. http://xxx.lanl.gov/abs/hep-th/0303063 10. David, J.R., Gava, E., Narain, K.S.: Konishi anomaly approach to gravitational F-terms. http://xxx.lanl.gov/abs/hep-th/0304227 11. Alday, L.F., Cirafici, M., David, J.R., Gava, E., Narain, K.S.: Gravitational F-terms through anomaly equations and deformed chiral rings. http://xxx.lanl.gov/abs/hep-th/0305217 12. Ita, H., Nieder, H., Oz, Y.: Gravitational F-terms of N = 1 supersymmetric SU(N) gauge theories. http://xxx.lanl.gov/abs/hep-th/0309041 13. Dijkgraaf, R., Grisaru, M.T., Ooguri, H., Vafa, C., Zanon, D.: Planar gravitational corrections for supersymmetric gauge theories. http://xxx.lanl.gov/abs/hep-th/0310061 14. Fradkin, E.S., Tseytlin, A.A.: Conformal supergravity. Phys. Rept. 119, 233–362 (1985) 15. Konishi, K.: Anomalous supersymmetry transformation of some composite operators in SQCD. Phys. Lett. B135, 439 (1984) 16. Konishi, K.-I., Shizuya, K.-I.: Functional integral approach to chiral anomalies in supersymmetric gauge theories. Nuovo Cim. A90, 111 (1985) 17. Brandhuber, A., Ita, H., Nieder, H., Oz, Y., Romelsberger, C.: Chiral rings, superpotentials and the vacuum structure of N = 1 supersymmetric gauge theories. http://xxx.lanl.gov/abs/hep-th/0303001 18. Christensen, S.M., Duff, M.J.: Axial and conformal anomalies for arbitrary spin in gravity and supergravity. Phys. Lett. B76, 571 (1978) 19. McArthur, I.N.: Super b(4) coefficients in supergravity. Class. Quant. Grav. 1, 245 (1984) Communicated by N.A. Nekrasov
Commun. Math. Phys. 254, 191–220 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1249-6
Communications in
Mathematical Physics
Parabolic Presentations of the Yangian Y (gln ) Jonathan Brundan, Alexander Kleshchev Department of Mathematics, University of Oregon, Eugene, OR 97403, USA. E-mail:
[email protected],
[email protected] Received: 22 January 2004 / Accepted: 28 July 2004 Published online: 2 December 2004 – © Springer-Verlag 2004
Abstract: We introduce some new presentations for the Yangian associated to the Lie algebra gln . These presentations are parametrized by tuples of positive integers summing to n. At one extreme, for the tuple (n), the presentation is the usual RTT presentation of Yn . At the other extreme, for the tuple (1n ), the presentation is closely related to Drinfeld’s presentation. In general, the presentations are useful for understanding the structure of the standard parabolic subalgebras of Yn .
1. Introduction Let Yn = Y (gln ) denote the Yangian associated to the Lie algebra gln over the ground field C; see e.g. [D1, CP, ch.12] and [MNO]. In this article, we record some new presentations of Yn that are adapted to standard parabolic subalgebras. Let us formulate the main result of the article precisely right away, even though the relations appearing in the statement look rather scary. (Note that in §§5–6 most of these formulae are written down in a more concise way in terms of generating series.) Let glν = glν1 ⊕ · · · ⊕ glνm be a standard Levi subalgebra of gln , so ν = (ν1 , . . . , νm ) is a tuple of positive integers summing to n. Theorem A. The algebra Yn is generated by elements {Da;i,j , D a;i,j }1≤a≤m,1≤i,j ≤νa ,r≥0 , (r)
(r)
(r)
(r)
{Ea;i,j }1≤a<m,1≤i≤νa ,1≤j ≤νa+1 ,r≥1 and {Fa;i,j }1≤a<m,1≤i≤νa+1 ,1≤j ≤νa ,r≥1 subject only to the relations
Research partially supported by the NSF (grant no. DMS-0139019).
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Da;i,j = δi,j , r
(1.1)
Da;i,p D a;p,j = −δr,0 δi,j , (t)
(r−t)
(1.2)
t=0 (r)
(s)
min(r,s)−1
(s)
t=0 r+s−1
(s)
t=0 r−1
[Da;i,j , Db;h,k ] = δa,b (r)
[Ea;i,j , Fb;h,k ] = δa,b (r)
[Da;i,j , Eb;h,k ] = δa,b
(r+s−1−t)
Da;i,k
(t)
(t)
(r+s−1−t)
Da;h,j − Da;i,k Da;h,j
(1.3)
,
(t) D (r+s−1−t) , D a;i,k a+1;h,j
(t)
(r+s−1−t)
Da;i,p Ea;p,k
(1.4)
δh,j − δa,b+1
t=0
r−1
(t)
(r+s−1−t)
Db+1;i,k Eb;h,j
,
t=0
(1.5) (r)
(s)
[Da;i,j , Fb;h,k ] = δa,b+1
r−1
(r+s−1−t)
Fb;i,k
(t)
Db+1;h,j − δa,b δi,k
t=0
r−1
(r+s−1−t)
Fa;h,p
(t)
Da;p,j ,
t=0
(1.6) (r)
(s)
s−1
(s)
t=1 r−1
[Ea;i,j , Ea;h,k ] = (r)
[Fa;i,j , Fa;h,k ] =
(t)
(r+s−1−t)
Ea;i,k Ea;h,j (r+s−1−t)
Fa;i,k
(t)
−
Fa;h,j −
r−1 t=1 s−1
(t)
(r+s−1−t)
,
(1.7)
Fa;h,j ,
(1.8)
Ea;i,k Ea;h,j (r+s−1−t)
Fa;i,k
(t)
t=1 t=1 (r) (s+1) (r+1) (s) (r) (s) [Ea;i,j , Ea+1;h,k ] − [Ea;i,j , Ea+1;h,k ] = −Ea;i,q Ea+1;q,k δh,j , (r+1) (s) (r) (s+1) (s) (r) [Fa;i,j , Fa+1;h,k ] − [Fa;i,j , Fa+1;h,k ] = −δi,k Fa+1;h,q Fa;q,j , (s) (r) [Ea;i,j , Eb;h,k ] = 0 if b > a + 1 or if b = a + 1 and h = j , (r) (s) [Fa;i,j , Fb;h,k ] = 0 if b > a + 1 or if b = a + 1 and i = k, (s) (t) (s) (r) (t) (r) if |a − b| = 1, [Ea;i,j , [Ea;h,k , Eb;f,g ]] + [Ea;i,j , [Ea;h,k , Eb;f,g ]] = 0 (r) (s) (t) (s) (r) (t) if |a − b| = 1, [Fa;i,j , [Fa;h,k , Fb;f,g ]] + [Fa;i,j , [Fa;h,k , Fb;f,g ]] = 0
(1.9) (1.10) (1.11) (1.12) (1.13) (1.14)
for all admissible a, b, f, g, h, i, j, k, r, s, t. (By convention the index p resp. q appearing here should be summed over 1, . . . , νa resp. 1, . . . , νa+1 .) In the special case ν = (n), this presentation is the RTT presentation of Yn originating in the work of Faddeev, Reshetikhin and Takhtadzhyan [FRT], while if ν = (1n ) the presentation is a variation on Drinfeld’s presentation from [D2]; see Remark 5.12 for the precise relationship. One reason that Drinfeld’s presentation is important is because it allows one to define subalgebras of Yn which play the role of the Cartan subalgebra and Borel subalgebra in classical Lie theory. Our presentations are well-suited to defining standard Levi and parabolic subalgebras. In the notation of Theorem A, let Yν , (r) (r) Yν+ and Yν− denote the subalgebras of Yn generated by all the Da;i,j ’s, the Ea;i,j ’s or (r)
the Fa;i,j ’s, respectively. Then, Yν = Y (glν ) is the standard Levi subalgebra of Yn ,
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isomorphic to Yν1 ⊗ · · · ⊗ Yνm . The standard parabolic subalgebras Yν and Yν of Yn are the subalgebras generated by Yν , Yν+ and by Yν , Yν− respectively; see also Remark 6.2 where these subalgebras are defined directly in terms of the Drinfeld presentation. The next theorem gives a PBW basis for each of these algebras. To write it down, (r) (r) define elements Ea,b;i,j and Fa,b;j,i for 1 ≤ a < b ≤ m and 1 ≤ i ≤ νa , 1 ≤ j ≤ νb (r)
(r)
(r)
(r)
by setting Ea,a+1;i,j := Ea;i,j , Fa,a+1;j,i := Fa;j,i and then inductively defining (r)
(r)
(1)
Ea,b;i,j := [Ea,b−1;i,k , Eb−1;k,j ],
(r)
(1)
(r)
Fa,b;j,i := [Fb−1;j,k , Fa,b−1;k,i ],
if b > a + 1, where 1 ≤ k ≤ νb−1 . The relations imply that these definitions are independent of the particular choice of k; see (6.9). (r)
Theorem B. (i) The set of all monomials in {Da;i,j }a=1,...,m,1≤i,j ≤νa ,r≥1 taken in some fixed order forms a basis for Yν . (r) (ii) The set of all monomials in {Ea,b;i,j }1≤a
Notation. Throughout the article, we work over the ground field C. We write Mn for the associative algebra of all n × n matrices over C, and gln for the corresponding Lie algebra. The ij -matrix unit is denoted ei,j .
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2. RTT Presentation To define the Yangian Yn = Y (gln ) we use the RTT formalism; see [ES, ch. 11] or [FRT]. Our basic reference for this material in the case of the Yangian is [MNO, §1]. Let R(u) = u −
n
ei,j ⊗ ej,i ∈ Mn ⊗ Mn [u]
(2.1)
i,j =1
denote Yang’s R-matrix with parameter u. This satisfies the QYBE with spectral parameters: R [1,2] (u − v)R [1,3] (u)R [2,3] (v) = R [2,3] (v)R [1,3] (u)R [1,2] (u − v),
(2.2)
equality in Mn ⊗ Mn ⊗ Mn [u, v]. The superscripts in square brackets here and later on indicate the embedding of a smaller tensor into a bigger tensor, inserting the identity into all other tensor positions. Now, Yn is defined to be the associative algebra on generators (r) {Ti,j }1≤i,j ≤n,r≥1 subject to certain relations. To write down these relations, let Ti,j (u) :=
Ti,j u−r ∈ Yn [[u−1 ]] (r)
r≥0 (0)
where Ti,j := δi,j , and T (u) :=
n
ei,j ⊗ Ti,j (u) ∈ Mn ⊗ Yn [[u−1 ]].
(2.3)
i,j =1
We often think of T (u) as an n × n matrix with ij -entry Ti,j (u). Now the relations are given by the equation R [1,2] (u − v)T [1,3] (u)T [2,3] (v) = T [2,3] (v)T [1,3] (u)R [1,2] (u − v),
(2.4)
equality in Mn ⊗ Mn ⊗ Yn ((u−1 , v −1 )) (the localization of Mn ⊗ Mn ⊗ Yn [[u−1 , v −1 ]] at the multiplicative set consisting of the non-zero elements of C[[u−1 , v −1 ]]). Equating ei,j ⊗ eh,k ⊗?-coefficients on either side of (2.4), the relations are equivalent to (u − v)[Ti,j (u), Th,k (v)] = Th,j (u)Ti,k (v) − Th,j (v)Ti,k (u).
(2.5)
Swapping i with h, j with k and u with v, we get equivalently that (u − v)[Ti,j (u), Th,k (v)] = Ti,k (v)Th,j (u) − Ti,k (u)Th,j (v).
(2.6)
Yet another formulation of the relations is given by (r)
(s)
[Ti,j , Th,k ] =
min(r,s)−1
(r+s−1−t)
Ti,k
(t)
(t)
(r+s−1−t)
Th,j − Ti,k Th,j
(2.7)
t=0
for every 1 ≤ h, i, j, k ≤ n and r, s ≥ 0; see [MNO, Proposition 1.2]. Using (2.4), one checks that the following are (anti)automorphisms of Yn ; see [MNO, Proposition 1.12].
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(A1) (Translation) For c ∈ C, let ηc : Yn → Yn be the automorphism defined from r−1 (r) s (r−s) . ηc[2] (T (u)) = T (u + c), i.e. ηc (Ti,j ) = r−1 s=0 s (−c) Ti,j (A2) (Multiplication by a power series) For f (u) ∈ 1 + u−1 C[[u−1 ]], let µf : Yn → (r) Yn be the automorphism defined from µ[2] f (T (u)) = f (u)T (u), i.e. µf (Ti,j ) = r (r−s) if f (u) = s≥0 as u−s . s=0 as Ti,j (A3) (Sign change) Let σ : Yn → Yn be the antiautomorphism of order 2 defined from (r) (r) σ [2] (T (u)) = T (−u), i.e. σ (Ti,j ) = (−1)r Ti,j . (A4) (Transposition) Let τ : Yn → Yn be the antiautomorphism of order 2 defined from (r) (r) τ [2] (T (u)) = (T (u))t (transpose matrix), i.e. τ (Ti,j ) = Tj,i . (A5) (Inversion) Let ω : Yn → Yn be the automorphism of order 2 defined from the equation ω[2] (T (u)) = T (−u)−1 . The Yangian Yn is a Hopf algebra with comultiplication : Yn → Yn ⊗ Yn , counit ε : Yn → C and antipode S : Yn → Yn defined from
[2] (T (u)) = T [1,2] (u)T [1,3] (u),
ε [2] (T (u)) = I,
S [2] (T (u)) = T (u)−1 ,
equalities written in Mn ⊗ Yn ⊗ Yn [[u−1 ]], Mn [[u−1 ]] and Mn ⊗ Yn [[u−1 ]] respectively. Note that S = ω ◦ σ . The involutions ω and σ do not commute, so S is not of order 2; a precise description of S 2 is given in [MNO, Theorem 5.11] or Corollary 8.4 below. (r) Since it arises quite often, we let Ti,j (u) = r≥0 Ti,j u−r := −S(Ti,j (u)), so T(u) :=
n
ei,j ⊗ Ti,j (u) = −T (u)−1 .
(2.8)
i,j =1
To work out commutation relations between Ti,j (u) and Th,k (v), it is useful to rewrite the RTT presentation in the form T[2,3] (v)R [1,2] (u − v)T [1,3] (u) = T [1,3] (u)R [1,2] (u − v)T[2,3] (v).
(2.9)
We record [NT, Lemma 1.1]: Lemma 2.1. Given i = k and h = j , Ti,j and Th,k commute for all r, s ≥ 1. (r)
(s)
Proof. Compute the ei,j ⊗ eh,k -coefficients on each side of (2.9).
We often work with the canonical filtration F0 Yn ⊆ F1 Yn ⊆ F2 Yn ⊆ · · ·
(2.10)
(r)
on Yn defined by declaring that the generator Ti,j is of degree r for each r ≥ 1, i.e. Fd Yn (r )
(r )
is the span of all monomials of the form Ti1 ,j1 1 · · · Tis ,js s with total degree r1 + · · · + rs at most d. It is easy to see using (2.7) that the associated graded algebra gr Yn is commutative. From this one deduces by induction on d that Fd Yn is already spanned by the set of (r) all monomials of total degree at most d in the elements {Ti,j }1≤i,j ≤n,r≥1 taken in some fixed order. In fact it is known that such ordered monomials are linearly independent, (r) hence the set of all monomials in the elements {Ti,j }1≤i,j ≤n,r≥1 taken in some fixed order gives a basis for Yn ; see [MNO, Corollary 1.23] or Corollary 3.2 below. In other
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words, the associated graded algebra gr Yn is the free commutative algebra on generators (r) {gr r Ti,j }1≤i,j ≤n,r≥1 . There is a second important filtration which we call the loop filtration L0 Yn ⊆ L1 Yn ⊆ L2 Yn ⊆ · · ·
(2.11)
(r)
defined by declaring that the generator Ti,j is of degree (r −1) for each r ≥ 1. We denote the associated graded algebra by gr L Yn . Let gln [t] denote the Lie algebra gln ⊗ C[t] with basis {ei,j t r }1≤i,j ≤n,r≥0 . By the relations (2.7), there is a surjective homomorphism (r+1) for each 1 ≤ i, j ≤ n, r ≥ 0. By the U (gln [t]) gr L Yn mapping ei,j t r to gr Lr Ti,j PBW theorem for Yn described in the previous paragraph, this map is actually an isomorphism, hence gr L Yn ∼ = U (gln [t]); see also [MNO, Theorem 1.26] where this argument is explained in more detail.
3. PBW Theorem In this section, we give a proof of the PBW theorem for Yn different from the one in [MNO]. It was inspired by the realization of the Yangian found in [BR]. Let U (gln ) denote the universal enveloping algebra of the Lie algebra gln . We have the evaluation homomorphism (1) Ti,j
κ1 : Yn → U (gln ),
→
ei,j if r = 1, 0 if r > 1.
(3.1)
More generally for l ≥ 1, consider the homomorphism κl := κ1 ⊗ · · · ⊗ κ1 ◦ (l) : Yn → U (gln )⊗l ,
(3.2)
where (l) : Yn → Yn⊗l denotes the l th iterated comultiplication. We define the algebra [s] for the element Yn,l to be the image κl (Yn ) of Yn under this homomorphism. Writing ei,j ⊗(s−1) ⊗(l−s) ⊗l 1 ⊗ ei,j ⊗ 1 ∈ U (gln ) , we have by the definition of that (r)
κl (Ti,j ) =
1≤s1 <···<sr ≤l 1≤i0 ,··· ,ir ≤n i0 =i,ir =j
r] ei[s0 1,i]1 ei[s1 2,i]2 · · · ei[sr−1 ,ir .
(3.3)
(r)
In particular, we see from this that κl (Ti,j ) = 0 for all r > l. (r)
Theorem 3.1. The set of all monomials in the elements {κl (Ti,j )}1≤i,j ≤n,r=1,...,l taken in some fixed order forms a basis for Yn,l . Proof. It is obvious that such monomials span Yn,l , so we just have to show that they are linearly independent. Consider the standard filtration F0 U (gln )⊗l ⊆ F1 U (gln )⊗l ⊆ F2 U (gln )⊗l ⊆ · · ·
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[r] on U (gln )⊗l , so each generator ei,j is of degree 1. The associated graded algebra [r] [r] := gr 1 ei,j . Let yi,j := gr U (gln )⊗l is the free polynomial algebra on generators xi,j (r)
(r)
gr r κl (Ti,j ), i.e. (r)
yi,j =
1≤s1 <···<sr ≤l 1≤i0 ,··· ,ir ≤n i0 =i,ir =j
r] xi[s0 ,i1 ]1 xi[s1 ,i2 ]2 · · · xi[sr−1 ,ir .
(r)
To complete the proof of the theorem, we show that the elements {yi,j }1≤i,j ≤n,r=1,...,l are algebraically independent. Let us identify gr U (gln )⊗l with the coordinate algebra C[Mn×l ] of the affine variety [r] is the function picking out Mn×l of l-tuples (A1 , . . . , Al ) of n × n matrices, so that xi,j the ij -entry of the rth matrix Ar . Let θ : Mn×l → Mn×l be the morphism defined by (A1 , . . . , Al ) → (B1 , . . . , Bl ), where Br is the rth elementary symmetric function er (A1 , . . . , Al ) := As1 · · · Asr 1≤s1 <···<sr ≤l [r] in the matrices A1 , . . . , Al . The comorphism θ ∗ maps xi,j to yi,j . So to show that the (r)
yi,j are algebraically independent, we need to show that θ ∗ is injective, i.e. that θ is a dominant morphism of affine varieties. For this it suffices to show that the differential of θ is surjective at some point x ∈ Mn×l . Pick pairwise distinct scalars c1 , . . . , cl ∈ C and consider x := (c1 In , . . . , cl In ). Identifying the tangent space Tx (Mn×l ) with the vector space Mn⊕l , a calculation shows that the differential dθx maps (A1 , . . . , Al ) to (B1 , . . . , Bl ) where (r)
Br =
l
er−1 (c1 , . . . , cs , . . . , cl )As .
s=1
Here, er−1 (c1 , . . . , cs , . . . , cl ) denotes the (r − 1)th elementary symmetric function in the scalars c1 , . . . , cl excluding cs . We just need to show this linear map is surjective, for which it clearly suffices to consider the case n = 1. But in that case its determinant
is the Vandermonde determinant 1≤r<s≤l (cs − cr ), so it is non-zero by the choice of the scalars c1 , . . . , cl .
(r)
Corollary 3.2. The set of all monomials in the elements {Ti,j }1≤i,j ≤n,r≥1 taken in some fixed order forms a basis for Yn . Proof. We have already observed that such monomials span Yn . The fact that they are linearly independent follows from Theorem 3.1 by taking sufficiently large l.
Corollary 3.3. The kernel of κl : Yn Yn,l is the two-sided ideal of Yn generated by (r) the elements {Ti,j }1≤i,j ≤n,r>l . (r)
Proof. Let I denote the two-sided ideal of Yn generated by {Ti,j }1≤i,j ≤n,r>l . It is obvious that κl induces a map κ¯ l : Yn /I Yn,l . Since Yn /I is spanned by the set of all (r) monomials in the elements {Ti,j + I }1≤i,j ≤n,r=1,...,l taken in some fixed order, Theorem 3.1 now implies that κ¯ l is an isomorphism.
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The first of these corollaries proves the PBW theorem for Yn . The second corollary shows that the algebra Yn,l is the Yangian of level l introduced by Cherednik [C1, C2]. Moreover, by Corollary 3.3, the maps κl induce an inverse system Yn,1 Yn,2 · · · of filtered algebras, where each Yn,l is filtered by the canonical filtration defined by (r) declaring that the generators κl (Ti,j ) are of degree r. It is easy to see using Theorem 3.1 and Corollary 3.2 that the Yangian Yn is the inverse limit lim Yn,l of this system taken in ← − the category of filtered algebras. This gives a concrete realization of the Yangian. 4. Levi Subalgebras Our exposition is biased towards the standard embedding Yn → Yn+1 under which (r) Ti,j ∈ Yn maps to the element with the same name in Yn+1 . We warn the reader that (r) the element Ti,j ∈ Yn does not map to the element with the same name in Yn+1 ! The standard embeddings define a tower of algebras Y1 ⊂ Y2 ⊂ Y3 ⊂ · · ·
(4.1)
which will be implicit in our work from now on. Since most of the automorphisms of the Yangian defined in §2 do not commute with the standard embeddings, we sometimes add a subscript to clarify notation; for example we write ωn : Yn → Yn for the automorphism ω if confusion seems likely. For m ≥ 0, we let ϕm : Yn → Ym+n denote the obvious injective algebra homomor(r) (r) phism mapping Ti,j ∈ Yn to Tm+i,m+j ∈ Ym+n for each 1 ≤ i, j ≤ n, r ≥ 1. Following [NT], we define another injective algebra homomorphism ψm : Yn → Ym+n by ψm := ωm+n ◦ ϕm ◦ ωn .
(4.2)
(r) (r) Observe that ψm maps Ti,j ∈ Yn to Tm+i,m+j ∈ Ym+n . So the subalgebra ψm (Yn ) of (r) Ym+n is generated by the elements {Tm+i,m+j }1≤i,j ≤n,r≥1 . Given this, the following lemma is an immediate consequence of Lemma 2.1.
Lemma 4.1. The subalgebras Ym and ψm (Yn ) of Ym+n centralize each other. Let us give another description of the map ψm in terms of the quasi-determinants of Gelfand and Retakh; see e.g. [GKLLRT, §2.2]. Suppose that A, B, C and D are m × m, m × n, n × m and n × n matrices respectively with entries in some ring R. Assuming that the matrix A is invertible, we define A B −1 (4.3) C D := D − CA B. Then: Lemma 4.2. For any 1 ≤ i, j ≤ n, T1,m+j (u) T1,1 (u) · · · T1,m (u) .. .. .. .. . . . . ψm (Ti,j (u)) = Tm,m+j (u) Tm,1 (u) · · · Tm,m (u) T m+i,1 (u) · · · Tm+i,m (u) Tm+i,m+j (u)
.
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Proof. Let T (u) denote the matrix Ti,j (u) 1≤i,j ≤n with entries in Yn [[u−1 ]] as usual and let T(u) := −T (u)−1 . Also define the matrices B(u) = Ti,j (u) 1≤i≤m,m+1≤j ≤m+n , A(u) = Ti,j (u) 1≤i,j ≤m , C(u) = Ti,j (u) m+1≤i≤m+n,1≤j ≤m , D(u) = Ti,j (u) m+1≤i,j ≤m+n with entries in Ym+n [[u−1 ]] and let
−1 A(u) B(u) A(u) B(u) := − . C(u) D(u) C(u) D(u) By block multiplication, one checks the classical identities −1 = − A(u) − B(u)D(u)−1 C(u) A(u) , −1 , B(u) = A(u)−1 B(u) D(u) − C(u)A(u)−1 B(u) −1 , C(u) = D(u)−1 C(u) A(u) − B(u)D(u)−1 C(u) −1 . D(u) = − D(u) − C(u)A(u)−1 B(u) Hence, it maps T (u) Now, by its definition, the homomorphism ψm maps T(u) to D(u). −1 −1 to −D(u) = D(u) − C(u)A(u) B(u). The lemma follows from this on computing ij -entries.
The description of ψm (Ti,j (u)) given by Lemma 4.2 does not depend on n. This means that the maps ψm are compatible with the standard embeddings, in the sense that the following diagram commutes: Y1 ψm
−−−−→
Y2 ψm
−−−−→
Y3 ψm
−−−−→ · · · (4.4)
Ym+1 −−−−→ Ym+2 −−−−→ Ym+3 −−−−→ · · · where the horizontal maps are standard embeddings. So our notation for the map ψm is unambiguous as n varies. We also note that ψm ◦ ψm = ψm+m
(4.5)
for any m, m ≥ 0, which is an obvious consequence of our original definition. Now we can define the standard Levi subalgebras of Yn . Given a tuple ν = (ν1 , . . . , νm) of positive integers summing to n, define Yν to be the subalgebra Yν := Yν1 ψν1 (Yν2 )ψν1 +ν2 (Yν3 ) · · · ψν1 +···+νm−1 (Yνm )
(4.6)
of Yn . For a = 1, . . . , m and 1 ≤ i, j ≤ νa , we let (r) Da;i,j u−r := ψν1 +···+νa−1 (Ti,j (u)). Da;i,j (u) =
(4.7)
r≥0
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By Lemma 4.1 and induction on m, the various “blocks” of Yν centralize each other, hence the map ¯ ν1 ⊗ ¯ · · · ⊗ψ ¯ ν1 +···+νm−1 : Yν1 ⊗ Yν2 ⊗ · · · ⊗ Yνm → Yν ψ0 ⊗ψ (r)
is an algebra isomorphism. This means that the elements {Da;i,j }1≤a≤m,1≤i,j ≤νa ,r≥1 generate Yν subject only to the following relations: (r) (s) [Da;i,j , Db;h,k ]
= δa,b
min(r,s)−1
(r+s−1−t)
Da;i,k
(t)
(t)
(r+s−1−t)
Da;h,j − Da;i,k Da;h,j
,
(4.8)
t=0 (0) where Da;i,j := δi,j . The special case ν = (1n ) is particularly important: Y(1n ) ∼ = Y1 ⊗ · · · ⊗ Y1 is a commutative subalgebra of Yn which plays the role of Cartan subalgebra.
5. Drinfeld Presentation In this section, we introduce the Drinfeld generators of Yn following the approach of [DF] and [I, Appendix B]. Since the leading minors of the matrix T (u) are invertible, it possesses a Gauss factorization T (u) = F (u)D(u)E(u)
(5.1)
for unique matrices
D1 (u) 0 · · · 0 0 D2 (u) · · · 0 D(u) = , .. .. .. ... . . . 0 0 · · · Dn (u)
· · · E1,n (u) 1 0 ··· 0 · · · E2,n (u) 1 ··· 0 F1,2 (u) , F (u) = . .. .. .. .. . . .. . . . . . . ··· 1 F1,n (u) F2,n (u) · · · 1 (r) (r) This defines the power series Di (u) = r≥0 Di u−r , Ei,j (u) = r≥1 Ei,j u−r and (r) −r (r) −r := Ei,i+1 (u) and Fi (u) = Fi,j (u) = r≥1 Fi,j u . Let Ei (u) = r≥1 Ei u (r) −r i (u) = r≥0 D (r) u−r := −Di (u)−1 . := Fi,i+1 (u) for short. Also let D r≥1 Fi u i In terms of quasi-determinants, we have the following more explicit descriptions; see [GR1, Theorem 4.4] or [GR2, Theorem 2.2.6]. 1 E1,2 (u) 1 0 E(u) = .. ... . 0 0
T1,1 (u) · · · T1,i−1 (u) T1,i (u) .. .. .. .. . . . . , Di (u) = Ti−1,1 (u) · · · Ti−1,i−1 (u) Ti−1,i (u) T (u) · · · T Ti,i (u) i,1 i,i−1 (u)
(5.2)
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T1,1 (u) · · · T1,i−1 (u) T1,j (u) .. .. .. .. −1 . . . . , Ei,j (u) = Di (u) Ti−1,1 (u) · · · Ti−1,i−1 (u) Ti−1,j (u) T (u) · · · T Ti,j (u) i,1 i,i−1 (u)
(5.3)
T1,1 (u) · · · T1,i−1 (u) T1,i (u) .. .. .. .. . . . . Di (u)−1 . Fi,j (u) = Ti−1,1 (u) · · · Ti−1,i−1 (u) Ti−1,i (u) T (u) · · · T Tj,i (u) j,1 j,i−1 (u)
(5.4)
(1)
(1)
(1)
(1)
(r)
(1)
Since Ej −1 = Tj −1,j and Fj −1 = Tj,j −1 , it follows easily that (r)
Ei,j = [Ei,j −1 , Ej −1 ],
(r)
(1)
(r)
Fi,j = [Fj −1 , Fi,j −1 ]
(5.5)
for i + 1 < j ≤ n. Comparing (5.2)–(5.4) with Lemma 4.2 we deduce: Lemma 5.1. For all admissible i, we have that (i) Di (u) = ψi−1 (D1 (u)) = ψi−1 (T1,1 (u)); (ii) Ei (u) = ψi−1 (E1 (u)) = ψi−1 (T1,1 (u)−1 T1,2 (u)); (iii) Fi (u) = ψi−1 (F1 (u)) = ψi−1 (T2,1 (u)T1,1 (u)−1 ). (r)
In particular Lemma 5.1(i) shows that the elements Di here are the same as the ele(r) + ments denoted Di;1,1 from §4, so they generate the Cartan subalgebra Y(1n ) . Also let Y(1 n)
− resp. Y(1 n ) denote the subalgebra of Yn generated by the elements {Ei }i=1,...,n−1,r≥1 (r)
+ resp. {Fi }i=1,...,n−1,r≥1 . In view of (5.5), all the elements Ei,j belong to Y(1 n ) and all the (r)
(r)
− elements Fi,j belong to Y(1 n ) . By applying the antiautomorphism τ to the factorization (5.1), one checks: (r)
τ (Ei,j (u)) = Fi,j (u),
τ (Fi,j (u)) = Ei,j (u),
τ (Di (u)) = Di (u).
(5.6)
+ − Hence, τ fixes Y(1n ) elementwise and interchanges the subalgebras Y(1 n ) and Y(1n ) . Now we state the main theorem of the section. This is essentially due to Drinfeld [D2]; see the remark at the end of the section for the precise relationship.
}1≤i≤n,r≥0 and Theorem 5.2. The algebra Yn is generated by the elements {Di , D i (r) (r) {Ei , Fi }1≤i
(r)
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Di r
= 1,
(5.7)
(t) (r−t) Di D = −δr,0 , i
(5.8)
t=0 (r)
(s)
(r)
(s)
[Di , Dj ] = 0,
(5.9)
[Ei , Fj ] = δi,j
r+s−1
(t) D (r+s−1−t) , D i i+1
(5.10)
t=0 (r)
(s)
r−1
(s)
t=0 r−1
[Di , Ej ] = (δi,j − δi,j +1 ) (r)
[Di , Fj ] = (δi,j +1 − δi,j )
(t)
(r+s−1−t)
,
(5.11)
Di ,
(5.12)
Di E j
(r+s−1−t)
Fj
(t)
t=0
(r)
(s)
s−1 t=1
t=1
(s)
r−1
s−1
[Ei , Ei ] = (r)
[Fi , Fi ] = (r)
(t)
(r+s−1−t)
Fi
t=1 (r+1)
(s+1)
[Ei , Ei+1 ] − [Ei (r+1)
[Fi
(s)
(r+s−1−t)
Ei E i
(r)
(t)
Fi
− −
r−1
(t)
(r+s−1−t)
,
(5.13)
Fi ,
(5.14)
Ei E i
(r+s−1−t)
Fi
(t)
t=1 (r) (s)
(s)
, Ei+1 ] = −Ei Ei+1 , (s+1)
(s)
(5.15)
(r)
, Fi+1 ] − [Fi , Fi+1 ] = −Fi+1 Fi ,
(r) (s) [Ei , Ej ] = 0 (r) (s) [Fi , Fj ] = 0 (r) (s) (t) (s) (r) (t) [Ei , [Ei , Ej ]] + [Ei , [Ei , Ej ]] (r) (s) (t) (s) (r) (t) [Fi , [Fi , Fj ]] + [Fi , [Fi , Fj ]]
(5.16)
if |i − j | > 1,
(5.17)
if |i − j | > 1,
(5.18)
=0
if |i − j | = 1,
(5.19)
=0
if |i − j | = 1,
(5.20)
for all admissible i, j, r, s, t. Remark 5.3. The relations (5.13) and (5.14) are equivalent to the relations (r)
(s+1)
[Ei , Ei
(r+1)
] − [Ei
(s)
(r)
(s)
(s)
(r)
, Ei ] = Ei Ei + Ei Ei ,
(r+1) (s) (r) (s+1) [Fi , Fi ] − [Fi , Fi ]
=
(r) (s) Fi Fi
(s) (r) + F i Fi ,
(5.21) (5.22)
respectively. In the remainder of the section, we are going to write down a proof, since we could not find one in the literature. There are two parts to the proof: first we must show that all these relations are satisfied in Yn ; second we must show that we have found enough relations. Let us begin with some reductions to the first part of the proof. We have already noted (r) that the elements {Di }i=1,...,n,r≥1 commute, hence the relations (5.7)–(5.9) hold. Also (r) (s) by Lemma 5.1, Di ∈ ψi−1 (Y1 ) and Ej ∈ ψj −1 (Y2 ), so Lemma 4.1 implies that
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(5.11) holds if either i < j or i > j + 1. Similar reasoning shows that (5.10) holds if |i − j | > 1 and (5.17)–(5.18) hold always. Having made these remarks, Lemma 5.1 and (5.6) reduces the verification of all the remaining relations to checking the following special cases: (5.10) with i = 1, j = 1 or i = 2, j = 1; (5.11) with i = 1, j = 1 or i = 2, j = 1; (5.13) with i = 1; (5.15) with i = 1; (5.19) with i = 2, j = 1 or i = 1, j = 2. Lemma 5.4. The following identities hold in Y2 ((u−1 , v −1 )): (i) (u − v)[D1 (u), E1 (v)] = D1 (u)(E1 (v) − E1 (u)); 2 (v)] = (E1 (u) − E1 (v))D 2 (v); (ii) (u − v)[E1 (u), D 1 (v)D2 (v) − D 1 (u)D2 (u); (iii) (u − v)[E1 (u), F1 (v)] = D (iv) (u − v)[E1 (u), E1 (v)] = (E1 (v) − E1 (u))2 . Proof. Compute the e1,1 ⊗ e1,2 -, e1,2 ⊗ e2,2 -, e1,2 ⊗ e2,1 - and e1,2 ⊗ e1,2 -coefficients on each side of (2.9) and rearrange the resulting four equations to obtain the following identities: (i) (u − v)[T1,1 (u), T1,2 (v)] = T1,1 (u)T1,2 (v) + T1,2 (u)T2,2 (v); (ii) (u − v)[T1,2 (u), T2,2 (v)] = T1,1 (u)T1,2 (v) + T1,2 (u)T2,2 (v); (iii) (u − v)[T1,2 (u), T2,1 (v)] = T1,1 (u)T1,1 (v) + T1,2 (u)T2,1 (v) − T2,1 (v)T1,2 (u) − T2,2 (v)T2,2 (u); (iv) [T1,2 (u), T1,2 (v)] = 0. We also note that
T1,1 (u) T1,2 (u) D1 (u)E1 (u) D1 (u) = T2,1 (u) T2,2 (u) F1 (u)D1 (u) D2 (u) + F1 (u)D1 (u)E1 (u) and that
T1,1 (v) T1,2 (v) T2,1 (v) T2,2 (v)
=
1 (v) + E1 (v)D 2 (v)F1 (v) −E1 (v)D 2 (v) D . 2 (v) 2 (v)F1 (v) D −D
Substituting from these into (i) and using the known fact that D1 (u) commutes with 2 (v) gives the identity D 2 (v) = D1 (u)(E1 (v) − E1 (u))D 2 (v). (u − v)[D1 (u), E1 (v)]D Multiplying on the right by D2 (v) gives (i). The deduction of (ii) from (ii) is entirely similar. Next we deduce (iii) from (iii) . Rewriting (i) and (ii) using τ gives that 2 (v) + (u − v − 1)E1 (u)D 2 (v) = (u − v)D 2 (v)E1 (u), E1 (v)D F1 (u)D1 (u) + (u − v − 1)F1 (v)D1 (u) = (u − v)D1 (u)F1 (v). Also rearranging (iii) gives 1 (v) + D1 (u)(E1 (v)D 2 (v) + (u − v − 1)E1 (u)D 2 (v))F1 (v) D1 (u)D 2 (v) + D 2 (v)(F1 (u)D1 (u) + (u − v − 1)F1 (v)D1 (u))E1 (u). = D2 (u)D
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Now substituting the first two of these identities into the third and multiplying on the 1 (u)D2 (v) gives (iii). left by D Finally we must deduce (iv). By (iv) , we have that 2 (v) = E1 (v)D1 (u)D 2 (v)E1 (u). D1 (u)E1 (u)E1 (v)D 2 (v) Multiply both sides by (u − v)2 and use (i) and (ii) to move D1 (u) to the left and D 2 (v)’s to get to the right, then cancel the leading D1 (u)’s and trailing D (u − v)2 E1 (u)E1 (v) = ((u − v)E1 (v) − E1 (v) +E1 (u)) ((u − v)E1 (u) + E1 (v) − E1 (u)) . Hence, (iv) (u − v)2 [E1 (u), E1 (v)] = (E1 (v) − E1 (u))(E1 (u) − E1 (v)) + (u − v)E1 (v) (E1 (v) − E1 (u)) + (u − v)(E1 (u) − E1 (v))E1 (u). Now subtract (u − v)[E1 (u), E1 (v)] from both sides of (iv) to deduce that (u − v)(u − v − 1)[E1 (u), E1 (v)] = (u − v − 1)(E1 (v) − E1 (u))2 . Hence (iv) follows on dividing both sides by (u − v − 1).
Lemma 5.5. The following identities hold in Y3 ((u−1 , v −1 )): (i) [E1 (u), F2 (v)] = 0; (ii) (u − v)[E1 (u), E2 (v)] = E1 (u)E2 (v) − E1 (v)E2 (v) − E1,3 (u) + E1,3 (v); (iii) [E1,3 (u), E2 (v)] = E2 (v)[E1 (u), E2 (v)]; (iv) [E1 (u), E1,3 (v) − E1 (v)E2 (v)] = −[E1 (u), E2 (v)]E1 (u). Proof. Arguing as in the proof of the previous lemma, we compute the e1,2 ⊗ e3,2 -, e1,2 ⊗ e2,3 -, e1,3 ⊗ e2,3 - and e1,2 ⊗ e1,3 -coefficients of (2.9) respectively to obtain the identities 3 (v)F2 (v)] = 0 (i) [D1 (u)E1 (u), D 3 (v)] = D1 (u)(E1 (u)E2 (v) − E1 (v)E2 (v) (ii) (u − v)[D1 (u)E1 (u), E2 (v)D + E1,3 (v) − E1,3 (u))D3 (v); 3 (v)] = 0; (iii) [D1 (u)E1,3 (u), E2 (v)D 3 (v)] = 0. (iv) [D1 (u)E1 (u), (E1,3 (v) − E1 (v)E2 (v))D Using commutation relations already derived, (i) and (ii) follow easily from (i) and (ii) . To prove (iii), we need one more identity. By Lemma 5.4(ii), we know that 3 (v). 3 (v), E2 (u)] = (E2 (v) − E2 (u))D (u − v)[D 3 (v), E (1) ] = E2 (v)D 3 (v). Hence, recalling (5.5), Considering u0 -coefficients gives [D 2 3 (v)] = [[E1 (u), E (1) ], D 3 (v)] = [E1 (u), [E (1) , D 3 (v)]] [E1,3 (u), D 2 2 3 (v)] = −[E1 (u), E2 (v)]D 3 (v). = −[E1 (u), E2 (v)D Now take (iii) , cancel the leading D1 (u) and then simplify to get 3 (v)] = [E1,3 (u), E2 (v)]D 3 (v) + E2 (v)[E1,3 (u), D 3 (v)] [E1,3 (u), E2 (v)D 3 (v) = 0. = [E1,3 (u), E2 (v)] − E2 (v)[E1 (u), E2 (v)] D
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This proves (iii). For (iv), note to start with by considering the u0 -coefficients of (ii) that (1) (1) [E1 , E2 (v)] = E1,3 (v) − E1 (v)E2 (v). Lemma 5.4(i) implies that [D1 (u), E1 ] = D1 (u)E1 (u). Now compute: (1)
(1)
[D1 (u), E1,3 (v) − E1 (v)E2 (v)] = [D1 (u), [E1 , E2 (v)]] = [[D1 (u), E1 ], E2 (v)] = [D1 (u)E1 (u), E2 (v)] = D1 (u)[E1 (u), E2 (v)]. Using this identity to rewrite (iv) we get [D1 (u)E1 (u), E1,3 (v) − E1 (v)E2 (v)] = D1 (u)[E1 (u), E2 (v)]E1 (u) + D1 (u)[E1 (u), E1,3 (v) − E1 (v)E2 (v)] = 0. Now (iv) follows on cancelling D1 (u).
Lemma 5.6. The following relations hold: (i) [[E1 (u), E2 (v)], E2 (v)] = 0; (ii) [E1 (u), [E1 (u), E2 (v)]] = 0. Proof. (i) Compute using Lemma 5.5(ii) and (iii): (u − v)[[E1 (u), E2 (v)], E2 (v)] = [E1 (u)E2 (v) − E1 (v)E2 (v) − E1,3 (u) + E1,3 (v), E2 (v)] = [E1 (u), E2 (v)]E2 (v) − [E1 (v), E2 (v)]E2 (v) +E2 (v)[E1 (v), E2 (v)] − E2 (v)[E1 (u), E2 (v)] = [[E1 (u), E2 (v)], E2 (v)] − [[E1 (v), E2 (v)], E2 (v)]. We conclude that (u−v−1)[[E1 (u), E2 (v)], E2 (v)] = −[[E1 (v), E2 (v)], E2 (v)]. Now let u = v + 1 to deduce that the right-hand side equals zero, then divide by (u − v − 1) to prove the lemma. (ii) Similar calculation using Lemma 5.5(iv) instead of (iii).
Lemma 5.7. The following relations hold: (i) [[E1 (u), E2 (v)], E2 (w)] + [[E1 (u), E2 (w)], E2 (v)] = 0; (ii) [E1 (u), [E1 (v), E2 (w)]] + [E1 (v), [E1 (u), E2 (w)]] = 0. Proof. (i) We show that the expression (u − v)(u − w)(v − w)[[E1 (u), E2 (v)], E2 (w)] is symmetric in v and w. By Lemma 5.5(ii) it equals (u − w)(v − w)[E1 (u)E2 (v) − E1 (v)E2 (v) + E1,3 (v) − E1,3 (u), E2 (w)]. Using Lemmas 5.5(iii) and 5.6(i) this equals (u − w)(v − w)[E1 (u), E2 (w)]E2 (v) + (u − w)(v − w)E1 (u)[E2 (v), E2 (w)] −(u − w)(v − w)[E1 (v), E2 (w)]E2 (v) − (u − w)(v − w)E1 (v)[E2 (v), E2 (w)] +(u − w)(v − w)[E1 (v), E2 (w)]E2 (w)−(u − w)(v − w)[E1 (u), E2 (w)]E2 (w).
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Now use Lemmas 5.5 (ii) and 5.4 (iv) to expand the commutators once more to get (v − w)(E1 (u)E2 (w)E2 (v) − E1 (w)E2 (w)E2 (v) + E1,3 (w)E2 (v) − E1,3 (u)E2 (v)) +(u − w)(E1 (u)E2 (v)2 − E1 (u)E2 (v)E2 (w) − E1 (u)E2 (w)E2 (v) + E1 (u)E2 (w)2 ) −(u − w)(E1 (v)E2 (w)E2 (v) − E1 (w)E2 (w)E2 (v) + E1,3 (w)E2 (v) − E1,3 (v)E2 (v)) −(u − w)(E1 (v)E2 (v)2 − E1 (v)E2 (v)E2 (w) − E1 (v)E2 (w)E2 (v) + E1 (v)E2 (w)2 ) +(u − w)(E1 (v)E2 (w)E2 (w) − E1 (w)E2 (w)E2 (w) + E1,3 (w)E2 (w) − E1,3 (v)E2 (w)) −(v − w)(E1 (u)E2 (w)E2 (w) − E1 (w)E2 (w)E2 (w) + E1,3 (w)E2 (w) − E1,3 (u)E2 (w)).
Now open the parentheses and check that the resulting expression is symmetric in v and w to complete the proof. (ii) A similar calculation using Lemma 5.5(iv) instead of (iii) and Lemma 5.6(ii) instead of (i) shows that the expression (u − v)(u − w)(v − w)[E1 (u), [E1 (v), E2 (w)]] is symmetric in u and v.
Now we can verify the remaining relations needed for the first part of the proof. Note that (r+s−1) (E1 (v) − E1 (u))/(u − v) = E1 u−r v −s . (5.23) r,s≥1
Using this, divide both sides of the identity from Lemma 5.4(i) by (u − v) and equate u−r v −s -coefficients on both sides to prove (5.11) with i = 1, j = 1. Next, multiplying Lemma 5.4(ii) on the left and right by D2 (v) then swapping u and v gives the identity (u − v)[D2 (u), E1 (v)] = −D2 (u)(E1 (v) − E1 (u)).
(5.24)
Now argue using (5.23) again to deduce (5.11) with i = 2, j = 1 from this. Similarly one gets (5.10) with i = 1, j = 1 from Lemma 5.4(iii), (5.10) with i = 2, j = 1 from Lemma 5.5(i), (5.13) with i = 1 from Lemma 5.4(iv), (5.15) with i = 1 from Lemma 5.5(ii), (5.19) with i = 2, j = 1 from Lemma 5.7(i) and (5.19) with i = 1, j = 2 from Lemma 5.7(ii). n denote the algebra with genNow we consider the second part of the proof. Let Y erators and relations as in the statement of Theorem 5.2. For 1 ≤ i < j ≤ n, define (r) (r) n by Eqs. (5.5). Let Y (1n ) resp. Y +n resp. Y −n denote the subalelements Ei,j , Fi,j ∈ Y (1 ) (1 ) n generated by the elements {D (r) }i=1,...,n,r≥1 resp. {E (r) }i=1,...,n−1,r≥1 resp. gebra of Y (r)
i
i
{Fi }i=1,...,n−1,r≥1 . Define an ascending filtration +n ⊆ L1 Y +n ⊆ · · · L0 Y (1 ) (1 ) +n by declaring that the generator E (r) is of degree (r −1), i.e. Ld Y +n is the span of on Y i (1 ) (1 ) +n denote the assoall monomials in these generators of total degree at most d. Let gr L Y (1 ) (r+1) +n for each 1 ≤ i < j ≤ n ciated graded algebra, and let ei,j ;r := gr Lr Ei,j ∈ gr L Y (1 ) and r ≥ 0. Lemma 5.8. For 1 ≤ i < j ≤ n, 1 ≤ h < k ≤ n and r, s ≥ 0, we have that [ei,j ;r , eh,k;s ] = ei,k;r+s δh,j − δi,k eh,j ;r+s . n , we have easily that Proof. By the defining relations for Y
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(i) [ei,i+1;r , ej,j +1;s ] = 0 if |i − j | = 1; (ii) [ei,i+1;r+1 , ej,j +1;s ] = [ei,i+1;r , ej,j +1;s+1 ] if |i − j | = 1; (iii) [ei,i+1;r , [ei,i+1;s , ej,j +1;t ]] = −[ei,i+1;s , [ei,i+1;r , ej,j +1;t ]] if |i − j | = 1. We also have by definition that (iv) ei,j ;r = [ei,j −1;r , ej −1,j ;0 ] for j > i + 1. Now we consider seven cases. (1) j < h. Obviously, [ei,j ;r , eh,k;s ] = 0. (2) j = h. By (ii) and (iv), [ej −1,j ;r , ej,j +1;s ] = ej −1,j +1;r+s . Now bracket with ej +1,j +2;0 , . . . , ek−1,k;0 to deduce that [ej −1,j ;r , ej,k;s ] = ej −1,k;r+s . Finally bracket with ej −2,j −1;0 , . . . , ei,i+1;0 . (3) i < h, j = k. Let us just show that [e1,3;r , e2,3;s ] = 0, since the general case is an easy consequence. Note that by (iii), [e1,3;r , e2,3;s ] = [[e1,2;r , e2,3;0 ], e2,3;s ] = −[[e1,2;r , e2,3;s ], e2,3;0 ]. By (ii) this equals −[[e1,2;r+s , e2,3;0 ], e2,3;0 ] which is zero by (iii). (4) i = h, j < k. Similar to (3). (5) i = h, j = k. If j = i + 1, we are done by (i); otherwise, [ei,j ;r , ei,j ;s ] = [[ei,j −1;r , ej −1,j ;0 ], ei,j ;s ] = [[ei,j −1;r , ei,j ;s ], ej −1,j ;0 ] + [ei,j −1;r , [ej −1,j ;0 , ei,j ;s ]] which is zero by (3) and (4). (6) i < h < j < k. We just show [e1,3;r , e2,4;s ] = 0. It equals [[e1,2;r , e2,3;0 ], [e2,3;0 , e3,4;s ]] = [e2,3;0 , [[e1,2;r , e2,3;0 ], e3,4;s ]] = [e2,3;0 , [e1,2;r , [e2,3;0 , e3,4;s ]]] = [[e2,3;0 , e1,2;r ], [e2,3;0 , e3,4;s ]] = −[[e1,2;r , e2,3;0 ], [e2,3;0 , e3,4;s ]]. Hence it is zero. (7) i < h, k < j . We just show [e2,3;r , e1,4;s ] = 0. It equals [e2,3;r , [e1,3;s , e3,4;0 ]] = [e1,3;s , e2,4;r ], which is zero by (6).
n is spanned by the set of monomials in {D (r) }i=1,...,n,r≥1 ∪ Lemma 5.9. The algebra Y i (r) (r) {Ei,j , Fi,j }1≤i<j ≤n,r≥1 , taken in some fixed order so that F ’s come before D’s and D’s come before E’s. +n Proof. Using Lemma 5.8, one shows easily that the associated graded algebra gr L Y (1 ) is spanned by the set of all ordered monomials in the elements {ei,j ;r }1≤i<j ≤n,r≥0 taken +n itself is spanned by the corresponding monomials in in some fixed order. Hence Y (1 ) (r) n , there is an antiautomorphism τ of {E }1≤i<j ≤n,r≥1 . By the defining relations for Y i,j
n fixing each D (r) and interchanging each E (r) with F (r) . It follows on applying τ Y i i,j i,j (r) −n , while that the set of monomials in {F }1≤i<j ≤n taken in some fixed order span Y i,j
(1 )
(r) (1n ) . Since obviously the ordered monomials in the elements {Di }i=1,...,n,r≥1 span Y − + (1n ) ⊗ Y n →Y n is n ⊗Y by the defining relations the natural multiplication map Y (1 ) (1 ) surjective, the lemma follows.
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Now, the first part of the proof of Theorem 5.2 above implies that there is a surjective n to the elements n → Yn sending D (r) , E (r) , F (r) ∈ Y algebra homomorphism θ : Y i i,j i,j with the same name in Yn . To complete the proof of Theorem 5.2 we need to show that θ is an isomorphism. This follows immediately from Lemma 5.10 below, since it shows n from Lemma 5.9 are linearly independent that the images of the monomials that span Y in Yn itself. (r)
(r)
(r)
Lemma 5.10. The set of monomials in {Di }i=1,...,n,r≥1 ∪{Ei,j , Fi,j }1≤i<j ≤n,r≥1 taken in some fixed order is linearly independent in Yn . Proof. As explained at the end of §2, we can identify the associated graded algebra (r+1) is identified with ei,j t r . It is easy to see from gr L Yn with U (gln [t]), so that gr Lr Ti,j (r+1)
(r+1)
(r+1)
resp. gr Lr Ei,j resp. gr Lr Fi,j is (5.2)–(5.4) that under this identification gr Lr Di r r r identified with ei,i t resp. ei,j t resp. ej,i t . Hence by the PBW theorem for U (gln [t]), the set of all monomials in (r+1)
{gr Lr Di
(r+1)
}i=1,...,n,r≥0 ∪ {gr Lr Ei,j
(r+1)
, gr Lr Fi,j
}1≤i<j ≤n,r≥0
taken in some fixed order forms a basis for gr L Yn . The lemma follows easily.
This completes the proof of Theorem 5.2. Let us also state the following theorem which was obtained in the course of the above proof; cf. [L]. (r)
Theorem 5.11. (i) The set of all monomials in {Di }i=1,...,n,r≥1 taken in some fixed order form a basis for Y(1n ) . (r) (ii) The set of all monomials in {Ei,j }1≤i<j ≤n,r≥1 taken in some fixed order form a + basis for Y(1 n). (r)
(iii) The set of all monomials in {Fi,j }1≤i<j ≤n,r≥1 taken in some fixed order form a − basis for Y(1 n). (r)
(r)
(r)
(iv) The set of all monomials in {Di }i=1,...,n,r≥1 ∪ {Ei,j , Fi,j }1≤i<j ≤n,r≥1 taken in some fixed order form a basis for Yn . Remark 5.12. Let us explain the relationship between the presentation given in Theorem 5.2 and Drinfeld’s presentation from [D2], since there are some additional shifts in u. Actually, the latter is a presentation for the subalgebra Y (sln ) = {x ∈ Yn | µf (x) = x for all f (u) ∈ 1 + C[[u−1 ]]}; ± for i = 1, . . . , n − 1 and k ≥ 0 from the see [MNO, Definition 2.14]. Define κi,k , ξi,k equations i u − i−1 Di+1 u − i−1 , κi,k u−k−1 := 1 + D (5.25) κi (u) = 2 2 k≥0
ξi+ (u)
=
k≥0
ξi− (u)
=
k≥0
+ −k−1 ξi,k u := Ei u −
i−1 2
− −k−1 ξi,k u := Fi u −
i−1 2
,
(5.26)
.
(5.27)
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One can check by equating coefficients in the identities from Lemmas 5.4, 5.5 and 5.7 that these elements generate Y (sln ) subject to the Drinfeld relations, namely: [κi,k , κj,l ] = 0, + − , ξj,l ] [ξi,k ± [κi,0 , ξj,l ]
(5.28)
= δi,j κi,k+l ,
(5.29)
± ±ai,j ξj,l ,
(5.30)
=
ai,j ± ± ± ± [κi,k , ξj,l+1 ] − [κi,k+1 , ξj,l ]=± + ξj,l κi,k ), (κi,k ξj,l 2 ai,j ± ± ± ± ± ± ± ± [ξi,k , ξj,l+1 ] − [ξi,k+1 , ξj,l ]=± ξi,k ), (ξ ξ + ξj,l 2 i,k j,l ± ± ± ± i = j, N = 1 − ai,j ⇒ Sym[ξi,k , [ξi,k , · · · [ξi,k , ξj,l ] · · · ]] = 0, 1 2 N
(5.31) (5.32) (5.33)
where (ai,j )1≤i,j
i u − i−1 − v − 2i − 21 Ei u − i−1 2 2 , Ei+1 v − 2 i i i = Ei u − i−1 2 Ei+1 v − 2 − Ei v − 2 Ei+1 v − 2 −Ei,i+2 u − i−1 + Ei,i+2 v − 2i 2 i − 21 Ei u − i−1 2 , Ei+1 v − 2 + + = 21 ξi+ (u)ξi+1 (v) + ξi+1 (v)ξi+ (u) −Ei v − 2i Ei+1 v − 2i + Ei,i+2 v − 2i . −Ei,i+2 u − i−1 2
+ (u − v)[ξi+ (u), ξi+1 (v)] =
Now equate u−k−1 v −l−1 -coefficients on both sides to get + + + + + + + + [ξi,k+1 , ξi+1,l ] − [ξi,k , ξi+1,l+1 ] = 21 ξi,k ξi+1,l + ξi+1,l ξi,k as required. (Beware: the relations (5.28)–(5.33) are actually not exactly the same as the + − relations in [D2] — one needs to swap ξi,k and ξi,k and replace κi,k with −κi,k to get those. The reason for the difference is that we have chosen to work with the opposite presentation to Drinfeld throughout.)
6. Parabolic Subalgebras Now we are ready to prove Theorems A and B stated in the introduction. Fix throughout the section a tuple ν = (ν1 , . . . , νm ) of non-negative integers summing to n. Note there is going to be some overlap between the notation here and that of the previous section, which is the special case ν = (1n ) of the present definitions. When necessary, we will add an additional superscript ν to our notation to avoid any ambiguity as ν varies. Factor the n × n matrix T (u) as T (u) = F (u)D(u)E(u)
(6.1)
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for unique block matrices
Iν1 0 E(u) = ...
D1 (u) 0 · · · 0 0 0 D2 (u) · · · D(u) = , .. .. .. ... . . . 0 0 · · · Dm (u) Iν1 E1,2 (u) · · · E1,m (u) Iν2 · · · E2,m (u) F1,2 (u) , F (u) = .. .. .. .. . . . .
0 Iν2 .. .
··· ··· .. .
0 0 , .. .
0 0 · · · Iνm F1,m (u) F2,m (u) · · · Iνm where Da (u) = (Da;i,j (u))1≤i,j ≤νa , Ea,b (u) = (Ea,b;i,j (u))1≤i≤νa ,1≤j ≤νb and Fa,b (u) = (Fa,b;i,j (u))1≤i≤νb ,1≤j ≤νa are νa ×νa , νa ×νb and νb ×νa matrices, respectively. Also a (u) = (D a;i,j (u))1≤i,j ≤νa by D a (u) := −Da (u)−1 . The define the νa × νa matrix D (r) a;i,j (u) = entries of these matrices define power series Da;i,j (u) = r≥0 Da;i,j u−r , D (r) (r) −r and F −r (r) −r a,b;i,j (u) = r≥0 Da;i,j u , Ea,b;i,j (u) = r≥1 Ea,b;i,j u r≥1 Fa,b;i,j u . (r) (r) We let Ea;i,j (u) = r≥1 Ea;i,j u−r := Ea,a+1;i,j (u) and Fa;i,j (u) = r≥1 Fa;i,j u−r := Fa,a+1;i,j (u) for short. Like before, there are explicit descriptions of all these elements in terms of quasideterminants. To write them down, write the matrix T (u) in block form as ν T1,1 (u) · · · ν T1,m (u) .. T (u) = ... , . ··· νT
m,1 (u)
· · · ν Tm,m (u)
where ν Ta,b (u) is a νa × νb matrix. Then, recalling the notation (4.3), ν T1,1 (u) · · · ν T1,a−1 (u) ν T1,a (u) .. .. .. .. . . . . , Da (u) = ν ν ν Ta−1,1 (u) · · · Ta−1,a−1 (u) Ta−1,a (u) ν T (u) · · · ν T ν T (u) a,a a,1 a,a−1 (u)
(6.2)
ν T1,1 (u) · · · ν T1,a−1 (u) ν T1,b (u) .. .. .. .. −1 . . . . , Ea,b (u) = Da (u) ν ν ν Ta−1,1 (u) · · · Ta−1,a−1 (u) Ta−1,b (u) ν T (u) · · · ν T ν T (u) a,b a,1 a,a−1 (u)
(6.3)
ν T1,1 (u) · · · ν T1,a−1 (u) ν T1,a (u) . . . .. .. .. .. . Da (u)−1 . Fa,b (u) = ν ν ν Ta−1,1 (u) · · · Ta−1,a−1 (u) Ta−1,a (u) ν T (u) · · · ν T ν T (u) b,a b,1 b,a−1 (u)
(6.4)
It follows in particular from these descriptions that for b > a + 1 and 1 ≤ i ≤ νa , 1 ≤ j ≤ νb , (r)
(r)
(1)
Ea,b;i,j = [Ea,b−1;i,k , Eb−1;k,j ],
(r)
(1)
(r)
Fa,b;j,i = [Fb−1;j,k , Fa,b−1;k,i ],
for any 1 ≤ k ≤ νb−1 . We also get the analogue of Lemma 5.1:
(6.5)
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Lemma 6.1. Fix a ≥ 1 and let ν¯ := (νa , νa+1 , . . . , νm ). Then, for all admissible i, j , (i) ν Da;i,j (u) = ψν1 +···+νa−1 (ν¯ D1;i,j (u)); (ii) ν Ea;i,j (u) = ψν1 +···+νa−1 (ν¯ E1;i,j (u)); (iii) ν Fa;i,j (u) = ψν1 +···+νa−1 (ν¯ F1;i,j (u)). (r)
In particular Lemma 6.1(i) shows that the elements Da;i,j here are the same as the generators of the standard Levi subalgebra Yν introduced at the end of §4, so they satisfy the relations (4.8). We also let Yν+ resp. Yν− denote the subalgebra generated (r) (r) by {Ea;i,j }1≤a<m,1≤i≤νa ,1≤j ≤νa+1 ,r≥1 resp. {Fa;i,j }1≤a<m,1≤i≤νa+1 ,1≤j ≤νa ,r≥1 . The antiautomorphism τ has the properties τ (Da;i,j (u)) = Da;j,i (u), τ (Ea,b;i,j (u)) = Fa,b;j,i (u), τ (Fa,b;i,j (u)) = Ea,b;j,i (u),
(6.6) (6.7) (6.8)
so it leaves Yν invariant and interchanges Yν+ and Yν− . Finally, as in the introduction, set Yν := Yν Yν+ and Yν := Yν− Yν , giving the standard parabolic subalgebras of Yn of shape ν.
Remark 6.2. Here is an alternative definition of the algebras Yν .Yν and Yν just in terms of the Drinfeld generators from Theorem 5.2. First, Yν is the subalgebra of Yn gener(r) (r) (r) ated by all {Di }1≤i≤n,r≥1 together with {Ei , Fi }r≥1 for i ∈ {1, . . . , n} − {ν1 , ν1 + ν2 , . . . , ν1 + · · · + νm }. Then Yν resp. Yν is the subalgebra generated by Yν together (r) (r) with all remaining elements of {Ei }1≤i
(r)
(r)
We have now defined the elements Da;i,j , Ea;i,j and Fa;i,j appearing in Theorems A and B stated in the introduction. We are ready to explain the proofs of these theorems. Actually, the argument runs almost exactly parallel to the proofs of Theorems 5.2 and 5.11 given in the previous section. As before, there are two parts: first, to show all the relations (1.1)–(1.14) from Theorem A hold; second, to show we have enough relations by constructing the PBW bases described in Theorem B. For the first part, one uses Lemma 6.1, (4.8) and (6.6)–(6.8) to reduce the problem to checking the following special cases: (1.4) with a = 1, b = 1 or a = 2, b = 1; (1.5) with a = 1, b = 1 or a = 2, b = 1; (1.7) with a = 1; (1.9) with a = 1; (1.11) with a = 1, b = 2; (1.13) with a = 2, b = 1 or a = 1, b = 2. These special cases may be deduced from the following four lemmas by equating coefficients. Note these lemmas are the exact analogues of Lemmas 5.4–5.7. Lemma 6.3. Suppose m = 2, i.e. ν = (ν1 , ν2 ). The following identities hold for all admissible h, i, j, k: (i) (u − v)[D1;i,j (u), E1;h,k (v)] = D1;i,p (u)(E1;p,k (v) − E1;p,k (u))δh,j ; 2;h,k (v)] = (E1;i,q (u) − E1;i,q (v))D 2;q,k (v)δh,j ; (ii) (u − v)[E1;i,j (u), D 1;i,k (v)D2;h,j (v) − D 1;i,k (u)D2;h,j (u); (iii) (u − v)[E1;i,j (u), F1;h,k (v)] = D (iv) (u − v)[E1;i,j (u), E1;h,k (v)] = (E1;i,k (v) − E1;i,k (u))(E1;h,j (v) − E1;h,j (u)). (Here, p resp. q should be summed over 1, . . . , ν1 resp. 1, . . . , ν2 .)
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Proof. Compute the ei,j ⊗eh,ν1 +k -, ei,ν1 +j ⊗eν1 +h,ν1 +k -, ei,ν1 +j ⊗eν1 +h,k - and ei,ν1 +j ⊗ eh,ν1 +k -coefficients on each side of (2.9) and then rearrange the resulting identities like we did in the proof of Lemma 5.4 above to obtain: 2;q,k (v)] = (i) (u − v)[D1;i,j (u), E1;h,q (v)D 2;q,k (v)δh,j ; D1;i,p (u)(E1;p,q (v) − E1;p,q (u))D (ii) (u − v)[D1;i,p (u)E1;p,j (u), D2;h,k (v)] = 2;q,k (v)δh,j ; D1;i,p (u)(E1;p,q (u) − E1;p,q (v))D (iii) (u − v)[D 1;i,p (u)E1;p,j (u), D2;h,q (v)F1;q,k (v)] = 2;h,q (v) D2;q,j (u) + (F1;q,p (u) − F1;q,p (v))D1;p,p (u)E1;p ,j (u) δi,k D 1;p,k (v) + (E1;p,q (v) − E1;p,q (u))D 2;q,q (v)F1;q ,k (v) δh,j ; − D1;i,p (u) D (iv) [D1;i,p (u)E1;p,j (u), E1;h,q (v)D2;q,k (v)] = 0. (Here, p, p resp. q, q should also be summed over 1, . . . , ν1 resp. 1, . . . , ν2 .) Now (i), (ii) and (iii) are deduced from (i) , (ii) and (iii) by simplifying exactly like we did in the proof of Lemma 5.4. It turns out to be more difficult than before to deduce (iv) from (iv) so we explain this part of the argument more carefully. As before, one rewrites (iv) using (i) and (ii) to obtain: (iv) (u−v)2 [E1;i,j (u), E1;h,k (v)] = (E1;i,j (v)−E1;i,j (u))(E1;h,k (u)−E1;h,k (v))+ (u − v)E1;h,j (v)(E1;i,k (v) − E1;i,k (u)) + (u − v)(E1;i,k (u) − E1;i,k (v))E1;h,j (u). Now we deduce (iv) from this. For a power series X in Yn [[u−1 , v −1 ]], let us write {X}d for the homogeneous component of X of total degree d in the variables u−1 and v −1 . We show by induction on d = 1, 2, . . . that (u − v){[E1;i,j (u), E1;h,k (v)]}d+1 = {(E1;i,k (v) − E1;i,k (u))(E1;h,j (v) − E1;h,j (u))}d . For the base case d = 1, applying {.}0 to (iv) shows (u−v)2 {[E1;i,j (u), E1;h,k (v)]}2 = 0, hence (u−v){[E1;i,j (u), E1;h,k (v)]}2 = 0 as required. For the induction step, assume the statement is true for d > 1. Apply {.}d to (iv) to get that (u − v)2 {[E1;i,j (u), E1;h,k (v)]}d+2 = (u − v){E1;h,j (v)(E1;i,k (v) − E1;i,k (u))}d+1 −(u − v){(E1;i,k (v) − E1;i,k (u))E1;h,j (u)}d+1 − {(E1;i,j (v)−E1;i,j (u))(E1;h,k (v)−E1;h,k (u))}d . Now use the induction hypothesis, together with the identity {[E1;h,j (v), E1;i,k (v)]}d+1 = 0 which follows by dividing both sides of the induction hypothesis by (u − v) then setting u = v, to rewrite the right hand side to deduce that (u − v)2 {[E1;i,j (u), E1;h,k (v)]}d+2 = (u − v){(E1;i,k (v) − E1;i,k (u))E1;h,j (v)}d+1 −(u − v){(E1;i,k (v) − E1;i,k (u))E1;h,j (u)}d+1 . Dividing both sides by (u − v) completes the proof of the induction step.
Lemma 6.4. Suppose m = 3, i.e. ν = (ν1 , ν2 , ν3 ). The following identities hold for all admissible g, h, i, j, k: (i) [E1;i,j (u), F2;h,k (v)] = 0; (ii) (u − v)[E1;i,j (u), E2;h,k (v)] = (E1;i,q (u)E2;q,k (v) − E1;i,q (v)E2;q,k (v) − E1,3;i,k (u) + E1,3;i,k (v))δh,j ;
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(iii) [E1,3;i,j (u), E2;h,k (v)] = E2;h,j (v)[E1;i,g (u), E2;g,k (v)]; (iv) [E1;i,j (u), E1,3;h,k (v)−E1;h,q (v)E2;q,k (v)] = −[E1;i,g (u), E2;g,k (v)]E1;h,j (u). (Here, q should be summed over 1, . . . , ν2 .) Proof. One computes the ei,ν1 +j ⊗eν1 +ν2 +h,ν1 +k -, ei,ν1 +j ⊗eν1 +h,ν1 +ν2 +k -, ei,ν1 +ν2 +j ⊗ eν1 +h,ν1 +ν2 +k - and ei,ν1 +j ⊗eh,ν1 +ν2 +k -coefficients of (2.9) respectively like in the proof of Lemma 5.5 to obtain the identities 3;h,r (v)F2;r,k (v)] = 0; (i) [D1;i,p (u)E1;p,j (u), D 3;r,k (v)] = D1;i,p (u)(E1;p,q (u)E2;q,r (v) (ii) (u−v)[D1;i,p (u)E1;p,j (u), E2;h,r (v)D 3;r,k (v)δh,j ; − E1;p,q (v)E2;q,r (v) + E1,3;p,r (v) − E1,3;p,r (u))D 3;r,k (v)] = 0; (iii) [D1;i,p (u)E1,3;p,j (u), E2;h,r (v)D 3;r,k (v)] = 0. (iv) [D1;i,p (u)E1;p,j (u), (E1,3;h,r (v) − E1;h,q (v)E2;q,r (v))D (Here, p, q and r sum over 1, . . . , ν1 , 1, . . . , ν2 and 1, . . . , ν3 respectively.) Now (i)–(iv) are deduced from (i) –(iv) by copying the arguments from the proof of Lemma 5.5.
Lemma 6.5. Suppose m = 3, i.e. ν = (ν1 , ν2 , ν3 ). The following identities hold for all admissible f, g, h, i, j, k: (i) [[E1;i,j (u), E2;h,k (v)], E2;f,g (v)] = 0; (ii) [E1;i,j (u), [E1;h,k (u), E2;f,g (v)]] = 0. Proof. Dividing both sides of Lemma 6.3(iv) by (u − v) then setting v = u shows that [Ea;i,j (u), Ea;h,k (u)] = 0. Given this and Lemma 6.4(ii), (i) is obvious unless f = h = j and (ii) is obvious unless f = k = j . Now the proof in these cases is completed exactly like the proof of Lemma 5.6.
Lemma 6.6. Suppose m = 3, i.e. ν = (ν1 , ν2 , ν3 ). The following identities hold for all admissible f, g, h, i, j, k: (i) [[E1;i,j (u), E2;h,k (v)], E2;f,g (w)] + [[E1;i,j (u), E2;h,k (w)], E2;f,g (v)] = 0; (ii) [E1;i,j (u), [E1;h,k (v), E2;f,g (w)]] + [E1;i,j (v), [E1;h,k (u), E2;f,g (w)]] = 0. Proof. Show that (u − v)(u − w)(v − w)[[E1;i,j (u), E2;j,k (v)], E2;f,g (w)] is symmetric in v and w and that (u − v)(u − w)(v − w)[E1;i,j (u), [E1;h,k (v), E2;k,g (w)]] is symmetric in u and v, following the argument of Lemma 5.7 exactly.
n denote the algebra with generaNow we consider the second part of the proof. Let Y (r) (r) tors and relations as in the statement of Theorem A. Define elements Ea,b;i,j , Fa,b;j,i ∈ n by Eqs. (6.5). We need to check that these definitions are independent of the particular Y (r) (s) choice of k. Well, given 1 ≤ k, k ≤ νb−1 with k = k , we have that [Ea,b−1;i,k , Eb−1;k ,j ] (1)
= 0 by (1.11). Bracketing with Db−1;k,k and using (1.5), one deduces that (r)
(s)
(r)
(s)
[Ea,b−1;i,k , Eb−1;k,j ] = [Ea,b−1;i,k , Eb−1;k ,j ]
(6.9)
(r)
as required to verify that the definition of the elements Ea,b;i,j is independent of the (r)
choice of k. A similar argument shows that the definition of the elements Fa,b;j,i is independent of k too.
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ν , Y ν+ and Y ν− denote the subalgebras of Y n generated by the D’s, E’s and Let Y F ’s respectively. By the first part of the proof, there is a surjective homomorphism n → Yn sending Y ν onto Yν and Y ν± onto Yν± . We just need to show that θ is an θ :Y isomorphism. This is done just like in the previous section by exhibiting a set of monon whose image in Yn is linearly independent. We just explain the key mials that span Y step, namely, the analogue of Lemma 5.8 allowing one to construct the spanning set for ν+ . Given this, the rest of our earlier argument extends without further complication to Y complete the proof. Define a filtration ν+ ⊆ L1 Y ν+ ⊆ · · · L0 Y ν+ by declaring that the generators E (r) are of degree (r − 1). Let gr L Y ν+ denote of Y a;i,j the associated graded algebra. Letting na := ν1 + · · · + νa−1 for short, define ν+ ena +i,nb +j ;r := gr Lr Ea,b;i,j ∈ gr L Y (r+1)
for each 1 ≤ a < b ≤ m, 1 ≤ i ≤ νa , 1 ≤ j ≤ νb and r ≥ 0. Then: Lemma 6.7. For 1 ≤ a < b ≤ m, 1 ≤ c < d ≤ m, r, s ≥ 0 and all admissible h, i, j, k, we have that [ena +i,nb +j ;r , enc +h,nd +k;s ] = ena +i,nd +k;r+s δnc +h,nb +j − δna +i,nd +k enc +h,nb +j ;r+s . Proof. Like in the proof of Lemma 5.8, we split into seven cases: (1) b < c; (2) b = c; (3) a < c, b = d; (4) a = c, b < d; (5) a = c, b = d; (6) a < c < b < d; (7) a < c; d < b. Since the analysis of each of the cases is very similar to Lemma 5.8, we just illustrate the idea with the two hardest situations, both of which require the Serre relations. First we check for case (3) that [en1 +i,n3 +j ;r , en2 +h,n3 +k;s ] = 0. For any 1 ≤ g ≤ ν2 , ν that we have by (6.5) and the images of the relations (1.9) and (1.13) in gr L Y [en1 +i,n3 +j ;r , en2 +h,n3 +k;s ] = [[en1 +i,n2 +g;r , en2 +g,n3 +j ;0 ], en2 +h,n3 +k;s ] = −[[en1 +i,n2 +g;r , en2 +g,n3 +j ;s ], en2 +h,n3 +k;0 ] = −[[en1 +i,n2 +g;r+s , en2 +g,n3 +j ;0 ], en2 +h,n3 +k;0 ] = 0. Second we check for case (6) that [en1 +i,n3 +j ;r , en2 +h,n4 +k;s ] = 0. By the case (1), (6.5), (1.9) and (1.13), we have that [en1 +i,n3 +j ;r , en2 +h,n4 +k;s ] = [[en1 +i,n2 +h;r , en2 +h,n3 +j ;0 ], [en2 +h,n3 +j ;0 , en3 +j,n4 +k;s ]] = [en2 +h,n3 +j ;0 , [[en1 +i,n2 +h;r , en2 +h,n3 +j ;0 ], en3 +j,n4 +k;s ]] = [en2 +h,n3 +j ;0 , [en1 +i,n2 +h;r , [en2 +h,n3 +j ;0 , en3 +j,n4 +k;s ]]] = [[en2 +h,n3 +j ;0 , en1 +i,n2 +h;r ], [en2 +h,n3 +j ;0 , en3 +j,n4 +k;s ]] = −[[en1 +i,n2 +h;r , en2 +h,n3 +j ;0 ], [en2 +h,n3 +j ;0 , en3 +j,n4 +k;s ]] = −[en1 +i,n3 +j ;r , en2 +h,n4 +k;s ].
Hence it is zero.
This completes the proof of Theorems A and B.
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7. Centers and Centralizers In this section, we compute the centralizer in Yn of the standard Levi subalgebra Yν . The argument depends on the following auxiliary lemma, which is a generalization of [MNO, Proposition 2.12]; the proof given here is based on the argument in loc. cit.. Lemma 7.1. Let h be a reductive subalgebra of a finite dimensional Lie algebra g over C. Let c be the centralizer of h in g. Then, the centralizer of U (h[t]) in U (g[t]) is equal to U (c[t]). (We believe that the word “reductive” is unnecessary here, but we did not find a proof without it.) Proof. The symmetrization map S(g[t]) → U (g[t]) is an isomorphism of h[t]-modules. Using this, it suffices to show that the space of invariants of h[t] acting on S(g[t]) is S(c[t]). Since h is reductive, we can pick an ad h-stable complement c to c in g. Let x1 , . . . , xm be a basis for c and let xm+1 , . . . , xn be a basis for c. Let z be an h[t]-invariant in S(g[t]). Define h ≥ 0 to be minimal such that z has the form z= zd (x1 t h )d1 · · · (xm t h )dm d
summing over d = (d1 , . . . , dm ) with d1 , . . . , dm ≥ 0, where the coefficients zd are polynomials in the variables xi t k for i = 1, . . . , m and 0 ≤ k < h together with the k variables x i t for i = m + 1, . . . , n and k ≥ 0. Pick a basis y1 , . . . , yr for h and let [yi , xj ] = m k=1 ci,j,k xk for each j = 1, . . . , m. Acting on z with yi t ∈ h[t] and taking the coefficient of xk t h+1 gives the equation d
zd
m
ci,j,k dj (x1 t h )d1 · · · (xj t h )dj −1 · · · (xm t h )dm = 0,
j =1
for each i = 1, . . . , r and k = 1, . . . , m. Now fix d = (d1 , . . . , dm ) with d1 , . . . , dm ≥ 0. Taking the coefficient of (x1 t h )d1 · · · (xm t h )dm in our equation gives m
ci,j,k (dj + 1)zd+δj = 0
(i = 1, . . . , r, k = 1, . . . , m),
j =1
where d + δj denotes the tuple (d1 , . . . , dj + 1, . . . , dm ). Since h has no non-trivial invariants in c , the system of linear equations [yi , m j =1 λj xj ] = 0 (i = 1, . . . , r) has only the trivial solution λ1 = · · · = λm = 0. Equivalently, the system of equations m
ci,j,k λj = 0
(i = 1, . . . , r, k = 1, . . . , m)
j =1
has only the trivial solution λ1 = · · · = λm = 0 too. We deduce that (dj + 1)zd+δj = 0 for each j = 1, . . . , m. Hence zd = 0 for all non-zero d, which implies by the minimality of the choice of h that h = 0, hence that z ∈ S(c[t]).
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Now, working once more in terms of the usual Drinfeld generators from §5, define Cn (u) = Cn(r) u−r := D1 (u)D2 (u − 1) · · · Dn (u − n + 1). (7.1) r≥0 (r)
The importance of the elements Cn is due to the following theorem; cf. [MNO, Theorem 2.13]. We remark that this theorem implies in particular that the commutative subalgebra Y(1n ) of Yn is generated by the centers Z(Y1 ), Z(Y2 ), . . . , Z(Yn ) of the nested subalgebras Y1 ⊂ Y2 ⊂ · · · ⊂ Yn from (4.1). (1)
(2)
Theorem 7.2. The elements Cn , Cn , . . . are algebraically independent and generate the center Z(Yn ). (r)
Proof. First we check that the Cn are central. For this, it suffices to show that [Di (u)Di+1 (u − 1), Ei (v)] = 0 = [Di (u)Di+1 (u − 1), Fi (v)] for each i = 1, . . . , n − 1. Actually we just need to check the first equality, since the second then follows on applying τ . By Lemma 5.4(i), (u − v)Ei (v)Di (u) = (u − v − 1)Di (u)Ei (v) + Di (u)Ei (u). By (5.24), (u − v − 1)Ei (v)Di+1 (u − 1) = (u − v)Di+1 (u − 1)Ei (v) − Di+1 (u − 1)Ei (u − 1). Hence, setting v = u, Ei (u)Di+1 (u − 1) = Di+1 (u − 1)Ei (u − 1). Now calculate (u − v)Ei (v)Di (u)Di+1 (u − 1) using these identities to show that it equals (u − v)Di (u)Di+1 (u − 1)Ei (v). Hence [Di (u)Di+1 (u − 1), Ei (v)] = 0. Now we complete the proof by following the argument of [MNO, Theorem 2.13]. Recall the filtration (2.11) of Yn , with associated graded algebra gr L Yn = U (gln [t]). Let z = e1,1 + · · · + en,n ∈ gln . One checks from the definition (7.1) that gr Lr−1 Cn(r) = zt r−1 . By Lemma 7.1 (taking h = g = gln ) the center of U (gln [t]) is freely generated by the elements {zt r | r ≥ 0}. The theorem now follows on combining these two observations (r)
with the fact already proved that each Cn belongs to Z(Yn ). We now use essentially the same argument to prove the following theorem, which is a well known variation on a result of Olshanskii [O, §2.1]. Theorem 7.3. The centralizer of Ym in Ym+n is equal to Z(Ym )ψm (Yn ). Proof. Lemma 4.1 shows that Z(Ym )ψm (Yn ) centralizes Ym , so we just need to show that the centralizer is no larger. Consider the associated graded algebra gr L Ym+n = U (glm+n [t]). Since (r) gr Lr−1 ψm (Ti,j ) = em+i,m+j t r−1 , we have that gr L Ym = U (glm [t]) (where glm is embedded into the top left corner of glm+n ) and gr L ψm (Yn ) = U (gln [t]) (where gln is embedded into the bottom right corner of glm+n ). By Lemma 7.1, the centralizer of U (glm [t]) in U (glm+n [t]) is equal to Z(U (glm [t]))U (gln [t]). The theorem follows.
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Corollary 7.4. Let ν = (ν1 , . . . , νm ) be a tuple of non-negative integers summing to n. The centralizer of the Levi subalgebra Yν in Yn is equal to Z(Yν ). Proof. Proceed by induction on m, the case m = 1 being vacuous. By (4.5) Yν = Yν1 ψν1 (Yν¯ ) where ν¯ = (ν2 , . . . , νm ). By the theorem, the centralizer in Yn of Yν1 is Z(Yν1 )ψν1 (Yν2 +···+νm ). By induction, the centralizer in Yν2 +···+νm of Yν¯ is Z(Yν¯ ). Hence, the centralizer of Yν in Yn is Z(Yν1 )ψν1 (Z(Yν¯ )) = Z(Yν ). (Alternatively one can prove the corollary directly using Lemma 7.1 once more.)
Corollary 7.5. Y(1n ) is a maximal commutative subalgebra of Yn . Proof. By the previous corollary, Y(1n ) is its own centralizer.
8. Quantum Determinants In the literature, Drinfeld generators are usually expressed in terms of quantum determinants, rather than the quasi-determinants used up to now. In this section we complete the picture by relating quasi-determinants to quantum determinants. We begin by introducing quantum determinants following [MNO, §2]. Fix d ≥ 1 and let Ad ∈ Mn⊗d denote the antisymmetrization operator, i.e. the endomorphism v1 ⊗ · · · ⊗ vd → sgn(π )vπ1 ⊗ · · · ⊗ vπd π∈Sd
of the natural space (Cn )⊗d that Mn⊗d acts on. Note that A2d = (d!)Ad . We have the following fundamental identity: A[1,...,d] T [1,d+1] (u)T [2,d+1] (u − 1) · · · T [d,d+1] (u − d + 1) = d T [d,d+1] (u − d + 1) · · · T [2,d+1] (u − 1)T [1,d+1] (u)A[1,...,d] , d
(8.1)
equality written in Mn⊗d ⊗ Yn [[u−1 ]]; see [MNO, Proposition 2.4]. For tuples i = (i1 , . . . , id ) and j = (j1 , . . . , jd ) of integers from {1, . . . , n}, the quantum determinant Ti,j (u) ∈ Yn [[u−1 ]] is defined to be the coefficient of ei,j = ei1 ,j1 ⊗ · · · ⊗ eid ,jd ∈ Mn⊗d on either side of Eq. (8.1). Explicit computation using the left and the right-hand sides of (8.1) respectively gives that Ti,j (u) = sgn(π )Tiπ 1 ,j1 (u)Tiπ 2 ,j2 (u − 1) · · · Tiπ d ,jd (u − d + 1) (8.2) π∈Sd
=
sgn(π )Tid ,jπ d (u − d + 1) · · · Ti2 ,jπ 2 (u − 1)Ti1 ,jπ 1 (u),
(8.3)
π∈Sd
where Sd is the symmetric group. It is obvious from these formulae that Ti·π,j (u) = sgn(π )Ti,j (u) = Ti,j ·π (u)
(8.4)
for any permutation π ∈ Sd (acting naturally on the tuples i, j by place permutation). Using (8.4) one obtains further variations on the formulae (8.2)–(8.3) as in [MNO, Remark 2.8], for instance: Ti,j (u) = sgn(π )Ti1 ,jπ 1 (u − d + 1)Ti2 ,jπ 2 (u − d + 2) · · · Tid ,jπ d (u). (8.5) π∈Sd
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The following properties of quantum determinants are easily derived from (8.2)–(8.4), τ (Ti,j (u)) = Tj ,i (u), σ (Ti,j (u)) = Ti,j (−u + d − 1).
(8.6) (8.7)
In the special case i = j = (1, . . . , n), we denote the quantum determinant Ti,j (u) instead by Cn (u), i.e. Cn (u) := T(1,...,n),(1,...,n) (u).
(8.8)
We will show in Theorem 8.6 below that this agrees with the definition (7.1), hence the coefficients of the series Cn (u) generate the center of Yn , but we do not know this yet. The next few results taken from [NT] describe the effect of the maps and S on quantum determinants. Actually we do not need the first of these here, but include it for the sake of completeness. Lemma 8.1. Let i, j be d-tuples of distinct integers from {1, . . . , n}. Then, Ti,k (u) ⊗ Tk,j (u),
(Ti,j (u)) = k
where the sum is over all k = (k1 , . . . , kd ) with 1 ≤ k1 < · · · < kd ≤ n. Proof. See [NT, Proposition 1.11].
Lemma 8.2. Let i, j be d-tuples of distinct integers from {1, . . . , n}. Choose i = (id+1 , . . . , in ) and j = (jd+1 , . . . , jn ) so that {i1 , . . . , in } = {j1 , . . . , jn } = {1, . . . , n}, and let ε denote the sign of the permutation (i1 , . . . , in ) → (j1 , . . . , jn ). Then, ω(Ti,j (u)) = εCn (−u + n − 1)−1 Tj ,i (−u + n − 1). Proof. This is proved in [NT, Lemma 1.5] but for the opposite algebra, so we repeat the argument once more. By the identity (8.1) and the definition (8.8), we have that T [1,n+1] (u)T [2,n+1] (u − 1) · · · T [n,n+1] (u − n + 1) = Cn (u)[n+1] A[1,...,n] ; A[1,...,n] n n see also [MNO, Proposition 2.5]. Hence, T [1,n+1] (u) · · · T [d,n+1] (u − d + 1) = A[1,...,n] n T[n,n+1] (u − n + 1) · · · T[d+1,n+1] (u − d). (−1)n−d Cn (u)[n+1] A[1,...,n] n Now equate the e(i1 ,...,id ,in ,...,id+1 ),(j1 ,...,jd ,in ,...,id+1 ) -coefficients on each side and use (8.2) to deduce that Ti,j (u) = εCn (u)ω(Tj ,i (−u + n − 1)). The lemma follows on making some obvious substitutions.
Corollary 8.3. In the notation of Lemma 8.2, S(Ti,j (u)) = εCn (u + n − d)−1 Tj ,i (u + n − d). Proof. Recalling that S = ω ◦ σ , this is a consequence of (8.7) and Lemma 8.2. Corollary 8.4. S 2 (Ti,j (u)) = Cn (u + n)−1 Ti,j (u + n)Cn (u + n − 1).
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Now we describe the embedding ψm : Yn → Ym+n from §4 in terms of quantum determinants. Lemma 8.5. Let i, j be d-tuples of distinct integers from {1, . . . , n}. Then, ψm (Ti,j (u)) = Cm (u + m)−1 Tm#i,m#j (u + m), where m#i denotes the (m+d)-tuple (1, . . . , m, m+i1 , . . . , m+id ) and m#j is defined similarly. Proof. Calculate using (4.2) and Lemma 8.2 twice.
Now we can verify that Cn (u) as defined in this section is the same as the earlier definition (7.1). Note this theorem is theYangian analogue of [GKLLRT, Theorem 7.24]. Theorem 8.6. Cn (u) = D1 (u)D2 (u − 1) · · · Dn (u − n + 1). Proof. Recalling that Di (u) = ψi−1 (T1,1 (u)), Lemma 8.5 implies that Di (u) = Ci−1 (u + i − 1)−1 Ci (u + i − 1). The lemma follows easily from this by induction.
Finally, we apply Lemma 8.5 once more to express the Drinfeld generators from §5 in terms of quantum determinants; cf. [I, Theorem B.15] or [Cr, Proposition 3.2]. Theorem 8.7. For i ≥ 1, (i) Di (u) = T(1,...,i−1),(1,...,i−1) (u + i − 1)−1 T(1,...,i),(1,...,i) (u + i − 1); (ii) Ei (u) = T(1,...,i),(1,...,i) (u + i − 1)−1 T(1,...,i),(1,...,i−1,i+1) (u + i − 1); (iii) Fi (u) = T(1,...,i−1,i+1),(1,...,i) (u + i − 1)T(1,...,i),(1,...,i) (u + i − 1)−1 . Proof. Calculate using Lemmas 5.1 and 8.5 and the formulae (5.6) and (8.6).
Remark 8.8. Using Theorem 8.7, we can also express the generating functions κi (u) and ξi± (u) from Remark 5.12 in terms of quantum determinants: κi (u) = 1 − ai (u)−1 ai (u + 1)−1 ai−1 u + 21 ai+1 u + 21 ,
ξi+ (u) = ai (u)−1 bi (u), ξi− (u) = ci (u)ai (u)−1 ,
where ai (u) = T(1,...,i),(1,...,i) u + i−1 , bi (u) = T(1,...,i),(1,...,i−1,i+1) u + i−1 and 2 2 i−1 ci (u) = T(1,...,i−1,i+1),(1,...,i) u + 2 . Allowing for the fact that we are working with the opposite presentation to Drinfeld’s (cf. Remark 5.12) this is the “simpler isomorphism” recorded immediately before [D2, Theorem 2]. Acknowledgement. The second author would like to thank Arun Ram for stimulating conversations.
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References [BR]
Briot, C., Ragoucy, E.: RTT presentation of finite W -algebras. J. Phys. A 34, 7287–7310 (2001) [BK] Brundan, J., Kleshchev, A.: Shifted Yangians and finite W -algebras. Adv. Math., to appear [CP] Chari, V., Pressley, A.: A guide to quantum groups. Cambridge: CUP, 1994 [C1] Cherednik, I.: A new interpretation of Gelfand-Tzetlin bases. Duke Math. J. 54, 563–577 (1987) [C2] Cherednik, I.: Quantum groups as hidden symmetries of classic representation theory, In: “Differential geometric methods in theoretical physics (Chester, 1988)”, Singapore World Sci. Publishing, 1989, pp. 47–54 [Cr] Cramp´e, N.: Hopf structure of the Yangian Y (sln ) in the Drinfeld realization. J. Math. Phys. 45, 434–447 (2004) [D] Ding, J.: Partial Gauss decomposition, Uq (gln−1 ) in Uq (gln ) and the Zamolodchikov algebra. J. Phys. A 32, 671–676 (1999) [DF] Ding, J., Frenkel, I.: Isomorphism of two realizations of quantum affine algebra Uq (gl(n)). Commun. Math. Phys. 156, 277–300 (1993) [D1] Drinfeld, V.: Hopf algebras and the quantum Yang-Baxter equation. Soviet Math. Dokl. 32, 254–258 (1985) [D2] Drinfeld, V.: A new realization of Yangians and quantized affine algebras. Soviet Math. Dokl. 36, 212–216 (1988) [ES] Etingof, P., Schiffmann, O.: Lectures on quantum groups, Lectures in Mathematical Physics, Boston, MA: International Press, 1998 [FRT] Faddeev, L., Reshetikhin, N., Takhtadzhyan, L.: Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1, 193–225 (1990) [GKLLRT] Gelfand, I., Krob, D., Lascoux, A., Leclerc, B., Retakh, V., Thibon, J.-Y.: Non-commutative symmetric functions. Adv. Math. 112, no. 2, 218–348 (1995) [GR1] Gelfand, I., Retakh, V.: Theory of non-commutative determinants and characteristic functions of graphs. Funct. Anal. Appl. 26, 231–246 (1992) [GR2] Gelfand, I., Retakh, V.: Quasideterminants, I, Selecta Math. 3, 517–546 (1997) [I] Iohara, K.: Bosonic representations of Yangian double DY (g) with g = glN , slN . J. Phys. A 29, 4593–4621 (1996) [L] Levendorskii, S.: On PBW bases for Yangians. Lett. Math. Phys. 27, 37–42 (1993) [MNO] Molev, A., Nazarov, M., Olshanskii, G.: Yangians and classical Lie algebras. Russ. Math. Surveys 51, 205–282 (1996) [NT] Nazarov, M., Tarasov, V.: Representations of Yangians with Gelfand-Zetlin bases. J. Reine Angew. Math. 496, 181–212 (1998) [O] Olshanskii, G.: Representations of infinite dimensional classical groups, limits of enveloping algebras and Yangians. Adv. in Soviet Math. 2, 1–66 (1991) Communicated by L. Takhtajan
Commun. Math. Phys. 254, 221–253 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1202-8
Communications in
Mathematical Physics
Closed and Open Conformal Field Theories and Their Anomalies Po Hu1 , Igor Kriz2 1 2
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. E-mail:
[email protected] Mathematics Department, University of Michigan, Ann Arbor, MI 48109-1109, USA. E-mail:
[email protected]
Received: 26 January 2004 / Accepted: 7 April 2004 Published online: 14 October 2004 – © Springer-Verlag 2004
Abstract: We describe a formalism allowing a completely mathematical rigorous approach to closed and open conformal field theories with general anomaly. We also propose a way of formalizing modular functors with positive and negative parts, and outline some connections with other topics, in particular elliptic cohomology. 1. Introduction The main purpose of this paper is to give rigorous mathematical foundations for investigating closed and closed/open conformal field theories (CFT’s) and their anomalies. In physics, the topic of closed/open CFT has been extensively discussed in the literature (see e.g. [8, 22, 23, 7]). Our investigation was originally inspired by two sources: Edward Witten (cf. [41]) proposed a general program for using K-theory to classify stable D-branes in string theory. On the other hand, G. Moore and G. Segal [25] obtained a mathematically rigorous approach to classifying D-branes in the case of 2-dimensional topological quantum field theory (TQFT) (see also [26] for excellent detailed lectures on this topic). We attempted to consider a case in between, namely D-branes in conformal field theory. We point out that G. Moore in [26] has also considered this direction, and Yi-Zhi Huang and L. Kong [18] in parallel with our investigation developed a vertex operator algebra approach to open CFT, so there is overlap with existing work. However, we will end up exploring the topic in a somewhat different light, as will become apparent below. In particular, our investigation will lead us to a new approach to anomaly in rational CFT (RCFT), and D-brane categories, using 2- and 3-vector spaces. This will also lead to constructions which relate to certain mathematical topics, such as foundation of elliptic cohomology. Even in this direction, however, substantial inroads have already been made in the literature, in particular [30, 12, 13]. The main contribution of the present paper therefore is that we set out to proceed with complete mathematical rigor.
The authors were supported by the NSF.
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There are many reasons for D-branes to be easier in CFT than in string theory. First of all, one does not have to insist on anomaly cancellation, and can investigate the anomaly instead. Here by anomaly we mean a certain indeterminacy (usually finite-dimensional), allowed in the correlation functions of a CFT (much more precise discussion will be given later: this is, in some sense, the main subject of the present paper). Another reason is that CFT makes good sense even without supersymmetry, while string theory does not. (In fact, in this paper, we restrict attention to non-supersymmetric CFT, although mostly just in the interest of simplicity.) Most importantly, however, any CFT approach to string theory amounts to looking at the complicated string moduli space only through the eyes of one tangent space, which is a substantial simplification. We will make some comments on this relationship between CFT and string theory in Sect. 3 below. We should point out that in this paper, we restrict attention to CFT’s defined in oriented surfaces. This is not a restriction on our formalism, which works in the unoriented case as well, but it simplifies the discussion somewhat. However, interesting phenomena certainly arise when considering CFT’s in the non-oriented worldsheet context, even for example in the case of the critical Ising model [32, 13]. On the other hand, CFT is incomparably more complicated than 2-dimensional TQFT. Because of this, in fact, a theorem classifying D-branes (as outlined in [25] for TQFT) seems, at the present time, out of reach for CFT. However, an exact mathematical definition of the entire structure of closed/open CFT is a reasonable goal which we do undertake here. To this end, we use our formalism of stacks of lax commutative monoids with cancellation (SLCMC’s), developed in [16], and reviewed in the Appendix. In Sect. 2 below, we shall also describe analogues of some of the concepts of [25] for CFT’s, and observe some interesting new phenomena. For example, one may ask what is the correct generalization of the category of modules over the algebra corresponding to the closed sector of 2-dimensional TQFT in defining a D-brane category. We will see fairly quickly how going in this direction leads to 2-vector spaces in the case of CFT. In Sect. 4, we give a basic example, the free bosonic CFT (=linear σ -model), and show how to obtain the D-brane modules corresponding to Von Neumann and Dirichlet boundary conditions for open strings. As we will see, however, even in this basic case, a substantial complication is giving a mathematically rigorous treatment of the convergence issues of the CFT. Up to this point, we suppressed the discussion of anomaly, by assuming that anomaly is 1-dimensional. However, there is an obvious suggestion: a parallel between the set of D-branes of a closed/open CFT, and the set of labels of a modular functor of RCFT, see [27–29, 35]. It therefore seems we should look for axioms for the most general possible kind of anomaly for closed/open CFT, which would include sets of both D-branes and modular functor labels. There are, however, further clues which suggest that the notion of “sets” in this context is too restrictive. Notably, the free C-vector space CS on the set of labels S of a modular functor is the well known Verlinde algebra [40]. But the multiplication rule of the Verlinde algebra uses only dimensions of vector spaces involved in the modular functor, so it seems that if one wants to consider the spaces themselves, it is that one should consider, instead of CS, the free 2-vector space on S). Is it possible to axiomatize modular functors for RCFT’s in a way which uses 2-vector spaces in place of sets of labels? In Sect. 5, we answer this last question in the affirmative. This is rather interesting, because it leads to other questions: the authors [16] previously proposed RCFT as a possible tool for geometrically modelling elliptic cohomology, while Baas-Dundas-Rognes
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[2] obtained a version of elliptic cohomology based on 2-vector spaces. Is there a connection? Observations made in [2] show that the right environment of such discussion would be a suitable group completion of the symmetric bimonoidal category C2 of vector spaces, while simultaneously noting the necessary difficulty of any such construction. Nevertheless, we propose in Sect. 6, such group completion, despite major technical difficulties. Our construction involves super-vector spaces, and thus suggests further connections with P. Deligne’s observation [19] that the modulars functor of bc-systems must be considered as super-vector spaces, and with the work of Stolz-Teichner [38], who, in their approach to elliptic objects, also noticed the role of fermions and, in effect, what amounts to 1-dimensional super-modular functors. In D-brane theory, this same construction allows axiomatization of anti-D-branes with 1-dimensional anomaly. However, let us return to D-branes. Is it possible to formulate axioms for general anomalies of closed/open CFT’s analogous to the 2-vector space approach to modular functors? We give, again, an affirmative answer, although another surprise awaits us here: while the “set of labels” of a modular functor was naturally a 2-vector space, the “set of D-branes” of a closed/open CFT must be a 3-vector space! We discuss this, and propose axioms for a general anomaly of closed/open CFT in Sect. 7. An intriguing problem is to extend the group completion approach of Sect. 6 to the case of general closed/open CFT anomaly, which would give an axiomatization of anti-D-branes in that context. 2. Closed/Open CFT’s with 1-Dimensional Anomaly, D-Brane Modules and D-Brane Cohomology There is substantial physical literature on the subject of D-branes (see e.g. [8, 22, 23, 7]). In this paper, we shall discuss a mathematically rigorous approach to D-branes in non-supersymmetric CFT’s. Moreover, in Sects. 2-4, we shall restrict attention to 1dimensional anomaly allowed both on the closed CFT and the D-brane. More advanced settings will be left to the later sections. We begin by defining the stack of lax commutative monoids with cancellation (SLCMC) corresponding to oriented open/closed string (more precisely conformal field) theory. SLCMC’s were introduced in [16], but to make this paper self-contained, we review all the relevant definitions in the Appendix. We consider a set L. This is not our set of labels, it is the set of D-branes. Our set of labels consists of K = L × L which we will call open labels and we will put K = K {1}, where 1 is the closed label. We will now define the SLCMC D of closed/open worldsheets, which we will need for oriented closed/open CFT. We shall first define the LCMC of its sections over a point. As usual (see [16]), the underlying lax commutative monoid is the category of finite sets labelled by a certain set K, not necessarily finite. Before describing the exactly correct analytic and conformal structure, we first specify that these are compact oriented surfaces (2-manifolds) X together with homeomorphic embeddings ci : S 1 → ∂X, dj : I → ∂X with disjoint images. Moreover, each ci is labelled with 1, and each dj is labelled by one of the open labels K . Moreover, each connected component of ∂X − I m(ci ) − I m(dj ) (which we shall call D-brane components) is labelled with an element of L, and each dj is labelled with the pair (1 , 2 ) ∈ L × L of D-branes which the beginning point and endpoint of dj abut. The ci ’s and dj ’s are considered inbound or outbound depending on the usual comparison of their orientation with the orientation of X.
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It is now time to describe the smoothness and conformal structure on X. To this end, we simply say that X is a smooth complex 1-manifold with analytic (real) boundary and corners; this means that the interior of X is a complex 1-manifold, and the neighbourhood of a boundary point x of X is modeled by a chart whose source is either an open subset of the halfplane H = {z ∈ C|I m(z) ≥ 0}, where 0 maps to x or an open subset of the quadrant K = {z ∈ C|I m(z) ≥ 0, Re(z) ≥ 0}, where 0 maps to x; the transition maps are (locally) holomorphic maps which extend biholomorphically to an open neighbourhood of 0 in C. The points of the boundary whose neighbourhoods are modelled by open neighborhoods of 0 in K are called corners. Further, we specify exactly which points are the corners of an open/closed string world sheet: we require that the corners be precisely the endpoints of the images of open string boundary parametrizations. We also require that open as well as closed string parametrizations be real-analytic diffeomorphisms onto their image; this completes the definition of the objects. An isomorphism X → Y is a diffeomorphism which preserves complex structure, D-brane labels, is smooth on the interior, and commutes with the ci , dj (which we shall call parametrization components - note that the set of parametrization components is not ordered, so an automorphism may switch them). Now to define the SLCMC D, the main issue is fixing the Grothendieck topology. We use simply finite-dimensional smooth manifolds with open covers. As usual, the underlying stack of lax commutative monoids is the stack of covering spaces with locally constant K-labels (analogously to [16]). Sections of D over M are smooth manifolds fibered over M, where the fibers are elements of D, and the structure varies smoothly in the obvious sense. It is important, however, to note that it does not seem possible to define this stack over the Grothendieck topology of complex manifolds and open covers; in other words, it does not appear possible to discuss chiral CFT’s with D-branes. To see this, we consider the following Example. The moduli space of elliptic curves E with an unparametrized hole (i.e. one closed D-brane component with a given label). It is easily seen that the moduli space of such worldsheets is the ray (0, ∞), i.e. not a complex manifold. To see this, the key point is to notice that the invariant I m(τ ) of the elliptic curve F obtained by attaching a unit disk to E along the D-brane component does not depend on its parametrization. Intuitively, this seems plausible since I m(τ ) is the “volume”. To rigorize the argument, we first recall that if one cuts the elliptic curve along a non-separating curve, then I m(τ ) can be characterized as the “thickness” of the resulting annulus (every annulus is conformally equivalent to a unique annulus of the form S 1 × [0, r] for some boundary parametrization; r is the thickness). But now any reparametrization of the D-brane component c of E is a composition of reparametrizations which are identity outside of a certain small interval J ⊂ c. Thus, it suffices to show that the invariant I m(τ ) does not change under such reparametrizations. However, we can find a smooth non-separating curve d ⊃ J in F ; then cutting F along d, the so-called change of parametrization of c becomes simply a change of parametrization of one of the boundary components of F ; we already know that does not affect thickness. By a K-labelled closed/open CFT with 1-dimensional anomaly (Hi )i∈K we shall mean a CFT with 1-dimensional modular functor on the SLCMC D over the stack of lax monoids SK , with target in the SLCMC H K . This means a lax morphism of SLCMC’s D˜ → (H i )i∈K where D˜ is a C× -central extension of D. These concepts were defined in [16] (see the Appendix for a review). When K is not mentioned (i.e. we speak of just a closed/open
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CFT), we shall assume that there is only one D-brane, i.e. K = {1, m}, where m is the open label. Now, however, we would like to ask a more fundamental question: what are Dbranes? The answer “elements of the set L associated with a K-labelled closed/open CFT” is clearly not a satisfactory one. For one thing, D-branes should form an additive category, not a set. Thus, let us look to Moore-Segal ([25], Sect. 3) for a guideline. There, in the case of a 2-dimensional topological quantum field theory (TQFT), a candidate for an additive category of D-branes is proposed (at least in the semisimple case): the category of modules over the closed sector algebra. Let us review these results of [25] in more detail: a (2-dimensional closed TQFT) is a Poincar´e algebra C, which is the same thing as a commutative Frobenius algebra. The first theorem of [25], Sect. 3 is that a closed/open 2-dimensional TQFT is equivalent to the following additional set of data: A (not necessary commutative) Frobenius algebra O and a map of algebras C→O satisfying the Cardy condition, which asserts equality between the operations corresponding to cutting an annulus which has one parametrization component and one Dbrane component on each boundary component in two ways: either by an open string (curve) connecting the two D-brane components, or by a closed string (curve) separating the two boundary components. The second theorem of [25], Sect. 3 states that when C is semisimple, then O = EndC (M) for some C-module M. Therefore, the category of C-modules is the correct candidate for a category of D-branes. Which of these concepts have we extended to CFT so far? The algebra O is no problem: it corresponds simply to a closed/open CFT with one D-brane. However, what is the analogue of a C-module M? We shall explain why we think it may be too naive to simply search for some suitable concept of such a C-module (e.g. some modification of VOA module) which would do the job. The key point is that even in the case of TQFT, C-modules are only the right answer when C is semisimple. When C is not semisimple, it is not obvious how to make, for a general C-module M, the algebra EndC (M) into an open sector of a closed/open TQFT. It is the Cardy condition which causes trouble. When C is semisimple, the simple summands themselves are open sectors of closed/open TQFT’s, and moreover, they can be summed, because semisimplicity makes 0 the only possible choice for mixed sectors. Now for a (non-chiral) CFT, semisimplicity seems like a completely unnatural assumption. For example, for a 1-dimensional free bosonic CFT (see Sect. 4 below), we have different irreducible Dirichlet branes corresponding to points of spacetime, which certainly have non-trivial, and interesting, mixed sectors. In conclusion, therefore, for a closed CFT H , there does not seem to be a satisfactory notion of “H -module” M which would always make (H, M ⊗ M ∗ ) (or some similar construction) into a closed/open CFT: the reason is that it doesn’t quite work even for TQFT’s, in the general case. On the other hand, there is something we can do. Suppose (H, R) is a closed/open CFT (R is the open sector) and V is a finite-dimensional vector space over C. Then there is a canonical way of making (H, V ⊗ R ⊗ V ∗ ) into a closed/open CFT with one D-brane: in the open CFT operations, simply insert units and traces C → V ⊗ V ∗ , V ⊗V ∗ → C wherever suggested by the diagrams. This is like putting a |V |-dimensional linear Chan-Paton charge on the ends of the open strings. Therefore, if we characterized a D-brane D by the closed/open CFT (H, R), then (H, V ⊗ R ⊗ V ∗ ) should represent
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V ⊗ D. This gives as an important clue: the category of D-branes should be a lax module over the symmetric bimonoidal category C2 of finite-dimensional complex vector spaces (see the Appendix, and also, for more detail, [9]). At the same time, however, this presents a puzzle, namely that there is no such obvious candidate for the sum: If the closed/open CFT (H, R) represents a D-brane D and (H, S) represents E, what closed/open CFT represents D ⊕ E? The problem is the same as above: there doesn’t seem to be enough information to recover the mixed open sector of strings stretched between the D-branes D and E. It seems at this point that we have no choice but to axiomatize the whole structure we wish to have, including a category of D-branes which is a lax module over C2 . However, even writing the axioms correctly is a challenge. We will give the answer in full generality (not assuming that the anomaly is necessarily 1-dimensional) in Sect. 7 below. But since we already gave a substantial discussion of this topic in this section, let us work out the solution here in the case of a 1-dimensional anomaly. This case is of particular interest from the point of view of string quantization, and it is also a case in which there is a substantial simplification. So we already know that we want a “D-brane category” A which is a lax module over C2 . For simplicity, let us further require that this be a finite-dimensional free module, i.e. equivalent to a sum of finitely many copies of C2 . Now the closed sector H is a Hilbert space and the open sector R is an object of the category ilb . A ⊗ C 2 A ∗ ⊗C 2 C H 2 ilb is the symmetric bimonoidal category of Hilbert spaces (see Sect. 5 below), Here CH 2 and ⊗C2 denotes lax extension of scalars in the obvious sense (see [9] for reference). From this data, we can construct an SLCMC, which we denote by C(A, H, R) and which we will now describe. As usual, there is no difficulty with extending the construction to families, so we will limit ourselves to the LCMC, i.e. to sections over one point. Even before getting into that, however, there is another wrinkle which we must mention (see also Sect. 7 below). To keep track of which open parametrization components share a boundary component, and in what order and orientation, one must introduce a separate SLCMC of all such configurations, which we will call incidence graphs. Therefore, more precisely, an incidence graph will encode (1) a set of closed parametrization components together with their orientations, (2) a set of (unlabelled) closed D-brane boundary components, (3) cyclically ordered sets of open parametrization components in the same boundary component, together with their orientations. Now our construction will come with a map of SLCMC’s, C(A, H, R) → .
To use the same notation as in Sect. 7, we will decorate sets S of boundary components with two indices, the first of which will specify closed or open label (1 or m), and the second will specify inbound or outbound orientation (in or out). Then the set of sections of C(A, H, R) (over a point) over a given configuration of is ˆ ˆ H ∗⊗ ˆ ˆ R ⊗ ˆ ˆ H. trcyclic ˆ R ∗ ⊗ Sm,in
Sm,out
S1,in
S1,out
Here trcyclic denotes the tensor product over C2 of copies of the canonical functor tr : A∗ ⊗C2 A → C2
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over all pairs of ends of open strings which share a D-brane in the given configuration of (note that the orientations are arranged in such a way that one copy of A∗ and one copy of A always arise in such case). In view of this discussion, we can define a closed/open CFT with D-brane category A with 1-dimensional anomaly as a (lax) map of SLCMC’s over D˜ {1,m} → C(A, H, R), where D˜ {1,m} is a C× -central extension of the SLCMC D{1,m} . We will see that the case of general anomaly (where different D-branes can have different anomalies, possibly multidimensional) is still more complicated, by yet another level. However, we leave this to Sect. 7 below. 3. Conformal Field Theory and String Theory We will now consider the relationship between conformal field theory and string theory, and the way it reflects on our investigation. As a standing reference on string concepts, we recommend one of the standard textbooks on the subject, e.g. [14] or [31]. One added feature of string theory is, of course, supersymmetry, but we shall soon see that this turns out not to be the only complication. We will, therefore, begin our discussion with bosonic string theory (superstrings will enter later). The essential point of string quantization is that conformal field theory quantizes parametrized strings, while physical strings should be unparametrized. Now to pass from parametrized strings to unparametrized, one needs a way to quantize the complex structure. This problem is analogous to gauge fixing in gauge theory. In fact, this is more than just an analogy: from a strictly worldsheet point of view, conformal field theory is indeed a 2-dimensional quantum field theory satisfying Schwinger axioms, and can be viewed as a gauge theory in a certain sense; however, we do not need to pursue this here. The important point is that in string theory, complex structure gauge is needed to produce a consistent theory: conformal field theory is anomalous and, in Minkowski space, contains states of negative norm. The modern approach to gauge fixing in gauge theory, and to string quantization, is through Fadeev-Popov ghosts and BRST cohomology. In the string theory case, we start with a CFT Hm , the (matter CFT). In this case, BRST cohomology is essentially a semiinfinite version of Lie algebra cohomology of the complexified Witt algebra (viewed as a “Lie algebra of the semigroup of annuli”) with coefficients in Hm . To be precise about this, we must describe the semi-infinite analogue of the complex (g) for a Lie algebra g where g is the Witt algebra. As it turns out, this semi-infinite Lie complex is also a CFT which is denoted as Hgh and called the Fadeev-Popov ghost CFT. A mathematical description is outlined in [35], and given in more detail in [19]. In the chiral CFT case, Hgh is a Hilbert completion (with a chosen Hilbert structure) of (bn |n < 0) ⊗ (cn , n ≤ 0).
(1)
In the physical case, both chiralities are present, and Hgh is a Hilbert completion of the tensor product of (1) with its complex conjugate. To understand why this is a semi-infinite Lie complex of the Witt algebra, we write the generators of (1) as d , dz cn = z−n−2 (dz)2 .
bn = z−n+1
(2)
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So, the bn ’s are vector fields on S 1 (elements of the Witt algebra) and the cn ’s are dual to the b−n ’s. An ordinary Lie complex would be the exterior algebra on the duals cn , n ∈ Z. However, a special feature of CFT is a choice of vacuum which allows us, even before gauge fixing, to define correlation functions which are finite, albeit anomalous: this is the mathematical structure known as Segal-type CFT, which we have axiomatized in [16] and here. However, this choice of vacuum of Hm is what prompts the “semi-infinite approach”, where the exterior generators of Hgh are not all of the cn ’s, but half of the cn ’s and half of the bn ’s, as in (1). Now it turns out that for our further discussion, it will be important to know explicitly one part of the Virasoro action on Hgh , namely the conformal weights, or eigenvectors of L0 . One may guess that bn , cn should be eigenvectors of conformal weight −n, but it turns out that one must decrease the conformal weights of the entire complex (1) by 1, so the correct conformal weight of a monomial in the bn ’s and cn ’s is −k − 1, where k is the sum of the subscripts of the bn ’s and cn ’s in the monomial. In fact, it turns out that the vacuum of the ghost theory, i.e. the element of Hgh assigned to the unit disk, is b−1 c0 .
(3)
Now the ghost CFT has an anomaly which is described by a 1-dimensional modular functor L which has central charge −26 in the chiral case (see [35, 19]) and (−26, −26) in the physical case. (In the chiral case, there is an additional complication that L must be considered a super-modular functor, see [19] and Sect. 5 below.) A CFT Hm is called critical if it has anomaly described by the 1-dimensional modular functor L⊗−1 . The 26th power of the 1-dimensional free bosonic CFT described (briefly) in the next section is critical in the physical sense (with both chiralities). Now for a critical CFT Hm , there is a certain differential Q (called the BRST differential) on the (non-anomalous) CFT ˆ gh . H = Hm ⊗H
(4)
In the chiral case, one has explicitly Q=
r∈Z
m LH r c−r −
1 (r − s) : c−r c−s br+s : −c0 2 r,s∈Z
(5)
(see [5], formula (4.59)). Here Lm r are the Virasoro generators acting on Hm , and cn , bn , n ∈ Z are now understood as operators on Hgh in the standard way (see [5]). In the non-chiral case, one must add to (4) its complex conjugate. Q is a differential, which means that QQ = 0.
(6)
The cohomological dimension is called the ghost number. The cn ’s have ghost number 1, the bn have ghost number −1, so the ghost number degree of Q is +1. We shall fix the ghost number as an algebra grading, so 1 has ghost number 0, but other conventions also exist. What is even more interesting than (6), however, is that Q turns H into a “differential graded CFT”. If we use the usual notation where we write for a CFT, as a lax morphism of SLCMC’s, X → UX ,
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then we may define a differential graded CFT by the relation (1 ⊗ ... ⊗ Q ⊗ ...1)UX = 0,
(7)
with the correct sign convention. For simplicity, we assumed in (7) that all boundary components of X are outbound, the adjoint operator Q∗ is used on inbound boundary components. Physically, (7) is due to the fact that Q is a conserved charge corresponding to the Noether current of a supersymmetry (called BRST symmetry) of the Lagrangian of H (see [5]). In any case, we now see that the BRST cohomology H = H ∗ (H, Q)
(8)
is a non-anomalous CFT. Infinitesimally, this implies that [Q, Ln ] = 0, where Ln are the standard Virasoro algebra (in our case in fact Witt algebra) generators. However, more is true. In fact, one has Ln = [Q, bn ], so Qx = 0 ⇒ Ln x ∈ I m(Q).
(9)
Because of (9), Ln actually act trivially on H , so H is in fact a TQFT (which means that UX only depends on the topological type of X). Therefore, our machinery would certainly seem to apply to H , in fact so would that of Moore-Segal [25]. There are, however, two difficulties. First of all, Hm may not actually be a CFT as we defined it because of convergence problems. For example, when Hm is the free bosonic CFT on the (25, 1)-dimensional Minkowski space, the inner product on the space Hm is indefinite, so this space cannot be Hilbert-completed with respect to its inner product. This is more than a technical difficulty: in physical language, this is the cause of the 1-loop divergence of bosonic string theory. In our language, this means that the state space of our would-be TQFT is infinite-dimensional, so UE for an elliptic curve E is infinity, or more precisely undefined. So there isn’t, in fact, any variant of the (25, 1)-dimensional free bosonic CFT for which the machinery outlined above would work mathematically and produce a true TQFT. In physics, this is an argument why the free bosonic string theory is not physical, and one must consider superstring theory. Our definition of CFT works on the SLCMC of superconformal surfaces, but the convergence problems persist, i.e. again, for the free (9, 1)-dimensional super-CFT, the BRST cohomology would be TQFT is infinitedimensional. Physicists argue that the (infinite) even and odd parts of the TQFT are “of equal dimension”, and thus the 1-loop amplitude vanishes (a part of the “non-renormalization theorem”). However, we do not know how to make this precise mathematically. There is another, more interesting caveat, namely that H is actually not exactly the object one wants to consider as the physical spectrum of string theory. Working, for simplicity, in the bosonic case, one usually restricts to states of ghost number 0, which, at least in the free case, is isomorphic to the quotient H0 of the submodule Z0 ⊂ Hm of states x ∈ Hm satisfying Ln x = 0 for n > 0 and L0 x = x, by the submodule B0
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of states of null norm (the Goddard-Thorn no ghost theorem). In bosonic string theory, the vacuum of Hm is in H0 , but this is a tachyon, which is not the vacuum of H : the vacuum in H is b−1 , as remarked above, and has ghost number −1. In superstring theory, the tachyon is factored out by the so-called GSO projection, while the vacuum of course persists, and has also ghost number −1, but a different name, due to the different structure of the theory, which we have no time to discuss here. One may ask what mathematical structure there is on H0 itself. Here the answer depends strongly on whether we work chirally or not. In the chiral case, Borcherds [4] noticed that H0 is a Lie algebra. The Lie algebra structure comes from [u, v] = u0 v where u0 is the residue of the vertex operator Y (u, z). The Jacobi identity follows immediately from the vertex operator algebra Jacobi identity. From CFT point of view, this operation is analogous to the Lie bracket in Batalin-Vilikovisky algebras. However, when both chiralities are present (which is the case we are interested in), the rabbit hole goes deeper than that. First note that the CFT vertex operator is not holomorphic, and curve integrals do not seem to be the right operations to consider. Instead, elements of Z0 are operator-valued (1, 1)-forms, and therefore can be naturally integrated over worldsheets. Indeed, one can see that integration of an element of Z0 over worldsheets produces an infinitesimal deformation of CFT. Elements of B0 also deform the CFT, but only by a gauge transformation, so elements of H0 give rise to infinitesimal deformations of string theory. We may therefore (despite potentially serious convergence problems) wish to consider a moduli space M of string theories, to which H0 is a tangent space at one point. In fact, points of the curved space M should be the true states of string theory, while the points of the tangent space H0 are only an approximation. In the physical theory, one conjectures that the space M contains all of the 5 original superstring theories as states, and a continuum of states in between. As seen even by studying the basic example of toroidal spacetime, some states in M differ only by “boundary conditions on open strings”, and such conditions are called D-branes. When there is a well defined spacetime manifold X, D-branes as a rule are associated with submanifolds of X with some additional structure. These, however, are classical and not quantum objects (cf. Polchinski [31]), so that approach also has its drawbacks. While rigorous mathematical attempts to define and investigate D-branes from the manifold point of view have (with some success) also been made in the literature exist, the “tangent” CFT approximation which we consider here is, in some sense, complementary. Finding a mathematical theory which would unify both points of view is an even much more complex task, which we do not undertake here. 4. An Example: The 1-Dimensional Free Scalar CFT We shall now give the standard examples of D-branes in the free bosonic CFT in dimension 1 (which is the CFT description of the linear σ -model). Unfortunately, even for this most basic CFT, a mathematically rigorous description of its convergence issues is nowhere to be found in the literature. The best outline we know of is given in [35]. A good first guess for the free (bosonic) field theory state space is, analogously with the lattice theories (see [16]) ˆ < zn , zn |n > 0 > . ˆ Sym H = L2 (R, C)⊗
(10)
Here L2 (R, C) denotes L2 -functions with respect to the Gaussian measure. The quantum number associated with this space is the momentum.
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To be more precise, (10) should be a Heisenberg representation of a certain infinitedimensional Heisenberg group. To describe it, we start with the topological vector space V of all harmonic functions on S 1 or, more precisely, harmonic functions on small open sets in C which contain S 1 and, say, the topology of uniform convergence in an open set containing S 1 . Thus, V is topologically generated by the functions zn , zn , n ∈ Z and
ln ||z||.
To define the Heisenberg group, we would like to find a C× -valued cocycle on V which would be invariant under the action of Diff + (S 1 ). However, similarly as in the case of lattice theories, we do not know any such cocycle. Instead, one considers the space U of harmonic C-valued functions on (an open domain containing) the unit interval I . The point is that the harmonic functions on I break up into holomorphic and antiholomorphic parts; a topological basis of the holomorphic part is given by the elements zn , n ∈ Z, ln(z), and a topological basis of the antiholomorphic parts is given by their complex conjugates. Therefore, the holomorphic and antiholomorphic parts U+ and U− of U have well defined winding numbers which can be added to a total winding number; let V ⊂ U be the set of functions of total winding number 0. Then the map exp(?) = e2πi? gives a projection V → V whose kernel consists of the constant functions. Now to get the free field theory, one proceeds analogously to lattice theories (see [16]), specifying a cocycle on U . We shall specify separately cocycles on both U+ and U− . However, because the integrality condition is replaced by equality of winding numbers on U+ and U− , we have more freedom in choosing the cocycle. For example, we can put, on both U+ and U− ,
1 1 c(f, g) = exp (11) f dg − f g(0) + f g , 2 S1 2 (where f is the winding number). The effect of this is that if we apply the cocycle to lifts of two harmonic functions f , g on a worldsheet to its universal cover, whose restriction to boundary components are fi , gi (as is done in [16] for the lattice theories), the Greene formula implies that n 1 c(f, g) = exp ( ( fi gj + gi fj ) + fi gi ) 4 i<j i=1 n n 1 = exp ( fi gi ) = 1. 4 i=1 i=1
(12)
Thus, the situation is simpler than in the case of lattice theory. Now similarly as in the case of lattice theory, the cocycle c we have constructed, when restricted to V , is trivial on the constant functions, so we get a canonical map C → V˜
(13)
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(where ?˜ denotes the Heisenberg group with respect to a given cocycle). Similarly as in the case of lattice theories, in fact, c(f, g) = 0 for f, g ∈ V , f constant, so the subgroup (13) is normal, so the desired Heisenberg group can be defined by V˜ = V˜ /C.
(14)
Then H should be the Heisenberg representation of the right real form of (14). Note, however, that the above construction comes with no obvious natural choice of real form. Let us postpone the discussion of this issue, as we shall see it is related to the convergence issues of the CFT. Now, conformal field theory structure is specified as usual: looking at the Heisenberg representation H∂X of the central extension V˜∂X of the space V∂X of harmonic functions on the boundary of a worldsheet X, we have already constructed a canonical splitting of the pullback of the central extension to the subspace VX of harmonic functions on X; we would like to define the field theory operator associated with X as the vector space of invariants of H∂X with respect to VX . A usual “density argument” (cf. [35, 16, 33]) shows that the invariant vector space HX is always at most 1-dimensional. In more detail, if we denote by H arm(X) the space of harmonic functions on X and by H arm(∂X) the space of harmonic functions on a small neighborhood of ∂X, and also by D the unit disk, then, by restriction, we may form the double coset space H arm(X)\H arm(∂X)/
H arm(D)
(15)
∂
(the product is over boundary components of X). Then (15) is isomorphic to H 1 (X, H arm) ∼ = C, where H arm is the sheaf of harmonic functions and X is the worldsheet obtained from X by gluing unit disks on the boundary components. Since H can be interpreted as a (completed) space of functions on H arm(∂X)/
H arm(D),
(16)
∂
the identification of (15) shows that only functions on the orbits C have a chance to be H arm(X)-fixed points. However, studying further the constant functions in H arm(X), we see that only functions supported on {0} ⊂ C have a chance of being fixed points. These observations also point to a difficulty with a Hilbert space model for H . What kind of reasonable Hilbert space functions on R contain distributions supported on a single point? Now recall the reason why the Hilbert space is not yet fixed: we haven’t fixed the real structure on V . One clue for such real structure is that, from the desired interpretation of H as functions on (16), A = H arm(D) ⊂ V (cf. [33], Section 9.5). Thus, should be our “Lagrangian subspace” so that H = Sym(A), we can define the real structure on V by specifying the inner product on A. The choice enjoying the desired invariances is f, g = f dg. (17)
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But then this is only a semidefinite Hermitian product on A, where constants have norm 0! The above discussion shows that this is more than a technical difficulty. To get the field operator UX to converge, we must take the inner product (17), which leads to H =
k∈R
n , zn |n > 0 Symz
( is the Cartesian product). Physically, the quantum number k is the momentum. Then the notion of “Hilbert product” of copies of H and “trace” must be adjusted. For more details, see the Appendix. With these choices, convergence of UX can be proven similarly as in [16] for lattice theories (e.g. using boson-fermion correspondence at 0 momentum), so the free bosonic CFT is rigorous. This convergence problem does not arise if we consider the σ -model on a compact torus instead of flat Euclidean space. For completeness, we note that we haven’t discussed inbound boundary components, and closed worldsheets. The former topic offers no new phenomena and can be treated simply by reversing the sign of the cocycle. Discussing closed worldsheets amounts really to discussing in detail the anomaly, which is H 1 (X, H arm), analogously to the fermionic case treated in [19]. Now we want to give examples of simple elementary D-branes in the free CFT. The above discussion shows that we would have to work in compact spacetime (a torus) to make the examples fit the scheme proposed in Sect. 2 literally. However, we elect instead to stick to the flat spacetime R, where the situation seems more fundamental. It must be then understood, however, that the notion of closed/open CFT in this situation must also be generalized in a way analogous to closed CFT (as discussed in the Appendix), to solve the convergence issue. Consider the 1/2-disk B consisting of elements of D with non-negative imaginary part. We consider B an open string worldsheet where the real boundary elements are the D-brane component, and the open string component is parametrized by the map eπit . Then we can consider the space of all harmonic functions on the boundary of B which obey a suitable boundary condition on the D-brane component. The boundary conditions allowable first of all must be conformally invariant. The most obvious such condition is that the derivative of the function in question be 0 in the direction of a certain vector u ∈ S 1 , I m(u) > 0 or u = 1. A priori, all of those conditions are allowable. However, if we want to follow the methods we used above to describe closed free CFT, additional conditions are needed. Namely, we need the vector space of functions satisfying the condition to have a central extension which is a Heisenberg group. Moreover, to get consistency of open and closed CFT, we need the Heisenberg group to be obtained by restriction of the cocycle (11), and the real structure must also be induced from the closed CFT real structure. We will see however that the real structure is incompatible with the closed CFT real structure unless u = 1 or u = i. In effect, let f be a holomorphic function which sends R to R (e.g. a polynomial with real coefficients). Then uf − uf has zero derivative in the direction u on R (as z → uf (z/u) − uf (z/u) has zero derivative in the direction of u on the line z ∈ uR). Here by ? we mean the usual complex conjugation. But now recall that in the real structure involved in defining closed CFT, the complex conjugate of zn is z−n , so the complex conjugate of uzn − uzn is uz−n − uz−n , which is not of the form uf − uf for a holomorphic function f sending R to R, unless u = (±)1 or u = (±)i.
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Now for u = i, the functions on B which we get are zn + zn , n ∈ Z, ln ||z||.
(18)
(Note the singularities on the D-brane components of ∂B- those needn’t bother us.) The cocycle is defined by the same formula as (11), although one integrates over the parametrization component only (the resulting factor of 1/2 is related to the squaring relation between the open and closed string couplings). The corresponding Heisenberg representation (with the same discussion of convergence at momentum quantum numbers as in the closed case) is the von Neumann open string sector, or the 1-dimensional “D0-brane” of the 1-dimensional free bosonic CFT. Together with the closed sector, this defines (up to the convergence-related modifications) a closed/open CFT with 1dimensional anomaly in the sense of Sect. 2 (the vacua are obtained completely analogously as in the closed case). When u = 1, the functions satisfying the boundary condition (which is the Dirichlet condition) are zn − zn , n ∈ Z − {0}, 1, 1 z ln . 2 z Although at first everything looks analogous as in the von Neumann case, there is an important difference: if we define again the cocycle by (11) integrated over the parametrization component of B (which we must to get consistency), this time the cocycle is degenerate with kernel 1, arg(z). arg(z) =
This is a case of spontaneous symmetry breaking: we get a 2-dimensional continuum of irreducible representations, one for each weight of 1 and arg(z) (which are independent real numbers). To name these sectors, we must figure out what these quantum numbers mean. To this end, in turn, we must review our recipe for defining a closed/open CFT from these Heisenberg representations. To get consistency, the recipe must be the same as in the closed case: for a closed/open worldsheet X, take the tensor product H∂X of state spaces corresponding to the parametrization components of X (suitably completed, as above). By the representations we constructed, the group H arm(X) of harmonic functions on X acts on H∂X (the central extension given by our cocycle splits on H arm(X) canonically, as above). The scalar multiples of the vacuum in H∂X now form the space of invariants H arm(X) . The discussion is analogous to the closed case, and we omit the details. H∂X However, the sector numbers we are interested in correspond to weights, on each open parametrization component c of X, of functions which have given values α, β on the endpoints of c (this is linear in α, β). Note that we can get any such pair of numbers α, β by taking a linear combination of 1 and arg(z). Now consider, on X, harmonic functions which are constant on the D-brane components of X: such functions must act by identity (have weight 0). We see from this that the fixed point space will be 0 unless weights corresponding to the two endpoints of each open D-brane component coincide. Therefore, these weights can be interpreted as position coordinates of the two endpoints of the open string, in other words position coordinates of the (instanton) D-branes. The
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2-dimensional continuum of state spaces we got describes all the possible mixed sectors Hab of open strings ending on any two of these D-branes with positions a, b ∈ R. Convergence issues work out after a discussion analogous to the closed case. To build a closed/open CFT with D-brane category in the sense of Sect. 2 from the 1-dimensional free boson, note that we have restricted attention to D-brane categories which are finite-dimensional free lax C2 -modules. Therefore, we must select finitely many positions of instanton D-branes a1 , ..., aN (and the D0-brane, if we wish). The D-branes can then be summed using the mixed sectors in the obvious way. The mixed sectors between the D0 and D(−1)-branes are 0. As in the closed case, the Hilbert tensor product and trace must be modified to get proper convergence behavior. We have not axiomatized anti-D-branes here, but that can be done using the formalism of Sect. 6, at the cost of increasing technical difficulty substantially. 5. CFT Anomaly via 2-Vector Spaces and Elliptic Cohomology In this section, we give a new definition of modular functor which generalizes the definition given in [16]. Consider a free finitely generated lax module M over the lax semiring C2 (the category of finite-dimensional C-vector spaces; it is convenient to let the morphisms of C2 be all linear maps; thereby, C2 is not a groupoid, and we have to use the version of lax algebra theory which works over categories - see the appendix; of course, it is possible to consider the subcategory C× 2 of C2 whose morphisms are isomorphisms). ilb , the C -algebra of Hilbert spaces with the Hilbert tensor product. Consider further CH 2 2 ilb . Now consider H ∈ MH ilb (the symbol ∈ means a We put MH ilb = M ⊗C2 CH 1 1 2 map of lax C2 -modules C2 →?; in our case, this is the same thing as H ∈ Obj (MH ilb )). We shall define two LCMC’s C(M) and C(M, H ) (underlying LCM of sets) which are, in standard ways, extended into SLCMC’s over the Grothendieck category of finitedimensional smooth complex manifolds. The LCMC’s are constructed as follows: the objects of C(M) over the pair of finite sets (S, T ) are 1-elements of M⊗S ⊗ M∗⊗T ; the morphisms are 2-isomorphisms. Here M∗ is the dual lax C2 -module of M, whose objects are (lax) morphisms of lax C2 -modules M → C2 and morphisms are natural isomorphisms compatible with the operations. The gluing maps are given by trace over C2 , i.e. the evaluation morphism M ⊗ M∗ → C2 . An object of C(M, H ) over (S, T ) consists of an object M of C(M), and 2ˆ ˆ ˆ whose image consists of trace class elements: ⊗H ∗⊗T morphism u : M →2 H ⊗S (Choosing a basis of M, a 1-element of MH ilb becomes a collection of Hilbert spaces, ˆ ˆ so ⊗-powers of H are collections of ⊗-powers; an element is trace class if each of its components is trace class. For generalizations beyond the trace class context, see Remarks in the Appendix.) Here H ∗ ∈1 M∗ H ilb is defined by putting, for V ∈1 M, H ∗ (V ) = H omMH ilb (H, V ). Morphisms are commutative diagrams of the obvious sort. To define gluing operations, note that ˆ
ˆ ∗⊗(T +U ) u : M →2 H ⊗(S+U ) ⊗H ˆ ˆ ˆ induces a 2-morphism tr(u) : (1 ⊗ tr1 )M →2 H ⊗S , where tr1 : M⊗U ⊗ ⊗H ∗⊗T ∗⊗U M → C2 is the evaluation morphism; it is defined by using the canonical morphism tr2 : tr1 (H ⊗ H ∗ ) → C. To define the corresponding SLCMC’s, (which we denote by the same symbols), as usual, it suffices to define sections over pairs of constant covering spaces (U ×S, U ×T ) of a complex manifold U . For defining C(M), we need a concept of a holomorphically
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varying 1-element of M. To this end, we denote by H ol(U, C2 ) the lax commutative monoid of finite-dimensional holomorphic bundles on U . Then the concept we need is MU ∈1 MU := M ⊗C2 H ol(U, C2 ). This determines the SLCMC C(M). To define C(M, H ), note that H does not depend on U , so in the case of the constant covering space described above, we simply need a ˆ
ˆ
ˆ U∗⊗T , uU : MU →2 HU⊗S ⊗H where HU is the constant H -bundle on U , and the 2-morphism means a morphism of holomorphic bundles. Definition. A modular functor on an SLCMC C with labels M is a (lax) morphism of SLCMC’s φ : C → C(M). A CFT on C with modular functor on labels M with state space H is a (lax) morphism of SLCMC’s : C → C(M, H ). For a very detailed discussion of issues related to laxness, see [9]. Now note that C(?) is a 2-functor from the 2-category C2 − mod of lax C2 -modules (1-morphisms are equivalences of C2 -modules, and 2-morphisms are natural isomorphisms compatible with C2 -module structure) into the 2-category of SLCMC’s. Similarly, C(?, ?) is a 2-functor from the 2-category C2 − mod∗ of pairs M, H , where M is a C2 -module, and H ∈1 MH ilb . Here 1-morphisms in C2 − mod, C2 − mod∗ are equivalences of lax C2 -modules, 2-morphisms are natural isomorphisms compatible with C2 -module structure. (In C2 − mod∗ , 1-morphisms (M, H ) → (N , K) are 1-morphisms φ : M → N in C2 − mod together with a 2-isomorphism λ : φ(H ) → K; a 2-morphism φ → ψ is a 2-morphism in C2 − mod which induces an isomorphism φ(H ) → ψ(H ) commuting with the λ’s.) We use this to build a 2-category of CF T ’s as a “comma 2-category”. The objects are tuples M, H, where, is a CFT on C with labels M and state space H , 1-morphisms are tuples , , f, ι, where , are CFT’s with labels M, N and state spaces H , K, f is a C2 −mod∗ - 1-morphism (M, H ) → (N , K) and ι is a natural isomorphism f (MX ) →2 NX , where MX , NX are the 1-elements of M, N assigned to X ∈ Obj C by , which commutes with SLCMC structure maps and the u’s assigned by , (we have used the notation of sections over a point, but we mean this in the stack sense for sections over any complex manifold U ). 2-morphisms , , f, ι → , , g, κ are given by 2-isomorphisms f → g in C2 − mod∗ which commute with ι, κ (hence the u’s). Next, we shall show that is a symmetric monoidal 2-category. This means that we have a lax 2-functor ⊕ with the same coherence 1-isomorphisms as in a symmetric monoidal category, but coherence diagrams commute up to 2-cells; the 2-cells, in turn, are required to satisfy all commutations valid for the trivial 2-cells of coherence diagrams in an ordinary symmetric monoidal category. Thus, the main point is to construct the 2-functor ⊕. Suppose we have two objects M, H, and N , K, of . Then their sum is M ⊕ N , H ⊕ K, ⊕ : The first component is the direct sum in C2 − mod. H ⊕ K is the direct sum induced by that functor on 1-morphisms. The symbol ⊕ , however, has to be defined explicitly. For simplicity, we shall restrict to sections over a point. Then, the data which remains to be defined, for an object X of the source SLCMC, is M ⊕ N,
(19)
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and u : M ⊕ N →2 (H ⊕ K)⊗S ⊗ (H ⊕ K)∗⊗T .
(20)
Of this data, for X connected, (19) is, again, the direct sum induced by the direct sum of C2 -modules on 1-morphisms, composed with the canonical map M⊗S ⊗ M∗⊗T ⊕ N ⊗S ⊗ N ∗⊗T → (M ⊕ N )⊗S ⊗ (M ⊕ N )∗⊗T .
(21)
Analogously, (20) is given by the ⊕ of C2 -modules on 2-morphisms, composed with (21). For X non-connected, note that we are forced to define all data by applying the tensor product to the data on connected components. This definition extends to 1morphisms and 2-morphisms in a standard way to produce a symmetric monoidal 2category of CFT’s . Note that we may not always wish to work with the whole , but with some symmetric monoidal sub-2-category ; for example, we may take direct ⊕-sums of copies of a given CFT. Now there is an infinite loop space machine for 2-categories: for example, Segal’s machine. Segal’s machine is supposed to construct an F-space, which is a functor from the category F of finite sets with base point ∗ and based maps into spaces (alternately, one can think of this as a category of partial maps); it is also required that the functor (called F-space) B be special, which means that the product map from B(n) to the product of copies of B(1) by the maps which send all numbers in {1, ..., n} except i into the basepoint be an equivalence. Now to produce a special F-space from a symmetric monoidal category C, simply consider the category C(n) which is a category of diagrams, whose objects are tuples (xT ) of objects of C indexed by non-empty subsets of S, together with isomorphisms
x{i} ∼ = xT .
(22)
i∈T
Morphisms are commutative diagrams of the obvious kind (see [37]). Now C(?) is a functor from F into categories, so applying the classifying space gives the requisite F-space. It is special by basic theorems about the homotopy types of classifying spaces. However, now note that the same definition (22) in the case of a symmetric monoidal 2-category C gives a 2-category C(n). The only difference is that on 1-morphisms, we do not consider merely diagrams commutative on the nose, but up to 2-cells and 2morphisms are systems of 2-cells which further commute with the 2-cells thus introduced. With that, however, C(?) becomes a (strict) functor from F into 2-categories. So, we are done if we can produce a functorial classifying space construction B2 on 2-categories, and show that the F-space B2 C(?) is special. The latter is a straightforward exercise which we omit. For the former, however, we remark that to define a classifying space of a 2-category C, we can first form the bar construction B1 = B(Mor(C)), i.e. the bar construction on 2-morphisms. However, if C is lax, then B1 is not a category, but composition is defined with respect to a contractible operad (without permutations). The operad D is as follows: the space D(n) is the standard (n − 1)-simplex and the composition is given by joining: D(k) × D(n1 ) × ... × D(nk ) → D(n1 ) ∗ ... ∗ D(nk ) = D(n1 + ... + nk ).
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Nevertheless, it is well known that such A∞ -categories still have a classifying space functor (for example, one can “rectify” them by push-forward change of operads to the one point operad without permutations, which encodes associativity). Thus, we have produced a symmetric monoidal 2-category of CFT’s, and an infinite loop space machine for such a case. Therefore, we have an infinite loop space E. This is related to the kind of construction used in [16] to give a candidate for an elliptic-like cohomology theory and seems like an improvement in the sense that it gives a model for the additive infinite loop structure (a “free” construction was used in [16]). However, Baas-Dundas-Rognes [2] point out that this kind of construction is naive. The problem is that there are not enough isomorphisms of free lax C2 -modules: they are essentially just permutation matrices composed with diagonal matrices of line bundles. In [2], a solution to this problem is proposed, conjecturally calculating the algebraic K-theory of C2 . The point is to consider, instead of invertible matrices of finitedimensional vector spaces, numerically invertible matrices, which means that the corresponding matrix of dimensions of the entry vector spaces is invertible. Unfortunately, this approach does not seem satisfactory for the purposes of CFT: along with an iso f : M → N , we need to consider also the inverse M∗ → N ∗ of the adjoint morphism N ∗ → M∗ ; there is no candidate for such inverse when f is only numerically invertible. Another clue that something else is needed is the following example of bc-systems, whose anomaly, it seems, can only be expressed by considering “modular functors with positive and negative parts”. Example. Consider the chiral bc-system of α -forms, α ∈ Z (see also Sect. 3 above). The bc-system was first considered mathematically by Segal [35], but the observation that the super-modular functor formalism is needed to capture its properties is due to P. Deligne ([19]). In the case, the state space of the bc-system is the “fermionic Fock space” ˆ + ⊕ H − ), Fα = (H
(23)
where H =< zn dzα |n ∈ Z > and H+ is, say, the subspace < zn dzα |n ≥ 0 >. We select some real form to make this a positive definite Hilbert space (cf. [16], Chap. 2). Then the modular functor is 1-dimensional, and is given by the determinant line of α X, the space of holomorphic α-forms on a worldsheet X. The reason why a super-modular functor is needed here is that we are dealing with Grassmannians, and signs must be introduced when permuting odd-degree variables for the CFT to be consistent; no such signs, however, occur in CFT’s with 1-dimensional anomaly as considered above (see [19] for more details). Thus, it seems that C2 in our definition of modular functors and CFT’s should be replaced by some sort of “group completion” which would involve Z/2-graded vector spaces. A candidate for such a construction is given in the next section, although we will see that this comes at the price of substantially increasing technical difficulty. 6. The Group Completion of C2 ˆ 2 of C2 over As argued above, it would be desirable to have a group completion C which we could do the analogues of all of our constructions as suggested by BaasDundas-Rognes [2]: this would give approaches to axiomatizing CFT with positive and
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negative-dimensional modular functors, as well as anti-D-branes with 1-dimensional anomaly. In this section, we propose such a construction. However, as also pointed out in [2], any such construction is necessarily accompanied by substantial difficulties. The first problem is to even define what we mean by “group completion”. It is easy to show that for a lax commutative ring R, BR is always an Eilenberg-MacLane space (and hence cannot be used for our purposes), but there is strong evidence that a large class of weaker categorical notions of “weakly group-complete” lax commutative semirings suffer from the same problem [39]. We take an alternate approach of introducing topology into the picture. This means we construct a model of a topological lax commutative semiring Cˆ2 where there is an object −1 so that 1 ⊕ (−1) is in the same connected component of 0. While this approach does seem to lead to a viable definition, one must overcome a variety of technical difficulties caused by the additional topology. The first issue is what is the appropriate 2-category T Cat of topological categories? The point is that requiring functors to be continuous on objects appears to restrict too much the notion of equivalence of topological categories, and consequently alter their lax colimits. To remedy this situation, we define 1-morphisms C → D in T Cat to be of the form F
C G
/D (24)
C
where F is a continuous functor and G is a partition which we define as follows: A partition is given by a topological space X and a continuous map f : X → Obj (C) such that the topology on Obj (C) is induced by f (we work in the category of weakly Hausdorff compactly generated topological spaces). Then we have Obj (C ) = X, Mor(C ) is a pullback of the form / Mor(C)
Mor(C ) X×X
S×T
/ Obj (C) × Obj (C).
f ×f
Two functors F1 , F2 as in (24) are considered equal if they coincide on a common partition, i.e. we have a commutative diagram C2 O ?? ?? F2 G2 H2 ?? ?? F / C ^> C ?D >> >> H1 G1 >> F1 C1 where H1 , H2 are partitions. Composition is defined by pullback in the usual way, using the following
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Lemma 1. If we have a pullback Y
j
/ X f
h
Y
g
(25)
/X
in the category of compactly generated weakly Hausdorff spaces such that f induces the (compactly generated) topology on X, then h induces the (compactly generated) topology on Y . Proof. Suppose V ⊂ Y , v ∈ (Y − V ) ∩ Cl(V ), h−1 (V ) closed. Then there exists K compact where v is a limit point of K ∩ V . So, we may replace V by K ∩ V and assume Cl(V ) = K = Y . Next, let Z = g(K), so Z is compact. Case 1. g(V ) = Z. Then there exists T ⊂ X compact, T ∩ f −1 (g(V )) not closed. Consider the pullback /T T g −1 f (T )
/ f (T ).
Then T is compact since g is proper. But then j (T ∩ h−1 (V )) = f −1 (g(V )) ∩ T , so T ∩ h−1 (V ) cannot be closed in T (since j |T is closed, T being compact). This is a contradiction. Case 2. g(V ) = Z. Then in particular, there exists z ∈ V , g(v) = g(z ) =: z. Now we may assume that v is a limit point of V ∩ g −1 ({z}).
(26)
Indeed, otherwise, since K = Y is compact weakly Hausdorff, it is normal, hence regular and there exist U, W open in K, U ∩ W = ∅, v ∈ U , Cl(V ∩ g −1 ({z})) ⊂ W . But then we may replace Y by Y − V (and X by g(Y − V )), and we are back to Case 1. So we may assume (26). But then we may replace X by {z} and Y by g −1 ({z}). But then (25) is a product, in which case the statement of the lemma is obviously true (a product projection induces the topology on its target in the compactly generated weakly Hausdorff category). Thus, we have a contradiction again. Now 2-morphisms in T Cat are defined as follows: we can assume we have two 1-morphisms F1 , F2 given as C G
F1 ,F2
/D
C
where G is a partition. Then a 2-morphism is given by a partition C
G
/ C
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and a continuous natural transformation F 1 G → F2 G . Again, two 2-morphisms are identified if they coincide after pullback via a partition, similarly as above. This completes the definition of the 2-category T Cat. This 2-category is defined in such a way that it has lax limits defined in the same way as in Cat [9]. (Lax limit is given as the category whose objects and morphisms are lax cones from a point or an arrow to a diagram with the topology induced from the product; by “lax”, we always mean “up to coherences which are iso”.) Next, one may discuss lax monads in T Cat, which, in our definition, are lax functors C : T Cat → T Cat with lax natural transformations µ : CC → C, η : I d → C which are associative and unital up to coherence isos with commutative coherence diagram the same as for lax monoids. For a lax monad C in T Cat we then have a category of lax C-algebras whose objects are objects M of T Cat together with a functor θ : CM → C satisfying associativity and unitality up to coherence isos with commutative coherence diagrams of the same form as those for categories with lax action of a lax monoid. Then lax algebras over a lax monad in T Cat form a 2-category which has lax limits created by the forgetful functor to T Cat. We may be interested in lax algebras over a strict monad, for example the monad associated with a theory T . One example of a lax monad whose lax algebras we are interested in is gotten from a 2-theory (, T ) and a lax T -algebra I . Then we can define a lax monad C,I not over T Cat, but over the k category T cat I of strict functors I k → T Cat. In effect, C,I (X) has C,I (X)i =
(m)((γ1 , . . . , γp ); γ ) ×
p
Xγi (j1 , . . . , jm ),
i=1
where the coproduct is indexed over m, (j1 , . . ., jm ) ∈ I , γ ∈ T (m)k , γ (j1 , . . . , jm ) = i, γ1 , . . . , γp ∈ T (m)k . Then lax C,I -algebras are precisely lax algebras over (, T ) with underlying lax T -algebra I . ˆ 2 with an object −1 such We are now ready to describe a topological lax semiring C that 1 ⊕ (−1) is in the same connected component as 0. First consider the lax semiring sC2 of pairs (V+ , V− ), V+ , V− ∈ Obj (C2 ) with the lax C2 -module structure given by C2 ⊕ C2 and multiplication (V+ , V− ) ⊗ (W+ , W− ) = (V+ ⊗ W+ ⊕ V− ⊗ W− , V+ ⊗ W− ⊕ V− ⊗ W+ ). Now in sC2 , consider the full subcategory J on pairs (V+ , V− ), where dim(V+ ) = dim(V− ). Then J is a lax sC2 -module, and J ⊕ C2 is a lax commutative C2 -algebra with a lax commutative C2 -algebra morphisms J ⊕ C2 → C2 (an augmentation) and J ⊕ C2 → sC2 (the inclusion). Thus, we have a lax simplicial commutative C2 -algebra (=lax functor Op → lax commutative C2 -algebras) BC2 (C2 , J ⊕ C2 , sC2 ).
(27)
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ˆ 2 to be the realization of (27). This needs some explaining, namely we We propose C must define realization. We shall describe a lax realization functor A· → |A| from lax simplicial commutative C2 -algebras to topological commutative C2 -algebras (i.e. commutative C2 -algebras in T Cat). We want to mimic as closely as possible the strict construction. This means that we will define (27) as the lax simplicial realization in the 2-category of lax C2 -modules, which we must define. First, let, for a space X, C2 X be the free lax C2 -module on X (objects and morphisms are finite formal linear combinations with coefficients in objects and morphisms of C2 , and the topology is induced from the topologies of finite powers of X). We want the realization |A| to be the lax coequalizer of ⊕ (C2 m ⊗ An ) → → ⊕ (C2 n ⊗ An )
m,n
(28)
n
(the arrows are the usual two arrows coming from lax simplicial structure, ⊕, ⊗ are over C2 ). To construct the lax coequalizer (28), we can take the objects of ⊕ (C2 m ⊗ An ) ⊕ ⊕ (C2 n ⊗ An ). m,n
(29)
n
To get morphisms, we take the morphisms of (29), and adjoin isomorphisms between all source and target objects of the arrows in (28). Take the free topological category spanned by these morphisms, modulo the obvious commutative diagrams required. This gives us a category with the lax C2 -module (29) as a subcategory. The free construction we must then perform is applying the strict left adjoint to the forgetful functor from the category of lax C2 modules with lax submodule (29) on the same set of objects (taking only functors which are identity on objects) to the category of categories with subcategory (29) on the same set of objects (taking only functors which are identity on objects). As usual, the functors are strict because objects and coherences are already specified. This completes the construction of the lax simplicial realization (27). One must still prove that this is a lax C2 -algebra, but this is accomplished analogously as in the strict case, using the shuffle map (and the morphism definition (24) to assure continuity). Now topological SLCMC’s C(M), C(M, H ) for a finitely generated free topologiˆ 2 -module M are defined analogously as over C2 . (Since the underlying stack cal lax C of covering spaces I = Set does not change, LCMC’s can be described as lax algebras over a lax monad in T Cat as above, and therefore stacks over a Grothendieck topology can be defined to be, as usual, contravariant functors which take Grothendieck covers to lax limits.) However, the topology would be of little use if we simply took for our definition of modular functor a lax morphism of SLCMC’s from C to C(M) (similarly for CFT’s). Instead, the corresponding “derived notion” is appropriate. This means that we should consider lax morphisms of topological SLCMC’s B(C,S , C,S , C) → C(M),
(30)
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where denotes the 2-theory of LCMC’s, and S the lax commutative monoid of finite sets, as above. The left hand side is obtained by taking the bar construction sectionwise and then applying the lax left adjoint to the forgetful functor from SLCMC’s to pre-stacks of LCMC’s. It still remains to define a realization functor from lax simplicial lax C,S -algebras to topological lax C,S -algebras. Analogously as in the case of BC2 , however, we may 2 simply lax-realize in the 2-category (T Cat)Set (where it is easy to construct lax colimits, cf. [9]), and use the lax analogue of Milnor’s map |A| × |B| → |A × B| to obtain lax C,S -algebra structure on the realization. We omit the details. 7. The General Anomaly for Open-Closed CFT In this section, we shall apply the principles of Sect. 5 to propose a general definition of open-closed CFT with both multiple D-branes and multi-dimensional conformal anomaly (although without any group completion). We shall see, however, that this is necessarily even much more complicated than what we have done in Sect. 5. We have already argued that neither the “set of D-branes” nor the “set of labels” should be sets. Rather, they should be higher vector spaces. However, on a boundary component of the worldsheet where several open parametrization components are present, we need to take traces of “sets of labels” over “sets of D-branes”. This suggests that our model of “set of D-branes” must be one categorical level above our notion of “set of labels”. Therefore, we propose that the “set of D-branes” be a 3-vector space A. When dealing with 3-vector spaces, note that they are 2-categories. 3-vector spaces are, by definition, 2-lax modules over the 2-lax commutative semiring C3 . We must, of course, define these notions. On generalizing from lax to 2-lax structures, we find it easiest to follow the approach of [17]. Let T be a theory. Then let T h(T ) be the free theory on T , with the canonical projection of theories φ : T h(T ) → T . Let G be a groupoid with objects T h(T ) and one isomorphism x → y for every x, y ∈ T h(T ) which satisfy φ(x) = φ(y). Then (T h(T ), G) is a theory (strictly) enriched over categories and a lax T -algebra is the same thing as a strict (T h(T ), G)-algebra enriched over categories. Now to go to the next level, consider the forgetful functor U : Theories enriched over groupoids → P, where P is the (strict) category of pairs (T , G) where T is a theory, G is a graph with objects T . Then let F be the left adjoint of U . Notice that F is the identity on objects T , so we may write F (T , G) = (T , F (G)). Now we have a map of theories enriched over groupoids: ψ : (T h(T ), F (G)) → (T h(T ), G). Therefore, we may create a 2-category (T h(T ), F (G), H ) by putting exactly one 2-isomorphism between every α, β ∈ F (G) with ψ(α) = ψ(β). Then the 2-category (T h(T ), F (G), H ) is naturally a theory (strictly) enriched over 2-categories, and a 2-lax T -algebra is a 2-category which is a strict (T h(T ), F (G), H )-algebra enriched over 2-categories. (Obviously, one may proceed further in the same way to define even higher laxness, but we shall not need that here.) One remark to be made is that theories, strictly speaking, do not model universal algebras which are modelled on more than one set, such as a module over a ring (which is
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modelled over two sets). However, algebras modelled over k sets can be easily included in the formalism by modifying the concept of theory to a category with objects Nk (i.e. k-tuples of natural numbers) with the axiom that for all a, b ∈ Nk , a +b is the categorical product of a, b. This is the concept of multisorted theories (see [9] for more details and references). All of our constructions generalize to this context. Now to identify the categorical levels with the levels we considered before, we will denote objects of 3-vector spaces by ∈0 and morphisms of 3-vector spaces by →0 . Thus, a 0-morphism C3 → C3 (where C3 is the lax symmetric monoidal category of 2-vector spaces) is a 2-vector space, and the notations →1 , →2 of such 2-vector spaces will coincide with the notations we used above. Now given the 3-vector space A (“the set of D-branes”), we must introduce the “set of labels” for anomalies. The “set of closed labels” will be, as before, a 2-vector space, which we will denote by C. The “set of open labels” will be an object of the form O ∈0 A ⊗C3 A∗ . (We remark here that A∗ for a 3-vector space A is defined analogously as in the case of 2-vector spaces.) Now we would like to define an SLCMC C(A; C, O).All our SLCMC’s in this section shall have two labels, 1 and m (closed and open). However, note that there is another subtlety we must provide for, namely that the set of all possible incidence graphs whose vertices are open and closed parametrization and D-brane components and edges describe their incidence relations with the obvious conditions (e.g. all vertices have degree 2, etc.) is itself an SLCMC, and in order to correctly keep track of incidences on the boundary, we must consider SLCMC’s over (see also the end of Sect. 2 above). We shall only describe sections of C(A; C, O) over a given object G of over a single point, over four given sets S1,in , S1,out , Sm,in , Sm,out of inbound and outbound closed and open “components”. Let P denote the set of closed D-brane components of G. Before making the definition, note that we have canonical dual 0-morphisms C3
η
/ A ⊗C A ∗ 3
/ C3 .
(31)
(If no further discussion is made, (31) requires a finiteness assumption about A.) Their composition is a 2-vector space which we shall denote by tr0 A. The set of sections of C(A; C, O) are 1-elements M ∈1
S1,in
P
C∗ ⊗
C⊗
S1,out
tr0 A ⊗ tr0,cyclic (
Sm,in
O∗
⊗
(32) O).
Sm,out
Here the tensor products are over C2 , and tr0,cyclic denotes composition with the tensor product of the appropriate number of ’s; note that although not explicitly written, the definition of tr0,cyclic makes use of all of the structure of G. Now in order to give (32) a structure of LCMC, one must show an appropriate gluing property, but this is analogous to our discussion for closed CFT’s. Now let, as above, D be the SLCMC of closed-open worldsheet with one closed and one open label. Then an open-closed CFT anomaly (modular functor) is a map of SLCMC’s over D → C(A; C, O).
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Now to define open-closed CFT, we must add the “Hilbert spaces”. The “closed Hilbert space” is, as above, H ∈1 C H ilb . The “open Hilbert space” is a 1-morphism K : η →1 OH ilb , where η is as in (31). We shall now define an SLCMC C(A; O, C; H, K) over . As above, we shall specialize to sections over a single point and single object of , with the same notation as above. Then a section consists of a section (32) of C(A; C, O) and a 2-morphism M →2
ˆ H ∗⊗ ˆ ˆ ˆ H⊗ ηtr0 A S1,in
P
S1,out
ˆ 1,cyclic ( ˆ K ∗ ⊗ ˆ ˆ K). ⊗tr Sm,in
Sm,out
Here tr1,cyclic is given by the structure of 2-category, and ηtr0 A is the canonical “unit” 1-element of tr0 A. To be more precise, write, in (31), η(C2 ) =
n
Vi ⊗C2 φi ,
i=1
so tr0 A =
n
φi Vi .
i=1
But then one can show
j
φi Vj = δi C2 ,
so we have tr0 A =
n
C2 ,
i=1
and we can write ηtr0 A =
n i=1
C ∈1
n
C2 .
i=1
Of course, such discussion reveals the weaknesses of the higher vector space formalism, and the desirability to really work, again, in a suitable higher group completion. However, we do not work out that approach here. 8. Appendix: Stacks of Lax Commutative Monoids with Cancellation To make this paper self-contained, we review here the basic definitions [16] related to stacks of lax commutative monoids with cancellation (SLCMC’s). We must begin by defining lax algebras. The formalism we use is theories according to Lawvere, and their extension which we call 2-theories. Recall first that a theory according to Lawvere [20]
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is a category T with objects N (the set of all natural numbers 0, 1, 2, ...) such that n is the product of n copies of 1. Categories of algebraic structures given by a set of operations and relations on one set X can be encoded by a theory, where Tn = H om(n, 1) is the set of all n-ary operations of the algebraic structure (including all possible compositions, repetitions of one or more variables, etc.). Definition. A 2-theory consists of a natural number k, a theory T and a (strict) contravariant functor from T to the category of categories (and functors) with the following properties. Let T k be a category with the same objects as T , and H omT k (m, n) = H omT (m, n)×k (obvious composition). Then Obj ((m)) =
H omT k (m, n),
n
on morphisms, is given by precomposition on Obj ((m)), and γ ∈ H omT k (m, n) is the product, in (m), of the n-tuple γ1 , ..., γn ∈ H omT k (m, 1) with which it is identified by the fact that T is a theory. We also speak of a 2-theory fibered over the theory T . The example relevant to CFT is the 2-theory of commutative monoids with cancellation. T is the theory of commutative monoids with an operation +, and k = 2. The 2-theory has three generating operations, addition (or disjoint union) + : Xa,c × Xb,d → Xa+b,c+d , unit 0 ∈ X0,0 and cancellation (or gluing) ?ˇ : Xa+c,b+c → Xa,b . The axioms are commutativity, associativity and unitality for +, 0, transitivity for ?ˇ and distributivity of ?ˇ under +. To get further, one needs to define algebras and lax algebras over theories and 2theories. An algebra over a theory T is a set I together with, for each γ ∈ Tn , a map γ : I ×n → I , satisfying appropriate axioms. These axioms can be written out explicitly, but a quick way to encode them is to notice that for a set I , we have the endomorphism theory End(I ), where End(I )(n) = Map(I ×n , I ), and we may simply say that a structure of T -algebra on I is given by a map of theories T → End(I ). To define an algebra over a 2-theory fibered over a theory T , we must first have an algebra I over the theory T (the ‘indexing theory’). This gives us, for γ ∈ H omT k (m, 1), a k-tuple of maps γ : I ×m → I . In an algebra over the 2-theory, we have, in addition, a map X : I ×k → Sets.
(33)
For a morphism in φ ∈ Mor() from (γ1 , . . . , γn ) ∈ H omT k (m, n) to γ ∈ H omT k (m, 1), we have, for each choice i1 , . . . , im of elements of I , maps φ : X(w1 (i1 , . . . , im )) × · · · × X(wn (i1 , . . . , im )) → X(w(i1 , . . . , im )),
(34)
satisfying appropriate axioms. Once again, we can avoid writing them down explicitly by defining the endomorphism 2-theory. Consider a set I and a map X : I k → Sets. To such data there is assigned a 2-theory End(X) fibered over the theory End(I ): let (w; w1 , . . . , wn )
Closed/Open Conformal Field Theories
247
consist of the set of all possible simultaneous choices of maps X(w1 (i1 , . . . , im )) × · · · × X(wn (i1 , . . . , im )) → X(w(i1 , . . . , im )),
(35)
where ij range over elements of I . A structure of an algebra over the 2-theory fibered over T is given by a morphism of 2-theories (, T ) → (End(X), End(I )). A lax algebra over a theory is a category I , with maps γ which are functors. We do not, however, require that these maps define a strict morphism from T to the endomorphism theory of I . Instead, this is only true up to certain natural isomorphisms, which we call coherence isomorphisms, which in turn are required to satisfy certain commutative diagrams, which are called coherence diagrams. This is, of course, always the case when defining lax algebras of any kind. But now the benefit of introducing theories is that the coherences and coherence diagrams always have the same shape. To be more precise, recall that the notion of theory itself is an algebraic structure which can be encoded by the sequence of sets T (n), and certain operations on these sets satisfying certain identities. Denoting the set of operations defining theories by G (for ‘generators’), and identities by R (for ‘relations’), we observe that the set of coherence isomorphisms we must require for lax T -algebras is always in bijective correspondence with G, while the set of coherence diagrams needed is in bijective correspondence with R! The concept of lax algebra over a 2-theory is defined in a similar fashion, but one important point is that one doesn’t want to consider the most general possible type of laxness (since that would lead to a 3-category). Rather, one starts with a lax algebra I over the indexing theory, and a strict functor X : I ×k → Categories; appropriate coherence isomorphisms and diagrams then follow in the same way as in the case of lax algebras over a theory (are indexed by operations and identities of 2-theories interpreted as a ‘universal algebras’ – see [16]). The lax commutative monoid we most frequently consider is the groupoid S of finite sets and isomorphisms (the operation is disjoint union). More generally, we often consider a set of labels K and the lax commutative monoid of SK of sets A labelled by K, i.e. maps A → K. Again, the operation is disjoint union. The example of lax commutative monoid with cancellation fibered over S considered in [16] is the groupoid C of worldsheets or rigged surfaces. These are 2-dimensional smooth manifolds with smooth boundary; further, each boundary component is parametrized by a smooth diffeomorphism with S 1 , and the surface has a complex structure with respect to which the boundary parametrization is analytic. Morphisms are biholomorphic diffeomorphisms commuting with boundary parametrization. Addition is disjoint union, and cancellation is gluing of boundary components. Similarly, again, one can consider the LCMC CK of worldsheets with K-labelled boundary components, which is an LCMC over SK . To complete the picture, one needs to consider stacks. We note that lax algebras over a theory and lax algebras (in our sense) over a 2-theory form 2-categories which have lax limits of strict diagrams (see Fiore [9]). For older references, which however work in slightly different contexts (and with different terminology), see Borceux [3] or [9]. For the 2-category structure, 1-morphisms are lax morphisms of lax algebras (functors such that there is a natural coherence isomorphism for every element of G), and 2-morphisms are natural isomorphisms which commute with the operations given by the theory (or
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2-theory). Now for any 2-category C with lax limits, and every Grothendieck topology B, we can define B-stacks over C: they are simply contravariant functors B → C which turn Grothendieck covers into lax limits. Note that such stacks then themselves form a 2-category with respect to stack versions of the same 1-morphisms and 2-morphisms. Now to turn C into a stack of lax commutative monoids with cancellation, we must first specify the Grothendieck topology. Note that there are two choices of the topology, either just (finite-dimensional) smooth manifolds and open covers (non-chiral setting) or finite-dimensional complex manifolds and open covers (chiral setting). As remarked in Sect. 2 above, however, D-branes can only be considered in the non-chiral setting. To define the stack, one must first define the underlying stack of lax commutative monoids; the answer is simply the stack of covering spaces with finitely many sheets. Now one must define smooth or holomorphic families of worldsheets. We shall only make the definition in the holomorphic case, the smooth case is analogous. The most convenient way to make this precise is to consider, for a worldsheet X, the complex manifold Y obtained by gluing, locally, solid cylinders to the boundary components of X. Then, a holomorphic family of rigged surfaces X over a finite dimensional complex manifold B is a holomorphic map q:Y →B transverse to every point, such that dim(Y ) = dim(B) + 1 and B is covered by open sets Ui for each of which there are given holomorphic regular inclusions si,c : D × Ui → Y with
q ◦ si,c = I dUi ,
where c runs through some indexing set Ci . Further, if Ui ∩ Uj = ∅, we require that there be a bijection ι : Ci → Cj such that si,c |D×(Ui ∩Uj ) = sj,ι(c) |D×(Ui ∩Uj ) . Then we let X =Y −(
si,c ((D − S 1 ) × Ui )).
i c∈Ci
Then the fiber of X over each b ∈ B is a rigged surface, which vary holomorphically in b, in the sense we want. (Note that the reason the maps sc cannot be defined globally in B is that it is possible for a non-trivial loop in π1 (B) to permute the boundary components of X.) The treatment of CK is analogous. As a rule, we shall use the same symbol for the SLCMC’s C, CK as for the corresponding LCMC’s (their sections over a point). We are done with the review of SLCMC’s, but we shall still briefly cover CFT’s, as defined in [16]. Although this definition is subsumed by Sect. 5 above, the reader might still find the more elementary definition useful while reading the earlier sections. Let H1 , ..., Hn be complex (separable) Hilbert spaces. Then on H1 ⊗ · · · ⊗ Hn , there is a natural inner product a1 ⊗ · · · ⊗ an , b1 ⊗ · · · ⊗ bn = a1 , b1 a2 , b2 · · · an , bn . The Hilbert completion of this inner product space is called the Hilbert tensor product ˆ · · · ⊗H ˆ n. H1 ⊗
(36)
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Now an element of (36) is called trace class if there exist unit vectors eij ∈ H, where j = 1, . . . , n and i runs through some countable indexing set I such that x=
µi (ei1 ⊗ · · · ⊗ ein )
i∈I
and
|µi | < ∞.
i∈I
The vector subspace of (36) of vectors of trace class will be denoted by H1 · · · Hn .
(37)
Note that (37) is not a Hilbert space. We have, however, canonical maps : (H1 · · · Hn ) ⊗ (Hn+1 · · · Hm+n ) → H1 · · · Hm+n and, if H∗ denotes the dual Hilbert space to a complex Hilbert space H, tr : H H∗ H1 · · · Hn → H1 · · · Hn . This allows us to define a particular example of stack of LCMC’s based on H, which we will call H. The underlying stack of lax commutative monoids (T -algebras) is S. Now let B ∈ B. Let s, t be sections of the stack S over B, i.e. covering spaces of B with finitely many sheets. Then we have an infinite-dimensional holomorphic bundle over B, (H∗ )s Ht .
(38)
What we mean by that is that there is a well defined sheaf of holomorphic sections of (38) (note that it suffices to understand the case when s, t are constant covering spaces, which is obvious). Now a section of H over a pair of sections s, t of S is a global section of (38) over b; the only automorphisms of these sections covering I ds × I dt are identities. The operation +, ?ˇ are given by the operations , tr (see above). We can also define a variation of this LCMC for the case of labels indexed over a finite set K. We need a collection of Hilbert spaces HK = {Hk |k ∈ K}. Then we shall define a stack of LCMC’s HK . The underlying stack of T -algebras (commutative monoids) is SK . Let s, t be sections of SK over B ∈ B. The place of (38) is taken by ∗ s t (HK ) HK .
(39)
By the sheaf of holomorphic section of (39) when B is a point we mean that -powers of Hk (or Hk∗ ) for each label k ∈ K are taken according to the number of points of (t) (resp. (s)); when s and t are constant covering spaces B, the space of sections of (39) is simply the set of holomorphically varied elements of the spaces of sections over points of B (which are identified). This is generalized to the case of general s, t in the obvious way (using functoriality with respect to permutations of coordinates). As above, the only automorphisms of these sections covering I ds × I dt are identities.
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Remark. For technical reasons (different types of convergence), the above setup involving Hilbert spaces and trace class elements is sometimes insufficient (see Example below and Sect. 4 above). Because of that, it is beneficial to generalize to a context where H simply means any SLCMC over S (resp. SK in the labelled case) whose spaces of sections are vector spaces, the operation ?ˇ is linear, and the operation + is bilinear. We shall further assume that H is a sheaf in the sense that the only endomorphisms over the identity in S (resp. SK ) is the identity. Example. Consider the free bosonic CFT (10) discussed in Sect. 4. As remarked above, the description 10 is actually already not quite right: the vacuum state is to be an eigenstate of momentum 0, but there is no non-zero function in L2 (R, C) with support in the set {0}. For the same reason, we also find that the operator UAq associated with the standard annulus Aq is not trace class as defined (since, for example, 1 is a limit point of the spectrum of UAq ). This is the usual problem in quantum mechanics. In the present setting, a solution along the lines of the Remark can be obtained as follows: Let F be the bosonic Fock space, i.e. n , zn , n > 0. F = Symz Then the sections of H over (s, t) (over a point) are elements f ∈
k∈R|s|+|t|
ˆ
ˆ
ˆ ∗⊗|s| F ⊗|t| ⊗F
(the product is a categorical product of vector spaces) which have the property that for every pair of injections i : u → s, j : u → t and every map k : (s − i(u)) (t − j (u)) → R,
we have
µ(φi,j,k (x)) < ∞, x∈Ru Rs t is defined by y(i(r)) =
where φi,j,k (x) = f (y), y ∈ y(j (r)) = x(r), y(r) = k(r) ˆ ˆ ˆ ∗⊗|s| , ⊗F for r ∈ (s − i(u)) (t − j (u)) and, for z ∈ F ⊗|t|
µ(z) = inf ({
|ai | | z =
ai
⊗ ex , ||ex || = 1 for all x}, x∈s
and gluing along i, j is defined by fˇ(k) =
t
x∈Ru
tr(φi,j,k (x)).
(40)
The expression (40) is always defined because of the condition imposed, and the condition is preserved by the gluing operation by Fubini’s theorem. Now we can define an abstract CFT based on an SLCMC D with underlying stack of lax commutative monoids (SLCM) S simply as a 1-morphism of SLCMC’s, over I dS , D → H. A similar definition applies if D has underlying SLCM SK , with H replaced by HK .
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However, this notion still is not definitive in the sense that it does not capture anomaly. In Sect. 5 above, we give the most general definition of modular functor, but it is useful to review a direct definition from [16] at least in one special case, namely the 1-dimensional anomaly. To this end, we give the definition of C× -central extension (or, equivalently, 1-dimensional modular functor) on an LCMC D. This is a strict morphism of stacks of LCMC’s ψ : D˜ → D
(41)
over I d on the underlying stacks of LCM’s with the following additional structure (for simplicity, let us just work in the holomorphic (chiral) setting): For each object B of B, and each pair of sections s, t of S over B, and each section α of D over s, t, B, B → B, ψ −1 (α|B )
(42)
B
is the space of sections of a complex holomorphic line bundle over B. with varying Furthermore, functoriality maps supplied by the structure of a stack of LCMC’s on D˜ are linear maps on these holomorphic line bundles. Regarding the operation +, we require that the map induced by + ψ −1 (α|B ) × ψ −1 (β|B ) → ψ −1 ((α + β)|B )
(43)
be a bilinear map, which induces an isomorphism of holomorphic line bundles ψ −1 (α|B ) ⊗OB ψ −1 (β|B ) → ψ −1 ((α + β)|B )
(44)
(OB is the holomorphic structure sheaf on B). ˇ we simply require that if α is a section of D over s+u, t +u, Regarding the operation ?, B, where u is another section of S over B, and αˇ is the section over s, t, B which is obtained by applying the operation ?ˇ to α, then the map of holomorphic line bundles coming from LCMC structure ψ −1 (α|B ) → ψ −1 (α| ˇ B )
(45)
(B → B) is an isomorphism of holomorphic line bundles. By a CFT with 1-dimensional modular functor over D with underlying stack S we shall mean a CFT φ : D˜ → H
(46)
(where D˜ is a C× -central extension of D which has the property that φ is a linear map on the spaces of sections (42). Similarly in case D has underlying stack SK , we simply replace S by SK everywhere throughout the definition. It is appropriate to comment on a weaker kind of morphism of lax algebras where we do not require that the coherence maps be iso. By a pseudomorphism of lax T -algebras (and similarly in the cases of lax , T -algebra and their stacks) we shall mean a functor f :X→Y together with morphisms (called cross-morphism, not necessarily iso) γ (f, ..., f ) → f γ
(47)
which commute with all the coherences in the lax T -algebra sense (we shall refer to these required commutative diagrams as cross-diagrams).
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Remark. Although the anomaly of the linear σ -model considered in Sect. 4 is 1-dimensional, in Sect. 5 we considered higher-dimensional anomalies. It is therefore appropriate to reconcile the above remark concerning generalizing the SLCMC H to cases when the Hibert/trace class model fails due to non-convergence with our discussion of higherdimensional modular functors via 2-vector spaces. In other words, what is the right generalization of H ∈ Obj (MH ilb ) in C(M, H )? The main point is that the Hilbert ˆ ˆ ˆ should be replaced by a “vector space indexed over M” ⊗H ∗⊗s tensor powers H ⊗t which depends only on s, t and have appropriate designated “trace maps”. The category MV ect of vector spaces indexed over M is defined as M ⊗C2 CV2 ect , where CV2 ect is the lax commutative semiring of C-vector spaces (not necessarily finitely dimensional). Now H ∈ Obj (MV ect ), we may consider a pseudomorphism of SLCMC’s over S (see above) h : S 2 → C(MV ect ). Here by C(MV ect ) we mean the analogous construction as C(M), but with the duals taken over CV ect , so (MV ect )∗ =def (M∗ )V ect . Then, an SLCMC C(M, h) is defined as follows. Sections over a point over σ ∈ Obj (S)2 consist of a section M of C(M) over σ and a 2-morphism M → h(σ ). Stacking, and the necessary verifications, are completed in the usual way. Acknowledgement. We thank G. Moore for pointing out to us his lectures [26]. We would also like to think N. A. Baas and J. Rognes for their helpful mathematical comments, and for detecting mistakes in the original version of this paper.
References 1. Ando, M.: Power operations in elliptic cohomology and representations of loop groups. Trans. AMS 352, 5619–5666 (2000) 2. Baas, N.A., Dundas, B.I., Rognes, J.: Two-vector bundles and forms of elliptic cohomology. To appear in Segal Proceedings, Cambridge University Press 3. Borceux, F.: Handbook of categorical algebra 1–2, Encyclopedia of Mathematics and its Applications, Cambridge: Cambridge University Press, pp.50–52 4. Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebra. Invent. Math. 109, 405–444 (1992) 5. d’Hoker, E.: String theory, In: Quantum fields and strings: a course for mathematicians, Vol. 2, Providence RI: AMS and IAS, 1999, pp. 807–1012 6. Deligne, P., Freed, D.: Notes on Supersymmetry (following J. Bernstein) In: Quantum fields and strings, a course for mathematicians, Vol. 1, Providence RI: AMS, 1999, pp. 41–98 7. Diaconescu, D.E.: Enhanced D-brane categories from string field theory. JHEP 0106, 16 (2001) 8. Douglas, M.R.: D-branes, categories and N = 1 SUSY. J. Math. Phys. 42, 2818–2843 (2001) 9. Fiore, T.: Lax limits, lax adjoints and lax algebras: the categorical foundations of conformal field theory. To appear 10. Frenkel, I.: Vertex algebras and algebraic curves. Seminaire Bourbaki 1999–2000, Asterisque 276, 299–339 (2002) 11. Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the monster. Pure and applied Mathematics, Vol. 134, London–NewYork: Academic Press, 1999 12. Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators I: Partition functions. Nucl. Phys. B 646, 353 (2002)
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13. Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators II: Unoriented world sheets. Nucl. Phys. B 678, 511 (2004) 14. Green, M.B., Schwartz, J.H., Witten, E.: Superstring theory. Vol. 1,2, Cambridge: Cambridge University Press, 1988 15. Horava, P.: Equivariant Topological Sigma Models. Nucl. Phys. B 418, 571–602 (1994) 16. Hu, P., Kriz, I.: Conformal field theory and elliptic cohomology. Advances in Mathematics 189(2), 325–412 (2004) 17. Hu, P., Kriz, I., Voronov, A.A.: On Kontsevich’s Hochschild cohomology conjecture. http://arxiv.org/abs/amth.AT/0309369, 2003 18. Huang, Y.Z., Kong, L.: Open-string vertex algebras, tensor categories and operads. Comm. Math. Phys. 250, 433–471 (2004) 19. Kriz, I.: On spin and modularity in conformal field theory. Ann. Sci. de ENS 36, 57–112 (2003) 20. Lawvere, W.F.: Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A. 50, 869–87 (1963) 21. Lazaroiu, C.I.: On the structure of open-closed topological field theory in two-dimensions. Nucl. Phys. B 603, 497–530 (2001) 22. Lazaroiu, C.I.: Generalized complexes and string field theory. JHEP 06, 52 (2001) 23. Lazaroiu, C.I.: Unitarity, D-brane dynamics and D-brane categories. JHEP 12, 31 (2001) 24. Lewellen, D.: Sewing constraints for conformal field theories on surfaces with boundaries. Nucl. Phys. B 372, 654 (1992) 25. Moore, G.: Some Comments on Branes, G-flux, and K-theory. Int. J. Mod. Phys. A 16, 936–944 (2001) 26. Moore, G.: Lectures on branes, K-theory and RR-charges. http://www.physics.rutgers.edu/∼gmoore/day1/12.html 27. Moore, G., Seiberg, N.: Classical and Quantum Conformal Field Theory. Commun. Math. Phys. 123, 177–254 (1989) 28. Moore, G., Seiberg, N.: Taming the conformal ZOO. Phys. Lett. B 220, 422–430 (1989) 29. Moore, G. Seiberg, N.: Lectures on RCFT. In: H.C. Lee (ed.), Physics, Geometry and Topology, RiverEdge, World Scientific, 1990, pp. 263–361 30. Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8, 177 (2003) 31. Polchinski, J.: String theory. Vols. 1,2, Cambridge: Cambridge Univ. Press, 1999 32. Pradisi, G., Sagnotti, A., Stanev, Y.A.: Planar duality in SU (2) WZW models. Phys. Lett. B 354, 279 (1995) 33. Pressley, A., Segal, G.: Loop groups. Oxford: Oxford University Press, 1986 34. Segal, G.: Elliptic cohomology. Seminaire Bourbaki 1987/88, Asterisque 161–162, Exp. No, 695, (1988) 4, 187–201 (1989) 35. Segal, G.: The definition of conformal field theory. Preprint, 1987 36. Segal, G.: ITP lectures. http://doug-pc.itp.ucsb.edu/online/geom99/, 1999 37. Segal, G.: Categories and cohomology theories. Topology 13, 293–312 (1974) 38. Stolz, S., Teichner, P.: What is an elliptic object?, In: U. Tillmann (ed.), Proc. of 2002 Oxford Symp. in Honour of G.Segal, Cambridge: Cambridge Univ. Press, 2004 39. Thomason, R.W.: Beware the phony multiplication on Quillen’s A−1 A. Proc. AMS 80(4), 569–573 (1980) 40. Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360–376 (1988) 41. Witten, E.: Overview of K-theory applied to strings. Int. J. Mod. Phys. A 16, 693–706 (2001) Communicated by M.R. Douglas
Commun. Math. Phys. 254, 255 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1239-8
Communications in
Mathematical Physics
Erratum
The Norm Convergence of the Trotter–Kato Product Formula with Error Bound Takashi Ichinose1 , Hideo Tamura2 1
Department of Mathematics, Faculty of Science, Kanazawa University, Kanazawa, 920–1192, Japan. E-mail:
[email protected] 2 Department of Mathematics, Faculty of Science, Okayama University, Okayama, 700–8530, Japan. E-mail:
[email protected] Received: 10 June 2003 / Accepted: 19 August 2004 Erratum published online: 2 December 2004 – © Springer-Verlag 2004 Commun. Math. Phys. 217, 489–502 (2001)
It was kindly pointed out by Vidmantas Bentkus that there is a small gap, in the proof of Lemma 2.1, for the case where C is strictly positive, i.e. C ≥ η for some constant η > 0. We have to establish an estimate e−tSε − e−tC ≤ Mt −1 ε α for every ε > 0 with a constant M independent of t and ε. To do so, we need to prove that for Sε = ε−1 (1 − F (ε)), the inverse Sε−1 exists and is uniformly bounded for every ε > 0. The proof given in the paper is correct for sufficiently small ε > 0, up to a certain ε0 > 0, because we can show Sε−1 is uniformly bounded with Sε−1 < 2/η for all positive ε ≤ ε0 . However, we need to supplement that with a proof for the large ε case. To this end, in the statement of Lemma 2.1, we should have further assumed on F (t), in the case C is strictly positive, that for this ε0 there exists δ0 = δ0 (ε0 ) > 0 such that F (t) ≤ 1 − δ0 (ε0 ) for every t ≥ ε0 , or that for every ε > 0 there exists δ = δ(ε) > 0 such that F (t) ≤ 1 − δ(ε) for every t ≥ ε. Hence we can easily see that for these large ε, Sε is uniformly bounded with Sε−1 < ε0 /δ0 . Such an additional assumption on F (t) in Lemma 2.1, for the case C is strictly positive, does not affect the rest of the proof of the main Theorem. Communicated by M. Aizenman
Commun. Math. Phys. 254, 257–287 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1203-7
Communications in
Mathematical Physics
Brownian Directed Polymers in Random Environment Francis Comets1, , Nobuo Yoshida2, 1 2
Universit´e Paris 7, Math´ematiques, Case 7012, 2 place Jussieu, 75251 Paris, France. E-mail:
[email protected] Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan. E-mail:
[email protected]
Received: 5 March 2003 / Accepted: 17 May 2004 Published online: 14 October 2004 – © Springer-Verlag 2004
Abstract: We study the thermodynamics of a continuous model of directed polymers in random environment. The environment is given by a space-time Poisson point process, whereas the polymer is defined in terms of the Brownian motion. We mainly discuss: (i) The normalized partition function, its positivity in the limit which characterizes the phase diagram of the model. (ii) The existence of quenched Lyapunov exponent, its positivity, and its agreement with the annealed Lyapunov exponent; (iii) The longitudinal fluctuation of the free energy, some of its relations with the overlap between replicas and with the transversal fluctuation of the path. The model considered here, enables us to use stochastic calculus, with respect to both Brownian motion and Poisson process, leading to handy formulas for fluctuations analysis and qualitative properties of the phase diagram. We also relate our model to some formulation of the Kardar-Parisi-Zhang equation, more precisely, the stochastic heat equation. Our fluctuation results are interpreted as bounds on various exponents and provide a circumstantial evidence of super-diffusivity in dimension one. We also obtain an almost sure large deviation principle for the polymer measure. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Brownian directed polymers in random environment . . . 1.2 Connection to the Kardar-Parisi-Zhang equation . . . . . . . . 1.3 Other related models . . . . . . . . . . . . . . . . . . . . . . 2. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The normalized partition function and its positivity in the limit 2.2 The quenched Lyapunov exponent . . . . . . . . . . . . . . .
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Partially supported by CNRS (UMR 7599 Probabilit´es et Mod`eles Al´eatoires) Partially supported by JSPS Grant-in-Aid for Scientific Research, Wakatekenkyuu (B) 14740071
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2.3 The replica overlap . . . . . . . . . . . 2.4 Fluctuation results . . . . . . . . . . . . 3. Some Preliminaries from Stochastic Calculus 4. Proof of Theorem 2.1.1 (b), (c) . . . . . . . . 4.1 Proof of part (b) . . . . . . . . . . . . . 4.2 Proof of part (c) . . . . . . . . . . . . . 5. Proof of Theorem 2.4.1 . . . . . . . . . . . . 5.1 Proof of part (a) . . . . . . . . . . . . . 5.2 Proof of part (b) . . . . . . . . . . . . . 5.3 Proof of Corollary 2.4.2 . . . . . . . . . 5.4 Proof of Corollary 2.4.3 . . . . . . . . . 6. Proof of Theorem 2.2.1 . . . . . . . . . . . . 6.1 Proof of part (a) . . . . . . . . . . . . . 6.2 Proof of part (b) . . . . . . . . . . . . . 7. Proof of Theorem 2.2.2 and Theorem 2.3.1 . . 7.1 Differentiating the averaged free energy 7.2 Proof of Theorem 2.2.2 (a) . . . . . . . 7.3 Proof of Theorem 2.2.2 (b) . . . . . . . 7.4 Proof of Theorem 2.2.2 (c) . . . . . . . 7.5 Proof of Theorem 2.3.1 . . . . . . . . . 8. Proof of Theorem 2.3.2 . . . . . . . . . . . . 9. Proof of Theorem 2.4.4 . . . . . . . . . . . . 10. The Stochastic Heat Equation . . . . . . . . .
F. Comets, N. Yoshida
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1. Introduction Directed polymers in random environment can be thought of as paths of stochastic processes interacting with a quenched disorder (impurities), depending on both time and space. Roughly, individual paths are not only weighted according to their length, but also according to random impurities that they meet along their route, with a larger influence as the temperature is decreased. A physical example is the interface in the 2-dimensional Ising model with random bonds [13], within the Solid-On-Solid approximation – where the interface can be parametrized by one coordinate. The heuristic picture at low temperature is that typical paths are pinned down to clouds of favorable impurities. With the relevant clouds located at a large distance, the polymer behaves superdiffusively. Similarly, the free energy essentially depends on the characteristics of the relevant clouds, exhibits large fluctuations, as well as other thermodynamic quantities. Directed polymers in random environment, at positive or zero temperature, relate – even better, can sometimes be exactly mapped – to a number of interesting models of growing random surfaces (last passage oriented percolation, ballistic deposition, polynuclear growth, low temperature Ising models), and non equilibrium dynamics (totally asymmetric simple exclusion, population dynamics in random environment). We refer to the survey paper [18] by Krug and Spohn for a detailed account on these models and their relations. On the other hand, stochastic calculus provides a number of natural tools for studying random processes and their fluctuations. In this paper, we introduce and study a model which allows using such tools as Doob-Meyer’s martingale decomposition and Itˆo’s formula.
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1.1. The Brownian directed polymers in random environment. The model we consider in this paper is defined in terms of Brownian motion and of a Poisson random measure. Before introducing the polymer measure, we first fix some notations. In what follows, R+ = [0, ∞), R− = (−∞, 0], d denotes a positive integer and B(R+ × Rd ) the class of Borel sets in R+ × Rd . • The Brownian motion: Let ({ωt }t≥0 , {P x }x∈Rd ) denote a d-dimensional standard Brownian motion. Specifically, we let the measurable space (, F) be the path space C(R+ → Rd ) with the cylindrical σ -field, and P x be the Wiener measure on (, F) such that P x {ω0 = x} = 1. • The space-time Poisson random measure: Let η denote the Poisson random measure on R+ × Rd with the unit intensity, defined on a probability space (M, G, Q). Then, η is an integer valued random measure characterized by the following property: If A1 , . . . , An ∈ B(R+ × Rd ) are disjoint and bounded, then n n |Aj |kj Q {η(Aj ) = kj } = exp(−|Aj |) (1.1) for k1 , . . . , kn ∈ N. kj ! j =1
j =1
Here, | · | denotes the Lebesgue measure in R1+d . For t > 0, it is natural and convenient to introduce ηt (A) = η(A ∩ ((0, t]×Rd )) ,
A ∈ B(R+ × Rd )
(1.2)
and the sub σ -field Gt = σ [ηt (A) ; A ∈ B(R+ × Rd )] .
(1.3)
• The polymer measure: We let Vt denote a “tube”around the graph {(s, ωs )}0<s≤t of the Brownian path, Vt = Vt (ω) = {(s, x) ; s ∈ (0, t], x ∈ U (ωs )},
(1.4)
where U (x) ⊂ Rd is the closed ball with the unit volume, centered at x ∈ Rd . For any t > 0 and x ∈ Rd , define a probability measure µxt on the path space (, F) µxt (dω) =
exp (βη(Vt )) x P (dω), Ztx
(1.5)
where β ∈ R is a parameter and Ztx = P x [exp (βη(Vt ))] .
(1.6)
Under the measure µxt , the graph {(s, ωs )}0≤s≤t may be interpreted as a polymer chain living in the (1 + d)-dimensional space, constrained to stretch in the direction of the first coordinate (t-axis). At the heuristic level, the polymer measure is governed by the formal Hamiltonian 1 t η βHt (ω) = |ω˙ s |2 ds − β # points (s, x) in η : s ≤ t, x ∈ U (ωs ) (1.7) 2 0 on the path space. Since this Hamiltonian is parametrized by η, the polymer measure µxt is random. The path ω is attracted to Poisson points when β > 0, and repelled by them when β < 0. The sets {s} × U (x) with (s, x) a point of the Poisson field η, appear as
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“rewards” in the first case, and “soft obstacles” in the second one. Note that the obstacles stretch in the transverse direction (x-hyperplane): This is a key technical point, allowing a simple use of stochastic calculus with respect to the Poisson field. In this paper, we address the question of understanding the large time behavior of the transversal motion (ωt )t≥0 under the polymer measures (µxt )t≥0 , in particular, how its fluctuation in large time scale is affected by the random environment η. As is the general rule in statistical mechanics, much information will be obtained by investigating the asymptotic behavior of the partition function Ztx . Let us finish the definition of the model with some remarks on the notation we use. An important parameter is λ = λ(β) = eβ − 1 ∈ (−1, ∞) ,
(1.8)
which is in fact the logarithmic moment generating function of a mean-one Poisson distribution. When we want to stress the dependence of λ on β ∈ R, we will use the notation λ(β). But otherwise, we will simply write λ. We will denote by P , µt , Zt , . . . , the quantities P x , µxt , Ztx , . . . with x = 0. Note that (Ztx )t≥0 has the same distribution as (Zt )t≥0 . For this reason, we will state and prove results on (Ztx )t≥0 only for the case x = 0 with simpler notation, however without loss of generality. 1.2. Connection to the Kardar-Parisi-Zhang equation. Another strong motivation for the present model is its relation to some stochastic partial differential equations. To describe the connection, it is necessary to relativize the partition function, by specifying x→y the ending point of the Brownian motion at time t. For 0 ≤ s < t, let Ps→t be the distribution of the Brownian bridge starting at point x at time s and ending at y at time t. Define x→y
Ztx (y) = gt (y − x)P0→t [exp (βη(Vt ))] ,
(1.9)
with gt (x) = (2π t)−d/2 exp{−|x|2 /2t} the Gaussian density. Then, by definition of the Brownian bridge, Ztx =
Rd
Ztx (y)dy .
Similar to the Feynman-Kac formula, we will show the following stochastic heat equation (SHE) with multiplicative noise in a weak sense, dZtx (y) = 21 y Ztx (y)dt + λZtx− (y)η(dt × U (y)) ,
t ≥ 0, x, y ∈ Rd ,
(1.10)
where dZtx (y) denotes the time differential and y = ( ∂y∂ 1 )2 +· · ·+( ∂y∂ d )2 the Laplacian operator. (SHE) will be properly formulated and be proved in Sect. 10. In the literature, this equation has been extensively considered in the case of a Gaussian driving noise, instead of the Poisson process η here. Although we are able to prove (1.10) only in the weak sense, let us now pretend that (1.10) is true for all y ∈ Rd . We would then see from Itˆo’s formula that the function ht (y) = ln Ztx (y) solves the Kardar-Parisi-Zhang equation (KPZ):
dht (y) = 21 ht (y) + |∇ht (y)|2 dt + β η(dt ×U (y)) . We observe that, since h has jumps in the space variable y, the non-linearity makes the precise meaning of this equation somewhat knotty. We will not address this equation in
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the present paper, but we make a few comments. This equation was introduced in [16] to describe the long scale behavior of growing interfaces. More precisely, the fluctuations in the KPZ equation –driven by a δ-correlated, Gaussian noise–are believed to be non-standard, and universal, i.e., the same as in a large class of microscopic models. See [18] for a detailed review of kinetic roughening of growth models within the physics literature, in particular to Sect. 5 for the status of this equation. In dimension d = 1, Bertini and Giacomin [1] proved that the KPZ equation comes as the limit of renormalized fluctuations for two microscopic models: the weakly asymmetric exclusion process, and the related Solid-On-Solid interface model.
1.3. Other related models. The model we introduce in this paper has a number of close relatives in the literature. We now mention some of them. • Simple random walk model of directed polymers. This model was originally introduced in physics literature [13] to mimic the phase boundary of the Ising model subject to random impurities. Later on, the model reached the mathematics community [15, 3], where it was reformulated in terms of the d-dimensional simple random walk (ωn )n≥1 and of i.i.d. random variables {η(n, x) ; (n, x) ∈ N × Zd }. The energy of this simple random walk model is given by βHnη (ω) = −β
n
η(j, ωj ).
(1.11)
j =1
Therefore, the Brownian motion model described by (1.7) can be thought of as a natural transposition of (1.11) into continuum setting. The model (1.11) has already been studied for more than a decade and by many authors. See for example [15, 3, 25, 23, 5, 7]. • Gaussian random walk model of directed polymers [22, 20]. The Hamiltonian of this model takes the same form as (1.11). However, the random walk (ωn )n≥1 here is the summation of independent Gaussian random variables in Rd and the random field {η(n, x) ; (n, x) ∈ N × Rd } has certain correlation in x variables. A major technical advantage in working with the Gaussian random walk rather than the simple random walk is the applicability of a Girsanov-type path transformation, which plays a key role in analyzing this model. • Crossing Brownian motion in a soft Poissonian potential [27, 30–32]. The model investigated there is also described in terms of Brownian motion and of Poisson points. The main difference is that the Brownian motion there is “undirected”, in other words, the d-dimensional Brownian motion travels through the Poisson points distributed in the space Rd , not in space-time as in ours. However, as crossings of the Brownian motion are enforced in some direction, backtracks of the path become scarcer, and the model starts to resemble the directed one. This is the case when the Brownian motion has sufficiently large drift, or when it is conditoned to hit a distant set before being killed by the obstacles. The directional Lyapunov exponent (point-to-hyperplane) in [27, 30] corresponds to the large-time limit of t −1 ln Zt in the present paper. • First and last passage percolation [17, 21, 19]. The first (resp. last) passage percolation can be thought of as an analogue of directed polymers at β = −∞ (resp. β = +∞). In fact, it is expected and even partly vindicated that the properties of the path with minimal/maximal passage time has a similar feature to the typical paths under the polymer measure.
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2. Main Results 2.1. The normalized partition function and its positivity in the limit. Let us begin by introducing an important martingale on (M, G, Q) ((2.1) below). In fact, the large time behavior of this martingale somehow characterizes the phase diagram of this model and for this reason, many of the results in this paper can be best understood from the viewpoint of this martingale. For any fixed path ω, the process {η(Vt )}t≥0 has independent, Poissonian increments, hence it is itself a standard Poisson process on the half-line, and {exp(βη(Vt ) − λt)}t≥0 is its exponential martingale. Therefore, the normalized partition function Wt = e−λt Zt ,
t ≥0
(2.1)
is itself a mean-one, right-continuous and left-limited, positive martingale on (M, G, Q), with respect to the filtration (Gt )t≥0 defined by (1.3). In particular, the following limit exists Q-a.s.: def.
W∞ = lim Wt . t ∞
(2.2)
Since exp(βη(Vt )) > 0 Q-a.s. for all 0 ≤ t < ∞ and all ω ∈ , the event {W∞ = 0} is measurable with respect to the tail σ -field σ [η|[t,∞)×Rd ] , t≥1
and therefore by Kolmogorov’s 0-1 law, we only have the two contrasting situations: Q{W∞ = 0} = 1,
(2.3)
Q{W∞ > 0} = 1.
(2.4)
or
Loosely speaking, the presence of the random environment is supposed to make qualitative difference in the large time behavior of the Brownian polymer in the former case (2.3) (the strong disorder phase), while it does not in the latter case (2.4) (the weak disorder phase). The phase structure of this model is described as follows. Theorem 2.1.1. (a) For all d ≥ 1, there is a finite β1 (d) > 0 such that (2.3) holds for β ∈ (β1 (d), ∞). (b) For d = 1, 2, (2.3) holds whenever β = 0. (c) For d ≥ 3, there is β0 (d) > 0 such that lim β0 (d) = ∞ and that (2.4) holds for β ∈ (−∞, β0 (d)).
d ∞
Theorem 2.1.1 in particular shows the existence of the phase transition in d ≥ 3, from the weak disorder phase to the strong disorder phase. In Theorem 2.2.2 below, we will capture this phase transition in terms of the quenched Lyapunov exponent. Theorem 2.1.1(a) follows from a stronger result (2.9) we present later on. The other parts of Theorem 2.1.1 are proved in Sect. 4.
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Remark 2.1.1. • Theorem 2.1.1(a) and (b) for the simple random walk model can be found in [7, Theorem 2.3 (b), Prop. 2.4 (a)]. Theorem 2.1.1(c) for the simple random walk model is also known, see [3, Lemma 2] and [25, Lemma 1]. • We emphasize at this point that we have chosen (for simplicity) unit values for the volume of the ball U (·), for the intensity of the Poisson process and for the diffusion coefficient of the Brownian motion. Our proof that (2.4) holds for all negative β, as stated in Theorem 2.1.1(c), crucially relies on this choice. In contrast, the other statements in Theorem 2.1.1(a)–(c) are not qualitatively affected by this choice. 2.2. The quenched Lyapunov exponent. We now state the result on the existence of the quenched Lyapunov exponent. Theorem 2.2.1. Let d ≥ 1 and β ∈ R be arbitrary. (a) There exists ψ(β) ∈ R+ such that 1 ψ(β) = lim − ln Zt + λ(β), t ∞ t
Q-a.s. and in L2 (Q).
(2.5)
(b) The function ψ(β) − λ(β) is concave on R. Hence, the function β → ψ(β) is (β) and ψ (β) locally Lipschitz continuous and has right and left derivatives ψ+ − for all β ∈ R such that −∞ < ψ+ (β) ≤ ψ− (β) < ∞.
(2.6)
(β) < ψ (β)} is at most Moreover, the set of non-differentiability {β ∈ R ; ψ+ − countable.
We call ψ(β) the quenched Lyapunov exponent. The quantity ψ(β) is the exponent for the decay of the martingale Wt as t ∞ (cf. (2.1)). Equivalently, but on more physical ground, ψ(β) is the difference between the annealed free energy t −1 ln Q[Zt ], and the quenched free energy t −1 Q[ln Zt ] in the thermodynamic limit t ∞. It is reasonably expected, and even confirmed partly in this paper (see Theorem 2.3.1 and Theorem 2.3.2 below for example) that the positivity of ψ makes the large time behavior of the Brownian polymer dramatically different from that of the original Brownian motion, while the polymer behaves somewhat the like usual Brownian motion when ψ = 0. The proof of Theorem 2.2.1 is given in Sect. 6 and goes roughly as follows. We first show the existence of the limit 1 def. ψ(β) = lim − Q[ln Zt ] + λ(β) . (2.7) t ∞ t The existence of (2.7) is in fact a consequence of a simple super-additivity argument. We will then derive (2.5) from Theorem 2.4.1 below, which deals with the fluctuation of ln Zt . We next study the Lyapunov exponent ψ(β) as a function of β ∈ R and gather some information on the phase diagram. Theorem 2.2.2. Let d ≥ 1 be arbitrary. (a) The function ψ(β) is non-decreasing in β ∈ R+ , and is non-increasing in β ∈ R− . Moreover, 0 ≤ ψ(β) ≤ λ(β) − β for all β ∈ R.
(2.8)
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F. Comets, N. Yoshida
(b) ψ(β) = eβ − O(eβ/2 ) as β ∞.
(2.9)
(c) There exist critical values βc+ = βc+ (d), βc− = βc− (d) with −∞ ≤ βc− ≤ 0 ≤ βc+ < +∞, such that (β) = 0 if β ∈ [βc− , βc+ ] ∩ R, ψ(β) = ψ±
ψ(β) > 0, if β ∈
R\[βc− , βc+ ].
(2.10) (2.11)
(d) For d ≥ 3, βc− (d) = −∞, βc+ (d) > 0 and lim βc+ (d) = ∞. d ∞
Remark 2.2.1. • Theorem 2.2.2 shows that the sign of β makes the drastic difference in the behavior of ψ(β); it grows exponentially fast as β ∞, while the growth is at most linear when β −∞. The contrast is even sharper if d ≥ 3, where ψ(β) = 0 for all negative β. • We see from (2.10) and (2.11) that, if βc− (d) < βc+ (d) (resp. −∞ < βc− (d) < βc+ (d)), a phase transition occurs at βc+ (d) (resp., βc− (d)), in the sense that ψ is nonanalytic there. Moreover, from (2.10) it will follow also that ψ is differentiable at this point with zero derivative, meaning that the phase transition there is at least of second order. Note that the phase transition does occur at βc+ (d) if d ≥ 3. • As in Remark 2.1.1, we emphasize that our proof that βc− (d) = −∞ is specific to unit values for the parameters of the model. The proofs of Theorem 2.2.2(a)–(c) is given in Sect. 7. Theorem 2.2.2(d) follows from Theorem 2.1.1(c). 2.3. The replica overlap. We now characterize the critical values βc± (d) in terms of replica overlaps of two independent polymers. On the product space (2 , F ⊗2 ), we consider the probability measure µx,x = t x (µt )⊗2 (dω, d ω), that we will view as the distribution of the couple (ω, ω) with ω an independent copy of ω with law µxt . We introduce random variables Itx and Jtx , t ≥ 0, x ∈ Rd , given by ωt )|] , Itx = µx,x t [|U (ωt ) ∩ U ( Jtx = µx,x ω)|] = t [|Vt (ω) ∩ Vt (
(2.12)
t 0
µx,x ωs )|] ds . t [|U (ωs ) ∩ U (
(2.13)
(We use the same notation | · | for the Lebesgue measure on Rd and on R1+d , and also t to denote the Euclidean norm in the sequel.) Then, both Jtx and 0 Isx ds are interpreted as the expected volume of the overlap in time [0, t] of tubes around two independent polymer paths in the same (fixed) environment. In particular, the fraction 1 |Vt (ω) ∩ Vt ( ω)|, (2.14) t is a natural transposition to our setting, of the so-called replica overlap often discussed in the context of disordered systems, e.g. mean field spin glass, and also of directed polymers on trees [9]. Its relevance to us is explained by the following results, which relates the asymptotics of the overlap to the critical values for β. Rt (ω, ω) =
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Theorem 2.3.1. For all β ∈ [βc− , βc+ ] ∩ R, 1 Q[Jt ] = 0 . t ∞ t lim
On the other hand, with
1 BJ = β ∈ R : lim Q[Jt ] > 0 t ∞ t
(2.15)
1
, BJ = β ∈ R : lim Q[Jt ] > 0 , t ∞ t
we have BJ ⊃ β > 0 ; ψ+ (β) > 0 ∪ β < 0 ; ψ− (β) < 0 ,
J ⊂ β > 0 ; ψ− B (β) > 0 ∪ β < 0 ; ψ+ (β) < 0 ⊂ β > 0 ; ψ+ (β) > 0 ∪ β < 0 ; ψ− (β) < 0 ∪ β ∈ R : ψ+ (β) < ψ− (β) .
(2.16) (2.17)
Therefore, βc+
1 = sup β ≥ 0 : ∀β ∈ [0, β ], lim Q[Jt ] = 0 t ∞ t
J ∩ R+ ) , = inf(BJ ∩ R+ ) = inf(B
(2.18) (2.19)
and similarly, 1 βc− = inf β ≤ 0 : ∀β ∈ [β , 0], lim Q[Jt ] = 0 t ∞ t
= sup(BJ ∩ R− ) = sup(BJ ∩ R− ) .
(2.20)
The proof of Theorem 2.3.1 is given in Sect. 7.5. Theorem 2.3.2. Let β = 0. Then, {W∞ = 0} =
∞
Is ds = ∞ ,
Q-a.s.
Moreover, if Q{W∞ = 0} = 1, then there exist c1 , c2 ∈ (0, ∞) such that t t c1 Is ds ≤ λt − ln Zt ≤ c2 Is ds for large t, Q-a.s. 0
(2.21)
0
(2.22)
0
In particular, 1 t Is ds = 0 if β ∈ [βc− (d), βc+ (d)] ∩ R, t ∞ t 0 1 t lim Is ds > 0 if β ∈ R\[βc− (d), βc+ (d)]. t ∞ t 0 lim
The proof of Theorem 2.3.2 is given in Sect. 8.
(2.23) (2.24)
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Remark 2.3.1. Analogous results of Theorem 2.3.2 for the simple random walk model for directed polymers can be found in [5] and in [7]; Theorem 2.3.1 has a counterpart in the case of a Gaussian environment, which can be seen from Lemma 7.1 in [5]. As in the discrete case, we can interpret the results from the present subsection, in terms of localization for the path. Indeed, we will prove in Sect. 8, that for some constant c1 = c1 (d) ∈ (0, 1), ωs )|] ≤ sup µxt [ωs ∈ U (y)] . (2.25) c1 sup µxt [ωs ∈ U (y)]2 ≤ µx,x t [|U (ωs ) ∩ U ( y∈Rd
y∈Rd
The maximum appearing in the above bounds should be viewed as the probability of the favorite “location” for ωs , under the polymer measure µxt . Both Theorem 2.3.1 and Theorem 2.3.2 are precise statements that the polymer localizes in the strong disorder regime in a few specific corridors of width O(1), but spreads out in a diffuse way in the weak disorder regime. 2.4. Fluctuation results. We now state the following estimate for the longitudinal fluctuation of the free energy. Theorem 2.4.1. Let d ≥ 1 and β ∈ R be arbitrary. (a) With QGs the conditional expectation under Q given Gs , we have 2 dsdx QGs ln (1 + λµt {U (ωs ) x}) . Var Q (ln Zt ) = Q [0,t]×Rd
As a consequence, the following inequalities hold:
2 2 Q dsdx QGs µt {U (ωs ) x} , Var Q (ln Zt ) ≥ c− [0,t]×Rd
2 2 Q dsdx QGs µt {U (ωs ) x} , Var Q (ln Zt ) ≤ c+ [0,t]×Rd
(2.26)
(2.27) (2.28)
and 2 Q[Jt ] Var Q (ln Zt ) ≤ c+
(2.29)
2 t, ≤ c+
(2.30)
where c− = 1 − e−|β| and c+ = e|β| − 1. 2 exp(c ), (b) With c = c+ + u2 1 Q {|ln Zt − Q[ln Zt ]| > u} ≤ 2 exp − (u ∧ ) , 2 ct
u≥0.
(2.31)
The formula (2.26) is analogous to (3.2) in [21]. Here, it is obtained rather easily, thanks to the power of stochastic calculus. The formula (2.29) shows that the fluctuations of ln Zt are small when the overlap is small (cf. (2.15)). The proof of Theorem 2.4.1 and those of the following corollaries are given in Sect. 5. Corollary 2.4.2. Let d ≥ 1, β ∈ R and ε > 0 be arbitrary. Then, as t ∞, ln Zt − Q[ln Zt ] = O(t
1+ε 2
), Q-a.s.
(2.32)
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Corollary 2.4.3. Let β = 0. For ξ > 0 and C > 0, there exists c1 = c1 (d, C) ∈ (0, ∞) such that 2 (2.33) lim t −(1−dξ ) Var Q (ln Zt ) ≥ c1 lim inf Qµt {|ωs | ≤ C + Ct ξ } t ∞
t ∞ 0≤s≤t
≥ c1 lim t ∞
2
Qµt { sup |ωs | ≤ C + Ct } ξ
. (2.34)
0≤s≤t
Remark 2.4.1. Let us interpret Theorem 2.4.1 and its corollaries in terms of critical exponents. Let us write ξ(d) for the “wandering exponent”, i.e., the critical exponent for the transversal fluctuation of the path, and χ (d) for critical exponent for the longitudinal fluctuation of the free energy. Their definitions are roughly that sup |ωs | ≈ t ξ(d) and ln Zt − Q[ln Zt ] ≈ t χ(d) as t ∞.
(2.35)
0≤s≤t
There are various ways to define rigorously these exponents, e.g. (0.6) and (0.10-11) in [30], (2.4) and (2.6-7-8) in [23]. Although the equivalence between these specific definitions are not always clear, the common idea behind all the definitions is described by (2.35). The polymer is said to be diffusive if ξ(d) = 1/2 and super-diffusive if ξ(d) > 1/2. The latter situation is of particular interest. We see from (2.30) that χ (d) ≤ 1/2, while Corolary 2.4.3 suggests that χ (d) ≥ (1 − dξ(d))/2 ,
(2.36)
for non-zero β. (Note that this inequality fails to hold for β = 0.) See also Theorem 2.4.4 and Remark 2.4.4 below for further considerations on the exponents. Remark 2.4.2. Critical exponents similar to the ones discussed above are investigated in the context of various other models and in a large number of papers. In particular, it is conjectured in physics literature that χ (d) = 2ξ(d) − 1, d ≥ 1, χ (1) = 1/3, ξ(1) = 2/3 for all β = 0.
(2.37) (2.38)
See, e.g., [13],[11, (3.4),(5.11),(5.12)], [18, (5.19),(5.28)]. We now mention a few articles which we think are especially relevant to us.1 • M. Piza [23] studies critical exponents for the simple random walk model for directed polymers. There he proves various relations between χ (d) and ξ(d) including an analogue of (2.36). He also proves that a certain curvature assumption on the free energy (p. 589, “Definition” in that paper) implies bounds ξ(d) ≤ 3/4 and χ (1) ≥ 1/8. It seems difficult to check this assumption in general. However, the large deviation principle (2.40)-(2.41) below with ξ = 1, means that the assumption is satisfied in our model: More precisely, the minimizer θ = 0 of the (quadratic) rate function I in (2.40)– corresponding to the direction of the diagonal in Piza’s framework–is a “direction of curvature” in the sense of [23]. • M. Petermann [22] proves for the Gaussian random walk model of directed polymers that ξ(1) ≥ 3/5, while O. Mejane [20] proves ξ(d) ≤ 3/4 for all d ≥ 1. 1 We warn the reader that the following quotations are quite rough, since we totally disregard the differences in the specific definitions of the exponents χ(d) and ξ(d) in these articles.
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• M. W¨uthrich studies critical exponents for crossing Brownian motion in a soft Poissonian potential [30–32]. There he obtains ξ(d) ≤ 3/4, ξ(1) ≥ 3/5, χ (1) ≥ 1/8 and various other relations between χ (d) and ξ(d) including an analogue of (2.36). We stress that his model is undirected, so the dimension there corresponds to 1 + d in our model. Also, his techniques do not seem to be immediately transportable to our model, since they depend quite heavily on the spatial invariance under rotation, which makes very precise information on the quenched Lyapunov exponent available [27, Chap. 5]. We have the following large deviation principle for the transversal fluctuation of the Brownian polymer. Theorem 2.4.4. Let tn be a positive sequence tending to infinity as n → ∞, let χ ≥ 0 be such that (2.39) Q | ln Ztn − Q[ln Ztn ]| > tnχ < ∞ , n≥1
and let ξ > (1 + χ )/2. Then,
−ξ (a) The large deviation principle for µtn tn ωtn ∈ · , n ∞ holds Q-a.s., with the rate function I (θ ) = |θ |2 /2 2ξ −1 : There exists an event Mξ with Q(Mξ ) = 1 such that, for and the speed tn any η ∈ Mξ and for any Borel set B ⊂ Rd , − inf◦ I − o(1) ≤ tn−(2ξ −1) ln µtn tn−ξ ωtn ∈ B ≤ − inf I + o(1), as n ∞. B
B
(2.40) As a consequence, for any ε > 0,
lim −tn−(2ξ −1) ln µtn |ωtn | ≥ εtnξ = ε 2 /2,
n ∞
Q-a.s.
(2.41)
(b) Assume that lim (tnχ ∧ tn2χ−1 )/ ln n = ∞. Then, for d ≥ 1 and β ∈ R, (2.39) n ∞
holds true with any 1/2 < χ and hence (2.40) and (2.41) hold for all ξ > 3/4. (c) In the particular case ξ = 1, then (2.40) and (2.41) hold true without taking the subsequence, i.e., replacing tn and n ∞ in these statements by t and t ∞, respectively. Remark 2.4.3. The proof of Theorem 2.4.4 given in Sect. 9, roughly goes as follows. We first make use of Girsanov’s formula to compute the cost (under the polymer measure) for the Brownian path to deviate away from the origin. Then, we use the G¨artner-Ellis-Baldi theorem [8, p. 44,Theorem 2.3.6] to conclude (2.40). We also mention that our proof of Theorem 2.40 works for the model studied in [22, 20] (cf. Remark 2.4.2), yielding the analogues of (2.40) and (2.41). Remark 2.4.4. In terms of critical exponents, Theorem 2.4.4 suggests that χ (d) ≥ 2ξ(d) − 1, ξ(d) ≤ 3/4,
(2.42) (2.43)
consistent with the conjectures (2.37) and (2.38). If we combine the statements (2.36) and (2.43), we then get χ (1) ≥ 1/8. If we now insert this into the conjecture (2.37), we then obtain ξ(1) ≥ 9/16 > 1/2, i.e., the polymer is super-diffusive in dimension d = 1, for all non-zero β.
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3. Some Preliminaries from Stochastic Calculus As we will see later in some proofs, it is convenient for us to assume that the Poisson random measure is canonically realized. For this reason, we henceforth specify the space (M, G) as follows. We take M = η; integer valued measures on R+ × Rd : ∀t > 0, sup r −d η([0, t] × [−r, r]d ) < ∞ , r>0
G = σ [η(A) ; A ∈ B(R+ × Rd ) ], where a generic element η ∈ M is regarded as the identity mapping id(η) = η on M. Recall the definitions (1.2) and (1.3). For a measurable function f (s, x, η) such that Q[|f (s, x, η)|]dsdx < ∞, [0,t]×Rd
we define the compensated integral f (s, x, η)ηt (dsdx) = f (s, x, η)ηt (dsdx) −
[0,t]×Rd
f (s, x, η)dsdx.
(3.1)
If the integrand (f (s, x, η))s≥0 is predictable in the sense of [14, p. 61, Definition 3.3], then (3.1) defines a martingale on (M, G, Q) for the filtration (Gt ; t ≥ 0). If, moreover, [0,t]×Rd
Q[f (s, x, η)2 ]dsdx < ∞,
then this martingale is square-integrable, and its predictable bracket f 2 dsdx f dη = t
[0,t]×Rd
(3.2)
is such that ( f dηt )2 − f dηt is a martingale on (M, G, Q). The following short hand notation will frequently be used in the sequel: ζt = ζt (ω, η) = exp (βη(Vt )) , χt,x = χt,x (ω) = 1{x ∈ U (ωt )} .
(3.3) (3.4)
The latter will not be confused with the exponent χ (d) from Remark 2.4.1. It is useful to note that χt,x dx = 1 , t ≥ 0 . (3.5) Rd
With these notations, we have Lemma 3.0.5.
η(Vt ) =
(3.6)
χs,x ηt (dsdx)
=
χs,x ηt (dsdx) + t , ζt = 1 + λ ηt (dsdx)ζs− χs,x ,
(3.7) for all ω ∈ ,
(3.8)
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Ztx = 1 + λ
ηt (dsdy)P x [ζs− χs,y ],
(3.9)
Wt = 1 + λ
ηt (dsdx) Ws− µs− [χs,x ] .
(3.10)
Proof. The first two identities are obvious. The three last ones follow from direct applications of Itˆo’s formula. (Note that (3.8) simply reduces to a telescopic sum.) 4. Proof of Theorem 2.1.1 (b), (c) 4.1. Proof of part (b). The argument we present here is based on [7, Proof of Prop. 1.3 (b)]. For (2.3), it is enough to prove that some fractional moment vanishes: lim Q[Wtθ ] = 0
(4.1)
t ∞
for some θ ∈ (0, 1). We first state a technical lemma, which is a generalization of Gronwall’s inequality. Lemma 4.1.1. Let u ∈ C 1 (R+ → R) and v, w ∈ C(R+ → R) be such that d u(t) ≤ −v(t)u(t) + w(t), dt
Then, with V (t) =
for all t > 0.
(4.2)
t
v(s)ds, t V (s) u(t) ≤ u(0) + w(s)e ds e−V (t) , 0
for all t > 0.
(4.3)
0
In particular, when v, w are non-negative, it holds t −V (t) u(t) ≤ u(0)e + w(s)ds,
t > 0.
(4.4)
0
Proof. We write u(t) for the right-hand side of (4.3). Then, u(0) = u(0),
d u(t) = w(t) − v(t)u(t), dt
for all t > 0,
and therefore,
d d
V (t) [u(t)−u(t)]e = (u(t)−u(t)) + v(t) (u(t)−u(t)) dt dt ×eV (t) ≤ 0,
for all t > 0.
By integration, this implies u(t) ≤ u(t) for all t > 0. All the statements follow easily.
We now present the following key lemma. Lemma 4.1.2. For θ ∈ (0, 1), there exists c0 = c0 (θ, β) > 0 such that for any ⊂ Rd , d ! θ" c0 ! θ " 2c0 Q Wt ≤ − Q Wt + P (U (ωt ) ⊂ )θ . dt || ||
(4.5)
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Let us postpone the proof of this lemma for a moment to complete the proof of (4.1). In what follows, ci , i = 1, 2 denote universal constants. For d = 1, set = (−t 2/3 , t 2/3 ]. Then,
P (U (ωt ) ⊂ ) = P (|ωt | ≥ t 2/3 − 21 ) ≤ c1 exp −t 1/3 /c1 , ! " and hence by (4.5), u(t) = Q Wtθ satisfies
c0 d u(t) ≤ − 2/3 u(t) + exp −θt 1/3 /c1 dt 2t for large t. We then have by Lemma 4.1.1 that t
t ∞ c0 t −2/3 s ds + exp −θs 1/3 /c1 ds −→ 0, u(t) ≤ exp − 2 t/2 t/2 which implies (4.1) for d = 1. For d = 2, we set
2
# # = − γ t ln t, γ t ln t
with γ > 0. We then see in a similar way as above that for large t, t t ! θ" 1 ds θγ exp − ln s ds, + Q Wt ≤ exp − 4γ ln t s ln s c2 ln t which, if θγ c2 > 1, goes to zero as t ∞. This proves (4.1) for d = 2. We now turn to the proof of Lemma 4.1.2. Lemma 4.1.3. For θ ∈ [0, 1] and ⊂ Rd , " ! " ! ||Q Wtθ It ≥ Q Wtθ − 2P (U (ωt ) ⊂ )θ ,
(4.6)
where It is defined by (2.12). Proof. Note first that | ∩ U (ωt )| = 1 − |U (ωt )\| ≥ 1 − 1{U (ωt ) ⊂ }. We then use the Schwarz inequality and the above observation as follows: µt [χt,x ]2 dx ||It ≥ || ≥
2
µt [χt,x ]dx
= µt [| ∩ U (ωt )|]2 ≥ (1 − µt {U (ωt ) ⊂ })2 ≥ 1 − 2µt {U (ωt ) ⊂ } ≥ 1 − 2µt {U (ωt ) ⊂ }θ .
(4.7)
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Note also that
F. Comets, N. Yoshida
! " Q Wtθ µt {U (ωt ) ⊂ }θ ≤ Q [Wt µt {U (ωt ) ⊂ }]θ = P {U (ωt ) ⊂ }θ .
(4.8)
We now use (4.7) and then (4.8) to conclude (4.6): " ! " ! " ! ||Q Wtθ It ≥ Q Wtθ − 2Q Wtθ µt {U (ωt ) ⊂ }θ ! " ≥ Q Wtθ − 2P {U (ωt ) ⊂ }θ . Proof of Lemma 4.1.2. From (3.9), Itˆo’s formula and from µs (χs,x )dx = 1, we have
! "θ θ Wtθ = 1 + Ws− Wsθ ds µs − (1 + λχs,x ) − 1 ηt (dsdx) − θλ (0,t]
! "θ θ = 1 + Ws− µs − (1 + λχs,x ) − 1 ηt (dsdx) − Wsθ f λµs [χs,x ] dsdx, (0,t]×Rd
where we have defined a function f : (−1, ∞) → [0, ∞) by f (u) = 1 + θ u − (1 + u)θ . Therefore, Q
!
Wtθ
"
=1−
(0,t]×Rd
! " dsdx Q Wsθ f λµs [χs,x ] .
It is clear that there are constants ci = ci (θ, β) ∈ (0, ∞) such that c1 u2 ≤ f (u) ≤ c2 u2 for all u ∈ (−1, |λ|], and hence
(4.9)
! " d ! θ" dx Q Wtθ f λµt [χt,x ] Q Wt = − dt Rd 2 ≤ −c1 λ dx Q Wtθ µt [χt,x ]2 Rd ! " = −c1 λ2 Q Wtθ It .
Now (4.5) follows from (4.6).
4.2. Proof of part (c). The next proposition provides a condition for the martingale Wt = e−λt Zt to converge in L2 , and hence for Q{W∞ = 0} = 1. To state the proposition, let us introduce the Bessel function as usual, Jν (γ ) = (γ /2)ν
k≥0
(−γ 2 /4)k , γ ≥ 0, ν > −1. k!(ν + k + 1)
We write γd for the smallest positive zero of J d−4 . Note then that (γ /2)− 2 for γ ∈ [0, γd ).
d−4 2
J d−4 (γ ) > 0 2
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Proposition 4.2.1.
(a) sup Q[Wt2 ] t≥0
2 ∞ λ = P exp |U (0) ∩ U (ωs )|ds 2 0 ∞ χs,0 ds . ≤ P exp 2λ2
(4.10) (4.11)
0
If d ≥ 3 and 2|λ|rd < γd ,
(4.12)
1/d √ / π stands for the radius of U (0), then, where rd = d+2 2 ∞ 1 < ∞. χs,0 ds = P exp 2λ2 d−4 d−2 − 0 2 (|λ|rd ) 2 J d−4 (2|λ|rd ) 2 (4.13) In particular, sup Q[Wt2 ] < ∞ if d ≥ 3 and (4.12) holds.
(b) 2rd < γd for all d ≥ 3, and hence supt≥0 Q[Wt2 ] < ∞ if β ∈ (−∞, ln 1 + Moreover, lim (γd /rd ) = ∞. t≥0
γd 2rd
).
d ∞
x x = introduced in Sect. 2.3, we let P x, Proof. (a) Consistently with the notation µx, t x
x P ⊗ P . Under this measure, ω and ω are independent Brownian motions starting respectively from x and x . As in [3, 25], we start by writing
ω, η)], Wt (η)2 = e−2λt P 0,0 [ζt (ω, η)ζt ( so that ω, η)]] Q[Wt2 ] = e−2λt P 0,0 [Q[ζt (ω, η)ζt (
(4.14)
ω, η)], using by Fubini’s theorem. For (ω, ω) ∈ 2 , we compute e−2λt Q[ζt (ω, η)ζt ( (1.1), (1.8), and observing that λ2 (β) = λ(2β) − 2λ(β), ω, η)] e−2λt Q[ζt (ω, η)ζt ( = Q[exp (2βη(Vt (ω) ∩ Vt ( ω)) + βη(Vt (ω) Vt ( ω)) − 2λt)] = exp (λ(2β)|Vt (ω) ∩ Vt ( ω)| + λ|Vt (ω) Vt ( ω)| − 2λt)
2 = exp λ |Vt (ω) ∩ Vt ( ω)|
ω)| exp λ2 |V∞ (ω) ∩ V∞ ( as t ∞, by monotone convergence. Now, ∞ |V∞ (ω) ∩ V∞ ( ω)| = |U (ωt ) ∩ U ( ωt )|dt 0 ∞ |U (0) ∩ U ( ωt − ωt )|dt = 0
(4.15)
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∞
=
0
= ≤
1 2 1 2
law
= 2
|U (0) ∩ U (ω2t )|dt ∞
0 ∞
|U (0) ∩ U (ωt )|dt
(4.16)
1{ωt ∈ 2U (0)}dt.
0∞ χt,0 dt.
(4.17)
0
We get (4.10) and (4.11) from (4.14), (4.15), (4.16) and (4.17). At this point, a standard way to bound the expectation on the right-hand side of (4.11) would be via Khas’minskii’s lemma (cf. Remark 4.2.1 below). Here, we take a different route to get the exact formula (4.13) and thereby proceed to part (b). Now, the exponent of the integrand on the right-hand-side of (4.11) is nothing but the occupation time for the Bessel process in the interval [0, rd ]. Therefore, (4.13) follows from a formula for the Laplace transform of the occupation time [4, p. 376] via analytic continuation. (b) The formula (4.13) shows that the expectation is finite if (4.12) holds. On the other hand, it is known that [28, pp. 486] γ3 = π/2, γ4 = 2.404 . . . , γ5 = π, # γd ≥ 21 d(d − 4), d ≥ 5.
(4.18) (4.19)
It is then, not difficult to see from (4.18), (4.19), and the direct estimation of rd that 2rd < γd for all d ≥ 3 and that lim (γd /rd ) = ∞. d ∞
Remark 4.2.1. The expectation on the right-hand-side of (4.11) can be bounded also by using Khas’minskii’s lemma as follows. For d ≥ 3, the Brownian motion is transient, and ! ∞ " c := sup P x 0 χs,0 ds < ∞ . x∈Rd
By Khas’minskii’s lemma (e.g., [27, p. 8, Lemma 2.1]), this implies that ! ∞ " sup P x exp a 0 χs,0 ds < (1 − ac)−1 , if ac < 1 , x∈Rd
from which, we see the convergence of the expectation on the right-hand-side of (4.11) when λ2 /2 < c−1 , i.e., when |β| is small enough. 5. Proof of Theorem 2.4.1 5.1. Proof of part (a). We first prove a Clark-Ocone type representation formula ((5.2) below. See also [29, Lemma 1.3].) Define the functional F : M → R, by F (η) = ln Zt ,
η∈M,
and its increment Ds,x F (η) = F (η + δs,x ) − F (η) P [eβη(Vt ) eβχs,x ] P [eβη(Vt ) ] = ln 1 + λµt [χs,x ] .
= ln
(5.1)
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Let t > 0, that we will keep fixed all through this section. We introduce a martingale (Yt,s )s∈[0,t] by Yt,s = QGs [Yt ] ,
with Yt = ln Zt − Q[ln Zt ] = F (ηt ) − Q[F (ηt )] ,
where QGs
is the conditional expectation given Gs . The random variable Yt is the one of interest in (2.31) and (2.26), and it is natural to introduce its Doob martingale (Yt,s )s∈[0,t] for an easier approach. Similarly to the proof of Proposition 2.1 in [6] of the basic concentration property for the Sherrington-Kirkpatrick model, we will use stochastic calculus, but for jump processes instead of Brownian functionals. Observe that, by independence and homogeneity of the Poisson increments, F (ηs + m)ρt−s (dm) , QGs [F (ηt )] = QGs [F (ηs + [ηt − ηs ])] = M
with ρt−s the law of ηt−s . Hence, for 0 ≤ s < s + h ≤ t, Yt,s+h = [F (ηs+h + m) − F (ηs + m)]ρt−s−h (dm) + M
M
F (ηs + m)ρt−s−h (dm) ,
and therefore, the stochastic differential of Yt,s with respect to s is given by ∂ η(dsdx) Ds,x F (ηs − + m)ρt−s (dm) − ds dYt,s = ∂r Rd M × F (ηs + m)ρr (dm)|r=t−s . M
But the factor of ds in the last term is equal to d Gs Q [F (ηt )] = dx Ds,x F (ηs − + m)ρt−s (dm) dt Rd M according to (3.1). Finally, the martingale (Yt,s )s∈[0,t] can be written as the compensated stochastic integral (5.2) Yt,s = ηs (dudx)ht (u− , u, x) , where
ht (s, u, x) = QGs [Du,x F (ηt )] = QGs [ln 1 + λµt [χu,x ] ]. It follows from (3.2) and (5.2) that Var(ln Zt ) = Q[Yt2 ] = Q dsdxht (s, s, x)2 , [0,t]×Rd
that is (2.26). Note that for u ∈ [0, 1] and β = 0, ln(1 + λu) = (λ/|λ|)| ln(1 + λu)| , c− u ≤ | ln(1 + λu)| ≤ c+ u.
(5.3)
This shows (2.27) and (2.28). Using Jensen’s inequality on the right-hand-side of (2.28),
2 2 Q dsdx QGs µt [χs,x ] Var(ln Zt ) ≤ c+ [0,t]×Rd 2 2 ≤ c+ Q dsdx µt [χs,x ] = This completes the proof of (a).
[0,t]×Rd 2 c+ Q[Jt ].
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5.2. Proof of part (b). In this proof, we set ϕ(v) = ev −v−1 for the notational simplicity. For the stochastic integral (5.2), it is a standard matter to see that, for a ∈ R, Mt,s = exp aYt,s − dudx ϕ (aht (u, u, x)) , 0 ≤ s ≤ t, (5.4) [0,s]×Rd
is a martingale. It follows from (5.3) and Jensen’s inequality that
2 dudx ht (u, u, x)2 ≤ c+ dudx QGu µt [χu,x ]2 [0,t]×Rd [0,t]×Rd 2 dudx QGu µt [χu,x ] ≤ c+ [0,t]×Rd
=
2 t. c+
Using this together with the inequality |ϕ(v)| ≤ e|v| v 2 /2, we obtain for a ∈ [−1, 1], $ $ $ $ $ dudx ϕ (aht (u, u, x))$$ ≤ Ca 2 t/2, $ [0,t]×Rd
where C = c+ ec+ . By the Markov inequality and the martingale property, we have for all a ∈ (0, 1] and u > 0, Q[Yt > u] ≤ exp Ca 2 t/2 − au , and hence, Q[Yt > u] ≤ exp
min
a∈(0,1]
Ca t/2 − au 2
1 u2 = exp − (u ∧ ) 2 Ct
.
Performing the same way with lower deviations, we obtain the desired estimate (2.31).
5.3. Proof of Corollary 2.4.2. The asymptotic bound (2.32) along a sequence t = 1, 2, . . . can be obtained by (2.31) and the Borel-Cantelli lemma. But we see from the following lemma that we do not need to take a subsequence. Lemma 5.3.1. Define δt (h) =
(t,t+h]×Rd
µs− [χs,x ]η(dsdx)
for a fixed h > 0. Then, for any ε > 0, δt (h) = O(t
1+ε 2
), Q-a.s.
(5.5)
Moreover, for 0 ≤ s ≤ h, −λ(|β|)δt (h) ≤ ln
Zt+s ≤ λ(|β|)δt (h). Zt
(5.6)
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Proof. We have δt (h) = Mt+h − Mt + h, where Mt = ηt (dsdx)µs− [χs,x ] − t. Then, M t = following facts:
t
0 Is ds
≤ t. The Q-a.s.bound in (5.5) is a consequence of the two
lim Mt exists and is finite if M ∞ < ∞,
(5.7)
t ∞
1+ε
lim Mt / M t 2 = 0 if M ∞ = ∞.
(5.8)
t ∞
These facts for the discrete martingales are standard (e.g. [10, p. 255, (4.9),(4.10)]). It is not difficult to adapt the proof for the discrete setting to our case. On the other hand, we have by (3.9) and Itˆo’s formula that (5.9) ln Zt = ηt (dudx) ln(1 + λµu− [χu,x ]), and hence that ln
Zt+s = Zt
from which (5.6) is obvious.
(t,t+s]×Rd
η(dudx) ln(1 + λµu− [χu,x ]),
5.4. Proof of Corollary 2.4.3. We set t = {x ∈ Rd ; |x| ≤ r + C + Ct ξ }, where r denotes the radius of U (0). We then see from (2.27) and Jensen’s inequality that
2 −2 dsdx QGs µt [χs,x ] λ(−|β|) Var Q (ln Zt ) ≥ Q [0,t]×Rd 2 ≥ dsdx Qµt [χs,x ] ≥
[0,t]×Rd t
2 dx Qµt [χs,x ]
ds 0
1 ≥ |t |
t t
ds (Qµt [|t ∩ U (ωs )|])2 .
0
Observe at this point, that λ(−|β|)2 > 0 since β > 0. Now, note that Qµt [|t ∩ U (ωs )|] ≥ Qµt {U (ωs ) ⊂ t } ≥ Qµt {|ωs | ≤ C + Ct ξ }. Therefore, with some c1 = c1 (d, C) ∈ (0, ∞), lim t t ∞
−(1−dξ )
Var Q (ln Zt ) ≥ c1 lim t
−1
t ∞
≥ c1 lim inf
t ∞ 0≤s≤t
t
2
Qµt {|ωs | ≤ C + Ct ξ }
0
2
Qµt {|ωs | ≤ C + Ct ξ }
.
ds
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6. Proof of Theorem 2.2.1 6.1. Proof of part (a). We first prove the existence of the limit (2.7). It is convenient to introduce x x Z(s,s+t] = P exp β (6.1) η(dudy)χu−s,y , t, s ≥ 0. (s,s+t]×Rd
x Note that Z(s,s+t] has the same law as Zt and is independent of Gs . We have by the Markov property that x , Zs+t = Zs µs {ωs ∈ dx}Z(s,s+t]
and then by Jensen’s inequality that ln Zs+t ≥ ln Zs +
x µs {ωs ∈ dx} ln Z(s,s+t] .
Taking the expectation and using independence, we obtain x µs {ωs ∈ dx}Q[ln Z(s,s+t] |Gs ] Q[ln Zs+t ] ≥ Q[ln Zs ] + Q = Q[ln Zs ] + Q[ln Zt ], i.e., Q[ln Zt ] is super-additive. Hence the following limit exists by the sub-additive lemma. 1 1 def. ψ(β) = lim − Q[ln Zt ] + λ(β) = inf − Q[ln Zt ] + λ(β). t>0 t t ∞ t Moreover, ψ(β) ≥ 0, since Q[ln Zt ] ≤ ln Q[Zt ] ≤ λt. We next prove (2.5). The convergence in L2 (Q) follows from (2.7) and (2.30). By Lemma 5.3.1, the almost sure convergence follows from the convergence along a sequence t = 1, 2, . . . . But this is a simple consequence of (2.7), (2.31) and the Borel-Cantelli lemma. 6.2. Proof of part (b). We now introduce short-hand notations: ϕt (β) =
1 Q ln Zt and ψt (β) = λ(β) − ϕt (β). t
(6.2)
It is easy to see that ϕt (·) is convex. By (6.2), the limit ϕ(β) = limt ∞ ϕt (β) exists, convex in β, and ϕ(β) = λ(β) − ψ(β). Thus, the function ψ inherits all the properties stated in Theorem 2.2.1 (b) from the convexity of ϕ. For example, we have by the convexity of ϕ, ϕ− (β) ≤ lim ϕt (β) ≤ lim ϕt (β) ≤ ϕ+ (β), t ∞
t ∞
and hence (β) ≤ lim ψt (β) ≤ lim ψt (β) ≤ ψ− (β). ψ+ t ∞
t ∞
(6.3)
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7. Proof of Theorem 2.2.2 and Theorem 2.3.1 7.1. Differentiating the averaged free energy. It is straightforward to check that, for a Poisson variable Y with parameter θ, the identity EYf (Y ) = θEf (Y + 1) holds for all f : N → R+ . The following statement is the analogous property for the Poisson point process, which is useful here. Proposition 7.1.1. For h : [0, t] × Rd × M → R+ a measurable function, we have ! " Q h(s, x; ηt )ηt (dsdx) = dsdxQ h(s, x; ηt + δs,x ) . [0,t]×Rd
Proof. Recall the (shifted) Palm measure Qs,x of the point process ηt , which can be thought of as the law of ηt “given that ηt {(s, x)} = 1”: By definition of the Palm measure, Q[ h(s, x; ηt )ηt (dsdx)] = dsdx h(s, x; η)Qs,x (dη). [0,t]×Rd
M
By Slivnyak’s theorem [26, p. 50] for the Poisson point process ηt , the Palm measure Qs,x is the law of ηt + δs,x , hence the right-hand-side of the above formula is equal to the right-hand-side of the desired formula. Recall the notations (6.2). We now use Proposition 7.1.1 to prove the following Lemma 7.1.2. For all β ∈ R, tψt (β) = λeβ
[0,t]×Rd
dsdx Q
[µt (χs,x )]2 . 1 + λµt (χs,x )
(7.1)
Hence, λQ[Jt ] ≤ tψt (β) ≤ eβ λQ[Jt ]. Proof. We see from (6.2) that 1 ψt (β) = λ (β) − ϕt (β) = eβ − Q[µt (η(Vt ))]. t By Fubini’s theorem, and by Proposition 7.1.1, Q[µt (η(Vt ))] = Q ηt (dsdx)µt [χs,x ] P [χs,x eβη(Vt ) ] = Q ηt (dsdx) P [eβη(Vt ) ] P [χs,x eβ(η+δs,x )(Vt ) ] =Q dsdx P [eβ(η+δs,x )(Vt ) ] [0,t]×Rd eβ P [χs,x eβη(Vt ) ] dsdx =Q P [(λχs,x + 1)eβη(Vt ) ] [0,t]×Rd µt [χs,x ] dsdx . = eβ Q 1 + λµt [χs,x ] [0,t]×Rd
(7.2)
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Since t =
[0,t]×Rd
dsdxµt [χs,x ], it follows that
tψt (β)
=e Q
dsdx µt [χs,x ] −
β
= λeβ
[0,t]×Rd
[0,t]×Rd
dsdx Q
−
µt [χs,x ] 1 + λµt [χs,x ]
[µt (χs,x )]2 , 1 + λµt (χs,x ) +
i.e., (7.1). Now, since e−β ≤ 1 +λµt (χs,x ) ≤ eβ , where β ± = max{0, ±β}, (7.1) implies (7.2) by definition of Jt = [0,t]×Rd dsdx[µt (χs,x )]2 . 7.2. Proof of Theorem 2.2.2 (a). We see from (7.2) and (6.3) that there are ci = ci (β) ∈ (0, ∞), i = 1, 2 such that the following hold: 1 1 (β). Q[Jt ] ≤ lim Q[Jt ] ≤ c2 ψ− t ∞ t t t ∞
If β > 0, then c1 ψ+ (β) ≤ lim
If β < 0, then
(7.3)
1 1 (β). (7.4) Q[Jt ] ≤ lim Q[Jt ] ≤ −c2 ψ+ t ∞ t t ∞ t
− c1 ψ− (β) ≤ lim
(β) for β ∈ R , and ψ (β) ≤ 0 for β ∈ R , from which the These imply that 0 ≤ ψ− + − + monotonicity of ψ on R± follows. Next, we obtain by Jensen’s inequality and Fubini’s theorem
Q[ln Zt ] = Q[ln P [exp βη(Vt )]] ≥ Q[P [βη(Vt )]] = βt, which implies that ψ(β) ≤ λ(β) − β.
7.3. Proof of Theorem 2.2.2 (b). We first explain the strategy of the proof. With a parameter γ > 0 which we will introduce later on, we will define a set Mt,γ ⊂ M such that the following hold for β large enough: lim Q[Mt,γ ] = 1
(7.5)
Q[Zt : Mt,γ ] ≤ 2 exp C1 tλ1/2 ,
(7.6)
t ∞
and
where C1 is a constant. We can then conclude (2.9) as follows. Observe that we have ! " " ! | Q ln Zt |Mt,γ − Q [ln Zt ] | ≤ Q | ln Zt − Q[ln Zt ]| | Mt,γ ≤ Q [| ln Zt − Q[ln Zt ]|] /Q[Mt,γ ] √ = O( t)
(7.7)
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by the concentration property (2.31) and (7.5). Therefore, by (7.5), (7.6), (7.7) and Jensen’s inequality, 1 Q [ln Zt ] t→∞ t " 1 ! = lim Q ln Zt | Mt,γ t→∞ t " ! 1 ≤ lim inf ln Q Zt | Mt,γ t→∞ t " ! 1 = lim inf ln Q Zt ; Mt,γ t→∞ t ≤ C1 λ1/2 .
λ(β) − ψ(β) = lim
This, together with (2.8), shows (2.9). Let us now turn to the construction of the set Mt,γ alluded to above. For a, t > 0 and {f, g} ⊂ , we denote by ρt (f, g) = sups∈[0,t] |f (s) − g(s)| the distance induced by the sup-norm on C([0, t] → Rd ) and by Kt,a the set of absolutely continuous function f : R → Rd , such that f (0) = 0 and 1 t ˙ 2 |f (s)| ds ≤ a . t 0 Lemma 7.3.1. There exist a finite constant a0 > 0 and a function b(a) > 0 defined for a > a0 , such that P [ρt (ω, Kt,a ) ≥ 1] ≤ exp −b(a)t for large t, and
lim inf b(a)/a = 1/2 . a→∞
The lemma is a simplified formulation of Lemma 2 in [12], which states, via the scaling property of Brownian motion, that for 4a ≥ e, at 1 1 P [ρt (ω, Kt,a ) ≥ 1] ≤ 2 exp − + ln(4a)[1 + ln(4a)] + ln(4a) . 2 2 2 We continue the construction of Mt,γ . We first cover the compact set Kt,a with finite numbers of unit ρt -balls. The point here is that the number of the balls we need is bounded from above explicitly in terms of a and t as we explain now. By a result of Birman and Solomjak, Theorem 5.2 in [2] (taking there p = 2, α = 1, q = ∞, m = 1, ω = 1), for all ε > 0, the set K1,1 can be covered by a number smaller than exp{C(d)ε −1 } of ρ1 -balls with radius ε. Since, for a, t > 0, a map f → g, g(u) = (ta 1/2 )−1 f (ut) defines a bijection from Kt,a to K1,1 , it follows that we can find fi ∈ Kt,a , 1 ≤ i ≤ i0 ≤ exp{C(d)ta 1/2 }, such that % Kt,a ⊂ f ∈ : ρt (f, fi ) ≤ 1 . i≤i0
We next introduce a tube Vt (f, R) ⊂ R+ × Rd (t > 0, R > 0) around the graph of a function f : [0, t] → Rd by Vt (f, R) = {(s, x) ; s ∈ (0, t], |x − f (s)| ≤ R}.
(7.8)
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For γ > 1 to be chosen later, we consider the set Mt,γ of “good environments”: Mt,γ =
i0
η ∈ M ; η(Vt (fi , r + 2)) ≤ γ
r+2 d t , r
i=1
with r = rd the radius of U (0). Since η(Vt (fi , r + 2)) has Poisson distribution with d t, we have, see Cram´er’s theorem [8], for all t > 0, mean r+2 r d Q[Mct,γ ] ≤ Q[η(Vt (fi , r + 2)) > γ r+2 t] r i≤i0
r+2 d
≤ exp{−t[
r
λ∗ (γ ) − C(d)a 1/2 ]},
(7.9)
λ∗ (u)
= supβ∈R {βu − λ(β)} = u ln u − u + 1, u > 0. We will eventually make where this term small by choosing our parameters. We estimate the expectation of Zt on the set Mt,γ by Q Zt ; Mt,γ = Q P [ζt ; ρt (ω, Kt,a ) < 1]; Mt,γ +Q P [ζt ; ρt (ω, Kt,a ) ≥ 1]; Mt,γ ≤ Q exp β max η[Vt (fi , r + 2)] ; Mt,γ i≤i0 +P Q[ζt ]; ρt (ω, Kt,a ) ≥ 1 d t} + exp{λt − b(a)t}, (7.10) ≤ exp{βγ r+2 r using Lemma 7.3.1. Choose now a = 4λ and γ = 2C(d)λ1/2 /β. For large β, we have a > a0 , γ > 1 and b(a) ≥ λ. Then, there exist finite, positive constants β0 , C1 , such that for β > β0 , we have (7.5) due to (7.9), while (7.6) can be obtained from (7.10). 7.4. Proof of Theorem 2.2.2 (c). We define βc+ = sup{β ≥ 0 ; ψ(β) = 0}, βc− = inf{β ≤ 0 ; ψ(β) = 0} Then, all the desired properties of βc± follow from what we have already seen (ψ is monotone, continuous on R± and diverges as β ∞), except the differentiability at β ∈ {βc− , βc+ } ∩ R. But this is easy to see. Suppose for example that βc+ < ∞. Then, (β + ) ≤ 0 ≤ ψ (β + ), since β + is a minimal point of ψ. This, together with (2.6) ψ− c + c c (β + ) = 0. proves ψ± c 7.5. Proof of Theorem 2.3.1. All the statements are consequences of (7.3), (7.4) and the following observations: β 0 < ψ(β) = ψ+ (γ )dγ for β > βc+ , βc+ β
0 < ψ(β) =
βc−
ψ− (γ )dγ for β < βc− .
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8. Proof of Theorem 2.3.2 To conclude (2.21) and (2.22), it is enough to show the following (8.1) and (8.2): ∞ {W∞ = 0} ⊂ Is ds = ∞ , Q-a.s. (8.1) 0
There are c1 , c2 ∈ (0, ∞) such that ∞ t Is ds = ∞ ⊂ c1 Is ds 0
0
t
≤ λt −ln Zt ≤ c2
Is ds for large enough t ,
Q-a.s. (8.2)
0
The proof of (8.1) and (8.2) is based on Doob’s decomposition with respect to the filtration (Gt , t ≥ 0), λt − ln Zt = Mt + At . In view of (5.9), the martingale part Mt and the increasing part At are given by Mt = − ηt (dsdx) ln(1 + λµs− [χs,x ]), At = ϕ(λµs [χs,x ])dsdx,
(8.3)
[0,t)×Rd
where ϕ(u) = u − ln(1 + u) ≥ 0, −1 < u. It is clear that there are constants ci = ci (β) ∈ (0, ∞) such that c1 u2 ≤ ϕ(u) ≤ c2 u2 , and hence
ln2 (1 + u) ≤ c2 u2 for all u ∈ [λ ∧ 0, λ ∨ 0],
t
c1
t
Is ds ≤ At ≤ c2
0
(8.4)
Is ds,
(8.5)
Is ds.
(8.6)
0 t
M t ≤ c2 0
Note also that we have (5.7) and (5.8) for the martingale Mt defined by (8.3). We now conclude (8.1) from (8.5), (8.6) and (5.7) as follows (the equalities and the inclusions here being understood as Q-a.s.): ∞ Is ds < ∞ = {A∞ < ∞} 0
= {A∞ < ∞, M ∞ < ∞} ⊂ {A∞ < ∞, lim Mt exists and is finite} t ∞
⊂ {W∞ > 0}. Finally we prove (8.2). By (8.5), it is enough to show that λt − ln Zt {A∞ = ∞} ⊂ lim = 1 , Q-a.s. t ∞ At
(8.7)
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Thus, let us suppose that A∞ = ∞. If M ∞ < ∞, then lim Mt exists and is finite t ∞
and therefore (8.7) holds. If, on the contrary, M ∞ = ∞, then
λt − ln Zt Mt M t = + 1 → 1 Q-a.s. At M t At by (8.5), (8.6) and (5.8). This completes the proof of Theorem 2.3.2. We end this section with the proof of the inequalities (2.25). Since x,x µt [|U (ωs ) ∩ U ( ωs )|] = µxt [ωs ∈ U (z)]2 dz , Rd
the right-hand-side inequality follows immediately from (3.5). To prove the left-handside, we introduce a smaller ball 21 U (0) = { 21 z ; z ∈ U (0)}. By the Schwarz and the triangle inequalities, 2 $1 $−1 x 2 x $ $ µ [ωs ∈ U (z)] dz ≥ U (0) µ [ωs ∈ U (z)] dz Rd
t
2
1 y+ 2 U (0)
≥2
d 1 y+ 2 U (0)
µxt
t
!
ωs ∈ y +
"
1 2 U (0)
2 dz
! "2 = 2−d µxt ωs ∈ y + 21 U (0) . By additivity of µxt ,
" ! sup µxt [ωs ∈ U (y)] ≤ c2 sup µxt ωs ∈ y + 21 U (0) ,
y∈Rd
y∈Rd
with c2 = c2 (d) the minimal number of translates of 21 U (0) necessary to cover U (0). Combining these two inequalities, we finish the proof. 9. Proof of Theorem 2.4.4 We define t (θ ) = ln µt [exp(θ ·ωt )], θ ∈ Rd . We will prove that, for each ξ > (1+χ )/2, there is an event Mξ ∈ G with Q(Mξ ) = 1 such that lim tn−(2ξ −1) tn (tnξ −1 θ ) = |θ |2 /2,
n ∞
(9.1)
for all η ∈ Mξ and θ ∈ Rd and that lim t −1 t (θ ) = |θ |2 /2,
n ∞
(9.2)
for all η ∈ M1 and θ ∈ Rd . Then the theorem follows from the G¨artner-Ellis-Baldi theorem [8, p. 44,Theorem 2.3.6]. The set Mξ should be independent of the choice of θ ∈ Rd for the G¨artner-Ellis-Baldi theorem to be applied. However, we first fix θ ∈ Rd and prove (9.1) and (9.2) on an event Mξ,θ of Q-measure one, depending on θ . To do so, we define a transformation Tθ on R+ × Rd by Tθ (t, x) = (t, x + tθ ) ,
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and, for t ≥ 0, the function t : s → (s ∧ t)θ. We will abuse the notation slightly and write Tθ also for the induced transformation η → η ◦ (Tθ )−1 on M. By Girsanov’s formula, the process ω = ω − t is a Brownian motion under the probability P , P (dω) = exp(θ · ωt − t|θ |2 /2)P (dω). We therefore have that P [ζt (ω, η) exp(θ · ωt − t|θ |2 /2)] = P [exp (βη[Vt (ω + t )])] = P [exp (βη[Tθ (Vt )])] = Zt ◦ T−θ , i.e., t (θ ) = t|θ |2 /2 + ln Zt ◦ T−θ − ln Zt .
(9.3)
Observe that Zt ◦ T−t ξ −1 θ has the same distribution as Zt for each fixed t > 0. Therefore, we see from (2.39) and the Borel-Cantelli lemma that 1
lim 2ξ −1 ln Ztn ◦ T−t ξ −1 θ − Q[ln Ztn ] = 0, Q-a.s. (9.4) n n ∞ t n Observe also that the process (Zt ◦ T−θ )t≥0 has the same distribution as (Zt )t≥0 , and hence by (2.32), lim
t ∞
1 (ln Zt ◦ T−θ − Q[ln Zt ]) = 0, t
Q-a.s.
(9.5)
Combining (9.3) and (9.4), we see that for each θ ∈ Rd , there is a set Mξ,θ ∈ G with Q(Mξ,θ ) = 1 such that (9.1) holds for η ∈ Mξ,θ . We now define Mξ = ∩θ∈Qd Mξ,θ and prove that the set Mξ has the desired property. This can be done with the help −(2ξ −1) ξ −1 of convex analysis as follows. For any η ∈ Mξ , {θ → tn tn (tn θ)}n≥1 is a sequence of convex (and hence continuous) functions which converge to |θ |2 /2 for all θ ∈ Qd . Then, by [24, p. 90, Theorem 10.8], the sequence converges for all θ ∈ Rd and the convergence is locally uniform, which imply that (9.1) holds for all θ ∈ Rd . This ends the proof of (a). A similar argument, based on (2.31), (9.3) and (9.5) proves that (9.2) holds for all η ∈ M1 and θ ∈ Rd . We have proved the statement (c). Finally, (b) follows from (9.3) and (2.31). 10. The Stochastic Heat Equation We prove (1.10) in the following formulation. Proposition 10.0.1. For every compactly supported2 test function ∈ C 2 (Rd ), it holds Q-a.s., t x 1 Zt (y)(y)dy = (x) + 2 ds Zsx (y) (y)dy d Rd R 0 +λ (y)dy Zsx− (y)η(ds × U (y)) (10.1) Rd
for all t ≥ 0, x, y ∈ 2
Rd .
In particular,
It is enough to assume that is first and second derivatives.
C2
Ztx (y)dy
(0,t]
→ δx weakly as t 0.
and that it is polynomially bounded at infinity together with its
286
F. Comets, N. Yoshida
Proof. Note that Rd Ztx (y)(y)dy = P x [ζt (ωt )]. We obtain by first applying Itˆo’s formula to ζt (ωt ), and then by taking P x -expectation that Q-a.s., Ztx (y)(y)dy d R t = (x) + 21 P x [ζs (ωs )]ds + λ η(dsdz)P x [ζs− χs,z (ωs )] = (x) +
1 2
0
ds 0
(0,t]×Rd
t
Rd
Zsx (y) (y)dy
+λ
(y)dy (0,t]
x η(ds × U (y))Zs− (y),
which proves (10.1). Here, we have used Fubini’s theorem on the last line. But this can easily be justified. For example, t x Q η(dsdz)P [ζs− χs,z |(ωs )|] = Q[ζs− ]P x [|(ωs )|]ds (0,t]×Rd
0 t
=
eλs P x [|(ωs )|]ds < ∞.
0
Acknowledgements. The authors would like to thank Tokuzo Shiga for his careful reading of an earlier version of the manuscript, and two anonymous referees for indicating some obscure points in the manuscript, for suggesting improvements and references.
References 1. Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1997) ˇ Solomjak, M. Z.: Piecewise polynomial approximations of functions of classes Wp α . 2. Birman, M.S., (Russian) Mat. Sb. (N.S.) 73(115), 331–355 (1967) English translation: Math. USSR-Sb. 2, 295–317 (1967) 3. Bolthausen, E.: A note on diffusion of directed polymers in a random environment. Commun. Math. Phys. 123, 529–534 (1989) 4. Borodin, A.N., Salminen, P.: Handbook of Brownian Motion–Facts and Formulae. 2nd Ed., BaselBoston-Berlin: Birkh¨auser Verlag 2002 5. Carmona, P., HuY.: On the partition function of a directed polymer in a random environment. Probab. Theory Related Fields 124, 431–457 (2002) 6. Comets, F.: The martingale method for mean-field disordered systems at high temperature. In: Mathematical aspects of spin glasses and neural networks, Progr. Probab. 41, Birkh¨auser, Boston: 1998 pp. 91–113, 7. Comets, F., Shiga, T., Yoshida, N.: Directed Polymers in Random Environment: Path Localization and Strong Disorder. Bernoulli 9, 705–723 (2003) 8. Dembo, A., Zeitouni, O.: Large Deviation Techniques and Applications. 2nd Ed. Berlin-HeidelbergNew York: Springer Verlag, 1998. 9. Derrida, B., Spohn, H.: Polymers on disordered trees, spin glasses, and traveling waves. J. Statist. Phys. 51, 817–840 (1988) 10. Durrett, R.: Probability-Theory and Examples. 2nd Ed., Pacific Grove, CA: Duxbury Press, 1995 11. Fisher, D.S., Huse, D.A.: Directed paths in random potential. Phys. Rev. B 43, 10,728–10,742 (1991) 12. Goodman, V., Kuelbs, J.: Rates of clustering in Strassen’s LIL for Brownian motion. J. Theoret. Probab. 4, 285–309 (1991) 13. Huse, D.A., Henley, C.L.: Pinning and roughening of domain wall in Ising systems due to random impurities. Phys. Rev. Lett. 54, 2708–2711 (1985) 14. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes (2nd ed.),Amsterdam / Kodansha,Tokyo: North-Holland, 1989 15. Imbrie, J.Z., Spencer, T.: Diffusion of directed polymer in a random environment. J. Stat. Phys. 52(3–4), 609–626 (1998)
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16. Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamical scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986) ´ ´ de Probabilit´es de Saint-Flour XIV, 17. Kesten, H.: Aspect of first passage percolation. In: Ecole d’Ete Springer Lecture Notes in Mathematics 1180, Berlin-Heidelberg-New York, 1986, pp. 126–263 18. Krug, H., Spohn, H.: Kinetic roughening of growing surfaces. In: Solids Far from Equilibrium, C. Godr`eche, ed., Cambridge: Cambridge University Press, 1991 19. Licea, C., Newman, C., Piza, M.: Superdiffusivity in first-passage percolation. Probab. Theory Related Fields 106(4), 559–591 (1996) 20. Mejane, O.: Upper bound of a volume exponent for directed polymers in a random environment. Ann. Inst. H. Poincar´e Probab. Statist. 40, 299–308 (2004) 21. Newman, C., Piza, M.: Divergence of shape fluctuations in two dimensions. Ann. Probab. 23(3), 977–1005 (1995) 22. Petermann, M.: Superdiffusivity of directed polymers in random environment. Ph.D. Thesis Univ. Z¨urich (2000) 23. Piza, M.S.T.: Directed polymers in a random environment: some results on fluctuations. J. Statist. Phys. 89(3-4), 581–603 (1997) 24. Rockafeller, R.T.: Convex Analysis. Princeton, NJ: Princeton University Press, 1970 25. Song, R., Zhou, X.Y.: A remark on diffusion on directed polymers in random environment. J. Statist. Phys. 85(1-2), 277–289 (1996) 26. Stoyan, D. Kendall, W.S., Mecke, J.: Stochastic Geometry and its Applications. New York: John Wiley & Sons, 1987 27. Sznitman, A.-S.: Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics, Berlin-Heidelberg-New York: Springer, 1998 28. Watson, G.N.: A Treatise on the Theory of Bessel Functions. 2nd ed., Cambridge: Cambridge University Press, 1958 29. Wu, Liming.: A new modified logarthmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118, 428–438 (2000) 30. W¨uthrich, M.V.: Scaling identity for crossing Brownian motion in a Poissonian potential. Probab. Theory Related Fields 112(3), 299–319 (1998) 31. W¨uthrich, M.V.: Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential. Ann. Probab. 26(3), 1000–1015 (1998) 32. W¨uthrich, M.V.: Fluctuation results for Brownian motion in a Poissonian potential. Ann. Inst. H. Poincar´e Probab. Statist. 34(3), 279–308 (1998) Communicated by H. Spohn
Commun. Math. Phys. 254, 289–322 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1205-5
Communications in
Mathematical Physics
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation for Wave Beams in Turbulent Media Albert C. Fannjiang Department of Mathematics, University of California at Davis, Davis, CA 95616, USA. E-mail:
[email protected] Received: 27 March 2003 / Accepted: 11 May 2004 Published online: 20 October 2004 – © Springer-Verlag 2004
Abstract: Starting with the Wigner distribution formulation for beam wave propagation in H¨older continuous non-Gaussian random refractive index fields we show that the wave beam regime naturally leads to the white-noise scaling limit and converges to a Gaussian white-noise model which is characterized by the martingale problem associated to a stochastic differential-integral equation of the Itˆo type. In the simultaneous geometrical optics the convergence to the Gaussian white-noise model for the Liouville equation is also established if the ultraviolet cutoff or the Fresnel number vanishes sufficiently slowly. The advantage of the Gaussian white-noise model is that its n-point correlation functions are governed by closed form equations.
1. Introduction Laser beam propagation in the turbulent atmosphere is governed by the classical wave equation with a randomly inhomogeneous refractive index field n(z, x) = n(1 ¯ + n(z, ˜ x)),
(z, x) ∈ R3 ,
where n¯ is the mean and n(x) ˜ is the fluctuation of the refractive index field. We seek the solution of the form E(t, z, x) = (z, x) exp [i n(kz ¯ − wt)] + c.c., where E is the (scalar) electric field, k and w = kc0 /n¯ are the carrier wavenumber and frequency, respectively, with c0 being the wave speed in vacuum. Here and below z and x denote the variables in the longitudinal and transverse directions of the wave beam, respectively. In the paraxial approximation [24], the modulation is approximated by the solution of the parabolic wave equation which after nondimensionalization with respect to some The research is supported in part by The Centennial Fellowship from American Mathematical Society, the UC Davis Chancellor’s Fellowship and U.S. National Science Foundation grant DMS 0306659.
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reference lengths Lz and Lx in the longitudinal and transverse directions, respectively, has this form ∂ γ i k˜ ˜ + + k˜ 2 k0 Lz n(zL z , xLx ) = 0, ∂z 2 (0, x) = 0 (x) ∈ L2 (Rd ), d = 2 (1) where k˜ = k/k0 is the normalized wavenumber with respect to the central wavenumber k0 and γ is the Fresnel number γ =
Lz . k0 L2x
A widely used model for the fluctuating refractive index field n˜ is a spatially homogeneous random field (usually assumed to be Gaussian) with the spatial structure function Dn (|x|) = E[n( ˜ x + ·) − n(·)] ˜ 2 = Cn2 |x|2/3 , x = (z, x) ∈ R
d+1
,
|x| ∈ (0 , L0 ),
d = 2,
where 0 and L0 are the inner and outer scales, respectively. Here and below E stands for ensemble average. The refractive index structure function has a spectral representation ∞ x|) sin (| k|| 2 d|k|, Dn (|x|) = 8π (2) n (|k|) 1− k ∈ Rd+1 |k| | k|| x | 0 with the Kolmogorov spectral density = 0.033Cn2 |k| −11/3 , n (|k|)
∈ (0 , L0 ). |k|
(3)
Here the structure parameter Cn2 depends in general on the temperature gradient on the scales larger than L0 . See, e.g., [21, 16, 5] for more sophisticated models of turbulent refractive index fields. In this paper we will consider a general class of spectral density parametrized by H ∈ (0, 1) and satisfying the upper bound −2 ≤ K(L−2 +|k| 2 )−H −1/2−d/2 1+−2 |k|2 (k) , k = (ξ, k) ∈ Rd+1 , d = 2 (4) 0 0 for some positive constant K < ∞. L0 and 0 in (4) are the infrared and ultraviolet cutoffs. The ultraviolet cutoff is physically due to dissipation on the small scales which normally results in a Gaussian decay factor [21]. We are particularly interested in the regime where the ratio L0 /0 is large as in the high Reynolds number turbulent atmosphere. Let us introduce the non-dimensional parameters that are pertinent to our scaling: Lx Lx Lx , η= , ρ= . ε= Lz L0 0 In terms of the parameters and the power-law spectrum in (4) we rewrite (1) as i k˜
∂ ε γ z k˜ 2 µ + ε + V ( 2 , x) ε = 0, ∂z 2 γ ε ε
ε (0, x) = 0 (x)
(5)
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
291
with µ=
σ LH x , ε3
(6)
where σ is the standard variation of the homogeneous field n(z, ˜ x) and V is the normalized refractive index field with a spectral density satisfying the upper bound −2 ≤ K(η2 + |k| 2 )−H −1/2−d/2 1 + ρ −2 |k|2 η,ρ (k) , k ∈ Rd+1 ,
H ∈ (0, 1)
(7)
for some positive constant K. The generalized von K´arm´an spectral density [10, 21] = 2H −1 (H + d + 1 )η2H π −(d+1)/2 (η2 + |k| 2 )−H −1/2−d/2 vk (k) 2 corresponds to the isotropic covariance function Bvk (x) = E V (x + ·)V (·) = |ηx|H KH (η|x|),
(8)
x = (z, x) ∈ Rd+1 ,
where KH is a Bessel function of the third kind given by
∞ et + e−t eH t + e−H t exp −z dt. KH (z) = 2 2 0 For H = 1/2 we have the exponential covariance function Bvk (x) = exp [−η|x|]. The additional ultraviolet cutoff imposed in the upper bound (7) would then give rises to the covariance function B(x) = G Bvk (x), where G is the inverse Fourier transform of the cutoffs. For high Reynolds number one has L0 /0 = ρ/η 1 and thus a wide range of scales in the power spectrum (7). Note that in the worst case scenario the refractive index field loses spatial differentiability as ρ → ∞ and homogeneity as η → 0. The Gaussian field with the von K´arm´an spectral density (8) has H as the upper limit of the H¨older exponent of the sample field. The Kolmogorov spectrum has the exponent H = 1/3. Since our result does not depend on d we hereafter take it to be any positive integer. Although we do not assume isotropic spectral densities, the spectral density always satisfies the basic symmetry: (η,ρ) (ξ, k) = (η,ρ) (−ξ, k) = (η,ρ) (ξ, −k),
∀(ξ, k) ∈ Rd+1 .
(9)
In other words, the spectral density is invariant under change of sign in any component of the argument because the underlying stochastic process is real-valued. We also assume that Vz (x) ≡ V (z, x) is a centered, square-integrable, z-stationary and x-homogeneous process with the (partial) spectral representation z (dp), (10) Vz (x) = exp (ip · x)V
292
A.C. Fannjiang
z (dp) is the z−stationary orthogonal spectral measure satisfying where the process V
(w, p)dw dpdq. (11) E Vˆz (dp)Vˆz (dq) = δ(p + q) We do not assume the Gaussian property but instead a sub-Gaussian property (see Sect. 3.1 for precise statements). If the observation scales Lz and Lx are the longitudinal and transverse scales, respectively, of the wave beam then ε 1 corresponds to a long, narrow wave beam. The white-noise scaling then corresponds to ε → 0 with a fixed µ. For convenience we set µ = 1. The white-noise scaling limit ε → 0 of Eq. (5) is analyzed in [3, 4, 11]. The limit γ → 0 corresponds to the geometrical optics limit. In this paper we study the higher moments behavior in both white-noise and geometrical optics limits by considering the Wigner transform of the modulation function. Our method is also suitable for the situation where deterministic large-scale inhomogeneities are present. One type of slowly varying, large-scale inhomogeneities is multiplicative and can be modeled by a bounded smooth deterministic function µ = µ(z, x) due to variability of any one of the three factors in (6) (see, e.g., [5, 2] for models with slowly varying σ ). The second type is additive and can be modeled by adding a smooth background V0 (z, x). Altogether we can treat the random refractive index field of the general type µ(z, x) z V 2,x V0 (z, x) + ε ε with a bounded smooth deterministic modulation µ(z, x) and background V0 (z, x). We describe the results in Sect. 2.3 but omit the details of the argument for simplicity of presentation. As the small-scale turbulent fluctuations are invariably embedded in a structure determined by large-scale geophysics this generalization is necessary for practical application of the scaling limits. 1.1. Wigner distribution and Wigner-Moyal equation. The Wigner transform of ε , called the Wigner distribution, is defined as γy γy 1 ε −ip·y ε ε∗ z, x − e z, x + Wz (x, p) = dy. (12) (2π)d 2 2 One has the following bounds from (12) Wzε ∞ ≤ (2γ π)−d ε (z, ·) 22 ,
Wzε 2 = (2γ π )−d/2 ε (z, ·) 22
[13, 15, 20]. The Wigner distribution has many important properties. For instance, it is real and its p-integral is the modulus square of the function φ, W ε (x, p)dp = | ε (x)|2 , (13) Rd
so we may think of W (x, p) as wave number-resolved mass density. Additionally, its x-integral is d 2π ε |2 (p/γ ). W ε (x, p)dx = | γ Rd
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
The energy flux is expressed through W ε (x, p) as 1 (∇ ∗ − ∗ ∇) = pW ε (x, p)dp 2i Rd and its second moment in p is |p|2 W (x, p)dp = |∇ ε (x)|2 .
293
(14)
(15)
In view of these properties it is tempting to think of the Wigner distribution as a phasespace probability density, which is unfortunately not the case, since it is not everywhere non-negative. Nevertheless, the Wigner distribution is a useful tool for analyzing the evolution of wave energy in the phase space. Moreover, in the recent development of time reversal of waves in which a part of the waves is received, phase-conjugated and then back-propagated toward the source the refocused wave field is given by a Wigner distribution of mixed-state type (see (25) below) [7, 23, 12]. The Wigner distribution, written as Wzε (x, p) = W ε (z, x, p), satisfies an evolution equation, called the Wigner-Moyal equation, ∂Wzε k˜ p + · ∇x Wzε + Lεz Wzε = 0 ∂z ε k˜ with the initial data 1 W0 (x, k) = (2π)d
eik·y 0 (x −
γy ∗ γy )0 (x + )dy , 2 2
(16)
(17)
where the operator Lεz is formally given as ( z , dq) Lεz Wzε = i eiq·x γ −1 Wzε (x, p + γ q/2) − Wzε (x, p − γ q/2) V ε2
−1 ( z , dq) . = 2γ −1 Wzε (x, γ q/2)Im e−i2γ p·x eiq·x V (18) ε2 We will use the following definition of the Fourier transform and inversion: 1 Ff (p) = e−ix·p f (x)dx, (2π)d F −1 g(x) = eip·x g(p)dp. When making a partial (inverse) Fourier transform on a phase-space function we will write F1 (resp. F1−1 ) and F2 (resp. F2−1 ) to denote the (resp. inverse) transform w.r.t. x and p respectively. A useful way of analyzing Lεz Wzε as formally given in (18) is to look at its partial inverse Fourier transform F2−1 Lεz Wzε (x, y) acting on F2−1 Wzε (x, y) ≡ eip·y Wzε (x, p) dp = ε (x + γ y/2) ε∗ (x − γ y/2) in the following completely local manner: F2−1 Lεz Wzε (x, y) = −iγ −1 δγ Vzε (x, y)F2−1 Wzε (x, y),
(19)
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where δγ Vzε (x, y) ≡ Vzε (x + γ y/2) − Vzε (x − γ y/2), Vzε (x) = Vz/ε2 (x).
(20) (21)
Hereby we define for every realization of Vzε the operator Lεz to act on a phase-space test function θ as
Lεz θ(x, p) ≡ −iγ −1 F2 δγ Vzε (x, y)F2−1 θ(x, y)
(22)
with the difference operator δγ given by (20) for any test function θ ∈ S, where S = θ (x, p) ∈ L2 (R2d ); F2−1 θ (x, y) ∈ Cc∞ (R2d ) . We note that Lεz is skew-symmetric and real (i.e. mapping real-valued functions to realvalued functions). In this paper we consider the weak formulation of the Wigner-Moyal equation: To find Wzε ∈ D([0, ∞); L2 (R2d )) such that Wzε 2 ≤ W0 2 , ∀z > 0, and
Wzε , θ − W0 , θ = k˜ −1
k˜ Wsε , p · ∇x θ ds + ε
z 0
z
0
Wsε , Lεs θ ds,
∀θ ∈ S. (23)
Remark 1. Since Eq. (23) is linear , the existence of weak solutions can be established straightforwardly by the weak- compactness argument. Let us briefly comment on this. Without loss of generality we set ε = 1. First, we introduce truncation N < ∞, VN (z, x) = IN V (z, x), where IN is the characteristic function of the set {|V (z, x)| < N}. Clearly, for such bounded VN the corresponding operator Lεz is a bounded skew-adjoint operator on L2 (R2d ). Hence the corresponding Wigner-Moyal equation gives rise to a unique group (N) of unitary maps on L2 . Let us denote the solution by Wz . Passing to the limit N → ∞ by selecting a weakly convergent subsequence we obtain a L2 -weak solution for the Wigner-Moyal equation with the truncation removed if V is locally square-integrable as is assumed here. The limiting solution Wz has a L2 -norm equal to or less than that of W0 . (N)
Moreover, from Eq. (23), it is easy to see that Wz , θ is equi-continuous on any compact subset of z ∈ R.By the Arzela-Ascoli Lemma, Wz , θ is z-continuous almost (N) surely. Because Wz , θ is adapted to the filtration of Vz and the convergence is almost sure, the resulting solution Wz is adapted to the filtration of Vz . We will not address the uniqueness of solution for the Wigner-Moyal equation (23) but we will show that as ε → 0 any sequence of weak solutions to Eq. (23) converges in a suitable sense to the unique solution of a martingale problem (see Theorem 1 and 2).
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
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1.2. Liouville equation. In the geometrical optics limit γ → 0, if one takes the usual WKB-type initial condition (0, x) = A0 (x)eiS(x)/γ , then the Wigner distribution formally tends to the WKB-type distribution W0 (x, p) = |A0 |2 δ(p − ∇S(x))
(24)
which satisfies F2−1 W0 ∈ L∞ (R2d ). It has been shown [6] that the primitive WKB-type distribution (24) can not arise from the geometrical optics limit (γ → 0) from any pure state Wigner distribution as given by (17) but rather from a mixed state Wigner distribution of the form W0 (x, k) =
1 (2π)d
eik·y 0 (x −
γy γy ; α)0∗ (x + ; α)dydP (α) , 2 2
(25)
where P (α) is a probability distribution of a family of states 0 (·, α) parametrized by α. The mixed state Wigner distributions generally give rise to a smeared initial condition, i.e. W0 (x, p) ∈ L2 (R2d ) even in the geometrical optics limit. This, instead of the WKB type, is the kind of initial conditions considered in this paper. When acting on the test function space S, Lεz as given by (22) has the following limit:
lim Lεz θ (x, p) = −F2 ∇x Vzε (x) · iyF2−1 θ (x, y) = −∇x Vz (x) · ∇p θ(x, p) (26)
γ →0
in the L2 -sense for all θ ∈ S and all locally square-integrable Vz . Hence the WignerMoyal equation (23) formally becomes in the limit γ → 0 the Liouville equation in the weak formulation ε Wz , θ − W0 , θ = k˜ −1
z 0
Wsε , p · ∇x θ ds
k˜ z ε − Ws , ∇x Vsε · ∇p θ ds, ε 0
∀θ ∈ S.
(27)
The same weak- compactness argument as described in Remark 1 establishes the existence of L2 -weak solution of the Liouville equation except now that the operator (26) is unbounded and requires local square integrability of ∇Vz (·). We will show that as ε → 0 any sequence of weak solutions of the Wigner-Moyal equation with any L2 -initial condition converge as ε, γ → 0 in a suitable sense to the unique solution of a martingale problem associated with the Gaussian white-noise model of the Liouville equation (see Theorem 2). In addition to the limit ε → 0 we shall also let ρ → ∞ and η → 0 simultaneously. We first study the case ρ → ∞, but η fixed, as ε → 0. This means that the Fresnel length is comparable to the outer scale. Then we study the narrow beam regime η → 0, where the Fresnel length is in the middle of the inertial-convective subrange.
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2. Formulation 2.1. Martingale formulation. The tightness result (see below) implies that for L2 initial data the limiting measure P is supported in L2 ([0, z0 ]; L2 (R2d )). For tightness as well as identification of the limit, the following infinitesimal operator Aε will play an important role. Let Vzε ≡ V (z/ε 2 , ·) and z0 < ∞ be any positive number. Let Fzε be the σ -algebras generated by {Vsε , s ≤ t} and Eεz the corresponding conditional expectation w.r.t. Fzε . Let Mε be the space of a measurable function adapted to {Fzε , z ∈ R} such that supz
z,δ>0
lim E|fzδ − gz | = 0,
∀t.
δ→0
Consider a special class of admissible functions fz = f ( Wzε , θ ), fz = f ( Wzε , θ ), ∀f ∈ C ∞ (R). We have the following expression from (23) and the chain rule: ˜ 1 k Aε fz = fz (28) Wzε , p · ∇x θ + Wzε , Lεz θ . ε k˜ A main property of Aε is that z Aε fs ds fz − 0
Also,
is a Fzε -martingale,
Eεs fz − fs =
z s
Eεs Aε fτ dτ
∀f ∈ D(Aε ).
∀s < z
a.s.
(29)
(30)
(see [18]). Note that the process Wzε is not Markovian and Aε is not its generator. We denote by A the infinitesimal operator corresponding to the unscaled process Vz (·) = V (z, ·). 2.2. White-noise models. Now we formulate the solutions for the Gaussian white-noise model as the solutions to the corresponding martingale problem: Find the law Q on Z = D([0, ∞); L2w (R2d )) such that for ζ ∈ Z and Wz (ω) ≡ ζ (z), z ≥ 0 we have that Q W0 (ω) = W0 ∈ L2 (R2d ) = 1 and that
z 1 2 ˜
Ws , p · ∇x θ + k Ws , Q0 θ f ( Wz , θ) − f ( Ws , θ ) k˜ 0 2 ˜ +k f ( Ws , θ ) Ws , Kθ Ws ds is a martingale for each f ∈ C ∞ (R) with
K θ Ws =
Q(θ ⊗ θ )(x, p, y, q)Ws (y, q) dydq.
(31)
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297
Here, in the case of the white-noise model for the Wigner-Moyal equation (Theorem 1), the covariance operators Q, Q0 are defined as −2 Q0 θ = ∞ −2θ(x, p) + θ(x, p − γ q) η (q)γ + θ (x, p + γ q) dq, (32) −2 θ(x, p − γ q /2) Q(θ ⊗ θ )(x, p, y, q) = eiq ·(x−y) ∞ η (q )γ − θ (x, p + γ q /2) × θ (y, q − γ q /2) − θ(y, q + γ q /2) dq (33) and, in the case of the white-noise model for the Liouville equation (Theorem 2), Q0 θ (x, p) = ∇p · ρη (q)q ⊗ q dq · ∇p θ(x, p), (34)
Q(θ ⊗ θ )(x, p, y, q) = ∇p θ (x, p)· eiq ·(x−y) ρη (q )q ⊗ q dq · ∇q θ(y, q), η ≥ 0, ρ < ∞,
(35)
with the spectral density ∞ η (q) given by ρ ∞ η (q) = lim η (q) ≡ lim η,ρ (0, q), ρ→∞
ρ→∞
η ≥ 0.
Note that the operators Q and Q0 are well-defined for any test function θ ∈ S in the former case for any H ∈ (0, 1), η > 0 or η = 0, H ∈ (0, 1/2), and in the latter case for H ∈ (0, 1), 0η < ρ < ∞ or H ∈ (0, 1/2), 0 = η < ρ < ∞ or H ∈ (1/2, 1), 0 < η < ρ = ∞. That the martingale problem as formulated with the special class of test functions is sufficient to characterize the law Q follows from the uniqueness result discussed in Sect. 2.4. To see that (31)–(33) is square-integrable and well-defined for any L2 (R2d )-valued process Wz , we apply F2−1 to (31) and obtain
−2 iγ q ·x /2 −iγ q ·x /2 e F2−1 Kθ Ws (x, x ) = F2−1 θ (x, x ) eiq ·(x−y) ∞ (q )γ − e η × θ (y, q − γ q /2) − θ (y, q + γ q /2) Wz (y, q)dydqdq (36) −d −1 = (2π) F2 θ (x, x ) F2−1 θ (y, y )F2−1 Wz (y, −y )
−2 iγ q ·x /2 −iγ q ·x /2 e × eiq ·(x−y) ∞ (q )γ − e η
× eiγ q ·y /2 − e−iγ q ·y /2 q dydy . (37) The integral on the right side of (36) is bounded over compact sets of (x, x ) because firstly θ ∈ S, Wz ∈ L2 (R2d ), and secondly the function
iγ q ·x /2 −iγ q ·x /2 iγ q ·y /2 −iγ q ·y /2 e (q ) e − e − e ∞ η
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A.C. Fannjiang
is integrable in q ∈ Rd and the associated integral is bounded over compact sets of x for any H ∈ (0, 1), η > 0 or η = 0, H < 1/2. Hence the function on the right side of (36) has a compact support and is square-integrable. Similarly, one can show that (32)–(35) is well defined for H ∈ (0, 1), ρ < ∞ or H > 1/2, ρ = ∞. In view of the martingale problem the white-noise model is an infinite-dimensional Markov process whose generator when applied to the special class of test functions fz has the form
¯ z ≡ fs 1 Wz , p · ∇x θ + k˜ 2 A¯ 1 (Wz ) + k˜ 2 fz A¯ 2 (Wz ). Af k˜ This Markov process Wz can also be formulated as weak solutions to the Itˆo’s equation −1 ˜ z Wz , W0 (x) ∈ L2 (R2d ) dWz = (38) p · ∇x + k˜ 2 Q0 Wz dz + kdB k˜ or as the Stratonovich’s equation dWz =
−1 ˜ z ◦ Wz , p · ∇x + kdB k˜
W0 (x) ∈ L2 (R2d ),
where B z is the operator-valued Brownian motion with the covariance operator Q, i.e. E dB z θ (x, p)d B¯z θ (y, q) = δ(z − z )Q(θ ⊗ θ)(x, p, y, q)dzdz . Equation (38) should be solved in the space D([0, ∞); L2w (R2d )), namely, to find Wz ∈ D([0, ∞); L2w (R2d )) such that for all θ ∈ L2 (R2d ), 1 d Wz , θ = Wz , p · ∇x + k˜ 2 Q0 θ dz+ k˜ Wz , dB zθ , W0 (x) ∈ L2 (R2d ). (39) k˜ Our results show that the weak solution to (39) exists, is unique and satisfies the L2 -bound Wz 2 ≤ W0 2 (cf. Theorem 1, 2, Remark 1, 2 and Sect. 2.4). In view of (33), (32), (34) and (35) we can interpret the white-noise limit ε → 0 as giving rise to a white-noise-in-z potential Vz∗ whose spectral density is bounded from above by K ∗ (η2 + |k|2 )−H
∗ −d/2
for some constant K ∗ < ∞ with the effective H¨older exponent H∗ = H + 1/2 by observing that
lim Lεz θ (x, p) = −iF2 γ −1 δγ Vz∗ (x, y)F2−1 θ(x, y) , ∀θ ∈ S, (40) ε→0
lim Lεz θ (x, p) = ∇x Vz∗ (x) · ∇p θ (x, p),
ε,γ →0
in the mean square sense.
∀θ ∈ S
(41)
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299
2.3. White-noise models with large-scale inhomogeneities. First we consider the case of deterministic, large-scale inhomogeneities of a multiplicative type which has µ, given by (6), as a bounded smooth function µ = µ(z, x). The resulting limiting process can be described analogously as above except with the term ∞ η replaced by ∞ ∞ η (k) −→ µ(z, x)µ(z, y)η (k), 2 ∞ ∞ η (k) −→ µ (z, x)η (k),
in
in
Q,
Q0 .
As a consequence the operator Q0 is no longer of convolution type. Next we add a slowly varying smooth deterministic background V0 (z, x) to the rapidly fluctuating field ε−1 µ(z, x)V (ε −2 z, x). Namely we have V0 (z, x) +
z µ(z, x) V ( 2 , x) ε ε
as the potential term in the parabolic wave equation (5). The resulting martingale problem has an additional term z k˜ Ws , L0 θ ds −
(42)
0
in the martingale formulation where L0 θ has the form 0 (z, dq) L0 θ (x, p) = i eiq·x γ −1 θ (x, p + γ q/2) − θ(x, p − γ q/2) V
≡ −iγ −1 F2 (V0 (x + γ y/2) − V0 (x − γ y/2))F2−1 θ(x, y)
(43)
for γ > 0 fixed in the limit, and the form L0 θ (x, p) = −∇x V0 (z, x) · ∇p θ(x, p)
(44)
in the case of γ → 0. 2.4. Multiple-point correlation functions of the limiting model. The martingale solutions of the limiting models are uniquely determined by their n-point correlation functions which satisfy a closed set of evolution equations. Using the function f (r) = r n in the martingale formulation and taking expectation, we arrive after some algebra at the following equation: n n 1 ∂F (n) = pj · ∇xj F (n) + k˜ 2 Q0 (xj , pj )F (n) ∂z k˜ j =1 j =1
+k˜ 2
n
Q(xj , pj , xk , pk )F (n)
j,k=1 j =k
for the n−point correlation function F (n) (z, x1 , p1 , . . . , xn , pn ) ≡ E Wz (x1 , p1 ) · · · Wz (xn , pn ) ,
(45)
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A.C. Fannjiang
where Q0 (xj , pj ) is the operator Q0 acting on the variables (xj , pj ) and Q(xj , pj , xk , pk ) is the operator Q acting on the variables (xj , pj , xk , pk ), namely Q(xj , pj , xk , pk )F (n) (
n
(xi , pi ))
i=1
Wz (xi , pi ) eiq·(xj −xk ) (η,∞) (0, q)γ −2 =E i=j,k
×[Wz (xj , pj − γ q/2) − Wz (xj , pj + γ q/2)] × [Wz (xk , pk − γ q/2) − Wz (xk , pk + γ q/2)] dq} . Equation (45) can be more conveniently written as n n ∂F (n) 1 pj · ∇xj F (n) + k˜ 2 Q(xj , pj , xk , pk )F (n) = ∂z k˜ j =1 j,k=1
(46)
with the identification Q(xj , pj , xj , pj ) = Q0 (xj , pj ). The operator Qsum =
n
Q(xj , pj , xk , pk )
(47)
j,k=1
is a non-positive symmetric operator. We note that the mean Wigner distribution can be exactly solved for from Eq. (46) for n = 1 [12] and has a number of interesting applications in optics including time reversal. The 2nd moment equation n = 2 is related to the problem of scintillation [24] (see, e.g., [5]). The uniqueness for Eq. (45) with any initial data F (n) (z = 0, x1 , p1 , . . . , xn , pn ) = E W0 (x1 , p1 ) · · · W0 (xn , pn ) , W0 ∈ L2 (R2d ) in the case of the Wigner-Moyal equation can be easily established by observing that the operator given by (47) is self-adjoint. For instance, for n = 2, we have that F2−1 QF (2) (x1 , y1 , x2 , y2 ) = F2−1 Q(x1 , y1 , x2 , y2 )F2−1 F (2) (x1 , y1 , x2 , y2 ), where
F2−1 F (2) (x1 , y1 , x2 , y2 ) = E F2−1 Wz (x1 , y1 )F2−1 Wz (x2 , y2 )
and F2−1 Q(x1 , y1 , x2 , y2 ) is the function
eiq·(x1 −x2 ) (η,∞) (0, q)γ −2 eiγ y1 ·q/2 − e−iγ y1 ·q/2 eiγ y2 ·q/2 − e−iγ y2 ·q/2 dq −2 = −8γ cos q · (x1 − x2 ) (η,∞) (0, q) sin γ y1 · q/2 sin γ y2 · q/2 dq. Namely, in the (xj , yj ) variables, the operator Qsum becomes the multiplication by a function which is dominated by the “diagonal terms” with j = k, −1 (2) −2 F2 Q0 F (xj , yj ) = −8γ (η,∞) (0, q) sin2 γ yj · q/2 dq and hence is non-positive. Therefore Qsum is a non-positive self-adjoint operator on L2 . The case with n > 2 is similar.
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
301
Each of the operators on the right side of (46) generates a unique C0 -semigroup of contractions on L2 (R2nd ) and, by the product formula, their sum generates a unique C0 -semigroup of contractions on L2 (R2nd ). Standard theory for linear equations then yields the uniqueness result for the weak solution of (46). In the case of the Liouville equation, Eq. (46) can be more explicitly written as the Fokker-Planck equation on the phase space n n 1 ∂F (n) pj · ∇xj F (n) + k˜ 2 D(xj − xk ) : ∇pj ∇pk F (n) = ∂z k˜ j =1 j,k=1
with
D(xj − xk ) =
(48)
eiq·(xj −xk ) ρη (q)q ⊗ qdq
with η ≥ 0. In the worst case scenario allowed by the bound (7) (cf. (8)) the diffusion coefficient D(0) diverges as ρ → ∞ (but well-defined as η → 0) when H < 1/2. When H > 1/2 then the limit ρ → 0 poses no difficulty. Moreover the diffusion operator n
D(xj − xk ) : ∇pj ∇pk
j,k=1
is an essentially self-adjoint positive operator on Cc∞ (R2nd ) ⊂ L2 (R2nd ) due to the subLipschitz growth of the square-root of D(xk − xk ) at large |xj |, |xk | [8]. The uniqueness follows from the same argument as in the previous case. 3. Assumptions and Main Theorems 3.1. Assumptions and properties of the refractive index field. As mentioned in the introduction, we assume that Vz (x) is a square-integrable, z-stationary, x-homogeneous process with a spectral density satisfying the upper bound (7). Let r(t) be a non-negative (random or deterministic) function such that % % % % %E Ez [Vs (x)]Ez [Vt (y)] % = %E Ez [Vs (x)]Vt (y) % ≤ r(s −z)r(t −z)E Vz2 , ∀s, t ≥ z, ∀x, y ∈ Rd . (49) An obvious candidate for r(t) is the correlation coefficient defined as follows. Let Fz and Fz+ be the sigma-algebras generated by {Vs : ∀s ≤ z} and {Vs : ∀s ≥ z}, respectively. The correlation coefficient rη,ρ (t) is given by rη,ρ (t) =
sup h∈Fz E[h]=0,E[h2 ]=1
sup
+ g∈Fz+t E[g]=0,E[g 2 ]=1
E [hg] .
(50)
Lemma 1. The correlation coefficient rη,ρ (t) as given by (50) satisfies the inequality (49).
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A.C. Fannjiang
Proof. Let hs (x) = Ez [Vs (x)],
gt (x) = Vt (x).
Clearly hs ∈ L2 (P , , Fz ), gt ∈ ∈ L2 (P , , Ft+ ), and their second moments are uniformly bounded in x since E[h2s ](x) ≤ E[gs2 ](x), 2 E[gs ](x) = (ξ, q)dξ dq. From the definition (50) we have
|E[hs (x)ht (y)]| = |E [hs gt ]| ≤ rη,ρ (t − z)E1/2 h2s (x) E1/2 gt2 .
Hence by setting s = t first and the Cauchy-Schwartz inequality we have 2 E h2s x)] ≤ rη,ρ (s − z)E[gt2 ], E hs (x)ht (y) ≤ rη,ρ (t − z)rη,ρ (s − z)E[gt2 ], ∀s, t ≥ z, ∀x, y. We assume Assumption 1. The function r(t) in (49) satisfies ∞ ∞ r(s)r(t)dsdt < ∞. 0
0
Corollary 1. The formula V˜z (x) =
∞
Ez [Vs (x)] ds
(51)
z
defines a square-integrable z-stationary, x-homogeneous process. Proof. Let ω ∈ denote the random element and τx , x = (z, x) ∈ Rd+1 the translation operator acting on . Then without loss of generality we may assume that there exists a square-integrable function V defined on such that Vz (x, ω) = V (τx ω). It suffices to show that the second moment of ∞ V˜ (ω) ≡ E0 V (τ(s,0) ω) ds 0
is finite since V˜z (x, ω) = V˜ (τx ω),
∀x = (z, x) ∈ Rd+1 .
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
303
To this end we have
∞ ∞
2 ˜ E V =E E0 [Vs (0)]E0 [Vt (0)]dsdt 0 ∞ 0 ∞
=E E0 [Vs (0)]Vt (0)dsdt 0 0 ∞ ∞ r(s)r(t)dsdtE[V02 ] ≤
0
which is finite by Assumption 1.
0
One can adopt other alternative mixing coefficients to get the above results and higher order moment estimates, see Appendix A. Hereafter we will mainly focus on the correlation coefficient as it is most convenient to work within the Gaussian case and we shall write explicitly the dependence of the correlation coefficient on η, ρ as rη,ρ (t). In the Gaussian case the correlation coefficient rη,ρ (t) equals the linear correlation coefficient given by rη,ρ (t) = sup R(t − τ1 − τ2 , k)g1 (τ1 , k)g2 (τ2 , k)dkdτ1 dτ2 , (52) g1 ,g2
where
R(t, k) =
eitξ (η,ρ) (ξ, k)dξ,
and the supremum is taken over all g1 , g2 ∈ L2 (Rd+1 ) which are supported on (−∞, 0]× Rd and satisfy the constraint R(t − t , k)g1 (t, k)g¯ 1 (t , k)dtdt dk = R(t − t , k)g2 (t, k)g¯ 2 (t , k)dtdt dk = 1. (53) Alternatively, by the Paley-Wiener theorem we can write rη,ρ (t) = sup eiξ t f1 (ξ, k)f2 (ξ, k)η,ρ (ξ, k)dξ dk,
(54)
f1 ,f2
where f1 , f2 are elements of the Hardy space H2 of L2 ((η,ρ) dξ dk)-valued analytic functions in the upper half ξ -space satisfying the normalization condition |fj (ξ, k)|2 (η,ρ) (ξ, k)dξ dk = 1, j = 1, 2. There are various criteria for the decay rate of the linear correlation coefficients, see [17]. Corollary 2. If Vz is a Gaussian random field and its linear correlation coefficient rη,ρ (t) is integrable, then V˜z is also Gaussian and hence possesses finite moments of all orders. This follows from the fact that the mapping from Vz to V˜z is a bounded linear operator on the Gaussian space.
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A.C. Fannjiang
The main property of V˜z as a random function is that AV˜z = −Vz ,
a.s.
z ∈ R.
(55)
Since A commutes with the shift in x so the appearance of x in Eq. (55) is suppressed. We have the following simple relation
1 izλξ lim E V˜zλ (x)Vzλ (y) = lim ei(x−y)·p e − 1 (η,ρ) (ξ, p)dξ dp λ→∞ λ→∞ iξ (56) = π ei(x−y)·p (η,ρ) (0, p)dp, ∀z. Define the covariance functions
B˜ z (x − y) ≡ E V˜z (x)V˜z (y)
and write B˜ z (x) =
˜ z (k)dk, eik·x
˜ z (k) is its spectral density function. where By the properties of the orthogonal projection Ez [·], we know that
ˆ ˆ ˆ ˆ E Ez [V (A)]Ez [V (A)] ≤ E V (A)V (A) = (η,ρ) (ξ, k)dξ dk
(57)
A
for every Borel set A ⊂ Rd+1 . Assumption 2. For any η > 0,
∞
Rη = lim sup ρ→∞
rη,ρ (t)dt < ∞
0
such that lim sup ηRη < ∞. η→0
For Gaussian fields with the generalized von K´arm´an spectrum (8), a straightforward scaling argument with (54) shows that rη,∞ (t) = r1,∞ (ηt), hence Rη = η−1 R1 . This motivates Assumption 2. Set ˜ εz (k) ≡ ˜ ε−2 z (ξ, k) which is the spectral density of V˜zε (x) ≡ V˜z/ε2 (x).
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
305
Define analogously to (22)
L˜ εz θ (x, p) ≡ −iγ −1 F2 δγ V˜zε (x, y)F2−1 θ(x, y)
(58)
with δγ V˜zε (x, y) ≡ V˜zε (x + γ y/2) − V˜zε (x − γ y/2). Lemma 2 (Appendix B). For each z0 < ∞ there exists a positive constant C˜ < ∞ such that % %2−2H 2 ˜ 2 %%min (γ −1 , ρ)%% (y) ≤ Cγ , sup E δγ Vzε |z|≤z0 |y|≤L
2 ˜ −2−2H , sup E V˜zε (x) ≤ Cη
|z|≤z0
sup E
|z|≤z0 |y|≤L
2
˜ −2 γ 2 | min (ρ, γ −1 )|2−2H , (y) ≤ Cη δγ V˜zε
%
2 %% % ε % ˜ −2 γ 2 ρ 1−H | min (ρ, γ −1 )|1−H , sup %∇y E δγ V˜z (y)%% ≤ Cη |z|≤z 0 |y|≤L
˜ −2 ρ 4−2H , sup E p · ∇x (L˜ εz θ ) 22 ≤ Cη
|z|≤z0
θ ∈S
for all H ∈ (0, 1), ε, γ , η ≤ 1 ≤ ρ, x, y ∈ Rd , where the constant C˜ depends only on z0 , L and θ . We also need to know the first few moments the random fields involved. The case of Gaussian fields motivates the following assumption of the 6th order sub-Gaussian property. Assumption 3. 4 2 sup E δγ Vzε (y) ≤ C1 sup E2 δγ Vzε (y),
(59)
4
2 sup E δγ V˜zε (y) ≤ C2 sup E2 δγ V˜zε (y),
(60)
|y|≤L
|y|≤L
sup E
|y|≤L
|y|≤L
2 δγ Vzε
δγ V˜zε
4
|y|≤L
& (y) ≤ C3 & ×
sup E
|y|≤L
2 δγ Vzε (y)
sup E2 δγ V˜zε
|y|≤L
'
'
2 (y)
for all L < ∞, where the constants C1 , C2 and C3 are independent of ε, η, ρ, γ . From (22) and (58) we can form the iteration of operators Lεz L˜ εz ,
Lεz L˜ εz θ (x, p) = −γ −2 F2 δγ Vzε (x, y)δγ V˜zε (x, y)F2−1 θ(x, y) .
(61)
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A.C. Fannjiang
The operator Lεz L˜ εz θ is well-defined if δγ Vzε and δγ V˜zε are locally square-integrable. Other iterations of Lεz and L˜ εz allowed by Assumption 3 can be similarly constructed. The following estimates can be obtained from Lemma 2 and Assumption 3. Corollary 3 (Appendix C).
E Lεz θ (x, p)L˜ εz θ (y, q) 22 ≤ C η−2 | min (ρ, γ −1 )|4−4H ,
E Lεz L˜ εz θ 22 ≤ C η−2 | min (ρ, γ −1 )|4−4H ,
E L˜ εz L˜ εz θ 22 ≤ C η−4 | min (ρ, γ −1 )|4−4H , (2 ( ( ( E (Lεz L˜ εz L˜ εz θ ( ≤ C η−4 | min (ρ, γ −1 )|6−6H , 2
where the constant C is independent of ρ, η, γ and L is the radius of the ball containing the support of F2−1 θ . Assumption 4. For every θ ∈ S, there exists a random constant C5 such that C5 sup δγ V˜zε F2−1θ 4 ≤ √ sup E1/2 |δγ V˜zε (x, y)|2 , ε z∈[0,z0 ] z
∀θ ∈ S, ε, η, γ ≤ 1 ≤ ρ (62)
|x|,|y|≤L
with C5 possessing finite moments and depending only on θ, z0 , where L is the radius of the ball containing the support of F2−1 θ , cf. Lemma 2 and (63). For a Gaussian random field, Assumption 4 is readily satisfied by Lemma 2 and Borell’s inequality [1] sup δγ V˜zε F2−1 θ 4 ≤ F2−1 θ 4 sup |δγ V˜zε (x, y)|
z
z∈[0,z0 ] |x|,|y|≤L
≤ C5 log
z 0
ε2
sup E1/2 |δγ V˜zε (x, y)|2 ,
∀η, γ ≤ 1 ≤ ρ,(63)
z∈[0,z0 ] |x|,|y|≤L
where the random constant C5 has a Gaussian-like tail. Note that with γ or ρ held fixed the first term on the right side of (62) is always O(1). Compared to the corresponding condition (63) for the Gaussian field condition (62) allows for a certain degree of intermittency in the refractive index field. As we have seen above, most of the assumptions here are motivated by the Gaussian case and we have formulated them in such a way as to allow a significant level of non-Gaussian fluctuation.
3.2. Main theorems. Theorem 1. Let Vzε be a z-stationary, x-homogeneous, almost surely locally bounded random process with the spectral density satisfying the bound (7) and Assumptions 1, 2, 3, 4. Let γ > 0 be fixed.
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
307
(i) Let η be fixed and ρ be fixed or tend to ∞ as ε → 0 such that lim ερ 2−H = 0.
ε→0
(64)
Then the weak solution W ε of the Wigner-Moyal equation with the initial condition W0 ∈ L2 (R2d ) converges in law in the space D([0, ∞); L2w (R2d )) of L2 -valued right continuous processes with left limits endowed with the Skorohod topology to that of the corresponding Gaussian white-noise model with the covariance operators Q and Q0 as given by (33) and (32), respectively (see also (42) and (43)). The statement holds true for any H ∈ (0, 1). (ii) Suppose additionally that H < 1/2 and η = η(ε) → 0 such that lim εη−1 (η−1 + ρ 2−H ) = 0.
ε→0
(65)
Then the same convergence holds true. Here and below L2w (R2d ) is the space of square integrable functions on the phase space R2d endowed with the weak topology. The next theorem concerns a similar convergence to the solution of a Gaussian whitenoise model for the Liouville equation. Theorem 2. Let Vzε be a z-stationary, x-homogeneous, almost surely smooth, locally bounded random process with the spectral density satisfying the bound (7) and Assumptions 1, 2, 3, 4. Let γ = γ (ε) → 0 as ε → 0. Then under any of the following three sets of conditions: (i) ρ < ∞ and η > 0 held fixed; (ii) H > 1/2, η > 0 fixed and ρ = ρ(ε) → ∞ as ε → 0 such that lim ερ 2−H = 0;
ε→0
(66)
(iii) H < 1/2, ρ < ∞ fixed and η = η(ε) → 0 such that lim εη−2 = 0;
ε→0
(67)
the weak solutions W ε of the Wigner-Moyal equation (16) with the initial condition W0 ∈ L2 (R2d ) converge in distribution in the space D([0, ∞); L2w (R2d )) to the martingale solution of the Liouville equation of the Gaussian white-noise model with the covariance operators Q and Q0 as given by (34) and (35), respectively (see also (42) and (44)). It is worthwhile to point out that the stochastic geometrical optics limit (Theorem √ 2) puts restriction more on the aspect ratio ε 2 of the wave beam than on the ratio rρ between the Fresnel length and the inner scale as commonly assumed in the literature (see for example [22]). Note also that the Kolmogorov value H = 1/3 is covered by the regimes of Theorem 1 and Theorem 2(i), (iii). Remark 2. Both Theorem 1 and 2 can be viewed as a construction (and the convergence) of approximate solutions (via Remark 1) to the Gaussian white-noise models which are widely used in practical applications [24, 5].
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A.C. Fannjiang
4. Proof of Theorem 1 and 2 4.1. Tightness. In the sequel we will adopt the following notation: fz ≡ f ( Wzε , θ ), fz ≡ f ( Wzε , θ ), fz ≡ f ( Wzε , θ ), ∀f ∈ C ∞ (R). (68) Namely, the prime stands for the differentiation w.r.t. the original argument (not z) of f, f , etc. Let L denote the radius of the ball containing the support of F2−1 θ. Let all the constants c, c , c1 , c2 , . . . etc. in the sequel be independent of ρ, η, γ and ε and depend only on z0 , θ, W0 2 and f . First we note that since S is dense in L2 (R2d ) and Wzε 2 ≤ W0 2 , ∀z > 0, the tightness of the family of L2 (R2d )-valued processes {W ε , 0 < ε < 1} in D([0, ∞); L2w (R2d ) is equivalent to the tightness of the family in D([0, ∞); S ) as distribution-valued processes. According to [14], a family of processes {W ε , 0 < ε < 1} ⊂ D([0, ∞); S ) is tight if and only if for every test function θ ∈ S the family of processes { W ε , θ , 0 < ε < 1} ⊂ D([0, ∞); R) is tight. With this remark we can now use the tightness criterion of [19] (Chap. 3, Theorem 4) for finite dimensional processes, namely, we will prove: Firstly, (69) lim lim sup P{ sup | Wzε , θ | ≥ N } = 0, ∀z0 < ∞. N→∞
ε→0
z
C ∞ (R)
Secondly, for each f ∈ there is a sequence fzε ∈ D(Aε ) such that for each ε ε z0 < ∞{A fz , 0 < ε < 1, 0 < z < z0 } is uniformly integrable and lim P{ sup |fzε − f ( Wzε , θ )| ≥ δ} = 0, ∀δ > 0. (70) ε→0
z
Then it follows that the laws of { W ε , θ , 0 < ε < 1} are tight in the space of D([0, ∞); R) and hence {Wzε } is tight in D([0, ∞); L2w (R2d )). Condition (69) is satisfied because the L2 -norm is preserved. ε + f ε + f ε . First we We shall construct a test function of the form fzε = fz + f1,z 2,z 3,z ε construct the first perturbation f1,z . Let V˜zε = V˜z/ε2 . Recall that Aε V˜zε = −ε −2 Vzε . Let k˜ ∞ ε ε ε fz Wz , Ez Ls θ ds ε z ∞ −1 ε ˜ z F −1 Wzε , γ −1 δγ E [V ]dsF θ = kεf z s 2 2 z ˜ z F −1 Wzε , γ −1 δγ V˜zε F −1 θ = kεf 2 2 ε ε ˜ z Wz , L˜ z θ = kεf
ε ≡ f1,z
be the 1st perturbation of fz .
(71)
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
309
Proposition 1. ε lim sup E|f1,z | = 0,
ε→0 z
ε lim sup |f1,z | = 0 in probability .
ε→0 z
Proof. First ε |] ≤ ε f ∞ W0 2 E L˜ εz θ 2 E[|f1,z
≤ cε f ∞ W0 2
sup E1/2
|x|,|y|≤L
2 γ −1 δγ V˜zε (x, y)
= O εη−1 | min (ρ, γ −1 )|1−H which is of the following order of magnitude: ε, if η, ρ held fixed ε, if γ , η held fixed εη−1 , if γ or ρ held fixed ε| min (ρ, γ −1 )|1−H , if η is held fixed,
(72)
(73)
(74)
and vanishes in the respective regimes. Secondly, we have ε sup |f1,z | ≤ ε f ∞ W0 2 sup γ −1 δγ V˜zε F2−1 θ 2
z
z
≤ cε
sup E
1/2
1/2
|x|,|y|≤L
1/2 −1
=cε
η
|γ −1 δγ V˜zε (x, y)|2
| min (ρ, γ −1 )|1−H
(75)
by Assumption 4, with a random constant c possessing finite moments. The right side of (75) is of the following order of magnitude: 1/2 if η, ρ held fixed ε , 1/2 ε , if γ , η held fixed (76) 1/2 η−1 , ε if ρ or γ held fixed 1/2 if η is held fixed, ε | min (ρ, γ −1 )|1−H , which vanishes in the respective regimes. The right side of (75) now converges to zero in probability by a simple application of Chebyshev’s inequality and (65). A straightforward calculation yields * ε ε ε ˜ z Wz , p · ∇ + A f1 = −kεf k˜
+ k˜ ε ˜ ε L Lz θ ε z
k˜ ˜ z Wzε , Aε θ Wzε , L˜ εz θ , − fz Wzε , Lεz θ + kεf ε where Aε θ denotes 1 k˜ Aε θ = − p · ∇x θ − Lεz θ ε k˜
310
A.C. Fannjiang
cf. (28). Hence 1 ε Aε fz + f1,z = fz Wzε , p · ∇x θ + k˜ 2 fz Wzε , Lεz L˜ εz θ + k˜ 2 fz Wzε , Lεz θ Wzε , L˜ εz θ k˜ +ε fz Wzε , p · ∇x L˜ εz θ + fz Wzε , p · ∇x θ Wzε , L˜ εz θ = Aε1 (z) + Aε2 (z) + Aε3 (z) + R1ε (z), where Aε2 (z) and Aε3 (z) are the coupling terms. Proposition 2. lim sup E|R1ε (z)| = 0.
ε→0 z
Proof. By Lemma 2 we have
|R1ε | ≤ ε f ∞ W0 22 p · ∇x θ 2 L˜ εz θ 2 + p · ∇x (L˜ εz θ) 2 = O η−1 (| min (ρ, γ −1 )|1−H + ρ 2−H ) , which is of the following order of magnitude: ε, ερ 2−H , εη−1 , εη −1 ρ 2−H , ε(| min (ρ, γ −1 )|1−H + ρ 2−H ), and vanishes in the respective regimes.
if η, ρ held fixed if η, γ held fixed if ρ is held fixed if γ held fixed if η held fixed
and
(78)
ε , f ε . Let We introduce the next perturbations f2,z 3,z (1) A2 (φ) ≡ φ(x, p)Q1 (θ ⊗ θ )(x, p, y, q)φ(y, q) dxdp dydq, (1) A1 (φ) ≡ Q1 θ (x, p)φ(x, p) dxdp,
where
(77)
Q1 (θ ⊗ θ )(x, p, y, q) = E Lεz θ (x, p)L˜ εz θ(y, q) ,
(79) (80)
(81)
Q1 θ (x, p) = E Lεz L˜ εz θ (x, p) ,
where the operator L˜ εz is defined as in (58). Note that Q1 θ and Q1 θ are O(1) terms because of (56). Clearly, we have (1) A2 (φ) = E φ, Lεz θ φ, L˜ εz θ . (82)
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
Define ε f2,z ≡ k˜ 2 fz ε f3,z ≡ k˜ 2 fz
Let
∞
z ∞ z
311
(1) Eεz Wzε , Lεs θ Wzε , L˜ εs θ − A2 (Wzε ) ds, Eεz
(1) Wzε , Lεs L˜ εs θ − A3 (Wzε ) ds.
Q2 (θ ⊗ θ )(x, p, y, q) ≡ E L˜ εz θ (x, p)L˜ εz θ(y, q)
and
Q2 θ (x, p) = E L˜ εz L˜ εz θ (x, p) .
Let
(2)
A2 (φ) ≡ (2)
A1 (φ) ≡
φ(x, p)Q2 (θ ⊗ θ )(x, p, y, q)φ(y, q) dxdp dydq,
(83)
Q2 θ (x, p)φ(x, p) dx dp,
(84)
we then have
ε 2 k˜ 2 ε ˜ ε 2 (2) fz Wz , Lz θ − A2 (Wzε ) , 2
2 ε k˜ 2 ε ˜ ε ˜ ε (2) = fz Wz , Lz Lz θ − A3 (Wzε ) . 2
ε = f2,z
(85)
ε f3,z
(86)
Proposition 3. ε lim sup E|fj,z | = 0,
ε→0 z
ε lim sup |fj,z | = 0,
ε→0 z
j = 2, 3.
Proof. We have the bounds
(2) ε | ≤ sup ε 2 k˜ 2 f ∞ W0 22 E L˜ εz θ 22 + E[A2 (Wzε )] , sup E|f2,z z
(2) ε sup E|f3,z | ≤ sup ε 2 k˜ 2 f ∞ W0 2 E L˜ εz L˜ εz θ 2 + E[A1 (Wzε )] . z
z
The first term can be estimated as in (74); the second term can be estimated as in (74) by using (62). ε |, j = 2, 3, we have As for estimating supz
(2) ε sup |f2,z | ≤ sup ε 2 k˜ 2 f ∞ W0 22 L˜ εz θ 22 + A2 (Wzε ) , z
(2) ε sup |f3,z | ≤ sup ε 2 k˜ 2 f ∞ W0 2 L˜ εz L˜ εz θ 2 + A1 (Wzε ) . z
z
Using the assumption (62) we can estimate the right side of the above as in (76).
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A.C. Fannjiang
We have
(1) ε = k˜ 2 fz − Wzε , Lεz θ Wzε , L˜ εz θ + A2 (Wzε ) + R2ε (z), Aε f2,z
(1) Aε f ε = k˜ 2 fz − Wzε , Lεz L˜ εz θ + A (Wzε ) + R ε (z), 3,z
3
3
with
2 2 ˜ ˜ k 1 k (2) ε 2 ε ε ε ε ˜ε ε Wz , Lz θ − A2 (Wz ) W , p · ∇x θ + Wz , Lz θ f R2 (z) = ε 2 z k˜ z ε 1 k˜ 2 ˜ 2 ε ˜ε ε ε ε ε ˜ε ˜ +ε k fz Wz , Lz θ Wz , p · ∇x (Lz θ ) + Wz , Lz Lz θ ε k˜ k˜ (2) ε 2 ˜2 1 ε ε ε (2) ε −ε k fz (87) Wz , p · ∇x (Gθ Wz ) + Wz , Lz Gθ Wz , ε k˜ (2)
where Gθ denotes the operator (2) Gθ φ ≡ Q2 (θ ⊗ θ )(x, p, y, q)φ(y, q) dydq. Similarly R3ε (z)
k˜ 1 ε ε ε ε ε ε ε = Wz , p · ∇x (L˜ z L˜ z θ ) + Wz , Lz L˜ z L˜ z θ ε k˜
2 ˜ ˜ k 1 k (2) +ε2 fz Wzε , L˜ εz L˜ εz θ − A1 (Wzε ) Wzε , p · ∇x θ + Wzε , Lεz θ 2 ε k˜ k˜ ε ε ε 2 ˜2 1 −ε k fz (88) Wz , p · ∇x (Q2 θ ) + Wz , Lz Q2 θ . ε k˜ ε2 k˜ 2 fz
Proposition 4. lim sup E|R2ε (z)| = 0,
ε→0 z
lim sup E|R3ε (z)| = 0.
ε→0 z
Proof. Part of the argument is analogous to that given for Proposition 3. The additional estimates that we need to consider are the following: In R2ε (87): % % % % (2) sup ε 2 E % Wzε , p · ∇x (Gθ Wzε ) % z
( −1 −1 ε ε ε ( ˜ ˜ × E δγ Vz (x, y)δγ Vz (x , y ) F2 θ (x , y )F2 Wz (x , y )dx dy ( 2 (
2 ( −1 ˜ε ≤ cε2 γ −2 W0 2 E ( (∇y · ∇x F2 θ (x, y)E δγ Vz (x, y) ( % % % −1 −1 ε % ( × %F2 θ(x , y )F2 Wz (x , y )% dx dy ( ( 2
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
313
( (
2 ( ( ( ( ( −1 −1 ε ( −1 ε ( ˜ ∇ ≤ cε2 γ −2 W0 2 ( · ∇ F θ E δ E θ F W V (F γ z z( 2 2 ( y x 2 ( 2 2 (
2 ( ( ( −1 ˜ε ( ≤ cε2 γ −2 θ 2 W0 22 ( (∇y · ∇x F2 θ E δγ Vz ( 2 (
2 ( ( −1 ( 2 2 −1 ( ≤ c θ 2 W0 2 ε γ ([F2 ∇x · ∇x θ ](x, y)E δγ V˜zε (y)( ( 2 (
2 ( ( −1 ( 2 2 −2 ( ε +ac θ 2 W0 2 ε γ ([F2 ∇x θ ](x, y) · ∇y E δγ V˜z (y)( ( ≤
c θ 2 W0 22 ε 2 γ −1
sup E
|y|≤L
+c θ 2 W0 22 ε 2 γ −2
δγ V˜zε
2
2
(y)
%
2 %% % ε % ˜ sup %∇y E δγ Vz (y)%%
|y|≤L
≤ O ε2 η−2 γ | min (ρ, γ −1 )|2−2H + ε 2 η−2 ρ 1−H | min (ρ, γ −1 )|1−H
by Lemma 2 where L is the radius of the ball containing the support of θ . Further delineation yields the following order-of-magnitude estimates: 2 ε 2 1−H ε ρ ε2 η−2 ρ 1−H ε2 η−2 2 1−H | min (ρ, γ −1 )|1−H ε ρ
if η, ρ held fixed if η, γ held fixed if γ held fixed if ρ held fixed if η held fixed.
Consider the next term: % % % % (2) sup εE % Wzε , Lεz Gθ Wzε % z
( −1 −1 ε ε ε ( ˜ ˜ × E δγ Vz (x, y)δγ Vz (x , y ) F2 θ (x , y )F2 Wz (x , y )dx dy ( 2 (
2 ( −1 ε ˜ε ≤ cε2 γ −3 W0 2 E ( (δγ Vz (x, y)F2 θ (x, y)E δγ Vz (x, y) ( % % % −1 −1 ε % ( × %F2 θ (x , y )F2 Wz (x , y )% dx dy ( ( 2 (
2 ( ( ( −1 2 −3 2 ( ε ε ( ˜ ≤ cε γ θ 2 W0 2 E (δγ Vz (x, y)F2 θ E δγ Vz ( 2 2 −2 −1 3−3H ≤ O ε η | min (ρ, γ )| by Corollary 3.
314
A.C. Fannjiang
In R3ε (88): , ( % (2 % % ε ε ˜ε ˜ε % ( ( sup εE % Wz , Lz Lz Lz θ % ≤ ε W0 2 sup E (Lεz L˜ εz L˜ εz θ ( 2 z
= O εη−2 | min (ρ, γ −1 )|3−3H , by (62) and Lemma 2. The preceding two terms can be estimated from above by the following order of magnitude: ε if ρ and η held fixed ε if γ and η held fixed −2 εη if γ or ρ held fixed ε| min (ρ, γ −1 )|3−3H if η held fixed; - % % % %2 ε2 E % Wzε , p · ∇x (Q2 θ ) % ≤ ε 2 E % Wzε , p · ∇x (Q2 θ) % ( (
2 ( ( −1 2 −2 ε ( ˜ ≤ cε γ W0 2 (∇y · ∇x E δγ Vz (x, y) F2 θ(x, y)( ( 2 % %
2 % % = O ε2 γ −2 E|y|≤L %%∇y E δγ V˜zε (y)%% = O ε2 η−2 ρ 1−H | min (ρ, γ −1 )|1−H (89) which in the various regimes has the following order of magnitude: 2 ε if ρ and η held fixed if γ and η held fixed ε 2 ρ 1−H if γ held fixed ε 2 η−2 ρ 1−H 2 η−2 ε if ρ held fixed 2 1−H | min (ρ, γ −1 )|1−H if η held fixed; ε ρ - % % % %2 εE % Wzε , Lεz Q2 θ % ≤ ε E % Wzε , Lεz Q2 θ % ( (
2 ( ( −1 ε ε ( ˜ ≤ cε 2 γ −3 W0 2 E ( δ V (x, y)E δ (x, y) F θ(x, y) V γ z 2 ( γ z ( 2 & ' %2 % % % 2 % % = O ε2 γ −3 sup E %δγ V˜zε % (y)E1/2 %δγ Vzε % (y)
|y|≤L
= O ε2 η−2 | min (ρ, γ −1 )|3−3H by Lemma 2.
(90)
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
315
ε + f ε + f ε . We have Consider the test function fzε = fz + f1,z 2,z 3,z
Aε fzε =
1 ε (1) (1) fz Wz , p · ∇x θ + k˜ 2 fz A2 (Wzε ) + k˜ 2 f A1 (Wzε ) k˜ +R2ε (z) + R3ε (z) + R1ε (z).
(91)
Set R ε (z) = R1ε (z) + R2ε (z) + R3ε (z).
(92)
It follows from Propositions 2 and 4 that lim sup E|R ε (z)| = 0.
ε→0 z
For the tightness it remains to show Proposition 5. {Aε fzε } are uniformly integrable. Proof. Indeed, each term in the expression (91) is uniformly integrable. We only need to be concerned with terms in R ε (z) since other terms are obviously uniformly integrable because Wzε is uniformly bounded in the square norm. But since the previous estimates establish the uniform boundedness of the second moments of the corresponding terms, the uniform integrability of the terms follow. 4.2. Identification of the limit. Our strategy is to show directly that in passing to the weak limit the limiting process solves the martingale problem formulated in Sect. 2.1. The uniqueness of the martingale solution mentioned in Sect. 2.4 then identifies the limiting process as the unique L2 (R2d )-valued solution to the initial value problem of the stochastic PDE (38). Recall that for any C 2 -function f , z Mzε (θ ) = fzε − Aε fsε ds 0 z 1 ε ε = fz + f1 (z) + f2ε (z) + f3ε (z) − fz Wz , p · ∇x θ ds 0 k˜ z z
2 (1) ε (1) ε ˜ − k fs A2 (Ws ) + fs A1 (Ws ) ds − R ε (s) ds (93) 0
0
is a martingale. The martingale property implies that for any finite sequence 0 < z1 < z2 < z3 < ... < zn ≤ z, C 2 -function f and bounded continuous function h with compact support, we have ε / . (θ ) − Mzε (θ ) = 0, E h Wzε1 , θ , Wzε2 , θ , ..., Wzεn , θ Mz+s ∀s > 0, z1 ≤ z2 ≤ · · · ≤ zn ≤ z. (94) Let
¯ z ≡ fs 1 Wz , p · ∇x θ + k˜ 2 A¯ 1 (Wz ) + k˜ 2 fz A¯ 2 (Wz ), Af k˜
316
A.C. Fannjiang
where (1) A¯ 2 (φ) = lim A2 (φ) =
ρ→∞
(1) A¯ 1 (φ) = lim A1 (θ ) = ρ→∞
Q(θ ⊗ θ )(x, p, y, q)φ(x, p)φ(y, q)dxdpdydq, (95) Q0 (θ )(x, p)φ(x, p)dxdp,
(96)
where Q(θ ⊗θ ) and Q0 (θ ) are given by (33) and (32), respectively. For ρ → ∞, γ → 0 as ε → 0 the limits in (95) are not well-defined unless H ∈ (0, 1/2) in the worst case scenario allowed by (7). Likewise, the convergence does not hold for H ∈ [1/2, 1) when η → 0 in the worst case scenario allowed by (7). For each possible limit process in D([0, ∞); L2w (R2d )) there is at most a countable set of discontinuous points with a positive probability and we consider all the finite set {z1 , ..., zn } in (94) to be outside of the set of discontinuity. In view of the results of Propositions 1, 2, 3, 4 we see that fzε and Aε fzε in (93) can ¯ z , respectively, modulo an error that vanishes as ε → 0. With be replaced by fz and Af this and the tightness of {Wzε } we can pass to the limit ε → 0 in (94). We see that the limiting process satisfies the martingale property that . / E h Wz1 , θ , Wz2 , θ , ..., Wzn , θ Mz+s (θ ) − Mz (θ ) = 0, ∀s > 0, where
z
Mz (θ ) = fz −
¯ s ds. Af
(97)
0
Then it follows that E Mz+s (θ ) − Mz (θ )|Wu , u ≤ z = 0,
∀z, s > 0
which proves that M z (θ ) is a martingale. Note that Wzε , θ is uniformly bounded: % ε % % W , θ % ≤ W0 2 θ 2 z so we have the convergence of the second moment 2 lim E Wzε , θ = E Wz , θ 2 . ε→0
Using f (r) = r and r 2 in (97) we see that
z 1
Ws , p · ∇x θ − k˜ 2 A¯ 3 (Ws ) ds Mz(1) (θ ) = Wz , θ − k˜ 0 is a martingale with the quadratic variation z
M (1) (θ ), M (1) (θ ) = k˜ 2 A¯ 2 (Ws ) ds = k˜ 2 z
where Kθ is defined as in (31).
0
z
Ws , Kθ Ws ds,
0
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
317
˜z Appendix A. Mixing Coefficients and Moment Estimates for V Let Fz and Fz+ be the sigma-algebras generated by {Vs : ∀s ≤ z} and {Vs : ∀s ≥ z}, respectively. Consider the strong mixing coefficient α(t) = sup
sup |P (AB) − P (A)P (B)|
+ B∈F z A∈Fz+t
=
1 sup E |P (A|Fz ) − P (A)| 2 A∈F + z+t
which can be used to bound the first order moment: 1/q , E |E Vs |Fz | ≤ 8α(s − z)1/p E|Vs |q
p−1 + q −1 = 1
∀s > z,
([9], Cor. 2.4). Hence the integrability of α(t) implies that V˜z has a finite first order moment. To bound the higher order moments of V˜z one can consider, for example, the general Lp -mixing coefficients φp (t) = sup E1/p |P (A|Fz ) − P (A)|p , p ∈ [1, ∞) A∈Fz+t
=
sup
sup
+ h∈Lp (P ,Fz+t )
g∈Lq (P ,Fz ) Eg q =1,Eg=0
E[hg],
p−1 + q −1 = 1,
p ∈ [1, ∞).
We note that α(t) = φ1 (t) and for p = ∞, φ∞ (t) = sup
+ A∈Ft+z
sup |P (A|B) − P (A)|,
∀t ≥ 0
B∈Fz P (B)>0
= sup ess-supω |P (A|Fz ) − P (A)| + A∈Ft+z
≡ φ(t) is called the uniform mixing coefficient [9]. In terms of φp one has the following estimate vp
q
|E [h1 h2 ] − E[h1 ]E[h2 ]| ≤ 2min (q,2) φp (t)1/u E1/(vp) [h2 ]E1/q [h1 ]
(98)
for u, v, p, q ∈ [1, ∞], u−1 + v −1 = 1, p−1 + q −1 = 1 and real-valued h1 ∈ + Lq (, Fz , P ), h2 ∈ Lvp (, Fz+t , P ) (see [9], Prop. 2.2). In particular, for q > 2, v = q/p, q
q
|E [h1 h2 ]−E[h1 ]E[h2 ]| ≤ 4φp (t)(q−p)/q E1/q [h2 ]E1/q [h1 ],
p−1 + q −1 = 1 (99)
by which, along with the H¨older inequality, we can bound the second moment of V˜z as follows: First we observe that for s, τ ≥ z and h1 = Ez (Vs ), h2 = Vτ , E Ez [Vs (x)]Ez [Vτ (x)] q q = E Ez [Vs (x)]Vτ (x) ≤ 4φp (τ − z)(q−p)/q E1/q [Vz ]E1/q [Ez [Vs ]].
318
A.C. Fannjiang
By setting s = τ first and the Cauchy-Schwartz inequality we have
q E E2z [Vs ] ≤ 4φp (s − z)(q−p)/q E2/q [Vz ], E Ez [Vs (x)]Ez [Vτ (x)] ≤ 4φp (s − z)(q−p)/(2q) φp q
×(τ − z)(q−p)/(2q) E2/q [Vz ], Hence E[V˜z2 ] ≤ 2
∞ ∞
z
z
E Ez [Vτ ]Ez [Vs ] dsdτ + 2
∞
q
≤ 8E2/q [Vz ]
0 ∞
≤ 8E
1/3
∞ ∞
[Vz6 ]
0
0
2
s,τ≥ z.
E [E0 [Vτ ]E0 [Vs ]] dsdτ
0
φp (t)(q−p)/(2q) dt 2/5 φ6/5 (t)dt
2
2/5
which is finite if φ6/5 (t) is integrable (if Vz is assumed to have a finite 6th order moment). When Vz is almost surely bounded, the preceding calculation with p = 1, q = ∞ becomes ∞ 2 q 1/2 E[V˜z2 ] ≤ 8 lim E1/q [Vz ] φ1 (t)dt q→∞
0
1/2
which is finite when φ1 (t) is integrable. In order to bound higher order moments in the non-Gaussian case, one can assume the integrability of the uniform mixing coefficient φ(t) ≡ φ∞ (t). Then we have |P (A|Fz ) − P (A)| ≤ φ(s − z),
∀A ∈ Fs ,
s ≥ z,
and for p ∈ [1, ∞), p−1 + q −1 = 1, % % %E Vs |Fz % ≤ 21/p φ 1/p (s − z) E Vsq |Fz + E Vsq 1/q
(100)
(cf. [9], Prop. 2.6). Using (100) and the H¨older inequality repeatedly we obtain ∞
p p ∞ p E Vs |Fz ds ≤c φ(s)ds E[Vs ]. (101) E 0
z
Hence the integrability of φ(t) implies that V˜z given by (51) has a finite moment of any order p < ∞ if Vz has a finite moment of order p. In summary we have Proposition 1. (i) Assume that E[Vz ] < ∞, p ∈ [1, ∞). If the uniform (L∞ -) mixing coefficient φ∞ (t) of Vz is integrable then V˜z has finite moments of order p. (ii) Assume that E[Vz6 ] < ∞. If the 2/5-power of the L6/5 -mixing coefficient φ6/5 (t) is integrable, then V˜z has finite second moment. (iii) Assume Vz is almost surely bounded. If the square-root of the alpha- (L1 -) mixing coefficient φ1 (t) is integrable then V˜z has finite second moment. p
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
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Appendix B. Proof of Lemma 2 2 (i) Estimation of sup |z|≤z0 E δγ Vzε (y) : We have that for γρ ≤ 1, |y|≤L
sup E
|z|≤z0
2
δγ Vzε (x, y)
= sup
|z|≤z0
4| sin (γ y · k/2)|2 (η,ρ) (ξ, k)dξ dk
≤ sup
|z|≤z0
|γ y · k|2 (η,ρ) (ξ, k)dξ dk
≤ c0 γ |y| sup 2
2
|z|≤z0 |ξ |≤ρ
≤ c1 γ |y| sup 2
2
|z|≤z0 |ξ |≤ρ
≤ c2 γ 2 |y|2
|k|≤ρ
(η2 + |k|2 + |ξ |2 )−H −(d+1)/2 |k|d+1 d|k|dξ
(η2 + |ξ |2 )−H +1/2 dξ
|ξ |−2H +1 dξ + ρ 2−2H .
|ξ |∈(η,ρ)
≤ c3 γ 2 |y|2 η2−2H
For ργ ≥ 1 we divide the domain of integration into I0 = {|k| ≤ γ −1 } and I1 = {|k| ≥ γ −1 } and estimate their contributions separately. For I0 the upper bound is similar to the above, namely, we have sup
|z|≤z0 I0
4| sin (γ y · k/2)|2 (η,ρ) (ξ, k)dξ dk ≤ c4 γ 2 |y|2 η2−2H + γ −2+2H .
For I1 we have instead that sup 4| sin (γ y · k/2)|2 (η,ρ) (ξ, k)dξ dk |z|≤z0 I1
≤ 4 sup
|z|≤z0 I1
(η,ρ) (ξ, k)dξ dk
≤ c5 sup
|z|≤z0 |ξ |∈(γ −1 ,ρ)
≤ c6
|ξ |∈(γ −1 ,ρ)
(η2 + |ξ |2 )−H −1/2 dξ
|ξ |−2H −1 dξ
≤ c7 γ 2H + ρ −2H . Put together, the upper bound becomes sup E
|z|≤z0 |x|,|y|≤L
2
δγ Vzε (x, y)
% %2−2H % % , ≤ c8 γ 2 %min (γ −1 , ρ)%
γ , η ≤ 1 ≤ ρ.
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2 (ii) Estimation of sup|z|≤z0 E V˜zε (x) : It follows from the argument for Corollary 1 and Assumption 2 that 2
2 ∞ ε ˜ E Vz (x) ≤ rη,ρ (t)dt E[Vzε ]2 0
(iii) Estimation of sup |z|≤z0 |y|≤L
≤ cη−2 η−2H . 2
ε ˜ (y): First note that the correlation coefficient E δ γ Vz
for δγ V˜zε is bounded from above by crη,ρ (t) for some constant c > 0. Then we have as in (i), (ii) that ∞ 2
2 ε ˜ E δγ Vz (x) ≤ c1 rη,ρ (t)dt E[δγ Vzε ]2 0
% %2−2H % γ %min (γ −1 , ρ)% . %
2 %% % %∇y E δγ V˜ ε (y)%: By the Cauchy-Schwartz inequality and z % % ≤ c2 η
(iv) Estimation of sup |z|≤z0 |y|≤L
−2 2 %
the preceding calculation we have %
2 %% % ε % sup %∇y E δγ V˜z (y)%% |z|≤z 0 |y|≤L
,
,
2
2 ε ε ˜ ˜ ≤ c1 ∇x V (x + γ y/2) + ∇x V (x − γ y/2) E δγ V˜ ε (x, y) ∞ 2 2 2 ≤ c3 rη,ρ (t)dt γ E1/2 ∇x V ε E1/2 δγ Vzε (x, y) γ 2E
0
≤ c4 η−2 γ 2 ρ 1−H | min (ρ, γ −1 )|1−H . (v) Estimation of sup|z|≤z0 E p · ∇x (L˜ εz θ ) 22 : A similar line of reasoning and a straightforward spectral calculation yield that E p · ∇x (L˜ εz θ ) 22 = E ∇y · ∇x γ −1 δγ V˜zε F2−1 θ 22 ≤ c1 E ∇x2 V˜zε F2−1 θ 22
2 ≤ c2 η−2 E ∇x2 Vzε ≤ c3 η−2 ρ 4−2H . Appendix C. Proof of Corollary 3 By the Cauchy-Schwartz inequality we have the following calculation:
E Lεz θ (x, p)L˜ εz θ(y, q) 22 ( ( (2
(2 ( (2 ( ( ( ( ≤ C1 (E Lεz θ (x, p)L˜ εz θ (y, q) ( + E (Lεz θ (x, p)( E (L˜ εz θ(y, q)( 2
2
( (2
( ( = C1 γ −4 (E δγ Vzε (x, x )δγ V˜zε (y, y ) F2−1 θ(x, x )F2−1 θ(y, y )( 2
2
White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation
321
( % (2 0 %
( % ( % −1 ( % ˜ ε %2 ( % (2 ( −1 ( ε %2 ( + (E δγ Vz F2 θ ( (E %δγ Vz % F2 θ ( & =O
2
'
2
%2 % % %2 % % sup E %δγ Vzε % (y)E %δγ V˜zε % (y)
|y|≤L
and
E Lεz L˜ εz θ 22 ( 2
(2
2 ( ( −1 −4 ε 2 ε ˜ε ε ˜ ≤ C1 γ E δγ V z E δ γ V z F2 θ dxdy + (E Lz Lz θ(x, p) ( 2
2 2 2 F2−1 θ dxdy = C1 γ −4 E δγ Vzε E δγ V˜zε ( (2
( ( + (E δγ Vzε δγ V˜zε F2−1 θ (x, y)( 2 & ' % % 2 % %2 % % ε ε = O sup E %δγ V % (y)E %δγ V˜ % (y) |y|≤L
z
z
and
E L˜ εz L˜ εz θ 22 ( 2
(2
2
2 ( ( ≤ C2 γ −4 E δγ V˜zε E δγ V˜zε F2−1 θ dxdy + (E L˜ εz L˜ εz θ(x, p) ( 2 1 0 ( (2
2 2
2 ( ( −1 −1 −4 ε ε ε = C2 γ E δγ V˜z F2 θ dxdy + (E δγ V˜z δγ V˜z F2 θ(x, y)( & =O
2
'
% %2 % % sup E2 %δγ V˜zε % (y) ,
|y|≤L
where C1 , C1 , C2 are constants independent of ρ, η, γ and L is the radius of the ball containing the support of F2−1 θ . Similarly we have that ' & ( % %2 % ( %2 ( ε ˜ ε ˜ ε (2 % 2% ε ε E (Lz Lz Lz θ ( = O sup E %δγ V˜z % E %δγ Vz % . 2
|y|≤L
Acknowledgement. I thank the referee for a careful reading of the manuscript and the suggestions which lead to improvement of the presentation. I also benefited from the inspiring program of the Mathematical Geophysics Summer School at Stanford University, 2002.
References 1. Adler, R.J.: An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. Lecture Notes, Monograph Series, vol. 12, Hayward, CA: Institute of Mathematical Statistics, 1990
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2. Andrew, L.C.: An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere. J. Mod. Opt. 39, 1849–1853 (1992) 3. Bailly, F., Clouet, J.P., Fouque, J.-P.: Parabolic and Gaussian white noise approximation for wave propagation in random media. SIAM J. Appl. Math. 56(5), 1445–1470 (1996) 4. Bailly, F., Fouque, J.-P.: High frequency wave propagation in random media. Preprint, 1997 5. Beran, M.J., Oz-Vogt, J.: Imaging through turbulence in the atmosphere. Prog. Optics 33, 319–388 (1994) 6. Berry, M.V.: Semi-classical mechanics in phase space: a study of Wigner’s function. Philos. Trans. R. Soc. A 287, 237–271 (1977) 7. Blomgren, P., Papanicolaou, G., Zhao, H.: Super Resolution in Time Reversal Acoustics. J. Acoust. Soc. Am. 111, 230–248, (2002) 8. Davies, E.B.: L1 properties of second order elliptic operators. Bull. London Math. Soc. 17, 417–436 (1985) 9. Ethier, S.N., Kurtz, T.G.: Markov Processes – Characterization and Convergence. New York: John Wiley and Sons, 1986 10. Falkovich, G., Gawedzki, G., Vergassola, M.: Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975 (2001) 11. Fannjiang, A., Solna, K.: Scaling limits for beam wave propagation in atmospheric turbulence. Stoch. Dyn 4(1), 135–151 (2004) 12. Fannjiang, A., Solna, K.: Propagation and time reversal of wave beams in atmospheric turbulence. SIAM J. Multiscale Modeling Comp., in press 13. Folland, G.B.: Harmonic Analysis in Phase Space. Princeton, VJ: Princeton University Press, 1989 14. Fouque, J.-P.: La convergence en loi pour les processus a` valeurs dans un espace nucl´eaire. Ann. Inst. Henri Poincar´e 20, 225–245 (1984) 15. Gerard, P., Markowich, P.A., Mauser, N.J., Poupaud, F.: Homogenization limits and Wigner transforms. Commun. Pure Appl. Math. L, 323–379 (1997) 16. Hill, R.J.: Models of the scalar spectrum for turbulent advection. J. Fluid Mech. 88, 541–562 (1978) 17. Ibragimov, I.A., Rozanov, Y.A.: Gaussian Random Processes. New York: Springer-Verlag, 1978 18. Kurtz, T.G.: Semigroups of conditional shifts and approximations of Markov processes. Ann. Prob. 3(4), 618–642 (1975) 19. Kushner, H.J.:: Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory. Cambridge, Massachusetts: MIT Press, 1984 20. Lions, P.L., Paul, T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9, 553–618 (1993) 21. Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics, Vol. 1 &2. Cambridge, MA: MIT Press, 1975 22. Manning, R.M.: Stochastic Electromagnetic Image Propagation and Adaptive Compensation. New York: McGraw-Hill, 1993 23. Papanicolaou, G., Ryzhik, L., Solna, K. : The parabolic approximation and time reversal. Matem. Contemp. 23, 139–159 (2002) 24. Strohbehn, J.W.: Laser Beam Propagation in the Atmosphere. Berlin: Springer-Verlag, 1978 Communicated by P. Constantin
Commun. Math. Phys. 254, 323–341 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1154-z
Communications in
Mathematical Physics
A Local Index Formula for the Quantum Sphere Sergey Neshveyev , Lars Tuset Mathematics Institute, University of Oslo, PB 1053 Blindern, Oslo 0316, Norway. E-mail:
[email protected];
[email protected] Received: 7 November 2003 / Accepted: 10 February 2004 Published online: 27 August 2004 – © Springer-Verlag 2004
Abstract: For the Dirac operator D on the standard quantum sphere we obtain an asymptotic expansion of the SUq (2)-equivariant entire cyclic cocycle corresponding to 1 ε 2 D when evaluated on the element k 2 ∈ Uq (su2 ). The constant term of this expansion is a twisted cyclic cocycle which up to a scalar coincides with the volume form and computes the quantum as well as the classical Fredholm indices. Introduction The geometry of q-deformed spaces, see e.g. [KS], and the theory of non-commutative geometry of Connes [C1] have developed considerably over the last twenty years. Crucial in establishing connections between these two areas is the construction of Dirac operators which give rise to reasonable differential calculi. Other aspects of the theory include computations of cyclic cohomology and analysis of the index theorem. Work in this direction has progressed furthest for spaces like SUq (2) and the quantum spheres, see e.g. [CP, DS, SW, MNW1, MNW2, C2]. The purpose of this paper is to show that, although the spectral triple associated with the Dirac operator [DS] on the homogeneous sphere Sq2 of Podle´s has properties quite different from the classical one, the machine of non-commutative geometry by Connes, when gently tuned, works perfectly well also in this case. Specifically, we obtain an index formula for Sq2 . Concerning the differences from the classical case, we mention that the ζ -function 2 of the Dirac operator has infinitely many poles on vertical lines and the traces Tr(e−tD ) of the heat operators tend to infinity slower than t −p for any p > 0. More importantly, the spectral triple does not satisfy the regularity assumption, a condition which is often overshadowed by other assumptions such as boundedness of commutators, but which is
Partially supported by the Norwegian Research Council. Supported by the SUP-program of the Norwegian Research Council.
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crucial for a detailed analysis. This can mean that the Dirac operator is, in fact, not the right one, and one can try to construct another operator with the same Fredholm module. However, in this process one will most likely lose the SUq (2)-equivariance [DS]. What one gains is not so clear: with the absence of Getzler’s symbol calculus and of a Wodzicki-type geometric description of the Dixmier trace, the computation of the indices in terms of the Dixmier trace remains for the moment a non-trivial problem. So we will stick to the standard Dirac operator. The associated spectral triple is SUq (2)-equivariant and thus, via the JLO-cocycle, defines the Chern character in the equivariant entire cyclic cohomology of Sq2 . We evaluate this cocycle on k 2 ∈ Uq (su2 ). The resulting object is an entire twisted cyclic cocycle, and its pairing with the equivariant K-theory computes the quantum Fredholm index, that is, the difference of quantum dimensions of the kernel and the cokernel [NT]. The possibility of detecting cocycles by evaluating them on k 2 was pointed out by Connes in [C2]. As was promised in [NT] such evaluations are much easier to compute. The philosophical reason for this is that the quantum dimension is intrinsically associated with the tensor category of finite dimensional corepresentations of a compact quantum group. On the technical side, the evaluation on k 2 gives an immediate connection to the Haar state, as was already remarked in [G, SW], and thus serves as a replacement for the trace theorem by Connes on equality of the Dixmier trace and the Wodzicki residue. Note also that for q-deformations the classical index can be recovered from the quantum one: if one can prove that the Fredholm operators depend continuously on q and has a polynomial formula in q and q −1 for the quantum index, the classical index is obtained by simply setting q = 1. The evaluation on k 2 does not, however, solve all problems: the spectral triple is still non-regular, and while the quantum traces of the heat operators are better controlled than the classical ones, they possess a strange oscillating behavior near zero. The non-regularity of the spectral triple can be illuminated by saying that though the principal symbol of |D| is scalar-valued, the operator |D|z T |D|−z does not necessarily have the same symbol as T . The symbol is, however, computable, and this is the key property allowing to apply the ideas of the proof of the local index formula of Connes-Moscovici [CM2] to our situation. It seems that the development of a full-scaled pseudo-differential calculus for the Dirac operators on q-deformed spaces is the main prerequisite for the analysis of more general examples, such as the ones in [Kr]. The Dirac operator on the quantum sphere is defined using the standard differential calculus, and in the course of our analysis we use very little beyond basic properties of this calculus and some information on the spectrum of the Dirac operator. In this respect the fact that Connes’ non-commutative geometry can be applied successfully to the study of a q-deformed space is more important than the final result for the quantum sphere. 1. Cyclic Cohomology In this section we formulate some results of non-commutative geometry in the Hopf algebra equivariant setting. Fortunately, as the Hopf algebra equivariant cyclic cohomology theory is now available [AK, NT], a simple argument involving crossed products allows to transfer such results from the non-equivariant case without having to prove each one of them from scratch. We use the same notation and make the same assumptions as in [NT]. So let (H, ) be a Hopf algebra over C with invertible antipode S and counit ε, and adapt the Sweedler notation (ω) = ω(0) ⊗ ω(1) . The results below are also true for the algebra of finitely
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supported functions on a discrete quantum group, which is of main interest for us. Let B be a unital right H-module algebra with right action of (H, ) denoted by . Consider n (B) of H-invariant n-cochains, so C n (B) consists of linear functionals on the space CH H ⊗(n+1) such that H⊗B f (S −1 (ω(0) )ηω(1) ; b0 ω(2) , . . . , bn ω(n+2) ) = ε(ω)f (η; b0 , . . . , bn ) n−1 for any ω, η ∈ H and b0 , . . . , bn ∈ B. The coboundary operator bn : CH (B) → n n n CH (B) and the cyclic operator λn : CH (B) → CH (B) are given by
(bn f )(ω; b0 , . . . , bn ) =
n−1
(−1)i f (ω; b0 , . . . , bi−1 , bi bi+1 , bi+2 , . . . , bn )
i=0
+(−1)n f (ω(0) ; bn ω(1) b0 , b1 , . . . , bn−1 ) and
(λn f )(ω; b0 , . . . , bn ) = (−1)n f (ω(0) ; bn ω(1) , b0 , . . . , bn−1 ).
• (B), b) is denoted by (C • (B), b) and its cohoThe subcomplex (Ker(ι − λ), b) of (CH H,λ • (B). There is a pairing ·, · : H C 2n (B) × K H (B) → R(H) mology is denoted by H CH 0 H of the even cyclic cohomology with equivariant K-theory, where R(H) is the space of H-invariant linear functionals on H with H acting on itself by
ηω = S −1 (ω(0) )ηω(1) . 2n (B) is a cyclic cocycle, then If p ∈ B is an H-invariant idempotent and f ∈ CH ,λ
[f ], [p](ω) =
1 f (ω; p, . . . , p). n!
(1.1)
More generally, if X is a finite dimensional (as a vector space) right H-module, then End(X) ⊗ B is a right H-module algebra with right action given by (T ⊗ b)ω = πX (ω(0) )T πX S −1 (ω(2) ) ⊗ bω(1) , where πX : H → End(X) is the anti-homomorphism defining the H-module structure n : C n (B) → C n (End(X) ⊗ B) given by on X. Then we have a map X H H n f )(ω; T0 ⊗ b0 , . . . , Tn ⊗ bn ) = f (ω(0) ; b0 , . . . , bn )Tr(πX S −1 (ω(1) )T0 . . . Tn ). ( X
Now suppose p ∈ End(X) ⊗ B is an H-invariant idempotent, so it defines an element of K0H (B). By definition we have 2n [f ], [p] = [ X f ], [p] 2n (B). for a cyclic cocycle f ∈ CH ,λ We will also need the (b, B)-bicomplex description of the periodic cyclic cohomoln (B) → C n (B), N = n λi , ogy. So consider the operator B = N B0 , where Nn : CH n i=0 n H n+1 n (B), B n = (−1)n s n (ι − λ ), so (B) → CH B0n : CH n n 0
(B0n f )(ω; b0 , . . . , bn ) = f (ω; 1, b0 , . . . , bn ) − (−1)n+1 f (ω; b0 , . . . , bn , 1). 0 (B) = ⊕∞ C 2n (B) and C 1 (B) = ⊕∞ C 2n+1 (B). Then we have a well-defined Set CH n=0 H n=0 H H complex
326
S. Neshveyev, L. Tuset b+B
b+B
b+B
1 0 1 0 CH (B) −→ CH (B) −→ CH (B) −→ CH (B). n (B), n = 0, 1. Then the The cohomology of this small complex is denoted by H PH formula
[(f2n )n ], [p](ω) =
∞ n=0
(−1)n
(2n)! 2n 1 ( X f2n )(ω; p − , p, . . . , p) n! 2
(1.2)
0 (B) × K H (B) → R(H) (this can be deduced from the non-equidefines a pairing H PH 0 variant case in the same way as in [NT]). The map 2n 0 CH (B) → CH (B), f →
(−1)n f, (2n)!
(1.3)
2n (B) → H P 0 (B) compatible with pairings (1.1) and induces a homomorphism H CH H (1.2) and which respects the periodicity operator in the sense that the images of f and Sf in H P 0 are equal. k (B) ⊂ ∞ C 2n+k (B), k = 0, 1, consisting of all Consider the subspaces CEH n=0 H cochains (f2n+k )n such that for any ω ∈ H and any finite subset F ⊂ B there exists C > 0 such that
|f2n+k (ω(0) ; b0 ω(1) , . . . , bj −1 ω(j ) , bj , . . . , b2n+k )| ≤
C n!
k (B) the cohomology for any n ≥ 0, 0 ≤ j ≤ 2n + k + 1 and bi ∈ F . Denote by H EH of the complex b+B
b+B
b+B
1 0 1 0 CEH (B) −→ CEH (B) −→ CEH (B) −→ CEH (B). 0 (B) with K H (B). The formula (1.2) still defines a pairing of H EH 0 Suppose we are given an equivariant even spectral triple. By this we mean a Hilbert space H with grading operator γ , an odd selfadjoint operator D with compact resolvent, and even representations of B and H on H such that D commutes with H, [D, b] is bounded for any b ∈ B and
bω = ω(0) bω(1) ∀b ∈ B, ∀ω ∈ H.
(1.4)
If the spectral triple is θ -summable in the sense that e−tD is trace-class for all t > 0, set 2 2 Ch2n (D)(ω; b0 , . . . , b2n ) = dtTr(γ ωb0 e−t0 D [D, b1 ]e−t1 D 2
2n
× . . . [D, b2n ]e−t2n D ), 2
where 2n dt means integration over the simplex 2n = {(t0 , . . . , t2n ) | ti ≥ 0, i ti = 0 (B) having all the usual properties of a JLO1}. Then (Ch2n (D))n is a cocycle in CEH cocycle [C1, CM1, GBVF]: (i) (Ch2n (tD))n and (Ch2n (sD))n are cohomologous for any s, t > 0; (ii) the pairing of K0H (B) with (Ch2n (tD))n computes the index map defined by D (so e.g. for an H-invariant idempotent p ∈ B we get [(Ch2n (D))n ], [p](ω) = Tr(ω|Ker p− Dp+ ) − Tr(ω|Ker p+ Dp− ));
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(iii) if the spectral triple is p-summable in the sense that |D|−p is a trace-class operator, F = D|D|−1 , and τF2m denotes the Chern character of the Fredholm module (H, F, γ ) in the equivariant cyclic cohomology for some fixed m, 2m ≥ p, so τF2m (ω; b0 , . . . , b2m ) =
(−1)m m!Tr(γ ωF [F, b0 ] . . . [F, b2m ]), 2
2m (B) → then (Ch2n (D))n is cohomologous to the image of τF2m under the map CH 0 CEH (B) given by (1.3).
As in the non-equivariant case, all these properties are consequences of the homotopy invariance of the cohomology class of (Ch2n (D))n meaning that if we have an H-invariant homotopy Dt such that e.g. D˙ t is bounded, then (Ch2n (D0 ))n and (Ch2n (D1 ))n are cohomologous (concerning (ii) see also [KL]). This in turn can be deduced from the non-equivariant case as follows, see the discussion at the end of [C2]. Consider the crossed product algebra B H, i.e. the vector space B ⊗ H with product (b ⊗ ω)(c ⊗ η) = bcS −1 (ω(1) ) ⊗ ω(0) η. n (B) → In the sequel we write bω instead of b ⊗ ω. Then we have a map n : CH n C (B H) given by 0 n 0 n . . . ω(0) ; b0 (ω(1) . . . ω(1) ), (n f )(b0 ω0 , . . . , bn ωn ) = f (ω(0) 1 n n b1 (ω(2) . . . ω(2) ), . . . , bn ω(n+1) ).
The definition of n is motivated by the following easily proved lemma. Lemma 1.1. Consider a covariant representation of B and H on H (that is, identity (1.4) holds). Suppose c0 , . . . , cn ∈ B(H ) commute with H. Then for any b0 , . . . , bn ∈ B and ω0 , . . . , ωn ∈ H we have 0 n 0 n c0 b0 ω0 . . . cn bn ωn = ω(0) . . . ω(0) c0 b0 (ω(1) . . . ω(1) ) 1 n n . ×c1 b1 (ω(2) . . . ω(2) ) . . . cn bn ω(n+1)
It is not difficult to check that the maps n constitute a morphism of cocyclic objects, so they induce maps for the various cyclic cohomology theories. Note also that if H is unital, or H has an approximate unit in an appropriate sense, then n is injective. So to prove a property for a cochain in an equivariant theory it is enough to establish the analogous property for the image of the cochain under . Now note that any equivariant spectral triple can be considered as a spectral triple for B H. It follows immediately from Lemma 1.1 that (2n (Ch2n (D))n is the JLO-cocycle for this spectral triple. This allows to deduce the properties of (Ch2n (D))n from the non-equivariant case. From now onwards we assume that (H, ) is the algebra of finitely supported funcˆ ) ˆ ), ˆ and as in [NT] we write (A, ˆ instead of tions on a discrete quantum group (A, ˆ (H, ). Denote by ρ ∈ M(A) the Woronowicz character f−1 . We will be interested in evaluating equivariant cocycles on ω = ρ. The following known lemma explains why such evaluations are easier to deal with.
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Lemma 1.2. Let U ∈ M(A ⊗ K(H )) be a unitary corepresentation of the dual compact group (A, ), and αU : B(H ) → M ⊗ B(H ) the coaction of the von Neumann closure (M, ) of (A, ) on B(H ), αU (x) = U ∗ (1 ⊗ x)U. Consider also the corresponding representation of Aˆ on H , so ωξ = (ω ⊗ ι)(U )ξ . Then the map a → Tr(·aρ) defines a one-to-one correspondence between positive elements a ∈ B(H )αU such that Tr(aρ) = 1 and normal αU -invariant states on B(H ). In particular, if B ⊂ B(H ) is an αU -invariant C∗ -subalgebra with a unique αU invariant state ϕ, then Tr(baρ) = ϕ(b)Tr(aρ) for any b ∈ B and any a ∈ B(H )α+U with Tr(aρ) < ∞. ˆ Note also that the fixed point algebra B(H )αU is precisely the commutant of A. ˆ one needs some care in dealing with this element. From now Since in general ρ ∈ / A, on we assume as in [NT] that we have a left coaction α of (A, ) on a C∗ -algebra B, ˆ and is the corresponding action of A, bω = (ω ⊗ ι)α(b), ˆ Then σz (b) = bρ z is a well-defined element of B for any b ∈ B and while B = BA. z ∈ C. The automorphism σ = σ1 will be called the twist. Now if we have an equivariant θ -summable spectral triple, for the expression Ch2n (D)(ρ; b0 , . . . , b2n ) to make sense it is enough to require that Tr(ρe−sD ) < ∞ ∀s > 0. 2
Indeed, since z → ρ −z [D, b]ρ z = [D, bρ z ] ∈ B(H ) is an analytic function, the H¨older inequality and the identity e−t0 D [D, b1 ]e−t1 D . . . [D, b2n ]e−t2n D ρ = e−t0 D ρ t0 [D, b1 ρ t0 ]e−t1 D 2 2 ×ρ t1 [D, b2 ρ t0 +t1 ]e−t2 D ρ t2 . . . [D, b2n ρ 1−t2n ]e−t2n D ρ t2n (1.5) 2
2
2
2
2
show that Ch2n (D)(ρ; b0 , . . . , b2n ) is well-defined. In fact, these elements form a cocycle in an appropriate cyclic cohomology theory, namely, in the entire twisted cyclic cohomology H Eσ0 (B), cf. [KMT, G]. This is immediate as such twisted theories are obtained by setting ω = ρ in the formulas above. Thus the space of n-cochains Cσn (B) becomes the space of linear functionals f on B ⊗(n+1) such that f (σ (b0 ), . . . , σ (bn )) = f (b0 , . . . , bn ), the cyclic operator is given by (λn f )(b0 , . . . , bn ) = (−1)n f (σ (bn ), b0 , . . . , bn−1 ), and so on. In particular, we still have pairings of K0H (B) with H Cσ2n (B), H Pσ0 (B) and H Eσ0 (B).
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2. Differential Calculus on the Quantum Sphere From now on (A, ) will denote the compact quantum group SUq (2) of Woronowicz [W], q ∈ (0, 1). So A is the universal unital C∗ -algebra with generators α and γ satisfying the relations α ∗ α + γ ∗ γ = 1, αα ∗ + q 2 γ ∗ γ = 1, γ ∗ γ = γ γ ∗ , αγ = qγ α, αγ ∗ = qγ ∗ α. The comultiplication is determined by the formulas (α) = α ⊗ α − qγ ∗ ⊗ γ , (γ ) = γ ⊗ α + α ∗ ⊗ γ . ˆ Recall that Consider the quantized universal enveloping algebra Uq (su2 ) ⊂ M(A). −1 it is the universal unital ∗-algebra generated by elements e, f, k, k satisfying the relations kk −1 = k −1 k = 1, ke = qek, kf = q −1 f k, ef − f e = k ∗ = k, e∗ = f.
k 2 − k −2 , q − q −1
ˆ with The algebra Uq (su2 ) is a Hopf subalgebra of M(A) ˆ ) = f ⊗ k −1 + k ⊗ f, ˆ ˆ (k) = k ⊗ k, (e) = e ⊗ k −1 + k ⊗ e, (f −1 −1 ˆ ˆ ) = −qf, ˆ = −q e, S(f S(k) = k , S(e) εˆ (k) = 1, εˆ (e) = εˆ (f ) = 0. The set I of equivalence classes of irreducible corepresentations of (A, ) is identified with the set 21 Z+ of non-negative half-integers. The fundamental corepresentation (s = 21 ) is defined by 1 1 α −qγ ∗ 2 2 U = (uij )ij = . (2.1) γ α∗ The corresponding representation of Uq (su2 ) is given by 1 01 q2 0 , e → . k → 1 00 0 q− 2
(2.2)
The formulas (2.1–2.2) completely determine the pairing between (A, ) and ˆ Note also that (Uq (su2 ), ). ρ = f−1 = k 2 .
(2.3)
ˆ define elements in A by For a ∈ A and ω ∈ M(A) ∂ω (a) = ω ∗ a = (ι ⊗ ω)(a) and aω = a ∗ ω = (ω ⊗ ι)(a). Then the modular property of the Haar state h can be expressed as h(a1 a2 ) = h(a2 f1 ∗ a1 ∗ f1 ) = h(a2 ∂k −2 (a1 k −2 )). For n ∈ Z set
n
An = {a ∈ A | ∂k (a) = q 2 a}.
(2.4)
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The norm closure B = C(Sq2 ) ⊂ A of A0 with the left coaction |B of (A, ) is the quantum homogeneous sphere of Podle´s [P1]. Note that B = A0 . The norm closure An of An is an analogue of the space of continuous sections of the line bundle over the sphere with winding number n. It is well-known that the standard differential calculus on the quantum sphere [P2] can be obtained from various covariant differential calculi on SUq (2), see e.g. [S]. Consider the 3D-calculus of Woronowicz [W, KS], that is, the left-covariant first order differential calculus (Aω−1 ⊕ Aω0 ⊕ Aω1 , d) of (A, ) with differential da = ∂f k −1 (a)ω−1 +
a − ∂k −4 (a) ω0 + ∂ek −1 (a)ω1 , 1 − q −2
and right A-module action on the left-invariant forms ω−1 , ω0 , ω1 given by ω0 a = ∂k −4 (a)ω0 and ωi a = ∂k −2 (a)ωi for i = ±1 and a ∈ A. The associated exterior differential algebra spanned by a0 da1 . . . dan , ai ∈ A, is completely described by the following rules: 2 ω−1 = ω02 = ω12 = 0, ω−1 ω1 = −q 2 ω1 ω−1 , ω0 ω1 = −q 4 ω1 ω0 , ω−1 ω0 = −q 4 ω0 ω−1 , dω−1 = (q 2 + q 4 )ω0 ω−1 , dω0 = −ω1 ω−1 , dω1 = (q 2 + q 4 )ω1 ω0 .
The standard differential calculus on Sq2 is then obtained by restricting the differential ∧ , d) for S 2 is the projective B-module given by d to B, so the exterior algebra (B q ∧ B = span{b0 db1 · · · dbn | bi ∈ B}, and we have the following concrete description. Theorem 2.1. Set 0,1 (B) = A−2 , 1,0 (B) = A2 and 1,1 (B) = B. Then (i) the de Rham complex on Sq2 is the graded differential algebra ∧ ∧0 ∧1 ∧2 = B ⊕ B ⊕ B = B ⊕ (0,1 (B) ⊕ 1,0 (B)) ⊕ 1,1 (B) B
with multiplication ∧ given by (a0,0 , a0,1 , a1,0 , a1,1 ) ∧ (b0,0 , b0,1 , b1,0 , b1,1 ) = (a0,0 b0,0 , a0,0 b0,1 + a0,1 b0,0 , a1,0 b0,0 + a0,0 b1,0 , a0,0 b1,1 −a0,1 b1,0 + q 2 a1,0 b0,1 + a1,1 b0,0 ) and differential d = ∂ + ∂¯ given by ∂(a0,0 , a0,1 , a1,0 , a1,1 ) = (0, 0, ∂e (a0,0 ), q∂e (a0,1 )), ¯ 0,0 , a0,1 , a1,0 , a1,1 ) = (0, ∂f (a0,0 ), 0, −q∂f (a1,0 )); ∂(a ∧2 by identifying ∧2 with B, then is closed (ii) if we define ω = h(ω) for ω ∈ B B ∧1 in the sense that dω = 0 for any ω ∈ B ; ∧ commuting with d, (iii) the twist σ of B extends uniquely to an automorphism of B which we again denote by σ , and then #ω#ω ω ∧ ω = (−1) σ (ω ) ∧ ω ∧ with #ω + #ω = 2. for any ω, ω ∈ B
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∧ = B + A ω Proof. Part (i) follows from the equality B −2 −1 + A2 ω1 + Bω1 ω−1 , ∧1 = BdB, to prove (ii) it is enough to show that which is easily checked. Since B db1 ∧ db2 = 0, that is
q 2 h(∂e (b1 )∂f (b2 )) = h(∂f (b1 )∂e (b2 )). We have
(2.5)
h(∂e (b1 )∂f (b2 )) = h(b1 ∂S(e)f (b2 )) = −q −1 h(b1 ∂ef (b2 )). ˆ
Similarly h(∂f (b1 )∂e (b2 )) = −qh(b1 ∂f e (b2 )). Since ∂ef = ∂f e on B, assertion (ii) is proved. The existence of an extension of σ is obvious: it comes from the automorphism a → ak 2 of A. Concerning the equality in part (iii), the only interesting case is when ω ∈ 1,0 (B) and ω ∈ 0,1 (B). In this case we have to prove that for a ∈ A2 and a ∈ A−2 we have q 2 h(aa ) = h(σ (a )a). But this is true, since h(σ (a )a) = h(a∂k −2 (a )) = q 2 h(aa ) by (2.4).
(2.6)
Define the volume form by τ (b0 , b1 , b2 ) = b0 db1 ∧ db2 = h(b0 (q 2 ∂e (b1 )∂f (b2 ) − ∂f (b1 )∂e (b2 ))). Set also τ1 (b0 , b1 , b2 ) = h(b0 ∂e (b1 )∂f (b2 )) and τ2 (b0 , b1 , b2 ) = h(b0 ∂f (b1 )∂e (b2 )). These forms were introduced in [SW], where part of the following proposition was proved. Proposition 2.2. We have that • (B), b); (i) τ is a cocycle in the twisted cyclic complex (Cσ,λ (ii) τ1 and τ2 are cocycles in the complex (Cσ• (B), b + B); (iii) the cocycle q 2 τ1 +τ2 ∈ Cσ0 (B) is a coboundary; in particular, the cocycles τ , 2q 2 τ1 and −2τ2 are cohomologous.
Proof. Part (i) is a standard consequence of the properties of the integral, cf [C1, KMT]. To prove (iii), set τ˜ (b0 , b1 ) = h(∂f (b0 )∂e (b1 )). Then (B0 τ˜ )(b0 ) = (B00 τ˜ )(b0 ) = τ˜ (b0 , 1) + τ˜ (1, σ (b0 )) = 0. On the other hand, using that ∂e and ∂f are derivations on B and that h(σ (b)a) = h(ab) for any a, b ∈ B, we get (b2 τ˜ )(b0 , b1 , b2 ) = h(b0 ∂f (b1 )∂e (b2 )) + h(∂f (σ (b2 ))b0 ∂e (b1 )) = τ2 (b0 , b1 , b2 ) + q 2 τ1 (b0 , b1 , b2 ), where we have used (2.6) with a = b0 ∂e (b1 ) and a = ∂f (a2 ). Thus q 2 τ1 + τ2 is the coboundary of τ˜ . Since τ = τ1 + τ2 and q 2 = 1, statement (ii) follows immediately from (i) and (iii). It can also be checked directly. Note that although τ is cyclic, this is not the case for τ1 and τ2 .
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3. The Dirac Operator Dirac operators on the quantum sphere have appeared in several papers, see e.g. [O, PS, DS, SW, M, Kr] and references therein. However, they all coincide by an important result of [DS] stating that the Dirac operator is essentially uniquely determined by the requirements of SUq (2)-invariance, boundedness of commutators and the first order condition. In the next paragraph and in Proposition 3.1 below we summarize the properties of the Dirac operator which we shall need. One of the easiest ways to construct the Dirac operator is to recall [F] that the spinor 0,i ⊗S, bundle on a classical K¨ahler manifold M of complex dimension m is ⊕m i=0 ∧ where√S is a square root of the canonical line bundle ∧m,0 , and then the Dirac operator D is 2(∂¯ + ∂¯ ∗ ). In view of the previous section these notions have rather straightforward analogues for the quantum sphere. So consider the space H = L2 (A, h) of the GNS-representation of A = C(SUq (2)) corresponding to the Haar state h. Then H = ⊕n∈Z L2 (An , h). The left actions a → ∂ω (a) and a → aSˆ −1 (ω) of Aˆ on A extend to ∗-representations ˆ In the of Aˆ on H . These are, in fact, the left and the right regular representations of A. sequel we will write ∂ω for the operators of the first representation and simply ω for the operators of the second one. Then the space of L2 -spinors and the Dirac operator are defined by 0 ∂e 2 2 H+ ⊕ H− = L (A1 , h) ⊕ L (A−1 , h) and D = , ∂f 0 respectively. Note that ∂f (An ) ⊂ An−2 and ∂e (An ) ⊂ An+2 . This shows that D is indeed an operator on H+ ⊕ H− . Proposition 3.1. We have
2 1 1 2 k − q − 2 k −1 q (i) D 2 = C, where C = f e + is the Casimir; q − q −1 1 0 q 2 ∂e (b) (ii) [D, b] = for any b ∈ B. 1 q − 2 ∂f (b) 0 n ∂ef 0 2 . Since ∂k |L2 (An ,h) = q 2 and ∂ef − ∂f e = (q − Proof. We have D = 0 ∂f e q −1 )−1 (∂k 2 − ∂k −2 ), we see that D 2 = ∂C . To prove (i) it remains to note that ∂C = C ˆ and S(C) ˆ on H as C is in the center of M(A) = C. Part (ii) follows from the identity
[∂e , a] = ∂e (a)∂k −1 + ∂k (a)∂e − a∂e , which is valid for any a ∈ A, and from a similar identity for ∂f .
We now want to develop a simple pseudo-differential calculus for our Dirac operator. More precisely, we want to compute the principal symbol of the operator of the form |D|2z db|D|−2z , where db = [D, b]. For this it is more convenient to consider a more general problem. The Casimir C defines a scale of Hilbert spaces Ht , t ∈ R, so that for t ≥ 0, Ht = Domain(C t/2 ), ξ t = C t/2 ξ .
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Set H∞ = ∩t Ht . Note that H∞ is a dense subspace of Ht for every t. We say that an operator T : H∞ → H∞ is of order r ∈ R if it extends by continuity to a bounded operator Ht → Ht−r for every t. In this case we write T ∈ op r . If T ∈ op 0 , we denote by T t the norm of the operator T : Ht → Ht . Recall that T t is a convex function of t. Note that C z ∈ op 2Re z . For F ⊂ Z, denote by pF the projection onto ⊕n∈F L2 (An , h) with respect to the decomposition H = ⊕n∈Z L2 (An , h). Proposition 3.2. We have (i) A ⊂ op0 and pF ∈ op0 for any F ⊂ Z; (ii) for x = apF with a ∈ An and F ⊂ Z finite, the function C z → x(z) = 1 (C z xC −z − q nz x)C 2 takes values in op 0 and has at most linear growth on vertical strips; more precisely, for any finite interval I ⊂ R, there exists λ > 0 such that x(z)t ≤ λ|z| for t ∈ I and Re z ∈ I . Proof. Since pF is a projection commuting with C, it is obvious that pF ∈ op 0 . It is also clear that to prove that A ⊂ op0 it is enough to consider generators of A. Similarly, if (ii) is proved for xi = ai pFi , ai ∈ Ani , i = 1, 2, and it is proved that a1 , a2 ∈ op0 , then x1 x2 = a1 a2 pF1 −n2 pF2 = a1 a2 p(F1 −n2 )∩F2 and 1
(C z x1 x2 C −z − q (n1 +n2 )z x1 x2 )C 2 1 1 1 1 = (C z x1 C −z − q n1 z x1 )C 2 (C − 2 +z x2 C 2 −z ) + q n1 z x1 (C z x2 C −z − q n2 z x2 )C 2 , so (ii) is also true for a1 a2 p(F1 −n2 )∩F2 . Analogously, if (ii) is true for apF with a ∈ An , then it is also true for a ∗ pF +n . It follows that to prove (i) and (ii) it is enough to consider generators of A and arbitrary finite F ⊂ Z. We will only consider the generator α, the proof for γ is similar. Consider the orthonormal basis {ξijs | s ∈ 21 Z+ , i, j = −s, . . . , s} 1
given by normalized matrix coefficients, ξijs = q i ds2 usij ξh , where ds = [2s + 1]q . Then α can be written as the sum of two operators α + and α − such that α + ξijs = αijs+ ξ
s+ 21 i− 21 ,j − 21
and α − ξijs = αijs− ξ
s− 21 i− 21 ,j − 21
.
What we have to know about the numbers αijs+ and αijs− is that they are of modulus ≤ 1 and if |j | ≤ m, then |αijs+ | ≤ λm q s for some λm depending only on m and q (in fact, we 1
1
have αijs+ = q 2 (ds ds+ 1 )− 2 q 2
2s+i+j 2
1
([s − i + 1]q [s − j + 1]q ) 2 , cf [V]). In our basis we
have also Cξijs = [s + 21 ]2q ξijs . It follows immediately that α + , α − ∈ op0 . Then (C z+t α − C −z−t − q z C t α − C −t )Cξijs 2z+2t 2t
1 [s] [s]q 1 2 q α s− ξ s−12 1 = s+ − qz ij 1 1 i− 2 ,j − 2 2 q [s + 2 ]q [s + 2 ]q 2t 2z
1 1 − q 2s 1 − q 2s 1 2 s− s− 2 = q z+t s + − 1 α ξ . 1 ij i− 2 ,j − 21 2 q 1 − q 2s+1 1 − q 2s+1
1 Using the simple estimate |(1 − β)z − 1| = |βz 0 (1 − βτ )z−1 dτ | ≤ β|z|(1 − β0 )Re z−1 for 0 ≤ β ≤ β0 < 1, we conclude that the operator (C z+t α − C −z−t − q z C t α − C −t )C is
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bounded, and moreover, if t and Re z lie in some finite interval then the operator norm can be estimated by some multiple of |z|. Thus the function z → (C z α − C −z − q z α − )C takes values in op0 and has at most linear growth on vertical strips. Then surely the 1 function z → (C z α − C −z − q z α − )C 2 has the same properties. Similarly the function 1 z → (C z α + C −z − q −z α + )C 2 takes values in op 0 and has at most linear growth on vertical strips. Then 1
1
(C z αpF C −z − q z αpF )C 2 = (C z α − C −z − q z α − )C 2 pF 1 +(C z α + C −z − q −z α + )C 2 pF 1 +(q −z − q z )α + pF C 2 . It remains to note that pF is the projection onto the space spanned by the vectors ξijs such that −2j ∈ F . Since |αijs+ | ≤ λF q s if −2j ∈ F for some λF , we conclude that 1
α + pF C 2 ∈ op0 .
It is worth noting that by analyzing the proof of Theorem B.1 in [CM2] the slightly weaker result that x(z) has polynomial growth on the vertical strip can be deduced using only the fact that Cx − q n xC ∈ op1 . This, in turn, follows from the identity ∂f e a − q n a∂f e = ∂f e (a)∂k −2 + ∂f k (a)∂k −1 e + ∂ke (a)∂f k −1 for a ∈ An . Thus a variant of Proposition 3.2 with polynomial growth (which would, in fact, be sufficient for our purposes) can be proved without any knowledge about the Clebsch-Gordan coefficients. q 0 on H+ ⊕ H− . Then for any b ∈ B Corollary 3.3. Consider the operator χ = 0 q −1 there exists an analytic function z → b(z) ∈ B(H+ ⊕ H− ) with at most linear growth on vertical strips such that |D|−2z db = dbχ 2z |D|−2z + b(z)|D|−2z−1 = χ −2z db|D|−2z + b(z)|D|−2z−1 . Proof. Since |D|2 = C and ∂f (b) ∈ A−2 , ∂e (b) ∈ A2 for b ∈ B = A0 , this is an immediate consequence of Propositions 3.1 and 3.2. Next we study the behavior of the heat operators. Consider the direct sum of irreducible corepresentations of (A, ) with spins s ∈ 21 + Z+ . Denote by H˜ the space of this corepresentation, and consider the corresponding representation of Aˆ on H˜ . Note that both H+ = L2 (A1 , h) and H− = L2 (A−1 , h) can be identified with H˜ (since in the notation of the proof of Proposition 3.2 the spaces H+ and H− are spanned by the vectors ξ s 1 and ξ s 1 , respectively). i,− 2
i, 2
Lemma 3.4. On H˜ we have that (i) the function ε → εTr(e−εC ρ) is continuous and bounded on (0, ∞); 2 (ii) lim ε(Tr(e−εC ρ) − q 2 Tr(e−εq C ρ)) = 0; ε→0+ 1 (iii) lim ε Tr(e−εtC ρ)dt = µ = q −1 − q. ε→0+
q2
Using the Karamata theorem (see e.g. [BGV]) it can be shown that the limit lim εTr(e−εC ρ) does not exist.
ε→0+
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2 Proof of Lemma 3.4. If f is a positive function then Tr(f (C)ρ) = ∞ n=1 [2n]q f ([n]q ). Since it is obvious that the function ε → εTr(e−εC ρ) is continuous and vanishes at −2n e−ε[n]2q infinity, to prove (i) it is thus enough to show that the function ε → ∞ n=1 εq is bounded near zero. As [n]2q ≥ (q −2n − 2)µ−2 , for q 2m ≤ ε ≤ q 2(m−1) we get ∞
εq −2n e−ε[n]q ≤ 2
n=1
∞
q −2(n−m+1) e−(q
n=1
∞
−2
≤ q −2 e2µ
−2(n−m) −2q 2m )µ−2
q −2n e−q
−2n µ−2
.
n=−∞
Thus (i) is proved. Since q −2n µ−2 ≥ [n]2q ≥ (q −2n − 2)µ−2 , we see also that lim ε(Tr(e−εC ρ) −
ε→0+
∞
q −2n µ−1 e−εq
−2n µ−2
) = 0.
n=1
Replacing ε by εµ2 , to prove (ii) and (iii) we thus have to show that lim ε(
ε→0+
∞
q −2n e−εq
−2n
− q2
n=1
q −2n e−εq
2 q −2n
)=0
n=1
and
lim ε
ε→0+
Both statements are obvious.
∞
1
q2
dt
∞
q −2n e−εtq
−2n
= 1.
n=1
4. Local Index Formula Let us first briefly recall the proof of the local index formula of Connes and Moscovici [CM2], which is a far-reaching generalization of the local index theorem of GilkeyAtiyah-Bott-Patodi [BGV]. Suppose we are given a p-summable even spectral triple for an algebra B. For each ε > 0 consider the JLO-cocycle corresponding to the Dirac 1 operator ε 2 D, so 2 2 2 ε ψ2n (b0 , . . . , b2n ) = εn dtTr(γ b0 e−εt0 D [D, b1 ]e−εt1 D . . . [D, b2n ]e−εt2n D ), 2n
and note that we always have the following estimate: p ε n− 2 , b2n )| ≤ Cp [D, bi ], Tr((1 + D 2 )− 2 )b0 (2n)! p
ε |ψ2n (b0 , . . .
2n
i=1
where Cp is a universal constant. In the course of proving the local index formula one ε , 0 ≤ 2n ≤ p, of the form provides a finite decomposition of ψ2n (2n) ε ψ2n = αk,l ε −pk (n) (log ε)l + o(1), (4.1) k,l
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where αk,l are 2n-cochains expressed in terms of the residues of certain ζ -functions. Since the pairing with K-theory is independent of ε, one concludes that it is given by (2n) the pairing with the cocycle (α0,0 )0≤2n≤p . The explicit form of this cocycle and the fact that it is cohomologous to (Ch2n (D))n is the main content of the local index formula. 2 To get (4.1) one needs to commute e−εtD through db. For this one first uses the identity 1 2 dz(z)|D|−2z (εt)−z , e−εtD = 2πi Cλ where Cλ = λ + iR, λ > 0. Then one invokes the expression b0 |D|−2z0 db1 |D|−2z1 . . . db2n |D|−2zn p−2n+1 = fk (b0 , . . . , b2n ; z0 , . . . , z2n )|D|−2(z0 +...+z2n )−k
(4.2)
k=0
given by the pseudo-differential calculus for D, where fk are analytic operator-valued functions with at most polynomial growth on vertical strips. Since the last term 2n−1 ε as O(ε 21 ). (k = p − 2n + 1) is of trace-class for Re zi ≥ 2(2n+1) , it contributes to ψ2n On the other hand, assuming that Tr(fk (b0 , . . . , b2n ; z0 , . . . , z2n )|D|−2(z0 +...+z2n )−k ) extends to a nicely behaved meromorphic function (note that a priori it is defined for p−k Re zi ≥ 2(2n+1) ), the contribution of the k th term can be estimated by counting the p−k n − δ < Re zi < 2(2n+1) for some δ > 0. residues of this function in the region 2n+1 We now want to apply this method to our situation. Although the decomposition (4.2) will be different from the one obtained in [CM2], the method itself works perfectly well. ε ) given by So consider the twisted entire cocycle (ψ2n n 1
ε ψ2n (b0 , . . . , b2n ) = Ch2n (ε 2 D)(ρ; b0 , . . . , b2n ).
Using Lemma 1.2 we get ψ0ε (b0 ) = Tr(γ b0 e−εD ρ) = h(b0 )Tr(e−εC ρ) − h(b0 )Tr(e−εC ρ) = 0, 2
where we think of the Casimir C as the operator acting on H˜ , see the end of Sect. 3. Consider ψ2ε . Fix b0 , b1 , b2 ∈ B and set ζ (z0 , z1 , z2 ) = Tr(γ b0 |D|−2z0 db1 |D|−2z1 db2 |D|−2z2 ρ). This function is well-defined and analytic in the region Re (z0 + z1 + z2 ) > 1. By Corollary 3.3, ζ (b0 , b1 , b2 ) = Tr(γ b0 db1 db2 χ 2z1 |D|−2(z0 +z1 +z2 ) ρ) + ζ˜ (z0 , z1 , z2 ), where ζ˜ (z0 , z1 , z2 ) = Tr(γ b0 db1 χ 2z0 b2 (z0 + z1 )|D|−2(z0 +z1 +z2 )−1 ρ) +Tr(γ b0 b1 (z0 )|D|−2(z0 +z1 )−1 db2 |D|−2z2 ρ).
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The function ζ˜ is holomorphic in the region Re (z0 + z1 + z2 ) > 21 . Moreover, if we fix λ, λ0 > 16 , λ > λ0 , then ζ˜ (z0 , z1 , z2 )(1 + |z0 |)−1 (1 + |z1 |)−1 is bounded in the region λ0 < Re zi < λ. Fix λ > 13 and 16 < λ0 < 13 . Then ε ψ2ε (b0 , b1 , b2 ) = dt dz(z0 )(z1 )(z2 ) (2πi)3 2 Cλ3 ×(εt0 )−z0 (εt1 )−z1 (εt2 )−z2 ζ (z0 , z1 , z2 ), where Cλ = λ + iR. If we consider the same expression for ζ˜ , we can replace integration over Cλ3 by integration over Cλ30 , since the integrand is holomorphic and vanishes at infinity. It follows that such an expression is O(ε 1−3λ0 ). Thus, as ε → 0, ε ε ψ2 (b0 , b1 , b2 ) = dt dz(z0 )(z1 )(z2 )(εt0 )−z0 (εt1 )−z1 (εt2 )−z2 (2πi)3 2 Cλ3 ×Tr(γ b0 db1 db2 χ 2z1 |D|−2(z0 +z1 +z2 ) ρ) + o(1) −2 2 dtTr(γ b0 db1 db2 e−ε(t0 +χ t1 +t2 )D ρ) + o(1) =ε 2 −2 2 =ε dtTr(γ b0 db1 db2 e−ε(1+(χ −1)t1 )D ρ) + o(1) 2 1
=ε
(1 − t)Tr(γ b0 db1 db2 e−ε(1+(χ
−2 −1)t)D 2
ρ)dt + o(1).
0
By Proposition 3.1(ii), we have
b0 ∂e (b1 )∂f (b2 ) 0 b0 db1 db2 = . 0 b0 ∂f (b1 )∂e (b2 )
Hence by Lemma 1.2 we get
ψ2ε (b0 , b1 , b2 ) = h(b0 ∂e (b1 )∂f (b2 ))ε
1
(1 − t)Tr(e−ε(1+(q
+o(1).
0
ρ)dt
0
−h(b0 ∂f (b1 )∂e (b2 ))ε We have 1 2 ε (1 − t)Tr(e−ε(1+(q −1)t)C ρ)dt = −
−2 −1)t)C
1
(1 − t)Tr(e−ε(1+(q
0
2 −1)t)C
ρ)dt (4.3)
ε (1 − q 2 )2
1 q2
(q 2 − t)Tr(e−εtC ρ)dt. (4.4)
On the other hand, 1 −2 ε (1 − t)Tr(e−ε(1+(q −1)t)C ρ)dt 0
=
ε (1 − q 2 )2
εq 2 = (1 − q 2 )2
1 q2 1 q2
(1 − t)Tr(e−εq
−2 tC
ρ)dt
(1 − t)Tr(e−εtC ρ)dt + o(1)
(4.5)
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by Lemma 3.4(ii). By Lemma 3.4(iii), ε (4.3-4.5) together we get ψ2ε
1 q2
Tr(e−εtC ρ)dt → q −1 (1−q 2 ). Thus putting
q ε = (τ1 + τ2 ) − (q 2 τ1 + τ2 ) 2 1−q (1 − q 2 )2
1 q2
Tr(e−εtC ρ)tdt + o(1).
ε with n > 1. Combining (1.5) with Lemmas 10.8 and 10.11 in Finally, consider ψ2n [GBVF] yields the following standard estimate
p ε n− 2 , b2n )| ≤ Cp max {[D, bi ρ t ]}, Tr(C − 2 ρ)b0 0≤t≤1 (2n)! p
ε (b0 , . . . |ψ2n
2n
i=1
where p is any number larger than 2. Thus we get the first part of the following theorem, which is the main result of the paper. ε ) in the twisted entire cyclic cohoTheorem 4.1. For ε > 0, consider the cocycle (ψ2n n mology of B given by 1
ε (b0 , . . . , b2n ) = Ch2n (ε 2 D)(ρ; b0 , . . . , b2n ). ψ2n
Then ψ0ε = 0; ψ2ε =
q ε (τ1 + τ2 ) − (q 2 τ1 + τ2 ) 2 1−q (1 − q 2 )2 1 × Tr(e−εtC ρ)tdt + o(1) as ε → 0;
ε ψ2n ≤ Cδ
q2 n−1−δ ε
(2n)!
for any δ > 0 and n > 1,
where the norm of a multi-linear form on B is defined using the norm max {bρ t + [D, bρ t ]}
0≤t≤1
SU (2)
on B. In particular, the map q−IndD on K0 q (C(Sq2 )) is given by the pairing with the twisted cyclic cocycle −q −1 τ , where τ (b0 , b1 , b2 ) = h(b0 (q 2 ∂e (b1 )∂f (b2 ) − ∂f (b1 ) ∂e (b2 ))). ε ) with K-theory does not depend on ε. Using the Karamata Proof. The pairing of (ψ2n n 1 theorem it can be shown that the limit limε→0+ ε q 2 Tr(e−εtC ρ)tdt does not exist. It follows that the cocycle q 2 τ1 + τ2 pairs trivially with K-theory. Thanks to Proposition 2.2(iii) we know even more, q 2 τ1 + τ2 is a coboundary. We also conclude that q 0 the pairing is given by the cocycle 1−q 2 (τ1 + τ2 ) ∈ Cσ (B), which is cohomologous to 1 2 (B) → C 0 (B), we see that τ . Recalling definition (1.3) of the map Cσ,λ the cocycle 2q σ −1 the pairing is defined by the twisted cyclic cocycle −q τ .
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We end the paper with an actual computation of indices. Observe first that the spaces An can be considered as equivariant Hilbert B-modules, and thus they define elements ˆ ˆ of K0A (B) which we denote by [An ]. As in the classical case, the group K0A (B) is a free abelian group with generators [An ], see [NT]. Consider now the module A1 and note that A1 = αB + γ B. The map T : H 1 ⊗ B → A1 given by 2
T (ξ− 1 ⊗ b) = qγ b and T (ξ 1 ⊗ b) = −αb, 2
2
where H 1 is the space of the spin 21 corepresentation of (A, ), see Sect. 2, is an equi2 variant partial isometry. Thus A1 is isomorphic to the equivariant Hilbert B-module p(H 1 ⊗ B) with projection 2
q 2 γ ∗ γ −αγ ∗ . p=T T = −γ α ∗ α ∗ α ∗
The explicit form of a projection corresponding to An for arbitrary n can be found ˆ and let in [HM]. Let Tr s be the trace on Aˆ defined by the spin s representation of A, φs be the corresponding q-trace, φs = Tr s (·ρ). By definition we have, for any twisted cyclic cocycle ϕ ∈ Cσ2 (B), that φ 1 (mi0 j0 mi1 j1 mi2 j2 )ϕ(pi0 j0 , pi1 j1 , pi2 j2 ) [ϕ], [p] = i0 ,i1 ,i2 j0 ,j1 ,j2
=
2
q −2i0 ϕ(pi0 i1 , pi1 i2 , pi2 i0 ).
i0 ,i1 ,i2
Using the formula h((γ ∗ γ )n ) = (1 − q 2 )(1 − q 2(n+1) )−1 for the Haar state, a lengthy but straightforward computation yields q−IndD ([A1 ]) = q−IndF ([A1 ]) = [−q −1 τ ], [p] = −1. This is enough to conclude that the equivariant index q−IndF ([A1 ]) equals −Tr 0 . To see this we shall use a continuity argument for q ∈ (0, 1). Write α(q), γ (q), and so on, to distinguish operators for different q. The spaces L2 (C(SUq (2)), h) can be identified for all q. We also identify the spaces H+ ⊕ H− of L2 -spinors. Note that F = D(q)|D(q)|−1 is independent of q (in the notation of the proof of Proposition 3.2 we have F ξ s 1 = ξ s 1 ). i,− 2
i, 2
The functions q → α(q) and q → γ (q) are norm-continuous as can easily be verified by looking at the Clebsch-Gordan coefficients. It follows that our quantum Bott projecˆ tions p(q) ∈ B(H 1 ⊕ (H+ ⊕ H− )) depend continuously on q. Let Is (q) ∈ A(q) be the 2 support of the spin s representation. Considered as operators on H 1 ⊕ (H+ ⊕ H− ) the 2 projections Is (q) depend continuously on q (and are, in fact, finite-rank operators). As the functions ms (q) = (2s + 1)−1 Ind(p(q)− Is (q)(1 ⊗ F )p(q)+ Is (q)) are continuous and integer-valued, they are constant. We have by definition ms Tr s . IndF ([p(q)]) = s
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Since
−1 = IndF ([p(q)])(ρ) =
ms [2s + 1]q ,
s
and the functions q → [n]q , n ∈ N, are linearly independent on any infinite set, we conclude that m0 = −1 and ms = 0 for s > 0. Thus IndF ([p]) = −Tr 0 = −ˆε. This, in turn, is sufficient in order to find the non-equivariant Chern character. Proposition 4.2. The image of the non-equivariant Chern character of our Fredholm module in H P 0 (B) coincides with the class of the cyclic 0-cocycle τ given by (1 − q 2n )−1 for n = m > 0, τ (α n−m γ m γ ∗ n ) = 0 otherwise, where we used the convention α k = (α ∗ )−k for k < 0. In particular, IndF ([An ])(1) = −n. Proof. The cocycle τ was found in [MNW2], and is one of the two generators of H P 0 (B) ∼ = C2 . Since the class of a cocycle in H P 0 (B) is completely determined by its pairing with [1] and [p], we conclude that the Chern character is cohomologous to τ . The equality [τ ], [An ] = −n was established in [H]. The fact that the non-equivariant Chern character is cohomologous to a 0-cocycle is natural as our spectral triple is ε-summable for any ε > 0. On the other hand, the spectral triple is (2 + ε, ρ)-summable in the sense of [NT], so twisted cyclic cohomology does not see the dimension drop and captures the volume form. We finally remark that IndF ([An ]) = −sign(n)Tr |n|−1 for n = 0. To prove this it 2
suffices to check that [−q −1 τ ], [An ] = −[n]q . Another possibility is to use the classical theory. To this end one just has to show that there are projections pn (q) representing [An ] with the property that Is (q)pn (q) depend continuously on q ∈ (0, 1] (the projections pn (q) themselves can be discontinuous at q = 1). In the classical case the operator 0 ∂e pn (1⊗D)pn is homotopic to the operator Dn = which acts on the Hilbert space ∂f 0 L2 (An+1 , h) ⊕ L2 (An−1 , h). Both operators are differential operators of order 1 with the same principal symbol, and the index of ∂f : L2 (An+1 , h) → L2 (An−1 , h) is given by the Borel-Weil-Bott theorem and can also easily be found by direct computations. Acknowledgement. The preparation of this paper was finished during the authors’ stay at Institute Mittag-Leffler in September 2003. They would like to express their gratitude to the staff at the institute and to the organizers of the year in "Noncommutative Geometry".
References [AK] [BGV] [CP] [C1] [C2]
Akbarpour, R., Khalkhali, M.: Equivariant cyclic cohomology of Hopf module algebras. Preprint, http://arxiv.org/abs/math.KT/0009236, To appear in K-Theory Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften, 298. Berlin: Springer-Verlag, 1992, pp. viii+369 pp. Chakraborty, P.S., Pal, A.: Equivariant spectral triples on the quantum SU(2) group. K-Theory 28(2), 107–126 (2003) Connes, A.: Noncommutative geometry. San Diego, CA: Academic Press, Inc. 1994 Connes, A.: Cyclic cohomology, quantum group symmetries and the local index gormula for SUq (2). Preprint, http://arxiv.org/abs/math.QA/0209142, 2002
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Connes, A., Moscovici, H.: Transgression and the Chern character of finite-dimensional Kcycles. Commun. Math. Phys. 155, 103–122 (1993) [CM2] Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5, 174–243 (1995) [DS] Dabrowski, L., Sitarz A.: Dirac operator on the standard Podle´s quantum sphere. Preprint, http://arxiv.org/abs/math.QA/0209048, 2002 [F] Friedrich, T.: Dirac operators in Riemannian geometry. Graduate Studies in Mathematics. 25, Providence, RI: AMS, 2000, pp. xvi+195 pp. [G] Goswami, D.: Twisted entire cyclic cohomology, J-L-O cocycles and equivariant spectral triples. To appear in Rev. Math. Phys. 2004 [GBVF] Gracia-Bondia, J.M., V´arilly, J.C., Figueroa, H.: Elements of noncommutative geometry. Boston, MA: Birkh¨auser Boston, Inc. 2001, xviii+685 pp [H] Hajac, P.: Bundles over quantum sphere and noncommutative index theorem. K-Theory 21, 141–150 (2000) [HM] Hajac, P., Majid, S.: Projective module description of the q-monopole. Commun. Math. Phys. 206, 247–264 (1999) [KL] Klimek, S., Lesniewski, A.: Chern character in equivariant entire cyclic cohomology. K-Theory 4, 219–226 (1991) [KS] Klimyk, A.: Schm¨udgen K. Quantum groups and their representations. Texts and Monographs in Physics. Berlin: Springer-Verlag, 1997, xx+552 pp [Kr] Kr¨ahmer, U., Dirac operators on quantum flag manifolds. Preprint, http://arxiv.org/abs/math.QA/0305071, 2003 [KMT] Kustermans, J., Murphy, G.J., Tuset L.: Differential calculi over quantum groups and twisted cyclic cocycles. J. Geom. Phys. 44, 570–594 (2003) [M] Majid, S.: Noncommutative Riemannian and Spin Geometry of the Standard q-Sphere. Preprint, http://arxiv.org/abs/math.QA/0307351, 2003 [MNW1] Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum SU(2). I. An algebraic viewpoint. K-Theory 4, 157–180 (1990) [MNW2] Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum two sphere of Podle´s. I. An algebraic viewpoint. K-Theory 5, 151–175 (1991) [NT] Neshveyev, S., Tuset, L.: Hopf algebra equivariant cyclic cohomology, K-theory and index formulas. K-Theory, 31, 323–343 (2004) [O] Owczarek, R.: Dirac operator on the Podle´s sphere. Int. J. Theor. Phys. 40, 163–170 (2001) [PS] Pinzul, A., Stern, A.: Dirac operator on the quantum sphere. Phys. Lett. B 512, 217–224 (2001) [P1] Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987) [P2] Podle´s, P.: The classification of differential structures on quantum 2-spheres. Commun. Math. Phys. 150, 167–179 (1992) [S] Schm¨udgen, K.: Commutator representations of differential calculi on the quantum group SUq (2). J. Geom. Phys. 31, 241–264 (1999) [SW] Schm¨udgen, K., Wagner, E.: Dirac operator and a twisted cyclic cocycle on the standard Podle´s quantum sphere. Preprint, http://arxiv.org/abs/math.QA/0305051, 2003 [V] Vaksman, L.L.: q-analogues of Clebsch-Gordan coefficients, and the algebra of functions on the quantum group SU(2). (Russian) Dokl. Akad. Nauk SSSR 306, 269–271 (1989); translation in Soviet Math. Dokl. 39, 467–470 (1989) [W] Woronowicz, S.L.: Twisted SU (2) group. An example of a noncommutative differential calculus. Publ. Res. Inst. Math. Sci. 23, 117–181 (1987) Communicated by A. Connes
Commun. Math. Phys. 254, 343–359 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1195-3
Communications in
Mathematical Physics
Conjugacies for Tiling Dynamical Systems Charles Holton , Charles Radin , Lorenzo Sadun Department of Mathematics, University of Texas, Austin, TX 78712, USA Received: 29 January 2004 / Accepted: 7 April 2004 Published online: 5 November 2004 – © Springer-Verlag 2004
Abstract: We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being of finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes. 1. Notation and Main Results We begin with a definition of tiling dynamical systems, in sufficient generality for this work. Let A be a nonempty finite collection of compact connected sets in the Euclidean space Ed , sets with dense interior and boundary of zero volume. Let X(A) be the set of all tilings of Ed by congruent copies, which we call tiles, of the elements of the “alphabet” A. We assume X(A) is nonempty, which is automatic for the special class of substitution tiling systems on which we will concentrate below. We label the “types” of tiles by the elements of A. We endow X(A) with the metric 1 m[x, y] ≡ sup mH [Bn ∩ ∂x, Bn ∩ ∂y], n≥1 n
(1)
where Bn denotes the open ball of radius n centered at the origin O of Ed , and ∂x the union of the boundaries of all tiles in x. (A ball centered at a is denoted Bn (a).) The Hausdorff metric mH is defined as follows. Given two compact subsets P and Q of Ed , mH [P , Q] = max{m(P ˜ , Q), m(Q, ˜ P )}, where m(P ˜ , Q) = sup inf p − q, p∈P q∈Q
with w denoting the usual Euclidean norm of w.
Research supported in part by NSF Vigre Grant DMS-0091946 Research supported in part by NSF Grant DMS-0071643 and Texas ARP Grant 003658-158
(2)
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Under the metric m two tilings are close if they agree, up to a small Euclidean motion, on a large ball centered at the origin. The converse is also true for tiling systems with finite local complexity (as defined below): closeness implies agreement, up to small Euclidean motion, on a large ball centered at the origin [see RaS1]. Although the metric m depends on the location of the origin, the topology induced by m is Euclidean invariant. A sequence of tilings converges in the metric m if and only if its restriction to every compact subset of Ed converges in mH . It is not hard to show [RW] that X(A) is compact and that the natural action of the connected Euclidean group GE on X(A), (g, x) ∈ GE × X(A) → g[x] ∈ X(A), is continuous. To include certain examples it is useful to generalize the above setup, to use what is sometimes called “colored tiles”. To make the generalization we assign a “color” from some finite set to each element of A, represented on each tile by a “color marking”, a line segment in the interior of the tile, of different length for different colors. We then redefine ∂x as the union of the tile boundaries and color markings in the tiling x. Definition 1. A tiling dynamical system is the action of GE on a closed, GE -invariant subset of X(A). We emphasize the close connection between such dynamical systems and subshifts. A subshift with Zd -action is the natural action of Zd on a compact, Zd -invariant subset d X of B Z , for some nonempty finite set B. If we associate with each element of B a “colored” unit cube in Ed , the face-to-face tilings of Ed by those arrays of such cubes corresponding to the subshift X gives a tiling dynamical system which is basically the suspension of the subshift X (but with rotations of the entire tiling also permitted). A significant difference between subshifts and tiling dynamical systems is that for (nontrivial) subshifts the group acts on a Cantor set, while the space is typically connected for interesting tiling systems. In fact, the spaces for different tiling systems need not be homeomorphic. A major objective in dynamics is the classification of interesting subclasses up to topological conjugacy. For the class of subshifts a central theorem, due to Curtis, Lyndon and Hedlund, shows that a topological conjugacy can be represented by a sliding block code (see [LM]). For tiling dynamical systems there is a natural analogue of such a representation for which we use the same term. (Such maps are called “local” in [P] and are closely related to mutual local derivability [BSJ].) Definition 2. A topological conjugacy ψ : XA → XA between tiling systems is a sliding block code if for every n > 0 there is n > 0 such that for every x, y ∈ XA such that Bn ∩ ∂x = Bn ∩ ∂y we have Bn ∩ ∂(ψx) = Bn ∩ ∂(ψy). Our first result is: Theorem 1. Within the subclass of substitution tiling systems with invertible substitution, every topological conjugacy is a sliding block code. Before defining the subclass of “substitution” tiling systems in general we present some relevant examples. A “Penrose” tiling of the plane, Fig. 1, can be made as follows. Consider the 4 (colored) tiles of Fig. 2. Divide each tile (also called a “tile of level 0”) into √ 2 or 3 pieces as in Fig. 2 and rescale by a linear factor of the golden mean τ = (1 + 5)/2 so that each piece is the same size as the original. This yields 4 collections of tiles that we call “tiles of level 1”. Subdividing each of these tiles and rescaling gives 4 collections of tiles that we call tiles of level 2. Repeating the process n times gives tiles of level n. A Penrose
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Fig. 1. A Penrose tiling
tiling is a tiling of the plane with the property that every finite subcollection of tiles is congruent to a subset of a tile of some level. A Penrose tiling has only 4 types of tiles, each appearing in 10 different orientations.
1
3 1
3
4
4
2 2 Fig. 2. The Penrose substitution
2 3
4
1 4
3
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√ A “pinwheel” tiling of the plane, Fig. 3, uses two basic tiles: a 1-2- 5 right triangle and its mirror image, as shown in Fig. 4 with their substitution rule [R1]. Notice that at the center of each tile of level 1 there is a tile of level 0 similar to the level 1 tile but rotated by an angle α = tan−1 (1/2). Thus the center tile of a tile of level n is rotated by nα relative to the tile. Since α is an irrational multiple of π, we see, using the fact that within a tile of level 2 there is a tile of level 0 similar and parallel to the level 2 tile, that this rotation never ends, and each tiling contains tiles in infinitely many distinct orientations. More generally, for any integers m < n we consider the “(m, n)-pinwheel” √ tilings defined (for m = 3, n = 4) by the substitution of Fig. 5, whose tiles are m-n- m2 + n2 right triangles. Like the ordinary pinwheel, such variant pinwheel tilings also necessarily have tiles in infinitely many distinct orientations. It is easy to construct explicit examples of Penrose and pinwheel tilings. Pick a tile to include the origin of the plane. Embed this tile in a tile of level 1 (there are several ways to do this). Embed that tile of level 1 in a tile of level 2, embed that in a tile of
Fig. 3. A pinwheel tiling
Fig. 4. The pinwheel substitution
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Fig. 5. The substitution for (3,4)-pinwheel tilings
level 3, and so on. The union of these tiles of all levels will cover an infinite region, typically—though not necessarily—the entire plane. In order to generalize from the above examples we need some further notation. A “patch” is a (finite or infinite) subset of a tiling x ∈ X(A); the set of all finite patches for A will be denoted WA . A “substitution function” φ is a map from WA to itself defined by a decomposition of each tile type, stretched linearly about the origin by a factor λφ > 1, into congruent copies of the original tiles. (Recall the Penrose and pinwheel examples.) We assume: i) For each k > 0 there are only a finite number of possible patches, up to Euclidean motion, obtained by taking a ball of radius k around any point inside a tile T of level n, φ n (T ), where T and n are arbitrary. (This is usually called “finite local complexity”.) ii) For each tile T ∈ A, φ(T ) contains at least one tile of each type. (This is usually called “primitivity”.) iii) For every tile T ∈ A there is nT ≥ 1 such that φ nT (T ) contains a tile of the same type as T and parallel to it. Condition i) is highly significant, the remaining conditions much less so. In interesting cases condition ii) can usually be obtained by replacing the substitution by a power of itself; note that this does not affect the tiling dynamical system at all. Condition iii) is related to the existence of a fixed point of the substitution; we know of no interesting examples of systems not satisfying this condition. Definition 3. For a given alphabet A of (possibly colored) tiles, and a substitution function φ, the “substitution tiling system” is the compact subspace Xφ ⊂ X(A), invariant under GE , of those tilings x such that every finite subpatch of x is congruent to a subpatch of φ n (T ) for some n > 0 and T ∈ A. The map φ extends naturally to a continuous map (again denoted φ) from Xφ into itself. There are two natural relaxations of this definition of substitution tiling systems. For any tiling system X and any positive constant λ, let λX denote the system of tilings obtained by rescaling each tiling in X by λ. If Xφ is a substitution tiling system, then Xφ and λφ Xφ are topologically conjugate, via a sliding block code that associates tiles (of level 0) of tilings in λφ Xφ with tiles of level 1 of tilings in Xφ . Definition 4. If a tiling system X has the property that, for some λ > 1, λX and X are topologically conjugate via a sliding block code, then X is a “pseudo-substitution tiling
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system”, and the map X → X obtained by first rescaling by λ and then applying the conjugacy is called a “pseudo-substitution”. If X and λX are topologically conjugate (not necessarily via a sliding block code), then X is a “quasi-substitution tiling system”. In the literature, pseudo-substitutions are sometimes called “improper substitutions” or “substitutions with amalgamation”, and their fixed points are called “pseudo-selfsimilar tilings”. In 2 dimensions, and with some additional assumptions, the categories of substitution tiling systems and of pseudo-substitution tiling systems are essentially identical, thanks to a construction of Priebe and Solomyak [PS] that converts a pseudoself-similar tiling into a self-similar tiling. It is generally believed that this construction can be generalized to higher dimensions, dropping some of the restrictive assumptions, in which case properties of substitution tiling systems, such as Theorem 1, can be expected to apply to pseudo-substitution tiling systems. However, the following example (see also [P]) shows that the conclusions of Theorem 1 do not apply to quasi-substitution tiling systems. Following [RS2], we consider suspensions of the 1-dimensional Fibonacci substitution subshift. That subshift is defined by the alphabet B = {0, 1} and the substitution of 0 by 1 and of 1 by the word 0 1. One can make a family of suspensions of this subshift by replacing 0 and 1 by marked closed intervals of any positive lengths. One gets a substitution tiling system if 0 is associated with a segment T0 of length √|T0 | = 1 and 1 is associated with a segment T1 of length the golden mean τ = 1 + 5/2, |T1 | = τ . When |T1 |/|T0 | = τ , the resulting tiling system is merely a quasi-substitution tiling system [CS]. It was proven in [RS2] that two different Fibonacci tiling systems, defined by T0 , T1 and T0 , T1 , are topologically conjugate if |T0 | + τ |T1 | = |T0 | + τ |T1 |, but that such a conjugacy cannot be a sliding block code. Definitions 3 and 4 yield spaces of tilings in which the tiles appear in all orientations, although tiles appear in at most countably many different orientations within any fixed tiling. Any Penrose tiling, for example, has tiles in only 10 distinct orientations, but the space of all Penrose tilings contains all rotated versions of any tiling. For each tiling x and each r > 0, consider the set of Euclidean motions g for which x and gx agree exactly on a ball of radius r around the origin. The subgroup of SO(d) generated by the rotational parts of the g’s is denoted GRO (r, x) and is called the relative orientation group of x. In [RS1] it was shown that GRO (r, x) is independent of r (and we henceforth write it as GRO (x)), and that the groups for different tilings x ∈ Xφ are related by inner automorphism of SO(d). There are no inner automorphisms of SO(2), so in 2 dimensions the group is exactly the same for all tilings x ∈ Xφ . For the Penrose tiling, GRO = Z10 is the set of rotations by multiples of 2π/10. For the pinwheel tiling, GRO is generated by rotations of π/2 and 2 tan−1 (1/2). For the (m, n)-pinwheel tiling, the group is generated by rotations by π/2 and 2 tan−1 (m/n). From Theorem 1 we deduce that the relative orientation group is a conjugacy invariant: Corollary 1. If ψ : X1 → X2 is a topological conjugacy between substitution tiling systems with invertible substitution, and if x ∈ X1 , then GRO (ψ(x)) = GRO (x). As an application, √ consider the (1,2)-pinwheel and (3,4)-pinwheel tilings. These have stretching factors 5 and 5, and so cannot be distinguished by the homeomorphism invariant of [ORS]. Moreover, their relative orientation groups are each isomorphic to Z4 ⊕ Z as abstract groups. However, these groups are different as subsets of SO(2) (one is an index-2 subgroup of the other), so the two tiling systems cannot be topologically conjugate.
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For each tiling x ∈ Xφ , consider the closure of the orbit of x under translations. Let Grel (x) be the maximal subgroup of GE that maps this orbit closure to itself. It is not hard to see that Grel (x) is the smallest subgroup of GE that contains the closure of GRO (x) in SO(d) and contains all translations. If the translation orbit of x is dense in Xφ , as with the pinwheel and its variants, then Grel (x) = GE . We should add that for tiling systems in which tiles only appear in finitely many orientations in any tiling, it can be convenient to consider the dynamical system of translations on the orbit closure of such a tiling. Although our convention is to allow all of the Euclidean group to act on tilings it is easy to obtain corresponding information about these more limited dynamical systems from our results. Our second result concerns what is often called the “recognizability” or “unique composition property” of the substitution: Theorem 2. If for a substitution function φ there is some tiling x in Xφ not fixed by any translation, for which the orbit under translation is dense in Xφ , then the extension of φ to Xφ is a homeomorphism. Theorem 2 complements a result of Solomyak [S], which dealt with tiling systems that had finite relative orientation groups. Solomyak’s result was itself a generalization of Moss´e’s work on 1-dimensional subshifts [Mos]. For tilings in 3 or more dimensions there is an additional case, where the relative orientation group is infinite but not dense in SO(d). We have constructed a pseudo-substitution tiling system in 3 dimensions, with GRO a dense subgroup of SO(2), for which the pseudo-substitution is not a homeomorphism. However, the recognizability of true substitutions in 3 or more dimensions remains open. Our third result concerns “finite type”. If X is a tiling space and r > 0, let Xr be the set of tilings for which every patch of radius r also appears in some tiling in X. If r1 > r2 , then Xr1 ⊆ Xr2 , and it is easy to show that ∩r Xr = X. If X = Xr for some finite r, then we say that X is of finite type. Roughly speaking, this means that the patterns in X are defined by local conditions, whose range is at most 2r. For subshifts, it is well known that being of finite type is an invariant of topological conjugacy. (See [RS2] for an explicit proof of this folk theorem.) We extend this to tiling systems: Theorem 3. Let X, Y be topologically conjugate tiling systems, each of finite local complexity. X is of finite type if and only if Y is of finite type. 2. Proofs of Theorems 1 and 2, and Some Related Results We begin with the proof of Theorem 2. We abbreviate Xφ by X and call a patch admissible if there is a tiling x ∈ X containing it. We assume φ has a fixed point in X, not fixed by any translation, whose orbit under translations is dense in X. (The existence of a periodic point for φ follows from iii), and we are free to replace φ with a higher power.) Let H : GE → GE be defined by φ(gx) = H (g)φ(x). The following four lemmas are proved with standard arguments, as sketched below. Lemma 1. The extension φ : X → X is surjective. Sketch of proof. Since X is compact, φ(X) is a closed subset of X. To see that it is dense, note that any admissible patch is a subset of some tile of level n. Thus, for any tiling φ(y) x ∈ X, and any r > 0, Brx = Br for some tiling y.
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Lemma 2. There is a constant C > 0 such that for every r > 0 and for every pair of admissible patches P , P with supp(P ) ⊂ Br and BCr ⊂ supp(P ) there exists g ∈ GE such that gP ⊂ P . Sketch of proof. By finite local complexity and primitivity, there exists N such that for x. every x ∈ X and for every prototile T , the patch φ N (T ) contains a congruent copy of Bm N+1 /m times the maximum diameter of a tile. Suppose r, P , P Take C greater than λ are as in the statement of the lemma. Let n be the least integer such that rλ−n ≤ m. Then P contains a tile of level N + n, and every tile of level N + n contains a congruent copy of P . k Lemma 3. If b1 , . . . , bk ∈ Ed and t = i=1 ai bi with ai ∈ N, then there exist t0 , t1 , . . . , t such that t0 = 0, t = t, and for each j = 1, . . . , , tj − tj −1 ∈ {±bi : i = 1, . . . , k},
(3)
and tj lies within (k/2) max1≤i≤k bi of the straight-line path from O to t. Sketch of proof. Each point along the straight-line path from O to t is a linear combination of the bi ’s with real coefficients. Round these coefficients to the nearest integer to get the sequence of tj ’s. Lemma 4. If Grel (y) = GE there is a constant D such that if P , P are admissible patches in y with supp(P ) ⊂ Br and BDr ⊂ supp(P ) then there exist α1 , . . . , αn ∈ SO(d) and t1 , . . . , tn ∈ Rd such that αi P + ti ⊂ P , i = 1, 2, . . . , n, and such that no proper subspace of Ed is invariant under all the αi . Sketch of proof. This is similar to Lemma 2. One can prove it first when P is a tile and then extend the result to larger patches by inverting the substitution. Let M = max{diam(T ) : T ∈ A}. Let us say that a patch P has period g ∈ GE if P ∪ gP is a patch, i.e., if P and gP agree where their supports overlap (we do not require that they actually overlap). Alternatively, P has period g ∈ GE if and only if whenever T ∈ P is such that gT ◦ intersects supp(P ) we also have gT ∈ P . Of course any subpatch of a patch of period g has period g. Lemma 5. If {b1 , . . . , bk−1 } ⊂ Ed is a basis for a lattice L and P is a patch having all periods in L and additional translational period bk such that Br ⊂ supp(P ), where r > (d + 1) max bi + 4M, 1≤i≤k
(4)
P . then b1 , . . . , bk is a lattice of periods for Br/2 P and T ◦ +t intersects supp(B P ), where t = a b +· · ·+a b Proof. Suppose T ∈ Br/2 1 1 k k r/2 with each ai ∈ N. Let b ∈ T ∩ Br/2 . Then b + t ∈ Br/2+2M . Let t0 , . . . , t be as in Lemma 3. The straight-line path from b to b + t lies in Br/2+2M so each b + tj is in Br . Thus, for each j we have (T ◦ + tj ) ∩ P = ∅ and by finite induction T + tj ∈ P . It P . follows that T + t ∈ Br/2
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Lemma 6. If k < d and {b1 , . . . , bk } is a basis for a lattice L and P is an admissible patch having all periods in L such that Br ⊂ supp(P ), where r > D((d + 1) max bi + 4M),
(5)
1≤i≤k
then there exists bk+1 ∈ Ed \ span(L) with bk+1 ≤ max1≤i≤k bi and such that P . b1 , . . . , bk+1 is a lattice of periods for Br/3D P Proof. Let P = Br/D−M . By Lemma 4, there exist α ∈ SO(d) and t ∈ Rd such that αP + t ⊂ P and α span(L) = span(L). It follows that P has all periods in L as well as all periods in α −1 L. Let bk+1 ∈ α −1 {b1 , . . . , bk } \ span(L). By Lemma 5, P b1 , . . . , bk+1 is a lattice of periods for Br/2D−M/2 . Since r/2D − M/2 > r/3D, the proof is complete.
We next show that a large patch cannot have periods which are small relative to the size of the patch. Proposition 1. There is a constant K > 0 such that if P is an admissible patch whose support contains a ball of radius r then every non-identity period g of P satisfies gb − b > Kr for some b ∈ supp(P ). Proof. It suffices to prove there is a K which satisfies the conclusion for all sufficiently large r. Recall the notation of m as the inner radius. No tiling in X has a translational period of magnitude less than m. Let K > 0 be less than each of the following: a) 4Cλ(λ + 1)(3D)d−1 (d + 4M/m)−1 , b) inf{α − I operator : I = α ∈ SO(d) and α fixes some element of X}, 1 c) 4C . Suppose there exist r > 4MC and a patch P having a non-identity period g ∈ GE with gb − b ≤ Kr for all b ∈ supp(P ), and such that Br ⊂ supp(P ). Let x ∈ X x be a fixed point for φ, let P = Br/C−M , and let h ∈ GE be such that hP ⊂ P . Then −1 h gh is a period for P and h−1 ghb − b ≤ Kr, for all b ∈ P .
(6)
Since φ(P ) ⊃ P and H (h−1 gh) is a period for φ(P ), H (h−1 gh)(h−1 gh)−1 is a period for P ∩ (h−1 gh)−1 P . We have x Br( ⊂ P ∩ (h−1 gh)−1 P 1 −K)−M
(7)
C
and x ). H (h−1 gh)(h−1 gh)−1 b − b ≤ (λ + 1)Kr, for all b ∈ supp(Br( 1 −K)−M
(8)
C
Now H (h−1 gh)(h−1 gh)−1 is a translation, say by b1 ∈ Rd , and if b1 = 0 then h−1 gh ∈ SO(d) and h−1 ghb = b. This last is impossible due to our choice of K, hence 0 < x1 ≤ (λ + 1)Kr.
(9)
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An application of Lemma 5 followed by d −1 applications of Lemma 6 (we will see in a moment that r is large enough for this) yields a d-dimensional lattice L = b1 , . . . , bd of periods for P = Brx , where bi ≤ (λ + 1)Kr,
i = 1, . . . , d,
(10)
and 1 1 r( − K) − M 2(3D)d−1 C r ≥ 4C(3D)d−1 ≥ (d + 4M/m)λ(λ + 1)Kr ≥ (d + 4M/m)λ max xi .
r =
(11)
1≤i≤d
Thus r > λ( di=1 xi + M), so the fundamental domain F = {t1 b1 + · · · + td bd : ti ∈ [0, 1] } for L is such that φ(F x ) ⊂ Brx . This implies that every tile in φ(F x ) is a translate by an element of L of a tile in F x , and it follows that all tiles in x are translates of tiles in F x , contradicting that x should have tiles in infinitely many different orientations. For a tile T and n ≥ 0 let Pn (T ) be the set of admissible patches P for which φ n (T ) ⊂ φ n (P ) and φ n (T ) ⊂ φ n (P ) for any proper subpatch P of P . Then each Pn (T ) is finite and {T } = P0 (T ) ⊂ P1 (T ) ⊂ · · · . Lemma 7. For each tile T there is a positive integer NT such that PNT (T ) = PNT +1 (T ) = · · · . Proof. Set P(T ) = ∪n≥0 Pn (T ) and let r > 0, y ∈ Ed such that Br (y) ⊂ T . By finite local complexity, P(T ) has only finitely many patches up to rigid motion, since every P ∈ P(T ) is of the form (T ◦ )x for some tiling x ∈ X. If P , gP ∈ Pn (T ) for some g ∈ GE with gb − b < Kr for all b ∈ P then H n (g) is a period for φ n (P ) which violates Proposition 1. Thus for each patch P ∈ P(T ) the set of g ∈ GE for which gP ∈ P(T ) is discrete and bounded, hence finite. It follows that P(T ) is finite, which is equivalent to the desired result. Lemma 8. Suppose Pn (T ) = Pn+1 (T ). If x ∈ X is such that φ n+1 (T ) ⊂ φ(x) then φ n (T ) ⊂ x. Proof. Let x ∈ X be such that φ n (x ) = x. Then φ n+1 (T ) ⊂ φ(x) = φ n+1 (x ), hence there exists P ∈ Pn+1 (T ) such that P ⊂ x . Since P ∈ Pn (T ) we have φ n (T ) ⊂ φ n (P ) ⊂ φ n (x ) = x. Proof of Theorem 2. Set N = max{NT : T ∈ A}. Let x ∈ X, and let x1 , x2 ∈ X be any tilings such that φ(x1 ) = φ N+1 (x2 ) = x.
(12)
We only need to show that x1 = φ N (x2 ). Let T ∈ x2 be any tile and let g ∈ GE be such that gT is a tile in A. Then φ N+1 (gT ) ⊂ φ N+1 (gx2 ) = φ(H N (g)x1 ). By Lemma 8, φ N (gT ) ⊂ H N (g)x1 , and hence φ N (T ) ⊂ x1 .
(13)
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Remarks. The preceding arguments show that Proposition 1 is tantamount to recognizability, even for substitutions that do not satisfy hypothesis iii) and hence do not have a fixed point. This equivalence will be used in the proof of Propositions 3 and 4, below. The existence of a fixed point and the fact that Grel = GE were used to prove Proposition 1. We believe the conclusion is false without the latter assumption. We now begin the proof of Theorem 1. For i = 1, 2, let φi be a substitution on alphabet Ai with linear scaling factor λi such that φi : Xφi → Xφi is a homeomorphism. Write Xi for Xφi and suppose ψ : (X1 , GE ) → (X2 , GE ) is a topological conjugacy. Notation. For r > 0, a ∈ Ed and a tiling y, Br (a)y denotes the patch of y consisting of y all tiles in y that intersect the open ball of radius r about a. We abbreviate Br (O)y as Br . Pick an “inner radius” m such that every tile in A1 ∪ A2 contains an open ball of radius m. For patch-valued functions P , Q on X1 we say P determines Q (or Q is determined by P ) if whenever x, y ∈ X1 and P (x) = P (y) we also have Q(x) = Q(y). It follows from the fact that φi : Xi → Xi is a homeomorphism that there is a “recx determines the patch ognizability radius” Di > 0 such that for x ∈ Xi the patch BD i consisting of tiles containing the origin in φi−1 (x). Lemma 9. There is a constant ρ > 0 such that if n ∈ N and r > λn2 ρ then for y ∈ X2 φ −n (y)
y
2 the patch Br/(2λ n ) is determined by the patch Br . 2
Proof. Take ρ = 2D2 /(λ2 − 1). A patch of radius r in y determines a patch of radius (r −D2 )/λ2 in φ2−1 (y), of radius [(r −D2 )/λ2 −D2 ]/λ2 in φ2−2 (y), and r/λn2 −D2 (λ−1 2 + −2 −n −n −n λ2 + · · · + λ2 ) in φ2 (y). This last radius is greater than rλ2 − D2 /(λ2 − 1), which in turn is at least r/2λn2 . Any element g ∈ GE can be written uniquely as the composition of a rotation and a translation, i.e., there exist unique α ∈ SO(d) and s ∈ Rd such that ga = αa + s for all a ∈ Ed
(14)
(g) = α − I operator + s.
(15)
and we set
Notation. For patch-valued functions P , Q on X1 the phrase “P determines Q up to motion by some g ∈ GE with (g) ≤ η” means that if x, y ∈ X1 are such that P (x) = P (y) then there exists g ∈ GE with (g) ≤ η such that Q(x) = gQ(y). Lemma 10. There exist a constant S0 > 0 and a function η : R+ → R+ such that y limr→∞ η(r) = 0 and if r > S0 , then for y ∈ X1 the patch Br determines the patch ψ(y) Br−S0 up to motion by some g ∈ GE with (g) ≤ η(r). y
Proof. By uniform continuity of ψ, there is a radius S0 such that the patch BS0 determines the tile at the origin of ψ(y) and its immediate neighbors, up to motion by less than m/2. Since ψ is a conjugacy, for any point a ∈ Ed , the patch BS0 (a)y determines the tile at a in ψ(y) and its nearest neighbors, up to a small motion. Applying this to all y ψ(y) points a ∈ Br−S0 , we have that the patch Br determines Br−S0 up to an overall small motion g. The bound on (g) follows from uniform continuity of ψ.
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For n ∈ N put
log λ1 k(n) = n log λ2
(16)
and set −k(n)
ψn = φ2
◦ ψ ◦ φ1n .
(17)
k(n)
Fix x ∈ X1 . Note that λn1 /λ2 ∈ [1, λ2 ), and if log λ1 / log λ2 is rational then the −k(n) ∞ )n=1 is periodic and 1 is in its range, while if log λ1 / log λ2 is irratiosequence (λn1 λ2 nal then the range of this sequence is dense in [1, λ2 ]. Thus there exists a subsequence ni such that ψni (x) converges, say to x ∈ X2 , and such that −k(ni )
λn1 i λ2
→1
as i → ∞.
(18)
Proposition 2. The sequence {ψni }∞ i=1 converges uniformly to a conjugacy ψ : (X1 , GE ) y → (X2 , GE ) such that, for r > ρ + S0 and for y ∈ X1 the patch Br determines the ψ (y) patch Br/2 .
Proof. Step 1. If g ∈ GE then ψni (gx) → gx as i → ∞. Indeed, if we write gx = αx + s with α ∈ SO(d) and s ∈ Rd then −k(ni )
◦ ψ)(φ1ni (gx))
−k(ni )
◦ ψ)(αφ1ni (x) + λn1 i s)
ψni (gx) = (φ2 = (φ2
−k(ni )
= φ2
(αψ[φ1ni (x)] + λn1 i s) −k(ni ) ni λ1 s,
= αψni (x) + λ2
(19)
which clearly converges to gx as i → ∞. g x
Step 2. Suppose g, g ∈ GE and r > ρ + S0 are such that Br = Br . For each i, gx
n
B
φ1 i (gx)
n λ1 i r
n
=B
φ1 i (g x)
n λ1 i r
(20)
.
By Lemma 10, there exists hi ∈ GE with (hi ) ≤ η(λn1 i r) such that n
B
hi ψ[φ1 i (gx)]
n λ1 i r−S0
n
=B
ψ[φ1 i (g x)]
n λ1 i r−S0
(21)
.
By Lemma 9 −k(ni )
n (hi ψ[φ1 i (gx)]) ni k(ni ) (λ1 r−S0 )/2λ2
B
φ2
=B
−k(ni )
n (ψ[φ1 i (g x)]) ni k(n ) (λ1 r−S0 )/2λ2 i
φ2
.
(22)
Let α, βi ∈ SO(d) and s, ti ∈ Rd be such that ga = αa + s and hi a = βi a + ti for all a ∈ Ed . We have −k(ni )
φ2
−k(ni ) ni −k(n ) λ1 βi s + λ2 i ti ,
(hi ψ[φ1ni (gx)]) = βi αψni (x) + λ2
(23)
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and this converges to gx as i → ∞. We know from Step 1 that −k(ni )
φ2
(ψ[φ1ni (g x)]) → g x as i → ∞.
(24)
Since λn1 i r − S0 k(n ) 2λ2 i
→
r 2
as i → ∞,
(25)
it follows that gx
g x
Br/2 = Br/2 .
(26)
Step 3. Define ψ : X1 → X2 as follows. Given y ∈ X1 let {gj }j be a sequence in GE such that gj x → y as j → ∞. There exist group elements hj tending to the h g x
y
identity in GE and positive real numbers rj tending to infinity such that Brjj j = Brj for each j. By Step 2, (hj gj x )j converges, hence so does (gj x )j , and we define ψ (y) = limj →∞ gj x. The existence of this limit ensures that ψ is well defined and continuous. We see from Step 1 that it is a conjugacy, and the sliding block code follows from Step 2. Step 4. We still need to show that ψni converges to ψ uniformly on X1 . For y ∈ X1 and r > 0 one can find g ∈ GE such that gx agrees with y on a ball of radius r about the origin. By linear repetitivity (see Lemma 2), we can always choose g such that (g) < Cr for some fixed constant C. By the triangle inequality, m[ψni (y), ψ (y)] ≤ m[ψni (y), ψni (gx)] + m[ψni (gx), gψni (x)] +m[gψni (x), ψ (gx)] + m[ψ (gx), ψ (y)].
(27)
Given , we will show that for i large (with estimates independent of y), and for the correct choice of g, each term on the right hand side is bounded by /4. The left hand side (which is independent of the choices made) is then bounded by . Since the estimates on i were independent of y, the left hand side goes to zero as i → ∞ at a rate that is independent of y. The argument of Step 2 shows that the maps ψni are uniformly continuous with estimates that are independent of i. As a result, the first term can be made small, independent of i (and y), by choosing r large enough. The last term is also small if r is large, since ψ is uniformly continuous. For fixed r (and hence fixed g), the second term is bounded −k(n ) by Cr|1 − λn1 i λ2 i |, which is small if i is large enough. Finally, the third term is small once i is big enough that ψni (x) and ψ (x) agree up to a small motion on a ball of radius Cr about the origin. −k(ni )
Proposition 3. For all sufficiently large i we have λn1 i λ2
= 1.
Proposition 4. There exists I ∈ N such that for all i ≥ I , for all y ∈ X1 , ψni (y) = ψ (y) + sy,i
(28)
for some sy,i ∈ Rd . Proof of Propositions 3 and 4. For fixed r, ε > 0 to be specified in the proof one can find δ > 0 such that if y, y ∈ X1 with d(ψni (y), ψ (y)) < δ then there exists gy,i ∈ GE
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with (gy,i ) < ε such that ψni (y) and gy,i ψ (y) agree on the ball of radius r centered at the origin. Choose I such that d(ψni (y), ψ (y)) < δ for all i ≥ I, y ∈ X1 . Let us consider i ≥ I. If r is chosen large enough and ε small enough, then gy,i is uniquely determined by the above conditions and varies continuously with y, for otherwise we would have a large patch with a small period, contradicting the recognizability hypotheses (see the remarks following the proof of Theorem 2). Let αy,i ∈ SO(d) and sy,i ∈ Rd denote the rotational and translational parts of gy,i , respectively. We have, for t ∈ Rd with t < r/λ2 , −k(ni )
ψni (y + t) = ψni (y) + λn1 i λ2
t,
(29)
and this agrees with −k(ni )
gy,i ψ (y) + λn1 i λ2
−k(ni )
t = gy,i ψ (y + t) + λn1 i λ2
t − αy,i t
(30)
on the ball of radius r − λ2 t about the origin. If r is large enough and ε small enough then this implies, for all t sufficiently small, for all y ∈ X1 , αy+t,i = αy,i
(31)
and −k(ni )
sy+t,i = sy,i + λn1 i λ2
t − αy,i t.
(32)
By continuity αy+t,i = αy,i for all t, and this implies that the above formula for sy+t,i holds for all t as well. Now sy+t,i < ε for all t, and this is only possible if αy,i is the −k(n ) identity and λn1 i λ2 i = 1. Thus ψni (y + t) = ψ (y) + sy,i + t. Corollary 2. For each i ≥ I and each y ∈ X1 , there exists gy,i in the center of Grel (y) such that ψni (y) = gy,i ψ (y). Furthermore, if y is in the closure of the translation orbit of y, then gy ,i = gy,i . Proof. Fix i and y. By Proposition 4 there exists a translation gy,i such that ψni (y) = gy,i ψ (y). Since ψni and ψ are conjugacies and all translations commute, we have gy,i = gy ,i for any y in the translation orbit of y. By continuity, this last equality holds for all y in the closure of the translation orbit of y. To show that gy,i is in the center of Grel (y), it suffices to show that αgy,i = gy,i for every α ∈ SO(d) ∩ Grel (y). Fix such α. By definition of Grel (y) there is a sequence hj ∈ Grel (y) such that hj y → y as j → ∞ and such that the rotational part αj of hj converges to α. If gy,i is translation by sy,i then ψ (y) + sy,i = ψni (y) = lim ψni (hj y) j →∞
= lim hj ψni (y) j →∞
= lim hj gy,i ψ (y) j →∞
= lim hj ψ (y) + αj sy,i j →∞
= lim ψ (hj y) + αj sy,i , j →∞
= ψ (y) + lim αj sy,i ,
(33)
j →∞
and hence αj sy,i → sy,i . It follows that gy,i commutes with α.
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Proposition 5. If two tilings y, y in a substitution tiling space Xφ agree on a single tile, then they are in the same translation orbit closure. Proof. Suppose that y, y ∈ Xφ are tilings in different translation orbit closures which agree on a tile; without loss of generality, we may assume that the interior of the tile contains the origin, and thus, d(φ n (y), φ n (y )) → 0 as n → ∞. There exists a tiling z in the translation orbit closure of y and a rotation α ∈ SO(d) such that φ(z) = αz. Let {ni } be an increasing sequence of integers such that α ni converges to the identity. Let β ∈ SO(d) be such that βz is in the translation orbit closure of y . Then, for each ni ≥ 1, α ni y is in the translation orbit closure of φ ni (y), and φ ni (βz) = βα ni z is in the translation orbit closure of φ ni (y ). Taking a limit as ni → ∞, it follows that the distance from the translation orbit closure of y to that of βz is zero, hence that y and βz, and therefore y , are in the same translation orbit closure. Remark. Using property iii), one can take α to be the identity, thereby simplifying the proof of Proposition 5. The above argument, however, shows that the conclusions of Proposition 5 apply even to substitutions that do not have a fixed point. Proof of Theorem 1. First we show that ψni is a sliding block code for i ≥ I . Suppose x, y ∈ X1 agree on a large ball around the origin, so that ψ (x) and ψ (y) agree on a (smaller) ball around the origin. By Proposition 5, x and y lie in the same translation orbit closure. However, ψn1 and ψ differ by a (fixed) translation on this orbit closure, so ψni (x) and ψni (y) agree on a (still smaller) ball around the origin. k(n ) Now note that ψ = φ2 i ψni φ1−ni is a composition of three maps, each of which is a sliding block code up to scaling. Each patch in ψ(x) is determined by a (much smaller) patch in ψni ◦ φ1−ni (x), which is determined by a patch in φ1−ni (x), which is determined by a (much larger) patch in x. Thus ψ is a sliding block code. Corollary 3. If the translation orbit of a tiling is dense in X1 , then ψ intertwines the actions of some powers of φ1 and φ2 . Proof. In this case Grel (y) of a tiling y has no center, so gi is the identity and ψni = ψ for all i ≥ I . But then −k(ni )
φ2
−k(ni+1 )
◦ ψ ◦ φ1ni = ψni = ψ = ψni+1 = φ2 k(ni+1 )
Multiplying on the left by φ2
(34)
and on the right by φ1−ni gives
k(ni+1 )−k(ni )
φ2
n
◦ ψ ◦ φ1 i+1 .
n
◦ ψ = ψ ◦ φ1 i+1
−ni
.
(35)
3. Proof of Theorem 3 Suppose that ψ : X → Y is a topological conjugacy and that Y is of finite type. By assumption, there exists a finite length r1 such that Yr1 = Y . As in the proof of Theorems 1 and 2, let D be a length greater than the diameter of any tile in either tiling system. By finite local complexity there exists a radius m > 0 such that every tile contains a ball of radius m and the only way to move two adjacent
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tiles a distance m or less and obtain an admissible local pattern is to move the pair by a rigid motion. Since ψ −1 is uniformly continuous, there is a length r2 such that, for each y
ψ −1 (y)
, up to a Euclidean motion that moves each point in B2D y ∈ Y , Br2 determines B2D by a distance of m/2 or less. Likewise, there is a length r3 such that, for each x ∈ X, ψ(x) Brx3 determines Br1 +r2 +3D up to a wiggle of size at most m/2. We claim that Xr3 = X, and thus that X is of finite type. For if x ∈ Xr3 , then every patch of radius r3 in x corresponds to an admissible patch in a tiling in X, and so determines a patch of a tiling in Y (up to a small motion). That is, the tiling x determines a combinatorial tiling of the tiles of the Y system, such that each patch of radius r1 + r2 + 3D is actually admissible. This local information can be stitched together to form an actual tiling y ∈ Yr1 +r2 +3D = Y . The tiling ψ −1 (y) is then a tiling in X. However, the combinatorial structures of x and ψ −1 (y) are the same, since Brx3 (a) detery
ψ −1 (y)
mines Br1 +r2 +3D up to a small rigid motion, which determines B2D up to small rigid motion. Since the tiles are rigid, this implies that x is obtained by applying a rigid motion to ψ −1 (y), and is thus in X. 4. Conclusions We have been concerned with topological conjugacy between tiling dynamical systems, emphasizing the geometric aspects by including systems in which the tiles appear, in each tiling, in infinitely many orientations, thus incorporating the rotation group in an essential way. Some of our results are restricted to a subclass of tiling dynamical systems, substitution systems, which can be thought of as incorporating an extra group action which represents a certain similarity: that is, not only have we extended the usual action of the translation group by the rotations, we have in fact extended further, to a subgroup of the conformal group. Our first result is to show that topological conjugacies between substitution systems with invertible substitutions are quite rigid. We show that any conjugacy for the Euclidean actions automatically extends to (some powers of) the similarities, and can be represented by the natural analogue of a sliding block code. Tiling dynamical systems are a geometric extension of subshifts, and these results all have significant geometric meaning. Our second result is that substitutions whose systems do not admit periodic tilings are recognizable, as long as the relative orientation group is either finite [S] or dense in SO(d). In particular, all nonperiodic substitutions in two dimensions are recognizable. This result can then be used to generalize constructions such as those in [PS] from the category of translationally finite tilings to the more general case where tiles can appear in arbitrary orientation. Part of the significance of substitution subshifts and substitution tiling systems for dimension d ≥ 2 arises from the fact that, quite generally (see [Moz], [G]), such a system carries a unique invariant measure and is measurably conjugate to some uniquely ergodic system of finite type. Actually, the proofs show more than measurable conjugacy; they show that off sets of measure zero the map is bicontinuous, and it is boundedly finite to one on the sets of measure zero. These associated finite type systems are also of geometric interest, as part of the general effort of understanding the symmetries of densest packings of bodies [R2]. Our third result is a step in this direction, showing that finite type is a topological property among tilings with finite local complexity, and not merely an artifact of the way one defines the tiles.
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It would be significant if Theorem 1, and the conjugacy invariants (the relative orientation groups) which follow from it, apply to such tiling dynamical systems. For instance, we noted above that the (1,2)-pinwheel and (3,4)-pinwheel systems cannot be topologically conjugate. We thus conclude with an open problem. Open Problem. Are the two finite type tiling systems, which are measurably conjugate to the (1,2)-pinwheel and (3,4)-pinwheel systems, topologically conjugate? Acknowledgement. The authors are grateful for the support of the Banff International Research Station, at which we formulated and proved Theorem 3.
References [BSJ] [CS] [G] [LM] [Mos] [Moz] [ORS] [P] [PS] [R1] [R2] [RS1] [RS2] [RW] [S]
Baake, M., Schlottmann, M., Jarvis, P.D.: Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability. J. Physics A 24, 4637–4654 (1991) Clark, A., Sadun, L.: When size matters: subshifts and their related tiling spaces. Ergodic Theory & Dynamical Systems 23, 1043–1057 (2001) Goodman-Strauss, C.: Matching rules and substitution tilings. Ann. Math. 147, 181–223 (1998) Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding, Cambridge: Cambridge University Press, 1995 Moss´e, B.: Puissances de mots et reconnaisabilit´e des point fixes d’une substitution. Theor. Comp. Sci. 99(2), 327–334 (1992) Mozes, S.: Tilings, substitution systems and dynamical systems generated by them. J. d’Analyse Math. 53, 139–186 (1989) Ormes, N., Radin, C., Sadun, L.: A homeomorphism invariant for substitution tiling spaces. Geometriae Dedicata 90, 153–182 (2002) Petersen, K.: Factor maps between tiling dynamical systems. Forum Math. 11, 503–512 (1999) Priebe, N., Solomyak, B.: Characterization of planar pseudo-self-similar tilings. Discrete Comput. Geom. 26, 289–306 (2001) Radin, C.: The pinwheel tilings of the plane. Ann. Math. 139, 661–702 (1994) Radin, C.: Orbits of orbs: sphere packing meets Penrose tilings. Amer. Math. Monthly 111, 137–149 (2004) Radin, C., Sadun, L.: An algebraic invariant for substitution tiling systems. Geometriae Dedicata 73, 21–37 (1998) Radin, C., Sadun, L.: Isomorphism of hierarchical structures. Ergodic Theory Dynam. Systems 21, 1239–1248 (2001) Radin, C., Wolff, M.: Space tilings and local isomorphism. Geometriae Dedicata 42, 355–360 (1992) Solomyak, B.: Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20, 265–279 (1998)
Communicated by G. Gallavotti
Commun. Math. Phys. 254, 361–366 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1161-0
Communications in
Mathematical Physics
On the Absolutely Continuous Spectrum of Multi-Dimensional Schr¨odinger Operators with Slowly Decaying Potentials Oleg Safronov Department of Mathematics, UAB, 1300 University Boulevard, Birmingham, AL 35294, USA. E-mail:
[email protected] Received: 2 February 2004 / Accepted: 4 March 2004 Published online: 3 September 2004 – © Springer-Verlag 2004
Abstract: We consider a multi-dimensional Schr¨odinger operator − + V in L2 (R d ) and find conditions on the potential V which guarantee that the absolutely continuous spectrum of this operator is essentially supported by the positive real line. We prove some results which go beyond the case L1 + Lp with p < d. 1. In this short paper we extend the method of Laptev-Naboko-Safronov [15]. New estimates for the discrete spectrum obtained in this paper lead to stronger results compared to [15]. The main technical tool of the paper [15] is the so called trace inequality, which relates properties of negative eigenvalues to the properties of the a.c. spectrum. Based on this relation, the technique of our paper allows one to improve the known bounds for the distribution function of the discrete spectrum, i.e. obtain new estimates for eigenvalues of the Schr¨odinger operator. Let us state our main assertion. Theorem 0.1. Let d ≥ 3 and let V ∈ L∞ (Rd ) be a real valued potential such that V (x) → 0 as |x| → ∞. Assume that V ∈ Ld+1 (Rd ), and for some positive number δ > 0 the Fourier transform of V satisfies the estimate |Vˆ (ξ )|2 dξ < ∞. |ξ |<δ
Then the absolutely continuous spectrum of the operator − + V is essentially supported by the positive real line R+ , i.e. the spectral projection corresponding to any set in R+ of positive Lebesgue measure is different from zero. This theorem almost coincides with the result of Deift and Killip [6] for the case d = 1 ( to be precise they do not impose the conditions V ∈ L∞ and V → 0). Note that Denissov [9] proved recently some results about the a.c. spectrum of Dirac operators. These results look quite optimal for the considered operator. However the same method The author is grateful to Gunter Stolz for useful discussions. The work was supported by the grant of NSF DMS-0245210.
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can not be applied to the operators having a bottom of the spectrum. (Probably this explains where the condition on the Fourier transform for small energies comes from !) Different relations between the discrete and continuous spectrum appeared several times before in the case of the one dimensional operator. We would like to mention a recent result of Damanik and Killip [4] which says that if one dimensional Schr¨odinger operators with potentials +V and −V have only finite number of eigenvalues then the positive spectrum is absolutely continuous. In connection to our paper, a natural question in dimension d = 1 is whether the spectrum would be still absolutely continuous if instead of finiteness of the negative spectrum we supposed that a certain eigenvalue sum converges for V and −V . It is interesting to compare the main theorem of Damanik-Killip [4] with the results of LNS [15] which allow to estimate the spectral measure of −+V in terms of eigenvalue sums. Therefore whenever we have a nice estimate for the eigenvalues of the operator, we can prove that the a.c. spectrum is essentially supported by (0, +∞). Proof. Let λj (V ) be the negative eigenvalues of − + V and Br be the ball of radius r in Rd , d ≥ 3, around the origin. We are going to prove that (d+1)/4 |λj (V )| ≤ C(r) |Vˆ (ξ )|2 dξ Br
j
|Vˆ (ξ )|2 dξ
+
Br
(d+1)/2
+
Rd
|V (x)|d+1 dx ,
∀V ∈ C0∞ (Rd ).
(0.1)
Therefore we can apply the following result of Laptev-Naboko-Safronov [15]. Lemma 0.2. Let V ∈ L∞ (Rd ) be a real valued potential such that V (x) → 0 as |x| → ∞. Assume that the quantity in the right-hand side of (0.1) is finite for V . Then the inequality (0.1) implies the fact that the absolutely continuous spectrum of the operator H + V = − + V is essentially supported by the positive real line R+ . First for any self-adjoint operator T and s > 0 we define n± (s, T ) as n± (s, T ) = rank E±T (s, +∞), where ET (·) denotes the spectral measure for T . We put also n(s, T ) = n+ (s 2 , T ∗ T ) if the operator T is not necessarily self-adjoint. Then the following relations hold: n± (s1 + s2 , T1 + T2 ) ≤ n± (s1 , T1 ) + n± (s2 , T2 ); n(s1 s2 , T1 T2 ) ≤ n(s1 , T1 ) + n(s2 , T2 ), s1 , s2 > 0. By the Birman-Schwinger principle, we can represent the Lieb-Thirring sum as an integral of the function of distribution for the spectrum of a certain compact self-adjoint operator +∞ 1 |λj | = √ n− (s, H + V ) ds 2 s 0 j +∞ 1 = √ n− (1, (H + s)−1/2 V (H + s)−1/2 ) ds. 2 s 0 Note also that if an orthogonal projection Q commutes with H , then n− (s, H +QV Q) ≤ n− (s, H + V ), s > 0.
On the A.C. Spectrum
363
Let E denote the spectral projection E = EH (0, δ), δ > 0 and let P = I − E. Then (H + s)−1/2 V (H + s)−1/2 = (H + s)−1/2 EV E(H + s)−1/2 +(H + s)−1/2 EV P (H + s)−1/2 + (H + s)−1/2 P V E(H + s)−1/2 +(H + s)−1/2 P V P (H + s)−1/2 .
(0.2)
Now, let us introduce the potential V0 whose Fourier transform is the characteristic function of the ball B2√δ multiplied by the Fourier transform of V . Note that then EV E = EV0 E. Therefore (H + s)−1/2 V (H + s)−1/2 = (H + s)−1/2 V0 (H + s)−1/2 + T (s),
(0.3)
where T (s) satisfies the estimate:
+∞ 1 |V |d+1 (x) dx + |V0 |d+1 (x) dx . (0.4) √ n(1, (T (s)) ds ≤ C(δ) s Rd Rd 0 It must be noted here that |V0 |p (x) dx ≤ Cδ,p Rd
√ |ξ |<2 δ
|Vˆ (ξ )|2 dξ
p/2
for any 2 < p < ∞. Indeed, to prove (0.4) we remark that ||(H +s)−1/2 P (H +s +δ)1/2 || < C is bounded uniformly in s. Therefore we have the following inequality for any fixed number t > 0: +∞ 1 √ n(t 2 , (H + s)−1/2 P V P (H + s)−1/2 ) ds 2 s 0 +∞ 1 ≤ √ n(C −1 t, (H + s + δ)−1/2 |V |1/2 ) ds. 2 s 0 The right-hand side can be estimated by |V |d+1 dx due to the Lieb-Thirring estimate (d+1)/2 |λj (δ − |V |)| ≤ C (|V | − δ)+ dx ≤ Cδ |V |d+1 dx. The estimate for
0
+∞
1 √ n(t, (H + s)−1/2 EV P (H + s)−1/2 ) ds 2 s
is more straightforward, since ||P (H + s)−1/2 || ≤ δ −1/2 . In this case we obtain +∞ 1 √ n(t, (H + s)−1/2 EV P (H + s)−1/2 ) ds 2 s 0 +∞ 1 ≤ √ n(c(δ)t, (H + s)−1/2 EV ) ds 2 s 0 ≤ |λj (−CV 2 )|, C = C(t, δ).
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The latter sum of eigenvalues can be estimated due to the Lieb-Thirring estimate 2 |λj (−V )| ≤ C |V |d+1 (x) dx. Finally we apply the Lieb-Thirring inequality for the sum of square roots of eigenvalues to conclude that +∞ 1 −1/2 −1/2 V0 (H + s) ) ds ≤ C0 |V0 |(d+1)/2 (x) dx √ n− (1, (H + s) s 0
(d+1)/4 2 |Vˆ | (x) dx , d ≥ 3. ≤ C(δ) B2√δ
This completes the proof of Theorem 0.1.
Remark 1. It follows from the proof that the second term in (0.1) can be omitted, i.e. (d+1)/4 2 ˆ |λj (V )| ≤ C(r) |V (ξ )| dξ + |V (x)|d+1 dx , Rd
Br
j
∀V ∈ C0∞ (Rd ), d ≥ 3. Indeed, instead of (0.3) and (0.4) one should use the inequality ∞ ds n− (1/2, (H + s)−1/2 EV E(H + s)−1/2 ) √ s 0 |λj (2V0 )| ≤2 j
≤ C(δ)
B2√δ
|Vˆ (ξ )|2 dξ
(d+1)/4
.
Remark 2. In the same way, one can prove for γ = 2 − d/2, 1 ≤ d ≤ 4, that the inequality
|λj (V )|γ ≤ C(r) |Vˆ (ξ )|2 dξ + |V (x)|4 dx , Br
j
Rd
holds for functions V ∈ C0∞ (Rd ). This estimate is especially interesting in the case d = 4, since it gives a bound for the number of negative eigenvalues of − + V . Corollary 0.3. Let d ≥ 3 and let V ∈ L∞ (Rd ) be a real valued potential such that V (x) → 0 as |x| → ∞. Assume that V ∈ Ld+1 (Rd ) and for some positive number δ > 0, 2 V (x + y) dy dx < ∞. Rd
|y|<δ
Then the absolutely continuous spectrum of the operator −+V is essentially supported by the positive real line R+ .
On the A.C. Spectrum
365
Corollary 0.4. Let d ≥ 3 and let V ∈ L∞ (Rd ) be a real valued function such that V (x) → 0 as |x| → ∞. Assume that V ∈ Ld+1 (Rd ) and the Fourier transform of V is square integrable near ξ0 ∈ Rd . Then the absolutely continuous spectrum of the operator − + cos(ξ0 x)V (x) is essentially supported by the positive real line R+ . Theorem 0.1 treats dimensions d ≥ 3 and coincides with Deift-Killip [6] for d = 1. So, what about the mysterious case d = 2? Since the potentials V0 = 0 do not produce any eigenvalues at all, we come to the following conclusion: Theorem 0.5. Let d = 2 and let V ∈ L∞ (R2 ) be a bounded real valued function, V (x) → 0 as |x| → ∞. Assume that V ∈ L3 (R2 ) and the Fourier transform of V is identically zero near the origin. Then the essential support of the absolutely continuous spectrum of the operator − + V (x) is the positive real line R+ . Finally we can use the opportunity of adding a function of the L(d+1)/2 -class. Theorem 0.6. Let d ≥ 2 and let Vj ∈ L∞ (Rd ) , j = 1, 2, be real valued functions such that Vj (x) → 0 as |x| → ∞. Assume that V1 ∈ Ld+1 (Rd ), V2 ∈ L(d+1)/2 (Rd ) and for some positive number δ > 0 the Fourier transform of V1 satisfies |Vˆ1 (ξ )|2 dξ < ∞, d ≥ 3, |ξ |<δ
or
Vˆ1 (ξ ) = 0, ∀|ξ | < δ, d = 2. Finally, let V = V1 + V2 . Then the absolutely continuous spectrum of the operator − + V is essentially supported by the positive real line R+ . Proof. Indeed, j
|λj (V )| =
+∞ 0
1 √ n− (s, H + V ) ds 2 s
+∞
1 √ n− (1, (H + s)−1/2 V (H + s)−1/2 ) ds 2 s 0 +∞ 1 ≤ √ n− (1/2, (H + s)−1/2 V1 (H + s)−1/2 ) ds 2 s 0 +∞ 1 + √ n− (1/2, (H + s)−1/2 V2 (H + s)−1/2 ) ds. 2 s 0 The first integral in the right-hand side can be estimated by using (0.1) and the second is bounded by C |V2 |(d+1)/2 dx.
=
2. We conclude the article with the remark that according to [15] one is able to give a quantitative characteristic of how continuous the positive spectrum of the operator is. For that purpose we consider the operator H∗ which is − + V with the Dirichlet boundary condition on the unit sphere ∂B1 . Theorem 0.7. Suppose that the function V (x) + ((d − 1)2 /4 − (d − 1)/2)|x|−2 vanishes inside the ball B1+ for some > 0. Then under assumptions of Theorem 0.1 there is a function f ∈ L2 (Rd \ B1 ) such that R d dλ log (EH∗ (−∞, λ)f, f ) √ > −∞, ∀R > 0. dλ λ 0 This theorem implies that dEH∗ /dλ can not vanish on a set of positive measure.
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References 1. Christ, M., Kiselev, A.: Absolutely continuous spectrum for one-dimensional Schr¨odinger operators with slowly decaying potentials: Some optimal results. J. Am. Math. Soc. 11, 771–797 (1998) 2. Damanik, D., Hundertmark, D., Killip, R., Simon, B.: Variational estimates for discrete Schr¨odinger operators with potentials of indefinite sign. Commun. Math. Phys. 238, 545-562 (2003) 3. Damanik, D., Hundertmark, D., Simon, B.: Bound states and the Szeg¨o condition for Jacobi matrices and Schr¨odinger operators. J. Funct. Anal. 205, 357-379 (2003) 4. Damanik, D., Killip, R.: Half-line Schr¨odinger operators with no bound states. To appear in Acta Math. 5. Damanik, D., Killip, R., Simon, B.: Necessary and sufficient conditions in the spectral theory of Jacobi matrices and Schr¨odinger operators. IMRN 22, 1087–1097 (2004) 6. Deift, P., Killip, R.: On the absolutely continuous spectrum of one-dimensional Schr¨odinger operators with square summable potentials. Commun. Math. Phys. 203, 341–347 (1999) 7. Denisov, S.A.: On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm-Liouville operators with square summable potentials. J. Diff. Eqs. 191, 90–104 (2003) 8. Denisov, S.A.: On the absolutely continuous spectrum of Dirac operator. To appear in Commun. PDE 9. Denisov, S.A.: Absolutely continuous spectrum of Schrodinger operators and Fourier transform of the potential. Russ. J. Math. Phys. 8(N1), 14–24 (2001) 10. Hundertmark, D., Simon, B.: Lieb-Thirring inequalities for Jacobi matrices. J. Aprox. Theory. 118 (1), 106–130 (2002) 11. Killip, R.: Perturbations of one-dimensional Schr¨odinger operators preserving the absolutely continuous spectrum. Int. Math. Res. Not. N. 38, 2029–2061 (2002) 12. Killip, R., Simon, B.: Sum rules for Jacobi matrices and their applications to spectral theory. Ann. Math. 158, 253–321 (2003) 13. Kiselev, A.: Absolutely continuous spectrum of one-dimensional Schr¨odinger operators and Jacobi matrices with slowly decreasing potentials. Commun. Math. Phys. 179, 377–400 (1996) 14. Kiselev, A.: Imbedded Singular Continuous Spectrum for Schr¨odinger Operators. Preprint 15. Laptev, A., Naboko, S., Safronov, O.: Absolutely continuous spectrum of Schr¨odinger operators with slowly decaying and oscillating potentials. Commun. Math. Phys., DOI 10.1007/s00220-004-11579, 2004 16. Lieb, E.H., Thirring, W.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975); Errata Ibid 35, 1116 (1975) 17. Lieb, E.H., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schr¨odinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann, Princeton, NJ: Princeton University Press, 1976, pp. 269–303 18. Remling, C.: The absolutely continuous spectrum of one-dimensional Schr¨odinger operators with decaying potentials. Commun. Math. Phys. 193, 151–170 (1998) Communicated by B. Simon
Commun. Math. Phys. 254, 367–400 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1220-6
Communications in
Mathematical Physics
Nonabelian Bundle Gerbes, Their Differential Geometry and Gauge Theory Paolo Aschieri1,2,3 , Luigi Cantini4,5 , Branislav Jurˇco2,3 1
Dipartimento di Scienze e Tecnologie Avanzate, Universit´a del Piemonte Orientale, and INFN, Corso Borsalino 54, 15100 Alessandria, Italy Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, 80805 M¨unchen, Germany Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. E-mail:
[email protected];
[email protected] 4 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa & INFN sezione di Pisa, Italy 5 Department of Physics, Queen Mary, University of London, Mile End Road, London E1 4NS, UK. E-mail:
[email protected]
2 3
Received: 5 February 2004 / Accepted: 29 April 2004 Published online: 17 November 2004 – © Springer-Verlag 2004
Abstract: Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of Yang-Mills theories with 2-form gauge potential. 1. Introduction Fibre bundles, besides being a central subject in geometry and topology, provide the mathematical framework for describing global aspects of Yang-Mills theories. Higher abelian gauge theories, i.e. gauge theories with abelian 2-form gauge potential appear naturally in string theory and field theory, and here too we have a corresponding mathematical structure, that of the abelian gerbe (in algebraic geometry) and of the abelian bundle gerbe (in differential geometry). Thus abelian bundle gerbes are a higher version of line bundles. Complex line bundles are geometric realizations of the integral 2nd cohomology classes H 2 (M, Z) on a manifold, i.e. the first Chern classes (whose de Rham representative is the field strength). Similarly, abelian (bundle) gerbes are the next level in realizing integral cohomology classes on a manifold; they are geometric realizations of the 3rd cohomology classes H 3 (M, Z). Thus the curvature 3-form of a 2-form gauge potential is the de Rham representative of a class in H 3 (M, Z). This class is called the Dixmier-Douady class [1, 2]; it topologically characterizes the abelian bundle gerbe in the same way that the first Chern class characterizes complex line bundles. One way of thinking about abelian gerbes is in terms of their local transition functions [3, 4]. Local “transition functions” of an abelian gerbe are complex line bundles on double overlaps of open sets satisfying cocycle conditions for tensor products over quadruple overlaps of open sets. The nice notion of abelian bundle gerbe [5] is related to this picture. Abelian gerbes and bundle gerbes can be equipped with additional structures, that of a connection 1-form and of curving (the 2-form gauge potential), and that
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of (bundle) gerbe modules (with or without connection and curving ). Their holonomy can be introduced and studied [3, 4, 6–9]. The equivalence class of an abelian gerbe with connection and curving is the Deligne class on the base manifold. The top part of the Deligne class is the class of the curvature, the Dixmier-Douady class. Abelian gerbes arise in a natural way in quantum field theory [10–12], where their appearance is due to the fact that one has to deal with abelian extensions of the group of gauge transformations; this is related to chiral anomalies. Gerbes and gerbe modules appear also very naturally in TQFT [13], in the WZW model [14] and in the description of D-brane anomalies in the nontrivial background 3-form H -field (identified with the Dixmier-Douday class) [15–17]. Coinciding (possibly infinitely many) D-branes are submanifolds “supporting” bundle gerbe modules [6] and can be classified by their (twisted) K-theory. The relation to the boundary conformal field theory description of D-branes is due to the identification of equivariant twisted K-theory with the Verlinde algebra [18, 19]. For the role of K-theory in D-brane physics see e.g. [20–22]. In this paper we study the nonabelian generalization of abelian bundle gerbes and their differential geometry, in other words we study higherYang-Mills fields. Nonabelian gerbes arose in the context of nonabelian cohomology [23, 1] (see [24] for a concise introduction), see also ([25]). Their differential geometry –from the algebraic geometry point of view– is discussed thoroughly in the recent work of Breen and Messing [26] (and their combinatorics in [27]). Our study on the other hand is from the differential geometry viewpoint. We show that nonabelian bundle gerbes connections and curvings are very natural concepts also in classical differential geometry. We believe that it is primarily in this context that these structures can appear and can be recognized in physics. It is for example in this context that one would like to have a formulation of Yang-Mills theory with higher forms. These theories should be relevant in order to describe coinciding NS5-branes with D2-branes ending on them. They should be also relevant in the study of M5-brane anomaly. We refer to [28–30] for some attempts in constructing higher gauge fields. Abelian bundle gerbes are constructed using line bundles and their products. One can also study U (1) bundle gerbes; here line bundles are replaced by their corresponding principal U (1) bundles. In the study of nonabelian bundle gerbes it is more convenient to work with nonabelian principal bundles than with vector bundles. Actually principal bundles with additional structures are needed. We call these objects (principal) bibundles and D-H bundles (D and H being Lie groups). Bibundles are fibre bundles (with fiber H ) which are at the same time left and right principal bundles (in a compatible way). They are the basic objects for constructing (principal) nonabelian bundle gerbes. The first part of this paper is therefore devoted to their description. In Sect. 2 we introduce bibundles, D-H bundles (i.e. principal D bundles with extra H structure) and study their products. In Sect. 3 we study the differential geometry of bibundles, in particular we define connections, covariant exterior derivatives and curvatures. These structures are generalizations of the corresponding structures on usual principal bundles. We thus describe them using a language very close to that of the classical reference books [31] or [32]. In particular a connection on a bibundle needs to satisfy a relaxed equivariance property, this is the price to be paid in order to incorporate nontrivially the additional bibundle structure. We are thus lead to introduce the notion of a 2-connection (a, A) on a bibundle. Products of bibundles with connections give a bibundle with connection only if the initial connections were compatible. We call this compatibility the summability conditions for 2-connections; a similar summability condition is established also for horizontal forms (e.g. 2-curvatures).
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In Sect. 4, using the product between bibundles we finally introduce (principal) bundle gerbes. Here too we first describe their structure (including stable equivalence) and then only later in Sect. 7 we describe their differential geometry. We start with the proper generalization of abelian bundle gerbes in the sense of Murray [5]; we then describe the relation to the Hitchin type presentation [3, 4], where similarly to the abelian case, nonabelian gerbes are described in terms of their “local transition functions” which are bibundles on double overlaps of open sets. The properties of the products of these ˇ bibundles over triple and quadruple overlaps define the gerbe and its nonabelian Cech 2-cocycle. Section 5 is devoted to the example of the lifting bundle gerbe associated with the group extension H → E → G. In this case the bundle gerbe with structure group H appears as an obstruction to lift to E a G-principal bundle P . Again by generalizing the abelian case, bundle gerbe modules are introduced in Sect. 6. Since we consider principal bibundles we obtain modules that are D-H bundles (compatible with the bundle gerbe structure). With each bundle gerbe there is canonically associated an Aut (H )-H bundle. In the lifting bundle gerbe example a module is given by the trivial E-H bundle. In Sect. 7 we introduce the notion of a bundle gerbe connection and prove that on a bundle gerbe a connection always exists. Bundle gerbe connections are then equivalently described as collections of local 2-connections on local bibundles (the “local transition functions of the bundle gerbe”) satisfying a nonabelian cocycle condition on triple overlaps of open sets. Given a bundle gerbe connection we immediately have a connection on the canonical bundle gerbe module can. We describe also the case of a bundle gerbe connection associated with an arbitrary bundle gerbe module. In particular we describe the bundle gerbe connection in the case of a lifting bundle gerbe. Finally in Sect. 8 we introduce the nonabelian curving b (the 2-form gauge potential) and the corresponding nonabelian curvature 3-form h. These forms are the nonabelian generalizations of the string theory B and H fields. 2. Principal Bibundles and Their Products Bibundles (bitorsors) were first studied by Grothendieck [33] and Giraud [1], their cohomology was studied in [34]. We here study these structures using the language of differential geometry. ˜ on the same base space M, one can consider Given two U (1) principal bundles E, E, ˜ the fiber product bundle E E, defined as the U (1) principal bundle on M whose fibers ˜ fibers. If we introduce a local description of E and E, ˜ are the product of the E and E, i ij ij with transition functions h and h˜ (relative to the covering {U } of M), then E E˜ has transition functions hij h˜ ij . In general, in order to multiply principal nonabelian bundles one needs extra structure. Let E and E˜ be H -principal bundles, we use the convention that H is acting on the bundles from the left. Then in order to define the H principal left bundle E E˜ we need also a right action of H on E. We thus arrive at the following Definition 1. An H principal bibundle E on the base space M is a bundle on M that is both a left H principal bundle and a right H principal bundle and where left and right H actions commute ∀ h, k ∈ H , ∀e ∈ E, (k e) h = k(e h) ; we denote with p : E → M the projection to the base space.
(1)
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Before introducing the product between principal bibundles we briefly study their structure. A morphism W between two principal bibundles E and E˜ is a morphism between the bundles E and E˜ compatible with both the left and the right action of H : W (k e h) = k W (e) ˜ h;
(2)
˜ As for morphisms between principal bundles on here ˜ is the right action of H on E. the same base space M, we have that morphisms between principal bibundles on M are isomorphisms. Trivial bibundles. Since we consider only principal bibundles we will frequently write bibundle for principal bibundle. The product bundle M × H , where left and right actions are the trivial ones on H [i.e. k (x, h) h = (x, khh )] is a bibundle. We say that a bibundle T is trivial if T is isomorphic to M × H . Proposition 2. We have that T is trivial as a bibundle iff it has a global central section, i.e. a global section σ that intertwines the left and the right action of H on T : ∀h ∈ H , ∀x ∈ M, h σ (x) = σ (x) h.
(3)
Proof. Let σ be a global section of T , define Wσ : M × H → T as Wσ (x, h) = h σ (x), then T and M × H are isomorphic as left principal bundles. The isomorphism Wσ is also a right principal bundle isomorphism iff (3) holds. Note also that the section σ is unique if H has trivial centre. An example of nontrivial bibundle is given by the trivial left bundle M × H equipped with the nontrivial right action (x, h) h = (x, hχ (h )), where χ is an outer automorphism of H . We thus see that bibundles are in general not locally trivial. Short exact sequences of groups provide examples of bibundles that are in general nontrivial as left bundles [cf. (112), (113)]. The ϕ map. We now further characterize the relation between left and right actions. Given a bibundle E, the map ϕ : E × H → H defined by ∀e ∈ E , ∀h ∈ H , ϕ e (h) e = e h
(4)
is well defined because the left action is free, and transitive on the fibers. For fixed e ∈ E it is also one-to-one since the right action is transitive and left and right actions are free. Using the compatibility between left and right actions it is not difficult to show that ϕ is equivariant w.r.t. the left action and that for fixed e ∈ E it is an automorphism of H : ϕ he (h ) = hϕ e (h )h−1 ,
(5)
ϕ e (hh ) = ϕ e (h)ϕ e (h ),
(6)
ϕ eh (h ) = ϕ e (hh h−1 ).
(7)
we also have
Vice versa, given a left bundle E with an equivariant map ϕ : E × H → H that restricts to an H automorphism ϕ e , we have that E is a bibundle with right action defined by (4). Using the ϕ map we have that a global section σ is a global central section (i.e. that a trivial left principal bundle is trivial as a bibundle) iff [cf. (3)], ∀x ∈ M and ∀h ∈ H ,
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ϕ σ (x) (h) = h.
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(8)
In particular, since e ∈ E can be always written as e = h σ , we see that ϕ e is always an inner automorphism, ϕ e (h) = ϕ h σ (h) = Adh (h) .
(9)
Vice versa, we have that Proposition 3. If H has trivial centre then an H bibundle E is trivial iff ϕ e is an inner automorphism for all e ∈ E. Proof. Consider the local sections t i : U i → E, since H has trivial centre the map k(t) : U i → H is uniquely defined by ϕ t i (h ) = Adk(t i ) h . From (5), ϕ ht i (h ) = −1
Adh Adk(t i ) h , and therefore the sections k(t i ) t i are central because they satisfy ϕ k(t i )−1 t i (h ) = h . In the intersections U ij = U i ∩ U j we have t i = hij t j and −1 i t
therefore k(t i )
−1 j t . We
= k(t j )
can thus construct a global central section.
Any principal bundle with H abelian is a principal bibundle in a trivial way, the map ϕ is given simply by ϕ e (h) = h. Now let us recall that a global section σ : M → E on a principal H -bundle E → M can be identified with an H -equivariant map σ : E → H . With our (left) conventions, ∀E ∈ E, e = σ (e)σ (x) . Notice, by the way, that if E is a trivial bibundle with a global section σ , then σ is bi-equivariant, i.e.: σ (heh ) = hσ (e)h iff σ is central. We apply this description of a global section of a left principal bundle to the following situation: Consider an H -bibundle E. Let us form Aut (H ) ×H E with the help of the canonical homomorphism Ad : H → Aut (H ). Then it is straightforward to check that σ : [η, e] → η ◦ ϕ e with η ∈ Aut (H ) is a global section of the left Aut (H )-bundle Aut (H ) ×H E. So Aut (H ) ×H E is trivial as a left Aut (H )-bundle. On the other hand if E is a left principal H -bundle such that Aut (H ) ×H E is a trivial left Aut (H )-bundle then it has a global section σ : Aut (H ) ×H E → Aut (H ) and the structure of an H -bibundle on E is given by ϕ e ≡ σ ([id, e]). We can thus characterize H -bibundles without mentioning their right H structure, Proposition 4. A left H -bundle E is an H -bibundle if and only if the (left) Aut (H )bundle Aut (H ) ×H E is trivial. Any trivial left H -bundle T can be given a trivial H -bibundle structure. We consider a trivialization of T , i.e. an isomorphism T → M × H and pull back the trivial right H -action on M × H to T . This just means that the global section of the left H -bundle T associated with the trivialization T → M × H , is by definition promoted to a global central section. Product of bibundles. In order to define the product bundle E E˜ we first consider the fiber product (Withney sum) bundle E ⊕ E˜ ≡ {(e, e) ˜ | p(e) = p( ˜ e)} ˜
(10)
with projection ρ : E ⊕ E˜ → M given by ρ(e, e) ˜ = p(e) = p( ˜ e). ˜ We now can define the product bundle E E˜ with base space M via the equivalence relation
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∀h ∈ H (e, he) ˜ ∼ (e h , e); ˜
(11)
we write [e, e] ˜ for the equivalence class and E E˜ ≡ E ⊕H E˜ ≡ { [e, e] ˜ }.
(12)
The projection p p˜ : E E˜ → M is given by pp[e, ˜ e] ˜ = p(e) = p( ˜ e). ˜ One can show that E E˜ is an H principal bundle; the action of H on E E˜ is inherited from that on E: h[e, e] ˜ = [he, e]. ˜ Concerning the product of sections we have that if s : U → E is a ˜ then section of E (with U ⊆ M), and s˜ : U → E˜ is a section of E, s˜ s ≡ [ s, ˜ s ] : U → E E˜
(13)
˜ is the corresponding section of E E. ˜ When also E is an H principal bibundle, with right action , ˜ then E E˜ is again an H principal bibundle with right action ˜ given by [e, e] ˜ ˜ h = [e, e˜ ˜ h].
(14)
It is easy to prove that the product between H principal bibundles is associative. Inverse bibundle. The inverse bibundle E −1 of E has by definition the same total space and base space of E but the left action and the right actions −1 are defined by h e−1 = (e h−1 )−1 , e−1 −1 h = (h−1 e)−1 ;
(15)
here e−1 and e are the same point of the total space, we write e−1 when the total space is endowed with the E −1 principal bibundle structure, we write e when the total space is endowed with the E principal bibundle structure. From Definition (15) it follows that he−1 = e−1 −1 ϕ e (h). Given the sections t i : U i → E of E we canonically have the −1 −1 : U i → E −1 of E −1 (here again t i (x) and t i (x) are the same point sections t i −1 of the total space). The section t i t i of E −1 E is central, i.e. it satisfies (3). We also −1 −1 have t i t i = t j t j in U ij ; we can thus define a canonical (natural) global central section I of E −1 E, thus showing that E −1 E is canonically trivial. Explicitly we have I [e −1 , e] = h with e h = e. Similarly for EE −1 . The space of isomorphism classes of H -bibundles on M [cf. (2)] can now be endowed with a group structure. The unit is the isomorphism class of the trivial product bundle M × H . The inverse of the class represented by E is the class represented by E −1 . Consider two isomorphic bibundles E and E on M. The choice of a specific isomorphism between E and E is equivalent to the choice of a global central section of the bibundle EE −1 , i.e. a global section that satisfies (3). Indeed, let f be a global section −1 of EE , given an element e ∈ E with base point x ∈ M, there is a unique element e −1 ∈ E −1 with base point x ∈ M such that [e, e −1 ] = f (x). Then the isomorphism E ∼ E is given by e → e ; it is trivially compatible with the right H -action, and it is compatible with the left H -action and because of the centrality of f . W
More generally let us consider two isomorphic left H -bundles E ∼ E which are not necessarily bibundles. Let us write a generic element (e, e ) ∈ E ⊕ E in the form (e, hW (e)) with a properly chosen h ∈ H . We introduce an equivalence relation on
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E ⊕ E by (e, hW (e)) ∼ (h e, hW (h e)). Then T = E ⊕ E / ∼ is a trivial left H -bundle with global section σ¯ ([e, hW (e)]) = h−1 (the left H -action is inherited from E). Recalling the comments after Proposition 4, we equip T with trivial H -bibundle structure and global central section σ¯ . Next we consider the product T E and observe that any element [[e, e1 ], e2 ] ∈ T E can be written as [[e, ˜ W (e)], ˜ W (e)] ˜ with a unique e˜ ∈ E. We thus have a canonical isomorphism between E and T E and therefore we write E = T E . Vice versa if T is a trivial bibundle with global central section σ¯ : T → H and E,E are left H -bundles and E = T E , i.e E is canonically isomorphic to T E , then W
we can consider the isomorphism E ∼ E defined by W ([t, e ]) = σ¯ (t)e (here [t, e ] is thought of as an element of E because of the identification E = T E ). It is then easy to see that the trivial bibundle with section given by this isomorphism W is canonically isomorphic to the initial bibundle T . We thus conclude that the choice of an isomorphism between two left H -bundles E and E is equivalent to the choice of a trivialization (the choice of a global central section) of the bibundle T , in formulae W
E ∼ E ⇐⇒ E = T E ,
(16)
where T has a given global central section. Local coordinates description. We recall that an atlas of charts for an H principal left bundle E with base space M is given by a covering {U i } of M, together with sections t i : U i → E (the sections t i determine isomorphisms between the restrictions of E to U i and the trivial bundles U i × H ). The transition functions hij : U ij → H are defined by t i = hij t j . They satisfy on U ij k the cocycle condition hij hj k = hik . −1
On U ij we have hij = hj i . A section s : U → E has local representatives {s i } where s i : U ∩ U i → H and in U ij we have s i hij = s j .
(17)
ϕ i ≡ ϕ t i : U i → Aut (H ),
(18)
If E is also a bibundle we set
and we then have ∀ h ∈ H , ϕ i (h)hij = hij ϕ j (h) , i.e. Adhij = ϕ i ◦ ϕ j
−1
.
(19)
We call the set {hij , ϕ i } of transition functions and ϕ i maps satisfying (19) a set of local data of E. A different atlas of E, i.e. a different choice of sections t i = r i t i where r i : U i → H (we can always refine the two atlases and thus choose a common covering {U i } of M), gives local data
h ij = r i hij r j
−1
,
(20)
ϕ = Adr i ◦ ϕ .
(21)
i
i
We thus define two sets of local data {hij , ϕ i } and {hij , ϕ i } to be equivalent if they are related by (20), (21).
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One can reconstruct an H -bibundle E from a given set of local data {hij , ϕ i } relative to a covering {U i } of M. For short we write E = {hij , ϕ i }. The total space of this bundle is the set of triples (x, h, i) where x ∈ U i , h ∈ H , modulo the equivalence relation (x, h, i) ∼ (x , h , j ) iff x = x and hhij = h . We denote the equivalence class by [x, h, i]. The left H action is h [x, h, i] = [x, h h, i]. The right action, given by [x, h, i] h = [x, hϕ i (h ), i] is well defined because of (19). The hij ’s are transition functions of the atlas given by the sections t i : U i → E, t i (x) = [x, 1, i], and we have ϕ t i = ϕ i . It is now not difficult to prove that equivalence classes of local data are in
−1
one-to-one correspondence with isomorphism classes of bibundles. [Hint: t i (r i t i ) is central and i independent.] Given two H bibundles E = {hij , ϕ i } and E˜ = {h˜ ij , ϕ˜ i } on the same base space M, the product bundle E E˜ has transition functions and left H -actions given by (we can ˜ always choose a covering {U i } of M common to E and E) E E˜ = {hij ϕ j (h˜ ij ) , ϕ i ◦ ϕ˜ i }.
(22)
If E˜ is not a bibundle the product E E˜ is still a well defined bundle with transition functions hij ϕ j (h˜ij ). Associativity of the product (22) is easily verified. One also shows that if s i , s˜ i : U ∩ U i → H are local representatives for the sections s : U → E and s˜ : U → E˜ then the local representative for the product section s˜ s : U → E E˜ is given by s i ϕ i (˜s i ) .
(23)
The inverse bundle of E = {hij , ϕ i } is E −1 = {ϕ j
−1
−1
−1
−1
(hij
−1
) , ϕi
−1
}
(24)
(we also have ϕ j (hij )−1 = ϕ i (hij ) ). If s : U → E is a section of E with −1 −1 representatives {s i } then s −1 : U → E −1 , has representatives {ϕ i (s i )}. A trivial bundle T with global central section t, in an atlas of charts subordinate to a cover U i of the base space M, reads T = {f i f j
−1
, Adf i } ,
(25) −1
where the section t ≡ f −1 has local representatives {f i } . For future reference notice −1 that T −1 = {f i f j , Adf i −1 } has global central section f = {f i }, and that E T −1 E −1 is trivial, E T −1 E −1 = {ϕ i (f i
−1
)ϕ j (f j ), Adϕ i (f i −1 ) } .
(26)
We denote by ϕ(f ) the global central section {ϕ i (f i )} of E T −1 E −1 . Given an arbitrary section s : U → E, we have, in U ϕ(f ) = s f s −1 . Proof. f s −1 = {f i Adf i −1 (ϕ i
−1
(s i
−1
))} = {ϕ i
−1
(27) (s i
−1
)f i } and therefore sf s −1 =
{ϕ i (f i )} = ϕ(f ). Property (27) is actually the defining property of ϕ(f ). Without using an atlas of charts, we define the global section ϕ(f ) of ET −1 E −1 to be that section that locally satisfies (27). The definition is well given because centrality of f implies that
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ϕ(f ) is independent from s. Centrality of the global section f also implies that ϕ(f ) is a global central section. If σ is the global central section of T , the corresponding global section σ of E T −1 E −1 is σ [e, t, e −1 ] = ϕe (σ (t))h with e = he . The pull-back of a bi-principal bundle is again a bi-principal bundle. It is also easy to verify that the pull-back commutes with the product. D-H bundles. We can generalize the notion of a bibundle by introducing the concept of a crossed module. We say that H is a crossed D-module [35] if there is a group homomorphism α : H → D and an action of D on H denoted as (d, h) → d h such that ∀h, h ∈ H ,
α(h)
h = hh h−1
(28)
and for all h ∈ H , d ∈ D, α( d h) = dα(h)d −1
(29)
holds true. Notice in particular that α(H ) is normal in D. The canonical homomorphism Ad : H → Aut (H ) and the canonical action of Aut (H ) on H define on H the structure of a crossed Aut (H )-module. Given a D-bundle Q we can use the homomorphism t : D → Aut (H ), t ◦ α = Ad to form Aut (H ) ×D Q. Definition 5. Consider a left D-bundle Q on M such that the Aut (H )-bundle Aut (H )×D Q is trivial. Let σ be a global section of Aut (H ) ×D Q. We call the couple (Q, σ ) a D-H bundle. Notice that if σ : Aut (H ) ×D Q [η, q] → σ ([η, q]) ∈ Aut (H ) is a global section of Aut (H ) ×D Q then i) the automorphism ψ q ∈ Aut (H ) defined by ψ q ≡ σ ([id, q])
(30)
is D-equivariant, ψ dq (h) =
d
ψ q (h),
(31)
ii) the homomorphism ξ q : H → D defined by ξ q (h) ≡ α ◦ ψ q (h)
(32)
gives a fiber preserving action q h ≡ ξ q (h)q of H on the right, commuting with the left D-action, i.e. ∀ h ∈ H, d ∈ D, q ∈ Q, (d q) h = d(q h) .
(33)
Vice versa we easily have Proposition 6. Let H be a crossed D-module. If Q is a left D bundle admitting a right fiber preserving H action commuting with the left D action, and the homomorphism ξ q : H → D, defined by q h = ξ q (h)q is of the form (32) with a D-equivariant ψ q ∈ Aut (H ) [cf. (31)], then Q is a D-H bundle.
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There is an obvious notion of an isomorphism between two D-H bundles (Q, σ ) and ˜ σ˜ ); it is an isomorphism between D-bundles Q and Q ˜ intertwining between σ and (Q, σ˜ . In the following we denote a D-H bundle (Q, σ ) simply as Q without spelling out explicitly the choice of a global section σ of Aut (H ) ×D Q. As in the previous section out of a given isomorphism we can construct a trivial D-bibundle Z with a global central ˜ and ZQ are canonically identified and we again write this as section z−1 such that Q ˜ Q = ZQ. The ψ map of Z is given by Adz−1 . Note that the product of a trivial D-bibundle Z and a D-H bundle Q is well-defined and gives again a D-H bundle. The trivial bundle M ×D → M, with right H -action given by (x, d)h = (x, dα(h)), is a D-H bundle, we have ψ (x,d) (h) = d h. A D-H bundle Q is trivial if it is isomorphic to M × D. Similarly to the case of a bibundle we have that a D-H bundle is trivial iff it has a global section σ which is central with respect to the left and the right actions of H on Q, σ (x) h = α(h)σ (x) .
(34)
The corresponding map σ : Q → D is then bi-equivariant σ (dq h) = d σ (q)α(h).
(35)
The pull-back of a D-H bundle is again a D-H bundle. The trivial bundle Aut (H ) ×H E (cf. Proposition 4) is an Aut (H )-H bundle. Proof. The left Aut (H ) and the right H actions commute, and they are related by [η, e]h = Adη(ϕ e (h)) [η, e]; we thus have ψ [η,e] = η ◦ ϕ e , which structures Aut (H ) ×H E into an Aut (H )-H bundle. Moreover σ ([η, e]) = η ◦ ϕ e is bi-equivariant, hence Aut (H ) ×H E is isomorphic to M × Aut (H ) as an Aut (H )-H bundle. More generally, we can use the left H -action on D given by the homomorphism α : H → D to associate to a bibundle E the bundle D ×H E. The H -automorphism ψ [d,e] defined by ψ [d,e] = d ϕ e (h) endows D ×H E with a D-H bundle structure. There is the following canonical construction associated with a D-H module. We use the D-action on H to form the associated bundle H ×D Q. Using the equivariance property (31) of ψ q we easily get the following proposition. Proposition 7. The associated bundle H ×D Q is a trivial H -bibundle with actions h [h, q] = [ψ q (h )h, q] and [h, q] h = [hψ q (h ), q], and with global central section given by σ¯ ([h, q]) = ψ −1 q (h). The local coordinate description of a D-H bundle Q is similar to that of a bibundle. We thus omit the details. We denote by d ij the transition functions of the left principal D-bundle Q. Instead of local maps (18) we now have local maps ψ i : U i → Aut (H ), such that (compare to (19)) dij
−1
h = ψ i ◦ ψ j (h).
(36)
The product QE of a D-H bundle Q with a H -bibundle E can be defined as in (11), (12). The result is again a D-H bundle. If Q is locally given by {d ij , ψ i } and H is locally given by {hij , ϕ i } then QE is locally given by {d ij ξ j (hij ), ψ i ◦ ϕ i }. Moreover
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−1
if Z = {zi zj , Adzi } is a trivial D-bibundle with section z−1 = {zi }, then the well−1 i defined D-H bundle ZQ is locally given by {zi d ij zj , z ◦ ψ i }. We have the following associativity property: (ZQ)E = Z(QE) ,
(37)
and the above products commute with pull-backs. Given a D-H bundle Q and a trivial H -bibundle T with section f −1 there exists a unique trivial D-bibundle ξ (T ) with section ξ (f −1 ) such that QT = ξ (T )Q ,
(38)
i.e. such that for any local section s of Q one has sf −1 = ξ (f −1 )s. The notations ξ (T ), ξ (f −1 ) are inferred from the local expressions of these formulae. Indeed, if locally −1 T = {f i f j , Adf i } and f = {f i }, then ξ (T ) = {ξ i (f i )ξ j (f j )−1 , Adξ i (f i ) } and ξ (f ) = {ξ i (f i )}. Finally, as was the case for bibundles, we can reconstruct a D-H bundle Q from a given set of local data {d ij , ψ i } relative to a covering {U i } of M. Equivalence of local data for D-H bundles is defined in such a way that isomorphic (equivalent) D-H bundles have equivalent local data, and vice versa. 3. Connection and Curvature on Principal Bibundles Since a bibundle E on M is a bundle on M that is both a left principal H -bundle and a right principal H -bundle, one could then define a connection on a bibundle to be a one-form a on E that is both a left and a right principal H -bundle connection. This definition [more precisely the requirement Ar = 0 in (49)] preserves the left-right symmetry property of the bibundle structure, but it turns out to be too restrictive, indeed not always a bibundle can be endowed with such a connection, and furthermore the corresponding curvature is valued in the center of H . If we insist in preserving the left-right symmetry structure we are thus led to generalize (relax) equivariance property of a connection and thus define the notion of connection. In this section we will see that a connection on a bibundle is a couple (a, A), where a is a one-form on E with values in Lie(H ) while A is a Lie(Aut (H )) valued one-form on M. In particular we see that if A = 0 then a is a left connection on E where E is considered just as a left principal bundle. We recall that a connection a on a left principal bundle E satisfies [31]. i) The pull-back of a on the fibers of E is the right invariant Maurer-Cartan one-form. Explicitly, let e ∈ E, let g(t) be a curve from some open interval (−ε, ε) of the real line into the group H with g(0) = 1H , and let [g(t)] denote the corresponding tangent vector in 1H and [g(t)e] the vertical vector based in e ∈ E. Then a[g(t)e] = −[g(t)] .
(39)
Equivalently a[g(t)e] = ζ[g(t)] , where ζ[g(t)] is the right-invariant vector field associated with [g(t)] ∈ Lie(H ), i.e. ζ[g(t)] |h = −[g(t)h]. ii) Under the left H -action we have the equivariance property ∗
l h a = Adh a, where l h denotes left multiplication by h ∈ H .
(40)
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Now property i) is compatible with the bibundle structure on E in the following sense, if a satisfies i) then −ϕ −1 (a) pulled back on the fibers is the left invariant Maurer-Cartan one-form −ϕ −1 (a)[eg(t)] = [g(t)] ,
(41)
here with abuse of notation we use the same symbol ϕ −1 for the map ϕ −1 : E ×H → H and its differential map ϕ −1 ∗ : E × Lie(H ) → Lie(H ). Property (41) is equivalent to a[g(t)e] = ξ[g(t)] , where ξ[g(t)] is the left-invariant vectorfield associated with [g(t)] ∈ Lie(H ), i.e. ξ[g(t)] |h = [hg(t)]. Property (41) is easily proven, −1 −ϕ −1 (a)[eg(t)] = −ϕ −1 e (a[ϕ e (g(t))e]) = ϕ e [ϕ e (g(t))] = [g(t)] . ∗ ∗ Similarly, on the vertical vectors vV of E we have r h a − a (vV ) = 0 , l h ϕ −1 (a)− ϕ −1 (a) (vV ) = 0 and ∗ l h a − Adh a (vV ) = 0 , (42) ∗ r h ϕ −1 (a) − Adh−1 ϕ −1 (a) (vV ) = 0 . (43)
On the other hand property ii) is not compatible with the bibundle structure, indeed if a satisfies (40) then it can be shown (see later) that −ϕ −1 (a) satisfies ∗
r h ϕ −1 (a) = Adh−1 ϕ −1 (a) − p ∗ T (h−1 ),
(44)
where T (h) is a given one-form on the base space M, and p : E → M. In order to preserve the left-right symmetry structure we are thus led to generalize (relax) the equivariance property ii) of a connection. Accordingly with (42) and (44) we thus require ∗
l h a = Adh a + p ∗ T (h),
(45)
where T (h) is a one-form on M. From (45) it follows T (hk) = T (h) + Adh T (k) ,
(46)
i.e., T is a 1-cocycle in the group cohomology of H with values in Lie(H ) ⊗ 1 (M). Of course if T is a coboundary, i.e. T (h) = hχ h−1 − χ with χ ∈ Lie(H ) ⊗ 1 (M), then a + χ is a connection. We thus see that Eq. (45) is a nontrivial generalization of the equivariance property only if the cohomology class of T is nontrivial. Given an element X ∈ Lie(Aut (H )), we can construct a corresponding 1-cocycle TX in the following way, TX (h) ≡ [hetX (h−1 )] , where [hetX (h−1 )] is the tangent vector to the curve hetX (h−1 ) at the point 1H ; if H is normal in Aut (H ) then etX (h−1 ) = etX h−1 e−tX and we simply have TX (h) = hXh−1 − X. Given a Lie(Aut (H ))-valued one-form A on M, we write A = Aρ X ρ , where {Xρ } is a basis of Lie(Aut (H )). We then define TA as TA ≡ Aρ TXρ .
(47)
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Obviously, p∗ TA = Tp∗A . Following these considerations we define Definition 8. A 2-connection on E is a couple (a, A) where: i) a is a Lie(H ) valued one-form on E such that its pull-back on the fibers of E is the right invariant Maurer-Cartan one-form, i.e. a satisfies (39), ii) A is a Lie(Aut (H )) valued one-form on M, iii) the couple (a, A) satisfies ∗
l h a = Adh a + p ∗ TA (h) .
(48)
This definition seems to break the left-right bibundle symmetry since, for example, only the left H action has been used. This is indeed not the case Theorem 9. If (a, A) is a 2-connection on E then (a r , Ar ), where a r ≡ −ϕ −1 (a), satisfies (39) and (48) with the left H action replaced by the right H action (and rightinvariant vectorfields replaced by left-invariant vectorfields), i.e. it satisfies (41) and ∗
r h a r = Adh−1 a r + p ∗ TAr (h−1 ) ,
(49)
here Ar is the one-form on M uniquely defined by the property p∗Ar = ϕ −1 (p ∗A + ada )ϕ + ϕ −1 dϕ .
(50)
Proof. First we observe that from (39) and (48) we have ∗
l h a = Adh a + p ∗ TA (h ) + h dh −1 ,
(51)
where now h = h (e), i.e. h is an H -valued function on the total space E. Setting h = ϕ(h), with h ∈ H we have ∗
∗
r h a = l ϕ(h) a = Adϕ(h) a + p ∗ TA (ϕ(h)) + ϕ(h)dϕ(h−1 ) = a + ϕ(TAr (h));
(52)
in equality (52) we have defined Ar ≡ ϕ −1 (p ∗A + ada )ϕ + ϕ −1 dϕ . Equality (52) holds because of the following properties of T , Tϕ −1 dϕ (h) = ϕ −1 ϕ(h)dϕ(h−1 ) , Tϕ −1 p∗Aϕ (h) = ϕ −1 Tp∗A (ϕ(h)) , Tada (h) = Adh a − a .
(53)
(54) (55) (56)
From (52), applying ϕ −1 and then using (7) one obtains ∗
r h a r = Adh−1 a r + TAr (h−1 ) .
(57)
Finally, comparing (43) with (57) we deduce that for all h ∈ H , TAr (h)(vV ) = 0, and this relation is equivalent to Ar (vV ) = 0. In order to prove that Ar = p∗Ar , where Ar is a one-form on M, we then just need to show that Ar is invariant under the H action, ∗ ∗ l h Ar = Ar . This is indeed the case because l h (ϕ −1 dϕ) = ϕ −1 Adh−1 dAdh ϕ = −1 ϕ dϕ, and because
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∗ ϕ −1 (p ∗A + ada )ϕ = ϕ −1 Adh−1 (p ∗A + l h ada )Adh ϕ = ϕ −1 Adh−1 p ∗AAdh + ada + adAdh−1 Tp∗A (h) ϕ
= ϕ −1 (p ∗A + ada )ϕ .
Notice that if (a, A) and (a , A ) are 2-connections on E then so is the affine sum
p∗ (λ)a + (1 − p ∗ λ)a , λA + (1 − λ)A
(58)
for any (smooth) function λ on M. As in the case of principal bundles we define a vector v ∈ Te E to be horizontal if a(v) = 0. The tangent space Te E is then decomposed in the direct sum of its horizontal and vertical subspaces; for all v ∈ Te E, we write v = Hv +Vv, where V v = [e−ta(v) e]. The space of horizontal vectors is however not invariant under the usual left H -action, indeed a(l h ∗ (Hv)) = TA (h)(v) , in this formula, as well as in the sequel, with abuse of notation TA stands for Tp∗A . Remark 10. It is possible to construct a new left H -action L∗ on T∗ E, that is compatible with the direct sum decomposition T∗ E = HT∗ E +VT∗ E. We first define, for all h ∈ H , LhA : T∗ E → VT∗ E , Te E v → [etTA (h)(v) he] ∈ VThe E ,
(59)
and notice that LhA on vertical vectors is zero, therefore LhA ◦ LhA = 0. We then consider the tangent space map, Lh∗ ≡ l∗h + LhA .
(60)
h k It is easy to see that Lhk ∗ = L∗ ◦ L∗ and therefore that L∗ defines an action of H on T∗ H . We also have ∗
Lh a = Adh a .
(61)
Finally the action Lh∗ preserves the horizontal and vertical decomposition T∗ E = HT∗ E + VT∗ E, indeed HLh∗ v = Lh∗ Hv , VLh∗ v = Lh∗Vv.
(62)
Proof. Let v = [γ (t)]. Then HLh∗ v = Hl∗h v = [hγ (t)] − [e−ta[hγ (t)] e] = [hγ (t)] + h∗ [et (l a)(v) he] = [hγ (t)]+[heta(v) e]+[etTA (h)(v) he] = Lh∗ (v +[eta(v) e]) = Lh∗ Hv . Curvature. An n-form ϑ is said to be horizontal if ϑ(u1 , u2 , . . . un ) = 0 whenever at least one of the vectors ui ∈ Te E is vertical. The exterior covariant derivative Dω of an n-form ω is the (n + 1)-horizontal form defined by
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Dω(v1 , v2 , . . . , vn+1 ) ≡ dω(Hv1 , Hv2 , . . . , Hvn+1 ) −(−1)n TA (ω)(Hv1 , Hv2 , . . . , Hvn+1 )
(63)
for all vi ∈ Te E and e ∈ E. In the above formula TA (ω) is defined by TA (ω) ≡ ωα ∧ TA∗ (X α ) ,
(64)
where TA∗ : Lie(H ) → Lie(H ) ⊗ 1 (E) is the differential of TA : H → Lie(H ) ⊗ 1 (E). If H is normal in Aut (H ) we simply have TA (ω) = ωρ ∧ p∗Aσ [X ρ , Xσ ] = [ω, p∗A], where now X ρ are generators of Lie(Aut (H )). The 2-curvature of the 2-connection (a, A) is given by the couple (k, K) ≡ Da , dA + A ∧ A . (65) We have the Cartan structural equation 1 k = da + [a, a] + TA (a) , 2
(66)
where 21 [a, a] = 21 a α ∧ a β [X α , Xβ ] = a ∧ a with X α ∈ Lie(H ). The proof of Eq. (66) is very similar to the usual proof of the Cartan structural equation for principal bundles. One has just to notice that the extra term TA (a) is necessary since da(Vv, Hu) = −a([Vv, Hu]) = TA∗ (a(Vv))(Hu) = −TA (a)(Vv, Hu). The 2-curvature (k, K) satisfies the following generalized equivariance property: ∗
l h k = Adh k + TK (h) ,
(67)
where with abuse of notation we have written TK (h) instead of Tp∗ K (h). We also have the Bianchi identities, dK + A ∧ K = 0 and Dk = 0 .
(68) ∗
Given an horizontal n-form ϑ on E that is -equivariant, i.e. that satisfies l h ϑ = Adh ϑ + T (h) , where is an n-form on M, we have the structural equation Dϑ = dϑ + [a, ϑ] + T (a) − (−1)n TA (ϑ),
(69)
where [a, ϑ] = a α ∧ ϑ β [X α , Xβ ] = a ∧ ϑ − (−1)n ϑ ∧ a. The proof is again similar to the usual one (where = 0) and is left to the reader. We also have that Dϑ is (d + [A, ])-equivariant, ∗
l h Dϑ = Adh ϑ + Td+[A,] (h) .
(70)
Combining (69) and (68) we obtain the explicit expression of the Bianchi identity dk + [a, k] + TK (a) − TA (k) = 0.
(71)
D 2 ϑ = [k, ϑ] + T (k) − (−1)n TK (ϑ) .
(72)
We also have
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As was the case for the 2-connection (a, A), also for the 2-curvature (k, K) we can have a formulation using the right H action instead of the left one. Indeed one can prove that if (k, K) is a 2-curvature then (k r , K r ) where k r = −ϕ −1 (k) , K r = ϕ −1 (K + adk )ϕ is the right 2-curvature associated with the right 2-connection (a r , Ar ). In other words we have that k r is horizontal and that k r = k a r , K r = KAr (for the proof we used TAr (ϕ −1 (X)) = ϕ −1 ([X, a]+TA (X))+dϕ −1 (X), X ∈ Lie(H )). We also have ∗
r h k r = Adh−1 k r + TK r (h−1 ) .
(73)
More in general consider the couple (ϑ, ) where ϑ, is an horizontal n-form on E that is -equivariant. Then we have the couple (ϑ r , r ), where ϑ r = −ϕ −1 (ϑ) is an horizontal n-form on E that is right r -equivariant, ∗
r h ϑ r = Adh−1 ϑ r + Tr (h−1 ),
(74)
with r = ϕ −1 ( + adϑ )ϕ. The pull-back of a 2-connection (or of a horizontal form) on a principal H -bibundle is a 2-connection (horizontal form) on the pulled back principal H -bibundle, moreover the exterior covariant derivative -and in particular the definition of 2-curvature- commutes with the pull-back operation. Local coordinates description.. Let’s consider the sections t i : U i → E subordinate to the covering {U i } of M. Let ι : H ×U i → p −1 (U i ) ⊂ E be the local trivialization of E induced by t i according to ι(x, h) = ht i (x), where x ∈ M. We define the one-forms on U i ⊂ M, ∗
ai = t i a ,
(75)
then, the local expression of a is ha i h−1 + TA (h) + hdh−1 , more precisely, ι∗ (a)(x,h) (vM , vH ) = ha i (x)h−1 (vM ) + TA(x) (h)(vM ) + hdh−1 (vH ) ,
(76)
where vM ,vH are respectively tangent vectors of U i ⊂ M at x, and of H at h, and where −hdh−1 denotes the Maurer—Cartan one-form on H evaluated at h ∈ H . Similarly the ∗ local expression for k is hk i h−1 + TK (h), where k i = t i k. i Using the sections {t } we also obtain an explicit expression for Ar , ∗
Ar = t i Ar = ϕi−1 (A + ad a i )ϕi + ϕi−1 dϕi . ∗
∗
(77)
Of course in U ij we have t i Ar = t j Ar , so that Ar is defined on all M. In U ij we also −1 −1 −1 have a i = hij a j hij + hij dhij + TA (hij ) and k i = hij k j hij + TK (hij ) .
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Sum of 2-connections. If the group H is abelian, on the product bundle E1 E2 there is the natural connection a 1 +a 2 obtained from the connections a 1 and a 2 on E1 and E2 . In this subsection we generalize to the nonabelian case the sum of connections. Consider the following diagram: / E2 E1 ⊕ EJ2 JJ π JJ ⊕ π1 JJ JJ % E1 E1 E2 π2
(78)
and let (a 1 , A2 ) be a 2-connection on E1 and (a 2 , A2 ) a 2-connection on E2 . Recalling the definition of the product E1 E2 , we see that the one-form on E1 ⊕ E2 π1∗ a 1 + ϕ 1 (π2∗ a 2 )
(79)
is the pull-back of a one-form on E1 E2 iff, for all v1 ∈ Te1 E, v2 ∈ Te2 E and h ∈ H , ∗ π1 a 1 + ϕ 1 (π2∗ a 2 ) (e ,e ) (v1 , v2 ) 1 2 = π1∗ a 1 + ϕ 1 (π2∗ a 2 ) (e h−1 ,he ) (r∗h v1 , l∗h v2 ) 1 2 ∗ ∗ + π1 a 1 + ϕ 1 (π2 a 2 ) (e h−1 ,he ) ([e1 h−1 (t)], [h(t)e2 ]), 1
2
where h(t) is an arbitrary curve in H with h(0) = 1H . Since a 1 and a 2 satisfy the CartanMaurer condition (39) the last addend vanishes identically and therefore the expression is equivalent to ∗ π1∗ a 1 + ϕ 1 (π2∗ a 2 ) = rl h π1∗ a 1 + ϕ 1 (π2∗ a 2 ) , (80) where rl h : E1 ⊕ E2 → E1 ⊕ E2 , (e1 , e2 ) → (e1 h−1 , he2 ) . Now, using (7), and then (52) we have −1 ∗ ∗ ∗ rl h π1∗ a 1 + ϕ 1 (π2∗ a 2 ) = π1∗ r h a 1 + ϕ 1 Adh−1 (π2∗ l h a 2 ) = π1∗ a 1 + ϕ 1 (π2∗ a 2 ) + ϕ 1 π1∗ TAr1 (h−1 ) + π2∗ Adh−1 TA2 (h) and the last addend vanishes iff A2 = A1 r .
(81)
In conclusion, when (81) holds, there exists a one-form on E1 E2 , denoted by a 1 +a 2 , such that ∗ π⊕ (a 1 +a 2 ) = π1∗ a 1 + ϕ 1 (π2∗ a 2 ).
(82)
From this expression it is easy to see that (a 1 +a 2 , A1 ) is a 2-connection on E1 E2 . We then say that (a 1 , A1 ) and (a 2 , A2 ) (or simply that a 1 and a 2 ) are summable and we write (a 1 , A1 ) + (a 2 , A2 ) = (a 1 +a 2 , A1 ) .
(83)
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Notice that the sum operation + thus defined is associative (and noncommutative). In other words, if a 1 and a 2 are summable, and if a 2 and a 3 are summable then a 1 +(a 2 +a 3 ) = (a 1 +a 2 )+a 3 and (a 1 +a 2 +a 3 , A1 ) is a 2-connection on E1 E2 E3 . We also have a summability criterion for the couples (ϑ 1 , 1 ) and (ϑ 2 , 2 ) where ϑ i , i = 1, 2 is an horizontal n-form on Ei that is i -equivariant. We have that (ϑ 1 , 1 )+ (ϑ 2 , 2 ) = (ϑ 1 +ϑ 2 , 1 ) where ∗ π⊕ (ϑ 1 +ϑ 2 ) = π1∗ ϑ 1 + ϕ 1 (π2∗ ϑ 2 )
(84)
is a well defined horizontal 1 -equivariant n-form on E1 E2 iff 2 = 1 r .
(85)
We have (Da 1 ϑ 1 , DA1 1 )+(Da 2 ϑ 2 , DA2 )2 = (Da 1 + a 2 (ϑ 1 +ϑ 2 ), DA1 1 ),
(86)
with obvious notation: Da ϑ = dϑ + [a, ϑ] + T (a) − (−1)n TA (ϑ) and DA = d + [A, ]. Also the summability of curvatures is a direct consequence of the summability of their corresponding connections. If (a 1 , A1 ) + (a 2 , A2 ) = (a 1 +a 2 , A1 ) then (k 1 , K1 ) + (k 2 , K2 ) = (k 1 +k 2 , K1 ) ,
(87)
k a1 + a2 = k 1 +k 2 .
(88)
and we also have
Summability is preserved under isomorphism, i.e. if a i are summable connections on Ei (i = 1, 2) and we have isomorphisms σi : Ei → Ei , then σi∗ (a i ) are summable and ∗ (a +a ), where we have considered the induced isomorphism σ1∗ (a 2 ) + σ2∗ (a 2 ) = σ12 1 2 σ12 ≡ σ1 σ2 : E1 E2 → E1 E2 . The same property holds for horizontal forms. 4. Nonabelian Bundle Gerbes Now that we have the notion of product of principal bibundles we can define nonabelian bundle gerbes generalizing the construction studied by Murray [5] (see also Hitchin [3] and [4]) in the abelian case. Consider a submersion ℘ : Y → M (i.e. a map onto with differential onto) we can always find a covering {Oα } of M with local sections σα : Oα → Y , i.e. ℘ ◦ σα = id. The manifold Y will always be equipped with the submersion ℘ : Y → M. We also consider Y [n] = Y ×M Y ×M Y . . . ×M Y the n-fold fiber product of Y , i.e. Y [n] ≡ {(y1 , . . . yn ) ∈ Y n | ℘ (y1 ) = ℘ (y2 ) = . . . ℘ (yn )}. ∗ (E) the H principal Given a H principal bibundle E over Y [2] we denote by E12 = p12 [3] [3] [2] bibundle on Y obtained as the pull-back of p12 : Y → Y (p12 is the identity on its first two arguments); similarly for E13 and E23 . Consider the quadruple (E, Y, M, f ), where the H principal bibundle on Y [3] , E12 E23 −1 −1 −1 E13 is trivial, and f is a global central section of E12 E23 E13 [i.e. f satisfies (3)]. Recalling the paragraph after formula (15) we can equivalently say that E12 E23 and E13 are isomorphic, the isomorphism being given by the global central section f −1 of −1 T ≡ E12 E23 E13 .
(89)
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We now consider Y [4] and the bundles E12 , E23 , E13 , E24 , E34 , E14 on Y [4] relative to the −1 −1 −1 projections p12 : Y [4] → Y [2] etc., and T123 , T124 , T134 relative to p123 : Y [4] → Y [3] , etc. Since the product of bundles commutes with the pull-back of bundles, we then have −1 −1 −1 −1 E12 (T234 E23 E34 ) = T134 (T123 E12 E23 )E34 = E14 T124
(90)
as bundles on Y [4] . The first identity in (90) is equivalent to −1 −1 −1 −1 −1 E12 T234 E12 = T134 T123 . T124
(91)
−1 −1 E12 and denote by Let us now consider the global central section f of T −1 = E13 E23 −1 −1 f 124 (f 234 , etc.) the global central section of T124 (T234 , etc.) obtained as the pull-back of f . Consistently with (91) we can require the condition
f 124 ϕ 12 (f 234 ) = f 134 f 123
(92)
−1 that in any open where, following the notation of (27), ϕ 12 (f 234 ) is the section of T234 −1 [4] U ⊂ Y equals s 12 f 234 s 12 , where s 12 : U → E12 is any section of E12 , in particular we can choose s 12 to be the pull-back of a section s of E.
Definition 11. A Bundle gerbe G is the quadruple (E, Y, M, f ) where the H prin−1 cipal bibundle on Y [3] , E12 E23 E13 is trivial and f is a global central section of −1 −1 that satisfies (92). E12 E23 E13 Recall that when H has trivial centre then the section f of T −1 is unique; it then follows that relation (92) is automatically satisfied because the bundle on the l.h.s. and the bundle on the r.h.s. of (91) admit just one global central section, respectively f 124 ϕ 12 (f 234 ) and f 134 f 123 . Therefore, if H has trivial centre, a bundle gerbe G is simply the triple −1 (E, Y, M), where E12 E23 E13 is trivial. Consider an H principal bibundle N over Z and let N1 = p1∗ (N ), N2 = p2∗ (N ), be the pull-back of N obtained respectively from p1 : Z [2] → Z and p2 : Z [2] → Z (p1 projects on the first component, p2 on the second). If (E, Z, M, f ) is a bundle gerbe also N1 EN2−1 , Z, M, ϕ 1 (f ) is a bundle gerbe. Here ϕ 1 (f ) is the canonical global central section of the bibundle N1 T −1 N1−1 and now N1 is the pull-back of N via p1 : Z [3] → Z; locally ϕ1 (f ) = s 1 f s −1 the pull-back of any local 1 , where s 1 is
−1 section s of N. Similarly also ηE, Z, M, −1 13 f ϕ 12 (23 )12 is a bundle gerbe if η
is a trivial bundle on Z [2] with global central section (as usual ϕ 12 (23 ) denotes the −1 −1 canonical section of E12 η23 E12 ). These observations lead to the following definition [36] Definition 12. Two bundle gerbes G = (E, Y, M, f ) and G = (E , Y , M, f ) are stably isomorphic if there exists a bibundle N over Z = Y ×M Y and a trivial bibundle η−1 over Z [2] with section such that
N1 q ∗ E N2−1 = η q ∗ E
(93)
and
∗ ϕ 1 (q ∗f ) = −1 13 q f ϕ 12 (23 )12 ,
(94)
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where q ∗ E and q ∗ E are the pull-back bundles relative to the projections q : Z [2] → Y [2] and q : Z [2] → Y [2] . Similarly q ∗f and q ∗f are the pull-back sections relative to the projections q : Z [3] → Y [3] and q : Z [3] → Y [3] . The relation of stable isomorphism is an equivalence relation. The bundle gerbe (E, Y, M, f ) is called trivial if it is stably isomorphic to the trivial bundle gerbe (Y × H, Y, M, 1); we thus have that E and N1−1 N2 are isomorphic as H -bibundles, i.e. E ∼ N1−1 N2
(95)
−1 −1 ≡ N E −1 N −1 . and that f = ϕ −1 2 1 (13 23 12 ), where is the global central section of η 1
Proposition 13. Consider a bundle gerbe G = (E, Y, M, f ) with submersion ℘ : Y → M; a new submersion ℘ : Y → M and a (smooth) map σ : Y → Y compatible with ℘ and ℘ (i.e. ℘ ◦ σ = ℘ ). The pull-back bundle gerbe σ ∗ G (with obvious abuse of notation) is given by (σ ∗ E, Y , M, σ ∗f ). We have that the bundle gerbes G and σ ∗ G are stably equivalent. Proof. Consider the following identity on Y [4] : −1 E11 E1 2 E22 = η12 E12 ,
(96)
−1 −1 −1 where η12 = T11 2 T122 so that η12 has section 12 = f 122 f 11 2 ; the labelling 1, 1 , 2, 2 instead of 1, 2, 3, 4 is just a convention. Multiplying three times (96) we obtain −1 −1 the following identity between trivial bundles on Y [6] E11 T1 2 3 E11 = η12 E12 η23 E12 −1 T123 η13 . The sections of (the inverses of) these bundles satisfy
ϕ 11 (f ) = −1 13 f ϕ 12 (23 )12 ,
(97)
thus E1 2 and E12 give stably equivalent bundle gerbes. Next we pull-back the bundles in (96) using (id, σ, id, σ ) : Z [2] → Y [4] where Z = Y ×M Y ; recalling that the product commutes with the pull-back we obtain relation (93) with η = (id, σ, id, σ )∗ η12 and N = (id, σ )∗ E. We also pull-back (97) with (id, σ, id, σ, id, σ ) : Z [3] → Y [6] and obtain formula (94). Theorem 14. Locally a bundle gerbe is always trivial: ∀x ∈ M there is an open O of x such that the bundle gerbe restricted to O is stably isomorphic to the trivial bundle gerbe (Y |[2] O × H, Y |O , O, 1). Here Y |O is Y restricted to O: Y |O = {y ∈ Y | ℘ (y) ∈ O ⊂ M}. Moreover in any sufficiently small open U of Y |[3] O one has −1
f = s 13 s 23 s −1 12
(98)
−1
−1 −1 with s −1 12 , s 23 and s 13 respectively sections of E12 , E23 and E13 that are pull-backs of sections of E.
Proof. Choose O ⊂ M such that there exists a section σ : O → Y |O . Define the maps [n+1] r[n] : Y |[n] , O → Y |O (y1 , . . . yn ) → (y1 , . . . yn , σ (℘ (yn )));
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notice that σ (℘ (y1 )) = σ (℘ (y2 )) . . . = σ (℘ (yn )). It is easy to check the following equalities between maps on Y |[2] O , p12 ◦ r[2] = id , p13 ◦ r[2] = r[1] ◦ p1 , p23 ◦ r[2] = r[1] ◦ p2 , and between maps on Y |[3] O , p123 ◦ r[3] = id , p124 ◦ r[3] = r[2] ◦ p12 , p234 ◦ r[3] = r[2] ◦ p23 , p134 ◦ r[3] = r[2] ◦ p13 .
(99)
−1 We now pull back with r[2] the identity E12 = T E13 E23 and obtain the following local trivialization of E ∗ E = r[2] (T ) N1 N2−1 , ∗ (E). Let U = U × U × U ⊂ Y |[3] , where N1 = p1∗ (N ), N2 = p2∗ (N ) and N = r[1] O O O where U, U , U are opens of Y |O that respectively admit the sections n : U → N , ∗ (f −1 )n n −1 : n : U → N , n : U → N . Consider the local sections s = r[2] 1 2 −1
−1
∗ (f −1 )n n : U × U → E, s = r ∗ (f −1 )n n : U ×O U → E, s = r[2] 2 3 1 3 [2] O U ×O U → E and pull them back to local sections s 12 of E12 , s 23 of E23 and s 13 of −1 E13 . Then (98) holds because, using (99), the product s 13 s 23 s −1 12 equals the pull-back −1 with r[3] of the section f 134 f 124 ϕ 12 (f 234 ) = f 123 [cf. (92)].
Local description. Locally we have the following description of a bundle gerbe; we choose an atlas of charts for the bundle E on Y [2] , i.e. sections t i : U i → E relative to a trivializing covering {U i } of Y [2] . We write E = {hij , ϕ i }. We choose also atlases ij ij i }, E i for the pull-back bundles E12 , E23 , E13 ; we write E12 = {h12 , ϕ12 23 = {h23 , ϕ23 }, ij i }, where these atlases are relative to a common trivializing covering {U i } E13 = {h13 , ϕ13 −1
−1
of Y [3] . It then follows that T = {f i f j , Adf i }, where {f i } are the local representatives for the section f −1 of T . We also consider atlases for the bundles on Y [4] that are relative to a common trivializing covering {U i } of Y [4] (with abuse of notation we denote with the same index i all these different coverings1 ). Then (89), that we rewrite as E12 E23 = T E13 , reads ij
j
ij
ij
h12 ϕ12 (h23 ) = f i h13 f j
−1
i i i , ϕ12 ◦ ϕ23 = Adf i ◦ ϕ13
(100)
and relation (92) reads i i i i i (f234 )f124 = f123 f134 . ϕ12
(101)
1
An explicit construction is for example obtained pulling back the atlas of E to the pull-back bun∗ (t i ) : dles on Y [3] and on Y [4] . The sections t i : U i → E induce the associated sections t i12 ≡ p12
i → E i [3] → Y [2] and U i ≡ p −1 (U i ). We then have E U12 12 where p12 : Y 12 = {h12 , ϕ12 } with 12 12 ij i ∗ ∗ ij i [3] h12 = p12 (h ), ϕ12 = p12 (ϕ ). Similarly for E13 , E23 . The Y covering given by the opens U I ≡ ij
i ∩ U i ∩ U i can then be used for a common trivialization of the E , E and E bunU ii i ≡ U12 12 13 23 23 13 dles; the respective sections are t I12 = t i12 | I , t I23 = t i23 | I , t I13 = t i13 | I ; similarly for the transition U
U
U
I , ϕ I , ϕ I . In U I we then have f −1 = f I functions hI12 , hI23 , hI13 and for ϕ12 23 13
−1 I I I −1 t 12 t 23 t 13 .
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Bundles and local data on M. Up to equivalence under stable isomorphisms, there is an alternative geometric description of bundle gerbes, in terms of bundles on M. Consider the sections σα : Oα → Y , relative to a covering {Oα } of M and consider also the induced sections (σα , σβ ) : Oαβ → Y [2] , (σα , σβ , σγ ) : Oαβγ → Y [3] . Denote by Eαβ , Tαβγ the pull-back of the H -bibundles E and T via (σα , σβ ) and (σα , σβ , σγ ). Denote also by f αβγ the pull-back of the section f . Then, following the Hitchin description of abelian gerbes, Definition 15. A gerbe is a collection {Eαβ } of H principal bibundles Eαβ on each Oαβ −1 are trivial, and such that on the triple intersections Oαβγ the product bundles Eαβ Eβγ Eαγ such that on the quadruple intersections Oαβγ δ we have f αβδ ϕ αβ (f βγ δ ) = f αγ δ f αβγ . } and {E } (we can always conWe also define two gerbes, given respectively by {Eαβ αβ sider a common covering {Oα } of M), to be stably equivalent if there exist bibundles Nα and trivial bibundles ηαβ with (global central) sections −1 αβ such that Nβ−1 = ηαβ Eαβ , Nα Eαβ
(102)
ϕ α (f αβγ ) = −1 αγ f αβγ ϕ αβ (βγ )αβ .
(103)
A local description of the Eαβ bundles in terms of the local data (100), (101) can be ij given considering the refinement {Oαi } of the {Oα } cover of M such that (σα , σβ )(Oαβ ) ⊂ ij k
U ij ⊂ Y [2] , the refinement {Oαi } such that (σα , σβ , σγ )(Oαβγ ) ⊂ U ij k ⊂ Y [3] , and similarly for Y [4] . We can then define the local data on M ij
ij
hαβ : Oαβ → H ij
i i ϕαβ : Oαβ → Aut (H ),
ij
i i = ϕ12 ◦ (σα , σβ ), hαβ = h12 ◦ (σα , σβ ) ϕαβ
(104)
and i i : Oαβγ → H, fαβγ i = f i ◦ (σα , σβ , σγ ) . fαβγ
(105)
j −1
ij
i } and T i It follows that Eαβ = {hαβ , ϕαβ αβγ = {fαβγ fαβγ , Adf i }. Moreover relations αβγ (100), (101) imply the relations between local data on M, ij
j
j −1
ij
i hαβ ϕαβ (hβγ ) = fαβγ hij αγ fαβγ , i i ϕαβ ◦ ϕβγ = Adf i
αβγ
i ◦ ϕαγ ,
(106)
i i i i i ϕαβ (fβγ δ )fαβδ = fαβγ fαγ δ .
(107)
ˇ We say that (107) define a nonabelian Cech 2-cocycle. From (102), (103) we see that ij i i ij i i two sets {hαβ , ϕαβ , fαβγ }, {hαβ , ϕαβ , fαβγ } of local data on M are stably isomorphic if ij
j
j −1
ij
ij
j −1
j j i hij α ϕα (hαβ ) ϕα ϕαβ ϕβ (hβ ) = αβ hαβ αβ −1 ϕαi ◦ ϕαβ ◦ ϕβi i
,
i = Adi ◦ ϕαβ ,
(109)
αβ
i
i i ϕαi (fαβγ ) = iαβ ϕαβ (iβγ )fαβγ iαγ−1 , ij
(110)
j −1
i = {h , ϕ } and η here Nα = {hα , ϕα } , Eαβ αβ = {αβ αβ , Adi }. αβ αβ ij
(108)
αβ
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We now compare the gerbe {Eαβ } obtained from a bundle gerbe G using the sections } obtained from G using a different choice of sections σα : Oα → Y to the gerbe {Eαβ σα : Oα → Y . We first pull back the bundles in (96) using (σα , σα , σβ , σβ ) : Oαβ → Y [4] ; recalling that the product commutes with the pull-back we obtain the following relation between bundles respectively on Oα , Oαβ , Oβ and on Oαβ , Oαβ , Nα Eαβ Nβ−1 = ηαβ Eαβ ;
here Nα equals the pull-back of E11 with (σα , σα ) : Oα → Y [2] . We then pull back (97) } and with (σα , σα , σβ , σβ , σγ , σγ ) : Oαβγ → Y [6] and obtain formula (103). Thus {Eαβ {Eαβ } are stably equivalent gerbes. We have therefore shown that the equivalence class of a gerbe (defined as a collection of bundles on Oαβ ⊂ M) is independent from the choice of sections σα : Oα → Y used to obtain it as pull-back from a bundle gerbe. It is now easy to prove that equivalence classes of bundle gerbes are in one to one correspondence with equivalence classes of gerbes {Eαβ }, and therefore with equivalence classes of local data on M. First of all we observe that a bundle gerbe G and its pull-back σ ∗ G = (σ ∗ E, Y , M, σ ∗f ) (cf. Theorem 13) give the same gerbe {Eαβ } if we use the sections σα : Oα → Y for σ ∗ G and the sections σ ◦σα : Oα → Y for G. It then follows that two stably equivalent bundle gerbes give two stably equivalent gerbes. In order to prove the converse we associate to each gerbe {Eαβ } a bundle gerbe and then we prove that on equivalence classes this operation is the inverse of the operation G → {Eαβ }. Given {Eαβ } we consider Y = Oα , the disjoint union of the opens Oα ⊂ M, with projection ℘ (x, α) = x. Then Y [2] is the disjoint union of the opens Oαβ , i.e. Y [2] = Oαβ = ∪Oα,β , where Oα,β = {(α, β, x) / x ∈ Oαβ }, similarly Y [3] = Oαβγ = ∪Oα,β,γ etc.. −1 We define E such that E|Oα,β = Eαβ and we define the section f −1 of T = E12 E23 E13 to be given by f −1 |Oα,β,γ = f −1 αβγ , thus (92) holds. We write (Eαβ , Oα , M, f αβγ ) for this bundle gerbe. If we pull it back with σα : Oα → Y , σα (x) = (x, α) we obtain the initial gerbe {Eαβ }. In order to conclude the proof we have to show that (Eαβ , Oα , M, f αβγ ) is stably isomorphic to the bundle gerbe G = (E, Y, M, f ) if {Eαβ } is obtained from G = (E, Y, M, f ) and sections σα : Oα → Y . This holds because (Eαβ , Oα , M, f αβγ ) = σ ∗ G with σ : Oα → Y given by σ |Oα = σα . We end this section with a comment on normalization. There is no loss in generality if we consider for all α, β and for all i, i i i = id , fααβ = 1 , fαββ = 1. ϕαα
(111)
i = Ad i (f i ) = f i Indeed first notice from (106) and (107) that ϕαα and ϕαα i fααα ααα ααβ i i i so that fααβ = fααα |Oαβ . Now, if fααα = 1 consider the stably equivalent local data j −1
−1
i i ≡ η E , where η i obtained from Eαβ αβ αβ αβ = {αβ αβ , Adi } with αβ = fααα |Oαβ .
αβ
i = id; from (110) we have f i From (109) we have ϕαα ααβ = 1, it then also follows
i = 1. fαββ
5. Nonabelian Gerbes from Groups Extensions We here associate a bundle gerbe on the manifold M to every group extension π
1→H →E→G→1
(112)
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and left principal G bundle P over M. We identify G with the coset H \E so that E is a left H principal bundle. E is naturally a bibundle, the right action too is given by the group law in E e h = eh = (ehe−1 )e,
(113)
thus ϕe (h) = ehe−1 . We denote by τ : P [2] → G, τ (p1 , p2 ) = g12 the map that associates to any two points p1 , p2 of P that live on the same fiber the unique element g12 ∈ G such that p1 = g12 p2 . Let E ≡ τ ∗ (E) be the pull-back of E on P [2] , explicitly E = {(p1 , p2 ; e) | π(e) = τ (p1 , p2 ) = g12 }. Similarly E12 = {(p1 , p2 , p3 ; e) | π(e) = τ (p1 , p2 ) = g12 }, for brevity of notations we set e12 ≡ (p1 , p2 , p3 ; e). Similarly with −1 −1 is a symbolic notation for a point of E13 . Recalling (15) we have E23 and E13 , while e13 −1 −1 −1 −1 −1 −1 (he)13 = (ek)13 = k −1 e13 , e13 h = k e13 ,
(114)
where k = e−1 he . We now consider the point −1 f −1 (p1 , p2 , p3 ) ≡ [e12 , e23 , (ee )−1 13 ] ∈ E12 E23 E13 ,
(115)
where the square bracket denotes, as in (12), the equivalence class under the H action2 . Expression (115) is well defined because π(ee ) = π(e)π(e ) = g12 g23 = g13 the last equality following from p1 = g12 p2 , p2 = g23 p3 , p1 = g13 p3 . Moreover f (p1 , p2 , p3 ) is independent from e and e , indeed let eˆ and eˆ be two other elements of E such that π(e) ˆ = π(e) , π(eˆ ) = π(e ); then eˆ = he, eˆ = h e with h, h ∈ H and −1 −1 [eˆ12 , eˆ23 , (eˆeˆ )13 ] = [h e12 , h e 23 , e −1 h −1 e−1 h−1 ee (ee )−1 13 ] = [e12 , e23 , (ee )13 ]. −1 This shows that (115) defines a global section f −1 of T ≡ E12 E23 E13 . Using the second relation in (114) we also have that f −1 is central so that T is a trivial bibundle. Finally (the inverse of) condition (92) is easily seen to hold and we conclude that (E, P , M, f ) is a bundle gerbe. It is the so-called lifting bundle gerbe. 6. Bundle Gerbes Modules The definition of a module for a nonabelian bundle gerbe is inspired by the abelian case [6]. Definition 16. Given an H-bundle gerbe (E, Y, M, f ), an E-module consists of a triple (Q, Z, z), where Q → Y is a D-H bundle, Z → Y [2] is a trivial D-bibundle and z is a global central section of Z −1 such that: i) On Y [2] Q1 E = ZQ2
(116)
and moreover ϕ 12 = ψ −1 1 ◦
−1 z¯ 12
◦ ψ 2.
(117)
ii) Equation (116) is compatible with the bundle gerbe structure of E, i.e. from (116) −1 we have Q1 T = Z12 Z23 Z13 Q1 on Y [3] and we require that z23 z12 = z13 ξ 1 (f )
(118)
holds true. ] ∈ E E It can be shown that a realization of the equivalence class [e12 , e23 12 23 is given by (p1 , p2 , p3 ; ee ), where ee is just the product in E. (We won’t use this property). 2
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Remark 17. Let us note that the pair (Z, z−1 ) and the pair (T , f −1 ) in the above definition give the isomorphisms z : Q1 E → Q2 , f : E12 E23 → E13
(119)
respectively of D-H bundles on Y [2] and of bibundles on Y [3] . Condition ii) in Definition 16 is then equivalent to the commutativity of the following diagram id 1 f
Q1 E12 E23 −−−−→ Q1 E13 z z12 id 3 13 Q2 E23
z23
−−−−→
(120)
Q3
Definition 18. We call two bundle gerbe modules (Q, Z, z) and (Q , Z , z ) (with the same crossed module structure) equivalent if: i) Q and Q are isomorphic as D-H bundles; we write Q = IQ where the D-bibundle ¯−1 I has global central section i −1 and ψ = i ◦ ψ , −1 ii) the global central sections z, z and i satisfy the condition z12 = i −1 2 z12 i 1 . Let us now assume that we have two stably equivalent bundle gerbes (E, Y, M, f ) and (E , Y , M, f ) with Y = Y . We have [cf. (93), (94)] η12 E12 = N1 E12 N2−1 and ϕ 1 (f ) = −1 13 f ϕ 12 (23 )12 . Let Q be an E-module and I a trivial D-bibundle with a = global central section i −1 . It is trivial to check that IQN is an E -module with Z12 −1 −1 I1 ξ 1 (η12 )Z12 I2 and z12 = i 2 z12 ξ (η12 )i 1 . It is now easy to compare modules of stably equivalent gerbes that in general have Y = Y . Proposition 19. Stably equivalent gerbes have the same equivalence classes of modules. Now we give the description of bundle gerbes modules in terms of local data on M. Let {EEαβ } be a gerbe in the sense of Definition 15. Definition 20. A module for the gerbe {Eαβ } is given by a collection {Qα } of D-H bundles such that on double intersections Oαβ there exist trivial D-bibundles Zαβ , −1 Qα Eαβ = Zαβ Qβ , with global central sections zαβ of Zαβ such that on triple intersections Oαβγ , zβγ zαβ = zαγ ξ α (f αβγ )
(121)
and on double intersections Oαβ ϕ αβ = ψ −1 α ◦
−1 z¯ αβ
◦ ψβ.
(122)
Canonical module. For each H -bundle gerbe (E, Y, M, f ) we have a canonical Emodule associated with it; it is constructed as follows. As a left Aut (H )-bundle the canonical module is simply the trivial bundle over Y . The right action of H is given by the canonical homomorphism Ad : H → Aut (H ). For (y, η) ∈ Y × Aut (H ) we have ξ (y,η) (h) = η ◦ Adh ◦ η−1 = Adη(h) and ψ (y,η) (h) = η(h). The Aut (H )-H bundle morphism z : (Y × Aut (H ))1 E → (Y × Aut (H ))2 is given in the following way. A generic element of (Y × Aut (H ))1 E is of the form [(y, y , (y, η)), e], where η ∈ Aut (H ), (y, y ) ∈ Y [2] and e ∈ E such that p1 ◦ p(e) = y and p2 ◦ p(e) = y . Here p is the projection p : E → Y [2] . We set
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z([(y, y , (y, η)), e]) = (y, y , (y , η ◦ ϕ e )). The commutativity of diagram (120) is equivalent to the following statement: η ◦ ϕ f [e1 ,e2 ] = η ◦ ϕ e1 ◦ ϕ e2 , and this is a consequence of the isomorphism of H -bibundles f : E12 E23 → E13 . We have
f ([e1 , e2 ]h) = (f [e1 , e2 ])h ,
but we also have f ([e1 , e2 )]h) = f (ϕ e1 ◦ ϕ e2 (h)[e1 , e2 ]) = ϕ e1 ◦ ϕ e2 (h)f ([e1 , e2 ]). On the other hand we can write (f [e1 , e2 ])h = ϕ f [e1 ,e2 ] (h)f [e1 , e2 ]. Hence
ϕ f [e1 ,e2 ] (h) = ϕ e1 ◦ ϕ e2 (h)
and the commutativity of diagram (120) follows. We denote the canonical module as can in the following. In the case of a bundle gerbe E associated with the lifting of a G-principal bundle P , as described in Sect. 5, we have another natural module. We follow the notation of Sect. 5. In the exact sequence of groups (112), π
1 → H → E → G → 1, H is a normal subgroup. This gives the group H the structure of a crossed E-module. The E-module Q is simply the trivial E-H bundle P × E → P . The D-H bundle ˜ 2) morphims z : Q1 E → Q2 is given by (recall p1 = π(e)p ˜ = (p1 , p2 , (p2 , ee)), ˜ z [(p1 , p2 , (p1 , e), (p1 , p2 , e)] which of course is compatible with the bundle gerbe structure of E. Due to the exact sequence (112) we do have a homomorphism E → Aut (H) and hence we have a map t : Y × E → Y × Aut (H ), which is a morphism between the modules compatible with the module structures, i.e. the following diagram is commutative: z12
Q1 E12 −−−−→ Q2 t t
(123)
z12
can1 E12 −−−−→ can2 More generally given any bundle gerbe E and an E-module Q we have the trivial Aut (H )H bundle Aut (H ) ×D Q (see Sect. 2). This gives a morphism t : Q → can.
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Now suppose that the bundle gerbe E is trivialized (stably equivalent to a trivial bundle gerbe) by E12 ∼ N1−1 N2 with N an H -bibundle on Y, hence a E-module satisfies Q2 ∼ Q1 E12 ∼ Q1 N1−1 N2 , hence Q2 N2−1 ∼ Q1 N1−1 .
(124)
It easily follows from (124) that QN −1 → Y gives descent data for a D-H bundle Q → M the bundle p ∗ (Q)N over M. Conversely given a D-H bundle Q is a E-module. This proves the following Proposition 21. For a trivial bundle gerbe (E, Y, M, f ) the category of E-modules is equivalent to the category of D-H bundles over the base space M. 7. Bundle Gerbe Connections Definition 22. A bundle gerbe connection on a bundle gerbe (E, Y, M, f ) is a 2-connection (a, A) on E → Y [2] such that a 12 +a 23 = f ∗ a 13 ,
(125)
or which is the same a 12 +a 23 +a r13 = f
−1
df + TA1 (f
−1
)
(126)
holds true. −1
−1
In the last equation f is the bi-equivariant map f : T → H associated with the global central section f −1 of T . Moreover we used that (a r , Ar ) is a right 2-connection on E and a left 2-connection on E −1 [cf. (15)]. Remark 23. It follows from (125) that for a bundle gerbe connection A12 = A13 must be satisfied, hence A is a pull-back via p1 on Y [2] of a one form defined on Y . We can set A1 ≡ A12 = A13 . Definition 21 contains implicitly the requirement that (a 12 , a 23 ) are summable, which means that Ar1 = A2 . More explicitly (see (53)): −1 −1 A1 + ada 12 = ϕ12 A2 ϕ12 + ϕ12 dϕ12 .
(127)
The affine sum of bundle gerbe connections is again a bundle gerbe connection. This is a consequence of the following affine property for sums of the 2-connection. If on the bibundles E1 and E2 we have two couples of summable connections (a 1 , a 2 ), (a 1 , a 2 ), then λa 1 + (1 − λ)a 1 is summable to λa 2 + (1 − λ)a 2 and the sum is given by (λa 1 + (1 − λ)a 1 ) + (λa 2 + (1 − λ)a 2 ) = λ(a 1 +a 2 ) + (1 − λ)(a 1 +a 2 ) .
(128)
We have the following theorem: Theorem 24. There exists a bundle gerbe connection (a, A) on each bundle gerbe (E, Y, M, f ).
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Proof. Let us assume for the moment the bundle gerbe to be trivial, E = N1−1 ZN2 with a bibundle N → Y and a trivial bibundle Z → Y [2] with global central section z−1 . ˜ where the Lie(Aut (H ))-valued one-form A˜ on Consider on Z the 2-connection (α, A), Y is the pull-back of a one-form on M. Here α is canonically determined by A˜ and z−1 , ˜ ˜ A). we have α = z¯ −1 d z¯ + TA (¯z−1 ). Next consider on N an arbitrary 2-connection (a, Since A˜ is the pull-back of a one form on M we have that the sum a = a˜ r1 +α+ a˜ 2 is well defined and that (a, A ≡ A˜ r ) is a 2-connection on E. Notice that under the canonical identification Z12 N2 N2−1 Z23 = Z12 Z23 we have the canonical identification α 12 + a˜ 2 + a˜ r2 +α 23 = α 12 +α 23 . The point here is that N2 N2−1 has the canonical sec¯ 1¯ −1 + TA (1) ¯ independently from tion 1 = [n, n−1 ], n ∈ N2 , and that a˜ 2 + a˜ r2 = 1d −1 −1 −1 −1 a˜ 2 . Then from E = N1 ZN2 we have E12 E23 E13 = N1 Z12 Z23 Z13 N1 and for the connections we have a 12 +a 23 +a r13 = a˜ r1 +α 12 +α 23 +α r13 + a˜ 1 .
(129)
We want to prove that the r.h.s. of this equation equals the canonical 2-connection −1 −1 f df + TA1 (f ) associated with the trivial bundle T with section f −1 . We first −1 observe that a similar property holds for the sections of Z12 Z23 Z13 and of T : f −1 = −1 −1 −1 −1 −1 −1 ϕ 1 (z12 z23 z13 ) ≡ n1 z12 z23 z13 n1 independently from the local section n1 of N1 . Then one can explicitly check that this relation implies the relation a˜ r1 +α 12 +α 23 −1 −1 + α r13 + a˜ 1 = f df + TA1 (f ). This proves the validity of the theorem in the case of a trivial bundle gerbe. According to Theorem 14 any gerbe is locally trivial, so we can use the affine property of bundle gerbe connections and a partition of unity subordinate to the covering {Oα } of M to extend the proof to arbitrary bundle gerbes. A natural question arises: can we construct a connection on the bundle gerbe (E, Y, M, f ) starting with: −1 ˇ • its nonabelian Cech cocycle f : T → H, ϕ : E × H → H • sections σα : Oα → Y • a partition of unity {ρα } subordinate to the covering {Oα } of M ?
The answer is positive. Let us describe the construction. First we use the local sections [3] via the map r [2] : [y, y ] → [σ (x), y, y ], where ℘ (y) = σα to map Y |[2] α α Oα to Y [2] ℘ (y ) = x, similarly rα[1] : Y |[1] Oα → Y . Next let us introduce the following H -valued one form a ∗ −1 a= . (130) ρα rα[2] ϕ −1 f d f 12 α
We easily find that
l a = Adh a h∗
+ p1∗
h
∗ ρα rα[1] ϕ −1 (d
−1
ϕ(h
)
.
α
The Lie(Aut (H ))-valued 1-form We set A=
∗
α
α
ρα rα[1] ϕ −1 d ϕ is, due to (5), well defined on Y . ∗
ρα rα[1] ϕ −1 d ϕ − d
(131)
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for the sought Lie(Aut (H ))-valued 1-form on Y . Using the cocycle property of f¯ and ϕ we easily have Proposition 25. Formulas (130) and (131) give a bundle gerbe connection. Using (88) we obtain that the 2-curvature (k, K) of the bundle gerbe 2-connection (a, A) satisfies k 12 +k 23 +k r13 = TK1 (f
−1
(132)
).
Connection on a lifting bundle gerbe. Let us now consider the example of a lifting bundle gerbe associated with an exact sequence of groups (112) and a G-principal bundle P → M on M. In this case, for any given connection A¯ on P we can construct a connection on the lifting bundle gerbe. Let us choose a section s : Lie(G) →Lie(E); ¯ and then consider the i.e a linear map such that π ◦ s = id. We first define A = s(A) ¯ and A2 = p ∗ s(A), ¯ where p1 Lie(E) valued one-forms on P [2] given by A1 = p1∗ s(A) 2 and p2 are respectively the projections onto the first and second factor of P [2] . We next consider the one-form a on E that on (p1 , p2 ; e) ∈ E is given by a ≡ eA2 e−1 + ede−1 − A1 ,
(133)
here A1 = p ∗ (A1 ) and A2 = p ∗ (A2 ), with p : E → P [2] . It is easy to see that π ∗ a = 0 and that therefore a is Lie(H ) valued; moreover (a, adA ) is a 2-connection on E. Recalling that on E we have ϕ (p1 ,p2 ;e) = Ade , it is now a straightforward check left to the reader to show that (a, adA ) is a connection on the lifting bundle gerbe. Connection on a module. Let us start discussing the case of the canonical module can = Aut (H ) × Y (see Sect. 6). Let (a, A) be a connection on our bundle gerbe (E, Y, M, f ). The Lie(Aut (H ))-valued one-form A on Y lifts canonically to the connection A˜ on can defined, for all (η, y) ∈ can, by A˜ = ηAη−1 +ηdη−1 . Let us consider the following diagram: /E can1 ⊕ KE KKK π KKK⊕ π1 KKK % can1 can1 E π2
(134) z
/ can2 .
As in the case of the bundle gerbe connection we can consider whether the Lie(Aut (H ))valued one-form A˜ 1 +ξ (a) that lives on can1 ⊕E is the pull-back under π⊕ of a one-form connection on can1 E. If this is the case then we say that A˜ 1 and a are summable and we denote by A˜ 1 +ada the resulting connection on can1 E. Let us recall that on can we have ξ (η,y) = Ad ◦ ψ (η,y) with ψ (η,y) (h) = η(h). It is now easy to check that A˜ 1 and a are summable and that their sum equals the pull-back under z of the connection A˜ 2 ; in formulae A˜ 1 +ada = z∗ A˜ 2 .
(135)
We also have that equality (135) is equivalent to the summability condition (127) for the bundle gerbe connection a. Thus (135) is a new interpretation of the summability condition (127).
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We now discuss connections on an arbitrary module (Q, Z, z) associated with a bundle gerbe (E, Y, M, f ) with connection (a, A). There are two natural requirements that a left connection AD on the left D-bundle Q has to satisfy in order to be a module D on Aut (H ) × Q has to be connection. The first one is that the induced connection A D ˜ equal (under the isomorphism σ ) to the connection A of can. This condition reads AD = ψAψ −1 + ψdψ −1 ,
(136)
where in the l.h.s. AD is thought to be Lie(Aut (H )) valued. In other words on Y we D = A, where σ is the global section of Aut (H ) × Q. require σ ∗ A D Next consider the diagram π2 /E Q1 ⊕ EI II π II ⊕ π1 II II $ Q1 Q1 E
(137) z
/ Q2 .
We denote by AD 1 +α(a) the well defined D-connection on Q1 E that pulled back on ∗ D Q1 ⊕ E equals π1∗ AD 1 + ξ (π2 a). It is not difficult to see that A1 is indeed summable to a if for all h ∈ H , α(TAD (h)) = α(Tψ Aψ −1 +ψdψ −1 (h)). This summability condition is thus implied by (136). The second requirement that AD has to satisfy in order to be a module connection is ∗ D AD 1 +α(a) = z A2 .
(138)
These conditions imply the summability condition (127) for the bundle gerbe connection a. Concerning the D-valued curvature KD = dAD + AD ∧ AD we have ∗ K2D . K1D +α(k a ) = z12
(139)
In terms of local data a gerbe connection consists of a collection of local 2-connections (a αβ , Aα ) on the local bibundles Eαβ → Oαβ . For simplicity we assume the covering {Oα } to be a good one. The explicit relations that the local maps fαβγ : Oαβγ → H , ϕαβ : Oαβ → Aut (H ) and the local representatives Aα , Kα , aαβ and kαβ (forms on Oα , Oαβ , etc.) satisfy are fαβγ fαγ δ = ϕαβ (fβγ δ )fαβδ ,
(140)
ϕαβ ϕβγ = Adfαβγ ϕαγ ,
(141)
−1 −1 + fαβγ d fαβγ + TAα (fαβγ ) , aαβ + ϕαβ (aβγ ) = fαβγ aαγ fαβγ
(142)
−1 −1 Aα + adaαβ = ϕαβ Aβ ϕαβ + ϕαβ d ϕαβ ,
(143)
−1 kαβ + ϕαβ (kβγ ) = fαβγ kαγ fαβγ + TKα (fαβγ ),
(144)
−1 Kα + adkαβ = ϕαβ Kβ ϕαβ .
(145)
and
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8. Curving In this section we introduce the curving two form b. This is achieved considering a gerbe stably equivalent to (E, Y, M, f ). The resulting equivariant H -valued 3-form h is then shown to be given in terms of a form on Y . This description applies equally well to the abelian case; there one can however impose an extra condition [namely the vanishing of (147)]. We also give an explicit general construction of the curving b in terms of a partition of unity. This construction depends only on the partition of unity, and in the abelian case it naturally reduces to the usual one that automatically encodes the vanishing of (147). Consider a bundle gerbe (E, Y, M, f ) with connection (a, A) and curvature (k a , KA ) and an H -bibundle N → Y with a 2-connection (c, A). Then we have a stably equivalent r1 gerbe (N1−1 EN2 , Y, M, ϕ −1 1 (f )) with connection (θ, A ) given by θ = cr11 +a+c2 .
(146)
Also we can consider a KA -equivariant horizontal 2-form b on N . Again on the bibundle N1−1 EN2 → Y [2] we get a well defined KAr1 -equivariant horizontal 2-form ˜ δ = br11 +k a +b2 .
(147)
δ is Contrary to the abelian case we cannot achieve ˜ δ = 0, unless KA is inner (remember ˜ always KAr1 -equivariant). Next we consider the equivariant horizontal H -valued 3-form h on N given by h = Dc b .
(148)
Because of the Bianchi identity dKA + [A, KA ] = 0 this is indeed an equivariant form on N . Obviously the horizontal form ϕ −1 (h) is invariant under the left H -action ∗
l h ϕ −1 (h) = ϕ −1 (h),
(149)
and therefore it projects to a well defined form on Y . Using now the property of the covariant derivative (86) and the Bianchi identity (68) we can write hr1 +h2 = Dθ ˜ δ.
(150)
Finally from (72) we get the Bianchi identity for h, Dc h = [k c , b] + TKA (k c ) − TKA (b) .
(151)
For the rest of this section we consider the special case where N is a trivial bibundle with global central section σ¯ and with 2-connection given by (c, A), where c is canonically given by σ¯ , c = σ¯ d σ¯ −1 + TA (σ¯ ) . Since the only H -bibundle N → Y that we can canonically associate to a generic bundle gerbe is the trivial one (see Proposition 7), the special case where N is trivial seems quite a natural case. In terms of local data curving is a collection {bα } of Kα -equivariant horizontal two forms on trivial H -bibundles Oα × H → Oα . Again we assume the covering Oα to be
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a good one and write out explicitly the relations to which the local representatives of bα and hα (forms on Oα ) are subject: kαβ + ϕαβ (bβ ) = bα + δαβ , δαβ + ϕαβ (δβγ ) =
(152)
−1 fαβγ δαγ fαβγ
+ Tνα (fαβγ ) ,
να ≡ Kα + adbα , hα = d bα − TAα (bα ) , ϕαβ (hβ ) = hα + dδαβ + [aαβ , δαβ ] + Tνα (aαβ ) − TAα (δαβ ),
(153) (154) (155) (156)
and the Bianchi identity d hα + TKA (bα ) = 0 .
(157)
Here we introduced δαβ = ϕα (δ˜αβ ) . Equations (140)–(145) and (152)–(157) are the same as those listed after Theorem 6.4 in [26]. We now consider the case Y = Oα ; this up to stable equivalence is always doable. Given a partition of unity {ρα } subordinate to the covering {Oα } of M, we have a natural choice for the H -valued curving 2-form b on Oα × H . It is the pull-back under the projection Oα × H → Oα of the 2-form
ρβ kαβ
(158)
β
on Y = Oα . In this case we have for the local H -valued 2-forms δαβ the following expression: δαβ =
−1 ργ (fαβγ kαγ fαβγ − kαγ + TKα (fαβγ ))
γ
=
ργ (kαβ + ϕαβ (kβγ ) − kαγ ) .
(159)
γ
We can now use Proposition 25 together with (158) in order to explicitly construct from ˇ the Cech cocycle (f , ϕ) an H -valued 3-form h. We conclude this final section by grouping together the global cocycle formulae that imply all the local expressions (140)–(145) and (152)–(157), f 124 ϕ 12 (f 234 ) = f 134 f 123 ,
(92)
a 12 +a 23 = f ∗ a 13 ,
(125)
˜ δ = br11 +k a +b2 ,
(147)
h = Dc b .
(148)
Acknowledgements. We have benefited from discussions with L. Breen, D. Husemoller, A. Alekseev, L. Castellani, J. Kalkkinen, J. Mickelsson, R. Minasian, D. Stevenson and R. Stora.
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References 1. Giraud, J.: Cohomologie non-ab´elienne. Grundlehren der mathematischen Wissenschaften 179, Berlin: Springer Verlag, 1971 2. Brylinski, J.L.: Loop Spaces, Characteristic Classes And Geometric Quantization. Progress in mathematics 107, Boston: Birkh¨auser, 1993 3. Hitchin, N.: Lectures on special Lagrangian submanifolds. http://arxiv.org/abs/math.dg/9907034, 1999 4. Chatterjee, D.: On Gerbs. http://www.ma.utexas.edu/users/hausel/hitchin/hitchinstudents/chatterjee. pdf, 1998 5. Murray, M.K.: Bundle gerbes. J. Lond. Math. Soc 2, 54, 403 (1996) 6. Bouwknegt, P., Carey, A.L., Mathai, V., Murray, M.K., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228, 17 (2002) 7. Carey, A.L., Johnson, S., Murray, M.K.: Holonomy on D-branes. http://arxiv.org/abs/hepth/0204199, 2002 8. Mackaay, M.: A note on the holonomy of connections in twisted bundles. Cah. Topol. Geom. Differ. Categ. 44, 39–62 (2003) 9. Mackaay, M., Picken, R.: Holonomy and parallel transport for Abelian gerbes. Adv. Math. 170, 287 (2002) 10. Carey, A., Mickelsson, J., Murray, M.: Index theory, gerbes, and Hamiltonian quantization. Commun. Math. Phys. 183, 707 (1997) 11. Carey, A., Mickelsson, J., Murray, M.: Bundle gerbes applied to quantum field theory. Rev. Math. Phys. 12, 65–90 (2000) 12. Carey, A.L., Mickelsson, J.: The universal gerbe, Dixmier-Douady class, and gauge theory. Lett. Math. Phys. 59, 47 (2002) 13. Picken, R.: TQFT’s and gerbes. Algebr. Geom. Toplo. 4, 243–272 (2004) 14. Gawedzki, K., Reis, N.: WZW branes and gerbes. Rev.Math.Phys. 14, 1281–1334 (2002) 15. Freed, D.S., Witten, E.: Anomalies in string theory with D-branes. http://arxiv.org/abs/hepth/9907189, 1999 16. Kapustin, A.: D-branes in a topologically nontrivial B-field. Adv. Theor. Math. Phys. 4, 127 (2000) 17. Bouwknegt, P., Mathai, V.: D-branes, B-fields and twisted K-theory. JHEP 0003, 007 (2000) 18. Mickelsson, J.: Gerbes, (twisted) K-theory and the supersymmetric WZW model. http:// arxiv.org/abs/hep-th/0206139, 2002 19. Freed, D.S., Hopkins, M.J., Teleman, C.: Twisted K-theory and Loop Group Representations I. http://arxiv.org/abs/math.AT/0312155, 2003 20. Witten, E.: D-branes and K-theory. JHEP 9812, 019 (1998) 21. Witten, E.: Overview of K-theory applied to strings. Int. J. Mod. Phys. A 16, 693 (2001) 22. Olsen, K., Szabo, R.J.: Constructing D-Branes from K-Theory. Adv. Theor. Math. Phys. 3, 889–1025 (1999) 23. Dedecker, P.: Sur la cohomologie nonab´eliene, I and II. Can. J. Math 12, 231–252 (1960) and 15, 84–93 (1963) 24. Moerdijk, I.: Introduction to the language of stacks and gerbes. http://arxiv.org/abs/math. AT/0212266, 2002 25. Moerdijk, I.: Lie Groupoids, Gerbes, and Non-Abelian Cohomology, K-Theory, 28, 207– 258 (2003) 26. Breen, L., Messing, W.: Differential Geometry of Gerbes. http://arxiv.org/abs/math.AG/0106083, 2001 27. Attal, R.: Combinatorics of Non-Abelian Gerbes with Connection and Curvature. http:// arxiv.org/abs/math-ph/0203056, 2002 28. Kalkkinen, J.: Non-Abelian gerbes from strings on a branched space-time. http://arxiv.org/abs/ hep-th/9910048, 1999 29. Baez, J.: Higher Yang-Mills theory. http://arxiv.org/abs/hep-th/0206130, 2002 30. Hofman, C.: Nonabelian 2-Forms. http://arxiv.org/abs/hep-th/0207017, 2002 31. Kobayashi, S., Nomizu, K.: Foundations od Differential Geometry, Volume I. New York: WileyInterscience 1996 32. Husemoller, D.: Fibre bundles. 3rd edition, Graduate Texts in Mathematics 50, Berlin: Springer Verlag, 1994 33. Grothendieck, A.: S´eminaire de G´eometri Alg´ebrique du Bois-Marie, 1967-69 (SGA 7), I. LNM 288, Berlin-Heidelberg-Newyork Springer-Verlag, 1972 34. Breen, L.: Bitorseurs et cohomologie non ab´eliene. In: The Grothendieck Festschrift I, Progress in Math. 86, Boston: Birkh¨auser 1990, pp. 401–476
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35. Brown, K.: Cohomology of Groups. Graduate Texts in Mathematics 50, Berlin: Springer Verlag, 1982 36. Murray, M.K., Stevenson, D.: Bundle gerbes: stable isomorphisms and local theory. J. Lon. Math. Soc 2, 62, 925 (2000) Communicated by M.R. Douglas
Commun. Math. Phys. 254, 401–423 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1221-5
Communications in
Mathematical Physics
Mirror Symmetric SU(3)-Structure Manifolds with NS Fluxes St´ephane Fidanza, Ruben Minasian, Alessandro Tomasiello Centre de Physique Th´eorique, Ecole Polytechnique, Unit´e mixte du CNRS et de l’EP, UMR 7644, 91128 Palaiseau Cedex, France Received: 5 February 2004 / Accepted: 15 May 2004 Published online: 11 November 2004 – © Springer-Verlag 2004
Abstract: When string theory is compactified on a six-dimensional manifold with a nontrivial NS flux turned on, mirror symmetry exchanges the flux with a purely geometrical composite NS form associated with lack of integrability of the complex structure on the mirror side. Considering a general class of T 3 -fibered geometries admitting SU(3) structure, we find an exchange of pure spinors (eiJ and ) in dual geometries under fiberwise T–duality, and study the transformations of the NS flux and the components of intrinsic torsion. A complementary study of action of twisted covariant derivatives on invariant spinors allows to extend our results to generic geometries and formulate a proposal for mirror symmetry in compactifications with NS flux. 1. Introduction Mirror symmetry is a pairing between different compactifications which give rise to the same four–dimensional effective theory. For Calabi–Yau compactifications it is well– understood and has played an important role, becoming arguably the most interesting mathematical application of string theory. More general compactifications with fluxes on manifolds which are not Ricci–flat have become the focus of much attention recently, and it would be important to extend to these at least partially the machinery which proved so useful for Calabi–Yaus. If we had to consider only supersymmetric vacua, our search would be premature. The conditions on fluxes and warping to compensate non–Ricci–flatness and preserve supersymmetry are well–understood for some types of fluxes. To some extent, as we review later, these conditions are even translated into mathematical requirements: the manifold has to have SU(3) structure and fall into a certain class in the mathematical classification of these objects. But the Bianchi identity becomes an equation for which there is no existence theorem in the literature, unlike the famous Yau theorem for Calabi–Yaus (not even the analogue of Calabi conjecture seems to have been formulated: this might be a task for string theory). If there is no singularity in the internal compact manifold,
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and the higher derivative terms are not taken into account, one can actually show even non–existence theorems. Fortunately mirror symmetry as a more general equivalence of effective theories, and not only of vacua, still makes sense. As emphasized in [1], to have supersymmetry of the effective action SU(3) structure is enough, without the extra requirements mentioned above, which ensure we are actually in a supersymmetric vacuum. Not only looking for mirror symmetric SU(3) manifolds makes sense, but it is sensible to expect that a formal advance in this direction might help to understand the still elusive problem of compactifications with fluxes. Much in this spirit, [1] (building on a comment in [2]) considered a particular case. Namely, an H flux on Calabi–Yau manifolds (without back–reaction: we are not dealing with a vacuum) get mapped to so–called half–flat manifolds, a particular class of SU(3) structure manifolds, without any H flux. The amount by which these half–flat manifolds fail to be Ricci–flat is measured by a certain quantity called intrinsic torsion. We can thus say that, in this example, H flux gets exchanged by mirror symmetry with components of the intrinsic torsion associated with lack of integrability of the complex structure. It is natural to wonder what happens in more general cases, when on both sides one has both H and intrinsic torsion. (As mentioned above, this for example is necessary in order to have supersymmetric vacua.) In the Calabi–Yau case, a concrete approach to mirror symmetry is the Strominger–Yau–Zaslow (SYZ) [3] conjecture. This states that i) every Calabi–Yau is a T 3 special lagrangian fibration over a three–dimensional base, and ii) mirror symmetry is T–duality along the three circles of the T 3 . It is natural to try and generalize this method to the present problem. Part i) of the conjecture came originally from considering moduli spaces of D–branes on Calabi–Yaus; generalizing this to background with fluxes does seem premature, and in any case we do not attempt it here, although later we will comment more on it. So we simply assume the manifold and flux we start with have this property, of admitting three Killing vectors. The idea is that the mirror transformations found in this class of examples will generalize to some extent to the most general case. Having assumed this, we perform T–duality along the three isometries at once. T– duality will preserve four–dimensional effective theories, but since eventually we hope this procedure could be extended to more general situations by including singular fibers as in SYZ, we want to show why this should be called mirror symmetry – for that matter, indeed, why is there any mirror symmetry at all. A good framework for answering this is Hitchin’s method based on Clifford(6,6) spinors [4]. As we review later in more detail, these are simply formal sums of forms on the manifold. Existence on a manifold of a Clifford(6,6) spinor without zeros which is also pure (annihilated by half of the gamma matrices) is the same as saying that there is a SU(3,3) structure on the manifold. (If the spinor is also closed, Hitchin calls these manifolds generalized Calabi–Yaus.) For a SU(3) structure, there are two pure spinors which are orthogonal and of unit norm. From this point of view it is natural to conjecture that mirror symmetry between two SU(3) structure manifolds exchanges these two pure spinors. We can be more explicit if we compare this Clifford(6,6) spinor definition of SU(3) structure with the more usual one, ¯ = (2J )3 /3!. existence of a two–form J and three–form obeying J ∧ = 0 and i∧ iJ In these terms the two pure spinors are e and . We can actually multiply the first spinor by eB leaving it pure [4]. So what we are claiming is eB+iJ ←→ .
(1.1)
The arrows here will be made precise in Sect. 3. In the Calabi–Yau case, this exchange is implicit in many applications of mirror symmetry. For example, the even periods and
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the D–brane charge can be written using eB+iJ , and its exchange with was used in mapping [5] stringy–corrected DUY equations [6] to the special lagrangian condition; eB+iJ was also used in formulating the concept of –stability [7]. With this in mind, we check that T–duality along T 3 (when it is possible) realizes the exchange (1.1), and for this reason we call it mirror symmetry. In this sense we have generalized part ii) of SYZ. However as it stands, (1.1) is hardly useful in predicting the mirror background starting from a particular six-manifold and NS flux. After having discussed and justified the method, we can schematically describe here our results. The usual quantities which measure non–Ricci–flatness of the SU(3) manifold are the five components of the intrinsic torsion (mentioned above) labeled as Wi , ¯ 3 ⊕ 3, ¯ 3 ⊕ 3¯ respectively. What is i = 1 . . . 5, in the representations 1 ⊕ 1, 8 ⊕ 8, 6 ⊕ 6, puzzling at first is that one does not see many ways of mirror pairing these representations, except for W4 and W5 which are two vectors. The answer is that the two mirrors have indeed two different SU(3) structures: the two SU(3) are differently embedded into Spin(6,6), because the fiber directions change from tangent bundle to cotangent bundle, roughly speaking. As a result, representations get actually mixed. What is preserved is the representations that these objects have once pulled back to the base manifold, which is untouched by T–duality. W2 and W3 get then split as W2 = w2s + w2a (8 → 5 ⊕ 3) and W3 = w3s + w3t (6 → 5 ⊕ 1), and we get W1 − iH1 ←→ −(W1 − iH1 ) , w¯ 2s ←→ w3s − ihs3 , w5 , w¯ 2a ←→ w4 − ih4 .
(1.2)
A more detailed discussion of these equations can be found in Sect. 4 (see in particular (4.11) for the precise statement). In (1.2) one can see that W1 , W3 , W4 get naturally complexified by the components of H in the corresponding representation. This is no surprise as these torsions appear, as we review later, in dJ , and the natural object in string theory is always B + iJ . In the present context, this arises rather due to the usual combination E = g + B of T–duality. As we will specify in Sect. 2, we mostly work with a purely base–fiber type B–field, which is not the most general form allowed by T 3 invariance. However, we will see that this is just a simplifying technical assumption, and may eventually be relaxed. Note also that (1.2) complements (1.1) in an essential way by specifying in a more practical fashion the data of the mirror background (the metric and the NS flux). In particular, it quantifies the exchange of components of the flux and the intrinsic torsion on mirror sides. We have also indicated in (1.2) that some of the components of the intrinsic torsion we begin with are actually related; so T 3 fibrations are not the most general SU(3) manifold. This in a way answers in the negative the question about generalizing part i) of SYZ: we are getting mirror symmetry only for a subclass of manifolds. In particular, supersymmetric vacua with only the three-form switched on are outside this class: indeed the conditions for these are [8] (reinterpreted in terms of torsions for example in [9] and [10]) W1 = W2 = 0 ,
W3 = ∗H3 ,
W4 = dφ = iH4 ,
W5 = 2dφ,
(1.3)
where we denoted by H4 and H3 the components in the representations 3 and 6 of SU(3) respectively, in analogy with the notation for torsions (see also Appendix A). So the next natural step would be to try and include more general classes, among which maybe
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supersymmetric vacua1 . In order to do so, it is natural to wonder to what extent the transformation rules can be put in a nicer form, and in particular be covariantized. It turns out that it is convenient to use spinors. Although also the previously mentioned Clifford(6,6) spinors can be used, here we mean a more conventional Clifford(6) spinor without zeros. One such spinor, call it , always exists on any SU(3) structure manifold and can actually be used to define it. It turns out that using a different basis for W ’s can be defined, which is diagonal under T–duality: elements of the basis transform picking a sign. The idea of this different spinorial basis for W ’s is roughly speaking the following. Usual W ’s are defined, as we review later, from dJ and d. Now, not only is equivalent to the pair J, , but the information contained in dJ and d can also be completely extracted from DM . Using SU(3) structure, this can be decomposed as DM = qM + i q˜M γ + iqMN γ N , where γ is the chiral gamma in six dimensions and the group representations inside the quantities qM , q˜M , qMN are in one-to-one correspondence with the W ’s. Switching to the spinorial basis accomplishes two things. First, it allows to capture the exchange of the pure spinors eiJ and and the exchange of their integrability properties simultaneously. More importantly, it allows to conjecture the six-dimensional covariantization of the mirror transformation (1.2), written in terms of the forms pulled back to the base of the T 3 fibration. Details can be found in Sect. 5. For the purposes of studying mirror symmetry/T–duality we will need first to introduce the covariant derivative twisted by the NS flux: H ˜ M γ + iQMN γ N , (1.4) DM = QM + i Q where, as we will see in detail, Q’s are obtained from q’s by complexifying certain components of the intrinsic torsion by the matching components of the flux (as in (1.2)). We will show that their restrictions to the base (denoted by hatted quantities) transform as ¯ˆ , ˆ ij −→ −Q Q ij
¯ˆ . ˆ i −→ −Q Q i
(1.5)
We will then argue that this simplification is due to the simple transformation of the ten–dimensional spinors under T–duality. Finally we will try, in Sect. 5.1, to collect these several points of view to argue that in general a rule like Qmn ←→ −Qmn¯ ,
¯m Qm ←→ −Q
should hold. This rule is consistent with what we found in the T 3 fibered case, and with the principle that supersymmetric vacua should map in supersymmetric vacua (not necessarily the same). There are however more checks that could be done if one understood better examples; we discuss this in Sect. 6. For example, in the case we mentioned above of compactifications with H only described by (1.3), one should understand moduli spaces and then check that a kind of exchange of complex and K¨ahler moduli (although, as we will argue, this has to be taken with a grain of salt) should happen. This might be interesting for the problem of fixing moduli. We end with a section on open problems. 1
There may actually be supersymmetric vacua involving T 3 fibrations, if other fields are included.
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2. Geometric Setting We start with an introductory section on T–duality, mainly to fix the notations. The six–dimensional manifold will be taken to be a T 3 fibration over a base B. Coordinates on the base will be denoted by (y 1 , y 2 , y 3 ), and on the fiber by (x 1 , x 2 , x 3 ). All the quantities will only depend on the y coordinates, so that the x directions are Killing vectors. We will use conventions for indices as follows: • • • • • •
i, j, k, . . . are used in the 3d y subspace, α, β, γ , . . . are used in the 3d x subspace, M, N, . . . are used in the total 6d space for real coordinates : dy M = (dy i , dx α ), m, n, . . . are used for holomorphic/antiholomorphic indices, A, B, C, . . . are indices in the total 3d complex frame space, a, b, c, . . . and a , b , c , . . . are used in the 3d real y and x frame spaces. Primes will be dropped quickly. We write then the most general metric and B field as 2 ds 2 = gij dy i dy j + hαβ eα eβ = GMN dy M dy N , 1 1 1 B2 = Bij dy i ∧ dy j + Bα ∧ (dx α + λα ) + Bαβ eα ∧ eβ , 2 2 2
(2.1) (2.2)
where λα = λαi dy i , Bα = Biα dy i and we have defined eα ≡ dx α + λα . Of course the vielbein reads (eia dy i , Vαa eα ), where
δab eia ejb = gij ,
δa b Vαa Vβb = hαβ ,
g ij eia ejb = δ ab ,
hαβ Vαa Vβb = δ a b ;
we also record that the inverse vielbein has instead the form eai (
∂ ∂ − λαi α ) , i ∂y ∂x
Vaα
∂ . ∂x α
(2.3)
T–duality along the three x α directions can be expressed conveniently in terms of the quantity E = g + B: Eij dy i dy j + Eiα dy i dx α + Eαi dx α dy i + Eαβ dx α dx β → Eij dy i dy j + E αβ (dx α + Eiα dy i )(dx β − Eβj dy j ) ;
(2.4)
notice that in this expression all the (implicit) tensor products are neither symmetrized nor antisymmetrized, for example dy i dy j = dy i ⊗ dy j . Also remark that in this expression we used the dy i , dx α basis instead of dy i , eα as virtually everywhere else. E αβ is the inverse of Eαβ and can be decomposed in symmetric and antisymmetric parts: 1 1 αβ hˆ = h 1 αβ αβ αβ h + B h − B . (2.5) = hˆ + Bˆ , where E = 1 1 h+B Bˆ = (−B) h+B h−B 2
with the convention that ω1 ∧ ω2 = ω1 ⊗ ω2 − ω2 ⊗ ω1
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The objects hˆ and Bˆ would also be called in other contexts H and . (In this paper H denotes instead the three–form field.) Using the relations Bˆ hˆ −1 = −h−1 B ,
hˆ −1 Bˆ = −Bh−1 ,
Bˆ hˆ −1 Bˆ = hˆ − h−1
one can show that the T–dual metric and B field can be obtained by the original ones (2.1), (2.2) by the substitutions hαβ ←→ hˆ αβ ;
Bαβ ←→ Bˆ αβ ;
Bα ←→ λα ,
(2.6)
and leaving the gij and Bij in variant. Notice that last equation in (2.6) means that the twisting of each of the three S 1 bundles gets exchanged with the B field. This fact played, for example, a role in a number of applications and was recently formalized in mathematical terms in [11]. We can also find the vielbein Vˆ aα of the T–dual metric hˆ αβ , that satisfies Vˆ aα Vˆ aβ = αβ hˆ : αβ βα 1 1 aα a a ˆ V = Vβ = Vβ (2.7) h+B h−B whose inverse is Vˆαa ≡ hˆ αβ Vˆ aβ = (h − B)αβ V aβ = V aβ (h + B)βα .
(2.8)
The T–duality transformations of the vielbein then are: Vαa ←→ Vˆ aα ;
V aα ←→ Vˆαa .
(2.9)
We will mostly work in the case when the B-field is purely of base–fiber type in frame indices. Transformation (2.6) shows that this condition is conserved by T–duality, while (2.5) reduces to hˆ αβ = hαβ . Consequently, Vˆ aα = V aα and Vˆαa = Vαa . T–duality then only amounts to moving fiber indices up and down (still exchanging Bα and λα though). For later use, we also define here the tensors defining the SU(3) structure. These would be a priori only a two–form J and a three–form satisfying J ∧ = 0 and ¯ = (2J )3 /3!, but here we define the structure in a more conventional way starting i∧ from an almost complex structure. The latter is defined by giving the (1, 0) vielbein
E A = ieia dy i + Vαa eα ,
(2.10) ¯
where A = a = a goes from 1 to 3. The corresponding (0,1) vielbein is E B = E B . This almost complex structure is in general not integrable, as (even after rescaling) it is not expressible as d of a complex coordinate, E A = α A dzA . However, with an abuse of language we will use the quantity j
dzj ≡ dy j − iVγj eγ = −iea E a ,
(2.11)
keeping in mind that there is no reason for an actual coordinate zi to exist. We also used in this expression the notation
Viα ≡ δaa eia Vαa = eia Vaα .
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The two–form J (sometimes called the fundamental form) is defined by J =
i i ¯ δAB E A ∧ E B = gij dzi ∧ d z¯ j = −Viα dy i ∧ eα . 2 2
(2.12)
The holomorphic 3-form reads instead = E1 ∧ E2 ∧ E3 =
1 i ABC E A ∧ E B ∧ E C = − ij k dzi ∧ dzj ∧ dzk , (2.13) 6 6
where ij k = abc eai ebj eck . The choices we are making for the SU(3) structure are inspired by the SYZ approach. As we stressed above, these choices reduce the structure group further and are thus not to be expected to be as general (not even locally) as the T 3 fibration structure was in the SYZ approach. In particular some unaesthetic features will arise later in the dual of the complex coordinates. Anyway, in Sect. 5.1 we will try to amend this loss of generality. 3. Mirror Symmetry as T–Duality We start in this section showing, as promised in the introduction, how eB+iJ and get exchanged by T–duality. First we do the easier case, in which there is no B field and λ twisting of the T 3 bundle. The basic idea is that can be written in a sense as an exponential of the almost complex structure JM N applied to a degenerate three–form ij k dy i dy j dy k , that can be thought of as the holomorphic three–form in the large complex structure limit. A way to be more explicit is the following. Expand from (2.13) using the expression for the holomorphic vielbein in (2.10). One obtains four terms, with dy 3 , dy 2 e, and so on. Define now the operation V ⊥ (·) by V ⊥ (eα1 . . . eαk ) =
1 α1 ...α3 eαk+1 . . . eα3 , (3 − k)!
k = 0...3 .
This is essentially a Hodge star on the fiber, except it sends a k-form in the fiber into a 3 − k-vector (a section of 3−k T ). Lower eα are indeed vectors ∂α ≡ ∂/∂x α . This operation is very similar to the T–duality transformation of spinors to be discussed shortly. Using this, on every component of the expansion in dy and e of , we get a sum of (k, k) tensors, namely k indices up and k down: those down are along the fiber. The sum can be expressed as an exponent of Viα eα dy i , which is the complex structure. T–duality is now easy to perform. According to (2.9) its action is simply to raise and lower the α index: the tangent bundle (in the fiber direction) of the starting manifold is equal to the cotangent bundle (again in the fiber direction) of the T–dual manifold. As a result the complex structure now gets mapped to Viα eα dy i , the fundamental two–form J . So we have gotten T (V ⊥ ) =
i iJ e . 3!
(3.1)
The case with B–field and λ is less trivial. Although this is not strictly required here, we find it already at this point helpful to think about this in terms of Clifford(d, d) spinors. So we make a brief intermezzo explaining these and then we get back to our computation. Much of this material is taken from [4].
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3.1. Clifford(d, d) spinors. Clifford algebra is usually defined on the tangent bundle (or cotangent) of a manifold using the metric. In physical notation this amounts to defining d gamma matrices which satisfy {γ M , γ N } = 2g MN , where g MN is the metric on the cotangent bundle of the manifold. On the SU(3) manifold there is moreover a well–known representation of this Clifford algebra, on 0,p forms. If we on the contrary forget about the metric (thus about the SO(d) structure), this algebra cannot be defined. If we consider, however, both the tangent and the cotangent bundles of the manifold at the same time, there is a natural pairing between them (namely contraction between a vector and a form, (dy M , ∂N ) = δ M N ), in which the metric does not enter. This “metric” on T ⊕ T ∗ is block–off–diagonal 01 10 and thus of signature (d, d). Concretely, what this means is that one has to define 2d independent gamma matrices, γ M , γM , that satisfy {γ M , γ N } = 0 = {γM , γN } and M . Even though the Clifford structure has been defined on T ⊕ T ∗ , for{γ M , γN } = δN tunately the algebra still has a representation in terms of the forms on the manifold. Only now we have twice the number of creators and annihilators, and instead of using simply (0, p) forms as before, we have to use forms of all possible degrees. On this space ⊕dp=1 p T ∗ , an explicit representation is γ M = dy M ∧ ,
γM = ι∂M .
(3.2)
In all this we stress again that we have to consider γM and γ M as independent: we cannot raise and lower indices using the metric. In this Clifford(d, d) algebra, however, the usual Clifford(d) is embedded: indeed a combination of wedge and contraction in (3.2) is the more conventional Clifford product, and if we use that we can raise and lower indices. As stated in the introduction, a pure spinor is one which is annihilated exactly by half of the gamma matrices. If we come back to the application we have in mind, both eiJ ¯ are pure: and (γ M − iJ MN γN )eiJ = 0 ,
¯ =0, (γ M − iJ M N γ N ) ¯ =0. (γM − iJM N γN )
(3.3)
¯ are more familiar if one expresses them The gammas that annihilate the pure spinor ¯ is one of the in holomorphic/antiholomorphic indices: γ m = γm¯ = 0. Indeed Clifford vacua for the Clifford(d) representation mentioned above (this is why we wrote ¯ rather than ). Let us also notice that the annihilators of the two the relations for Clifford(d, d) spinors in (3.3) become the same when we allow ourselves to raise and lower indices on gammas, that is, when we descend to Clifford(d): eiJ becomes then an alternative expression for a Clifford vacuum of Clifford(d). As already mentioned, this dual way of realizing Clifford(d) from Clifford(d, d) is obviously in the center of mirror symmetry-exchange of the K¨ahler form and the holomorphic three-form (or their non-integrable generalizations) is seen as different choices of Clifford vacuum.
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3.2. Back to mirror symmetry. In this section, the only parts of the above theory that we actually use here are the formulas for the annihilators (3.3), which of course could have been derived independently. This insight gives however a useful rule of thumb, in particular when dealing with eiJ , where we can save ourselves expanding the exponential as we did above. What we will do in the following will be to consider eiJ and as Clifford(6,6) spinors, and other forms acting on them as combinations of gamma matrices. This is of course not the only possibility. One might have included B in the definition of the pure spinor. Due to technical details in how T–duality works we preferred this way. Also, we will work here in the case Bαβ = 0. Let us consider for example the expression eB . Due to γ α = iγ i Viα , this equals α iB α e ∧V . If we act on this with the operator V ⊥ defined above, theαprefactor can be taken out (it does not contain any eα ). On we get V ⊥ () = e−iV eα as above; the only thing to notice is that eα is simply ∂α , as seen on (2.3). If we finally apply T–duality, α α V α eα → Vα eα ; putting it together with the inert factor eiBα ∧V = eiB ∧Vα , we have shown i α T (eiJ ) = V ⊥ (eB ) e−Bα λ . 3!
(3.4)
It is a little surprising that the B field has to be subtracted on the right-hand side rather than being already present on the left-hand side. In the same way we can also prove the more reassuring T () =
i ⊥ B iJ Bα λα . V (e e ) e 3!
(3.5)
The exchange eB+iJ ←→ as presented in (3.4) and (3.5) is not very aesthetically pleasing, however the exponents involving the T–duality anti-invariant Bα λα are easy to explain going back to (3.3). The condition of purity e.g. on is essentially dz ∧ = 0, and the holomorphic coordinates change under T–duality. The reason for this is that the dzi which we have defined above as dy i − iVγi eγ has a λ hidden inside eγ . Since λ gets exchanged with B due to (2.6), dz on the original manifold does not map exactly to dz, but dz −→ dz −α i(Bα V α − λα Vα ) shifting by another T–duality anti-invariant. Thus the role of e±Bα λ is to compensate for this change, preserving the condition for purity. The combinations eiJ and allow, as we have commented on in the introduction and as we will see further later on, to treat J and more symmetrically. The most symmetrical object one might imagine is actually the SU(3) invariant spinor itself. Given also the role that we anticipated it will have in torsions, one might wonder at this point if it is more convenient to use T–duality transformation of and forget all the rest. The problem is, so to speak, that the spinor is too symmetric. The transformation rule of the ten–dimensional spinors are known: in the case without Bαβ , we simply have ψ+ → ψ+ , ψ− → γf ψ− , where γf is the product of the three gammas in the fiber directions [12]. However, when we express ψ± in terms of the chirality projected ± of the six–dimensional spinor, γf + is actually − and all the information we get is that a IIA compactification has been exchanged with a IIB one. This means that the spinor is essentially on both the original and the T–dual manifold the pull–back of a spinor in the base. Still, using the familiar bilinear definitions for J and (5.2) and γf + = − , one can show the identities above in a different way.
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4. Intrinsic Torsions and Their Duals This section is the technical core of the paper. Here we define and compute intrinsic torsions for our T 3 fibered manifolds. As stressed in the introduction, these are not the most general SU(3) structure manifolds. Performing T–duality along the T 3 is then easy using (2.6) and (2.9).
4.1. Conventional definition of torsions. We do not aim here at reviewing intrinsic torsions on manifolds with G–structures as discussions already exist in the literature, see for example [13] and among recent physics papers [1, 9]. Here we give a good working definition. It is familiar that, if we are on a SU(3) holonomy manifold, not only J and are well defined, but also they are closed: dJ = 0 = d. If they are not, dJ and d give a good measure of how far the manifold is from having SU(3) holonomy. The usual definitions require to split them in SU(3) representations: 3 ¯ + W4 ∧ J + W 3 , dJ = − Im(W1 ) 2 d = W1 J 2 + W2 ∧ J + W¯ 5 ∧ ,
(4.1)
where the representations of the Wi are as follows: • • • • •
W1 is a complex zero–form in 1 ⊕ 1; W2 is a complex primitive two–form, so it lies in 8 ⊕ 8; ¯ W3 is a real primitive (2, 1) ⊕ (1, 2) form, so it lies in 6 ⊕ 6; ¯ W4 is a real one–form in 3 ⊕ 3; W5 is a complex (1, 0)–form (notice that in (4.1) the (0, 1) part drops out), so its ¯ degrees of freedom are again 3 ⊕ 3.
These Wi allow to classify quickly any SU(3) manifold. We will later define them in an alternative way using directly the spinor; that definition will be more natural for T–duality, but the W ’s are often better to analyze the type of the manifold. For example, notice that in (4.1) the exterior derivative d does not satisfy the usual rule d : p,q → p+1,q ⊕ p,q+1 . For an almost complex manifold as we have here, there are also (p + 2, q − 1) and (p − 1, q + 2) contributions. Hence in (4.1) the (3, 0) ⊕ (0, 3) ¯ and the (2, 2) part of d, which reads W1 J 2 + W2 ∧ J . So part of dJ , namely Im(W1 ), we know actually that W1 = W2 = 0 iff the manifold is complex. One can check indeed that the Nijenhuis tensor can be expressed in terms of W1 and W2 . Other examples of the use of these W ’s abound in the literature. Notice also that the information of dJ and d is a little redundant, as W1 appears in both. Before we start computing, notice that from this classical definition it would be not obvious to guess transformation laws for W ’s, other than some qualitative features. There are two vectors, but the 8 and the 6 are different representations. If one thinks already at this stage about decomposing in base representations, guessing becomes easier, but one feels rapidly the need for a more solid ground. One way, which we pursue in this section, is to compute blindly. The other way is to put J and on a more symmetrical basis, using the formalism of Clifford(d, d), or, which is another manifestation of the same idea, to actually use the SU(3) invariant spinor directly. We do this in the next section.
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4.2. Computations of torsions in the T 3 fibered case. We can now compute W ’s from the expressions (2.12) and (2.13). This is done by doing contractions, partial or total, appropriate to isolate the component of interest. For example W4 is computed contracting J dJ .3 First we give W1 , W4 , W5 , expressed in the holomorphic basis.4 Note that W4 is real and W5 holomorphic, so that W4 = wi4 dzi + c. c. and W5 = wi5 dzi , while W1 = w1 is a scalar.5 These components read: i ij k Viα [d(V − iλ)]αjk , 12 1 wk4 = − V αj [dVα ]j k , 4 1 5 wk = − Vαj [d(V + iλ)]αjk − hαβ ∂k hαβ , 4 w1 = −
(4.2) (4.3) (4.4)
where [d(·)]ij = 2∂[i (·)j ] . We now pass to W2 and W3 . W2 is a (1,1)-form, and W3 is a real (2, 1) ⊕ (1, 2), and are written as 2 W2 = wij dzi ∧ d z¯ j ,
W3 =
1 3 w dzi ∧ d z¯ j ∧ d z¯ k + c. c. 2 ij k
(4.5)
However, since the representation 6 can be expressed not only as a primitive (2, 1) form, but also as a symmetric tensor with two holomorphic indices, we will give this 3 = w 3 pq . latter expression for W3 . The way to pass from one to another is wij j ipq This is already a little in the spirit of the different basis for intrinsic torsion that we will give later. Furthermore, these two matrices with indices ij can actually be further 2 has a symmetric decomposed in representation theory of the SO(3) of the base. wij and an antisymmetric part; the symmetric part does not drop out, it only contributes to the dy ∧ e part; the antisymmetric part can be dualized to a three dimensional vector 3 = w3 w2i = 21 ij k wj2k . As for W3 , wij + 13 wt3 gij is already symmetric but has a trace {ij }0 part wt3 on the three-dimensional base (of course, it is traceless in six dimensions): 1 pqk [d(V − iλ)]αpq [2Vkα gij − 3Vj α gik − 3Viα gj k ], 24 1 wk2 = − Vαj [d(V − iλ)]αjk , 4
2 w{ij } =
1 pqk [dVα ]pq [2Vkα gij − 3Vjα gik − 3Viα gj k ], 24 1 wt3 = pqk [d(V − 3iλ)]αpq Vkα . 8
3 = w{ij }0
(4.6) (4.7)
(4.8) (4.9)
3 Complete expressions for all five components of the intrinsic torsion for a metric of the form (2.1) can be found in Appendix A. 2 4 In what follows, we will denote W ’s in complex coordinates as lower case w’s. For example, w ¯ ij 2 ¯ is Wmn¯ , even if we did not explicitly mark i and j as holomorphic and antiholomorphic indices in this expression. This is also true for the other components. 5 As already emphasized, one has to bear in mind that the almost complex structure is in general not integrable, so that dzi is not to be understood as the differential of a hypothetical coordinate zi .
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Before turning to the T–duality transformations of components of the intrinsic torsion and the flux, we observe that the conditions for a supersymmetric vacuum with H only (1.3) are not compatible in a nontrivial way with the expressions above, as we anticipated in the introduction. For example, demanding W1 = W2 = 0 sets λ and V to constants. 4.3. T–duality. It is now easy to see what the transformation rules of the W ’s are. Decomposing in base representations says essentially where to look. One sees immediately that the various three–dimensional vectors and symmetric matrices are all similar. Before spelling this out, one should however stress that the full six–dimensional quantities have a more complicated transformation rule. As explained in Sect. 3.2, due to the presence of λ in eγ , dzi = dy i − iVγi eγ on the original manifold does not map exactly to dz on the mirror side. With this important caveat in mind, let us proceed to give T–duality transformations. As we said, many of the expressions we have for W ’s are similar (see Appendix A). The differences are mainly because of Vα versus V α . This is already good, as these quantities are exchanged by T–duality (2.9). One also sees that some of the quantities contain λ, that after T–duality become B as we just recalled. So, we are led naturally to complexify some of the torsions adding dB projected in the appropriate representation. As these projections are verbatim those we did for dJ in the previous subsection, this step is trivial. Thus defining components for H as for other forms 6 3 ¯ + H4 ∧ J + H 3 H = − Im(H1 ) 2
(4.10)
we find the transformations: w1 − ih1 ←→ −(w1 − ih1 ) , 2 w{ij } ←→ (w3 + ih3 ){ij }0 , wk5 −
1 αβ h ∂k hαβ = w¯ k2 ←→ (w4 − ih4 )k , 4
(4.11)
describing the mixing of the components of the flux and of the intrinsic torsion under mirror symmetry. The central role in the mirror/T–duality transformation (4.11) is obviously played by W2 (a component of the torsion associated with the non-integrability of the complex structure). It splits in two different pieces upon restriction to the base and the respective mixing of the two parts of W2 with complexified H3 and H4 is an essential ingredient of the mirror map. We will now try to rederive and generalize to generic geometries these results from a different point of view, using spinors rather than differential forms. 5. Spinorial Basis The idea is that the same information we have in dJ and d are contained in DM . Doing the effort of reexpressing torsions in these terms pays off for several reasons. 6 The explicit expressions for components of H in the T 3 -fibered geometry can be found in Appendix A. We labeled these components so that they match the corresponding ones in dJ . H1 is then the 1 ⊕ 1 complex scalar, H3 the 6 ⊕ 6¯ real 3-form and H4 the 3 ⊕ 3¯ real 1-form.
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First of all, the combinations that appear in DM transform better. Second, they might be useful in future occasions to analyze the geometry behind a given supersymmetry transformation without even having to bother to construct bilinears. In particular, we can find from this approach immediately the conditions (1.3) for supersymmetric vacua with H . One proceeds in the following way. What we call in what follows is the SU(3) invariant spinor, which can be furthermore decomposed by chirality as + + − . Again, if we were on a manifold of SU(3) holonomy, we would have a covariantly constant spinor, DM = 0. This is not the case, but still decomposing DM into representations will give us a measure of how far we are from the SU(3) holonomy. The way of decomposing DM into representations is again implicit in the literature. On a SU(3) invariant manifold, a basis for spinors is given by ± and γM ± (or alternatively we can trade ± with and γ ). So, for example, anything else in Clifford algebra acting on , say γ M1 ...Mn , can be reexpressed in terms of this basis. Explicit formulas for this are known (see for example [6]; in [14] a complete set of these equations is provided, along with the simple group theoretical description of how to get them, for the case of seven–manifolds with G2 structure). We will not however need them here, it is enough to know that this decomposition can be done. Actually, with one exception: the relation γ M γ = iJ M N γ N can be used to eliminate one possible term. So we can write in general (5.1) DM = qM + i q˜M γ + iqMN γ N . The real q’s that we have defined in this equation are just another definition of intrinsic torsion. To see that they can be compared with the W ’s above, it suffices to use group theory. qM and q˜M are vectors, 3 ⊕ 3¯ ; as to qMN , it can be decomposed into ¯ ⊗2 = (6 ⊕ 3) ¯ ⊕ (6¯ ⊕ 3) ⊕ (8 ⊕ 1) ⊕ (8 ⊕ 1). We see that all the representations (3 ⊕ 3) of the W ’s are present. There is one redundancy, since we get three vectors (qM , q˜M and one from qMN ). The objects we get in this way are the same as the W ’s up to factors. Qualitatively we could stop here; in the present context we are actually interested in getting the factors, as they are important for being able to express q’s in terms of W ’s explicitly. This is done as follows. After having decomposed qMN as above, we can define J and as bilinears as † γMN γ = iJMN ,
−i † γMNP (1 + γ ) = MNP .
(5.2)
One can now compute their exterior derivative using (5.1). Comparing the result with (4.1) gives the desired coefficients. The result is 1 1 + P W1 GMN + W1− JMN + MNP (W5 − 2P W4 ) + c. c. 4 8 1 1 2,+ 2,− P 3 + −JM WP N + WMN + Im(WMN ) 4 8 1 1 W1 P¯MN + MNP (W¯ 5 − 2W4 )P = Re 2 4 i ¯ P 2 i 3 (5.3) + P M WP N − WMN , 2 8 = [W¯ 5 − P¯ W4 ]M , (5.4)
qMN =
qM + i q˜M
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3 where Wi = Wi+ − iWi− as usual in the literature, and we have defined WMN = 1 3 P Q and used a holomorphic projector P = (1 − iJ ). We had observed WMP N Q 2 in the previous section that the split of W2 in two parts upon restriction to the base is crucial in the mirror transformation. Here we can see that the split nature of W2 reveals itself in covariant six-dimensional expressions: W2+ and W2− enter respectively into the symmetric and antisymmetric parts of qMN . It is worth recording the same expression in holomorphic/antiholomorphic basis:
i 3 1 qmn = − wmn + mnp (w¯ 5 − 2w¯ 4 )p , 4 4
i 2 1 qmn¯ = − wm ¯ 1 gmn¯ . n¯ + w 4 4
(5.5)
And for the remaining vector: qm − i q˜m = (w5 − w4 )m .
(5.6)
The quantities we have defined so far would not be expected to behave nicely under T–duality, for the following simple reason. The transformation laws we have computed in (4.11) have, as one would expect also from the arguments in [1] and from (1.1), the feature of exchanging some torsions with H . Therefore we have to add a dependence on H to the covariant derivative in (5.1). Then also the q defined in (5.1) will change and (5.1) will become H ˜ M γ + iQMN γ N . DM = QM + i Q (5.7) We have defined D H (and as a consequence the Q’s) in such a way as to find good T–duality transformation properties afterwards. Not too surprisingly, we have found that the best definition is exactly the same as the one which appears in supergravity H ≡ (D + 1 H NP ). We find then supersymmetry transformations: DM M 8 MNP γ
1 1 QMN = Re (W1 + 3iH1 )P¯MN + MNP (W¯ 5 − 2(W4 + iH4 ))P 2 4 i i (5.8) + P¯MP WP2 N − (W 3 + iH 3 )MN . 2 8 So, adding H as D → D H complexifies W as W + iH , though at the end the Re in (5.8) makes the Q’s real. It should also be possible to write directly a formula for the (con)torsion, as an alternative to formulas for the q’s that we have given. The fact that H appears as HMNP γ NP tells us already that this formula will have a piece KMNP = dJMNP +. . . that will combine with H . As we will not need it here, we do not pursue this. Notice also that the G2 analogue of what we just did for H is discussed in detail in [14] for the G2 case. The fact that the natural combination for T–duality and for supersymmetry is the same will be useful later, when we will try to extend our results to the general case. Then this is also a good place to see that of course the conditions for supersymmetry in the case with H only (1.3) can be recovered from the spinor equation. To have supersymmetry it is enough that one chirality, say + , is annihilated by D H . We have the expressions H = (Q + i Q ˜ m )+ + iQmn γ n − , Dm + m
H = (Q − i Q ˜ m )− − iQmn¯ γ n¯ + , Dm − m
H = (Q + i Q n ˜ m¯ )+ + iQmn Dm m ¯ ¯ γ − , ¯ +
H = (Q − i Q ˜ m¯ )− − iQm¯ n¯ γ n¯ − . Dm m ¯ ¯ − (5.9)
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Notice that Qmn¯ and Qm¯ n¯ have disappeared from D H + , because − , being a Clifford vacuum, is annihilated by γ n¯ . From this one obtains directly that the complexified Qmn and Qmn ¯ have to vanish. These will say that the complexified W3 has to be purely antiholomorphic, which in more usual terms means of type (1, 2) (this is the condition W3 = ∗H3 ) and that W2 has to vanish. The vectors require a little more care because usually the dilaton is rescaled in the metric (as a warping) and in the spinor itself. More generally it is clear that one can use gamma matrices identities mentioned above to reduce the expression to a form like (5.1), and then use (5.4) or (5.5). For us the main advantage of having computed these quantities is to compare with T–duality transformations given in previous sections, although we will see shortly how these supersymmetry considerations can play a role in understanding the general case (without T 3 fibration structure). We can restrict the free index in (5.1) to be on the base, M = i, and furthermore apply a chirality projector ˆ i + + i Q ˆ ij γ j − , Di + = Q
¯ˆ + i Q ¯ˆ γ j , Di − = Q + i − ij
(5.10)
ˆ i and having introduced hatted quantities for restrictions to the base. The quantities Q ˆ Qij in these expressions turn out to transform neatly under T–duality: ¯ˆ , ˆ i −→ −Q Q i
¯ˆ , ˆ ij −→ −Q Q ij
(5.11)
with the expressions ˜ i = (W¯ 5 − 1 (W4 − iH4 ))i , ˆ i = Qi + i Q Q 2 ˆ ij = Qij − iQiα Vjα = 2P¯jM QiM Q
1 ¯ i ¯ iM ¯ W1 + 3i H(1) + P (W3 − iH3 )Mi gij = 4 12
i ¯2 1 ¯M − W{ij } − Pi (W3 − iH3 )Mj 4 4
k i 1 + ij k W5 − W4 − iH4 − W¯ 2 . 2 2
(5.12)
(5.13)
This means that at an effective level the rule tells us + ↔ − . We should remark that working so far with a finite-size T 3 -fiber, we have extra (nowhere vanishing) vector fields, and thus reduces structure. This may in particular allow to locally preserve supersymmetry even when conditions (1.3) are violated. Since when fibers degenerate this restricted stricture no longer exists, we avoided making explicit use of it, even though doing restrictions to the base manifolds implicitly uses the existence of a restricted structure. It is reasonable to expect that the results based on representations are valid over the entire moduli space, and thus next we turn to the six-dimensional covariantization of mirror transformation (4.11). 5.1. Approaches to the general case. At this point it is natural to wonder if we have enough information to simply guess what mirror symmetry should be in the general case. We have a precise set of transformation rules in the case of T 3 fibrations, and we also know that supersymmetric vacua should be sent to supersymmetric vacua. As we
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remarked above, T–duality is induced by an exchange of + with − . Since we also have γ m¯ − = 0, these two facts together would suggest the following proposal naturally generalizing (5.11): Qmn ←→ −Qmn¯ ,
¯m . Qm ←→ −Q
(5.14)
We noticed above that representations of W ’s do not match in such a way as to suggest immediately a transformation law. In the T–duality approach above this was solved by decomposing further in representations of the SO(3) of the base. The proposal (5.14), on the contrary, gets around this problem collecting together SU(3) representations rather than decomposing them further: qualitatively, 6 ⊕ 3¯ ↔ 8 ⊕ 1. Let us now check that this proposal for mirror symmetry agrees with T–duality and with supersymmetry, as we just required. First of all, (5.14) agrees with the exchange (5.11). Indeed we have ˆ P n¯ + P¯mP¯ Q ˆ P n¯ = 2Qmn¯ + 2Qm¯ n¯ ; ˆ M n¯ = Pm P Q Q
(5.15)
ˆ m. similarly one can consider the transformation of Qm = Q Turning now to supersymmetry, the two transformations in (5.14) induce simply H H + −→ −Dm − . Dm
(5.16)
So if only H is present we are sending D H + = 0 to −D H − = 0; in the latter case supersymmetry is of course still preserved. In this form the duality might seem a little tautological, in the sense that it sends a supersymmetric vacuum in another one in an obvious way. Compare however with the usual mirror symmetry: a Calabi–Yau is sent to another Calabi–Yau, and the nontriviality lies in the exchange of K¨ahler and complex structure moduli. This should be happening for vacua with H only as well, and in a sense this would be yet another check to do; we will comment on this in the next section. Coming back to checking compatibility with supersymmetry, the situation becomes more complicated with RR fluxes, because the latter also transform, and one would have to check that they do it in a way compatible with the one we are giving for geometry and H . This can be elaborated as follows. Just as the entire NS contribution to the covariant derivative of the invariant spinor got summarized in Q’s (see (5.8)), the RR contribution can be accounted for by introduction of similar objects, RM , R˜ M and RMN with a group decomposition matching that of Q’s. On supersymmetric backgrounds, the total action of the covariant derivative of the invariant spinor should be zero and thus R = −Q. Thus from this point of view the mirror transformation of the RR sector can also be brought to the form (5.14). From the other side, in the T 3 fibered case, one could use the known transformation rules of RR fields. From the above, it is clear that the natural way to do this check in general would be to consider RR fields not as sums of forms but as bispinors, expressing for example in terms of the latter supersymmetry transformations also. Even after all these motivations, the proposal (5.14) stands as a conjecture, and there would be other possible checks to be made. One possibility is to use again the formalism of Clifford(6,6) spinors. One can give an alternative definition of torsions, that we have not mentioned so far, using the Clifford(6,6) spinors eiJ and . Schematically one gets (2)
DM eiJ = qM eiJ + I m(qM · ) , (2)
(2)
DM = (qM + i q˜M ) + qM · eiJ . (5.17)
In these equations, qM · is the Clifford product of qMN using only the second index. These formulas seem indeed to be consistent with the general rule (5.14) given above.
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6. Applications and Examples In this section we analyze some simple consequences of the mirror symmetry transformation that we have proposed. Apart from the case in which only geometry and B-field are present, the situation will be different from the usual one for Calabi–Yau’s in that RR fluxes will transform, and so solutions with some types of fluxes switched on get mapped generically to solutions with other types of fluxes. On top of this we should also have the usual exchange of K¨ahler and complex structure moduli, in the sense of (1.1). Simple checks of both claims have been listed in the previous section; here we take these statements for granted and examine the consequences. The natural starting point is to check how the picture developed so far reduces to known cases. We start from a brief discussion of an example which has already been mentioned, and involves a CY manifold with B-field turned on. This case was considered in [1] in great detail. Since the intrinsic torsion vanishes on CY, we start from QMN built purely from components of H . The Qmn¯ gets a single contribution from H1 . If we follow [1] and look for a purely geometrical mirror, on the mirror side we may have non-zero W˜ 3 and W˜ 4 − W˜ 5 = 0. Looking at Qmn , we see that the reality of the remaining components of the flux ensures that on the mirror side only W˜ 1− and W˜ 2− survive. This agrees with [1] up to a conventional ± exchange. So we recover as a particular case the half-flat geometries and the G2 lifts discussed in [16, 17]. Note that neither the starting configuration, nor its mirror are vacua but rather domain walls. The simplest background is when the B-field is turned off and we just deal with Calabi–Yau geometry. This case was also discussed in Sect. 3, where we recover the exchange of the complex structure and (the exponentiated) K¨ahler form for mirror Calabi–Yau manifolds. An exchange of complex and K¨ahler moduli for a metric of the form (2.1) with λ = 0 and the integrability properties of its complex structure were studied in [15]. Here we easily see that the exchange of the eiJ and is accompanied by an exchange of their integrability conditions. Without turning RR fields on, we can also consider yet another possibility of vacua.7 These cases are to obey the conditions given in (1.3). In our language these conditions read Qmn = 0 = Qmn ¯ . What one gets by the proposal (5.14) is the condition Qmn¯ = 0 = Qm¯ n¯ , which is obviously isomorphic to it (see the comments in the previous section). Less trivial is the statement that complex and K¨ahler moduli are exchanged. To check this one would first of course have to know by what groups moduli spaces are computed. This check we will not be able to perform here, and we limit ourselves to some comments. First of all, in general moduli spaces of solutions with fluxes are likely not to be simply factorized in K¨ahler and complex part. This is because, unlike the Calabi–Yau case, the conditions are no longer dJ = 0 = d, but something involving torsions; and the definition of torsions (4.1) mixes and J . Also, in the Calabi–Yau case the fact that the conditions were of simple closure allowed to reduce the counting to a cohomology problem. In general, here, we are dealing with conditions involving projections Prep dJ and Prep d, where Prep is a projector on a certain representation. These conditions mean roughly that a form is closed “up to” a contribution from the other form, schematically dJ = operator(). In general it should be possible to restate this as the cohomology of a double complex. Coming back to the case with H only switched on, a preliminary 7 Here and in the parts with RR on also , the word vacuum should be understood with the usual grain of salt: no–go theorems force us to consider noncompact or singular cases, or to hope (in a less well–defined way) that some of the features analyzed here will survive after taking into account higher–derivatives corrections.
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analysis of moduli spaces was sketched in [18], following ideas in [19]. Indeed the H – twisted cohomology groups proposed there are total cohomologies of a double complex with ∂¯ and H 2,1 ∧ as differentials. We will unfortunately not say more on this here, but plan to come back to the issue in the future. For now we just observe that, in known examples, fluxes fix complex structure moduli. These considerations tell us that in a mirror picture K¨ahler moduli will be fixed. The Type B solution for IIB strings presented in [20] provides another related case of flux compactifications with the back-reaction taken into account. The metric now is conformally CY, and RR-fluxes are turned on as well. In addition, supersymmetry conservation imposes restrictions on H -flux, which now turns out to be primitive. We will not attempt here to present a complete analysis of the mirror transformation and will ignore the RR sector (which mirror symmetry maps to RR fields in Type IIA theory). Thus our starting data include W4 ∼ W5 and H4 . Note that this means in particular that ˜ mn = 0 on the mirror side. The two previous we have Qmn¯ = 0, and thus we need Q examples have this feature: we could either take H˜ = 0 and W˜ 3 = W˜ 4 − W˜ 5 = 0 or have H˜ with imaginary selfdual primitive part and geometry given by W˜ 3 = ∗H˜ 3 and 2W˜ 4 = W˜ 5 = 2d φ˜ = 2i H˜ 4 as in [8]. However it is different from previous cases Qmn = 0 and this results in additional non-integrability of the complex structure on the mirror side (in particular, W˜ 2− cannot be zero now). Of course, explicit constructions of such IIA string backgrounds would be of some interest. The last application we will discuss here concerns the possibility of lifting the SU(3) mirror symmetry picture to the G2 -structure case. We could start from IIA string theory in a monopole background and lift it to M-theory, using the explicit relations between the components of intrinsic torsion for SU(3) and G2 for U(1)-fibered manifolds. The components of the torsion for the representations 1, 7 and 27 get complexified by the corresponding representations of the G4 -flux. The analogy with the SU(3) case is rather close. There as well there was a number of components of the intrinsic torsion that get complexified by the H -flux; mirror symmetry then mixed these with the components corresponding to representations that are not contained in the flux (essentially 8 ⊕ 8 in that case, with some extra subtleties having to do with 3 ⊕ 3¯ appearing twice). In the G2 geometry, 14 is such a representation, and the corresponding component of the torsion is the lifting of W2− [17, 21], the component of SU(3)-torsion central in the exchange with the NS flux. Once more one would be hoping that going to the spinorial basis and writing for the invariant spinor the twisted covariant derivative will lead to a covariant expression for a mirror transformation for the G2 geometry. Indeed, as in (5.1) the torsion for the G2 -structure manifolds is also encoded in a covariant derivative DM = (qM + iqMN γ N ), where q’s are real. Then the eleven-dimensional supersymmetry transformations restricted to seven-dimensions twist the covariant derivative by a term 3i GM − 13 (2GδMN + GMN + 2GNM )γ N , where we have defined 1 1 1 G ≡ 4! GMNP Q (∗)MNP Q , GM ≡ 3! GMNP Q NP Q , GMN ≡ 3! GMP QR (∗)P QR N using the associative form . Putting all together we arrive at the twisted operator G DM = (QM + iQMN γ N )
which we can now use to extend the SU(3) mirror symmetry proposal. Indeed, the G2 analogue of (5.14) can be written as − Q+ [MN ] ←→ −Q[MN]
Q{MN } −→ −Q{MN} ,
(6.1)
where ± denote selfdual and antiselfdual representations respectively. Note that only the former is complexified by G-flux, and (6.1) exchanges 14 with 7+7. In view of this,
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we may go back to (5.14) and note that there as well, modulo the trace part, mirror symmetry can be thought of as an exchange of selfdual and antiselfdual matrices (`a la Hermitian Yang-Mills). 7. Discussion We conclude by mentioning some open technical and conceptual problems. Throughout the paper we have worked with a Bαβ = 0 case. Obviously, this choice simplifies greatly the T–duality transformation. The reason for this is most clear on the spinorial picture. As shown in [12], the only change in the simple T–duality transformations used above (see Sect. 3.2) occurs when the Bαβ component of the B-field is nonzero. In this case we have to use instead ψ+ → ψ+ ,
ψ− → eE γf ψ− ,
where E αβ is defined in (2.5). We have here a gamma matrix exponential of E ≡ 1 αβ 2 E γαβ which has the same form of the kappa–symmetry operator; in the power series expansion the products of all gamma matrices are antisymmetrized. Note that without a Bαβ component, there is a certain ambiguity in the choice of T–duality invariants (5.12). The ambiguity is in the complexification by H in Q’s. We have chosen everywhere the plus sign (and correspondingly T–dual expressions which become complex conjugates) for the following reason. The singlet representation allows a simple calculation even with a non-vanishing Bαβ . The result is then the first formula in (4.11), which fixes the ambiguity. For all other components we have chosen the complexification rule consistent with that of W1 , hence the choice of sign in the definition of the twisted covariant derivative (5.7). The T–duality rule for the spinors given above should allow to lift restrictions from the B-field and verify this explicitly. We would like to emphasize though that this restriction is of a technical nature-for a number of applications the B-field is generic enough. First, the H -flux contains all the representations it can. Second, in the holomorphic coordinate basis it is not hard to see that B is of generic type and contains both (1, 1) and (2, 0) components. The latter is important for several aspects of topological B-branes (see [22] for a recent discussion, in which also Clifford(d,d) spinors appear) and mirror symmetry [23]. Clearly there are two directions in which our results have to be extended. As mentioned many times we have worked with a T 3 fibration with finite-size fibers (and thus had a luxury of having extra vector fields without zeros) and most of our formulae explicitly involve restrictions to the base of the fibration. At the end we succeeded in finding a basis in which the mirror/T–duality transformations can be covariantized and written over the entire six-dimensional manifold. The final simple rule for the mirror transformation ¯m Qmn ←→ −Qmn¯ , Qm ←→ −Q is of group-theoretical nature, and we conjectured it to be true for general geometries, even without fibration structure at all, not even locally. In particular it should also work when there is a fibration but with singular fibers. From the other side, singular T 3 fibers hold the key to the SYZ picture, and would be extremely important to understand their fate in any generalization of SYZ. Finally, one would like to complete the picture by incorporating D-branes. A better understanding of submanifolds in generalized CY manifolds as well as vector bundles on these would be essential preliminaries. Extending the picture developed in [5] for
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the exchange of branes (a pair of calibration and bundle conditions) and T–duality to the generalized CY case would be of great interest. We may recall once more that in the SYZ picture both mirror manifolds appear as moduli spaces for D-branes wrapping (sub)manifolds. One may hope that eventually developing the picture of D-brane moduli spaces in geometries with NS fluxes may lead to refining the proposal for mirror symmetry presented here. Acknowledgements. We would like to thank Peter Kaste for participation in early stages of this work, and Mariana Gra˜na and Fawad Hassan for useful conversations. This work is supported in part by EU contract HPRN-CT-2000-00122 and by INTAS contracts 55-1-590 and 00-0334.
A. Intrinsic Torsion for T 3 -Fibered Manifolds The components of the intrinsic torsion are defined by 3 ¯ + W4 ∧ J + W3 , dJ = − Im(W1 ) 2 d = W1 J 2 + W2 ∧ J + W¯ 5 ∧ . They can be computed using contractions () with J and : 4 2 4i 1 J d = − dJ = ABC (E A ∧ E B ) dE C 3 3 3 i = − ij k Viα [d(V − iλ)]αjk , 12
W1 =
W2 = 4J [d − W1 J 2 − W¯ 5 ∧ ] 1 ij k = [d(V − iλ)]αjk [gpq Viα − 3gpi Vqα ]dzp ∧ d z¯ q , 12
W3 = dJ +
3 ¯ − W4 ∧ J Im(W1 ) 2
3 Viα [dλα ]j k dy i ∧ dy j ∧ dy k 8 1 3 − [dVα ]ik [ δjk δβα + Vjα Vβk − 2V kα Vjβ ] dy i ∧ dy j ∧ eβ 4 2 1 1 j j + [dλα ]j k [ Viα Vβ Vγk − δi hαβ Vγk ] dy i ∧ eβ ∧ eγ 4 2 1 − Vβi Vγj [dVα ]ij eα ∧ eβ ∧ eγ 8 1 = −i[dVα ]j k Viα + i[dVα ]ij Vkα + i[dVα ]ki Vjα 16 +i[dVα ]j l V lα gik − i[dVα ]kl V lα gij +[dλα ]j k Viα + [dλα ]ij Vkα + [dλα ]ki Vj α dzi ∧ d z¯ j ∧ d z¯ k + c. c.,
=
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W4 = 2J dJ = = (1,0)
W5
421
1 αk V [dVα ]j k dy j 2
1 αβ h [dhαβ − LgVα Vβ ], 2
¯ = − d 1 Vαk [d(V + iλ)]αjk + hαβ ∂j hαβ dzj = 4 1 = [hαβ LgV α V β ]j − iVαk [dλ]αjk dzj , 4
j
where dzj = dy j − iVγ eγ . In the last two expressions, we have used the Lie derivative L, which is defined by LX Y = [X, Y ] = [Xi ∂i Y j − Y i ∂i X j ]∂j ,
LX ω = [Xi ∂i ωj + ωi ∂j X i ]dy j , (A.1)
on the vector field Y and the 1-form ω, with respect to the vector field X. We wrote V β β and Vβ for the 1-forms Vj dy j and Vjβ dy j , while gV α and gVα are the vector fields V iα ∂i and Vαi ∂i . We also give here the components of the H field H = dB2 1 β = ∂k Bαβ dy k ∧ eα ∧ eβ + [∂k Biα − Bαβ ∂k λi ] dy k ∧ dy i ∧ eα 2 1 + [∂k Bij − ∂k Biα λαj + Biα ∂k λαj ] dy i ∧ dy j ∧ dy k . 2 As a 3-form, we project H on representations of SU(3) as we did for dJ : 3 ¯ + H4 ∧ J + H 3 . H = − Im(H1 ) 2 These components are computed with the same contractions used for W ’s: h1 = H1 = −
(A.2)
(A.3)
4i H 3
1 ij k α β Vi Vj ∂k Bαβ 12 i + ij k Viα [dBα − Bαβ dλβ ]j k 12 1 − ij k [∂k Bij − ∂k Biα λαj + Biα ∂k λαj ], 12 H4 = 2J H 1 1 = − V kα [dBα − Bαβ dλβ ]j k dy j − V kα ∂k Bαβ eβ 2 2 = h4k dzk + h¯ 4k d z¯ k , 1 jα β h4k = V [dBα − Bαβ dλβ ]j k − iV j α ∂j Bαβ Vk , 4 =
(A.4)
(A.5)
(A.6)
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H3 = H +
3 ¯ − H4 ∧ J Im(H1 ) 2
1 kβ iγ µ V V [∂k Biα − Bαµ ∂k λi ] eα ∧ eβ ∧ eγ 4 1 5 1 γ j jγ + ∂k Bαβ − V Vkα ∂j Bγβ − Vk Vα ∂j Bγβ dy k ∧ eα ∧ eβ 2 4 2 1 γ γ + [∂k Bij − ∂i Bj γ λk − ∂i λj Bkγ ] 8 j ×[Vβk Vαj dy i − Vβk Vαi dy j + Vαi Vβ dy k ] ∧ eα ∧ eβ
1 3 k α + [(dBα )ik − Bαβ (dλβ )ik ] δj δγ + Vjα Vγk − 2V kα Vj γ dy i ∧ dy j ∧ eγ 4 2 3 γ γ + [∂k Bij − ∂i Bj γ λk − ∂i λj Bkγ ] dy i ∧ dy j ∧ dy k 8 1 β − Vjα Vk ∂i Bαβ dy i ∧ dy j ∧ dy k 8 1 = h3ij k dzi ∧ d z¯ j ∧ d z¯ k + c. c. (A.7) 2
=
In analogy with w’s (see (4.6)-(4.9)) we have introduced h: h3ij = h3ipq pq j = h3{ij }0 +
1 3 h gij , 3 t
1 pqk [dBα ]pq [2Vkα gij − 3Vjα gik − 3Viα gj k ], 24 1 h3t = − pqk [dBα ]pq Vkα 8 i 1 − pqk [dB − (dBα ∧ λα + dλα ∧ Bα )]pqk . 8 2
h3{ij }0 = −
(A.8)
(A.9)
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13. Joyce, D.D.: Compact manifolds with special holonomy. Oxford: Oxford Univ. Press, 2000 14. Kaste, P., Minasian, R., Tomasiello, A.: Supersymmetric M-theory compactifications with fluxes on seven-manifolds and G-structures. JHEP 0307, 004 (2003) 15. Liu, J.T., Minasian, R.: U-branes and T**3 fibrations. Nucl. Phys. B 510, 538 (1998) 16. Hitchin, N.: Stable forms and special metrics. In: Global differential geometry: the mathematical legacy of Alfred Gray, Volume 288 of Contemp. Math. , Providence RI: AMS, 2001, pp. 70–89 17. Chiossi, S., Salamon, S.: The intrinsic torsion of SU(3) and G2 structures. In: Proc. conf. Differential Geometry, Valencia 2001, Singapore: World Scientific Publishing, 2002, pp. 115–133 18. Becker, K., Becker, M., Green, P.S., Dasgupta, K., Sharpe, E.: Compactifications of heterotic strings on non-Kaehler complex manifolds. II. Nucl. Phys. B 678, 19–100 (2004) 19. Rohm, R., Witten, E.: Annals Phys. 170, 454 (1986) 20. Grana, M., Polchinski, J.: Gauge / gravity duals with holomorphic dilaton. Phys. Rev. D 65 126005 (2002) 21. Kaste, P., Minasian, R., Petrini, M., Tomasiello, A.: Nontrivial RR two-form field strength and SU(3)-structure. Fortsch. Phys. 51, 764–768 (2003) 22. Kapustin, A.: Topological strings on noncommutative manifolds. Int. J. Geom. Math. Mod. Phys. 1, 49–81 (2004) 23. Kapustin, A., Orlov, D.: Vertex algebras, mirror symmetry, and D-branes: The case of complex tori. Commun. Math. Phys. 233, 79 (2003) Communicated by N.A. Nekrasov
Commun. Math. Phys. 254, 425–478 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1162-z
Communications in
Mathematical Physics
The Topological Vertex Mina Aganagic1 , Albrecht Klemm2 , Marcos Marino ˜ 3 , Cumrun Vafa1,4 1 2 3 4
Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA Humboldt-Universit¨at zu Berlin, Institut f¨ur Physik, 10115 Berlin, Germany Theory Division, CERN, Geneva 23, 1211 Switzerland California Institute of Technology, 452-48, Pasadena, CA 91125, USA
Received: 16 February 2004 / Accepted: 5 March 2004 Published online: 24 September 2004 – © Springer-Verlag 2004
Abstract: We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of K¨ahler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization. 1. Introduction Topological strings have been a focus of much interest since they were proposed more than a decade ago [1]. A central question has been how to compute the corresponding amplitudes. There have been two natural approaches available: i) using mirror symmetry to transform the problem to an easier one; ii) mathematical idea of localization. Both approaches can in principle yield answers to all genus amplitudes (at least in the noncompact case). However the computations get more and more involved as one goes to higher genera and neither method becomes very practical. Ever since the discovery of large N Chern-Simons/topological string duality [2] another approach has opened up: Chern-Simons amplitudes seem to give an efficient way to sum up all genus amplitudes. This idea was developed recently [3, 4] where it was shown that one can compute all genus A-model amplitudes on local toric 3-folds from its relation to Chern-Simons amplitudes. However in trying to obtain amplitudes in this way, one had often to take certain limits. The main aim of the present paper is to bring this line of thought to a natural conclusion by giving the direct answer for the topological string amplitudes, without any need to take any limits. Toric 3-folds are characterized by a graph which encodes where the cycles of a T 2 fibration degenerates. The vertices of this graph are generically trivalent. The
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computations in [3 and 4] were more natural in the context of tetravalent vertices of the toric graph. To obtain the generic situation of the trivalent graph, one had to take particular limits in the Calabi-Yau moduli space. Thus the basic goal is to directly capture the structure of the trivalent vertex. That there should be such a vertex has already been noted [5, 6]1 . In this paper we show how this can be achieved. The idea can be summarized, roughly, as putting many brane/anti-brane pairs which effectively chop off the Calabi-Yau to patches with trivial topology of C3 . Computing open topological string on C3 defines the cubic topological vertex. Gluing these together yields the closed topological string results (with or without additional D-branes). Thus the full amplitude can be obtained from a cubic field theory, where each Calabi-Yau corresponds to a Feynman graph with some fixed Schwinger times (determined by the K¨ahler class of the Calabi-Yau). This result can best be understood in the mirror picture as computation of the quantum Kodaira-Spencer theory [8]. The Kodaira-Spencer theory is, in this context of non-compact Calabi-Yau, captured locally by a chiral boson on a Riemann surface. The degrees of freedom on the brane get mapped, in this setup, to coherent states of the chiral boson, and the trivalent vertex gets identified with the quantum correlations of the chiral boson. Moreover, the brane in the B-model gets identified with the fermions of this chiral boson. Thus the fact that knowing amplitudes involving branes leads to closed string results translates to the statement that knowing amplitudes involving fermions leads via bosonization to the full answer for the chiral boson. The topological vertex gets mapped, in this setup, to a state in the three-fold tensor product of the Fock space of a single bosonic string oscillator. To leading order in string coupling and oscillator numbers this is a squeezed state as in the conventional approaches to the operator formulation in the Riemann surface. However the full topological vertex is far more complicated; the chiral scalar is not a conventional field. The full vertex involves infinitely many oscillator terms together with highly non-trivial gs dependence. Nevertheless, we find the following closed formula for this highly non-trivial vertex |C: 2 3 tn1 , tm , tp |C =
Q1 ,Q3
R1 R3t κR /2+κR /2 WR2t Q1 WR2 Qt3 3 trR1 V1 t q 2 1 Q3 WR2 0
NQ
trR2 V2 trR3 V3 , (1.1)
where
R1 R3t t 1 Q3
NQ
=
R
Rt
NQR1 R1 NQt R3 . 3
Here Ri , Qi are representations of U (N ), NRiRRkj is the number of times the representation Rk appears in the tensor product of representations Ri and Rj , R t denotes the representation whose Young Tableau is the transpose of that of R and WRQ = SRQ /S00 , where S is the S-matrix of the modular transformation of the characters of U (N )k WZW for fixed k +N = 2π i/gs and N → ∞. The tni are the coherent states of a single bosonic string oscillator and they are related to Vi by tni = tr(Vi )n in the fundamental representation, κR is related to the quadratic Casimir of the representation R and q = exp(gs ). This is obtained by considering certain amplitudes in the context of large N topological duality [2]. 1 This had also been noted in our discussions with D.-E. Diaconescu and A. Grassi. In particular, progress towards formulation of the vertex in terms of mathematical localization techniques has been made [7].
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The organization of this paper is as follows: In Sect. 2 we review the relevant facts about local toric Calabi-Yau threefolds including their T 2 fibration structure and its relation to (p, q) 5-branes. We also review mirror symmetry of these manifolds, where mirror geometry reduces, in the appropriate sense, to a Riemann surface. In Sect. 3 we discuss how the knowledge of A-model open topological string amplitudes on C3 with 3 sets of Lagrangian D-branes (defining a trivalent vertex) can be used to compute the A-model amplitudes for all toric Calabi-Yau threefolds with or without D-branes. In Sect. 4 we formulate the vertex in terms of a chiral bosonic oscillator in 1+1 dimension. In Sect. 5 we formulate the mirror B-model and discuss the interpretation of the vertex in this setup. In Sect. 6 we derive the complete expression for the cubic vertex using the large N topological duality in terms of certain Chern-Simons amplitudes. In Sect. 7 we explain how the vertex can be evaluated explicitly. In Sect. 8 we evaluate the vertex for low excitations and show that it passes some highly non-trivial tests. In Sect. 9 we apply our formalism to a number of examples. 2. Toric Geometry and Mirror Symmetry A smooth Calabi-Yau three-fold can be obtained by gluing together C3 patches in a way that is consistent with Ricci-flatness. For toric Calabi-Yau threefolds the gluing data and the resulting manifold are simple to describe. The toric Calabi-Yau 3-folds are special Lagrangian T 2 × R fibrations over the base R3 (they are also Lagrangian T 3 fibrations, but this will not be relevant for us). The geometry of the manifold is encoded in the one dimensional planar graph in the base that corresponds to the degeneration locus of the fibration. The edges of the graph are oriented straight lines labeled by vectors (p, q) ∈ Z2 , where the label corresponds to the generator of H1 (T 2 ) which is the shrinking cycle. Changing the orientation on each edge replaces (p, q) → (−p, −q) which does not change the Calabi-Yau geometry. The condition of being a Calabi-Yau is equivalent to the condition that on each vertex, if we choose the edges to be incoming with charges vi = (pi , qi ), one must have vi = 0. (2.1) i
If the local geometry of the threefold near the vertex is C3 , then the vertex is trivalent. Moreover, for any pair of incoming edges one has that |vi ∧ vj | = 1,
(2.2)
where ∧ denotes the symplectic product on H 1 (T 2 ). This condition ensures smoothness. The graph corresponding to C3 can be obtained as follows. Let zi be complex coordinates on C3 , i = 1, 2, 3. The base of the T 2 × R fibration is the image of moment maps rα (z) = |z1 |2 − |z3 |2 , rβ (z) = |z2 |2 − |z3 |2 , and rγ (z) = Im(z1 z2 z3 ). The special Lagrangian fibers are then generated by the action of the three “Hamiltonians” rα,β,γ on C3 via the standard symplectic form ω = i i dzi ∧ dzi on C3 and Poisson brackets, ∂ zi = { · r, zi }ω . In particular, the T 2 fiber is generated by circle actions (2.3) exp(iαrα + iβrβ ) : (z1 , z2 , z3 ) → eiα z1 , eiβ z2 , e−i(α+β) z3 , and rγ generates the real line R. We will call the cycle generated by rα the (0, 1) cycle, the (1, 0) cycle is generated by rβ .
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We have that the (0, 1) cycle degenerates over z1 = 0 = z3 . This subspace of C3 projects to the rα and rγ vanishing in the base and rβ ≥ 0, by their definition. Similarly over z2 = 0 = z3 , where (1, 0)-cycle degenerates, rβ and rγ vanish and rα ≥ 0, and the 1-cycle parameterized by α + β degenerates over z1 = 0 = z2 , where rα − rβ = 0 = rγ and rα ≤ 0 degenerate. To correlate the cycles unambiguously with the lines in the graph (up to (q, p) → (−q, −p)) we will let a (−q, p) cycle of the T 2 degenerate over an edge that corresponds to prα + qrβ = 0. The places in the base where T 2 fibers degenerate are correlated with the zero’s of the corresponding Hamiltonians. This yields the graph in Fig. 1 (drawn in the rγ = 0 plane). Above we have made a choice for generators of H1 (T 2 ) to be the 1-cycles generated by rα and rβ . Other choices will differ from this one by an SL(2, Z) transformation that acts on the T 2 . We can have rα generate a (p, q) 1-cycle and rβ the (t, s) 1-cycle where ps − qt = 1. This of course is a symmetry of C3 . However, when gluing different C3 ’s together, as we will discuss below, the relative choices will matter and will give rise to different geometries. 2.1. More general geometries. Other toric Calabi-Yau threefolds can be obtained by gluing together C3 ’s. First, one adds more coordinates e.g. z4 , . . . , zN+3 , so that flat patches are described by certain triples of the coordinates. Gluing different patches corresponds, in terms of the base, to identifying some of the coordinates by N linear relations: 2 A QA (2.4) i |zi | = t , i
where QA , A = 1, . . . N are integral charges satisfying QA i =0
(2.5)
i
which is the Calabi-Yau constraint. Finally, one divides the space of solutions to (2.5) by U (1)N action on z s, where the Ath U (1) acts on zi by zi → exp i QA i θA z i .
(0,1)
(1,0)
(−1,−1) Fig. 1. The degenerate locus of the T 2 × R fibration of C3 in the base R3 = (rα , rβ , rγ ). This locus is a graph . The labels (−pi , qi ) correspond to the cycles of T 2 which vanish over the corresponding edge
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The N parameters t A are K¨ahler moduli of the Calabi-Yau. The mathematical construction above arises in the physical context of the two-dimensional linear sigma model with N = (2, 2) supersymmetry on the Higgs branch [9]. The theory has N + 3 chiral fields, whose lowest components are z’s, which are charged under N vector multiplets with charges QA i . Equations (2.4) give minima of the D-term potential as solutions. Dividing by the U (1)N gauge group, the Higgs branch is a K¨ahler manifold, and when (2.5) holds, the theory flows to a two dimensional conformal sigma model in the IR. From the linear sigma model data described above, i.e. the set of N + 3 coordinates zi ’s and the D-term equations one can construct the graph corresponding to the toric Calabi-Yau manifold. First, we must find a decomposition of the set of all coordinates {zi }N+3 i=1 into triplets Uα = (zia , zja , zk a ) that correspond to the decomposition of X into C3 patches. We will describe this below in an example, but it should be clear how to do this in general. We can pick one of the C3 patches, say U1 and in this patch we get the Hamiltonians rα = |zi1 |2 − |zi3 |2 , rβ = |zi2 |2 − |zi3 |2 which generate the T 2 fiber in this patch. As it turns out, these can serve as global coordinates in the base R3 . Correspondingly, they generate a globally defined T 2 fiber2 . We can call the cycle generated by rα the (1, 0) cycle, and that generated by rβ the (0, 1) cycle. Equation (2.4) then can be used to find the action of rα,β on the other patches. Namely, in the Ua =1 patch, we can solve for all the other z’s in terms of zia , . . . , zka using (2.4), since this is by the definition what we mean by the Ua patch. The degenerate locus in this patch is then found analogously to the case of C3 above, where we use the rα and rβ as generators of the fiber globally. Example. O(−3) → P2 . A familiar example of a Calabi-Yau manifold, X, of this type is the O(−3) bundle over P2 . In this case, there are four coordinates z0 , z1 , z2 , z3 , and the D-term constraint is |z1 |2 + |z2 |2 + |z3 |2 − 3|z0 |2 = t.
(2.6)
There are three patches Ui defined by zi = 0, for i = 1, 2, 3, since at least one of these three coordinates must be non-zero in X. All of these three patches look like C3 . For example, for z3 = 0, we can “solve” for z3 in terms of the other three unconstrained coordinates which then parameterize C3 : U3 = (z0 , z1 , z2 ). Namely, in this patch, we can use (2.6) to solve for the absolute value of z3 , in terms of z0,1,2 , and moreover its phase can be gauged away by dividing with the U (1) action of the symplectic quotient construction: (z0 , z1 , z2 , z3 ) → (e−3iθ z0 , eiθ z1 , eiθ z2 , eiθ z3 ). We are left with the space of three unconstrained coordinates z0 , z1 , z2 as we claimed and this is of course C3 . A similar statement holds for the other two patches. Now let us construct the corresponding degeneration graph . Let the T 2 fiber in the U3 = (z0 , z1 , z2 ) patch be generated by rα and rβ , where rα = |z1 |2 − |z0 |2 and rβ = |z2 |2 −|z0 |2 . The graph of the degenerate fibers in the rα −rβ plane is the same as in our first C3 example, Fig. 1 (the third direction in the base, rγ is now given by the gauge invariant product rγ = Im(z0 z1 z2 z3 )). The same two Hamiltonians rα,β generate the action in the U2 = (z0 , z1 , z3 ) patch, where we use the (2.6) constraint to rewrite them as follows. Since both z0 and z1 are the coordinates of this patch rα does not change, 2
The third coordinate in the base is rγ = I m
N+3 i=1 zi
which is manifestly gauge invariant and
moreover, patch by patch, can be identified with the coordinate used in the C3 example above.
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rα = |z1 |2 − |z0 |2 . On the other hand, rβ changes as z2 is not a natural coordinate here, so instead we have rβ = t + 2|z0 |2 − |z1 |2 − |z3 |2 , and hence exp (iα rα + iβrβ ) : (z0 , z1 , z3 ) → ei(−α+2β) z0 , ei(α−β) z1 , e−iβ z3 . We see from the above that the fibers degenerate over three lines: i) rα + rβ = 0, and since z0 = 0 = z3 there, t ≥ rα ≥ 0, where the fact that we have to stop when rα = t comes from (2.6). Over this line (−1, 1) cycle degenerates. ii) There is a line over which a (−1, 2) cycle degenerates where z1 = 0 = z3 , 2rα + rβ = t, and t ≥ rβ ≥ 0 and finally, iii) There is a line over which rα = 0, t ≥ rβ ≥ 0, where z0 = 0 = z1 and (0, 1)-cycle degenerates. The U1 patch is similar, and we end up with the graph for O(−3) → P2 shown in Fig. 2. Since at least two of the z’s have to be zero for the fiber to degenerate, the graph lies in the rγ = 0 plane.
2.2. Toric algorithm for general geometries. The above way of constructing becomes cumbersome for more complicated geometries. There is an algorithm which does this efficiently. It is a standard construction in toric geometry and we will review it here. This is not meant to be didactic, so for a more thorough exposition see for example [10]. The algorithm is as follows. To each coordinate zi associate a vector v i in Z3 . The v i are chosen to satisfy an equation analogous to (2.4), i.e.
QA i = 0. i v
i
Since charges QA are integral, the equations can be solved. The Calabi-Yau condition, the A = 0 implies in fact that we can choose all the vectors to lie on a plane P , a unit Q i i distance from the origin, e.g. we can choose all the vi ’s to be of the form v i = (w i , 1), where w i is now a two-vector with integer entries. This provides an easy way to partition
Fig. 2. The graph of O(−3) → P2 . The manifold is built out of three C3 patches with the different orientations as in the figure. The transition functions correspond to SL(2, Z) transformations of the T 2 fibers as one goes from one patch to the next
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Fig. 3. The graph ˆ of O(−3) → P2 . The black points correspond to vectors w i . Its dual is the graph of degenerate fibers
the z’s into triplets that parameterize C3 patches. Namely, the z’s correspond to a collection of integral points on a plane P whose coordinates are w i , and this can be triangulated by considering triangles whose vertices are triplets of w’s. The triangulation that gives a good covering of X is such that all the triangles in P have unit area. This is in a sense a maximal triangulation. For example, for C3 we can take: w1 = (0, 0), w2 = (1, 0) and w3 = (0, 1) and the triangulation has a triangle with these vectors as vertices. For O(−3) → P2 we can take w0 = (0, 0), w1 = (−1, 0), w2 = (0, −1) and w3 = (1, 1) with 3 triangles corresponding to (w0 , w1 , w2 ), (w0 , w1 , w3 ) and (w0 , w2 , w3 ). In general the choice of the triangulation is not unique. There is an obvious SL(2, Z) action of the plane P , which is a symmetry of the closed string theory on X. But, in general there are also different possible triangulations of the same set of points, and these correspond to different phases in the K¨ahler moduli space. For a given choice of the K¨ahler parameters in (2.4) the allowed triangulation is such that the triplets of coordinates corresponding to every unit-volume triangle can all be simultaneously zero in X. We can think about this ˆ finding the graph describing triangulation as giving rise to a graph ˆ ∈ P . Given , the degeneration of the T 2 fibers is trivial – it is simply the dual graph in the sense that edges of ˆ are normals to the edges of and vice versa! In fact, running this algorithm backwards provides a fast way to associate a ˆ The vertices of this graph Calabi-Yau manifold to a graph . We first find a dual graph . are vectors wˆ i , with integer entries. Linear relations between the vectors vˆi = (wˆ i , 1) allow us to read off the charges QiA . To completely specify the geometry we also have to specify the K¨ahler form ω ∈ H 1,1 (X). As discussed above, this is captured by the moduli tA in the D-term equations (2.4) of the linear sigma model. In the formalism we will develop in the following sections we will need to know the areas of holomorphic curves in X which are fixed by the torus actions. It is in fact very easy to determine these directly from the graph . This however is easiest to physically motivate in terms of the (p, q) five-brane picture which we will discuss below.
2.3. Semi-compact theories. In the same spirit, we can also consider certain semi-compact models. Namely, the geometries discussed so far had only a T 2 subspace of the fiber compact. Here we show that we can also consider some models where four out of six
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M. Aganagic, A. Klemm, M. Mari˜no, C. Vafa
(1,−1)
(1,0) (0,1)
(1,1)
Fig. 4. The graph of the semi-compact O(−K) → P1 × P1 which arises by imposing identifications in the base R2 . The sizes of the two P1 ’s in the base, that are usually independent, must be equal here
directions in the geometry are compact. These geometries can be obtained by imposing identifications on two of the directions in the base corresponding to the plane of the graph . Clearly, not all toric Calabi-Yau manifolds will admit the compactifications, but only those with enough symmetry. For those that do, some of the moduli that exist in the non-compact geometry are frozen in the compact one, as they are not consistent with the identifications which we impose. For example, consider the graph corresponding to O(−K) → P1 × P1 . When the sizes of the two P1 ’s are equal we can consider a compactification that corresponds to identifying points related by: (rα , rβ ) ∼ (rα + 2πR, rβ + 2πR) ∼ (rα − 2π R, rβ + 2π R). The resulting geometry has a single K¨ahler modulus instead of the two that exist in the non-compact case. As in examples studied in [11] various degenerations of the graph and its different phases allowed by the charge conservation (2.1) correspond to geometric transitions of the full Calabi-Yau geometry. It is easy to see that the semi-compact models often have obstructions to existence of transitions that exist in the fully non-compact models. As the simplest example consider the semi-compact version of T ∗ S 3 . This corresponds to having a (1, 0) and (0, 1) cycle degenerating over the corresponding cycles of the “base” T 2 , at different values of rγ , i.e., the graph has two components, and the corresponding manifold has b3 = 1. This geometry, however, does not have a geometric transition O(−1) ⊕ O(−1) → P1 since the blowup-mode that gives the P1 a finite size is projected out in the semi-compact case. Such an obstruction to a transition from a single S3 is familiar from the fully compact Calabi-Yau manifolds. Once we discuss the (p, q) five-brane language, it will be manifest that these models have the same obstructions to resolutions of singularities as the compact manifolds do. This is related to the fact
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that the gauge theories obtained by compactifying string theory on these geometries are honestly 4-dimensional. 2.4. Relation to (p, q) 5-brane webs. We can connect the description of Calabi-Yau geometry by a duality to the web of (p, q) five branes [12]. This gives an intuitive picture of the geometry. The connection was derived in [13] and we will now review it. Recall that M-theory on T 2 is related to type IIB string theory on S1 . Since the Calabi-Yau manifolds we have been considering are T 2 fibered over B = R4 , we can relate geometric M theory compactification on the Calabi-Yau manifold X to type IIB on flat space B × S1 . However, due to the fact that T 2 is not fibered trivially, this is not related to the vacuum type IIB compactification. The local type of singularity over a line in the graph is the Taub-Nut space, where the (p, q) label denotes which cycle of the T 2 corresponds to the S1 of the Taub-Nut geometry. Under the duality, this local degeneration of X is mapped to the (p, q) five-brane that wraps the discriminant locus in the base space B, and lives on a point on the S1 . The fact that the (p, q) type of the five brane is correlated with its orientation in the base is a consequence of the BPS condition. More precisely, a configuration of five branes that preserves supersymmetry and 4 + 1 dimensional Lorentz invariance is pointlike in a fixed R2 subspace of the base. In the two remaining directions of the base, which we parameterized by (rα , rβ ) above, the five branes are lines where the equation of the (p, q) five brane is prα + qrβ = const. The (p, q) five-brane picture provides a simple way to read off the sizes of various holomorphic curves embedded in the Calabi-Yau X. For this paper, we will only need to know this for the curves which are invariant under the T 2 action. It is clear from the discussion in the previous subsections of this section that these are the curves in X which correspond to the edges of the graph . The duality of M-theory on X to the IIB with 5-branes relates the membranes wrapping holomorphic curves in X to (p, q) string webs ending on the 5-brane web. The masses of the corresponding BPS states get related by the duality. In the M-theory picture, the masses of BPS states are the K¨ahler volumes of the holomorphic curves, and in the IIB language they are the tensions×lengths of the corresponding strings. The curves that project to the edges of the graph correspond to strings that are within the five-branes themselves. These strings are instantons of the five-brane theory. As discussed in [12] the instanton of a (p, q) five brane is a string whose tension is I m(τ ) Tp,q = Ts , |pτ + q| χ where τ is the type IIB dilaton-axion field τ = 2π + gis , and the Ts is a tension of a fundamental string which is an instanton in an NS, or (1, 0), five-brane. Note that this is not a conventional free (r, s) string tension for any r, s (which would have been Ts r 2 + q 2 ), and correspondingly the instanton in general does not correspond to any free (p, q) string. This is because the action of an instanton (i.e. tension in this case, as the instanton is a string) is simply governed by the coefficient of the F ∧ ∗F term in the five-brane action and this is Ts /gs for a D5 brane and Tp,q for a (p, q) five-brane. Thus, knowing the length xof a (p, q) edge in , the area of a holomorphic curve corresponding to this edge is x/ p 2 + q 2 (we will take τ = i which is the square T 2 ). On the other hand, the slope of the five-brane in the R2 is correlated with its (p, q) type as we said above, and this allows us to read-off the lengths of all the edges in the graph in terms of the few independent ones which correspond to the K¨ahler moduli t A in (2.4). For example, suppose that the length of the horizontal edge in the graph of O(−3) → P2 is
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√ t. Then the length √ of the (1, −1) edge is t 2. However the tension of the corresponding instanton in 1/ 2 so the area of the holomorphic curve corresponding to this leg is t. Similarly, we find that the area of the curve corresponding to the (0, 1) leg is also t.
2.5. Mirror Symmetry and the dual B-model Geometries. Mirror manifolds of the local toric Calabi-Yau manifolds were derived in [14], by using T-duality in the linear sigma model in the previous section. The result of [14] is as follows. The mirror theory is a theory of variation of complex structure of a certain hypersurface Y which is given in terms of n + 3 dual variables y i [14] with the periodicity y i ∼ y i + 2π i. The variables y i are related to variables of the linear sigma model (2.4) as Re(y i ) = |zi |2 , so in particular, the D-term equation (2.4) is mirrored by i A QA i y =t .
(2.7)
(2.8)
i
Note that (2.8) has a three-dimensional family of solutions. One parameter is trivial and is given by y i → y i + c. Let us parameterize the two non-trivial families of solutions by u, v, and pick an inhomogenous solution. Then the hypersurface Y is given by [15] x x˜ = ey1 (u,v) + ey2 (u,v) + · · · + eyN +3 (u,v) ≡ P (u, v),
(2.9)
where yi (u, v) solve (2.8). The solutions to (2.8) are of the form yi = wiu u + wiv v + ti (t), for some vector w i = (wiu , wiv ) with integer entries. In fact this is the same vector that we associated to the coordinate zi in the previous section, when we discussed toric geometry. The monomials ew1 u+w2 v are in one to one correspondence with points of the ˆ graph . In the sections to follow a prominent role will be played by the Riemann surface X , obtained by setting x, x˜ to zero in (2.9),
X : 0 = P (u, v).
(2.10)
Note that this Riemann surface is closely related to the graph and it is in fact obtained by the fattening of its edges. For example, for the mirror of C3 we get eu + ev + 1 = 0
(2.11)
and this has three asymptotic regimes corresponding to u → ∞, where the equation of the Riemann surface is v = iπ. This is a long cylinder parameterized by u. Similarly, there is a long cylinder parameterized by v → ∞, where u = iπ and there is a third cylinder where u = v + iπ, and u, v → ∞, so that this Riemann surface corresponds to a sphere with three punctures. From “far away” the Riemann surface will look like the graph of C3 . Similarly, the Riemann surface X of any X has a degenerate limit where it looks like the graph . It is clear that by gluing various patches given by (2.11) dictated by the graph we can obtain the full Riemann surface X .
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3. Topological A-Model and the Vertex The amplitudes of the topological A-model localize on holomorphic maps from the worldsheet to the target space [1]. In particular the path integral defining the free energy of the theory reduces to a sum over the topological type of holomorphic maps from the worldsheet to the Calabi-Yau space X. Each term in the sum involves an integral over the moduli space M of that type of map, which leads to the so-called Gromov-Witten invariant of that map, weighted by the e−Area , where the area is that of the target space curve. In this paper we will find a very efficient way to calculate the A-model amplitudes on local toric Calabi-Yau manifolds described in the previous section, to all genera, exactly. The rough idea is to place Lagrangian D-brane/anti-D-brane pairs in appropriate places (one on each edge of the toric diagram) to cut the Calabi-Yau manifold X into patches which are C3 . They do not quite cut the Calabi-Yau in pieces as their dimension is too low, but all closed string worldsheet configurations will nevertheless cross them. More precisely, using toric actions the configurations can be made to pass through the lines of the toric graph [16]. Thus if we are interested in the closed string amplitudes we could use the D-branes as “tags” for when the closed string goes from one patch to another. Thus the open string amplitudes on each patch, glued together in an appropriate way, should have the full information about the closed string amplitudes. The idea is then as follows. Consider chopping the graph into tri-valent vertices by cutting each of the legs into two. Physically we can view this as placing a D-brane/ anti-D-brane pair. Each vertex corresponds to a C3 patch, as in Figs. 2 and 5. This cuts the P1 ’s which correspond to compact legs of into disks. Consider the maps g → X. The maps which contribute to the A-model amplitudes themselves project to (subgraph of) , and cutting the graphs cuts the maps as well, so we get Riemann surfaces with boundaries. We are led to consider the open topological string on C3 with three (stacks of) Lagrangian D-branes of the appropriate kind, one on each leg. From these data we should be able to obtain, by suitable gluing, closed string amplitudes on arbitrary toric Calabi-Yau threefolds. 3.1. The vertex as an open string amplitude. Consider again the description of the C3 in Sect. 2. The Lagrangian D-branes we need are in fact among the original examples of special Lagrangians of Harvey and Lawson [17]. The topology of all of the Lagrangians is C × S1 . In particular, they project to lines in the base R3 , and wrap the T 2 fiber. In the base, the three Lagrangians L1,2,3 are given by3 L1 : L2 : L3 :
rα = 0, rβ = r1∗ , rγ ≥ 0, rβ = 0, rα = r2∗ , rγ ≥ 0, rα − rβ = 0, rα = r3∗ , rγ ≥ 0.
(3.1)
In order not to have the boundary at rγ = 0, Li ’s are constrained to end on the graph , where one of the 1-cycles of the T 2 degenerates to S1 . The parameters ri∗ correspond to the moduli of Li ’s, and the “no boundary” constraint that we just mentioned is what constrains the number of the moduli to one. The Lagrangians are easily seen to intersect the fixed P1 ’s along S1 ’s so the boundaries of the maps can end on them. For example, a holomorphic disc ending on L1 is given by z1 = 0 = z3 , |z2 |2 ≤ r1∗ . 3 The Lagrangians are pointlike in the fiber generated by r . The fiber is parameterized by Re(z z z ) γ 1 2 3 and the Lagrangians are where this vanishes.
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Σ g=4
Fig. 5. The curves that are invariant under T 2 action in the O(−3) → P2 geometry. All the invariant curves are P1 ’s. The maps from g>0 that give a non-zero contribution to the A-model amplitudes are degenerate maps where the genus g > 0 parts of the curve are mapped to vertices
Now, consider the topological A-model string amplitude corresponding to some number of D-branes Ni on the i th Lagrangian Li on C3 . The partition function takes the form Z=
Ck (1) ,k (2) ,k (3)
k (1) ,k (2) ,k (3)
3 1 T rk (i) Vi , zk (i)
(3.2)
i=1
where Vi is the path ordered exponential of the Wilson-line on the i th D-brane, Vi = P exp[ A1 ] around the S1 , T rk V =
∞
(trV j )kj ,
j =1
and zk =
kj ! j kj .
j
= j kj is the Note that there are kj holes of winding number j so the sum h = |k| total number of holes on a fixed D-brane, and = j j kj is the total winding number. We have absorbed the modulus of the Lagrangian into the corresponding V which is complexified in string theory. The vertex amplitude, Ck (1) ,k (2) ,k (3) is naturally a function of the string coupling constant gs and, in the genus expansion, it contains information about maps from Riemann surfaces of arbitrary genera into C3 with boundaries on the D-branes, see Fig. 6. The vertex C is the basic object from which, by gluing, we should be able to obtain closed string amplitudes on arbitrary toric geometries. As we will see later, the vertex is naturally used to calculate general A-model amplitudes with boundaries as well.
The Topological Vertex
437
Σ
g=2,h=3
L2
L1 L3
Fig. 6. A C3 with three-stacks of Lagrangian D-branes of the type discussed in the text. The A-model amplitudes localize on holomorphic maps with boundaries where all the higher genus information is mapped to the vertex
3.2. Framing of the vertices. Because of the above considerations we are led to consider non-compact D-branes in C3 . Due to the non-compactness of the world-volume of D-branes, to fully specify the quantum theory we must specify the boundary conditions on the fields on the D-branes at infinity. This was discovered in [11] and is the closed string dual to the framing ambiguity of the Chern-Simons amplitudes [18]. To keep track of the boundary condition at infinity, we can use the following trick [19]. We modify the geometry in a way that makes the Lagrangian cycles wrapped by the D-branes compact, while not affecting the topological A-model amplitudes. We do so by introducing compact S3 cycles in the geometry by allowing the T 2 fiber to degenerate at additional locations in the base R3 , as in Fig. 7. The additional three lines Fi in the base correspond to degeneration of a fixed fi = (pi , qi ) cycle there. There are now compact special Lagrangian S3 cycles L˜ 1,2,3 which correspond to paths of the shortest distance between the graphs and Fi . For this cycle to be a non-degenerate S3 we need the following condition on the holonomy: fi ∧ vi = 1,
(3.3)
where vi corresponds to the H1 (T 2 ) class of the edges of the graph. Note that we have chosen a particular orientation for the framing so that the above product is always +1. Clearly, if fi is a solution to (3.3), so is fi − nvi for any integer n. This Z valued choice does affect the physics of the D-brane. To specify the theory on the D-brane fully, we
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F2
F1
F3 Fig. 7. Three stacks of D-branes on C3 . We have introduced graphs F1,2,3 to help us keep track of framing. Fi are straight lines in the base, corresponding to vectors fi in the text. Different choices of fi give different amplitudes. The choice in this figure is defined to be canonical framing
must specify a choice of framing [11], i.e., a choice of the integer n. This is a quantum ambiguity and only the relative values of n are meaningful. Given an (arbitrary) choice (0) of framing for the i th leg, i.e. a vector fi , the vector f (n) corresponds to a relative framing associated to an integer n if f (n) ∧ f (0) = n.
(3.4)
It is crucial for us to keep track of framing. The relevant object is a framed vertex, (f ,f2 ,f3 ) , k ,k (2) ,k (3)
C (1)1
where we specify the framing of the D-branes on the three legs. Without loss of generality we can take the vi to be v1 = (−1, −1), v2 = (0, 1), v3 = (1, 0), since any other choice is related to this one by an SL(2, Z) transformation. More generally we can introduce a vertex which depends on both vi and fi , but knowing the vertex for the canonical choice of vi with arbitrary framing fi is enough. Moreover, if we know the vertex in any one framing, the vertex in any other framing is related to it in a simple way [11, 20]. In order to describe this it is most convenient to go to the “representation basis” for the vertex which we will now turn to.
3.3. The vertex in the representation basis. Topological open string amplitudes can be written in terms of products of traces to various powers, as in (3.2). They can also be rewritten in the representation basis, and this can be done unambiguously in the limit where we take Ni → ∞ branes. We define the representation basis for the vertex by R1 ,R2 ,R3
f ,f ,f
CR11 ,R22 ,R33
3 i=1
Tr Ri Vi =
k (1) ,k (2) ,k (3)
f ,f2 ,f3 k ,k (2) ,k (3)
1 C (1)
3 i=1
1 T rk (i) Vi . zk (i)
The Topological Vertex
439
To obtain C in the representation basis defined above, we make use of Frobenius formula χR (C(k))Tr T rk V = RV , R
is the character of the symmetric group S of letters for the conjugacy where χR (C(k)) class C(k), in representation corresponding to the Young tableau of R. Using this we obtain f ,f ,f
CR11 ,R22 ,R33 =
k (1) ,k (2) ,k (3)
f ,f2 ,f3 k ,k (2) ,k (3)
1 C (1)
χR (C(k (i) )) i . zk (i)
(3.5)
i
Now we are ready to describe the framing dependence of the vertex. We have [20] f −n v ,f2 −n2 v2 ,f3 −n3 v3
CR11 ,R21,R13
= (−1)
i
ni (Ri )
q
i
ni κRi /2
f ,f ,f
CR11 ,R22 ,R33 ,
(3.6)
where κR is related to the quadratic Casimir CR of the representation R of U (N ) as κR = CR − N (R), and (R) is the number of boxes of the representation (which is the same as the total winding number in the k-basis). If the representation R is associated to a Young tableaux whose i th row has i boxes, (R) = i i , one has i (i − 2i + 1). (3.7) κR = i
3.4. Symmetries of the vertex. Consider an SL(2, Z) transformation that acts on the T 2 fiber of C3 , in the presence of D-branes. As already noted the vertex depends on three pairs (fi , vi ), where vi denotes the (p, q) structure of the edge and fi denotes the framing associated to that edge, and one has fi ∧ v i = 1 2 which means that (fi , vi ) forms an oriented basis for H1 (T ). Moreover, if we orient the edges inward towards the vertex, then i vi = 0. One also has that vi ∧ vj = ±1 for i = j . We can choose a cyclic ordering of vi according to the embedding of the corresponding vectors in R2 . In terms of this cyclic ordering we have
v2 ∧ v1 = v1 ∧ v3 = v3 ∧ v2 = 1. It is clear that an element g ∈ SL(2, Z) generates a symmetry of the vertex while replacing (fi , vi ) → (g · fi , g · vi ). There is one particularly natural choice of framing fi based on symmetry considerations, namely (see Fig. 7) (f1 , f2 , f3 ) = (v2 , v3 , v1 ). Note that this has the required property that fi ∧ vi = 1. For any given choice of vi cyclically ordered in this way, we shall call this the canonical framing and denote the corresponding vertex by C. Any other choice of framing, relative to this canonical choice, will be denoted by C n1 ,n2 ,n3 , where ni denote the amount of change in framing relative
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to the canonical choice. Let CR1 ,R2 ,R3 denote the vertex for the canonical framing for vi : (−1, −1), (0, 1), (1, 0). Then it follows that (f ,v )
CR1i,Ri2 ,R3 = (−1)
i
ni (Ri )
1
q2
i
ni κRi
CR1 ,R2 ,R3
(3.8)
where ni = fi ∧ vi+1 and i runs mod 3. With three D-branes on the legs of the vertex, the vertex amplitude CR1 ,R2 ,R3 is invariant under the Z3 subgroup of SL(2, Z) taking v1 → v2 ,
v2 → v3 ,
v3 → v1 .
Note that the condition that v3 → v1 follows from the first two from i vi = 0. Clearly there is such an SL(2, Z) transformation, because (v1 , v2 ) and (v2 , v3 ) form an oriented basis for H1 (T 2 ). For example for the simple choice of vi : (−1, −1), (0, 1), (1, 0) it is generated by T S −1 in the standard basis for generators of SL(2, Z), so we see that the vertex amplitude with canonical choice of framing, which is compatible with this cyclicity, has a cyclic symmetry, CR1 ,R2 ,R3 = CR3 ,R1 ,R2 = CR2 ,R3 ,R1 .
(3.9)
So far we have oriented edges of the vertex away from the vertex. In gluing vertices together we would need also to deal with arbitrary orientation of the edges. Suppose for example we take v1 → −v1 . What this does is to change the orientation of the circle on the corresponding D-brane. This is a parity operation on the D-brane, which changes the action to minus itself. Thus a genus g topological string amplitude with h boundaries on the corresponding D-brane (in the ‘t Hooft notation) gets modified by (−1)loops = (−1)2g−2+h = (−1)h . This can also be obtained by viewing the change of the sign of the action as replacing a topological brane by a topological anti-brane which replaces N → −N [21]. It is convenient to write how this modifies the vertex in the representation basis. This can be done using = (−1)|k|+(Q) χQt (C(k)) χQ (C(k)),
(3.10)
where (Q) denotes the number of boxes of representation Q. It follows that CR1 ,R2 ,R3 →v1 →−v1 (−1)(R1 ) CR t ,R2 ,R3 . 1
Similarly we can change any of the other vi → −vi . We have seen that the vertex has cyclic symmetry in the canonical framing. It is natural to ask what symmetry it has under permutation of any of the two representations. There is a symmetry of C3 that exchanges any pair of its coordinates, say z1 , z2 . This acts as orientation reversal on the world-volume of all three D-branes, as it acts on the T 2 fiber by exchanging (1, 0) and (0, 1) 1-cycles, and the T 2 is wrapped by all the D-branes. In addition the framings are shifted by one unit: We have ((f1 , v1 ), (f2 , v2 ), (f3 , v3 )) → ((f2 , v1 ), (f3 , v2 ), (f1 , v3 ))
The Topological Vertex
441
and from (3.8) it follows that the new framing is shifted by (f2 ∧ v2 , f3 ∧ v3 , f1 ∧ v1 ) = (v3 ∧ v2 , v1 ∧ v3 , v2 ∧ v1 ) = (1, 1, 1) (see also Fig. 8). From this it follows that CR1 ,R2 ,R3 = (−1)
i i
CR−1,−1,−1 t ,R t ,R t .
(3.11)
CR t ,R t ,R t .
(3.12)
1
3
2
Since κR t = −κR , we can write this as CR1 ,R2 ,R3 = q
i κRi /2
1
3
2
3.5. Gluing the vertices. In this section, we discuss how to glue open string amplitudes to obtain closed string amplitudes. Consider a leg of some graph , as in the Fig. 3. The leg will contribute to closed string amplitudes via holomorphic curves that map to the corresponding P1 . By cutting the curve in the middle of the leg, we obtain a product of open string amplitudes. Clearly, connected closed string graphs can give open string graphs that are disconnected, so the gluing must be done at the level of the partition function, schematically, Z() ∼ Z(L ) × Z(R ), where Z() is the amplitude corresponding to the graph and, by cutting one of its legs, the graph can be decomposed into L and R .
R2
R1
D
D
R3
D
R3
D A
R1
A
D
D
R2
R3
A
R2
R1 Fig. 8. Various symmetries of the three-point vertex. The figures in the top row are related by a Z3 subgroup of SL(2, Z). The figure in the bottom row is generated from the top left one by a symmetry of C3 that exchanges z1 and z2 . This also maps a D-brane (D) to an anti D-brane (A)
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Moreover, clearly the open string Riemann surfaces one gets in this way have a matching number of holes on the leg over which we glue, and also the winding numbers. Thus, the right-hand side of the above equation is in fact k
exp(−(k)t) Z(L )k Z(R )k . kj j kj !j
Above, t is the size of the relevant P1 . In gluing these we have to be careful that both gluing branes are defined with respect to the same framing. The combinatorial factor comes about because all holes with the same winding number are indistinguishable, and the factor of j for each hole of winding number j comes as the gluing respects the cyclic ordering of the j windings. In addition, we must remember that XL and XR (the manifolds corresponding to the graphs L,R ) come equipped with a choice of complex structures, and this induces natural orientation of boundaries of the two disks in XL,R . In order to glue the two disks into a P1 their boundaries must be oriented oppositely, which can be interpreted as putting branes versus anti-branes. As was already discussed this is equivalent to multiplying the amplitude by (−1)h , where h is the number of boundaries of the Riemann surface. This gives the gluing a nice physical interpretation: we put N D-branes on the relevant leg in XL and N anti D-branes in XR . The D-branes annihilate, so from the corresponding open string amplitudes we obtain the amplitude for closed strings on X. To summarize, Z() =
k
Z(L )k
(−1)|k| e−(k)t Z(R )k . zk
(3.13)
Obviously, (3.13) holds even in the presence of D-branes in X, where Z(X), etc. refer to amplitudes with D-branes. At the very least, this is true, as long as the D-branes are at locations away from the relevant leg, as all the considerations that led to (3.13) are purely local. We will return to this in the sections below. Note that in the representation basis the gluing operation is simply: Z(X) = Z(XL )Q (−1)Q e−(Q)t Z(XR )Qt , (3.14) Q
which follows from (3.10) and orthonormality of the characters 1 R (C(k)) = δRR . χR (C(k))χ zk k
3.6. The gluing algorithm for closed and open strings. Putting together all we have said so far, we can summarize the rules for computing closed string amplitudes from the cubic vertex and the gluing rules as follows: i) From the toric data described in Sect. 2, we can find the graph corresponding to the loci where the T 2 fibration degenerates. The edges of the graph are labeled by integral vectors vi that encode which cycle of the T 2 fiber degenerates over the i th edge. To each edge’s associated a representation Ri . ii) For smooth Calabi-Yau, the graph can be partitioned to trivalent vertices and corresponding C3 patches Ua , where a labels the vertices a = 1, 2, . . . .
The Topological Vertex
443
iii) This associates to every vertex an ordered triplet of vectors (vi , vj , vk ) by reading off the three edges that meet at the vertex in a counter-clockwise cyclic order – (vi , vj , vk ) is equivalent to (vj , vk , vi ). iv) If all the edges are incoming, we associate a factor CRi ,Rj ,Rk to the vertex Ua , otherwise we replace the corresponding representation by its transpose times (−1)(R) . v) Let the vertex Ua share the i th edge with the vertex Ub whose corresponding triple is (vi , vj , vk ). We can assume vi is outgoing at Ua and ingoing at Ub . We glue the amplitudes by summing over the representations on the i th edge as: CRj Rk Ri e−(Ri )ti (−1)(ni +1)(Ri ) q −ni κRi /2 CR t R R , (3.15) i
j
k
Ri
where the integer ni is defined as ni = |vk ∧ vk | and |vk ∧ vk | equals vk ∧ vk if both vk and vk are in(out)going, and −vk ∧ vk , otherwise. This factor reflects the fact that the framing over the i th edge should be the same on the two sides of gluings. The sign (−1)(ni +1)(Ri ) in (3.15) comes from the sign associated to the framing, and the one associated to the gluing in (3.14). vi) From the D-term equations (2.4), or the (p, q) 5-brane diagrams read off the lengths ti of the edges in terms of the K¨ahler moduli t A of X, ti = ti (t A ). Note that the edges of the graph are straight lines on the plane, with rational slope. To the i th edge in the (p ahler parameter i , qi ) direction of length xi in the plane, we associate a K¨ ti = xi / p 2 + q 2 . vii) For a non-compact edge of the graph the corresponding representation R is necessarily trivial, R = 0 (we will sometimes denote this also by R = ·). Naturally, the vertex can be used for calculating open string amplitudes on toric Calabi-Yau manifolds as well as the closed string ones. When we place the D-branes on the non-compact, outer edges of the graph , we simply modify the (vii) above to a sum over arbitrary representations R on the edge, where we weight the representations by TrR V and V is the holonomy on the corresponding D-brane. For D-branes on the inner edges, when we glue the maps on X from the maps on XL and XR , we must allow for maps with the boundaries on the D-brane. Suppose that we wish to calculate an amplitude corresponding to a D-brane on the i th edge of the toric graph. This modifies the gluing rule in (v) above as follows. v’) For a single D-brane on the i th edge, which is shared by vertices Ua and Ub in the setup of (v), we glue the amplitudes by summing over the representations on the i th R edge and representations QL i , Qi which “stop” on the D-brane from left and right as: CRj ,Rk ,Ri ⊗QL (−1)s(i) q f (i) e−L(i) CR t ⊗QR ,R ,R Tr QL Vi Tr QR Vi−1 , i
i
i
j
k
i
i
R Ri ,QL i ,Qi
(3.16) where we have collected the length, framing and sign factors in functionals L(i), f (i) and s(i) on this leg: R L(i) = (Ri ) ti + (QL i ) ri + (Qi ) (ti − ri ), f (i) = pi κRi ⊗QL /2 + (n + pi ) κR t ⊗QR /2, i
i
i
t R s(i) = (Ri ) + pi (Ri ⊗ QL i ) + (n + pi ) (Ri ⊗ Qi ).
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The piece of the edge to the left of the brane has length ri , while the right-hand side of the edge has length ti − ri . Vi is the holonomy on the D-brane. Note that e−r naturally complexifies Vi : changing the holonomy by Vi → Vi eiθ changes Tr R Vi to ei(r)θ Tr R V . The appearance of both Tr R Vi and Tr R Vi−1 reflects the fact that, along with open string instantons of area r and charge +1 ending on the D-brane, there are those of area t − r and charge −1, where the “charge” refers to how their boundaries couple to the holonomy on the D-brane world volume. The integer ni is defined as in (v) and the choice of an integer p corresponds to a choice of framing. This way of incorporating framing is natural. Namely, while for the closed string amplitudes only the relative framing in the left vs. right patch matters, corresponding to ni = |vk ∧ vk |, for the open string the absolute choice of framing matters: we pick a vector fi , which frames the i th leg both for the left and the right patch so that fi ∧ vi = 1. This corresponds to the choice of coordinate on the D-brane which does affect the open string amplitude. Then the framing of the left and the right vertex are ni + pi = |fi ∧ vk |,
pi = |fi ∧ vk |,
where |fi ∧ vk | is fi ∧ vk if vk and vi are both in(out) going in the vertex Ua , and equals −fi ∧ vk otherwise, and similarly with |fi ∧ vk |. If there is more than one stack of D-branes on the edge, say n stacks of them, we also must include contributions of n(n − 1)/2 massive open strings stretching between the D-branes. As shown in [22] the effect integrating out these strings is exp(−
∞ 1 (−1)R Tr R U1 Tr R t U2 . trU1m tr U2m ) = m
m=1
(3.17)
R
The relative minus sign in the exponent in (3.17) relative to that of [22] arises as follows. In the problem studied in [22] one had two D-branes intersecting on S1 (the S1 corresponds to the S1 factor in the D-brane world-volumes which are L = S1 × C) and the ground state was a boson. Here we have two D-branes whose world-volumes are parallel. There is one normalizable mode of the stretched string supported along an S1 , and it turns out to be a fermion. One way to see this is that, by changing complex structure which does not affect the A-model amplitudes, we can bring the branes to intersect on an S1 at the expense of turning one D-brane into an anti-D brane. The ground state of the string stretching between them is a fermion, as argued in [21]. For example, for stacks of m D-branes, we have CRj ,Rk ,Ri ⊗m QL (−1)s(i) e−L(i) q f (i) CR t ⊗m QR ,R ,R a=1
R Ri ,QL a,i ,Qa,i m
×
i
i,a
a=1
i,a
Tr QL Va Tr QR Va−1 , i,a
L(i) = ti (Ri ) +
m
L ra (QL i,a ) + (ti − ra ) (Qi,a ),
a=1
f (i) =
m
pi κRi ⊗QL /2 + (pi + ni ) κR t ⊗QR /2 , i,a
a=1
k
(3.18)
i,a
a=1
where
j
i
i,a
The Topological Vertex
s(i) = (Ri ) +
445 m
t R pi (Ri ⊗ QL ) + (p + n ) (R ⊗ Q ) i i i,a i,a . i
a=1
4. Chiral Bosonic Oscillator and the Vertex We have seen that the partition functions of the A-model on local toric 3-folds are computable from a set of gluing rules involving a cubic vertex and the propagator. The gluing rules are reminiscent of the construction of the partition function of bosons on a Riemann surface from the “pant diagram” and the tube propagators. We will show that this is not accidental. In fact, as we will argue, the vertex operator and the propagator we have obtained can be viewed as construction of the partition function of the mirror B-model whose geometry, as is well known, is captured by a Riemann surface. Towards this aim in this section we reformulate the vertex and the propagator we have obtained in terms of a free chiral boson on a Riemann surface. In particular, we will show that the winding basis can be identified with the Fock space of the chiral boson. In this connection the sewing rule gets mapped identically to the propagator of the chiral boson. Moreover the vertex gets identified with a state in the triple tensor product of the Hilbert space of the free boson. This vertex is highly non-trivial. Even in the classical limit it is more complicated than the usual vertex states one gets for a free boson on a Riemann surface (which is always given by a Bogoliubov transformation and can be represented as the exponential of quadratic monomials in the oscillator creation operators). In the next section we explain how to interpret the free chiral boson as the relevant field for the Kodaira-Spencer theory of gravity [8] in this local context (related ideas have appeared in [23, 24]).
4.1. Reformulation in terms of a chiral boson. There is a curious similarity between the winding number k basis and oscillator states of a free chiral boson. Recall the oscillator expansion of the chiral boson φ(u), ∂u φ(u) = jm emu , m =0
where [jm , jn ] = mδm+n,0 so that jm>0 is the annihilation operator and jm<0 the creation operator. The Hilbert space H of a free chiral boson on a circle is spanned by states of the form k m = |k j−m |0. m>0
We simply identify the vector k above with the vector of winding numbers. With this identification we can interpret the sewing as picking out an element P in the two-fold tensor product of the Hilbert space P ∈ H⊗2 , and the vertex as defining a state in the threefold tensor product Hilbert space C ∈ H⊗3 . It is natural to ask what these states are. We first turn to the propagator P . We will see that P is the conventional state of the free chiral boson on a cylinder. The path integral of a free chiral boson on a cylinder
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of length t is a state in the tensor product Hilbert space P ∈ H1 ⊗ H2 (where H1,2 are associated to the two boundaries) given by 1 |P = exp(−t j 1 j 2 )|01 ⊗ 0|2 . m −m m m>0
Expanding the exponential we get h (−1) ⊗ k|, e−(k)t |k zk k
where
zk =
km !mkm .
m
This is precisely the gluing rule we had discussed for A-model amplitudes! We now turn to the vertex C. As we said above, the vertex amplitude, as formulated in Sect. 3, 1 Z= Ck 1 k 2 k 3 T rk 1 V1 T rk 2 V2 T rk 3 V3 , i zki k i
can be written in terms of the state C in H3 of the chiral boson as 1 Z= T rk V1 T rk V2 T rk V3 k 1 | ⊗ k 2 | ⊗ k 3 |C. i zki k i
Note that, at the level of the answer, |C is given by |C = exp Fk 1 ,k 2 ,k 3 (gs ) j−k 1 j−k 2 j−k 3 |01 ⊗ |02 ⊗ |03 , k i
km where F is identified with the free energy of the topological string, and j±k = m>0 j±m , since “evaluating” the amplitude, i.e. performing contractions, amounts to replacing the creation operators in the free energy with V s, trV m ↔ j−m . It is natural to ask what the meaning of the three point vertex C ∈ H⊗3 is. One may at first think that this may be related to that of a free boson on a thrice punctured sphere. This is almost true. Namely it is a state associated with a sphere with three punctures, but the theory is that of a free scalar theory only to leading order in the oscillator expansion and in the gs → 0 limit. We will discuss this and its interpretation after we discuss the B-model interpretation of the chiral boson as describing the quantum field of the Kodaira-Spencer theory on the Riemann surface. We will explain why there are more oscillator terms in C, including the existence of non-trivial gs corrections. A full study of the vertex C from the B-model perspective will be done in [25]. For now, note that, most remarkably, the D-branes can be thought of as coherent states in the chiral boson theory! The vertex amplitude Z is computed by inserting
1 m V | = 0| exp trV jm m m>0
at each of the three punctures that give rise to the vertex state C, Z = V1 | ⊗ V2 | ⊗ V3 | C. We will explain this below.
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5. Local B-Model Mirror and the Quantum Kodaira-Spencer Theory We have seen that the vertex is naturally captured by the states of a chiral boson on a sphere with three punctures. In this section we will identify the chiral boson on each patch as the quantum field of the Kodaira-Spencer theory on the mirror B-model involving a Riemann surface. The modes of the chiral boson are affected, as we will explain, by the degrees of freedom (trV n ) on the B-branes. In particular we will explain, from the B-model perspective, why the open string amplitudes know about the closed string B-model amplitudes. Moreover we identify the branes in this setup as the fermions associated to the chiral bosons ψ(z) = eφ(z) . We use this picture to compute leading terms in the oscillator expansion of the vertex. Extension to the full vertex, from this perspective will appear elsewhere [25]. Moreover the gluing rules of the vertex can now be directly interpreted as computations of the Kodaira-Spencer theory in the operator formulation on the mirror Riemann surface. The target space of the B-model was interpreted in [8] as describing the quantum theory of complex deformation of the Calabi-Yau threefold. This in particular applies to the local Calabi-Yau case at hand. In particular, if we consider the A-model in the local toric case, as already discussed in Sect. 2, the mirror is given by a hypersurface in (x, x, ˜ u, v) ∈ C × C × C∗ × C∗ : x x˜ = F (u, v). Moreover F (u, v) can be obtained by gluing pant diagrams of the form eu + ev + 1 = 0.
(5.1)
The holomorphic 3-form is given by = dxdudv/x. As is well known in the local context, integration of over the non-trivial class of three cycles gets reduced to computation of a 1-form on the Riemann surface. The only non˜ x) → (eiθ x, ˜ e−iθ x)) trivial 3-cycles are formed by the S1 fibration (identified with (x, over a domain in the u, v plane bounded by the Riemann surface ( : F (u, v) = 0). The integral of over such cycles reduces to integrals of the meromorphic 1-form λ = udv on the 1-cycles of the non-compact Riemann surface . Note that λ is not globally well defined, and it makes sense only patch by patch. In particular if we had considered the upatch, integration of the 2-cycle fiber would have resulted in a 1-form −vdu. It is this lack of global definition of λ which lead to non-trivial interactions, to an otherwise free theory. The variations of the complex structure and the corresponding period integral get mapped to the variation of the complex structure of and the periods of the corresponding reduced 1-form λ on it. Note that there is a direct relation between changing the complex structure of and the choice of λ. In particular suppose we are in the v-patch defined by being centered at ev = 0; consider the complex deformation F → F (u, v) + δF (u, v). The above derivation for the reduced one form will still go through without any change, λ = udv but now u is a different function of v.
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Solving F (u, v) = 0 we would in principle get u = f (v) and under the complex deformation we have u = f (v) + δf and so the change in the 1-form λ is given by δλ = δf dv. Thus, as in [8] the basic quantum field gets identified with this variation. For this to be a good deformation of complex structure δf should be a meromorphic function of ev , i.e. ∂ v δf = 0.
(5.2)
To get an ordinary quantum field it is natural to write δf = ∂φv in which case (5.2) gets mapped to ∂∂φv = 0.
(5.3)
In terms of this scalar the variation of the 1-form is given by δλ = ∂φv ,
(5.4)
and Eq. (5.3) is sufficient for the condition of integrability of the complex structure, unlike the generic 3-fold complex structure deformation where the story is more complicated. Thus we have a free boson propagating on each patch. In the classical theory we can of course parameterize the deformation in any way we want; however for the quantum theory writing the variation this way is more natural. This is because the Kodaira-Spencer theory in the formulation of [8] has the kinetic term of the form 1 ω∂ −1 ∂ω, gs2 Calabi Y au where ω is a (2, 1)-form representing the change in the complex structure of Calabi-Yau. It is natural to write, at least for a patch, ω = ∂χ where χ is a (1, 1) form. In terms of χ the action would become 1 ∂χ ∂χ. gs2 Calabi Y au In the local context that we are discussing, χ gets identified with the φv above which is a scalar on , and we get the free scalar theory 1 ∂φ∂φ. gs2 Note that the anti-holomorphic piece of φ is a gauge artifact: Shifting φ by an anti-holomorphic function will not affect ∂φ and so does not change the 1-form λ. So φ should be viewed as a chiral boson. We can study the Kodaira-Spencer theory patch by patch by chiral fields φv . We will write the variation of the complex structure as u = f0 (v) + ∂v φv .
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We can also absorb f0 (v) as a classical vev for ∂v φv , which we will sometimes do. In the v−patch which is cylindrical we can write ∂ v φv = jn e−nv + gs2 j−n env , (5.5) n>0
where we have included factors of gs2 to account for the kinetic term of the scalar being 1/gs2 . In the quantum formulation j−n and jn are not independent, and correspond to creation and annihilation operators. To better understand this we will consider a coherent set of states given by replacing jn → tn . This is natural in this patch because changing the complex structure at the infinity of this patch corresponds to changing ∂v φv at ev → 0, and that is determined by the negative powers e−nv above. However now the positive powers of env are determined quantum mechanically. Let | denote the state created by the rest of the Riemann surface. Let the coherent states be defined by
1 |t = exp j−n tn |0. n n>0
Let us denote the partition function of the theory including the tn deformations by Z(t). Then we have Z(t) = exp( F (gs , tn )) = |t. To justify constructing the coherent state in terms of j−m alone, note that if we consider the expectation value of ∂v φv on the cylinder at infinity where e−v → ∞ then e−nv terms in (5.5) dominate. This can also be viewed as changing the 1-form λ at infinity by δλ = dv tn e−nv . n
On the other hand if we consider the expectation value of ∂v φv for the positive powers env it will not be zero. It might at first appear that in the classical limit gs → 0 this would be zero because of the explicit gs2 dependence in (5.5), but this is not the case. This is because F (gs , tn ) has a 1/gs2 term in the genus zero part given by F0 (tn ), so this survives in the limit. This will give us for the expectation value of ∂v φv in the limit gs → 0, 1 ∂F0 nv tn e−nv + n e , | ∂v φv |t = Z ∂tn n>0
which means that classically we have ∂v φv =
n>0
tn e−nv + n
∂F0 nv e . ∂tn
Here we need to clarify one important point: F is not an unambiguous function of tn ’s. This depends on how we choose the coordinates on each patch. Changing the coordinates, will give rise to a different function F (ti ). The difference between these results is the same as the Virasoro action. This dependence on the choice of the local coordinates on the Riemann surface will turn out to be related to the framing ambiguity. For us the
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Riemann surface comes with almost canonical coordinates involving combinations of u and v with du ∧ dv making sense in the full Calabi-Yau. In a given v-patch we will have the situation where z = ev → 0, i.e. v → −∞ is on the patch, and u → const. as z → 0. This almost uniquely fixes the coordinates except for an integer choice: The du ∧ dv is invariant under SL(2, Z). There is a subset of SL(2, Z), indexed by an integer n, which preserves the conditions we have put on each patch, namely u = u and v = v + nu. Note that the coefficient of v in v is 1 because we want z = O(z) as z → 0 in order to have a good coordinate. This transformation will give a new one form λ = u dv and a new coordinate
z = ev = enu z, where eu =
∞
i=0 ai z
i.
For example consider the pant Riemann surface e−u + ev + 1 = 0
in the v-patch, which includes v → −∞ (where u → iπ ). If we now change coordinates v = v + nu we will have z = z[(−1)n (1 + z)−n ]. So if we compute F in the z patch, in the new coordinate patch we will need to exponentiate an appropriate element of Virasoro algebra which changes the coordinates. Thus the choices of F is indexed by an integer n in each patch. In the interest of comparison with our A-model vertex we will now specialize to the case of the pant diagram, which is mirror to C3 , given by the Riemann surface eu1 + eu2 + eu3 = 0, where one variable is eliminated by rescaling the equation. This way of writing it, exhibits the cyclic symmetry between the three patches. A choice of coordinates that corresponds to the “standard framing of the vertex” discussed before and which preserves the Z3 symmetry is as to let u = u3 − u1 and v = u2 − u3 , and w = u1 − u2 then we have e−u + ev + 1 = 0 λ = vdu u − patch, e−v + ew + 1 = 0 λ = wdv v − patch, e−w + eu + 1 = 0 λ = udw w − patch.
(5.6)
Note that we also have the relation u + v + w = 0. Changing of the coordinates by framings (n1 , n2 , n3 ) is obtained by the choice of the coordinates u → u + n1 v, v → v + n2 w, w → w + n3 u. In each patch we can deform the defining equation by a chiral scalar, as discussed above. The corresponding scalars we call φu , φv , φw . The equation of the surface gets modified, when φi = 0 by following the deformation discussed in general above, and we get
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451
e−u + ev+∂u φu + 1 = 0, e−v + ew+∂v φv + 1 = 0, e−w + eu+∂w φw + 1 = 0, where in the classical limit ∂u φu =
tnu e−nu + n
∂F0 nu e , ∂tnu
tnv e−nv + n
∂F0 nv e , ∂tnv
n>0
∂ v φv =
n>0
∂w φw =
tnw e−nw + n
n>0
If we assume ∂φ’s are
small4 ,
∂F0 nw e . ∂tnw
we get the following equations:
e−u + ev + 1 + ev ∂u φu = 0 e−v + ew + 1 + ew ∂v φv = 0 e−w + eu + 1 + eu ∂w φw = 0
(i), (ii), (iii).
(5.7)
If we multiply Eq. (i) by e−v , and use the fact that u + v = −w we get ew + 1 + e−v + ∂u φu = 0. Comparing this with Eq. (ii) we learn that ∂u φu = ew ∂v φv . On the other hand, to leading order we have from Eq. (i), dv/du = e−u−v = ew , we thus have ∂u φ u =
dv ∂ v φv du
(5.8)
and by the Z3 cyclic symmetry similar equations with u → v → w → u. This in particular implies that in the classical limit, and to leading order in oscillators the three φ’s in the three patches can be viewed as coming from a global φ. This means that to leading order in the classical limit and in oscillators, the vertex operator should be the standard one coming from the well known techniques of operator formulation on Riemann surfaces [26–28] which lead to Bogoliubov transformations. Here we will digress to review this derivation. 4
This derivation of the small ∂φ limit of the vertex was suggested to us by Robbert Dijkgraaf.
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5.1. Bogoliubov transformation. Consider a chiral boson on a Riemann surface mirror to C3 . The Riemann surface is a sphere with three punctures, and the path integral on this gives a state in the tensor product of three free Hilbert spaces H1 ⊗ H2 ⊗ H3 corresponding to the punctures. Moreover, mirror symmetry provides us with a choice of complex structure on the punctured Riemann surface, and this gives a canonical choice of coordinates zi = eui near each puncture at zi = 0 and transition functions relating them. The transition functions between the patches give rise to Bogoliubov transformations that relate the three Hilbert spaces and these are sufficient to determine the ray in the Hilbert space to which the path integral corresponds. To do so we follow [27]. Note that the path integral of the chiral boson φ has an infinite dimensional group of symmetries corresponding to shifting φ → φ + f for any function f which is holomorphic on the punctured Riemann surface (a meromorphic function whose only poles are at the punctures). This gives rise to the conserved charge Q(f ) = f (zi )∂φ, i
zi =0
which must annihilate the path integral. In each of the three patches we have a different expansion for the chiral boson in terms of the local holomorphic coordinate, φ(zi ) =
1 (i) j zm . m −m i
m =0
The three patches are related by (5.6), where we put z1 = eu , z2 = ev and z3 = ew , so that the three patches are related by zi+1 +
1 + 1 = 0. zi (1)
Then, for example, a meromorphic function fm = z1m has expansion m z1m = Om,n z3−n , On,m z2−n = n n≥m m≥n where Om,n
m . = (−1) n m
The corresponding charge Q(f ) is given by m (2) (3) Q(fm(1) ) = jm(1) + On,m j−n . Om,n j−n + n n≥m n≥m (i)
The conditions that Q(fm ) annihilate the path integral suffice to determine it: On,m (1) (2) (2) (3) (3) (1) |Z = exp j−m j−n + j−m j−n + j−m j−n |01 ⊗ |02 ⊗ |03 . n m>0,n≥m
(5.9) Here we have suppressed the linear term in the j−m ’s corresponding to the fact that the vacuum |0 is not the ordinary vacuum, but ∂φ has a piece corresponding to the
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453
classical geometry, which we have been shifting away so far. Let us now restore it. The classical piece of the chiral boson in the u-patch, for example, is ∂φ(u)0 = v(u)du = log(1 + e−u )du. Shifting this away in the action g12 ∂φ∂φ gives a surface term s
1 gs
φ0 ∂φ = −
(−1)n n>0
n2
(1)
j−n ,
(5.10)
(after rescaling φ → gs φ), so this shifts the vacuum |01 to
(−1)−n (1) |01 , j |01 → exp − n2 gs −n n>0
and similarly in the other two patches. The state we have computed should be accurate to leading order in gs and up to quadratic terms in j−m = tm . The full vertex will have additional terms both in the gs corrections as well as in terms involving more tm ’s. This is because the derivation leading to a global chiral boson was valid only in this limit. A full discussion of this from the B-model perspective will be presented elsewhere [25]. From the B-model perspective if we know what the path-integral gives for the pant-diagram, then we can obtain any other amplitude by gluing. This is because the Kodaira-Spencer equation implies that in each patch φ is a chiral boson with the standard propagator. However we have to make sure that in the gluing the coordinate choices match–this is the same as making sure that the framings are compatible in the A-model computation of the vertex. Independently of how one computes this trivalent vertex, the knowledge of F = Fg (tnu , tnv , tnw )(gs )2g−2 (5.11) g
will capture arbitrary B-model local amplitudes, as everything can be obtained from this by gluing, as discussed in Sect. 2. We thus appear to have a system involving closed strings on the pant-diagram, in the operator formalism, capturing arbitrary local models. On the other hand we have given, motivated from the A-model considerations a similar gluing rule and a vertex C involving amplitudes of the A-branes on C3 , which is mirror to the pant diagram. To complete the circle of ideas we have to connect these two facts. The mirror of A-branes are B-branes, which on the Riemann surface get mapped to points on the pant diagram [19]. So the question, posed in a purely B-model context, is the following: What is the relation of the B-branes with the closed string Kodaira-Spencer amplitudes? We will explain how this works in the next section.
5.2. B-branes and closed string B-model. We will now argue how F defined in (5.11) in terms of operator formulation of closed string target theory, i.e. the quantum KodairaSpencer theory, can be equivalently phrased in terms of open string B-model amplitudes. To this end we will have to understand the effect of the B-brane on the closed string B-model. We will show that the insertion of a B-brane at the point z is equivalent, for the reduced Kodaira-Spencer theory, to the insertion of the fermionic field ψ(z) = eφ . In other words, fermions, which are usually viewed as a soliton of a chiral scalar can also be viewed as the B-branes, i.e., the soliton of the Kodaira-Spencer theory.
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Consider a B-model in the local patch given by v where ev → 0 is part of the patch. Consider placing many branes at the circumference of the cylinder given by v = vi 0. We ask how putting B-branes back reacts on the gravity B-model? In other words, how does the complex structure get modified by the B-branes? To answer this question we consider adding an extra brane, viewing it as a probe, and place it at v in this patch. We are interested in what the effect of many branes are on the probe. If the probe is placed at v 0 then the effect is simply given by summing over all the open strings stretching between them. This is the mirror of the computation of [22] and was discussed in the mirror setup in [29]. The effect on the free energy of the probe at v upon integrating out the stretched strings between branes at vi and the probe is δF = ∂φ =
1 1 e−n(v−vi ) = trV n e−nv , n n
(5.12)
n>0
i,n>0
where we have used the fact that evi are the eigenvalues of V . Note that for v 0 there is no contribution from the rest of the surface to the free energy of the probe. On the other hand we can ask which deformed geometry will give rise to this free energy probed by the brane. This is given by the computation in [19] where v v λ= ∂v φ = φ(v) + const, (5.13) gs F = where λ is the deformed 1-form on the surface and we have absorbed back into φ the classical piece of the 1-form. Thus placing a brane at v affects the Kodaira-Spencer action by the addition of φ(v). Let us consider how F changes for v 0. Let us call this singular part of φ by φ − . In particular the deformed one form λ− which dominates for v 0 gets identified with φ − (v) = gs F. We thus see that the free energy felt by the probe brane (5.12) is reproduced by the deformation 1 φ − (v) = trV n e−nv n n>0
which leads to the identification
tn = trV n .
This explains the observation we made before about the role of the chiral boson in the vertex we had obtained from A-model considerations. This suggests the following interpretation: To have a brane at the point v we add to the KS action the operator φ(v)/gs , or to the path integral the operator exp(φ(v)/gs ) (the 1/gs there is to remind us that the disk amplitude is proportional to 1/gs ). This leads to the same response in the free energy. We can redefine φ/gs → φ, which gets rid of the 1/gs2 in the kinetic term. We are thus led to identify the operator inserting the brane at z with the insertion in the path-integral of ψ(z) = eφ(z) , i.e. the fermion operator! (We have absorbed an i in the definition of φ in comparison with the conventional description of bosonization.) Connection between fermions and
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455
the D-branes was anticipated a while back [30] and this result makes this concrete. The anti-branes get identified with ψ † (z) = e−φ(z) . This is because the free energy will change sign for an anti-brane. As a check of this statement, note that the coherent state involving the branes at vi can be viewed as the state 1 envi j−n |0 = |ti exp(φ(vi ))|0 = exp n i i,n>0 with tn = i exp(nvi ). Note that in this expression we have normal ordered the operator. Not normal ordering it would have also led to the effect of the branes on each other, i.e. the result of integrating out the open string stretched between them. We will now present an additional argument why the fermion field is the B-brane operator. Consider a general Calabi-Yau threefold. Consider wrapping N B-branes over a (compact or non-compact) holomorphic curve C in Calabi-Yau. This curve is of real codimension 4 in the Calabi-Yau and is surrounded by a 3-cycle FC . Consider the integral IN = N , FC
where N is the holomorphic 3-form, corrected by the fact that there are N B-branes on C. Then we claim IN − I0 = Ngs . This is related to the mirror of the Chern-Simons/topological strings duality of [2]. For example consider N B-branes wrapping the P1 in O(−1) + O(−1) → P1 . Then there is an S3 surrounding P1 and the claim is that N = Ngs . S3
In particular under the large N duality this is consistent with the size of the S3 being given by Ngs . This statement should hold for compact or non-compact branes as it is a local question and it is our definition of the B-brane in terms of its coupling to the gravitational Kodaira-Spencer theory. In the context of the local model we are considering here, the B-brane wraps over a non-compact plane, which intersects the Riemann surface at a point. The integral of the holomorphic three-form around it reduces to the integral of the 1-form λ around this intersection point. So if we denote by B(z) the field creating a D-brane at point z on the Riemann surface, then λ(z ) B(z) = gs ∂z φ(z )B(z)dz = gs . z
z
This is indeed the correct OPE defining the fermionic field and we get the identification B(z) = ψ(z) = eφ(z) . Thus the trivalent topological vertex can be viewed as computing ψ(ui ) ψ(vj ) ψ(wk ) i
j
k
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in the closed string mirror formulation of the problem. It is then clear why the amplitudes involving brane can lead to a full reconstruction of the closed string amplitudes: This is simply the familiar bosonization! We can also consider mixed amplitudes with some branes left over in a closed string background. In the B-model all we need to do is to add certain fermion operators. Note that the framing ambiguity for the B-branes gets mapped to the fact the ψ(z) is a half-differential and so the amplitudes will depend on the coordinates chosen.
6. The Derivation of the Vertex Amplitude In this section we will provide a derivation of the three-point vertex in the A-model. It turns out to be more convenient to switch back to the representation basis in deriving the cubic vertex CR1 ,R2 ,R3 . We will derive this vertex using the large N topological duality [2] relating large N Chern-Simons amplitudes with those of closed topological strings. We will be able to compute the full vertex to all orders in gs and for arbitrary oscillator numbers. This will allow us to confirm, in Sect. 8, that this agrees, in the gs → 0 limit, with the linear and quadratic pieces of oscillators which we have computed from the perspective of the B-model in the previous section. As was conjectured in [2] and proven in [31], the topological open string A-model of N D-branes on S3 in Y = T ∗ S3 is the same as the topological A-model closed string theory on X = O(−1) ⊕ O(−1) → P1 . The large N duality is a geometric transition where the S3 and the D-branes disappear and get replaced by the P1 [2]. The string coupling constant is the same in both theories, and the size t of the P1 is identified with the ’t Hooft coupling t = Ngs . The open string theory, as was shown in [32], is the same as U (N) Chern-Simons theory on S3 , where the level k of the Chern-Simons is related 2πi to the string coupling as gs = k+N . Various aspects of this duality have been studied in [22, 33–35], in particular in [22] the duality was studied in the presence of non-compact D-branes. A variant of this is what we need here. Consider then T ∗ S3 with N D-branes on the S3 , but in addition N2 D-branes on one leg wrapping the Lagrangian L2 , and N1 and N3 D-branes on the other, wrapping L1 and L3 respectively, as depicted in Fig. 9. In the dual theory we end up with Y = O(−1) ⊕ O(−1) → P1 . The D-branes wrapping the compact cycle have disappeared, but the D-branes on the non-compact cycles are pushed through the transition. The resulting configuration is shown in the second picture in Fig. 9. The amplitude corresponding to D-branes on Y can easily be calculated using solvability of Chern-Simons theory and the large N transition. In the limit where N → ∞, the size of the P1 in Y grows, and zooming in on the vertex with the D-branes, we are left with C3 and the three D-branes. This is not exactly the configuration of D-branes that gives the three-point vertex, but it turns out to be close enough; we need to move the L1 Lagrangian D-branes through the vertex and put it on the other leg of C3 . We will explain below how this can be achieved. The open string theory on the S3 is U (N ) Chern-Simons theory with some matter fields coming from the three non-compact Lagrangians L1,2,3 . As shown in [3], there are bifundamental strings stretching between the S3 and L1,2,3 , and in addition there are strings between L1 and L3 . The ground state of all of these strings in the topological A-model is a bifundamental matter field, and integrating it out corresponds to inserting an annulus operator (3.17).
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457
L2
L2
Q2 Q1 3
S
L3
L1
L1
Q
L3
t
Q3
Fig. 9. The figure on the left corresponds to Chern-Simons theory on S3 with three source D-branes. Q1,2,3 , Q denote the bifundamental strings. In the large N limit, the S3 undergoes a geometric transition. The figure on the right depicts the large N dual geometry, with Lagrangians L1,2,3 , after the transition. The local patch where the D-branes are is a C3
Keeping track of the orientations, we have that Z(V1 , V2 , V3 ) =
1 S00
(−1)(Q1 ) Tr Q2 U Tr Qt ⊗Qt U 1
3
Q1 ,Q2 ,Q3 ,Q
× Tr Q1 V1 Tr Qt V1−1 Tr Q2 V2 Tr Q⊗Q3 V3 ,
(6.1)
where we have put the 1/S00 in front to compute the contribution to the partition function due to the branes, noting that S00 is the partition function of topological string on O(−1)⊕O(−1) → P1 . In the equation above the Vi is related to the holonomy on the i th stack of non-compact D-branes and U is holonomy on the S3 . The vacuum expectation value in (6.1) corresponds to a Hopf link with one of its components replaced by two unlinked unknots, as in Fig. 10, and evaluated on the S3 . The unnormalized expectation value is given by [18] Tr Q2 U Tr Qt ⊗Qt U = SQt ⊗Qt Q2 , 1
3
1
3
t
Q1
t
Q3
Q2
Fig. 10. This is a three-component link which is obtained from the Hopf link by “doubling” one of its components, i.e. by replacing it with two unlinked unknots. The labels denote representations, and the Chern-Simons invariant associated to the link is SQ Qt ⊗Qt /S00 2
1
2
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where S is the S-matrix of the U (N )k WZW model. On the other hand, it is well known that SQi ⊗Qj Ql = SQi Ql SQj Ql /S0Ql [36, 18]. Using this, we arrive at the following expression for (6.1):
Z(V1 , V2 , V3 ) =
(−1)(Q1 )
Q1 ,Q2 ,Q3 ,Q
SQt Q2 SQt Q2 1
3
S00 S0Q2
× Tr Q1 V1 Tr Qt V1−1 Tr Q2 V2 Tr Q⊗Q3 V3 .
(6.2)
By large N-duality the above amplitude is computed by the topological strings on X with three stacks of D-branes, as in [3], corresponding to the right figure in Fig. 7. In the limit where we send N or equivalently t = Ngs to infinity, Y becomes a C3 , and this is the limit we are interested in. All the N dependence in (6.2) is in the S−matrices, and in the following we will use WQi Qj to mean the limit of the matrix S-matrix as N goes to infinity, WQi Qj = limt→∞ SQj Qi /S00 which was introduced in [3]. So, in the limit where Y becomes C3 the amplitude in (6.2) simply becomes
Z(V1 , V2 , V3 ) =
(−1)(Q1 )
WQ2 Qt WQ2 Qt
Q1 ,Q2 ,Q3 ,Q
1
3
WQ2 0
× Tr Q1 V1 Tr Qt V1−1 Tr Q2 V2 Tr Q⊗Q3 V3 .
(6.3)
Our main interest is in the amplitude where the D-brane on L1 is on the first leg of C3 . This we will do by a suitable “analytic continuation”. The only part of the amplitude in (6.2) that can be affected by moving the D-brane there involves the representation labeled by Q1 , as this representation corresponds to the world-sheet instanton strings which may become massless in the process. The only other representation that could have been affected is the one labeled by Q, however the action of the corresponding strings is growing in the process, and as far as they are concerned, the above expressions are getting more and more reliable. At any rate, we will provide, in the following sections, strong evidence that this is correct. With this assumption, we only need to know what the transition means for the part of amplitude corresponding to WQ1 Q2 . To answer this, we may well study a simpler problem where we have only two stacks of D-branes, one wrapping L2 and one on L1 , to start with, as in Fig. 11. Now notice that this amplitude equals precisely Z(V1 , V2 ) = WQ2 Qt (−1)(Q1 ) Tr Q1 V1 Tr Q2 V2 . 1
Q1 ,Q2
By using the definition of the three-point vertex, and keeping track of framing, WQ2 Qt (−1)(Q1 ) should equal 1
0,0,−1 C0,Q = C0,Q2 ,Q1 (−1)(Q1 ) q −κQ1 /2 , 2 ,Q1
from which we conclude that C0,Q2 ,Q1 = WQ2 Qt q κQ1 /2 .
(6.4)
1
From this we can immediately find the amplitude corresponding to the second phase in Fig. 11. From the definition of the three-point vertex alone, this is CQ1 ,Q2 0 , but by cyclic symmetry (3.9) of the vertex and (6.4), CQ1 ,Q2 ,0 is the same as WQt Q1 q κQ2 /2 (the 2
The Topological Vertex
459
L2
L2
Q2
Q2
L1 Q1
L1
Q1
Fig. 11. The left- and the right-hand side of the figure describe two D-brane configurations in C3 , the latter obtained by moving the D-brane L1 . The amplitude corresponding to the configuration on the left 0,0,−1 0,0,0 is C0,Q ,Q and on the right it is CQ ,Q ,0 2
1
1
2
reader should recall that S, and hence W is symmetric). We conclude that in going from the left- to the right-hand side of Fig. 11 we must replace WQ2 Qt Tr Q1 V1 Tr Q2 V2 → WQt Q1 q κQ2 /2 Tr Q1 V1−1 Tr Q2 V2 , 1
2
in (6.3). The strings ending on the L1 D-brane in this new phase, are labeled with Q1 and will naturally have the same charge on the D-brane as the strings labeled with Q since now moving the D-brane affects both of their masses in the same way (recall that V is a phase of a complex field), and this is why we replace V1 by V1−1 in the above formula. Redefining V1 → V1−1 and collecting the coefficient of Tr R1 V1 Tr R2 V2 Tr R3 V3 in the partition function we compute CR0,0,−1 . Correspondingly, we get the following 1 ,R2 ,R3 expression for the three-point vertex in the canonical framing CR1 ,R2 ,R3 : CR1 ,R2 ,R3 =
Q1 ,Q3
R1 R3t t 1 Q3
where NQ R3t :
R1 R3t κR /2+κR /2 WR2t Q1 WR2 Qt3 3 , t q 2 1 Q3 WR2 0
NQ
(6.5)
counts the number of ways representations Q1 and Qt3 go into R1 and R1 R3t t 1 Q3
NQ
=
Q
Rt
R1 NQQ N 3. 1 QQt 3
One must be careful to note that NRiRRkj in the formula above are the ordinary tensor product coefficients, and not the Verlinde coefficients5 . We conjecture that the above expression (6.5) is the exact trivalent vertex amplitude. Here we have motivated this result based on the large N topological string duality, combined with certain plausible assumptions. As we discussed in Sect. 3 on general grounds, the vertex amplitude has a Z3 cyclic symmetry (3.9) and transforms simply 5 The careful reader should note that in writing (6.1) and (6.5), we have used the freedom to scale V to absorb a factor of (−1)(R) , into Tr R V .
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under exchanges of pairs of indices (3.12). Moreover, in the previous section, we have computed the leading piece of the vertex amplitude, corresponding to genus zero with up to two holes. The expression (6.5) has none of the symmetries of the vertex amplitude manifest, and checking agreement with (5.9) is highly involved. Nevertheless, the expression (6.5) passes all these checks. We will explicitly demonstrate this in Sect. 8. 7. Review of Chern-Simons and Topological String Amplitudes In order to work out some examples of closed and open string amplitudes from the three-point vertex, we need a more precise definition of the quantities appearing in (6.5). In this section we review these ingredients, as well as the integrality properties of open and closed string amplitudes. 7.1. Review of necessary Chern-Simons theory ingredients. In the evaluation of the amplitudes we will need the Chern-Simons invariants of the Hopf link in arbitrary representations of U (N). In this section we collect some formulae for these invariants. Recall that in terms of Chern-Simons variables, 2πi q = exp(gs ) = exp , λ = qN . (7.1) k+N In the duality with topological string theory [2], we have that t = Ngs , so λ = et . As a warmup, consider WR ≡ WR0 , which is related to the Chern-Simons invariant of the unknot in an arbitrary representation R. The invariant of the unknot is given by the quantum dimension of R: S0R = dimq R. (7.2) S00 The explicit expression for dimq R is as follows. Let R be a representation corresponding to a Young tableau with row lengths {µi }i=1,...,d(µ) , with µ1 ≥ µ2 ≥ · · · , and where d(µ) denotes the number of rows. Define the following q-numbers: [x] = q 2 − q − 2 , 1 x 1 x [x]λ = λ 2 q 2 − λ− 2 q − 2 . x
x
(7.3)
Then, the quantum dimension of R is given by dimq R =
µi −i d(µ) [v]λ [µi − µj + j − i] . µi v=−i+1 [j − i] [v − i + d(µ)] v=1 i=1 1≤i<j ≤d(µ)
(7.4)
1
The quantum dimension is a Laurent polynomial in λ± 2 whose coefficients are rational 1 functions of q ± 2 . We are interested in the leading power of λ in (7.4). As explained in [3], this power is /2, where = i µi is the total number of boxes in the representation R, and the 1 coefficient of this power is the rational function of q ± 2 , WR = q κR /4
1≤i<j ≤d(µ)
d(µ) µi [µi − µj + j − i] 1 , [j − i] [v − i + d(µ)] i=1 v=1
where κR is the framing factor introduced in Eq. (3.7).
(7.5)
The Topological Vertex
461
Let us now consider the Hopf link with linking number 1. Its invariant for representations R1 , R2 , WR1 R2 = SR1 R 2 /S00 is given by (WR1 R2 )U (N) = q 1 2 /N (WR1 R2 )SU (N) ,
(7.6)
where i is the total number of boxes in the Young tableau of Ri , i = 1, 2. The prefactor q 1 2 /N in (7.6) is a correction which was pointed out in [20], and is due to the fact that the vev WR1 ,R2 has to be computed in the theory with gauge group U (N ). The expression we will use for this invariant is the one obtained by Morton and Lukac in [37, 38]. Their formula is as follows. Let µ be a Young tableau, and let µ∨ denote its transposed tableau (remember that this tableau is obtained from µ by exchanging rows and columns). The Schur polynomial in the variables (x1 , . . . , xN ) corresponding to µ (which is the character of the diagonal SU (N ) matrix (x1 , . . . , xN ) in the representation corresponding to µ), will be denoted by sµ . They can be written in terms of elementary symmetric polynomials ei (x1 , . . . , xN ), i ≥ 1, as follows [39]: sµ = detMµ , where
(7.7)
Mµij = eµ∨i +j −i ,
Mµ is an r × r matrix, with r = d(µ∨ ). To evaluate sµ we put e0 = 1, ek = 0 for k < 0. The expression (7.7), known sometimes as the Jacobi-Trudy identity, can be formally extended to give the Schur polynomial sµ (E(t)) associated to any formal power series i E(t) = 1 + ∞ n=1 ai t . To obtain this, we simply use the Jacobi-Trudy formula (7.7), but where ei denote now the coefficients of the series E(t), i.e. ei = ai . Morton and Lukac define the series E∅ (t) as follows: E∅ (t) = 1 +
∞
cn t n ,
(7.8)
n=1
where the coefficients cn are defined by cn =
n 1 − λ−1 q i−1 i=1
q2 − 1
.
(7.9)
They also define a formal power series associated to a tableau µ, Eµ (t), as follows: Eµ (t) = E∅ (t)
d(µ) j =1
1 + q µj −j t . 1 + q −j t
(7.10)
One can then consider the Schur function of the power series (7.10), sµ (Eµ (t)), for any pair of tableaux µ, µ , by expanding Eµ (t) and substituting its coefficients in the JacobiTrudy formula (7.7). It turns out that this Schur function is essentially the invariant we were looking for. More precisely, one has 2
WR1 ,R2 (q, λ) = (dimq R1 )(λq) 2 sµ2 (Eµ1 (t)),
(7.11)
where µ1,2 are the tableaux corresponding to R1,2 , and 2 is the number of boxes of µ2 . More details and examples can be found in [37]. It is easy to see from (7.11) that the
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leading power in λ of WR1 ,R2 is (1 + 2 )/2, and its coefficient is given by the leading 1 coefficient of the quantum dimension, (7.5), times a rational function of q ± 2 that is given by: 2 WR1 R2 (q) = WR1 q 2 sµ2 Eµlead (t) , (7.12) 1 where d(µ)
Eµlead (t) = E∅lead (t)
1 + q µj −j t 1 + q −j t
(7.13)
tn . i i=1 (q − 1)
(7.14)
j =1
and E∅lead (t) = 1 +
∞
n
n=1
The above results are for knots and links in the standard framing. The framing can be incorporated as in [20], by simply multiplying the Chern-Simons invariant of a link with components in the representations R1 , . . . , RL , by the factor L
(−1)
α=1 pα α
1
q2
L
α=1 pα κRα
(7.15)
,
where pα , α = 1, . . . , L are integers labeling the choice of framing for each component. 7.2. Integrality of closed string amplitudes. In this and the following subsection we recall certain integrality properties that the topological A-model amplitudes possess on general grounds [40]. This allows one to formulate our answers in terms of certain integers which capture BPS degeneracies. The topological A-model free energy F (X) has the following structure: F (X) =
∞
2g−2
gs
Fg (t).
g=0
Here, Fg (t) is the free energy at genus g. It can be computed as a sum over two-homology classes of worldsheet instantons of genus g, Ng,Q e−Q·t , Fg (t) = Q
where the vector Q ∈ H2 (X, Z) labels the homology class, t is a vector of K¨ahler parameters, and Ng,Q are Gromov-Witten invariants. The free energy F (X) can be related to counting of certain BPS states on the Calabi-Yau manifold associated to D2 branes wrapping holomorphic curves in X [40]. The relation follows from the embedding of the topological A-model in type IIA string theory on X and its further embedding in M-theory. Moreover it relies on the target string interpretation of topological string amplitudes [8, 41]. This implies that the free energy has the following form [40]: F (X) =
∞
∞
n=1 Q∈H2 (X) g=0
g
nQ (2 sinh(ngs /2))2g−2
e−nQ·t . n
(7.16)
The Topological Vertex
463
In this formula, q = egs , g is related to an SU (2)L ⊂ SO(4) quantum number denoting the spin representation of the particle in 4 + 1 dimensions, Q · t is the mass of the BPS g state and nQ is an integer which counts the number of BPS states with quantum numbers Q and g. 7.3. Integrality of open string amplitudes. The free energy of open strings F (V ) is given by the logarithm of a partition function with the structure Zopen (V1 , . . . , VL ) =
Z(R1 ,...,RL )
L
Tr Rα Vα .
(7.17)
α=1
R1 ,...,RL
We define the generating function f(R1 ,...,RL ) (q, λ) through the following equation: F (V ) =
∞ n=1 R1 ,...,RL
1 f(R1 ,...,RL ) (q n , e−nt ) Tr Rα Vαn , n L
(7.18)
α=1
where Rα denote representations of U (M) and we are considering the limit M → ∞. In this limit we can exchange the basis consisting of the product of traces of powers in the fundamental representation, with the trace in arbitrary representations. It was shown in [22], following similar ideas in the closed string case [40], that the open topological strings compute the partition function of BPS domain walls in a related superstring theory. This led to the result that F (V ) has an integral expansion structure. This result was further refined in [34] where it was shown that the corresponding integral expansion leads to the following formula for f(R1 ,...,RL ) (q, λ): 1
1
f(R1 ,...,RL ) (q, λ) = (q 2 − q − 2 )L−2 ×
L
g≥0 Q R ,R ,...,R ,R α=1 1 1 L L
1
1
CRα Rα Rα SRα (q)N(R ,...,R ),g,Q (q 2 − q − 2 )2g e−Q·t . (7.19) 1
L
In this formula Rα , Rα , Rα label representations of the symmetric group S , which can be labeled by a Young tableau with a total of boxes. CR R R are the Clebsch-Gordon coefficients of the symmetric group, and the monomials SR (q) are defined as follows. If R is a hook representation (7.20) with boxes in total, and with − d boxes in the first row, then SR (q) = (−1)d q −
−1 2 +d
,
(7.21)
and it is zero otherwise. Finally, N(R1 ,...,RL ),g,Q are integers associated to open string amplitudes. They compute the net number of BPS domain walls of charge Q and spin g transforming in the representations Rα of U (M), where we are using the fact that representations of U (M) can also be labeled by Young tableaux. It is also useful to introduce a generating functional for these degeneracies as in [34]: 1 1 2g f(R1 ,...,RL ) (q, λ) = N(R1 ,...,RL ),g,Q q 2 − q − 2 e−Q·t . (7.22) g≥0 Q
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We then have the relation: 1
1
f(R1 ,...,RL ) (q, λ) = (q 2 − q − 2 )L−2
R1 ,...,RL
MR1 ,...,RL ;R1 ,...,RL (q)f(R1 ,...,RL ) (q, λ), (7.23)
where the matrix MR1 ,...,RL ;R1 ,...,RL (q) is given by MR1 ,...,RL ;R1 ,...,RL (q) =
L α=1 Rα
CRα Rα Rα SRα (q)
and it is invertible [34]. Finally, it is also useful sometimes to write BPS degeneracies in the winding number basis: L
n(k (1) ,...,k (L) ),g,Q =
χRα (C(k (α) ))N(R1 ,...,RL ),g,Q .
(7.24)
R1 ,...,RL α=1
Notice that the BPS degeneracies N(R1 ,...,RL ),g,Q in the representation basis are more fundamental than the degeneracies in the winding number basis, as emphasized in [34]. The f(R1 ,...,RL ) introduced in (7.18) can be extracted from Z(R1 ,...,RL ) through a procedure spelled out in detail in [33, 34, 42]. One has, for example, f
=Z
,
−Z
,
,· Z·,
(7.25)
.
It is also convenient to introduce the quantities Z(k (1) ,...,k (L) ) =
L
χRα (k (α) )Z(R1 ,...,RL ) ,
(7.26)
Rα α=1
in such a way that Zopen (V1 , . . . , VL ) =
Z(k (1) ,...,k (L) )
k (1) ,...,k (L)
L 1 ϒ (α) (Vα ). zk (α) k
(7.27)
α=1
We can now write the total free energy as: F (V ) =
∞
2g−2+h
gs
Fg,k (α) (t)
g=0 k (α)
where h =
α
L
ϒk (α) (Vα ),
(7.28)
α=1
hα is the total number of holes. We have then that ∞
2g−2+h
Fg,k (α) (t)gs
g=0
where (c) denotes the connected piece.
= L
1
α=1 zk (α)
(c) k
Z (α) ,
(7.29)
The Topological Vertex
465
8. The Vertex Amplitude In this section we use the apparatus developed above to calculate some values of the vertex amplitude. This will provide highly non-trivial checks that the vertex amplitude derived in Sect. 6 using large N-dualities is in fact the correct expression. As discussed in the previous section, we can extract the free energy of the vertex amplitude with fixed winding numbers as the connected part of Ck 1 k 2 k 3 : (n ,n2 ,n3 ) (g ) k ,k (2) ,k (3) s
F (1)1
= 3
1
α=1 zk (α)
(n ,n2 ,n3 ) . k ,k (2) ,k (3)
(C (c) ) (1)1
Consider the part of this amplitude corresponding to a single hole on each of the three stacks of D-branes. Since only the winding numbers remain to be specified, we can simply denote this by Fk,l,m , corresponding to k (i) , i = 1, 2, 3 with a single nonzero entry in positions k, l, m, respectively. Then, one has the following formulae: 1 [k + nk − 1]! , k [k]![nk]! (−1)l+1 [kl] k = , kl [k] l
(n,0,0)
=
Fk,0,0
(0,0,0)
Fk,l,0
where the q-number is [x] = q 2 − q − 2 , the q-factorial is given by x
x
[x]! = [x][x − 1] · · · [1], and finally the q-combinatorial number is defined as x [x]! = . y [x − y]![y]!
(8.1)
(8.2)
Note that the leading gs terms Fk,0,0 and Fk,m,0 are (n,0,0)
1 (k + nk − 1)! , gs k (k)!(nk)! (−1)l+1 k , = k l
Fg=0;k,0,0 = (0,0,0)
Fg=0;k,l,0
and these agree with (5.10) and (5.9) respectively up to a choice of coordinate, T rVim → (−1)m T rVim and the over-all sign of the free energy! Let us now look at some explicit values of the vertex (6.5) which we can easily compute using the explicit expressions for WR1 R2 , that we gave in the previous section. Using explicit evaluation of C one can verify that at least for a small number of boxes the highly non-trivial symmetry predictions (3.9) and (3.12) are indeed satisfied. We give here a list of values for the trivalent vertex up to five boxes in total. For the sake of space, we mostly list values which are not related by symmetries, although we have included some to make manifest the properties that we derived in Sect. 3. The dot · stands for the trivial representation. C
··
=
1 1 2
1
q − q− 2
,
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M. Aganagic, A. Klemm, M. Mari˜no, C. Vafa
C
·
C
=
C C C
q4 − q3 + q2 − q + 1 1
q 2 (q − 1)3 q3 − q2 + 1
=
·
··
q2 − q + 1 , C (q − 1)2 q , = (q − 1)(q 2 − 1) =
1
q 2 (q − 1)2 (q 2 − 1) 5 q2 = , (q − 1)2 (q 3 − 1)
··
··
q2 , (q − 1)(q 2 − 1)
=
3
,
C
·
=
q 2 (q 3 − q 2 + 1) , (q − 1)2 (q 2 − 1) 9
,
C
··
=
q2 , 2 (q − 1)(q − 1)(q 3 − 1) 3
C
··
q2 = , 2 (q − 1)(q − 1)(q 3 − 1)
q6 − q5 + q3 − q + 1 q6 − q5 + q3 − q + 1 , C , = 3 2 (q − 1) (q − 1) q(q − 1)3 (q 2 − 1) q 2 (q 4 − q 2 + 1) q(q 6 − q 5 − q 4 + 2q 3 − q + 1) = , C = , · (q − 1)2 (q 2 − 1)2 (q − 1)2 (q 2 − 1)2 =
C C
·
q 6 − q 5 + 2q 3 − q 2 − q + 1 q4 − q2 + 1 , C = , · 2 2 2 q(q − 1) (q − 1) (q − 1)2 (q 2 − 1)2 4 4 3 4 3 2 q (q − q + 1) q(q − q + q − q + 1) , C , = · = · (q − 1)2 (q 2 − 1)(q 3 − 1) (q − 1)3 (q 3 − 1) 4 q −q +1 , = · q(q − 1)2 (q 2 − 1)(q 3 − 1)
C
·
C C
=
q8 , (q − 1)(q 2 − 1)(q 3 − 1)(q 4 − 1) 4 q = , C ·· (q − 1)2 (q 2 − 1)(q 4 − 1)
C
··
C C
··
=
=
(q
− 1)(q 2
q5 , (q − 1)2 (q 2 − 1)(q 4 − 1) 3 q = , ·· (q − 1)2 (q 2 − 1)(q 4 − 1) C
··
=
q2 , − 1)(q 3 − 1)(q 4 − 1)
1
q 2 (q 8 − q 7 + q 5 − q 4 + q 3 − q + 1) = , (q − 1)3 (q 2 − 1)2
C
C
=
C
=
C C C
=
q 9 − q 8 − q 7 + 2q 6 − q 4 + q 3 − q + 1 1
,
3
,
q 2 (q − 1)3 (q 2 − 1)2 q 9 − q 8 + q 6 − q 5 + 2q 3 − q 2 − q + 1 q 2 (q − 1)3 (q 2 − 1)2 8 7 q − q + q5 − q4 + q3 − q + 1
, 3 q 2 (q − 1)3 (q 2 − 1)2 3 q 2 (q 8 − q 7 + q 4 − q + 1) = , (q − 1)3 (q 2 − 1)(q 3 − 1) q 8 − 2q 7 + 3q 6 − 3q 5 + 3q 4 − 3q 3 + 3q 2 − 2q + 1 = , 1 q 2 (q − 1)4 (q 3 − 1)
The Topological Vertex
=
C ·
=
C
·
=
C
·
=
·
=
C
C C C C C C C C
·
·
= =
·
=
·
=
·
=
·
·
= =
467
q8 − q7 + q4 − q + 1
, q (q − 1)3 (q 2 − 1)(q 3 − 1) 9 q 2 (q 5 − q 3 + 1) , (q − 1)2 (q 2 − 1)2 (q 3 − 1) 7 q 2 (q 8 − q 7 − q 6 + q 5 + q 4 − q 2 + 1) , (q − 1)2 (q 2 − 1)2 (q 3 − 1) 1 q 2 (q 7 − q 6 + q 4 − q + 1) , (q − 1)3 (q 2 − 1)(q 3 − 1) 1 q 2 (q 7 − q 6 + q 3 − q + 1) , (q − 1)3 (q 2 − 1)(q 3 − 1) q8 − q6 + q4 + q3 − q2 − q + 1 , 5 q 2 (q − 1)2 (q 2 − 1)2 (q 3 − 1) q5 − q2 + 1 , 1 q 2 (q − 1)2 (q 2 − 1)2 (q 3 − 1) 15 q 2 (q 5 − q 4 + 1) , (q − 1)2 (q 2 − 1)(q 3 − 1)(q 4 − 1) 7 q 2 (q 5 − q 4 + q 2 − q + 1) , (q − 1)3 (q 2 − 1)(q 4 − 1) 5 q 2 (q 4 − q 2 + 1) , (q − 1)2 (q 2 − 1)2 (q 3 − 1) 1 q 2 (q 5 − q 4 + q 3 − q + 1) , (q − 1)3 (q 2 − 1)(q 4 − 1) q5 − q + 1 , 3 q 2 (q − 1)2 (q 2 − 1)(q 3 − 1)(q 4 − 1) 3 2
25
C
··
q2 , = (q − 1)(q 2 − 1)(q 3 − 1)(q 4 − 1)(q 5 − 1) 17
C
··
C
··
C
··
q2 , 2 2 (q − 1) (q − 1)(q 3 − 1)(q 5 − 1) 13 q2 = , (q − 1)2 (q 2 − 1)(q 3 − 1)(q 4 − 1) 11 q2 , = (q − 1)2 (q 2 − 1)2 (q 5 − 1) =
9
C
··
=
q2 , 2 2 (q − 1) (q − 1)(q 3 − 1)(q 4 − 1) 7
C
··
q2 , = (q − 1)2 (q 2 − 1)(q 3 − 1)(q 5 − 1)
··
q2 . = 2 3 (q − 1)(q − 1)(q − 1)(q 4 − 1)(q 5 − 1)
5
C
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Note for example that: C
= q 3C
,
while on the other hand k = 6, so that this precisely agrees with (3.12). The vertex amplitude, being an open string amplitude, has to satisfy strong integrality requirements that we have reviewed in Sect. 6. In order to check this, we can compute the generating functionals for BPS states fR1 R2 R3 for arbitrary framings in the legs. If 1 1 we denote z = (q 2 − q − 2 )2 , one finds, for example: f , f
= (−1)n1 +n2 +n3 , n1 n1 2 2 − q− 2 n2 +n3 q = −(−1) 1 1 q 2 − q− 2 1 = −(−1)n2 +n3 n21 − (−1)n2 +n3 n21 (n21 − 1)z + · · · , 12 n1 +1 n1 +1 2 2 − q− 2 n2 +n3 q = −(−1) 1 1 q 2 − q− 2 1 = −(−1)n2 +n3 (1 + n1 )2 − (−1)n2 +n3 n1 (n1 + 1)2 (n1 + 2)z + · · · , 12
,
, ,
f ,
f ,
,
,
f , , f ,
,
f , ,
(−1)n3 2n2 (n2 − 1) + n1 (n22 − 3n2 + 2) + n21 (3n22 − n2 + 2) + · · · , 4 (−1)n3 n1 n2 (n2 − 1) + 2n2 (1 + n2 ) + n21 (3n22 + 5n2 + 4) + · · · , = 4 (−1)n3 4 − 6n2 + 6n22 + n1 (7n22 − 5n2 + 6) + n21 (3n22 − n2 + 2) + · · · , = 4 (−1)n3 4 + 6n2 + 6n22 + n1 (7n22 + 9n2 + 8) + n21 (3n22 + 5n2 + 4) + · · · , = 4 =
This passes the integrality check. 9. Examples of Open and Closed String Amplitudes From the Vertex In this section we compute various closed and open string amplitudes using the trivalent vertex. In the examples below we have made many checks of the vertex against amplitudes of closed and open string calculations using other means. In the closed string case the vertex can be checked, in principle to all genera, by comparison with mirror B-model calculations using holomorphic anomaly [8], see [43, 44, 29], as well as against A-model localisation calculations using the techniques of [16, 45, 46]. More directly one can compare the vertex with open string amplitudes. B-model calculations for the open string were so far only available for the disk following [19, 11]. A-model localization calculations for all genus open string amplitudes with a single stack of D-branes have been introduced in [47, 45]. At the operational level this is a minor modification of the localization procedure of the closed string case, and we provide a computer program which computes this for general toric configurations6 . Due to the extended combinatorics of the graphs indexing the fixed points of the torus action on 6 This program can be distributed on request. It requires the evaluation of 2d gravity correlation functions, which were implemented in Maple by Carel Faber, see also [48].
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the moduli space of stable maps [16, 45, 46], the computer calculation is very slow compared with the techniques developed in the present paper, in particular for amplitudes with higher genus and larger degree (w.r.t the compact K¨ahler classes). A-model calculations provide, on the other hand, expressions which describe all windings. While the results of the calculations have been checked for many cases, see in particular [45], the procedure has not been established rigorously and leaves interesting conceptual issues to be developed, in particular in regards to multiple stacks of branes. It should be possible to derive within the localization approach general expressions for the vertex for general windings and framings. Some attempts in this directions have already been made [7]. To begin with, it is useful to see how the vertex works in a few simple examples where the complete amplitudes are known.
9.1. Example I. The closed string amplitude in Fig. 12 can be written in terms of six trivalent vertices glued together. Two of them are of the kind we have already discussed. Using an SL(2, Z) transformation, we find the differently oriented trivalent vertex corresponding to Fig. 12. From this, and using the gluing rules above, we find that the closed string amplitude can be written as Z= (−1) i (Ri ) C··R1 e−(R1 )t1 C·R t R2 e−(R2 )r1 CR t ·R3 1
R1 ,··· ,R5 −(R3 )t2
×e
2
C·R t R4 e−(R4 )r2 CR t ·R5 e−(R5 )t3 C·R t · . 3
4
(9.1)
5
Note that to get the all genus answer up to degree n in any one of the five classes, we only need to perform the sum over the corresponding representation with up to n boxes. It is not difficult to check that this is the correct A-model amplitude on X, which is known to all genera. For example, one way to calculate the amplitude is to use mirror symmetry
t1
R1
r1 t2
R2
r2 t3
R3 R4
R5
Fig. 12. The left-hand side is the graph of a geometry containing a chain of P1 ’s with five independent classes in H2 . The right-hand side depicts a decomposition of this graph in terms of three-point vertices. All the vertices are obvious repetitions of the first two, moreover, all the vertex amplitudes equal CR1 R2 R3 with representation ordered cyclically, counter-clockwise. Some of the representations are set to be trivial, as corresponding legs are non-compact. The small figure in the upper left corner is the corresponding graph ˆ
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M. Aganagic, A. Klemm, M. Mari˜no, C. Vafa
to calculate the genus zero amplitude, and integrality to fix the full free energy. This is possible as there are no curves in X with genus higher than zero, so we find
Z = exp
n
NQ e−nQ·t
,
n(2 sinh(ngs /2))2
in X where NQ are the degeneracies of BPS states corresponding to a P1 in class Q whose values are as follows. First, there can be no BPS states corresponding to P1 ’s which are disconnected. For the chains of connected P1 ’s we have that: NQ = −1 if corresponds to the class of an odd number of connected P1 ’s in X (e.g. BPS states Q with masses r1 , r1 + t2 + r2 and t1 + r1 + t2 + r2 + t3 all have NQ = −1), NQ = 1 corresponds to a class of an even number of connected P1 ’s (e.g. r2 + t2 and if Q r1 + t2 +2 +t3 ). 9.2. Example II. It is also easy to see that our rules reproduce the O(−3) → P2 amplitudes computed in [3]. In [3], the all genus amplitude was computed using a quiver-type Chern-Simons theory with three nodes G = U (N1 ) × U (N2 ) × U (N3 ) and bifundamental matter-fields, in the Ni → ∞ limit. Using SL(2, Z) transformations and adjusting framings appropriately, it is easy to see that the amplitude corresponding to the graph in Fig. 13, when there are no branes in the outer legs, can be written as ZP2 = (−1) i (Ri ) e− i (Ri )t q i κRi C·R2 R t C·R1 R t C·R3 R t , (9.2) 3
2
1
R1 ,R2 ,R3
where t is the K¨ahler parameter of O(−3) → P2 . Using the fact that, e.g. C·R2 R t = WR2 R3 q −κR3 /2 , 3
the amplitude (9.2) precisely equals the amplitude obtained in [3] by related, but different methods. Q1
(−2)
R1
(−2)
R2
R(3−2) Q3
Q2
Fig. 13. The trivalent vertices for gluing to O(−3) → P2 amplitude (with D-branes on external legs). The superscript on the representations give the framing n in the corresponding propagator
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471
Q2
R (0)
Q1 Fig. 14. This shows O(−1) + O(−1) → P1 with D-branes on external legs
9.3. Example III. Another example where we can use the full vertex amplitude is shown in Fig. 14: two D-branes on the outer legs of O(−1) + O(−1) → P1 . According to the gluing rules we have Z(V1 , V2 ) = CQ1 Q2 R t (−1)(R) e−(R)t CR·· Tr Q1 V1 Tr Q2 V2 . (9.3) R,Q1 ,Q2
On the other hand, this amplitude corresponds to a Hopf link inside S3 with linking number −1, therefore we should have Q +Q 1 2 λ− 2 SQ1 Q2 Tr Q1 V1 Tr Q2 V2 , (9.4) Z(V1 , V2 ) = Q1 ,Q2
where λ = e−t . One can check that indeed (9.3) and (9.4) agree. Namely, we have that SQ1 Q2 = WQ1 Q2 (q, λ)S00 (λ), where WQ1 Q2 is the invariant calculated in (7.11), and S00 is the partition function of CS on S3 , which can be written as7 [2] ∞ −kt e S00 (e−t ) = exp − (9.5) . k k 2 − k=1 k q 2 − q 2 For example, for Q1 = R = , agreement requires that (1)
C
Q
= WQ W −
WQ W
(9.6)
,
(1)
where WQ1 Q2 is defined by the expansion WQ1 Q2 (q, λ) = λ
Q +Q 1 2 2
WQ1 Q2 (q) + λ
Q +Q 1 2 2
−1
(1)
WQ1 Q2 (q) + · · · .
(9.7)
7 There are pieces in the free energy which are finite polynomials in t which encode certain topological data in the compact case. In the non-compact case at hand they are ambiguous and in our vertex amplitudes we have naturally set them to zero.
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M. Aganagic, A. Klemm, M. Mari˜no, C. Vafa
Using the explicit formula (7.11), we find that C
= q κQ /2
Q
W
Qt W Q
WQ
+ WQ ,
(9.8)
in agreement with (6.5). 9.4. Example IV. Another non-trivial configuration involving the full trivalent vertex for which we have an immediate prediction is O(−3) → P2 with “outer” D-branes on the external legs. This corresponds to the amplitude (9.2) but where we allow non-trivial external representations on the trivalent vertices: CQ3 R2 R t CQ1 R1 R t CQ2 R3 R t (−1) i (Ri ) ZP2 (V1 , V2 , V3 ) = 3
Ri ,Qi − i (Ri )t i κRi
×e
q
2
1
Tr Q1 V1 Tr Q2 V2 Tr Q3 V3 .
(9.9)
This amplitude is the product of the closed string amplitude ZP2 given in (9.2), and the open string amplitude properly speaking, so we will write ZP2 (V1 , V2 , V3 ) ZP 2 = Z(Q1 ,Q2 ,Q3 ) (q, e−t )Tr Q1 V1 Tr Q2 V2 Tr Q3 V3 .
Zopen (V1 , V2 , V3 ) =
Qi
Notice that the amplitudes are completely symmetric in Q1 , Q2 , Q3 , as they should be by the symmetry of the geometry. The generating functions f(Q1 ,Q2 ,Q3 ) are computed from Z(Q1 ,Q2 ,Q3 ) . Let us denote 1 1 z = (q 2 − q − 2 )2 , y = e−t , so that N(R1 ,R2 ,R3 ),g,Q zg y Q , f(Q1 ,Q2 ,Q3 ) = g,Q
where N(R1 ,R2 ,R3 ),g,Q are the degeneracies of BPS states with the corresponding charges. We find, for the first few representations and up to degree five in the K¨ahler parameter, the following results: f ,·,· = 1 − 2 y + 5 y 2 − (32 + 9 z) y 3 + (286 + 288 z + 108 z2 + 14 z3 ) y 4 −(3038 + 6984 z + 7506 z2 + 4519 z3 + 1542 z4 + 276 z5 + 20 z6 ) y 5 + · · · , f , ,· = −y + 4 y 2 − (35 + 8 z) y 3 + (400 + 344 z + 112 z2 + 13 z3 ) y 4 −(5187 + 10504 z + 10036 z2 + 5434 z3 + 1691 z4 + 280 z5 + 19 z6 ) y 5 + · · · , f ,·,· = 7 y 3 − (110 + 68 z + 12 z2 ) y 4 +(1651 + 2938 z + 2353 z2 + 992 z3 + 212 z4 + 18 z5 ) y 5 + · · · , f ,·,· = y − 4 y 2 + (28 + 8 z2 ) y 3 − (290 + 276 z + 100 z2 + 13 z3 ) y 4 +(3536 + 7566 z + 7683 z2 + 4442 z3 + 1479 z4 + 262 z5 + 19 z6 ) y 5 + · · · , f , , = 3 y 2 − (36 + 7 z) y 3 + (531 + 396 z + 114 z2 + 12 z3 ) y 4 −(8472 + 15210 z + 13026 z2 + 6399 z3 + 1830 z4 + 282 z5 + 18 z6 ) y 5 + · · · , (9.10)
and so on. For representations involving only one nontrivial representation, the degeneracies obtained above agree with the ones obtained in B-model computations [11] (see also [49]) and in A-model computations through localization [45, 50].
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473
One can also compute the amplitudes in nontrivial framings, just by framing the trivalent vertex in the appropriate way. For a single nontrivial representation with framing n, we find N( ,·,·),g,d (n) = (−1)n N( ,·,·),g,d (0), and 1 1 1 − (−1)n − 2 n2 + −3 + 3(−1)n + 8 n2 − 2 n4 z + · · · 8 96 1 1 2 2 2 + n + n (n − 1) z + · · · y + (−1 + (−1)n − 14 n2 ) 12 4 1 + (3 − 3(−1)n + 8n2 − 14n4 ) z + · · · y 2 + · · · , 48 1 f ,·,· (n) = −1 + (−1)n − 4n − 2 n2 8 1 + 3 − 3 (−1)n + 8 n − 4 n2 − 8 n3 − 2 n4 z + · · · 96 1 2 2 + (n + 1) + n(1 + n) (n + 2) z + · · · y 12 1 + (−15 − (−1)n − 28 n − 14 n2 ) 4 1 n 2 3 4 + (−3 + 3 (−1) − 40 n − 76 n − 56 n − 14 n ) z + · · · y 2 + · · · . 48 (9.11)
f
,·,· (n)
=
9.5. Example V. So far we have considered open string amplitudes where D-branes were sitting on outer edges, but our formalism also allows to compute amplitudes with branes in inner edges, as we saw in (3.16). A simple example of such a situation is local P2 with an inner and an outer brane, as depicted in Fig. 15. The framings are as in Fig. 13. The
Fig. 15. The O(−3) → P2 with an outer brane and another brane in an inner edge
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M. Aganagic, A. Klemm, M. Mari˜no, C. Vafa
prediction for this amplitude is Z(V1 , V2 ) = (−1)s(Ri ,Qi ) q f (Ri ,Qi ) e−L(Ri ,Qi ) CQ1 R2 R3 ⊗Q2 Ri ,Qi
C·R t ⊗Q3 R t C·R1 R t TrQ1 V1 TrQ2 V2 TrQ3 V2−1 , 3
where L(Ri , Qi ) =
1
2
(Ri )t + +(Q1 )r1 + (Q2 )r2 + (Q3 )(t − r2 ),
i
f (Ri , Qi ) = κR1 + κR2 + nκQ1 /2 + pκR3 ⊗Q2 /2 + (p + 2)κR t ⊗Q3 /2, 3 s(Ri , Qi ) = (Ri ) + n(Q1 ) + p(R3 ⊗ Q2 ) + (p + 2)(R3t ⊗ Q3 ). i
The integers p and n correspond to the framing of the inner brane and the outer branes, respectively. p is related to the framing p in the B-model of [11] by p = −1 − p . We can again compute the generating functionals fˆ for degeneracies of BPS states for different representations. We present some results corresponding to p = −1, and n = 0 (for the inner brane, this is the zero framing of [11]) where we absorb e−ri in Vi : f·, ,· = −1 + y − (5 + z) y 2 + (40 + 31 z + 9 z2 + z3 ) y 3 −(399 + 743 z + 648 z2 + 322 z3 + 94 z4 + 15 z5 + z6 ) y 4 +(4524 + 16146 z + 29256 z2 + 33523 z3 + 26079 z4 + 14151 z5 + 5364 z6 +1390 z7 + 234 z8 + 23 z9 + z10 ) y 5 + · · · , f·,·, = −1 + 2 y − (12 + 3 z) y + (104 + 96 z + 33 z2 + 4 z3 ) y 3 −(1085 + 2328 z + 2334 z2 + 1315 z3 + 423 z4 + 72 z5 + 5 z6 ) y 4 −(12660 + 50874 z + 103683 z2 + 133002 z3 + 114732 z4 + 68040 z5 + 27711 z6 +7590 z7 + 1332 z8 + 135 z9 + 6 z10 ) y 5 + · · · , f , ,· = −1 + y − (6 + z) y 2 + (59 + 39 z + 10 z2 + z3 ) y 3 −(706 + 1152 z + 895 z2 + 403 z3 + 108 z4 + 16 z5 + z6 ) y 4 +(9372 + 29927 z + 48964 z2 + 51169 z3 + 36663 z4 + 18485 z5 + 6561 z6 + 1603 z7 +256 z8 + 24 z9 + z10 ) y 5 + · · · , f·, , = 2 y 2 − (46 + 30 z + 5 z2 ) y 3 + (852 + 1682 z + 1285 z2 + 536 z3 + 111 z4 + 9 z5 ) y 4 −(14848 + 55104 z + 101054 z2 + 113629 z3 + 83274 z4 + 40375 z5 + 12800 z6 +2544 z7 + 287 z8 + 14 z9 ) y 5 + · · · , (9.12)
and so on. The results for f·,R,· and f·,·,R correspond to inner branes with positive and negative winding numbers, respectively, and they agree with the B-model results of [11] in the case of disc amplitudes. For higher genus and/or number of holes, our results agree with those obtained through localization in [50]. The amplitudes with two nontrivial representations can also be obtained through localization, and in all cases we have found perfect agreement with the above results.
9.6. Example VI. We now consider more complicated examples of closed string amplitudes. Consider for example the toric diagram in Fig. 16. There are three K¨ahler parameters involved, s, t1 and t2 , as indicated in the figure. The amplitude is symmetric in t1 , t2 . When, say, t2 is taken to infinity, the resulting geometry is that of a local Hirzebruch surface F1 , where s corresponds to the K¨ahler parameter of the base in local
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475
(−1,2)
(0,1) (−1)
R3
(1)
R2
(−1,0)
(−2)
(−1,1)
(−1,1)
R4
(0)
R0
(−1,1)
R(1)5 (−2)
(1,0)
R1
(2,−1)
(0,−1) (−1)
R6
(1,−2)
Fig. 16. There are three K¨ahler parameters in the problem. The size of leg corresponding to R0 is s, and the sizes of R3 and R2 correspond to t1 , t2 , respectively
F1 , while t1 corresponds to the fiber. The amplitude for Fig. 16 can be computed by using our rules in Sect. 3, and the result is: Z(X) = (−1)(R0 )+(R1 )+(R4 ) q κR1 CR t ·R t q −κR2 /2 CR2 R3 R t q κR3 /2 CR t ·R t 1
R0...6 −κR4
q
2
0
CR4 ·R t q −κR5 /2 CR5 R6 R0 q κR6 /2 CR t ·R1 e−L(Ri ) , 6
5
3
4
(9.13)
where we wrote L(Ri ) = ((R0 ) + (R1 ) + (R4 ))s + ((R2 ) + (R6 ))t1 + ((R3 ) + (R5 ))t2 . (9.14) Notice that when t2 → ∞, (9.13) becomes the amplitude for local F1 , which was computed from Chern-Simons theory in [5, 6] with the techniques of [3]. We will write the answer in terms of a generating functional Fg of BPS degeneracies at genus g, as Fg (s, t1 , t2 ) =
∞
e−s F (t1 , t2 ), g
=0
where
g
F (t1 , t2 ) =
d1 ,d2
g
n,d1 ,d2 q1d1 q2d2 ,
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M. Aganagic, A. Klemm, M. Mari˜no, C. Vafa
and we have written qi = e−ti (these shouldn’t be confused with the Chern-Simons variable introduced before). We then find, up to order four in qi , the following results (symmetrization w.r.t. q1 , q2 is understood): F00 (t1 , t2 ) = −2q1 − 2q1 q2 + · · · , F10 (t1 , t2 ) = −1 − 3q1 − 5q12 − 7q13 − 9q14 − 4q1 q2 − 8q1 q22 − 12q1 q23 − 16q1 q24 − 9q12 q22 −15q12 q23 − 21q12 q24 − 16q13 q23 − 24q13 q24 − 25q14 q24 + · · · , 0 F2 (t1 , t2 ) = −6q12 − 32q13 − 110q14 − 10q1 q22 − 70q1 q23 − 270q1 q24 − 32q12 q22 − 126q12 q23 −456q12 q24 − 300q13 q23 − 784q13 q24 − 1584q14 q24 + · · · , 1 F2 (t1 , t2 ) = 9q13 + 68q14 + 16q1 q23 + 144q1 q24 + 21q12 q23 + 204q12 q24 + 59q13 q23 +297q13 q24 + 684q14 q24 + · · · , F22 (t1 , t2 ) = −12q14 − 22q1 q24 − 30q12 q24 − 36q13 q24 − 94q14 q24 + · · · , F30 (t1 , t2 ) = 27q13 + 286q14 + 64q1 q23 + 800q1 q24 + 25q12 q22 + 266q12 q23 + 1998q12 q24 +1332q13 q23 + 6260q13 q24 + 21070q14 q24 + · · · , 1 F3 (t1 , t2 ) = −10q13 − 288q14 − 18q1 q23 − 688q1 q24 −64q12 q23 − 1404q12 q24 − 516q13 q23 − 4372q13 q24 − 18498q14 q24 + · · · , 2 F3 (t1 , t2 ) = 108q14 + 224q1 q24 + 375q12 q24 + 49q13 q23 + 1168q13 q24 + 6837q14 q24 + · · · , F33 (t1 , t2 ) = −14q14 − 26q1 q24 − 36q12 q24 − 114q13 q24 − 1196q14 q24 + · · · , F34 (t1 , t2 ) = 81q14 q24 + · · · .
These numbers agree with the results obtained with localization8 . Notice that at genus 0 and for t2 → ∞, the above results coincide with the results for local F1 presented for example in [43]. One can find a similar result for the A2 fibration over P1 . In this case, the amplitude reads q 5κR1 /2 CR t ·R t q −κR2 /2 CR2 R3 R t q κR3 /2+3κR0 /2 CR t ·R t Z(X) = 1
R0...6 κR4 /2
q
2
0
3
CR4 ·R t q −κR5 /2 CR5 R6 R0 q κR6 /2 CR t ·R1 e−L(Ri ) , 5
6
4
(9.15)
where now L(Ri ) = ((R0 ) + (R1 ) + (R4 ))s +(4(R1 ) + (R2 ) + (R6 ))t1 + (2(R0 ) + 2(R1 ) + (R3 ) + (R5 ))t2 . Here, s corresponds to the K¨ahler parameter of the base of the fibration, and t1 , t2 correspond to the K¨ahler parameters of the fibers. Denoting the generating functional as before, we obtain in this case, F00 (t1 , t2 ) = −2q1 − 2q2 − 2q1 q2 , F10 (t1 , t2 ) = −1 − 2q2 − 4q22 − 6q23 − 8q24 − 2q1 q2 − 6q1 q22 −10q1 q23 − 14q1 q24 − 6q12 q22 − 12q12 q23 − 18q12 q24 −4q13 q22 − 12q13 q23 − 20q13 q24 − 6q14 q22 − 10q14 q23 − 20q14 q24 + · · · , 0 F2 (t1 , t2 ) = −6q23 − 32q24 − 10q1 q23 − 70q1 q24 − 12q12 q23 − 96q12 q24 −12q13 q23 − 110q13 q24 − 10q14 q23 − 112q14 q24 + · · · , 1 F2 (t1 , t2 ) = 9q14 + 16q1 q24 + 21q12 q24 + 24q13 q24 + 25q14 q24 + · · · , 8 In order to compare with the results using localization and mirror symmetry, we have redefined g g gs → igs and therefore nQ → (−1)g−1 nQ .
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and so on, again in agreement with the results for genus zero in [43]. We have also checked some of these results at higher genus with localization techniques. Acknowledgements. We would like to thank D.-E.Diaconescu, R. Dijkgraaf, J. Gomis, A. Grassi, A. Iqbal, A. Kapustin, S. Katz, V. Kazakov, I. Kostov, C-C. Liu, H. Ooguri, J. Schwarz, S. Shenker and E. Zaslow for valuable discussions (and the cap!). The research of MA and CV was supported in part by NSF grants PHY-9802709 and DMS-0074329. In addition, CV thanks the hospitality of the theory group at Caltech, where he is a Gordon Moore Distinguished Scholar. M.A. is grateful to the Caltech theory group for hospitality during part of this work. A.K. is supported in part by the DFG grant KL-1070/2-1.
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´ 28. Alvarez-Gaum´ e, L., G´omez, C., Moore, G.W., Vafa, C.: Strings In The Operator Formalism. Nucl. Phys. B 303, 455 (1988) 29. Aganagic, M., Klemm, A., Mari˜no, M., Vafa, C.: Matrix model as a mirror of Chern-Simons theory. JHEP 02, 010 (2004) 30. Shenker, S.: Private communication, 1995 31. Ooguri, H., Vafa, C.: Worldsheet derivation of a large N duality. Nucl. Phys. B 641, 3 (2002) 32. Witten, E.: Chern-Simons gauge theory as a string theory. In: The Floer memorial volume, Hofer, H., Taubes, C.H., Weinstein, A., Zehner, E. (eds.), Basel-Boston: Birkh¨auser, 1995, p. 637 33. Labastida, J.M.F., Mari˜no, M.: Polynomial invariants for torus knots and topological strings. Commun. Math. Phys. 217, 423 (2001) 34. Labastida, J.M.F., Mari˜no, M., Vafa, C.: Knots, links and branes at large N. JHEP 11, 007 (2000) 35. Ramadevi, P., Sarkar, T.: On link invariants and topological string amplitudes. Nucl. Phys. B 600, 487 (2001) 36. Verlinde, E.: Fusion rules and modular transformations in 2-D conformal field theory. Nucl. Phys. B 300, 360 (1988) 37. Morton, H.R., Lukac, S.G.: The HOMFLY polynomial of the decorated Hopf link. J. Knot Theory Ramif. 12, 395–416 (2003) 38. Lukac, S.G.: HOMFLY skeins and the Hopf link. Ph.D. Thesis, University of Liverpool, 2001 39. Macdonald, I.G.: Symmetric functions and Hall polynomials. 2nd edition, Oxford: Oxford University Press, 1995 40. Gopakumar, R., Vafa, C.: M-theory and topological strings, II. http://arxiv.org/abs/hep-th/9812127, 1998 41. Antoniadis, I., Gava, E., Narain, K.S., Taylor, T.R.: Topological amplitudes in string theory. Nucl. Phys. B 413, 162 (1994) 42. Labastida, J.M.F., Mari˜no, M.: A new point of view in the theory of knot and link invariants. J. Knot Theory Ramif, 11, 173 (2002) 43. Chiang, T.M., Klemm, A., Yau, S.T., Zaslow, E.: Local mirror symmetry: Calculations and interpretations. Adv. Theor. Math. Phys. 3, 495 (1999) 44. Hosono, S.: Counting BPS states via holomorphic anomaly equations. http://arxiv.org/abs/ hep-th/0206206, 2002 45. Graber, T., Zaslow, E.: Open-string Gromov-Witten invariants: calculations and a mirror ‘theorem’. http://arxiv.org/abs/hep-th/0109075, 2001 46. Klemm, A., Zaslow, E.: Local mirror symmetry at higher genus. In: Winter School on Mirror Symmetry, Vector bundles and Lagrangian Submanifolds, Providence, RI: American Mathematical Society, 2001, p. 183 47. Katz, S., Liu, C-C.: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. Adv. Theor. Math. Phys. 5, 1 (2002) 48. Faber, C.: Algorithms for computing intersection numbers of curves, with an application to the class of the locus of Jacobians. In: New trends in algebraic geometry, Cambridge: Cambridge Univ. Press, 1999 49. Lerche, W., Mayr, P.: On N = 1 mirror symmetry for open type II strings. http://arxiv.org/abs/ hep-th/0111113, 2001; Govindarajan, S., Jayaraman, T., Sarkar, T.: Disc instantons in linear sigma models. Nucl. Phys. B 646, 498 (2002) 50. Mayr, P.: Summing up open string instantons and N = 1 string amplitudes. http://arxiv.org/abs/ hep-th/0203237, 2002 Communicated by N. Nekrasov
Commun. Math. Phys. 254, 479–488 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1253-x
Communications in
Mathematical Physics
Hidden Structure of Symmetries O.I. Bogoyavlenskij Department of Mathematics, Queen’s University, Kingston, K7L 3N6 Canada Received: 19 February 2004 / Accepted: 24 May 2004 Published online: 2 December 2004 – © Springer-Verlag 2004
Abstract: A hidden additional algebraic structure is discovered for the Lie algebra of symmetries of any dynamical system V . The structure is based on the properties of the Lie derivative operator LV and on a hidden canonical flag structure in the eigenspaces of any linear operator. 1. Introduction As is known, the symmetries of dynamical systems and partial differential equations specify important structures of dynamics. In this paper, we show that for any dynamical system x˙ i = V i (x 1 , . . . , x N )
(1.1)
on a smooth manifold M N its Lie algebra of symmetries A itself enjoys a hidden algebraic structure. The structure is described by a canonical sequence of embedded subalgebras that are ideals in A and have special commutation relations. Similar flag structures exist also for the ring of first integrals R and for the complex of differential forms that are invariant with respect to system (1.1). The invariance of a geometric object Y with respect to the dynamical system (1.1) is defined by the equation LV Y = 0, where LV is the operator of the Lie derivative with respect to the vector field V (1.1). We introduce the hidden algebraic structures in the Lie algebras of symmetries A and in the rings of first integrals R which are based on some properties of the Lie derivative operator LV and show that they are manifestations of the hidden canonical flag structures in the eigenspaces of any linear operator. 2. Hidden Structure of Symmetries and First Integrals The Lie algebra of symmetries of system (1.1) consists of all vector fields U on M N that commute with V : [V , U ] = 0 [8, 9]. The Lie algebra A is infinite-dimensional for
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any system (1.1) that has a non-trivial first integral a(x) since for any smooth function F (z) there is a symmetry U = F (a(x))V . Theorem 1. For any dynamical system (1.1), the Lie algebra of symmetries A has a flag structure A ⊇ A 1 ⊇ A2 ⊇ · · · ,
(2.1)
where Lie subalgebras Ap are ideals in A. The inclusions [Ap , Aq ] ⊂ Ap+q
(2.2)
hold. The Lie subalgebras Ap are modules over the ring R of first integrals. The ring R has a flag structure R ⊇ R1 ⊇ R2 ⊇ · · · ,
(2.3)
where all subrings Rk are ideals in R. The inclusions Rk · R ⊂ Rk+
(2.4)
hold. The Lie subalgebras Ak (2.1) define differentiations of the rings R that satisfy the relations A(R ) ⊂ R ,
Ak (R) ⊂ Rk ,
Ak (R ) ⊂ Rk+ .
(2.5)
The subrings R ⊂ R define the R -module structures in Ak for which R · A ⊂ A ,
R · Ak ⊂ Ak+ .
(2.6)
Proof. As is known, an action of the Lie derivative operator LV on a vector field X and on a smooth function f (x) on the manifold M N has the form LV X = [V , X],
LV f (x) = V (f (x)).
Let Ap be the linear subspace of symmetries Up ∈ A that satisfy the equation p
Up = LV Xp = [V , [V , · · · [V , Xp ] · · · ], where Xp is some smooth vector field on M N . For any symmetry Up+1 ∈ Ap+1 we p+1 p have Up+1 = LV Xp+1 = LV (LV Xp+1 ). Hence Ap+1 ⊆ Ap . p q For any two symmetries Up = LV Xp ∈ Ap and Uq = LV Xq ∈ Aq , we have [Up , Uq ] =
p!q! p+q L [Xp , Xq ]. (p + q)! V
Indeed, Eq. (2.7) follows from the generalized Leibnitz formula LnV [X, Y ] =
n k=0
n! [Lk X, Ln−k V Y] k!(n − k)! V
(2.7)
Hidden Structure of Symmetries
481 p+1
q+1
for n = p + q and the equations LV Xp = 0 and LV Xq = 0. The equalities (2.7) prove the inclusions (2.2). Analogously for any symmetry U ∈ A we find [U, Up ] = p LV [U, Xp ], hence all subalgebras Ap are ideals in A. For any first integral a(x), the p equation LV a(x) = 0 and the Leibnitz formula yield a(x)Up = LV (a(x)Xp ). Hence R · Ap ⊂ Ap or all Lie subalgebras Ap are R-modules. Let Rk ⊂ R be a subspace of first integrals ak (x) that have the form ak (x) = LkV fk (x), where fk (x) are some smooth functions on M N . For any first integral ak+1 ∈ Rk+1 we k have ak+1 = Lk+1 V fk+1 = LV (LV fk+1 ). Hence Rk+1 ⊆ Rk . k Let Uk = LV Xk ∈ Ak and a = LV f ∈ R , and U ∈ A be any symmetry. Applying the generalized Leibnitz formula LnV (X(f )) =
n p=0
n! p n−p (L X)(LV f ) p!(n − p)! V
+1 for n = k + and using the equations Lk+1 V Xk = 0 and LV f = 0, we derive
Uk (a ) = (LkV Xk )(LV f ) =
k!! Lk+ (Xk (f )), (k + )! V
U (a ) = LV (U (f )).
Hence the inclusions (2.5) follow. Analogously, for a ∈ R, ak = LkV fk ∈ Rk and a = LV f ∈ R the Leibnitz formula implies aak = LkV (afk ),
ak a =
k!! Lk+ (fk f ). (k + )! V
Hence the subrings Rk are ideals in R and the inclusions (2.4) hold. An application of the generalized Leibnitz formula LnV (f X) =
n p=0
n! p n−p (L f )(LV X) p!(n − p)! V
for n = k + , U ∈ A, Uk = LkV Xk ∈ Ak and a = LV f ∈ R gives the equalities a · U = LV (f U ),
a · U k =
These formulae prove the inclusions (2.6).
k!! Lk+ (f Xk ). (k + )! V
Corollary 1. The quotient Lie algebras Ap /Ap+q are nilpotent for p, q ≥ 1 and are abelian for q ≤ p. If for some N the Lie algebra AN = 0 then all Lie algebras A1 , · · · , AN−1 are nilpotent and for q ≥ [(N + 1)/2] the Lie algebras Aq are abelian. If Lie algebra A1 contains a simple Lie algebra G then all Lie algebras Ap contain G and flag (2.1) is infinite. The proof evidently follows from the commutator relations (2.2) and from the identity [G, G] = G.
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Remark 1. In paper [5], Fuchssteiner introduced the vector fields Zk , k ≥ 2, that satisfy the equations [V , [V , · · · [V , Zk ] · · · ] = LkV Zk = 0, Lk−1 V Zk = 0 and had named them “symmetries of order k”. For k = 2, the vector fields Z2 satisfying equation [V , [V , Z2 ]] = 0, [V , Z2 ] = 0 were named “master symmetries” [5]. The vector fields Zk do not form Lie algebras. The symmetries U1 ∈ A1 are images of the “master symmetries” Z2 under the mapping LV : U1 = LV Z2 and symmetries Uk ∈ Ak are images of the “symmetries of order k + 1” Zk+1 under the mappings LkV : Uk = LkV Zk+1 . Note that the vector fields Zk can be treated also as the generalized zero-eigenvectors for the Lie derivative operator LV . Remark 2. In [2, 3], we introduced conformal symmetries of a dynamical system (1.1) as vector fields Uc satisfying the equation [V , Uc ] = a(x)V ,
(2.8)
where a(x) is an arbitrary first integral of system (1.1). The conformal symmetries Uc (2.8) form a Lie algebra Ac ⊃ A that is a module over the ring R. The Uc define diffeomorphisms that transform trajectories of system (1.1) into other trajectories with a modifed parametrization. First integrals a(x) that appear in Eq. (2.8) for different conformal symmetries Uc form a submodule in the ring R. A system (1.1) has a conformal symmetry Uc (2.8) if and only if a symmetry b(x)V belongs to the Lie subalgebra A1 for some first integral b(x). For the classical Kepler problem, the Lie subalgebra A1 consists of symmetries b(x)V and therefore all master symmetries are conformal symmetries, see Remark 3 below. 3. Examples Example 1. The Kepler problem has the Hamiltonian function H1 (p, q) =
1 GMm (p12 + p22 + p32 ) − . 2m q12 + q22 + q32
(3.1)
Using the Delaunay variables, Poincar´e constructed action-angle coordinates Ij , ϕj , where Hamiltonian H1 (p, q) < 0 has the form [9] H1 (I ) = −KI1−2 ,
K = G2 M 2 m3 /2.
In these coordinates, the Kepler problem dynamics is I˙1 = 0,
I˙2 = 0,
I˙3 = 0,
ϕ˙1 = 2KI1−3 ,
ϕ˙2 = 0,
ϕ˙3 = 0.
(3.2)
Thus the ring of first integrals R consists of arbitrary functions f (Ij , ϕ2 , ϕ3 ). It is evident that all symmetries U of system (3.2) are U = U2
∂ ∂ ∂ ∂ ∂ + U3 + f1 + f2 + f3 , ∂I2 ∂I3 ∂ϕ1 ∂ϕ2 ∂ϕ3
(3.3)
where U 2 , U 3 , f 1 , f 2 , f 3 (Ij , ϕ2 , ϕ3 ) are arbitrary first integrals. Applying Proposition 10 of [2], we find that system (3.2) has no symmetries U2 = L2V X2 (that means A2 = 0) and if LV X1 = U1 ∈ A1 then vector field X1 (a “master symmetry” [5]) has the form X1 = F 1 (Ij , ϕ2 , ϕ3 )
∂ + U, ∂I1
(3.4)
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where F 1 (Ij , ϕ2 , ϕ3 ) is any first integral and U is any symmetry (3.3). Hence we find U1 = LV X1 = [V , X1 ] = 6KF 1 I1−4 ∂/∂ϕ1 = 3F 1 I1−1 V .
(3.5)
Thus for the Kepler problem at H1 < 0, the formulae (3.3) and (3.5) prove that the Lie algebra A of symmetries (3.3) is a free module of rank 5 over the ring R and the Lie subalgebra A1 ⊂ A is a free module of rank 1, and A2 = 0. Remark 3. It is evident that Eq. (3.5) coincides with Eq. (2.8) with first integral a(x) = 3F 1 I1−1 . Hence for the Kepler problem all master symmetries (3.4) are conformal symmetries. Example 2. Let us show that the both flags (2.1) and (2.3) can be infinite. Consider a Hamiltonian system p˙ i = −
∂H , ∂qi
q˙i =
∂H , ∂pi
1 2 pi + (q1 , · · · , qn ), 2 n
H =
(3.6)
i=1
where potential (q1 , · · · , qn ) is a homogeneous function of degree −2, (λq1 , . . . , λqn ) = λ−2 (q1 , · · · , qn ). The equations F˙ = 2H,
H˙ = 0,
F = p1 q1 + · · · pn qn
are evident. Let V denote the vector field (3.6). We have H = LV F /2 ∈ R1 = 0 and hence H k ∈ Rk = 0. Evidently H k V is a nonzero symmetry of system (3.6) and H k V = (2k k!)−1 LkV (F k V ) ∈ Ak = 0 for all integers k > 0. Hence both flags (2.1) and (2.3) are infinite. For example, this is true for the integrable Calogero-Moser system (3.6) with potential (q1 , · · · , qn ) = i =j (qi − qj )−2 . Remark 4. For any system (1.1), the same method proves that flag (2.3) is either infinite or trivial (R1 = 0), and if flag (2.3) is infinite then flag (2.1) is either. The functions a(x) ∈ R1 satisfy the equations a(x) = LV f (x), LV a(x) = 0. Hence along trajectories x(t) of system (1.1) on which a(x(t)) = C = 0 we have f (x(t)) = Ct + C1 , where constant C = 0. Hence if R1 = 0 then system (1.1) either has trajectories entering its singular points in a finite time or some trajectories are unbounded. 4. Integrable Systems Non-Degenerate in the Poincar´e-Kolmogorov Sense Proposition 1. Let system (1.1) be any Hamiltonian system p˙ i = −
∂H , ∂qi
q˙i =
∂H , ∂pi
(4.1)
on a symplectic manifold M 2n which is integrable in Liouville’s sense and has compact invariant submanifolds. Then flag (2.1) has the form A ⊃ A1 ⊃ 0,
A2 = 0,
(4.2)
and the Lie subalgebra A1 is abelian. If the two Lie algebras coincide, A = A1 , then system (4.1) is non-degenerate in the Poincar´e-Kolmogorov sense [7, 9] everywhere where the action-angle coordinates are defined.
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Proof. In Theorem 10 of [2], we proved that for any integrable system (4.1) the equation L3V X = 0 implies L2V X = 0, where X is a vector field on M 2n . Hence any symmetry U2 = L2V X2 ∈ A2 necessarily vanishes and A2 = 0. Since [A1 , A1 ] ⊂ A2 we see that the Lie subalgebra A1 is abelian. Theorem on symmetries [2] proves that an integrable Hamiltonian system (4.1) is almost everywhere non-degenerate in the Poincar´e-Kolmogorov sense if and only if its Lie algebra of symmetries A is abelian. Since the equality A = A1 implies that A is abelian we obtain that the system (4.1) is non-degenerate almost everywhere. This means that in the action-angle variables Ij , ϕj the system has the form I˙j = 0,
ϕ˙j =
∂H (I ) , ∂Ij
det ||
∂ 2 H (I ) || = 0, ∂Ij ∂Ik
(4.3)
where the Hessian determinant is non-zero almost everywhere. Let us prove that if A = A1 then the determinant is everywhere non-zero. Indeed, any symmetry U of system (4.3) has the form U = nk=1 Sk (I )∂/∂ϕk with arbitrary functions Sk (I ) [2]. For any symmetry U1 = LV X1 , the vector field X1 is X1 = nj=1 fj (I )∂/∂Ij + U [2]. Hence we find U1 = [V , X1 ] = −
n k,j =1
fj (I )
∂ 2 H (I ) ∂ . ∂Ij ∂Ik ∂ϕk
(4.4)
Therefore the equality A = A1 implies that the Hessian matrix ∂ 2 H (I )/∂Ij ∂Ik is everywhere non-degenerate or the determinant in (4.3) is non-zero everywhere in the action-angle coordinates Ij , ϕj . Remark 5. Suppose that the Hessian matrix in (4.3) is degenerate at some N points xi where its rank is n − ri < n. Then the vector fields U1 (4.4) at the points xi are not arbitrary and belong to the subspaces of dimensions n−ri . Hence the Lie algebras A and A1 are different and the quotient Lie algebra A/A1 has dimension D ≥ r1 + · · · + rN . 5. Hidden Structure of Eigenspaces I. Let L be a linear operator in a linear space B over an arbitrary field K. Let λ be an eigenvalue of L and Bλ.0 its λ-eigenspace: x ∈ Bλ.0 if Lx = λx. We define a linear subspace Bλ.k ⊂ Bλ.0 as Bλ.k = Bλ.0 ∩ (L − λ)k B.
(5.1)
Since (L − λ)k+1 B ⊆ (L − λ)k B we have Bλ.k+1 ⊆ Bλ.k . Hence we obtain a canonical flag structure in every eigenspace Bλ.0 : Bλ.0 ⊇ Bλ.1 ⊇ Bλ.2 ⊇ · · · .
(5.2)
Remark 6. We note that the invariance equation LV Y = 0 is simultaneously the equation for the zero-eigenvectors of the Lie derivative operator LV . From this point of view, the flag structures (2.1) and (2.3) are special cases of the general flag structure (5.2) for L = LV and λ = 0. II. Let Bλi ⊇ Bλi .0 be the generalized λi -eigenspace that consists of all vectors x ∈ B that are annihilated by an operator (L − λi )k for some k ≥ 1 [4, 6]. Let Bλi .−p ⊆ Bλi
Hidden Structure of Symmetries
485
be the subspace annihilated by (L − λi )p+1 . It is evident that Bλi .−p−1 ⊃ Bλi .−p . By combining this with (5.2) we obtain a flag structure Bλi ⊃ · · · Bλi .−1 ⊃ Bλi .0 ⊇ Bλi .1 · · · ⊇ Bλi .k(i) ,
(5.3)
where Bλi .k(i) is the last non-zero subspace in (5.2) (assume it exists). Proposition 2. The operator (L − λi )p establishes an isomorphism Bλi .p = Bλi .0 ∩ (L − λi )p Bλi = Bλi .−p /Bλi .−(p−1) .
(5.4)
Symmetry principle for FLAG (5.3). The number of distinct subspaces Bλi .−k to the left of Bλi .0 is equal to the number of non-zero subspaces Bλi .k to the right of it. Proof. By the definition, the map (L − λi )p annihilates the subspace Bλi .−(p−1) and sends vectors of Bλi .−p into Bλi .p . If a vector x ∈ Bλi .p = Bλi .0 ∩ (L − λi )p Bλi then x = (L − λi )p y and (L − λi )x = 0. Hence (L − λi )p+1 y = 0 and y ∈ Bλi .−p . The isomorphism (5.4) implies that if Bλi .k(i) = 0 is the last non-zero subspace in the flag (5.2) then Bλi .−k(i)−1 = Bλi .−k(i) . Hence for all p > k(i) we have Bλi .−p = Bλi .−k(i) and therefore Bλi = Bλi .−k(i) . Since the subspaces Bλi .k = 0 for all 0 < k ≤ k(i), the isomorphism (5.4) implies that all subspaces Bλi .−k(i) ⊃ Bλi .−k(i)+1 ⊃ · · · Bλi .−1 ⊃ Bλi .0 are different. Hence flag (5.4) is symmetric with respect to the Bλi .0 . Example 3. Let us consider any dynamical system with quasi-periodic dynamics I˙1 = 0, · · · , I˙p = 0,
ϕ˙1 = ω1 (I ), · · · , ϕ˙q = ωq (I ),
(5.5)
in a toroidal domain O = Cr × Tq , where Cr is a ball of radius r in coordinates I1 , · · · , Ip and Tq is a q-dimensional torus with angular coordinates ϕ1 , · · · , ϕq . Let B be the space of smooth vector fields X on O and operator L = LV , the Lie derivative defined by system (5.5). In Theorem 10 of [2], we proved that for any system (5.5) equation L3V X = 0 implies L2V X = 0. Hence the generalized zero-eigenspace B0 is annihilated by operator L2V and therefore B0 = B0.−1 . Hence by the Symmetry principle the flag (5.3) takes the form B0.−1 ⊃ B0.0 ⊇ B0.1 . In the same Theorem 10 it is proved that for any system (5.5) the space of master symmetries B0.−1 is a Lie algebra and the two Lie algebras B0.−1 and B0.0 = A (the Lie algebra of symmetries) coincide if and only if ωk (I ) = const. III. Let the field K = C and operator L act in a complex space CN and λ1 , · · · , λr be its distinct eigenvalues. Let numbers pi1 ≥ pi2 ≥ · · · ≥ pin(i) be the dimensions of the Jordan λi -blocks in the Jordan normal form of the operator L. The following Segre characteristic [4] describes the Jordan blocks structure for the linear operator L: λ2 ··· λr λ1 {(p11 , · · · , p1n(1) ) (p21 , · · · , p2n(2) ) · · ·
(5.6) (pr1 , · · · prn(r) )}.
It is evident that dimension of the generalized eigenspace Bλi is di = pi1 + pi2 + · · · + pin(i) and Bλi is annihilated by (L − λi )pi1 . Hence Bλi = Bλi .pi1 −1 and the symmetry
486
O.I. Bogoyavlenskij
principle for flag (5.3) gives k(i) = pi1 − 1 where k(i) is the number of non-zero subspaces Bλi .k , k > 0. The partition di = pi1 + pi2 + · · · + pin(i) can be visualized from the Ferrers diagram [1] in Fig. 1. For the operator L in CN we assign another characteristic that describes the flag structures (5.2) for different eigenvalues λi : λ1 λ2 ··· {(q10 , · · · , q1k(1) ) (q20 , · · · , q2k(2) )
λr ···
(5.7) (qr0 , · · · qrk(r) )},
where qik = dim Bλi .k ; qi0 ≥ qi1 ≥ · · · ≥ qik(i) . The advantage of the characteristic (5.7) is that the numbers qik have invariant geometrical meaning: qik are the dimensions of the invariantly defined subspaces Bλi .k (5.1) of the eigenspaces Bλi .0 . The numbers qik = dim Bλi .k form the conjugate partition di = dim Bλi = qi0 + qi1 + · · · + qik(i) with respect to the partition di = pi1 + pi2 + · · · + pin(i) . Indeed, the isomorphisms (5.4) evidently imply the equality di = dim Bλi = qi0 + qi1 + · · · + qik(i) . It is evident that qik for k ≥ 0 is equal to the number of such Jordan λi -blocks that their sizes pi ≥ k + 1. This amounts to the definition of the conjugate partition [1] for the original one di = pi1 + pi2 + · · · + pin(i) . Figure 2 shows the corresponding Ferrers diagram. Hence Segre characteristic (5.6) and the characteristic (5.7) can be obtained from each other as the conjugate partitions di = pi1 + pi2 + · · · + pin(i) = qi0 + qi1 + · · · + qik(i) .
7 6 6 4 1
Fig. 1. Ferrers diagram for the Segre characteristic for an eigenvalue λi . Here pi1 = 7, pi2 = 6, pi3 = 6, pi4 = 4, pi5 = 1. The corresponding partition is di = dim Bλi = 24 = pi1 + pi2 + pi3 + pi4 + pi5
5
4
4
4
3
3
1
Fig. 2. Ferrers diagram for the conjugate partition di = dim Bλi = 24 = qi0 + qi1 + qi2 + qi3 + qi4 + qi5 + qi6 . Here qi0 = 5, qi1 = 4, qi2 = 4, qi3 = 4, qi4 = 3, qi5 = 3, qi6 = 1, and k(i) = 6
Hidden Structure of Symmetries
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Example 4. Let operator L in a basis e1 , · · · , eN have one Jordan block 0 λ1 0 . . . 1 λ1 . . A= . . λ1 0 0 . . . 0 1 λ1 Hence Segre characteristic is p11 = N . It is evident that the λ1 -eigenspace Bλ1 .0 = C1 is generated by vector eN . Since (B − λ1 )k eN−k = eN for k = 1, · · · , N − 1 we see that flag (5.2) has N − 1 subspaces Bλ1 .k isomorphic to C1 and Bλ1 .N = 0. Hence q10 = q11 = · · · = q1.N −1 = 1 and k(1) = N − 1 = p11 − 1. The generalized eigenspace Bλ1 is Bλ1 .−(N−1) = CN . Hence flag (5.3) takes the form CN ⊃ CN−1 ⊃ · · · C2 ⊃ C1 = C1 = · · · = C1 . The corresponding Ferrers diagram consists of one row of length N . 6. Conclusions We have introduced the hidden algebraic structure (2.1) – (2.2) in the Lie algebra A of symmetries of any dynamical system (1.1). The structure is based on the properties of the Lie derivative operator LV . A similar algebraic structure (2.3) – (2.4) exists also in the ring R of first integrals. The flag structure (2.1) has the form (4.2) for any Liouvilleintegrable Hamiltonian system (4.1) with compact invariant submanifolds, for example for the classical Kepler problem (3.1) at H < 0. For any Hamiltonian system (3.6) with a homogeneous potential function (q1 , · · · , qn ) of degree −2, both flags (2.1) and (2.3) are infinite. The flag structures (2.1), (2.3) are manifestations of the hidden canonical flag structures (5.2) in the eigenspaces Bλi .0 of any linear operator L in a linear space B. The structures are closely connected with the structure of the subspaces Bλi ⊂ B of generalized λi -eigenvectors. For operators L in a complex space CN , the numbers qik = dim Bλi .k for the flag (5.2) form a partition of the number dim Bλi = qi0 + · · · + qik(i) . The partition is conjugate to the classical Segre partition dim Bλi = pi1 + · · · + pin(i) [4, 6] defined by the Jordan blocks decomposition. Hence the flag structures (5.2) in the eigenspaces Bλi .0 contain all information about the Jordan normal form of the linear operator L in CN . References 1. Andrews, G.E.: The theory of partitions. London: Addison-Wesley Publishing Co., 1976 2. Bogoyavlenskij, O.I.: Theory of tensor invariants of integrable Hamiltonian systems. II. Theorem on symmetries and its applications. Commun. Math. Phys. 184, 301–365 (1997) 3. Bogoyavlenskij, O.I.: Conformal symmetries of dynamical systems and Poincar´e 1892 concept of iso-energetic non-degeneracy. C. R. Acad. Sci. Paris 326, 213–218 (1998) 4. Finkbeiner, D.T.: Introduction to matrices and linear transformations. San Francisco: W. H. Freeman and Co., 1978 5. Fuchssteiner, B.: Mastersymmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations. Progr. Theor. Phys. 70, 1508–1522 (1983) 6. Gantmacher, F.R.: The theory of matrices. 1. New York: Chelsea Publ. Co., 1960 7. Kolmogorov, A.N.: The general theory of dynamical systems and classical mechanics. In: Proc. Intern. Congr. Math. 1954, 1. Amsterdam: North-Holland Publ. Co., 1957, pp. 315–333
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8. Marsden, J.E., Ratiu, T.S.: Introduction to mechanics and symmetry. New York: Springer Verlag, 1999 9. Poincar´e, H.: Les m`etodes nouvelles de la m`echanique celeste. T. 1. Paris: Gauthier - Villars, 1892 Communicated by L. Takhtajan
Commun. Math. Phys. 254, 489–512 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1163-y
Communications in
Mathematical Physics
Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions John Gough Department of Computing & Mathematics, Nottingham-Trent University, Burton Street, Nottingham NG1 4BU, United Kingdom. E-mail:
[email protected] Received: 23 February 2004 / Accepted: 8 March 2004 Published online: 27 August 2004 – © Springer-Verlag 2004
Abstract: We consider a Markovian approximation, of weak coupling type, to an open system perturbation involving emission, absorption and scattering by reservoir quanta. The result is the general form for a quantum stochastic flow driven by creation, annihilation and gauge processes. A weak matrix limit is established for the convergence of the interaction-picture unitary to a unitary, adapted quantum stochastic process and of the Heisenberg dynamics to the corresponding quantum stochastic flow: the convergence strategy is similar to the quantum functional central limits introduced by Accardi, Frigerio and Lu [1]. The principal terms in the Dyson series expansions are identified and re-summed after the limit to obtain explicit quantum stochastic differential equations with renormalized coefficients. An extension of the Pul´e inequalities [2] allows uniform estimates for the Dyson series expansion for both the unitary operator and the Heisenberg evolution to be obtained. 1. Introduction In the interaction picture, the unitary Ut arising from a time-dependent perturbation Vt , is given by t exp −i Ut = T (1.1) ds Vs , 0
is Dyson’s time-ordering operation. A principal aim of quantum field theory is where T then to obtain a normal-ordered version of Ut . When Vt involves a sum of monomials of canonical quantum fields, we may use Feynman rules to expand Ut : we associate a vertex to each monomial, with the number of legs corresponding with the degree; we then construct the class F of Feynman diagrams consisting of such vertices with certain legs contracted (internal lines) and the remainder free (external lines); we then specify a rule for writing down an operator LG (t) which, for each G ∈ F, will be a
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normal-ordered product of the fields associated to the external lines of G. We then deter mine a development of the form Ut = G∈F LG (t). Now, if G can be decomposed as G1 LG2 , where N is Wick’s two disconnected sub-diagrams G1 and G2 , then LG = NL normal-ordering operation. This leads to a second presentation of Ut : exp Ut = N
G∈FC
(1.2)
LG (t) ,
where FC is the class of connected Feynman diagrams. If, in place of quantum fields, we considered quantum white noises, then the timeordered presentation corresponds to a Stratonovich form while the normal-ordered presentation corresponds to an Itˆo form. Our aim is not to justify this statement, for which there is ample support [3–5], but to prove an asymptotic result which, effectively, is an analogue of the Wong-Zakai theorem for classical stochastic processes. The interaction that we shall be interested in is given below as (1.8), and is quadratic in the reservoir creation/annihilation operator fields at± (λ): the corresponding connected Feynman diagrams will have at most two legs and therefore will be linear chains. These describe a reservoir quanta created, subsequently multiply-scattered (i.e., at several times annihilated and immediately re-created) and finally reabsorbed: external lines may also be present. We shall be interested, not in the S-matrix limit t → ∞, but in the more subtle van Hove [6], or weak coupling, limit where we rescale time as t/λ2 with λ a coupling strength parameter appearing in Vt and consider the limit λ → 0 with t fixed. The fields at± (λ) will converge, in a sense to be spelled out below, to quantum white noises: more correctly, integrated versions of these fields converge to the fundamental quantum stochastic processes of Hudson and Parthasarathy’s theory [7]. The van Hove limit turns out to have a dominant contribution from Feynman diagrams where there is no overlap in the time ranges of the individual connected subgraphs: these are the so-called type I terms. All other terms (type II) are suppressed. A similar feature is observed for the limit of the dynamical flow of observables.
1.1. The Classical Wong-Zakai Theorem. Wong and Zakai [8] studied Langevin type equations driven by differentiable noises ξt (λ) having correlation ξt (λ) ξs (λ) = 1 which became delta-correlated only in the limit λ → 0. They found that G t−s λ2 λ2 the limit dynamics was described by a stochastic differential equation taking the same form as the pre-limit equations in the Stratonovich calculus. Let us specialize to the flow on a symplectic manifold generated by a random Hamiltonian (λ)
ϒt
=H+
Fα ξtα (λ) ,
(1.3)
α
where H and Fα are smooth functions on phase space and ξtα (λ) are differentiable sto(λ) chastic processes converging to independent white noises. If xt is the phase
trajectory (λ)
starting from x0 then the evolution of functions is Jt
(λ)
(f ) := f xt
. In the limit
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λ → 0 we obtain, in accordance with the Wong-Zakai result, the Stratonovich-Fisk equation dJt (·) = Jt {·, H } dt + Jt {·, Fα } ◦ dBtα , (1.4) α
Btα
where are independent Wiener processes and the differential is of Stratonovich type: here we may view the motion as that governed by the formal Hamiltonian ϒt = H + α Fα ξtα , where ξtα are white noises. A general treatment of these problems using the van Hove limit is well-understood [9]. These are the stochastic flows that preserve the Poisson bracket structure [10]. Averaging with respect to the Wiener measure, we obtain the dynamical semigroup E [Jt (·)] ≡ exp {tL (·)}. From the Itˆo calculus, the generator will be the hypo-elliptic operator {{·, Fα } , Fα } + {·, H } L (·) = (1.5) α
which is already displayed in H¨ormander form. 1.2. Quantum Markov Approximations. It was first suggested by Spohn [11] that the weak coupling limit should be properly considered as a Markovian limit underscored by a functional central limit. The rigorous determination of irreversible semigroup evolutions has been given for specific models [12, 2]. (A detailed account of the derivation of the master equation for a class of quantum open systems is given in Davies’ book [13].) The form of the generator of quantum dynamical semigroups was deduced [14, 15] using the guiding principle that the semi-group be completely positive. Hudson and Parthasarathy [7] subsequently developed a quantum stochastic calculus giving an Itˆo theory of integration with respect to Bosonic Fock space processes and demonstrated how to construct dilations of the quantum dynamical semigroups mentioned above using a Fock space as auxiliary space. The program now is to begin with a microscopic model for a system-reservoir interaction and then obtain by some Markovian limit procedure, such as the weak coupling limit, a quantum stochastic evolution. It was first noted by von Waldenfels [16] that stochastic models successfully describe the weak coupling limit regime for the Wigner-Weisskopf atom. Later, Accardi, Frigerio and Lu [1] showed how to do this for an interaction of (λ) the type ϒt = E10 ⊗ at+ (λ) + E01 ⊗ at− (λ), where E10 and E01 are bounded, mutually adjoint operators on the system space hS and at± (λ) are creation/annihilation fields having a correlation + 1 t −s − at (λ) as (λ) = 2 G , (1.6) λ λ2 where G In the sense of Schwartz +∞ distributions, we have (.) is integrable. limλ→0 at− (λ) as+ (λ) = γ δ (t − s), where γ = −∞ dt G (t) is finite. We shall also take an interest in the constants ∞ 0 ∞ κ ≡ κ+ := dt G (t), κ− := dt G (t) and K := dt |G (t)| . (1.7) 0
−∞
∗
0
We shall assume that G (−t) = G (t) so that κ± ≡ ± iσ . Already in [1], several important steps were taken: to begin with, there is the anticipation of the limit algebraic 1 2γ
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structure by means of a quantum functional central limit theorem which captures the long time asymptotic behaviour; secondly, there is the identification of the principal, type I, terms in the Dyson series which survive the Markovian limit (they are the ones arising from only time-consecutive two-point contractions); finally, there is a rigorous estimate of the Dyson series expansion employing an argument due to Pul´e [2].
1.3. Statement of the problem. Our aim is to extend this result in [1] to the more general class of interactions (λ)
ϒt
= E11 ⊗ at+ (λ) at− (λ) + E10 ⊗ at+ (λ) + E01 ⊗ at− (λ) + E00 ⊗ 1 α β (1.8) = Eαβ ⊗ at+ (λ) at− (λ) .
(We introduce the summation convention that when the Greek indices α, β, . . . are repeated then we sum each index over the values 0 and 1 - moreover we understand the index α in [.]α to represent a power.) We require only the conditions that the system operators Eαβ are bounded with K E11 < 1, where K is the constant introduced in (1.7). The interaction includes a scattering term, E11 ⊗ at+ (λ) at− (λ), and a constant term. The terms involving E01 and E10 describe the emission and absorption of reservoir quanta and this component has been employed in models of laser interactions [17]. The constant term is of little consequence as we shall take it to commute with the free Hamiltonian. However, the scattering term is highly non-trivial: we have to contend with emission, multiple scatterings and absorption. This means that the number of terms in the Dyson series expansion of (λ) Ut
t (λ) = T exp −i ds ϒs
(1.9)
0
grows rapidly (in fact, as the Bell numbers of combinatorics [21] ). However, we are able to prove a uniform estimate of the Dyson series expansion by a generalization of the Pul´e inequalities, which we give in Sect. 7. We are then able to re-sum the series to obtain an adapted, unitary process Ut of Hudson-Parthasarathy type (Theorem 8.1). The type of limit involved is of a weak character and is often referred to as convergence in matrix elements. (λ) (λ)† (λ) We show that the Heisenberg evolution Jt (X) = Ut (X ⊗ 1R ) Ut likewise converges in weak matrix elements, for fixed bounded observables X ∈ B (hS ), to Jt (X) = Uλ† t (X ⊗ 1R ) Ut (Theorem 10.1). We are able to obtain the quantum stochastic differential equations satisfied by Ut and by the flow Jt . In particular, these equations will involve a gauge differential (due to the scattering) as well as creation, annihilation and time. In particular, we compute the Lindblad generator for the flow. We remark that interactions of the type (1.8) were considered previously in the case where the coefficients Eαβ were commuting operators [18], and Fermionic operators [19]. In the former case, a strong resolvent limit was established for the common spectral resolution, while in the latter, the anti-commutation relations kill off all but type I terms.
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2. Moments and Cumulants Let (h) be the (Bose) Fock space over the one-particle Hilbert h. The Fock vacuum will be denoted by and the exponential vector map by ε : h → (h). As usual ε (0) = . We denote the creation fields as A+ (·), the annihilation fields as A− (·) and the differential second quantization field as d (·), as standard. The Weyl operator with test function f is W (f ) := exp A+ (f ) − A− (f ) and we have the Weyl map W (·). As is well-known, the fields Q (·) = A+ (·) + A− (·) are Gaussian random fields when taken in the Fock vacuum state. More generally, we have [20] | exp it d (H ) + A+ (Hf ) + A− (Hf ) + f |Hf
f = exp eitx − 1 dµH (dx) , f
where H is self-adjoint on h with spectral measure µH for vector state f ∈ h. This time, f we are dealing with Poissonian fields. We remark that if µH = λδ1 , then we obtain a random variable with Poisson distribution of intensity λ > 0:
(it)n exp λ eit − 1 = S (n, m) λm . n! n m 1 m l+m n m The coefficients S (n, m) = m! l l are well-known combinatorial facl=1 (−1) tors: they are the Stirling numbers of the second kind [21] and they count the number of ways of partitioning a set of n items into m non-empty subsets. The expansion of Poissonian field moments in terms of cumulants, or more generally the expansion of Green’s functions in terms of their connected Green’s functions, can best be described in the language of partitions [22]. A partition of the integers {1, . . . , n} is a collection of non-empty, disjoint subsets (called parts) whose union is {1, . . . , n}. The set of all such partitions will be denoted as Pn : there will be S (n, m) partitions of {1, . . . , n} having exactly m parts and Bn = m S (n, m) partitions of {1, . . . , n} in total. Bn are called the Bell numbers [21]. Lemma 2.1. Let f1 , g1 , . . . , fn , gn ∈ h. Then α(n) − β(n) α(1) − β(1) | A+ (fn ) · · · A+ (f1 ) A (gn ) A (g1 ) α,β∈{0,1}n
=
gi(k) |fi(k−1) · · · gi(3) |fi(2) gi(2) |fi(1) ,
(2.1)
A∈Pn {i(1),...,i(k)}∈A
where we take the various sets (parts of the partition) {i (1) , . . . , i (k)} ∈ A to be ordered so that i (1) < i (2) < · · · < i (k) and if the set is a singleton it is given the factor of unity. Proof. If α (i) = 0, 1, then we have the absence, respectively presence, of the creator A+ (fi ). Likewise β (i) gives the absence or presence of the i th annihilator. Evidently we must have α (n) = 0 = β (1). α(i) Essentially we have a vacuum expectation of a product of n factors A+ (f1 ) − β(i) and this ultimately when put to normal order will be a sum of terms each A (g1 )
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of which is a product of pair contractions gi |fk where i > k. For a given term in the sum we write i ∼ k if gi |fk appears. An equivalence relation is determined by a set of contractions as follows: we always have i ≡ i and, more generally, we have i ≡ k if there exists a sequence j (1) , . . . j (r) such that either i ∼ j (1) ∼ j (2) ∼ · · · j (r) ∼ k or k ∼ j (1) ∼ j (2) ∼ · · · j (r) ∼ i. A partition A in Pn is then obtained by looking at the equivalence classes. (Singletons are just the unpaired labels.) The correspondence between the terms in the sum and the elements of Pn is one-to-one and the weight given to a particular partition A ∈ Pn is just the product of gi |fk ’s given in (2.1). We remark that can be considered as a special case of the expansion (2.1) G (x1 , . . . , xn ) = A∈Pn {i(1),... ,i(k)}∈A C xi(1) , · · · , xi(k) of an n-particle Green’s function G in terms of the connected Green’s functions C. Let us write P for the set ∪n Pn of finite partitions. ∞ each partition A ∈Pn we With associate a sequence of occupation numbers n = nj j =1 , where nj = 0, 1, 2, . . . counts the number of j -tuples making up A. In general, we set E (n) := j nj , N (n) := nj (2.2) j
j
so that if A ∈Pn leads to sequence n, then E (n) = n, while N (n) counts the number of parts making up the partition. We shall denote by Pn the set of all partitions having the same occupation number sequence n. Given a partition A ∈Pn we use the convention q (j, k, r) to label the r th element of the k th j -tuple. A simple example of a partition in Pn is given by selecting in order from {1, 2, . . . , E (n)} first of all n1 singletons, then n2 pairs, then n3 triples, etc. The labelling for this particular partition will be denoted as q¯ (., ., .) and explicitly we have q¯ (j, k, r) = l nl + (k − 1) nj + r. (2.3) l<j
Definition 2.2. We shall denote by S0n the collection of Pul´e permutations, that is, ρ ∈ Sn , E (n) = n, such that q = ρ ◦ q¯ again describes a partition in Pn . Specifically, S0n consists of all the permutations ρ for which the following requirements are met: i) the order of the individual j -tuples is preserved for each j ρ (q¯ (j, k, 1)) < ρ q¯ j, k , 1 ∀j, 1 ≤ k < k ≤ nj ;
(2.4)
ii) creation always precedes annihilation in time for any contraction pair ρ (q¯ (j, k, 1)) < ρ (q¯ (j, k, 2)) < · · · < ρ (q¯ (j, k, j ))
∀j, 1 ≤ k ≤ nj . (2.5)
In these notations we may rewrite the result of Lemma (2.1) as: Lemma 2.3. Let f1 , g1 , . . . , fn , gn ∈ h. Then α(n) − β(n) α(1) − β(1) · · · A+ (f1 ) | A+ (fn ) A (gn ) A (g1 ) α,β∈{0,1}n
=
E(n)=n n
nj j −1
|fρ(q(j,k,r)) gρ(q(j,k,r+1)) . ¯ ¯
ρ∈S0n j ≥2 k=1 r=1
(2.6)
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3. A Microscopic Model We shall consider a quantum mechanical system S (state space hS ) coupled to a Bose quantum field reservoir R over a one-particle space h1R (state space hR = h1R ). We shall take the reservoir to be in the Fock vacuum state . The interaction between the system and the reservoir will be given by the formal Hamiltonian (λ) H (λ) = HS ⊗ 1R + 1S ⊗ d HR1 + HInt ,
(3.1)
where the operators HS and HR1 are self-adjoint and bounded below on hS and h1R , respectively. The interaction is taken to be HInt = E11 ⊗ A+ (g) A− (g) + λE10 ⊗ A+ (g) + λE01 ⊗ A− (g) + λ2 E00 ⊗ 1R , (3.2) (λ)
† . where Eαβ are bounded operators on hS with E11 and E00 self-adjoint and E10 = E01 + − The operators A (g) and A (g) are the creation and annihilation operators with test function g ∈ h1R . (The parameter λ is real and will later emerge as a rescaling parameter in which we hope to obtain a Markovian limit.) We shall also assume the following harmonic relations
e+iτ HS Eαβ e−iτ HS = eiωτ (β−α) Eαβ , −iτ HR e+iτ HR A± = A± R (g) e R (θτ g) ,
(3.3)
where (θτ : τ ∈ R) will be the one-parameter group of unitaries on h1R with Stone generator HR1 . We transfer to the interaction picture with the help of the unitary U (τ, λ) = e+iτ (HS ⊗1R +1S ⊗HR ) e−iτ H . (λ)
(3.4)
In the weak coupling regime, we are interested in the behaviour at long time scales (λ) τ = t/λ2 and from our earlier specifications we see that Ut = U t/λ2 , λ satisfies the interaction picture Schr¨odinger equation ∂ (λ) (λ) U = −i ϒt (λ) Ut ∂t t
(3.5)
with ϒt (λ) as in (1.8). Here we meet the time-dependent rescaled reservoir fields 1 ∓iωt/λ2 ± e A θt/λ2 g . (3.7) λ 1 +∞ Specifically we have γ = −∞ dτ g, eiτ HR −ω g = 2π g, δ HR1 − ω g and κ+ = 1 1 1 g , where PV denotes the prin g = 2 γ − i PV g, 1 g, 1 i HR − ω − i0+ HR − ω ciple value part. at± (λ) :=
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4. Quantum Central Limit The limit λ → 0 for the above, the two-point function becomes delta-correlated. However, it is vital to have a mathematical framework in which to interpret the limit states and observables. For convenience we set
θτω := exp iτ HR1 − ω . (4.1) ∞ ! ! We assume the existence of a subspace, k ⊂ h1R for which −∞ ! fj , θυω fk ! du < ∞ whenever fj , fk ∈ k. (In reference [1], explicit examples of dense subspaces, k, are given and correspond to “mass-shell” Hilbert spaces.) The question of completeness can be addressed immediately: a sesquilinear form on k is defined by ∞
fj |fk := (4.2) fj , θυω fk du ≡ 2π fj , δ HR1 − ω fk , −∞
and we can quotient out the null elements for this form; the completed Hilbert space will again be denoted by k and (.|.) will be its inner product. The test vector g appearing in the interaction must belong to k so that the constant γ ≡ (g|g) is finite. Let W (·) be the Weyl map from h1R as before. We now fix fj ∈ k and 0 ≤ Sj < Tj < ∞ for certain indices j and introduce the rescaled operators # " Tj
Tj 1 1 ω ω . (4.3) du A± du θu/λ A± 2 fj λ (j ) := R θu/λ2 fj , Wλ (j ) := W λ Sj λ Sj Note that, with respect to our earlier notations (3.7), if fj = g then A± λ (j ) ≡ Tj ± (λ). The following result is proved as Lemma 3.2 in Accardi, Frigerio and du a u Sj Lu [1]. We write 1[S,T ] for the characteristic function of an interval [S, T ]. Lemma 4.1. For the fields introduced in (4.3), lim A− (j ) , A+ (k) = fj |fk 1[Sj ,Tj ] , 1[Sk ,Tk ] . λ λ λ→0 The right-hand side is the inner product fj ⊗ 1[Sj ,Tj ] , fk ⊗ 1[Sk ,Tk ] on the Hilbert space k ⊗ L2 R+ . This space is isomorphic in a natural way to the k -valued square integrable functions on R+ and we denote this space as L2 R+ , k . noise space for the limit λ → 0 will in fact be the Bose Fock space The appropriate
L2 R+ , k . Indeed, we have the following fact proved as Theorem 3.4 in [1]. Theorem 4.2. Let be the Fock vacuum for L2 R+ , k and let W (.) denote the usual Weyl mapping from L2 R+ , k into the unitaries on L2 R+ , k . Then lim | Wλ (1) . . . Wλ (k) = | W f1 ⊗ 1[S1 ,T1 ] . . . W fk ⊗ 1[Sk ,Tk ] λ→0
for arbitrary k and fj ∈ k and 0 ≤ Sj < Tj < ∞.
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(λ)
5. The Dyson Series Expansion of Ut
(λ) n The formal Dyson series development Ut = ∞ n=0 (−i) Dn (t, λ) involves the multiple time integrals Dn (t, λ) = dsn . . . ds1 ϒsn (λ) . . . ϒs1 (λ) . (5.1) n (t)
For σ ∈ Sn , we introduce the simplex σn (t) := (sn , . . . , s1 ) : t > sσ (n) > · · · > sσ (1) > 0
(5.2)
and n (t) in (5.1) is the simplex corresponding to the identity permutation. (λ) We consider matrix elements of the type φ1 ⊗ Wλ (1) | Ut φ2 ⊗ Wλ (2) with φj ∈ hS and Wλ (j ) as in (4.3). Substituting for the Dyson series, we find that the nth term can be rewritten as an expectation involving the vacuum state only: φ1 ⊗ Wλ (1) | ϒsn (λ) . . . ϒs1 (λ) φ2 ⊗ Wλ (2) = φ1 ⊗ | ϒ˜ sn (λ) . . . ϒ˜ s1 (λ) φ2 ⊗ Wλ (1) |Wλ (2) , (5.3) where ϒ˜ s (λ) is obtained from ϒs (λ) by the canonical translations at+ (λ) → at+ (λ) + h1 (t, λ) ; at− (λ) → at− (λ) + h∗2 (t, λ) with hj (t, λ) =
1 λ2
Tj Sj
ω ω du θu/λ 2 fj |θt/λ2 g .
(5.4)
(5.5)
That is, α β ϒ˜ s (λ) = E˜ αβ (t, λ) ⊗ at+ (λ) at− (λ) ,
(5.6)
E˜ 00 (t, λ) = E00 + E10 h1 (t, λ) + E01 h∗2 (t, λ) + E11 h1 (t, λ) h∗2 (t, λ) ; E˜ 10 (t, λ) = E10 + h∗2 (t, λ) E11 ; E˜ 01 (t, λ) = E01 + h1 (t, λ) E11 ; E˜ 11 (t, λ) = E11 .
(5.7)
where
In this way we see that the nth term in the Dyson series expansion of the matrix element is, up to the factor (−i)n Wλ (1) |Wλ (2) , dsn . . . ds1 φ1 | E˜ αn βn (sn , λ) . . . E˜ α1 β1 (s1 , λ) φ2 n (t) α β α β (5.8) × | as+n (λ) n as−n (λ) n . . . as+1 (λ) 1 as−1 (λ) 1 and our summation convention is now in place. The vacuum expectation can be computed using Lemmas (2.1) or (2.3). The resulting terms can be split into two types: type I will survive the λ → 0 limit; type II will not. They are distinguished as follows:
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Type I: Terms involving contractions of time consecutive annihilator-creator pairs only. (That is, under the time-ordered integral in (5.8), an annihilator as−j +1 (λ) must be contracted with the creator as+j (λ).) Type II: All others cases. The terminology used here is due to Accardi, Frigerio and Lu [1].
6. Principal Terms in the Dyson Series A standard technique in perturbative quantum field theory and quantum statistical mechanics is to develop a series expansion and argue on physical grounds that certain “principal terms” will exceed the other terms in order of magnitude [23]. Often it is possible to re-sum the principal terms to obtain a useful representation of the dominant behaviour. Mathematically, the problem comes down to showing that the remaining terms are negligible in the limiting physical regime being considered. Let n be a positive integer and m ∈ {0, . . . , n − 1}. Let { pj , qj }m j =1 be contraction pairs over indices {1, . . . , n} such that if P = {p1 , . . . , pm } and Q = {q1 , ..., qm } then P and Q are both non-degenerate subsets of size m and we require that pj < qj for each j and that Q be ordered so that q1 < ... < qm . We understand that (pj , qj )m j =1 is type I if qj = pj + 1 for each j and type II otherwise. The following result is an extension of Lemma 4.2 in Accardi, Frigerio and Lu [1] as now P ∩ Q need not be empty. Lemma 6.1. Let (pj , qj )m j =1 be a set of m pairs of contractions over indices {1, . . . , n}, then ! ! ! m !! ! γ m t n−m ! as− p (λ) as+ q (λ) !! ≤ . ds1 . . . dsn ! ( j) ( j) ! (n − m)! ! n (t) j =1
(6.1)
Moreover, as λ → 0, ds1 . . . dsn n (t)
m j =1
as− p (λ) as+ q (λ) ( j) ( j)
$ →
m t n−m κ+ (n−m)! ,
0,
type I . type II
(6.2)
Proof. Let q = q1 and set t (q) = [s (p) − s (q)] /λ2 then ! ! ! m !! ! ! as− p (λ) as+ q (λ) !! ds1 . . . dsn ! ( j) ( j) (t) ! ! n j =1 ! s(p)−λ2 t(q) 2 ! t s(q−2) s(p)/λ ! = ! ds (1)· · · ds (q − 1) dt (q) ds (q + 1) . . . ! 0 0 0 [s(p)−s(q−1)]/λ2 ! s(n−1) m !! ω ... as− p (λ) as+ q (λ) !! . ds (n) g, θt(q) g ( j) ( j) 0 ! j =2
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However, we have that s (p) − λ2 t (p) < s (q − 1) and so we obtain the bound ! s(q−2) ∞ s(p)−λ2 t(q) ! t ! ds (q − 1) dt (q) ds (q + 1) ! ds (1)· · · ! 0 0 −∞ 0 ! s(n−1) m !! ω ... as− p (λ) as+ q (λ) !! . ds (n) g, θt(q) g ( j) ( j) 0 ! j =2
And so, working inductively we obtain (6.1). Suppose now that the pairs are of type I, then p = q − 1 and so the lower limit of the t (q)-integral is zero. Consequently, we encounter the sequence of integrals s(q−2) s(q−1/λ2 s(q−1)−λ2 t (q) ··· ds(q − 1) dt (q) ds(q + 1) . . . g, θtω(q) g . . . ; 0
0
0
this occurs for each q-variable and so we recognize the limit as stated in (6.2) for type I terms. If the pairs are of type II, on the other hand, then let j = min {k : pk < qk − 1}; setting q = qk , we encounter the sequence of integrals s(q−2) s(p)/λ2 s(q−1) ω ··· ds (q − 1) dt (q) ds (q + 1) . . . g, θt(q) g ..., [s(p)−s(q−1)]/λ2
0
0
but now, with respect to the variables s(1), ..., s(p), ..., s(q − 1) we have that since 2 s (p) = s (q − 1), the lower limit [s (p) − s (q − 1)] /λ of the t (q)-integral is almost ω always negative and so, as t → g, θt g is continuous, we have the dominated convergence of the whole term to zero. Clearly type II terms do not contribute to the nth term in the series expansion in the limit. However, we must establish a uniform bound for all these terms when the sum over all terms is considered. We do this in the next section. Before proceeding let us remark that the expression (5.8) is bounded by Cαn βn . . . Cα1 β1 φ1 φ2 % & + αn − βn α1 − β1 + × asn (λ) . . . as1 (λ) as1 (λ) , (6.3) dsn . . . ds1 asn (λ) n (t)
where C11 = E11 , C10 = E10 + E11 h2 ; C01 = E01 + E11 h1 , (6.4) C00 = E00 + E10 h1 + E01 h2 + E11 h1 h2 , ! ω ! ! ω ! ∞ ∞ and h1 = −∞ du ! g|θu f1 !, h2 = −∞ du ! g|θu f2 !. Recall that we require that KC11 < 1 and that C = max {C11 , C10 , C01 , C00 } < ∞. We need to do some preliminary estimation. We employ the occupation numbers introduced in Sect. 2. The number of times that we will have (α, β) = (1, 1) in a particular term will be j >2 (j − 2) nj (that is, singletons and pairs have none, triples have
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one, quadruples have two, etc.) and this equals E (n) − 2N (n) + n1 . Therefore, we shall have E(n)−2N(n)+n1
Cαn βn . . . Cα1 β1 ≤ C11
C 2N(n)−n1 .
(6.5)
7. Generalized Pul´e Inequalities Putting all this together we get the bound Cαn βn . . . Cα1 β1 dsn . . . ds1 n (t) α β α β × | as+n (λ) n as−n (λ) n . . . as+1 (λ) 1 as−1 (λ) 1 ≤
E(n)=n
n
ρ∈S0n
E(n)−2N (n)+n1
C11
×
dsn . . . ds1 n (t)
C 2N(n)−n1
nj j −1
, Gλ sρ(q(j,k,r+1)) − sρ(q(j,k,r)) ¯ ¯
(7.1)
j ≥2 k=1 r=1
where we use the estimate (6.5) and we obtain the sum over all relevant terms by summing over all admissible permutations of the basic q¯ term. To estimate the simplicial integral we generalize an argument due to Pul´e (Lemma 3 of [2]). Let ρ˜ be the induced mapping on Rn obtained by permuting the Cartesian coordinates according to ρ ∈ S0n . Then the bound in (7.1) can be written as E(n)=n
E(n)−2N(n)+n1
C11
C 2N(n)−n1
nj j −1
n
×
dsn . . . ds1 R
, Gλ sq(j,k,r+1) − sq(j,k,r) ¯ ¯
(7.2)
j ≥2 k=1 r=1
˜ n (t) where R = ∪ ρ ˜ n (t) : ρ ∈ S0n . This is down to the fact that the image sets ρ will be distinct for different ρ ∈ S0n . Now the region, R, of integration is a subset of are ordered primarily by the index j and second[0, t]n for which the variables sq(j,k,1) ¯ arily by the index k. Moreover, each of the variables := sq(j,k,r+1) − sq(j,k,r) uq(j,k,r) ¯ ¯ ¯
(7.3)
are positive, ∀j ; k = 1, . . . nj ; r = 1, . . . , j − 1 . (These properties of R are implicit from the choice of the ordering q¯ and of the nature of the permutations ρ ∈ S0n .) Consider the change of variables ; uq(j,k,r) , (7.4) (s1 , . . . , sn ) → sq(j,k,1) ¯ ¯ where the ordering is first by the j , second by the k, and for the u’s finally by the r = 1, . . . , j −1. This defines a volume-preserving map which will take R into n1 (t)×
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n2 (t) × · · · × [0, ∞)n2 × [0, ∞)2n3 × · · · . From this we are able to find the upper estimate on (7.2) of the form E(n)=n
E(n)−2N(n)+n1
C11
C 2N(n)−n1
n
= ≤
E(n)=n
E(n)−2N(n)+n1
C11
n E(n)=n eAE(n)+BN(n)
n1 !n2 ! · · ·
n
(t ∨ 1)n1 (t ∨ 1)n2 ··· n1 ! n2 !
C 2N(n)−n1
'
∞
|Gλ (s)| ds
(n2 +2n2 +···
0
(t ∨ 1) K E(n)−N(n) n1 !n2 ! · · · N(n)
(7.5)
,
−2 where A = ln (KC11 ) and B = ln (t ∨ 1)+ln C 2 ∨ 1 +ln C11 ∨ 1 +ln K −1 ∨ 1 . The restriction to those sequences n with E (n) = n can be lifted and the following estimate for the entire series obtained: (A, B) =
eAE(n)+BN(n) n1 !n2 ! · · ·
n
A+B ∞ ∞ e(kA+B)nk e . = = exp nk ! 1 − eA
(7.6)
k=1 nk =0
The manipulations are familiar from, for example, the calculation of the grand canonical partition function for the free Bose gas [24]. The requirement for convergence is that eA < 1, or equivalently, that KC11 < 1. 8. Limit Transition Amplitudes We are now ready to re-sum the Dyson series. First of all, observe that the functions hj (t, λ) defined in (5.5) will have the limits hj (t) := lim hj (t, λ) = 1[Sj ,Tj ] fj |g . λ→0
(8.1)
Likewise, we obtain E˜ αβ (t) = limλ→0 E˜ αβ (t, λ) which will be just the expressions in (5.7) with the hj (t, λ) replaced by their limits. Explicitly, we have E˜ 11 (t) = E11 , E˜ 01 (t) = Eα1 [h1 (t)]α , β β E˜ 10 (t) = E1β h∗2 (t) , E˜ 00 (t) = [h1 (t)]α Eαβ h∗2 (t) .
(8.2)
Secondly, only type I terms will survive the limit. This means that, for the nth term in the Dyson series, the only sequences α1 , β1 , α2 , β2 , . . . , αn , βn appearing will be those for which 0 = αn = β1 and βl = αl+1 for l = 1, . . . , n − 1. Thirdly, we encounter the following limit of the two point function: Gλ (t − s). Let f and g be Schwartz functions, then we will have the limit
T
t2
dt2 0
0
dt1 Gλ (t2 − t1 ) f (t2 ) g (t1 ) → κ+
T
ds f (s) g (s) . 0
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Therefore, employing Lemma (2.3), we find
lim φ1 ⊗ Wλ (1) | U t/λ2 , λ φ2 ⊗ Wλ (2) λ→0 = W f1 ⊗ 1[S1 ,T1 ] | W f2 ⊗ 1[S2 ,T2 ] n−1 β × dsn · · · ds1 κ+ d+ (sl+1 − sl ) l (−i)n n
×
n (t)
l=1
φ1 | E˜ 0βn−1 (sn ) · · · E˜ β2 β1 (s2 ) E˜ β1 0 (s1 ) φ2 ,
(8.3)
β∈{0,1}n−1
where + we use the symbol d+ for a one-sided delta function: d+ (t − s) f (s) ds = f t . We now develop this series. Suppose that we have βk+1 = 0 = βk , that is, there are no β contractions to the kth term, then we encounter the factor E˜ 00 (s) = [h1 (s)]α Eαβ h∗2 (s) , where s = sk . Otherwise, if we have contractions on the terms associated to consecutive variables sk+r , . . . , sk+1 , sk and we assume that sk+r is not paired to sk+r+1 , nor sk to sk−1 : then we encounter the factor E˜ 01 (sk+r ) E˜ 11 (sk+r−1 ) · · · E˜ 11 (sk+1 ) E˜ 10 (sk ) with the variables sk+r , . . . , sk+1 , sk all forced equal to a common value s, say. This factor β will then be [h1 (sk )]α Eα1 (E11 )r−2 E1β h∗2 (sk ) . Now (8.3) involves a sum over all consecutive pairings: the corresponding partition will have all parts consisting of consecutive labels. We can list these parts in increasing order, say from 1 to m if there are m of them, and let rj be the size of the j th part. The m number of contractions will be βl and this will be n − m = j =1 rj − 1 . With these observations we see that (8.3) becomes
W f1 ⊗ 1[S1 ,T1 ] | W f2 ⊗ 1[S2 ,T2 ] ×
dsm · · · ds1 (−i)
m
j =1 rj
m (t)
m =n r1 +···+r
m
κ+
j =1
n
m
rm ,...r1 ≥1
(rj −1)
(r )
(r )
φ1 | Eαmm,βm · · · Eα11,β1 φ2
β β × [h1 (sm )]αm h∗2 (sm ) m · · · [h1 (s1 )]α1 h∗2 (s1 ) 1 ,
(8.4)
where we set (r) Eα,β
:=
Eαβ , r=1 . Eα1 (E11 )r−2 E1β , r ≥ 2
(8.5)
In the following, we shall encounter the coefficients Lαβ := −i
∞ r=1
(r)
(−iκ)r−1 Eα,β = −iEαβ − κEα1
1 E1β . 1 + iκE11
(8.6)
With respect to the representation L2 R+ , k ∼ introduce the four = k ⊗ L2 R+ , we fundamental operator processes (here χ[0,t] is the operator on L2 R+ corresponding
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503
to multiplication by 1[0,t] ) :
+ (creation)A10 g ⊗ 1[0,t] ; t =A (conservation)A11 t = d |g)(g| ⊗ χ[0,t] ; − (annihilation)A01 g ⊗ 1[0,t] ; t =A (time)A00 t = t.
(8.8)
These are thebasic quantum stochastic processes on the Hudson-Parthasarathy space
L2 R+ , k . We note that the quantum Itˆo table takes the concise form 1β
αβ
dAα1 t dAt = γ dAt
(8.9)
with all other pairs vanishing. Theorem 8.1. Suppose the system operators Eαβ are bounded with K E11 < 1 . Let φ1 , φ2 ∈ hS and f1 , f2 ∈ k. Then (λ) lim φ1 ⊗ Wλ (1) | Ut φ2 ⊗ Wλ (2) λ→0 = φ1 ⊗ W f1 ⊗ 1[S1 ,T1 ] | Ut φ2 ⊗ W f2 ⊗ 1[S2 ,T2 ] , where (Ut : t ≥ 0) is a unitary adapted quantum stochastic process on hS ⊗ (L2 (R+ , k)) satisfying the quantum stochastic differential equation αβ
dUt = Lαβ Ut ⊗ dAt
(8.10)
with U0 = 1 and where the coefficients are given by (8.6): L11 = −iE11 (1 + iκE11 )−1 , L01 = −iE01 (1 + iκE11 )−1 ,
L10 = −i(1 + iκE11 )−1 E10 , L00 = −iE00 − κE01 (1 + iκE11 )−1 E10 .
Proof. The quantum stochastic differential equation (8.10) takes the form dUt =
1 10 (W − 1) Ut ⊗ dA11 t + LUt ⊗ dAt γ 1 † −L† W Ut ⊗ dA01 − L + iH Ut ⊗ dA00 γ L t t , 2
where W =
1 − iκ− E11 (unitary) 1 + iκ+ E11
L = −i(1 + iκ+ E11 )−1 E10 (bounded) 1 E10 (self-adjoint). H = E00 + I m κ+ E01 1 + iκ+ E11
(8.11)
A fundamental result of quantum stochastic calculus [7] is that the process Ut defined as the solution of (8.11) with initial condition U0 = 1, exists and is an adapted, unitary process. With our summation convention in place, we have the chaotic expansion α(m)β(m) α(1)β(1) Ut = Lα(m)β(m) · · · Lα(1)β(1) ⊗ dAs(m) · · · dAs(1) (8.12) m≥0 m (t)
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and so φ1 ⊗ W f1 ⊗ 1[S1 ,T1 ] | Ut φ2 ⊗ W f2 ⊗ 1[S2 ,T2 ] can be expressed as φ1 |Lα(m)β(m) · · · Lα(2)β(2) Lα(1)β(1) φ2 W f1 ⊗ 1[S1 ,T1 ] |W f2 ⊗ 1[S2 ,T2 ]
β(m) β(1) · · · [h1 (s1 )]α(1) h∗2 (s1 ) . [h1 (sm )]α(m) h∗2 (sm )
dsm · · · ds1
×
m≥0
m (t)
By inspection, this evidently agrees with (8.4).
9. Dynamical Evolutions Let X be a bounded operator on the system state space hS . We define its Heisenberg evolute to be (λ)
Jt
(λ)†
(X) := Ut
(λ)
[X ⊗ 1R ] Ut .
(9.1)
In addition, what we term the co-evolute is defined to be (λ)
Kt
(λ)
(X) := Ut (λ)
(λ)†
[X ⊗ 1R ] Ut
(9.2)
.
(λ)
We wish to study the limits of Jt and Kt as quantum processes taken relative to the Fock vacuum state ∈ hR for the Bose reservoir. To this end, we note the developments (λ) n dsn · · · ds1 Xϒ (λ) ◦ · · · ◦ Xϒ (λ) (X ⊗ 1R ) , (9.3) Kt (X) = (−1) (λ)
Jt
(X) =
sn
n (t)
n
(−i)n+nˆ
dsn · · · ds1 n (t)
n,nˆ
s1
nˆ (t) (λ)
dtnˆ · · · dt1
(λ)
×ϒs(λ) · · · ϒs(λ) [X ⊗ 1R ] ϒtnˆ · · · ϒt1 , n 1
(9.4)
where XH (.) := 1i [., H ]. We note that the co-evolution has the simpler form when iterated. The evolution itself requires a separate expansion of the unitaries. (This disparity is related to the proof of unitarity for quantum stochastic processes in [7], where the isometric property requires some work while the co-isometric property is established immediately). In fact, the same (λ) inequalities as used to establish the convergence of Ut suffice for the co-evolution: in both cases we have a Picard iterated series. We remark that in [26] the co-evolution only is treated for emission/absorption interactions. We likewise have the expansion (λ) φ1 ⊗ Wλ (1) | Jt (X) φ2 ⊗ Wλ (2) = dsn · · · ds1 dtnˆ · · · dt1 (−i)n−nˆ n,nˆ
n (t)
nˆ (t)
× φ1 | E˜ α1 β1 (s1 , λ) . . . E˜ αn βn (sn , λ) X E˜ µnˆ νnˆ (snˆ , λ) . . . E˜ µ1 ν1 (s1 , λ) φ2 α β α β × | as+1 (λ) 1 as−1 (λ) 1 · · · as+n (λ) n as−n (λ) n ) *µnˆ ) *νnˆ µ ν × at+nˆ (λ) at−nˆ (λ) . . . at+1 (λ) 1 at−1 (λ) 1 . (9.5)
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The vacuum average of the reservoir operators can be expressed as a sum of products of two-point functions with each summand representable as a partition of n + nˆ vertices. The term in the series expansion at level n, nˆ is bounded by E(m)=n+ nˆ
m
ρ∈S0n+nˆ
E(m)−2N(m)+m1
C11
× n (t)×nˆ (t)
=
drn+nˆ . . . dr1
E(m)=n+ nˆ
m
ρ∈S0n+nˆ
× Rn+nˆ (t)
C 2N(m)−m1 X
nj j −1
Gλ rρ(q(j,k,r+1)) − rρ(q(j,k,r)) ¯ ¯
j ≥2 k=1 r=1 E(m)−2N(m)+m1
C11
drn+nˆ . . . dr1
C 2N(m)−m1 X
nj j −1
, (9.6) Gλ rρ(q(j,k,r+1)) − rρ(q(j,k,r)) ¯ ¯
j ≥2 k=1 r=1
˜ n (t) × nˆ (t) : ρ ∈ S0n+nˆ and we write rn+nˆ , . . . , r1 for where Rn+nˆ (t) = ∪ ρ the ordered collection (sn , · · · , s1 , tnˆ , · · · , t1 ). We distinguish three different cases: a) Contractions between vertices labelling s-variables; b) Contractions between vertices labelling t-variables; c) Contractions between vertices labelling an s and a t-variable. Let the number of contractions of type c) be denoted as k. This means that we have n−k of the s-variables and n−k ˆ of the t-variables left. A given partition of n+ nˆ vertices may then be described in terms of the k mixed contraction pairs and then a partition of the n − k s-variables and a partition of the nˆ − k t-variables. We have nk nkˆ ways to choose the k mixed contractions and so we are interested in the summability of n)= ˆ nˆ n∧nˆ E(n)=n E( n nˆ k k n n nˆ
nˆ
k
E(n)−2N (n)+n1 2N(n)−n1 E(n)−2N( ˆ n)+ ˆ nˆ 1 2N(n)− ×C11 C C11 C ˆ nˆ 1 nj j −1
× Rn+nˆ (t)
drn+nˆ . . . dr1
! ! !Gλ rq(j,k,r+1) !. − rq(j,k,r) ¯ ¯
(9.7)
j ≥2 k=1 r=1
We note the equalities E (m) = n+ nˆ = E (n)+E (n)−2k, ˆ N (m) = N (n)+N (n)+k ˆ and m1 = n1 + nˆ 1 . − rq(j,k,r) , where x = It is convenient to introduce new variables u(x) ≡ rq(j,k,r+1) ¯ ¯ a, b,c according will be E (n) − N (n) of type a), to the type of contraction. There E nˆ − N nˆ of type b) and k of type c). The u(a) -variables and the u(b) -variables are all positive by construction.
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The change of coordinates rn+nˆ , . . . , r1 → rq(j,k,1) , u(a) , u(b) , u(c) is volume ¯ preserving and maps Rn+nˆ (t) into k (t) × n1 (t) × n2 (t) × · · · × nˆ 1 (t) × nˆ 2 (t) × [0, ∞)n2 × [0, ∞)2n3 × · · · × [0, ∞)nˆ 2 × [0, ∞)2nˆ 3 × · · · × (−∞, ∞)k . Proceeding along the same lines as in the last section we obtain the estimate, with G = |κ+ + κ− |, eA− E(n)+BN(n) eA+ E (nˆ ) + BN nˆ k k k! n1 !n2 ! · · · nˆ 1 !nˆ 2 ! · · · n n,nˆ k nˆ (Gt)k E (n) + k E nˆ +k eA− E(n)+BN(n) eA+ E (nˆ ) + BN nˆ = k k nˆ 1 !nˆ 2 ! · · · k! n1 !n2 ! · · ·
n∧nˆ n nˆ (Gt)k
(Gt)k k
E (nˆ )=n−k ˆ
=
n,nˆ
k
=
E(n)n−k
(k!)
3
∂A−
k↑
∂A+
k↑
(A− , B) (A+ , B) ,
(9.8)
where x k↑ := (x + k) (x + k − 1) · · · (x + 1) is the rising factorial power. Here A± = ln (|κ± | C11 ). The expression is finite provided that the partition functions of (7.6), (A± , B), exist. The convergence is due to the presence of (k!)3 in the denominator of (9.8). We of course have A± = A numerically. To see this, let P be the “grand canonical ensemble” measure on the set of occupation numbers given by P [n] =
1 eAE(n)+BN(n) , (A, B) n1 !n2 ! · · ·
(9.9)
and let E be the corresponding expectation. Then the upper bound (3.4) can be equivalently written as (A, B)2
(2 |κ| t)k k
(k!)3
*2 ) E E k↑ .
(9.10)
Now the distribution of E = E (n) can be determined from the moment generating function A+B ) * (A + u, B) 1 − eu e E euE = = exp (A, B) 1 − eA eA+u − 1 which is convergent for u < −A. The factorial moment generator is then given by ( ' ∞ k ) * 1 v k↑ =E E E k! (1 − v)E k=0 and this converges for 0 < v < 1 − e−A . (Note that x k↑ = m s (k, m) x m , where the integers s (k, m) are the unsigned Stirling numbers of the first kind [21] and that
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= (1 − v)−E ). If we fix the value of v within the stated range, then we conclude from the ratio test that 1 E E k↑ 1 < . lim (k−1)↑ k→∞ k E E v k
1 k m m k! s (k, m) v E
This, of course, ensures the convergence of (9.10) and likewise (9.8).
10. The Convergence of the Heisenberg Evolution We now wish to determine the limit λ → 0 of (9.5). We have an integration over a double simplex region and the main features emerge from examining the vacuum expectation of the product of creation and annihilation operators. Evidently, the vacuum expectation can be decomposed as a sum over products of two point functions and it is here that Lemma (6.1) becomes important. What must happen for a term to survive the limit? If we have any contractions between vertices labelled by the t’s then the term will vanish if the times are not consecutive. The same is true for contractions between vertices labelled by the s’s. From our estimate in the previous section, we can ignore the terms that do not comply with this. As a result, contractions between the s’s, say, will come in time-consecutive blocks: for instance, we will typically have m blocks of sizes r1 , r2 , · · · , rm (these are integers 1,2,3,..., and m j =1 rj = n). With a similar situation for the t’s, we obtain the expansion (λ) φ1 ⊗ Wλ (1) | Jt (X) φ2 ⊗ Wλ (2) r=n l=nˆ n−nˆ = dsn · · · ds1 (−i) n (t) m,m ˆ r1 ,··· ,rm l1 ,··· ,lmˆ
n,nˆ
nˆ (t)
dtnˆ · · · dt1
(r ) (1) ˜ (rm ) s (m) , . . . sr(m) ; λ × φ1 | E˜ α11β1 s1 , . . . sr(1) ; λ . . . E αm βm 1 m 1
mˆ lmˆ ) m ˆ) ( ( ( ) (1) (1) ×X E˜ µmˆ νmˆ t1 , . . . , tlmˆ ; λ . . . E˜ µ(l11ν) 1 t1 , . . . , tl1 ; , λ φ2 ×
rj m
G∗λ
(j ) sk+1
j =1 k=1
a +(1) s1
×
ˆ=1 k=1 ˆ
a −(1) sr1
(λ) ,µmˆ '
× a +(mˆ ) (λ)
(β1 (λ) (νmˆ
a −(mˆ ) (λ)
tl
t1
m ˆ
ˆ m ˆ
l
(α1 '
'
× | +
(j ) − sk
ˆ
(ˆ) ( ) G∗λ t ˆ − t ˆ k+1
(αm '
'
···
a +(m) s1
k
a −(m) srm
(λ)
(βm (λ)
(µ1 ' (ν1 ' a −(1) (λ) . . . a +(1) (λ) tl
1
+negligible terms,
t1
(10.1)
where we relabel the times as sk
(j )
:= sr1 +···+rj −1 +k ,
1 ≤ k ≤ rj ;
(j ) tk
:= tl1 +···+lj −1 +k ,
1 ≤ k ≤ lj ;
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and introduce the block product of system operators
(r ) (j ) (j ) E˜ αβj s1 , . . . srj ; λ
(j ) (j ) (j ) (1) := E˜ α1 β1 s1 ; λ E˜ 11 s2 ; λ · · · E˜ 11 srj −1 ; λ E˜ 1β srj ; λ . We now examine the limit of (10.1). The estimate on the series expansion of the Heisenberg evolute given in the previous section shows that we can ignore the so-called negligible terms in (10.1). The limit is rather difficult to see at this stage. However, what we can do is to recast the expression that we claim will be the limit, (10.2) φ1 ⊗ W f1 ⊗ 1[S1 ,T1 ] | Jt (X) φ2 ⊗ W f2 ⊗ 1[S2 ,T2 ] , with Jt (X) = U†t (X ⊗ 1) Ut , in a more explicit form. Recall the chaotic expansion of the process Ut given in (8.12), the expression (10.2) then becomes (i) r− l (κ− ) r−m (κ+ ) l−mˆ m,m ˆ
m (t) mˆ (t) r1 ,··· ,rm l ,··· ,l 1 m ˆ
(l ) (r ) (r ) × φ1 | E˜ α11β1 . . . E˜ αmmβm X E˜ µmmˆˆ νmˆ . . . E˜ µ(l11ν) 1 φ2 % µ ν × W f1 ⊗ 1[S1 ,T1 ] | dAαsmm βm · · · dAαs11 β1 dAtmˆmˆ mˆ · · · µ ν ×dAt11 1
&
W f2 ⊗ 1[S2 ,T2 ] .
Now the expectation between the states W fj ⊗ 1[Sj ,Tj ] can be converted into an expectation between the Fock vacuum state if we make the following replacements: dA11 → dA11 + h∗2 dA10 + h1 dA01 + h1 h∗2 dA10 , dA10 → dA10 + h1 dA00 , dA01 → dA01 + h∗2 dA00 , dA00 → dA00 , where hj (t) = 1[Sj ,Tj ] fj |g as in (8.1). This leads to the development r−m (i) r− l κ ∗ (κ) l−mˆ m,m ˆ
(10.3)
m (t) mˆ (t) r1 ,··· ,rm l ,··· ,l 1 m ˆ
(l ) (r ) (r ) × φ1 | E˜ α11β1 (s1 ) . . . E˜ αmmβm (sm ) X E˜ µmmˆˆ νmˆ (tmˆ ) . . . E˜ µ(l11ν) 1 (t1 ) φ2 µ ν µ ν × | dAαsmm βm · · · dAαs11 β1 dAtmˆmˆ mˆ · · · dAt11 1 , (10.4) where the operators E˜ αb (t) are given by $ E˜ (t) , r=1 (r) αβ r−2 ˜ Eαβ (t) = . E˜ α1 (t) E˜ 11 (t) E˜ 1β (t) , r ≥ 2 (r)
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Again we note that the operators E˜ αβ (t) have been introduced in (8.2). It remains to be shown that the limit of (10.1) will be (10.4). We observe that (λ) lim φ1 ⊗ Wλ (1) | Jt (X) φ2 ⊗ Wλ (2)
λ→0
=
(−i)n−nˆ
n,nˆ
r=n l=nˆ m,m ˆ r1 ,··· ,rm l1 ,··· ,lmˆ
dsm · · · ds1 m (t)
mˆ (t)
dtmˆ · · · dt1
(l ) (r ) (r ) × φ1 | E˜ α11β1 (s1 ) . . . E˜ αmmβm (sm ) X E˜ µmmˆˆ νmˆ (tmˆ ) . . . E˜ µ(l11ν) 1 (t1 ) φ2
× (κ+ ) l−mˆ α β α β + × lim | as1 (λ) 1 as−1 (λ) 1 · · · as+m (λ) m as−m (λ) m λ→0 *µmˆ ) *νmˆ ) µ ν at−mˆ (λ) × at+mˆ (λ) . . . at+1 (λ) 1 at−1 (λ) 1 . × (κ− )
r−m
(10.5)
We now require the fact that
) * α1 ) * β1 α β as−1 (λ) dsm · · · ds1 dtmˆ · · · dt1 | as+1 (λ) · · · as+m (λ) m as−m (λ) m λ→0 R ) ) *µ ) *ν * µ1 ) * ν1 m ˆ m ˆ at− (λ) at−1 (λ) × at+ (λ) . . . at+1 (λ) f sm , . . . , s1 , tmˆ , . . . , t1 m ˆ m ˆ µ ν α β µ ν α β | d Asmm m · · · d As11 1 d At mˆ mˆ · · · d At11 1 f sm , . . . , s1 , tmˆ , . . . , t1 = lim
R
m ˆ
for f continuous and R a bounded region in m + m ˆ dimensions which is the union of simplices of the type (5.2). This is readily seen, of course, by expanding the -expectation as a sum of products of two-point functions and reassembling the limit in terms αβ of the -expectations of the processes At . This is evident from Theorems 4.2 and 6.1 quoted earlier and from the quantum Itˆo calculus [7]. We therefore see that the limit form as given in (10.5) agrees with the stated limit as represented in (10.1). This establishes the result. Theorem 10.1. Suppose that Eαβ are bounded with K E11 < 1, as before. Let φ1 , φ2 ∈ hS and f1 , f2 ∈ k. Then, for X ∈ B (hS ), (λ) lim φ1 ⊗ Wλ (1) | Jt (X) φ2 ⊗ Wλ (2) λ→0 = φ1 ⊗ W f1 ⊗ 1[S1 ,T1 ] | Jt (X) φ2 ⊗ W f2 ⊗ 1[S2 ,T2 ] . (λ)
(λ)
To summarize, the pre-limit flow Jt : B (hS ) → B (hS ⊗ hR ) given by Jt (X) := (λ) ⊗ 1R ) Ut converges in the sense of weak matrix elements, for fixed X ∈ B (hS ), to the limit process Jt (X) = U †t (X Ut . We find that (Jt )t≥0 determines ⊗ 1) a quantum stochastic flow on hS ⊗ L2 R+ , k and from the quantum stochastic calculus [7] we obtain the quantum Langevin, or stochastic Heisenberg, equation (λ)† Ut (X
αβ dJt (X) = Jt Lαβ (X) ⊗ dAt ,
(10.6)
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where L11 (X) =
1 † W XW − X ; γ
) * L10 (X) = W † [X, L] ; L01 (X) = − X, L† W ; γ ) † * γ L00 (X) = L , X L + L† [X, L] − i [X, H ] . 2 2
(10.7)
In particular, L00 is a generator of Lindblad type [15]. The analogous result will hold for the co-evolution. Though, as mentioned before, there is a more immediate proof using the original estimates. 11. Conclusions We began with a discussion of time-ordered versus normal ordered presentations of unitary operators relating to scattering dynamics. It is suggestive to write the limit unitary Ut as either t + α − β Ut = T exp −i , (11.1a) ds Eαβ ⊗ as as 0 t α − β exp or Ut = N . (11.1b) ds Lαβ ⊗ a+ as s 0
Here a± t are just symbols and we mean nothing more than that (11.1b) is the solution of (8.10) while (11.1a) reminds us that we have the limit generated by a perturbation α β (λ) ϒt = Eαβ ⊗ at+ (λ) at− (λ) . (Formally, of course, we might consider a± t as a limiting form of the fields at± (λ).) Remarkably, these identifications (11.1a, b) can be viewed as presentations of (1.1) and (1.2) if we supplement the operators a± t with the following white noise CCR: − + at , as = κ+ d+ (t − s) + κ− d− (t − s) , (11.2) where d± are future/past delta functions: we would have the right-hand side γ δ (t − s) if it was not for the fact that we invariably meet with simplicial integrals. The stochastic Schr¨odinger equation (8.10) can be written as β α Lαβ Ut a− dt dUt = a+ t t
(11.3)
α β which is in normal ordered form. If we understand that a+ Xαβ (t) a− dt means t t αβ Xαβ (t) ⊗ dAt then we recover the Hudson-Parthasarathy calculus. The product of two quantum stochastic integrals will have to be put into normal order, using (11.2) , but this will be equivalent to the usual quantum Itˆo rule with Itˆo table (8.9). Alternatively, we could consider the equation dUt = −iϒt Ut dt with ϒt = Eαβ ⊗ + α − β at : this is what is suggested by (11.1b). (The Hamiltonian ϒt plays an analat ogous role to the one encountered earlier for classical stochastic Hamiltonian flows leading to (1.4)). However, the expression ϒt Ut contains terms like a− t Ut which are
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out of normal order and so cannot be directly interpreted in the quantum Itˆo calculus. Nevertheless, the following purely formal manipulations can be used [5]: ' ( t t β − , 1 − i ϒ U ds = −iκ E1β a− d+ (t − s) Us ds a t , Ut = a − s s + t s 0
0
= −iκ+ E11 a− t Ut − iκ+ E10 Ut , leading to a− t Ut =
1 Ut a − t − iκ+ E10 Ut . 1 + iκ+ E11
(11.4)
(Similar manipulations have been performed separately for emission-absorption and for scattering interactions in [27].) By making the replacement (11.4), wherever it occurs, we obtain a proper normal ordered form and this turns out to be precisely (11.3). In the classical problem for the limit of the flow under the Hamiltonian (1.3), the canonical structure is never lost though we have to look to the Stratonovich calculus to see it. We similarly have that the canonical structure is retained in the quantum problem - and we even have a formal Hamiltonian ϒt - provided that we look at things in the appropriate way. References 1. Accardi, L., Frigerio, A., Lu, Y.G.: Weak coupling limit as a quantum functional central limit theorem. Commun. Math. Phys. 131, 537–570 (1990) 2. Pul´e, J.V.: The Bloch equations. Commun. Math. Phys. 38, 241–256 (1974) 3. Hudson, R.L., Streater R.F.: Itˆo’s formula is the chain rule with Wick ordering. Phys. Lett. 86, 277–279 (1982) 4. Kr´ee, P., Raczka, R.: Kernels and symbols of operators in quantum field theory. Ann. Inst. H. Poincar´e, Sect. A, 28, 41–73 (1978) 5. Gough, J.: Asymptotic stochastic transformations for non-linear quantum dynamical systems. Reports Math. Phys. 44(3), 313–338 (1999) 6. Van Hove, L.: Quantum mechanical perturbations giving rise to a statistical transport equation. Physica. 21, 617–640 (1955) 7. Hudson, R.L., Parthasarathy, K.R.: Quantum Itˆo’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984) 8. Wong, E., Zakai, M.: On the relationship between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3, 213–229 (1965) 9. Frigerio, A., Gorini V.: Diffusion processes, quantum dynamical semigroups, and the classical KMS condition. J. Math. Phys. 25(4), 1050–1065 (1984) 10. Bismut, J.M.: Mecanique Al´eatoire. Lecture Notes in Mathematics 866, Berlin: Springer-Verlag, 1981 11. Spohn, H.: Kinetic Equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 53, 569–615 (1980) 12. Davies, E.B.: Markovian master equations. Commun. Math. Phys. 39, 91–110 (1974) 13. Davies, E.B.: Quantum theory of Open Systems. London, New York: Academic Press, 1976 14. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups on N-level systems. J. Math. Phys. 17, 821–825 (1976) 15. Lindblad, G.: On the generators of completely positive semi-groups. Commun. Math. Phys. 48, 119–130 (1976) 16. Von Waldenfels, W.: Itˆo Solution of the Linear Quantum Stochastic Differential Equation Describing Light Emission and Absorption. Lecture Notes in Mathematics 1055, Berlin Heidelberg-New York: Springer, 1986, pp. 384–411 17. Spohn, H., Lebowitz, J.L.: Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 38, 109–142 (1978) 18. Chebotarev, A.N.: Symmetric form of the Hudson-Parthasarathy equation. Mat. Zametki 60(5), 725–750 (1996)
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19. 20. 21. 22.
Gough, J.: Noncommutative Markov approximations. Dokl. Akad. Nauk. 379(6), 112–116 (2001) Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Basel: Birkha¨user, 1992 Riordan, J.: An Introduction to Combinatorial Analysis. New York: Wiley, 1980 Rota, G.-C., Wallstrom, T.C.: Stochastic integrals: a combinatorial approach. Ann Probab. 25, 1257– 1283 (1997) Abrikosov, A.A., Gorkov, L.P., Dzyaloshinski, I.E.: Methods of Quantum field Theory in Statistical Physics. New York: Dover Publications, 1963 Pathria, K.R.: Statistical Mechanics. Oxford: Butterworth-Heinemann, 1972 Gough, J.: A new approach to non-commutative white noise analysis. C.R. Acad. Sci. Paris t. 326, s´erie I, 981–985 (1998) Accardi, L., Frigerio, A., Lu, Y.G.: The quantum weak coupling limit (II): Langevin equation and finite temperature case. Publ. RIMS Kyoto 31, 545–599 (1995) Accardi, L., Lu,Y.G., Volovich, I.V.: Quantum Theory and its Stochastic Limit. Berlin Heidelberg: Springer-Verlag, 2002
23. 24. 25. 26. 27.
Communicated by A. Connes
Commun. Math. Phys. 254, 513–563 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1270-9
Communications in
Mathematical Physics
Mathematics Underlying the F-Theory/Heterotic String Duality in Eight Dimensions Adrian Clingher1,, , John W. Morgan2, 1 2
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA Department of Mathematics, Columbia University, New York, NY 10027, USA. E-mail:
[email protected]
Received: 12 August 2003 / Accepted: 24 April 2004 Published online: 22 January 2005 – © Springer-Verlag 2005
Abstract: We give an analytic description of the moduli space of classical vacua for F-theory compactified on elliptic K3 surfaces, on open tubular regions near the two Type II boundary divisors. The structure of these open sets is related to the total spaces of certain holomorphic theta fibrations over the corresponding boundary divisors. As the two Type II divisors can be naturally identified as moduli spaces of elliptic curves and flat G-bundles with G = (E8 × E8 ) Z2 or Spin(32)/Z2 , one is led to an analytic isomorphism between these open domains and regions of the moduli spaces of heterotic string theory compactified on the two-torus corresponding to large volumes of the torus. This provides a proof for the classical version of F-theory/Heterotic String Duality in eight dimensions. A description of the Type II boundary points in terms of elliptic stable K3 surfaces is also given.
1. Introduction One of the dualities in string theory, the F-theory/heterotic string duality in eight dimensions [34, 31], predicts an interesting correspondence between two seemingly disparate geometrical objects. On one side of the duality there are elliptically fibered K3 surfaces with section. On the other side, one finds elliptic curves endowed with certain flat connections and complexified Kahler classes. The F-theory [30, 34] is a 12-dimensional string theory which generally exists on elliptically fibered ambient manifolds with section. Heterotic string theory, on the other hand, exists on a 10-dimensional space-time. In order to obtain effective 8-dimensional models, one compactifies the two theories along elliptic K3 surfaces and elliptic curves,
The first author was supported by NSF grants DMS-97-29992 and PHY-00-70928. Current address: Department of Mathematics, Stanford University, Stanford CA 94305, USA. E-mail:
[email protected] The second author was partially supported by NSF grant DMS-01-03877.
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respectively. The duality mentioned above predicts then that the two theories are equivalent at the quantum level. In particular, their moduli spaces of quantum vacua should be isomorphic. As it is generally believed, in certain ranges of parameters the quantum corrections should be small and the quantum vacua should be well approximated by classical vacua. This leads one to expect that the moduli spaces of classical vacua of the two theories should resemble each other, at least on regions corresponding to insignificant quantum effects. The classical vacua for the heterotic string theory compactified along a two-torus E (there are two distinct such theories, one with structure Lie group G1 = (E8 × E8 ) Z2 and the other with G2 = Spin(32)/Z2 ), consists of a flat Gi -connection on E, a flat metric and an extra one-dimensional field, the B-field. In the original physics formulation [24, 25], the B-field appears as a globally defined two-form B. The metric and B fit together to form the imaginary and, respectively, the real part of the so-called complexified Kahler class. Each triplet (A, g, B) determines a lattice of momenta L(A,g,B) (after K. Narain [24]) governing the associated physical theory. The lattices L(A,g,B) , turn out to be even, unimodular and of rank 20. They are well-defined up to O(2) × O(18) rotations and vary, according to the triplet parameter, in a fixed ambient real space R2,18 . The real group O(2, 18) acts transitively on the set of all L(A,g,B) and, in this light, one can regard the physical momenta as parameterized by the 36-dimensional real homogeneous space: O(2, 18)/O(2) × O(18).
(1)
One identifies then the configurations in (1) determining equivalent quantum theories. This amounts to factoring out the left-action of the group of integral isometries of the lattice. However, not all identifications so created are accounted for by classical geometry. Part of the -action models the so-called quantum corrections [2] and results in identifying momenta for pairs of triplets (A, g, B) which are not isomorphic from the geometric point of view. The quantum (Narain) moduli space of distinct heterotic string theories compactified on the two-torus appears as: quantum
Mhet
= \O(2, 18)/O(2) × O(18).
(2)
The above physics-inspired Narain construction has a major flaw, though. It does not provide a holomorphic description. Technically, one can endow the homogeneous quotient (2) with a natural complex structure, but holomorphic families of elliptic curves quantum and flat connections do not embed as holomorphic sub-varieties in Mhet . An outline of these issues is provided in the Appendix. In the recent years, it has been noted by a number of authors (see for example [35] or [8]) that, in order to fulfill various anomaly cancellation conditions required by heterotic string theory, the B-field has to be understood within a gerbe-like formalism. In [8], D. Freed introduces B as a cochain in differential cohomology. Taking this point of view, one can ask then for a description of the space of triplets (A, g, B) up to natural i geometric isomorphism. This is the moduli space MG het of classical vacua for Gi -heterotic string theory compactified over the two-torus. Freed’s approach can be used to i describe MG het in an explicit holomorphic framework. It was shown in [6] that: i Theorem 1. 1. The classical Gi -heterotic moduli space MG het can be given the structure of a 18-dimensional complex variety with orbifold singularities.
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i ∗ 2. MG het represents the total space of a holomorphic Seifert C -fibration i MG het → ME,Gi ,
(3)
where ME,Gi is the moduli space of isomorphism classes of pairs of elliptic curves and flat Gi -bundles. The holomorphic orbifold structure of ME,Gi is described in [15]. If one denotes by H the upper half-plane, and by the co-root lattice of Gi , then ME,Gi is represented by a quotient of H × C through the action of a discrete group. Under this description, the fibration (3) can be recovered from a well-known fibration with complex lines over ME,Gi . This is, roughly speaking, the line fibration supporting the holomorphic theta function: 1 πi(2(z,γ )+τ (γ ,γ )) BG : H × C → C, BG (τ, z) = , (4) e η(τ )16 γ ∈
where η(τ ) is Dedekind’s eta-function. In this setting, one can prove: Theorem 2 ([6]). The C∗ -fibration (3) is holomorphically identified with the complement of the zero-section in the complex line fibration induced by (4). i Hence, the heterotic classical moduli space MG het can be holomorphically identified with the total space of the theta fibration with the zero-section divisor removed. We turn now to the other side of the duality. The classical vacua for 8-dimensional Ftheory are simply elliptically fibered K3 surfaces with section. Using the period map and global Torelli theorem [4, 9], one can regard the moduli space MK3 of such structures as a moduli space of Hodge structures of weight two, i.e. as a quotient of an open 18-dimensional hermitian symmetric domain by an arithmetic group of integral automorphisms. In order to identify all equivalent classical vacua, there is one more factorization to be taken into account, identifying the complex conjugate structures. The classical 8-dimensional F-theory moduli space obtained, denoted MF , is then double-covered by MK3 and can be seen to be isomorphic to an arithmetic quotient of a symmetric domain:
MF \O(2, 18)/O(2) × O(18).
(5)
The identification between the above description and (2) is the usual physics literature formulation of the F-theory/heterotic string duality in eight dimensions. The goal of this paper is to establish a rigorous geometric comparison between the i classical moduli spaces MF and MG het . Our construction provides a natural holomorphic identification between these classical moduli spaces, exactly on the regions where physics predicts the quantum effects are insignificant. The paper is structured as follows. In Sect. 2 we review various facts pertaining to the construction of the moduli space MK3 of elliptic K3 surfaces with section. This space is not compact. However, using a special case of Mumford toroidal compactification [1, 11], one can perform an arithmetic partial compactification: MK3 ⊂ MK3 by adding two divisors at infinity D1 and D2 , related to the two possible kinds of Type II maximal rational parabolic subgroups of O(2, 18). The arithmetic machinery producing the partial compactification is reviewed in Sect. 2.3. In Sect. 3 we discuss the
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geometrical interpretation of the compactification. The points of D1 and D2 correspond to semi-stable degenerations of K3 surfaces given by either a union of two rational elliptic surfaces glued together along a smooth fiber or a union of rational surfaces glued along an elliptic curve with elliptic fibration degeneration into two rational curves meeting at two points. Each of the two configurations exhibits an elliptic curve E (the double curve of the degeneration) and endows this elliptic curve with a flat G-connection. For D1 the resulting Lie group G turns out to be G1 = (E8 × E8 ) Z2 , whereas for D2 one obtains G2 = Spin(32)/Z2 . In the second case the flat connections obtained carry “vector structure”, in the sense that they can be lifted to flat Spin(32)-connections. Under this geometrically defined correspondence, one obtains a holomorphic isomorphism: Di ME,Gi .
(6)
Next, each of the two types of parabolic groups determining the boundary components Di produces an infinite sheeted non-normal parabolic cover p : Pi → MK3 . The total space Pi fibers holomorphically π : Pi → Di over the corresponding divisor at infinity, all fibers being copies of C∗ . Under identification (6), one obtains therefore a diagram: π
Pi −→ Di ME,Gi ↓ MK3 .
(7)
It turns out that there exists an open subset Vi ⊂ Pi on which the restriction of the parabolic projection p : Pi → MK3 becomes a holomorphic isomorphism to an open neighborhood of infinity near the cusp Di in MK3 . Moreover, one can realize Vi as a punctured tubular neighborhood of the zero-section in the holomorphic complex line fibration naturally associated to π : Pi → ME,Gi .
(8)
Thus, a neighborhood of the boundary component Di in MK3 can then be identified with an open punctured neighborhood of the zero-section of the line fibration associated to (8). In Sect. 5 we give an explicit description of this line fibration. Theorem 3. The fibration with complex lines associated to (8) is holomorphically isomorphic with the theta C-fibration induced by (4). In the light of Theorems 1 and 2, there is then a holomorphic isomorphism of C∗ -fibrations, unique up to twisting with a unitary complex number: i Pi MG het ↓ ↓ ME,Gi = ME,Gi
(9)
i and that gives a natural explicit mathematical identification between the region in MG het corresponding to large volumes with a region of MF in the vicinity of the boundary component Di . These are exactly the regions that the physics duality predicts should be isomorphic. This paper belongs to a long project begun by the second author jointly with R. Friedman and E. Witten [13] in 1996 and continued jointly with R. Friedman afterwards. The initial aim of the project was to give precise mathematical descriptions of
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various moduli spaces of principal G-bundles over elliptic curves in order to verify conjectures arising out of the F-theory/heterotic string theory duality in physics. Building on this earlier work, the present paper and [6] establish the mathematical results allowing one to describe the duality completely when the two theories in question are compactified to eight dimensions. 2. Review of the Compactification Procedure A coarse moduli space MK3 for isomorphism classes of elliptically fibered K3 surfaces with section can be described using the period map. In this section we review the Type II partial compactification of MK3 . 2.1. Period space. It is well-known that any two K3 surfaces are diffeomorphic. The second cohomology group over integers is torsion-free of rank 22 and, when endowed with the symmetric bilinear form given by the cup product, is an even unimodular lattice of signature (3, 19). Up to isometry, there exists a unique lattice with these properties. We pick a lattice of this type and denote it by L. It happens then that for any K3 surface X there always exists an isometry: ϕ : H 2 (X, Z) → L.
(10)
Such a map is called a marking. An elliptic structure with section on X induces naturally two particular line bundles F, S ∈ Pic(X) corresponding to the elliptic fiber and section. Let f, s ∈ H 2 (X, Z) be the cohomology classes corresponding to F and S. These special classes intersect as f 2 = 0, f.s = 1, s 2 = −2 and therefore span a hyperbolic type sub-lattice Q inside H 2 (X, Z). The notion of marking can be adapted for this framework. Let H be a choice of hyperbolic sub-lattice in L. All such choices are equivalent under the action of the group of isometries of L. Choose a basis {F, S} for H with (F, F ) = 0 , (F, S) = 1 and (S, S) = −2. A marking ϕ as in (10) is said to be compatible with the elliptic structure if ϕ(f ) = F and ϕ(s) = S. In particular, a compatible marking transports the hyperbolic sub-lattice Q ⊂ H 2 (X, Z) isomorphically to H . Two marked pairs (X, ϕ) and (X , ϕ ) are called isomorphic if there exists an isomorphism of surfaces g : X → X such that ϕ = ϕ ◦ g∗. Let Lo be the sub-lattice of L orthogonal to H . The lattice Lo is even, unimodular, and of signature (2, 18). By standard arguments, a marked pair (X, ϕ) determines a polarized Hodge structure of weight two on Lo ⊗ C which is esentially determined by the period (2,0)-line [ω] ⊂ Lo ⊗ C. The periods satisfy the Hodge-Riemann bilinear relations (ω, ω) = 0, (ω, ω) ¯ > 0. The classifying space of polarized Hodge structures of weight two on Lo ⊗ C is then given by the period domain = {ω ∈ P (Lo ⊗Z C) |(ω, ω) = 0, (ω, ω) ¯ > 0}.
(11)
This is an open 18-dimensional complex analytic variety embedded inside the compact complex quadric: ∨ = {ω|(ω, ω) = 0} ⊂ P (Lo ⊗Z C) . One can equivalently regard the periods ω ∈ as space-like, oriented two-planes in Lo ⊗ R. The real Lie group O(2, 18) of real isometries of Lo ⊗ R acts then transitively
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on leading to a description of the period domain in the form of a symmetric bounded domain: O(2, 18)/SO(2) × O(18).
(12)
An elliptic structure with section on a K3 surface X can be naturally regarded as a (pseudo-ample) lattice polarization of X with a hyperbolic type lattice. Following arguments of [28, 21 and 26], one can prove then the existence of a fine moduli space of marked elliptically fibered K3 surfaces with section, which is a 18-dimensional complex manifold Mmark K3 . It follows then that a marked elliptic K3 surface with section is uniquely determined by its period. The period map: per : Mmark K3 →
(13)
is a holomorphic isomorphism. However, in this setting, the period ω ∈ clearly depends on the choice of marking. One removes the markings from the picture by dividing out the period domain by the action of the isometry group of the lattice. Let be the group of isometries of Lo . Two periods correspond to isomorphic marked surfaces if and only if they can be transformed one into the other through an isometry in . The arguments of the global Torelli theorem allow one to conclude that: MK3 = \
(14)
is a coarse moduli space for elliptic K3 surfaces with section, without regard to marking. Let us briefly analyze the quotient (14). First of all, MK3 is connected. The period domain consists of two connected components, corresponding to the choice of orientation in the set of positive two-planes in Lo ⊗ R. The two components are mapped into each other by complex conjugation. We choose either one and denote it by D. Thus = D D. However, there are isometries in which exchange D and D and therefore (14) is connected. Secondly, the space MK3 can be given a description as a quotient of a bounded symmetric domain by a discrete, arithmetically defined modular group. Indeed, the isomorphism (12) is -equivariant and therefore: \ \O(2, 18)/SO(2) × O(18).
2.2. The classical F-theory moduli space. The moduli space MF of classical vacua associated to F-theory compactified on a K3 surface is obtained by identifying complex conjugated structures in MK3 . Using identification (14) one obtains that: ˆ MF = \,
(15)
where the new group ˆ is the semi-direct product Z2 ⊂ Aut (Lo ⊗Z C) with the Z2 factor generated by complex conjugation, ˆ ˆ \O(2, 18)/SO(2) × O(18) \O(2, 18)/O(2) × O(18). (16) MF = \
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2.3. Arithmetic of compactification of MK3 . The moduli space MK3 is connected but not compact. There exists various arithmetic techniques aiming at compactifying \. The simplest one is the Baily-Borel procedure [3] which we briefly review next. Later, we shall turn our attention to a particular case of Mumford’s toroidal compactification [1] which plays a central role in the computation we undertake in this paper. The Baily-Borel procedure [3] introduces an auxiliary space ∗ with ⊂ ∗ ⊂ ∨ . The topological boundary of ⊂ ∨ decomposes into a disjoint union of closed analytic subsets, called boundary components. There are two types of such components. Some are zero-dimensional and are represented by the points in the real quadric ∨ ∩P (Lo ⊗Z R). The others are copies of P1 and are generated by the complexified images in P (Lo ⊗Z C) of the 2-dimensional isotropic subspaces of Lo ⊗Z R. Group theoretically, it can be seen that the stabilizer Stab(F ) = {g ∈ O ++ (2, 18)|gF = F } of a boundary component F is a proper maximal parabolic subgroup of O ++ (2, 18). A boundary component F is called then rational if its stabilizer Stab(F ) is defined over Q. The assignment P → FP with Stab(FP ) = P determines a bijective correspondence between the set of proper maximal rational parabolic subgroups of O ++ (2, 18) and the set of all rational boundary components. One defines then: ∗ FP , =∪ P
where the right union is made over all proper maximal rational parabolics. The action of extends naturally to ∗ . Moreover, one can endow ∗ with the Satake topology, under which the -action is continuous. The Baily-Borel compactification appears then as: (\)∗ = \∗ . def
(17)
The main features of this new quotient space are as follows (see [3] for details). The space (\)∗ is Hausdorff, compact, connected and can be given a structure of complex algebraic space. The quotient \ is embedded in (\)∗ as a Zariski open subset. If Ii (Lo ), i ∈ {1, 2} represents the set of primitive isotropic sub-lattices of rank i in Lo then the complement (\)∗ − \ consists of |\I1 (Lo )| points and |\I2 (Lo )| copies of PSL(2, Z)\H. Let us note that the complex conjugation involution on extends to ∗ . On boundary, it preserves the points and induces complex conjugation on the one-dimensional P1 ’s. The procedure provides therefore a compactification for the classical F-theory moduli space MF :
∗ ˆ ˆ \ ⊂ \ with boundary strata given by points and copies of (PSL(2, Z) Z2 ) \H with Z2 generated by τ → −τ¯ . It is known that Baily-Borel construction gives the minimal geometrically meaningful compactification of \ in the sense that it is dominated by any other geometric compactification. However, the disadvantage of the method is that the boundary has large
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codimension (it consists of only points and curves) and contains only partial geometrical information. One avoids these inconveniences by using a blow-up of the Baily-Borel construction, the toroidal compactification of Mumford [1]. This compactification, although not canonical in general, gives divisors as boundary components and carries significantly more geometrical information. The main arguments describing the construction, as presented in [11 and 9], are as follows. The Mumford boundary components associated to \ involve again the maximal rational parabolic subgroups of O(2, 18). These are stabilizers of non-trivial isotropic subspaces VQ ⊂ Lo ⊗Z Q. The lattice Lo has signature (2, 18), and hence, if VQ is isotropic then its dimension is either 2 or 1. If dim(VQ ) = 1, then the associated BailyBorel rational boundary component F is represented by just a point. Such a component is called of Type III. For dim(VQ ) = 2, the corresponding boundary component F is 1dimensional. In this case F is said to be of Type II. Each rational Baily-Borel component F will determine a Mumford boundary component B(F ). We shall be concerned here only with describing the components of Type II for which the construction is canonical. Let VQ be a rank-two isotropic lattice and F the associated Baily-Borel component. We denote: P (F ) = Stab (VR ) ⊂ O(2, 18), W (F ) = the unipotent radical ofP (F ), U (F ) = the center of W (F ). It turns out that U (F ) is 1-dimensional (also definable over Q) and the Lie algebra of its real form can be described as: u(F ) = {N ∈ Hom ((Lo )R , (Lo )R ) | Im (N ) ⊂ VR and ×(N a, b) + (a, N b) = 0, ∀a, b ∈ (Lo )R }.
(18)
One obtains that any N ∈ u(F ) satisfies N 2 = 0, Im(N ) = VR and Ker(N ) = VR⊥ . There is then an associated weight filtration: 0 ⊂ VR ⊂ VR⊥ ⊂ (Lo )R .
(19)
We pick a primitive integral endomorphism N ∈ u(F ) and consider the groups: U(N )C = {exp (λN ) |λ ∈ C}, U(N )Z = {exp (λN ) |λ ∈ Z} = U(N )C ∩ O ++ (2, 18; Z). The group U(N )C acts upon the extended period domain ∨ = {[z] ∈ P (Lo ⊗Z C) |(z, z) = 0}, providing an intermediate filtration ⊂ (F ) ⊂ ∨ , where (F ) = U(N )C · . One defines then the Mumford boundary component associated to F as the space of nilpotent orbits: B(F ) = (F )/ U(N )C .
(20)
(F )/ U(N )Z → B(F )
(21)
In this setting,
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is a holomorphic principal bundle with structure group U(N )C / U(N )Z C∗ . The inclusion / U(N )Z → (F )/ U(N )Z realizes / U(N )Z as an open subset in the total space of (21). Let then: (F )/ U(N )Z = ((F )/ U(N )Z ) ×C∗ C.
(22)
This amounts to gluing in the zero section in the C∗ -fibration (21). One defines then: def / U(N )Z = interior of the closure of / U(N )Z in (F )/ U(N )Z . Set-theoretically, one has: / U(N )Z = / U(N )Z B(F ). Finally: def = / U(N )Z = B(F ) , F
(23)
F
the union being performed over all rational Baily-Borel boundary components of Type II. This space inherits a topology. The arithmetic action of induces a closed discrete equivalence relation on (23). The quotient space, denoted by \, enjoys the following properties (see [1, 11] for details): Theorem 4. • \ is a quasi-projective analytic variety. • \ contains \ as a Zariski open dense subset. • The complement \ − \ consists of two irreducible divisors. These divisors are quotients of smooth spaces by finite group actions. We shall denote the two divisors by DE8 ⊕E8 and D16 . The reason for this terminology is the following. The two Type II divisors in question correspond to the two distinct orbits in \I2 (Lo ), where I2 (Lo ) is the set of primitive isotropic rank-two sub-lattices in Lo . One can identify the orbit to which a certain isotropic sub-lattice belongs using the following recipe. Let V ∈ I2 (Lo ). The quotient lattice V ⊥ /V is even, unimodular, negative-definite and has rank 16. It is known that, up to isomorphism, there exist only two lattices of this type: −(E8 ⊕ E8 ) and −16 . The two isomorphism classes perfectly differentiate the two orbits in \I2 (Lo ). There are therefore only two distinct Baily-Borel boundary curves in (\)∗ − \ and, accordingly, there are two Type II components in Mumford’s compactification. In fact, for each isotropic sub-lattice V there is a natural projection: B(F ) → F
(24)
defined by assigning to a nilpotent orbit { U(N )C ·ω} the complex line {ω}⊥ ∩VC ⊂ VC . We shall see the geometrical significance of (24) in the next section. At this point, we
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just note that these projections descend to maps from the Type II Mumford divisors to the two Baily-Borel boundary curves under \ → (\)∗ . As mentioned earlier, the main goal of this paper is to describe explicitly the structure of \ in a neighborhood of the two Type II divisors DE8 ⊕E8 and D16 . Our description will go along the following direction. Let F be a Type II Baily-Borel component and denote by F = P (F ) ∩ the stabilizer of the associated isotropic sub-lattice V . As subgroup of , the group F induces an equivalence relation on dominating the -one. One obtains therefore the following sequence of analytic projections: → F \ → \.
(25)
Then, as explained in Chapter 5 of [1]: Lemma 5. There exists an open subset UF ⊂ / U(N )Z ⊂ , tubular neighborhood of the Mumford boundary component B(F ) ⊂ such that on F · UF , the -equivalence reduces to F -equivalence. In the light of this lemma, the analytic projection: F \ (F · UF ) → \ ( · UF )
(26)
is an isomorphism. One has therefore an analytic identification between an open neighborhood of the Mumford divisor associated to F in \ and def
VF = F \ (F · UF ) ⊂ F \/ U(N )Z ⊂ F \(F )/ U(N )Z . But, as observed earlier, : (F )/ U(N )Z → B(F )
(27)
is a holomorphic line bundle. After factoring out the action of F , one obtains a holomorphic fibration with complex lines:
: F \(F )/ U(N )Z → F \B(F ).
(28)
One can regard then VF as an open punctured tubular neighborhood of the zero-section in (28). Based on the above arguments, one concludes that an open subset of the period domain \ which is a neighborhood of one of the two possible Type II divisors can be identified with an open punctured tubular neighborhood of the zero-section in the parabolic fibration (28). Therefore, in order to describe the structure of MK3 in the vicinity of one of the two Type II divisors DE8 ⊕E8 and D16 , it is essential to explicitly describe the holomorphic type of (28). We accomplish this task in Sect. 5. We finish this section with a note on the behavior of complex conjugation within the framework of the above construction. The complex conjugation on extends naturally to an involution of giving complex conjugation on each Type II Mumford boundary component B(F ). One can perform therefore a similar partial compactification: ˆ ˆ MF = \ ⊂ \
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ˆ ˆ with \ − \ consisting of two boundary divisors (obtained as quotients of the two Type II divisors of \ by complex conjugation). Open neighborhoods of MF near the boundary divisors are still described by open neighborhoods of the zero-section in the total space of the parabolic cover (28). 2.4. Boundary components and Hodge structures. One can give a Hodge theoretic interpretation for the boundary component B(F ). A period ω ∈ determines automatically a polarized Hodge structure of weight two on L ⊗Z C, corresponding geometrically to a marked elliptic K3 surface with section. Taking orthogonals with respect to the fixed hyperbolic sub-lattice H ⊂ L (which by construction consist of (1, 1)-cycles and is therefore orthogonal to the period line), one obtains a polarized Hodge structure of weight two on Lo ⊗Z C, 0 ⊂ {ω} ⊂ {ω}⊥ ⊂ Lo ⊗Z C.
(29)
Let then V ⊂ Lo be the primitive isotropic rank-two sub-lattice corresponding to the Type II Baily-Borel boundary component F . There is an induced weight filtration: 0 ⊂ VC ⊂ ( VC )⊥ ⊂ Lo ⊗ C.
(30)
Together, filtrations (29) and (30) yield a mixed Hodge structure on Lo ⊗Z C. Taking this point of view, one can regard the domain (F ) = U (N )C · as the space of mixed Hodge structures on the weight filtration (30). These structures are acted upon by the group U (N)C . The Type II Mumford boundary component B(F ) = (F )/U (N )C appears then as the space of nilpotent orbits of such mixed Hodge structures. There are three U (N )C -invariant graded pure Hodge structures associated to each nilpotent orbit in B(F ): 0 ⊂ {ω}⊥ ∩ VC ⊂ VC ,
0 ⊂ {ω} ∩ VC⊥ + VC /VC ⊂ {ω}⊥ ∩ VC⊥ + VC /VC ⊂ VC⊥ /VC ,
0 ⊂ {ω} + VC⊥ /VC⊥ ⊂ {ω}⊥ + VC⊥ /VC⊥ ⊂ (Lo )C /VC⊥ .
(31) (32) (33)
The first one, which we denote by H, is a pure Hodge structure of weight one induced on VC and is polarized with respect to a certain non-degenerate skew-symmetric bilinear form (·, ·)1 on V . Let (·, ·)3 be the bilinear form on Lo /V ⊥ given by (x, y)1 = (x, Ny) and (·, ·)1 be the form on V under which the isomorphism: N : Lo /V ⊥ → V
(34)
becomes an isometry. One has then (x, y)1 = ( x , y) for x, y ∈ V , where x is a lift of x to Lo . The bilinear form (·, ·)1 is non-degenerate and skew-symmetric. The space B(F ) of nilpotent U (N )C -orbits has two connected components and one can check that the Hodge structure (31) is polarized with respect to (·, ·)1 or −(·, ·)1 depending on the component the nilpotent orbit is part of. We agree to denote by B + (F ) the component for which (31) is polarized with respect to (·, ·)1 . Then B(F ) = B + (F ) B + (F ).
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The second graded Hodge structure, described in (32), has pure weight two and can be seen to be of type (1, 1). Finally, the third Hodge structure (33) has weight three, but one can check that, under the isomorphism (34), filtration (33) is just the (1, 1)-shift of Hodge structure (31). A mixed Hodge structure contains considerably more than the sum of its graded pieces. The first two graded parts are glued together by the extension of mixed Hodge structures: {0} → V → V ⊥ → V ⊥ /V → {0}.
(35)
In fact, one can check that the Hodge structure (31) together with the extension (35) completely determines the nilpotent orbit of ω. This gives a natural isomorphism between B(F ) and the space of equivalence classes of extensions of type (35). Such extensions of mixed Hodge structures have been studied by Carlson in [5]. They are classified, up to isomorphism, by an abelian group homomorphism: ψ : → J 1 (H),
(36)
where is the lattice V⊥ / V Z and J 1 (H) = VC / {ω}⊥ ∩ VC + VZ is the generalized Jacobian associated to the pure Hodge structure H described in (31). As mentioned before, has to be unimodular, even, negative-definite and of rank 16. One obtains then that, for a given Type II Baily-Borel component F , the Mumford boundary points lying in B(F ) can be identified with pairs (H, ψ) consisting of polarized Hodge structures H of weight one on VC together with homomorphisms ψ : → J 1 (E), where = V⊥ / V Z . The projection to the H-component (H, ψ) → H recovers exactly the projection:
B(F ) → F mentioned in the arithmetic discussion of the previous section. 3. Stable K3 Surfaces To this point we have described the partial compactification: \ ⊂ \
(37)
from a purely arithmetic point of view. In this section, we claim that the above compactification also has a geometrical interpretation. Namely, under the period map identification MK3 = \, (37) amounts to enlarging the moduli space MK3 by allowing certain explicit degenerations of elliptic K3 surfaces with section. Let 1 = (E8 ⊕ E8 ) and 2 = 16 be the two possible equivalence classes of unimodular, even, positive-definite lattices of rank16. We claim that there is an identification: elliptic Type II stable points on the Mumford ←→ K3 surfaces with section boundary divizor Di in − category i
and furthermore, the above correspondence can be regarded as a natural extension of the period map to the boundary.
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3.1. Definition and examples. Let us start by reviewing the notion of a Type II stable K3 surface (following [10, 11]) and the reason why these objects are natural geometrical candidates to be associated with the arithmetic Type II Mumford boundary points. Definition 6 ([11]). A Type II stable K3 surface is a surface with normal crossings Zo = X1 ∪ X2
(38)
satisfying the properties: • X1 and X2 are smooth rational surfaces. • X1 and X2 intersect with normal crossings and D = X1 ∩ X2 is a smooth elliptic curve. • D ∈ | − KXi | for i = 1, 2. • ND/X1 ⊗ ND/X2 = OD (d-semi-stability). Let us note that the above conditions imply that ωZo OZo , where ωZo is the d ualizing sheaf. Specializing the above definition, we say that a Type II stable K3 surface is endowed with an elliptic structure with section if Zo is in one of the following categories: (a) Both smooth rational surfaces Xi are endowed with elliptic fibrations Xi → P1 with sections Si ⊂ Xi . The double curve D is a smooth elliptic fiber on both sides. The two sections S1 and S2 meet D at the same point. (b) Both smooth rational surfaces Xi carry rulings defining maps Xi → P1 . The two restrictions on the double curve D agree, providing the same branched doublecover D → P1 . In addition X1 is endowed with a fixed section of the ruling, denoted So , disjoint from D. In short, a stable surface Zo is, in the case (a), the total space of an elliptic fibration X1 ∪ X2 → P1 ∪ P1 with a fixed section given by So = S1 ∪ S2 . In the case (b), Zo is the total space of a fibration X1 ∪ X2 → P1 whose generic fiber is a union of two smooth rational curves meeting at two points. The fixed rational curve S0 ⊂ X1 − D is a section for the fibration. For reasons to be clarified shortly, we shall sometimes refer to (a) and (b) as E8 ⊕ E8 and 16 categories, respectively. Two elliptic Type II stable K3 surfaces with section Zo and Zo are said to be isomorphic if there exists an isomorphism of analytic varieties f : Zo → Zo entering a commutative diagram (depending on the category): - - f f X1 ∪ X2 X1 ∪ [ XG2 X1 ∪ X2 X1 ∪ X ^ K2K B _ G s KK G ww ss GG w KK s w s GG KK w ss KK GG ww % ysss So So So {ww So # . P1 P1 ∪ P1
(39)
The reasons why one considers the configurations in (a)-(b) as elliptic structures with section on a stable K3 surface will be explained in Sect. 3.3. Let us next describe explicit examples of such configurations. Our construction pattern is as follows. Let E be a smooth elliptic curve. Consider p0 , q0 ∈ E and let ϕ1
E → P2
ϕ2
E → P2 ,
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be the projective embeddings determined by the linear systems |3p0 | and |3q0 |. Pick 18 more points p1 , p2 , · · · p18 (not necessarily distinct) on E and partition them into two ordered subsets {p1 , p2 , · · · pt } ∪ {pt+1 , pt+2 , · · · p18 }. Blow up the first copy of P2 at p1 , p2 , · · · pt (in the given order) and perform the same blow-up procedure on the second copy of P2 using the points pt+1 , pt+2 , · · · p18 . Let X1 and X2 be the resulting surfaces. A surface Zo with normal crossings is obtained by gluing X1 and X2 together along the proper transforms of ϕ1 (E) and ϕ2 (E) using, as gluing map, the isomorphism (ϕ2 )−1 ◦ ϕ1 . Definition 7. A collection {3p0 ; p1 , p2 , · · · pt ; 3q0 ; pt+1 · · · p18 }, with 3p0 and 3q0 considered as divisor classes in Pic(E), is called a special family if one of the following sets of conditions holds: (a) t = 9, p9 = p18 and OE (p1 + p2 + · · · + p8 + p9 ) = OE (9p0 ), OE (p10 + p11 + · · · + p18 ) = OE (9q0 ). (b) 2 ≤ t ≤ 17, p1 = p2 and OE (p1 + p2 + · · · + p18 ) = OE (9p0 + 9q0 ), OE (3p0 − p1 ) = OE (3q0 − pt+1 ). Let {3p0 ; p1 , p2 , · · · pt ; 3q0 ; pt+1 · · · p18 } be a special family on E. Denote by Zo (E; 3p0 ; p1 , p2 , · · · pt ; 3q0 ; pt+1 · · · p18 ) the surface with normal crossings constructed by the pattern described earlier. Theorem 8. The surface: Zo (E; 3p0 ; p1 , p2 , · · · pt ; 3q0 ; pt+1 · · · p18 ) is an elliptic Type II stable K3 surface with section. Moreover the surface falls in category (a) when the special family satisfies condition (a), and in category (b) when the special family satisfies condition (b). Proof. Assume that {3p0 ; p1 , p2 , · · · p9 ; 3q0 ; p10 · · · p18 } is a special family of type (a). Then, the double curve D of Zo is smooth elliptic and satisfies D ∈ | − KXi |, D 2 = 0. A computation involving the Riemann-Roch theorem leads to h0 (Xi , D) = 2. The linear system|D| is a base-point free pencil on each Xi and induces elliptic fibrations Xi → P1 . The exceptional curves E9 and E18 corresponding to p9 and p18 are sections in the two fibrations and they meet the double curve D at the same point. The d-stability condition on Zo is satisfied as both normal bundles ND/Xi are holomorphically trivial. We have therefore an explicit model Zo (E; 3p0 ; p1 , p2 , · · · p9 ; 3q0 ; p10 · · · p18 ) = X1 ∪ X2 → P1 ∪ P1 for an elliptic Type II stable K3 surface with section in the (a)-category. We treat now the case when {3p0 ; p1 , p2 , · · · pt ; 3q0 ; pt+1 · · · p18 } is a special family of type (b). Let H1 , H2 be the hyper-plane divisors of the two copies of P2 and denote by Ei the exceptional curve corresponding to pi . The linear systems |H1 − E1 |
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and |H2 − Et+1 | are base-point free pencils inducing rulings Xi → P1 . The restrictions of the two rulings agree on the double curve D, recovering the branched double cover E → P1 associated to the pencil |3p0 − p1 | = |3q0 − pt+1 |. Moreover, if one denotes by So the proper transform of E1 in X1 , then So is a smooth rational curve, with self-intersection −2, disjoint from D, and realizing a section of the ruling X1 → P1 . The d-semi-stability condition on Zo (E; 3p0 ; p1 , p2 , · · · pt ; 3q0 ; pt+1 · · · p18 ) is satisfied since the line bundle ND/X1 ⊗ ND/X2 is represented on E by the principal divisor 9p0 + 9q0 − p1 − p2 − · · · − p18 . We obtain therefore an elliptic Type II stable K3 surface with section Zo (E; 3p0 ; p1 , p2 , · · · pt ; 3q0 ; pt+1 · · · p18 ) = X1 ∪ X2 → P1 in the (b)-category.
The surfaces of Theorem 8 represent quite a large set of examples of elliptic Type II stable K3 surfaces with section. In fact, one can see that, up to certain explicit transformations, these surfaces actually exhaust all possibilities. Definition 9. Let Zo be an elliptic Type II stable K3 surface with section. A blowdown ρ : Z0 → P2 ∪ P2 consists of two sequences of applications: (n)
(n−1)
→ · · · → X1 → X1 ,
(m−1)
→ · · · → X2 → X2 ,
ρ1 : X1 = X1 → X1 (m)
ρ2 : X2 = X2
→ X2
(1)
(0)
(40)
(1)
(0)
(41)
such that: (0)
(0)
1. The surfaces X1 and X2 are copies of P2 . (l) (l−1) (l) is a contraction of an exceptional curve in Xi . 2. Each map Xi → Xi (n) (n−1) 3. If Zo is of type (a) then S1 , S2 are the exceptional curves associated to X1 → X1 (m) (m−1) and X2 → X2 . (l) (l−1) , l ≥ 2, 4. If Zo is of type (b) then the exceptional curve associated to Xi → Xi is a component of a reducible fiber of the ruling. Moreover, for i = 1, l ≥ 3 this exceptional curve is disjoint from So . Due to their specific construction pattern, the special surfaces Zo (E; 3p0 ; p1 , p2 , · · · pt ; 3q0 ; pt+1 · · · p18 ) carry a canonical blow-down. Furthermore, if a stable surface Zo admits a blow-down, then Zo is isomorphic to a special surface of Theorem 8. Indeed, let us assume a choice of blow-down ρ : Zo → P2 ∪ P2 . Choose p0 , q0 ∈ D such that 3p0 and 3q0 are hyper-plane section divisors for the embeddings: ρi
(0)
D → Xi → Xi , i = 1, 2. Let x1 , x2 , · · · , xn , y1 , y2 , · · · ym be the points of intersection between the exceptional (l) (l−1) and the double curve D. A cohomology calculation shows curves of Xi → Xi that m + n = 18. Then, one can see that {3p0 ; x1 , x2 , · · · , xn ; 3q0 ; y1 , y2 , · · · ym } is a special family and there is a canonical isomorphism: Zo Zo (D; 3p0 ; x1 , x2 , · · · , xn ; 3q0 ; y1 , y2 , · · · ym ) restricting to identity over D.
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Proposition 10. For any elliptic Type II stable K3 surface with section Zo in category (a), there exists a blow-down ρ : Zo → P2 ∪ P2 . Proof. As rational surfaces, both X1 and X2 have to dominate one of the geometrically ruled rational surfaces Fn , n ≥ 0. Since D meets all exceptional curves, the double curve has to be the proper transform of an effective anti-canonical divisor in Fn . Such divisors exist only if n ≤ 2. But F1 dominates P2 and F0 , F2 also dominate P2 after blowing up a point on an anti-canonical curve. Therefore, if Xi is neither Fo nor F2 (which is the case here since X1 , X2 are elliptic), one can always find blow-up sequences as in (40) and (41). Since D 2 = 0 on each Xi , it has to be that n = m = 9. Moreover, one can always choose the section components Si as the first exceptional curves to be contracted on each side. We have therefore a blow-down ρ : Z0 → P2 ∪ P2 as in Definition 9. Not all stable surfaces of category (b) admit blow-downs in the sense of Definition 9. X2 may be F0 or F2 and X1 may be F2 . None of these surfaces dominate P2 . However, it can be shown that any Zo of category (b) can be transformed, using certain explicit modifications, to a surface that admits blow-downs. An elementary modification of an elliptic Type II stable K3 surface with section Zo of category (b) consists of the blow-down of an exceptional curve C lying inside a fiber of the ruling Xi → P1 and disjoint from So , followed by the blow-up of the resulting point on the opposite rational surface. The resulting Zo is still an elliptic Type II K3 surface with section in category (b). Proposition 11. Any elliptic Type II stable K3 surface Zo with section of category (b) can be transformed, using elementary modifications, to a surface which admits a blowdown. Proof. We claim that, using elementary modifications, one can transform Zo to a new stable surface Zo such that X2 = F1 . Indeed, using an argument mentioned during the proof of Proposition 10, X2 is either F0 , or F2 or dominates F1 . If there is a blow-down X2 → F1 , then perform elementary transformations consisting of flipping successively to X1 the exceptional curves involved in the blow-down. The new X2 is clearly F1 . If X2 is rather a copy of F0 or F2 then choose an exceptional curve C sitting inside a fiber of the ruling on X1 (X1 and X2 cannot be simultaneously geometrically ruled). Let p be the point where C meets the double curve. Perform the elementary transform that takes C to X2 and flip back to X1 the proper transform of the initial rational fiber through p in X2 . The resulting X1 is then a copy of F1 . Contracting the unique section of negative self-intersection one obtains: (0)
X2 → X2
(42)
(0)
with X2 isomorphic to a projective space P2 . Assuming X2 = F1 , one has that X1 is ruled but not geometrically ruled. Let us then describe the blow-down process (n)
(n−1)
X1 = X1 → X1
(1)
(0)
→ · · · → X1 → X 1 .
(43)
We contract successively exceptional curves inside the reducible fibers of X1 , making sure that the exceptional curves in question do not intersect So . One can use this pro(2) cedure to reduce X1 to a new ruled surface X1 which has a unique reducible fiber F consisting of a union C1 ∪ C2 of two smooth exceptional curves. Pick the curve, among
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C1 , C2 , which intersects So and contract it. One obtains in this manner a projection (2) (1) (1) X1 → X1 with X1 geometrically ruled of type F1 . After contracting the image of (0) So , we are left with X1 which is a copy of P2 . Sequences (43) and (42) determine a blowdown ρ : Zo → P2 ∪ P2 . Summarizing the facts, every elliptic Type II stable K3 surface with section of category (a) is isomorphic to a surface Zo (D; 3p0 ; xp , p2 , · · · , p9 ; 3q0 ; p10 , p11 , · · · p18 ) with {3p0 ; xp , p2 , · · · , p9 ; 3q0 ; p10 , p11 , · · · p18 } a special family on D. Every stable surface of category (b) can be transformed, after elementary modifications to a surface Zo (D; 3p0 ; p1 , p2 , · · · , p17 ; 3q0 ; p18 ) associated to a special family {3p0 ; p1 , p2 , · · · , p17 ; 3q0 ; p18 }. 3.2. Stable periods and Torelli theorem. We are now in position to provide the formal connection between Type II elliptic stable K3 surfaces with section and Type II boundary points in the arithmetic partial compactification of \. This correspondence will be later justified geometrically as an extended period map, using the theory of K3 degenerations. Theorem 12. Let Zo be an elliptic Type II K3 surface with section. Denote by D the double curve. One can naturally associate to Zo a rank-sixteen unimodular even negative-definite lattice Zo together with an abelian group homomorphism: ψZo : Zo → Jac(D). Moreover, Zo is a lattice of type −(E8 ⊕ E8 ), for surfaces Zo in the (a)-category, and is of type −16 , for Zo in the (b)-category. Proof. We shall use a few known facts (see [11 and 12]) concerning the Hodge theory of a Type II stable K3 surface. The rank of H 2 (Zo , Z) is 21. The complex cohomology group H 2 (Zo , C) carries a canonical mixed Hodge structure of weight filtration: 0 ⊂ W1 ⊂ W2 = H 2 (Zo ).
(44)
The two associated graded Hodge structures involved satisfy: W1 H 1 (D) (isomorphism of Hodge structures),
W2 / W1 Ker H 2 (X1 ) ⊕ H 2 (X2 ) → H 2 (D) . One deduces that W2 / W1 has rank 19 and carries a pure Hodge structure of type(1, 1). The mixed structure on H 2 (Zo ) produces an extension of mixed Hodge structures: 0 → W1 → W2 → W2 / W1 → 0
(45)
which, according to Carlson [5], is classified by the associated abelian group homomorphism: Zo : ( W2 / W1 )Z → J 1 ( W1 ). ψ
(46)
Here J 1 ( W1 ) = W1 /F 1 W 1 + (W1 )Z is the generalized Jacobian associated to the Hodge structure on W1 . There is a purely geometrical description for(46). Since the
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Hodge structure on W1 is isomorphic to the geometrical weight-one Hodge structure of the double curve D, one has a natural identification: J 1 ( W1 ) Jac(D) = Pico (D). Moreover, since the two surfaces X1 , X2 are rational, any given cohomology class
[L] ∈ ( W2 / W1 )Z = Ker H 2 (X1 , Z) ⊕ H 2 (X2 , Z) → H 2 (D, Z) is uniquely represented by a pair of holomorphic line bundles L = (L1 , L2 ) ∈ Pic(X1 )× Pic(X2 ) satisfying L1 · D = L2 · D. The image of [L] under (46) can then be described as: Zo ([L]) = OD (L1 ) ⊗ OD (−L2 ) ∈ Pico (D) = Jac(D). ψ
(47)
Zo ([L]) = 0 for any cohomology class [L] representing a Cartier divisor In particular ψ on Zo . The lattice ( W2 / W1 )Z has rank 19 and is indefinite. However, the elliptic structure with section on Zo induces a series of Cartier divisors producing special cohomology classes. Firstly, the section So , which in case (a) is represented by two rational curves in X1 and X2 meeting D at the same point, while in case (b) is a unique rational curve in X1 disjoint from D, determines a Cartier divisor So on Zo . Secondly, the fiber on Zo , which in case (a) consists of elliptic fibers merging at D, while in case (b) consists of rulings on each Xi agreeing over the double curve, determines a Cartier divisor class Fo . Thirdly, let: ξ1 = OX1 (−D) ∈ Pic(X1 ), ξ2 = OX2 (D) ∈ Pic(X2 ). The d-stability condition assures us that the two line bundles agree over the double curve and therefore they can be seen to determine a line bundle ξo over Zo . The three Cartier divisors So , Fo , ξo on Zo determine integral cohomology classes: [So ], [Fo ], [ξo ] ∈ ( W2 / W1 )Z satisfying [So ]2 = −2, [Fo ]2 = 0, [ξo ]2 = 0, [So ].[Fo ] = 1, [So ].[ξo ] = 0, [Fo ].[ξo ] = 0. Denote by {[ξo ]}⊥ the sub-lattice of ( W2 / W1 )Z orthogonal to the class[ξo ]. Clearly all three elements [ξo ][So ] and [Fo ] belong to {[ξo ]}⊥ . Then, define: Zo ⊂ {[ξo ]}⊥ / (Z · [ξo ]) as the sub-lattice orthogonal to the equivalence classes induced by [So ] and [Fo ]. A simple observation involving the Hodge index theorem on X1 and X2 allows one to conclude that Zo is even, unimodular, negative-definite and of rank 16. As mentioned earlier, the extension homomorphism (47) vanishes on cohomology classes representing Cartier Zo vanishes an all [So ], [Fo ] and [ξo ]. Therefore, without losing divisors. In particular ψ geometrical information, one can descend (47) to an abelian group homomorphism: ψZo : Zo → Jac(D).
(48)
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The isomorphism type of the lattice Zo is characterized by the category to which the stable surface Zo belongs. Assume that Zo is a surface in the (a)-category. There is then a natural splitting Zo = 1Zo ⊕ 2Zo , where iZo = {γ ∈ H 2 (Xi , Z)|γ · [D] = 0, γ · [Si ] = 0}. Pick a blow-down ρ : Zo → P2 ∪ P2 as in Definition 9 and consider the associated classes: {H1 , H2 , E1 , · · · E18 } ⊂ H 2 (X1 , Z) ⊕ H 2 (X1 , Z) representing the proper transforms of a hyper-planes in P2 and the total transforms of the (1) (0) (2) (1) (9) exceptional curves associated to the blow-ups X1 → X1 , X1 → X1 , · · · X1 → (8) (1) (0) (2) (1) (9) (8) X1 , X2 → X2 , X2 → X2 , · · · X2 → X2 . Let α1 , α2 , · · · α8 , β1 , β2 , · · · β8 be the following sixteen elements in Zo : α1 = E1 − E2 , α2 = E2 − E3 , · · · , α7 = E7 − E8 , α8 = H1 − E1 − E2 − E3 , β1 = E10 − E11 , β2 = E11 − E12 , · · · , β7 = E16 − E17 , β8 = H2 − E10 − E11 − E12 . (49) One verifies that {α1 , α2 , · · · α8 } and {β1 , β2 , · · · β8 } are basis for 1Zo and 2Zo . Moreover, analyzing the intersection numbers, one finds out that, after changing the sign of the quadratic pairing, each of the two lines in (49) consists of a set of E8 simple roots. α1
•
α2
•
α3
•
α4
•
α5
α6
•
•
α8 •
α7
•
β1
•
β2
β3
•
•
β4
•
β5
β6
•
•
β7
•
.
β8 •
The lattice Zo is therefore isomorphic to− (E8 ⊕ E8 ). One can do a similar analysis when Zo is in category (b). Note that the
in the case isomorphism class of the pair Zo , ψZo does not change under elementary modifications. Indeed, a modification that flips an exceptional curve C from X1 to X2 induces an isometry: / H 2 (X , Z) ⊕ H 2 (X , Z)
H 2 (X1 , Z) ⊕ H 2 (X2 , Z)
1
H 2 (X1 , Z) ⊕ Z[C] ⊕ H 2 (X2 , Z)
2
(50)
/ H 2 (X , Z) ⊕ Z[C] ⊕ H 2 (X2 , Z). 1
This map sends [C] ∈ H 2 (X1 , Z) to −[C] ∈ H 2 (X2 , Z), [Fo ] to [Fo ], [So ] to [Fo ] and [ξo ] to [ξo ]. There is then an induced lattice isomorphism Zo Zo which clearly makes the diagram: / Z o KK s KK s s s % y ψZo ψ Jac(D) Zo
Zo
commutative.
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According to Proposition 11, Zo can be transformed, using elementary modifications, such that the resulting surface Zo admits a blow-down ρ : Zo → P2 ∪ P2 associated to a special family {3p0 ; p1 , p2 , · · · , p17 ; 3q0 ; p18 } on D. In such conditions, we define classes of cycles {γ1 , γ2 , · · · , γ16 } in Zo as follows: γ1 = H1 − E1 − E2 − E3 , γ2 = E3 − E4 , γ3 = E4 − E5 , · · · , γ14 = E15 − E16 , γ15 = E16 − E17 , γ16 = H2 − E18 + E16 + E17 .
(51)
One verifies that, after reversing the sign of the pairing, (51) is a root system of type D16 .
γ1
•
γ2
•
γ3
•
γ4
•
γ5
•
γ6
•
γ7
•
γ8
•
γ9
•
γ10
•
γ11
•
γ12
•
γ13
•
γ14
•
γ15
•
?? ? γ16
•
They determine therefore a sublattice of type −D16 inside Zo . The index of the sublattice is two with a generator of the quotient group being given by the class of: 2H1 − E1 − E2 − E17 + 2H2 − 3E18 . + The latticeZo is then of type−D16 and is therefore isomorphic to−16 .
(52)
We make now the connection with the arithmetic Mumford boundary points. In the notation of 2.3, assume that F is a Type II Bailey-Borel component for \, corresponding to the isotropic rank-two sub-lattice V ⊂ Lo , and B(F ) is the associated Type II Mumford boundary divisor. Let be the rank-sixteen lattice V ⊥ /V . Recall from 2.4 that B(F ) decomposes into two connected components B + (F ) B + (F ), and there is a bijective identification between boundary points in B + (F ) and pairs(H, ψ) consisting of weight-one Hodge structures on VC polarized with respect to the skewsymmetric form (·, ·)1 together with abelian group homomorphisms ψ : → J 1 (H). Let Zo be an elliptic Type II stable K3 surface with section as in Definition 6 (a)–(b). Attach to Zo a set of markings φ1 , φ2 consisting of isometries φ1 : H 1 (D, Z) → V , φ2 : Zo → .
(53)
The marking φ1 can be used to transport the geometrical weight-one Hodge structure W1 of D to a formal weight-one polarized Hodge structure H on V . There is then an induced isomorphism of abelian groups Jac(D) J 1 (H). This isomorphism, together with the marking φ2 , allows one to transport the homomorphism ψZo of Theorem 12 to a formal homomorphism ψ : → J 1 (H). In the light of the arguments in the previous paragraph, this procedure can be regarded as a period correspondence, associating to every marked elliptic Type II stable K3 surface with section (Zo , φ1 , φ2 ) in -category a marked stable period in the form of a pair (H, ψ) ∈ B + (F ). One can further refine this correspondence by removing the markings and considering the pairs (H, ψ) modulo the isometries of and V . Let us denote by SL2 (Z) the
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group of automorphisms of V preserving the skew-symmetric pairing (·, ·)1 . This group acts naturally on the set of weight-one Hodge structures on V polarized with respect to (·, ·)1 . Consider Aut() to be the group of isometries of lattice . The product group G = Aut() × SL2 (Z) acts then on the set of pairs (H, ψ) as:
(f, α).(H, ψ) = α(H), α ◦ ψ ◦ f −1 , where α : J 1 (H) → J 1 (α(H)) is the natural isomorphism induced by α. It is clear that, given a marked triplet (Zo , φ1 , φ2 ) inducing a marked pair (H, ψ), a variation of markings φ1 , φ2 or a change of Zo under an isomorphism as in (39) leaves (H, ψ) within the same G-orbit. Therefore, one can associate to any elliptic Type II stable K3 surface with section a well-defined stable period in G\B + (F ). Definition 13. Two elliptic Type II stable K3 surfaces with section in the same category: Zo = X1 ∪ X2 and Zo = X1 ∪ X2 are said to be equivalent if one of the following holds: 1. Zo and Zo are isomorphic (as in (39)). 2. Zo and Zo are both of category (a) and Zo is isomorphic to X2 ∪ X1 . 3. Zo and Zo are both of category (b) and can be made to be isomorphic by transforming each of them using a finite sequence of elementary modifications. , = E8 ⊕ E8 or 16 , be the coarse moduli spaces of equivalence Let then Mstable classes in category (a), respective (b). It can be easily seen that the stable period of a surface Zo = X1 ∪ X2 does not change when Zo gets replaced by X2 ∪ X1 (if Zo is of category (a) ) or when Zo gets transformed by an elementary modification. One has therefore a well-defined period map: per : Mstable → G\B + (F ).
(54)
Furthermore, as we shall see from the analysis in Sect. 5.2, if one denotes by + the index-two subgroup of consisting of isometries preserving the connected components of and defines F+ : = P (F ) ∩ + (recall that P (F ) is the rational parabolic subgroup associated to the Baily-Borel boundary component F ), then there exists a natural group isomorphism G F+ . Moreover, under this isomorphism, the action of G on B + (F ) reduces to the standard arithmetic action of P (F ) ∩ + . This produces a natural identification: G\B + (F ) F+ \B + (F ) = F \B(F ) = D . One can therefore interpret the stable period of an elliptic Type II stable K3 surface with section as a point of the arithmetic Mumford divisor D and hence regard (54) as a map per : Mstable → D . Theorem 14. The map (54) is an isomorphism. We prove this statement in two steps. To begin with, let us show that (54) is injective. Theorem 15. Two stable surfaces Zo and Zo of category (a), which have the same stable period, are equivalent.
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Proof. This follows from standard results concerning E8 del Pezzo surfaces (see [14, 7 and 22] for details). If Zo = X1 ∪ X2 and Zo = X1 ∪ X2 are stable surfaces of category (a), then, after contracting the sections, one obtains four E8 del Pezzo surfaces 1 , X 2 , X , X . Moreover, one has isomorphisms: X 1 2 ⊥ ⊥ 2 2 Zo = 1Zo ⊕ 2Zo [KX 1 ] ⊕ [KX 2 ] ⊂ H (X 1 , Z) ⊕ H (X 2 , Z), ⊥ ⊥ 2 2 Zo = 1Z ⊕ 2Z [KX ] ⊕ [KX ] ⊂ H (X 1 , Z) ⊕ H (X2 , Z). o
o
1
2
D), consisting of an E8 It was proved in [14] that the isomorphism class of a pair (X, del Pezzo surface X with an embedded smooth elliptic curve D, is determined by the ⊥ map [KX ] → Jac(D) modulo Weyl equivalence. Based on this argument, assuming that Zo and Zo determine the same stable period in G\B + (F ), it follows that there is an isomorphism of elliptic curves D D which extends to an isomorphism of stable surfaces of either X1 ∪ X2 X1 ∪ X2 form or X1 ∪ X2 X2 ∪ X1 form. We use different arguments for justifying the analog of Theorem 15 for stable surfaces of category (b). As shown earlier, given an elliptic Type II stable K3 surface with section Zo = X1 ∪ X2 of category (b), one can always transform Zo , by performing elementary modifications, to a new stable surface Zo such that X2 F1 . In this setting, there exist blowdowns ρ : Zo → P2 ∪P2 and each choice of such a blowdown induces a D16 simple root system {γ1 , γ2 , · · · γ16 } for Zo , as described in (51). The model Zo and the blowdown ρ are far from being unique. One can further transform Zo , using sequences of elementary modifications, to new surfaces Zo , satisfying X2 F1 , but not isomorphic to Zo . However, any modification from Zo to Zo induces a canonical isomorphism ϒ : Zo → Zo (see (50)) entering the commutative diagram: Zo
ϒ / Z o KKK s s KK% s s y s ψZo ψ Jac(D). Zo
Lemma 16. Let Zo be an elliptic Type II stable surface with section, of category (b). For any basis of simple roots S ⊂ Zo , there exists a sequence of elementary modifications transforming Zo to a new stable surface Zo = X1 ∪ X2 with X2 F1 and a blowdown ρ : Zo → P2 ∪ P2 such that the simple root system associated to ρ is ϒ (S). Proof. Any two sets of D16 simple roots can be transformed one into the other using a Weyl transformation. It suffices then to show that, given Zo with X2 F1 and fixing a blowdown ρ0 : Zo → P2 ∪ P2 with an associated set of simple roots S0 , for any Weyl transformation w ∈ W (Zo ), there exists a sequence of elementary modifications transforming Zo to Zo with X2 F1 and a blowdown ρ : Zo → P2 ∪ P2 , such that the simple root set associated to ρ is ϒ (w · S0 ). Let H1 , E1 , · · · , E17 and H2 , E18 be the hyper-plane sections and the total transforms of the exceptional curves associated to ρ0 . The simple root set S0 is: γ1 = H1 − E1 − E2 − E3 , γ2 = E3 − E4 , γ3 = E4 − E5 , · · · , γ14 = E15 − E16 , (55) γ15 = E16 − E17 , γ16 = H2 − E18 + E16 + E17 .
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We define ε1 , ε2 · · · , ε16 , elements of Zo ⊗ Q given by: ε1 =
1 1 (H2 − E18 ) + H1 − E1 − E2 , εl = (H2 − E18 ) + El+1 , 2 ≤ l ≤ 16. 2 2
The set {ε1 , ε2 · · · , ε16 } forms an orthonormal basis for 1Zo ⊗ Q (when changing the sign of the quadratic form) and the roots in S0 appear as: γl = εl − εl+1 , for 1 ≤ l ≤ 15, and γ16 = ε15 + ε16 . In this setting, it is known that the Weyl group W (Zo ) is generated by permutations of ε1 , ε2 · · · , ε16 and transformations tij , 1 ≤ i < j ≤ 16 taking εi , εj to −εi , −εj and leaving all other εl unchanged. In what follows, we shall indicate the elementary modifications and the change in blowdown sequence generating, at the level of roots, transpositions (εi , εj ). One can use a similar technique to treat the transformations tij . (l) We shall denote by X1 the surfaces obtained from X1 during the blowdown ρ0 , by El the corresponding contracting curves, and by pl the intersection points D ∩El . Start with (j +1) (17) X1 = X1 and contract successively E17 , · · · Ej +2 . The resulting surface is X1 . (j +1) is a chain C1 ∪ C2 ∪ · · · ∪ Ck of smooth rational The total transform of Ei+1 on X1 curves with self-intersection −2, with the exception of C1 which is exceptional. One has intersecting numbers Cl · Cl+1 = 1 for 1 ≤ l ≤ k − 1 and Cl · Cl = 0 otherwise. Flip C1 , C2 , · · · Ck−1 , successively, to X2 and then contract Ck . Then flip Ck−1 , Ck−2 , · · · C2 j ∪ X 2 . Next, if back. Flip Ej +1 to the right. Denote the resulting stable surface by X 1 j i ≥ 2 then contract successively the curves Ej , Ej −1 , · · · Ei+2 on X1 . Let the resulting i+1 . Flip Ej +1 back from X 2 and contract it on X i+1 . The resulting surface be denoted X 1 1 (i) surface is exactly X1 ∪ X2 . Keep then the rest of the blowdown intact and construct the upper part of the new blowdown ρ by retracing the steps and blowing up successively (i) the points pj +1 , pi+2 , pi+3 , · · · , pj , pi+1 , pj +2 , · · · p17 , on X1 . (j +1)
X1 = X1(17) → · · · → X1
↓ 1(j +1) X
(j )
→ X1
→ · · · → X1(i+1) ↓
X1
=
X (17) 1
→ ··· →
(j +1) X 1
→
(j ) X 1
→ ··· →
1(i+1) X
→ X1(i) → X1(i−1) → · · · → X1(0) P2
The new stable surface Zo = X1 ∪ X2 is obtained from Zo through such a sequence of elementary modifications and the simple root system associated to the blowdown ρ is ϒ (w · S0 ). The case i = 1 requires a slight modification of the above procedure. After obtain j +1 continue by contracting the proper transform of the line passing through p1 ing X 1 j ∪ X 2 . with multiplicity two. Flip Ej +1 to X2 . Denote the resulting stable surface by X 1 j Contract successively Ej , Ej −1 , · · · E3 on X1 . Flip Ej +1 back to the left. Contract the image of the proper transform of the line passing through p1 and pl+1 and then contract the image of the proper transform of the line passing through p1 and pl+1 . The resulting surface is a copy of P2 .
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We are then in position to justify the injectivity of the stable period map (54) for category (b) surfaces. Theorem 17. Two stable surfaces Zo and Zo of category (b), which have the same stable period, are equivalent. Proof. Since elementary modifications do not vary the stable period, we can assume that both X2 and X2 are copies of F1 . Choose a blowdown ρ : Zo → P2 ∪ P2 and denote by S ⊂ Zo the associated basis of simple roots. Let (φ1 , φ2 ), (φ1 , φ2 ) markings for Zo , Zo as in (53). Denote by(H, ψ), (H , ψ ) the induced marked periods. Since the stable periods of the two surfaces are identical, there must exist isometries α and f for V and , respectively, such that: H = α (H), ψ = α ◦ ψ ◦ f −1 .
Let S = (φ2 )−1 ◦ f ◦ φ2 (S). Then S is a basis of simple roots in Zo and, according to Lemma16, there exists a new stable surface Zo , obtained from Zo through a 2 2 sequence of elementary modifications, which admits a blowdown ρ : Z o → P ∪P such that the simple root basis S associated to ρ satisfies S = ϒ S . Let then and (3p0 ; p1 , p2 , p3 , · · · , (3p0 ; p1 , p2 , p3 , · · · , p16 , p17 ; 3q0 ; p18 ) p16 , p17 ; 3q0 ; p18 ) be the two special families on D and D induced by the blowdowns ρ and ρ , respectively. Fix base points on D and D and consider the induced identifications: D Jac(D) = J 1 (H), D Jac(D ) = J 1 (H ). Use these identifications to define an abelian group isomorphism: η: D
/ J 1 (H)
α
/ J 1 (H )
/ D,
(56)
and then constructη : D → D with η(p) = η(p) − η(p1 ) + p1 . It turns out then that the isomorphism η transports the special family(3p0 ; p1 , p2 , p3 , · · · , p16 , p17 ; 3q0 ; p18 )
, p ; 3q ; p . This implies that Z and Z are isomorto 3p0 ; p1 , p2 , p3 , · · · , p16 o o 17 0 18 phic which, in turn, implies that Zo and Zo are equivalent. One concludes from Theorems15 and 17 that the stable period map: per : Mstable → G\B + (F )
(57)
is injective. Let us then complete the proof of Theorem14: Theorem 18. The period map(57) is surjective. Proof. Let (H, ψ) be a pair in B + (F ). We show that there exists a marked surface (Zo , φ1 , φ2 ) with stable marked period(H, ψ). The Hodge-theoretic Jacobian J 1 (H) is itself a pointed elliptic curve endowed with a natural group structure. We agree to call it (E, p0 ) and denote by φ1 : H 1 (E, Z) V a marking that sends the geometrical Hodge structure of E to H. In particular, φ2 induces a group isomorphism E Jac(E) J 1 (H).
(58)
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If = E8 ⊕ E8 , pick a basis for {a1 , · · · a8 , b1 , · · · b8 } for such that {a1 , · · · a8 } and {b1 , · · · b8 } are E8 systems of simple roots. In what follows, we construct 19 points on E, denoted q0 , x1 , x2 · · · x9 , y1 , y2 · · · y9 . Choose x1 ∈ E such that: 3x1 = 2ψ(a1 ) + ψ(a2 ) − ψ(a8 ). Then construct p2 , · · · p9 , recursively, by the rule: xl = xl−1 − ψ(al−1 ), for2 ≤ l ≤ 8, x9 = −(x1 + x2 + · · · + x8 ). Then set y9 = x9 and: y1 = y9 + (7ψ(b1 ) + 6ψ(b2 ) + 5ψ(b3 ) + · · · + 2ψ(b6 ) + ψ(b7 )) −3 (2ψ(b1 ) + ψ(b2 ) − ψ(b8 )) . Construct then recursively yl = yl−1 + ψ(bl−1 ) for 2 ≤ l ≤ 8 and then pick q0 ∈ E such that: 3q0 = 2ψ(b1 ) + ψ(b2 ) − ψ(b8 ) + 3y1 . One verifies that(3p0 ; x1 , x2 , · · · x9 ; 3q0 ; y1 , y2 , · · · y9 ) is a special family of category (a) on E. The stable surface Zo = Zo (E; 3p0 ; x1 , x2 , · · · x9 ; 3q0 ; y1 , y2 , · · · y9 )
(59)
is then an elliptic Type II stable K3 surface with section, of category (a). Moreover, Zo comes endowed with a natural blow-down. Let {α1 , α2 , · · · α8 , β1 , β2 , · · · β8 } ⊂ Zo be the ordered set of simple roots associated to the respective blow-down. Then, under the isomorphism (58), ψZo (αi ) = ψ(ai ), ψZo (βi ) = ψ(bi ) for 1 ≤ i ≤ 8. In other words, if φ2 : Zo → is the marking sending {α1 , α2 , · · · α8 , β1 , β2 , · · · β8 } to {a1 , · · · a8 , b1 , · · · b8 }, then the diagram: Zo
ψZo
/ Jac(E)
ψ
/ J 1 (H)
φ1
is commutative. We conclude that the marked stable period of the marked triplet (Zo , φ1 , φ2 ) coincides with the pair(H, ψ). A similar procedure can be used if = 16 . Fix{c1 , c2 , · · · c16 } a basis of D16 simple roots in . We shall construct a set of 19 points q0 , p1 , p2 , · · · p18 in E. To begin with pick p1 ∈ E such that: 3p1 = − (2ψ(c1 ) + 2ψ(c2 ) + · · · + 2ψ(c13 ) + 2ψ(c14 ) + ψ(c15 ) + ψ(c16 )) . Define then p2 = p1 and construct recursively pl = pl−1 − ψ(cl−2 ) for 3 ≤ l ≤ 17. Pick then q0 ∈ E such that: 6q0 = 2p1 + p2 + p3 + · · · p17 .
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Finally, set p18 = p1 + 3q0 . It follows then that (3p0 ; p1 , p2 , · · · p17 ; 3q0 ; p18 ) is a special family of category (b) on E. Then, Zo = Zo (E; 3p0 ; p1 , p2 , · · · p17 ; 3q0 ; p18 )
(60)
is an elliptic Type II stable K3 surface with section endowed with a canonical blowdown. If {γ1 , γ2 , · · · γ16 } ⊂ Zo is the ordered set of simple roots associated to the respective blow-down, then, under the isomorphism(58), one has: ψZo (γl ) = ψ(cl ), 1 ≤ i ≤ 16. Therefore, if one sets a marking φ2 : Zo → such that the ordered basis {γ1 , γ2 , · · · γ16 } is sent to{c1 , c2 , · · · c16 }, then the marked stable period associated to(Zo , φ1 , φ2 ) is (H, ψ). 3.3. Stable surfaces as K3 degenerations. We have seen that the two Type II Mumford boundary divisors DE8 ⊕E8 involved in the partial compactification of \ can be regarded as moduli spaces of periods for elliptic Type II stable K3 surfaces with section in the E8 ⊕ E8 and 16 category, respectively. In this section we justify the presence of such surfaces from a geometrical point of view, as they appear naturally as central fibers for certain degenerations of K3 surfaces. Definition 19. A one-variable degeneration of elliptically fiberedK3 surfaces with section consists of a commutative diagram of analytic maps: / ~? ~ ~~ ~~ ~ ~ S
Z
π
(61)
where Z is a smooth three-fold,S is a smooth surface, is the unit disk. In addition, the structure requires the presence of an analytic section s : S → Z and of a line bundle F on Z, such that: • For every t ∈ ∗ , Zt is a smooth K3 surface, St is a smooth rational curve and the projection Zt → St is an elliptic fibration with section st : St → Zt . • The restriction of F on Zt coincides with the line bundle associated to the elliptic fiber in Zt → St . π
π
Two degenerations Z → and Z → as in (61) are said to be equivalent if one has a birational map α : Z → Z entering the commutative diagram: α
+ / o π ? `@@ π Z U @ @@ @@ @ + α S S
(62)
Z H s
s
and satisfying F = α ∗ F . One can think of an equivalence class of K3 degenerations as a punctured disc embedded in the moduli space MK3 . Intuitively, one can then regard the degenerated central
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fibers Zo as geometrical representatives for boundary points in a compactification of MK3 . A major difficulty appears here due to the fact that equivalent degenerations can have quite different central fibers. One tries to surmount this obstacle by restricting to more distinguished degenerations in the hope of obtaining a canonical model of central fiber for each degenerating equivalence class, which is a requirement for any attempt of geometrical partial compactification. Along this reasoning line (see [9, 27, 11, 20] π for details), we restrict ourself to degenerations Z → as in (61) which are semi-stable (meaning that the central fiber Zo is a surface with normal crossings) and satisfy KZ = OZ . We shall call these Kulikov degenerations. Since π1 (∗ ) Z, it is not necessarily possible to attach a consistent set of markings to the surfaces in such a Kulikov family. Attached to each degeneration, there is a monodromy operator:
(63) T ∈ Aut H 2 (Zt , Z) , which can be described explicitly as the Picard-Lefschetz transformation obtained by transporting cycles around the origin t = 0 in while preserving the classes representing the elliptic structure and section. The operator T is unipotent, meaning(T − I )3 = 0, which is equivalent to saying that its logarithm N = (T − I ) −
1 (T − I )2 2
(64)
is a nilpotent endomorphism of H 2 (Zt , Q) satisfying N 3 = 0. Complexifying the picture, one obtains a monodromy weight filtration:
{0} ⊂ Im N 2 ⊂ Im (N ) ∩ Ker (N ) ⊂ Im (N )
+ Ker (N ) ⊂ Ker N 2 ⊂ H 2 (Zt , C). (65) Moreover, as explained in [29], the degeneration data produces a mixed Hodge structure on H 2 (Zt , C) with weight filtration (65), the limiting mixed Hodge structure. With respect to this structure, the nilpotent endomorphism N becomes a morphism of mixed Hodge structures of type (−1, −1). Kulikov degenerations fall into three categories, denoted Type I, Type II and Type III, depending on whether N = 0, N 2 = 0 but N = 0, or N 3 = 0 but N 2 = 0. We shall restrict our attention here only to Type II families (N = 0 but N 2 = 0) as that will turn out to be the case relevant to our prior discussion. In this case N is always an integral endomorphism. The elliptic Type II stable K3 surfaces with section appear then naturally as central fibers for Type II degenerations with primitive N . π
Proposition 20. 1. Let Z → be a degeneration as in (61) which is Kulikov of Type II with primitive endomorphism N . The central fiber Zo is then an elliptic Type II stable K3 surface with section. 2. For every elliptic Type II stable K3 surface with section Zo , there exists a Type II π Kulikov degeneration Z → as in (61) with central fiber Zo . Proof. Both statements can be deduced easily from standard results on K3 degenerations (see [9 , 10 , 11 and 20]). Indeed, to prove the first part of the proposition, assume
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that Zo is a central fiber of a degeneration: / ? ~~ ~ ~ ~~ ~~ S
Z H s
π
(66)
as in(61), which is Kulikov of Type II with N primitive. Recall the following fact from [20]: Theorem 21. The central fiber of a Type II Kulikov degeneration of K3 surfaces is always a chain of smooth rational surfaces: Z0 = X1 ∪ X2 ∪ · · · ∪ Xr+1 . The surfaces V2 , V3 ,· · · Vr are smooth elliptic ruled. The chain contains only double curves and all double curves are smooth and elliptic. The integer r can be detected from the arithmetic of the degeneration by writing N = rNo with No primitive. Since we expect that N itself is primitive, it has to be that r = 1 and therefore the central fiber Zo is a union X1 ∪ X2 of two rational surfaces glued along an elliptic curve D. Let us analyze then the central fiber configuration: X1 ∪ X2 → So .
(67)
Since S → is a degeneration of smooth rational curves, So has to be a chain of rational curves. The map (67)is proper and its domain is a union of two irreducible varieties. Therefore, So cannot have more than two irreducible components. We divide then our discussion into two cases: 1. So is a union S1 ∪ S2 of two copies of P1 meeting at one point. 2. So is a smooth rational curve. In the first case, Si represents the image of Xi through (67) and D is the fiber above the common point. We have therefore two elliptic fibrations Xi → Si agreeing over the double curve. The section so : So → Zo allows us to regard S1 and S2 as two smooth rational curves embedded in X1 and X2 , respectively. The two curves S1 and S2 meet D at the same point. This is exactly the configuration required for Zo to be an elliptic Type II stable K3 surface with section of category (a). In the second case, the section So lies entirely inside one of two surfaces Vi . Assume So ⊂ V1 . Since So corresponds to a Cartier divisor on Zo , it cannot intersect the double curve D. Therefore So2 = −2. The projection (67) restricts to rulings: Vi → P 1 with the double curve D playing the role of a bi-section on each side. We obtain therefore that Zo is an elliptic Type II stable K3 surface with section of category (b). In order to prove the second assumption, we recall some facts pertaining to the deformation theory for stable K3 surfaces. Theorem 22 ([10]). Let Zo be a Type II stable K3 surface. 1. Zo is smoothable and appears as a central fiber in a Kulikov semi-stable degeneration.
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2. The space of first-order deformations of Zo :
T1Zo = Ext1 1Zo , OZo is 21-dimensional. 3. The versal deformation space of Zo looks like V1 ∪ V2 ⊂ T1Zo , where V1 and V2 are two smooth divisors meeting normally. The points of V1 correspond to locally trivial deformations of Zo . The points of V2 \V1 represent deformations of K3 surfaces. V1 ∪V2 corresponds to locally trivial deformations of Zo which remain d-semi-stable. Therefore, given an elliptic Type II stable K3 surface with section Zo , there are always plenty of smoothings of Zo . We just have to show that, on some of these deformations, the two Cartier divisors So and Fo can be extended on the three-fold. The obstruction to extending a Cartier divisor of the central fiber is measured by the Yoneda pairing [33]:
< ·, · > : Ext1 1Zo , OZo ⊗ H 1 Zo , 1Zo → Ext2 OZo , OZo = H 2 Zo , OZo , (68) which is non-degenerate for stable K3 surfaces. The Zariski tangent space to the smoothing component V2 is given by the hyper-plane (see [11]): σ ∈ T1Zo | < σ, [ξo ] >= 0 ⊂ T1Zo , where [ξo ] is the class in H 1 (Zo , 1Zo ) associated to the Cartier divisor ξo . The formal Zariski tangent space to the space of smoothings extending the elliptic structure and section is then given by: (69) σ ∈ T1Zo | < σ, [ξo ] >=< σ, [Fo ] >=< σ, [So ] >= 0 . Since Yoneda pairing is non-degenerate and [ξo ], [Fo ] and [So ] are independent in H 1 (Zo , 1Zo ), (69) is 18-dimensional. The space of versal deformations extending the elliptic structure and section has then a unique smoothing component V2 of dimension 18. The points V2 away from the discriminant locus correspond to deformations as in Definition19. Let us then present the stable period map (54) as a natural extension of the K3 period correspondence MK3 \. Assume that Z → is a degeneration of elliptic K3 surfaces with section, as in Definition19, which is Kulikov, Type II semi-stable, and has primitive endomorphism N . There is then a corresponding Griffiths’ period map (see [18]): : ∗ → \.
(70)
Following results of Mumford [1] and Schmid [29], one sees that(70) extends to a holomorphic map: : → \. (0). Choose a compatible marking Recall the construction of the boundary point H 2 (Zt , Z) L as in Sect. 2.1. The endomorphism N is integral and vanishes on both
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cohomology classes [Ft ], [St ] ∈ H 2 (Zt , Z) corresponding to the elliptic fiber and section in Zt → St . Therefore, it defines an isotropic rank-two sub-lattice V ⊂ Lo . Define = V ⊥ /V and let F be the Baily-Borel component associated to V . The monodromy weight filtration (65) associated to the degeneration Z → is just the complexification of: {0} ⊂ Im (N ) ⊂ Ker (N ) ⊂ H 2 (Zt , Z).
(71)
Taking the orthogonal part to the fiber and section classes [Ft ] and [St ], reduces (71) to: {0} ⊂ Im (N) ⊂ Ker (N ) ∩ {[Ft ], [St ]}⊥ ⊂ H 2 (Zt , Z) ∩ {[Ft ], [St ]}⊥ ,
(72)
which corresponds under the marking to: {0} ⊂ V ⊂ V ⊥ ⊂ Lo .
(73)
By the classical construction of Schmid [29], the family Z → induces a nilpotent orbit of limiting mixed Hodge structures with weight filtration (71). These structures descend, under the marking, to give a nilpotent orbit of polarized mixed Hodge structures on (73). The resulting U (N )C -orbit consists essentially of the decreasing filtrations: Lo ⊗ C ⊃ { exp(zN ) · ωt }⊥ ⊃ { exp(zN ) · ωt } ⊃ {0}, z ∈ C,
(74)
where {ωt } ⊂ Lo ⊗ C is the marked period line of Zt . But, as explained in Sect. 2.4, such a nilpotent orbit of Hodge structures is equivalent to a point on the Type II Mum (0) is the class of this point on the quotient boundary divisor ford component B + (F ). D ⊂ \ − \. Let then Zo be the central fiber of Z → . According to Proposition 20, Zo is an elliptic Type II stable K3 surface with section. (0) ∈ D coincides with the stable period of Zo . Theorem 23. The boundary point Proof. The Clemens-Schmid exact sequence (see [17] for details) allows one to relate the geometric mixed Hodge structure of Zo with the limiting mixed Hodge structure associated to the degeneration Z → . Essentially, this argument asserts the exactness of the following sequence of mixed Hodge structures: (−2,−2)
(3,3)
(−2,−2)
(3,3)
(0,0)
{0} → H 0 (Zt ) −→ H4 (Zo ) −→ H 2 (Zo ) −→ H 2 (Zt ) N
−→ H 2 (Zt ) −→ H2 (Zo ) −→ H 4 (Zt ) · · · . The sequence also holds as a cohomology sequence over Z.
(75)
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We use the notation from Sect. 3.2. The homology group H4 (Zo , Z) has rank two and is generated by the two fundamental cycles [Xi ] ∈ H4 (Zo , Z), i = 1, 2. Moreover, the image of [X1 ] and [X2 ] through the morphismH4 (Zo ) → H 2 (Zo ) is given by [ξo ] and−[ξo ]. The exactness of (75) allows one then to construct the following commutative diagram of polarized mixed Hodge structures:
0
0
0
0
/ W1
/ W1 H 2 (Zt )
/0
/ C[ξo ]
/ W2
/ W2 H 2 (Zt )
/0
/ W2 /W1
/ W2 H 2 (Zt )/W1 H 2 (Zt )
/0
0
0.
id
/ C[ξo ]
0
This leads to the following isomorphism of extensions of mixed Hodge structures:
(76)
0
0
0
/ W1
/ W1 H 2 (Zt )
/0
0
/ W2 /C[ξo ]
/ W2 H 2 (Zt )
/0
0
/ (W2 /W1 ) /C[ξo ]
/ W2 H 2 (Zt )/W1 H 2 (Zt )
/0
0
0.
Take then the orthogonal subspaces with respect to {[Fo ], [So ]} and {[Ft ], [St ]} in W2 /C[ξo ] and W2 H 2 (Zt ), respectively. One can make use of the marking to identify W1 H 2 (Zt ) VC , W2 H 2 (Zt ) ∩ {[Ft ], [St ]}⊥ = H 2 (Zt ) ∩ {[Ft ], [St ]}⊥ (VC )⊥ .
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In this setting, the diagram (76) becomes: (77)
0
0
0
/ W1
/ VC
/0
0
/ (W2 /C[ξo ]) ∩ {[Fo ], [So ]}⊥
/ (V )⊥ C
/0
0
/ ((W2 /W1 ) /C[ξo ]) ∩ {[Fo ], [So ]}⊥
/ V /(V )⊥ C C
/0
0
0.
Note that J 1 (W1 ) = Jac(D),
Zo = (W2 /W1 )Z /Z[ξo ] ∩ {[Fo ], [So ]}⊥ and the classifying homomorphism for the left hand-side extension of mixed Hodge structure is the abelian group homomorphism ψZo : Zo → Jac(D) associated to Zo by Theorem 12. As for the right hand-side extension, we have V ⊥ /V = and the weight-one Hodge structure on VC , denoted H, is given by the decreasing filtration VC ⊃ {ωt }⊥ ∩ VC ⊃ {0}. If ψ : → J 1 (H) is the classifying homomorphism, then the pair (H, ψ) represents (0) in B + (F ) (see the discussion in 2.4). The diagram (77) produces isometries ϕ1 : H 1 (D, Z) = (W1 )Z → V , ϕ2 : Zo → V ⊥ /V = which together define a marking for Zo . Moreover, the marking (ϕ1 , ϕ2 ) induces an isomorphism of elliptic curves Jac(D) ≈ J 1 (H) which enters a commutative diagram Zo
ϕ2
/
≈
/ J 1 (H).
ψZo
Jac(D)
(78)
ψ
(0) in D is exactly the stable period of Zo . This allows one to conclude that
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4. Boundary Components and Flat Bundles There exists a second geometric interpretation, more relevant from the point of view of heterotic/F-theory duality, for the boundary points on the two Type II Mumford divisors D with = E8 ⊕ E8 or = 16 . Recall that, given a Baily-Borel component F , D = F+ \B + (F ) and the points in B + (F ) are in one-to-one correspondence to pairs(H, ψ)of polarized weight-one mixed Hodge structures H on V together with abelian group homomorphisms ψ : = V ⊥ /V → J 1 (H). Such a pair is known to determine a flat G-connection over the elliptic curve E = J 1 (H). The Lie group G is (E8 × E8 ) Z2 if = E8 ⊕ E8 and Spin(32)/Z2 if = 16 . Let us briefly review the connection. For explicit details, see [15]. It is a standard fact that, given a compact Lie group G and a smooth two-torus E, there is a bijective correspondence between the equivalence classes of flat G-connections on E and their associated holonomy morphisms π1 (E) → G, up to conjugation. One can therefore formally identify a flat connection with a commuting pair of elements in G, up to simultaneous conjugation. Fix a maximal torus T ⊂ G. If G is simply connected, (in particular for G = (E8 × E8 ) Z2 ), it was shown that any given pair of commuting elements in G can be simultaneously conjugated in T . The same statement is true for G = Spin(32)/Z2 , providing that one considers only connections which can be lifted to Spin(32)-connections. In this way, a flat G-connection on E can be formally understood as an element in: Hom (π1 (E), T ) /W, where W is the Weyl group of G. The lattice plays the role of the lattice of the maximal torus T . In this framework: Hom (π1 (E), T ) Hom (π1 (E), U (1) ⊗ ) Hom (π1 (E), U (1)) ⊗ . The first factor of the last term above represents the set of gauge equivalence classes of flat hermitian line bundles over E. In the presence of a complex structure on E, one can identify Hom (π1 (E), U (1)) to Pico (E) which, in turn, is a complex torus isomorphic to E. There exists then a bijective correspondence between flat G-connections and points of the analytic quotient: E ⊗ /W
(79)
which one can see as the moduli space of flat G-bundles over E. Due to the unimodularity ∗ , each element in (79) can be regarded as a class of a morphism → E. Along this idea, one can associate to any Type II boundary point of B + (F ), a smooth elliptic curve E = J 1 (H) and a flat G-bundle. In Sect. 5.2 we shall show that all possible flat G-bundles are realized1 and that two points in B + (F ) determine equivalent pairs of elliptic curves and flat bundles exactly when they belong to the same F+ -orbit. This will lead to a holomorphic identification between the boundary divisors DE8 ⊕E8 , D16 and the moduli spaces ME,E8 ×E8 Z2 and ME, Spin(32)/Z2 of equivalence classes of pairs of elliptic curves and flat bundles, respectively. 1
For G = Spin(32)/Z2 , we only look at flat bundles liftable to Spin(32).
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5. Explicit Description of the Parabolic Cover In this section we give an explicit description of the two Type II boundary divisors D = F \B(F ) and identify precisely the holomorphic type of the parabolic fibrations given in (28):
: F \(F )/U (N )Z → F \B(F ).
(80)
This leads, following 2.3, to a description of the structure of MK3 in a neighborhood of D . 5.1. Fixing the parabolic group. Let F be a fixed Type II Baily-Borel boundary component for \. Denote by V the associated primitive isotropic rank-two sub-lattice of Lo and set, as in 2.3:
= − V ⊥ /V , P(F) = Stab (VR ) ⊂ O ++ (2, 18), W(F ) = the unipotent radical of P(F), U(F ) = the center of W(F) . It follows then that U(F ) is a one-dimensional Lie group with Lie algebra: u(F ) = { N ∈ EndR (Lo ⊗ R) | Im(N ) = VR and (N x, y) + (x, Ny) = 0}. Lemma 24. There exists a basis {A, B} for V such that the endomorphism N : Lo → Lo defined by: N (x) = (x, B)A − (x, A)B
(81)
is primitive, integral and belongs to u(F ). Proof. Let A be a primitive element of V . Due to unimodularity, there exists A ∈ Lo satisfying (A, A ) = 1. Pick B ∈ V , primitive, such that (B, A ) = 0. It follows that {A, B} forms a basis for V . Let then N be the endomorphism defined in (81). Pick C ∈ Lo such that(B, C) = 1 and define: B = C − (C, A )A − (C, A)A + (C, A)(A, A )A ∈ V . One verifies that N(A ) = −B and N (B ) = A. Therefore, N is primitive and Im(N ) = V . Moreover, since: (N x, y) = (x, B)(y, A) − (x, A)(y, B) = −(x, Ny) for any x, y ∈ Lo , the endomorphism N belongs to u(F ).
In order to facilitate future computations, we shall introduce a special coordinate system on Lo . The linearly independent family {A , B , A, B}, can be seen to provide a decomposition:
Mathematics Underlying F-Theory/Heterotic String Duality in 8 Dimensions
Lo Z · A ⊕ Z · B ⊕ (Z · A ⊕ Z · B) ⊕ . 2 2 Z Z
547
(82)
In this light, any element Lo (or Lo ⊗ C) can be written uniquely as: x1 A + x2 B + y1 A + y2 B + z. We convene therefore to regard the elements of Lo as triplets (x, y, z) with x = (x1 , x2 ) ∈ Z2 , y = (y1 , y2 ) ∈ Z2 and z ∈ . The quadratic pairing on Lo is recovered as:
(x, y, z), (x , y , z ) = x.y + x .y − (z, z ), where the first two dot-pairings on the left represent the standard Euclidean pairing on Z2 and (·, ·) is the pairing of . Under this rule, the isotropic lattice V corresponds to the space of triplets (0, y, 0) and the integral endomorphism N is given by N (x, y, z) = (0, T x, 0) with T : Z2 → Z2 is the standard skew-adjoint endomorphism T (x1 , x2 ) = (x2 , −x1 ). As in 2.3, we define the groups: U(N )C : = { exp (λN) |λ ∈ C}, U(N )Z : = { exp (λN) |λ ∈ Z}, leading to the sequence of inclusions: ⊂ (F ) = U(N )C · ⊂ ∨ . We shall use the newly introduced coordinate system to analyze these inclusions. Let ¯ This function is invariant r : Lo ⊗C → R be the function defined by r(ω) = −i(N ω, ω). under the action of U (N )C . In fact, if ω = (x, y, z) then r(ω) = 2 Im(x1 x2 ). Let ∨ = − (F ) ∪ 0 (F ) ∪ + (F ) be the decomposition of ∨ in subsets for which r(ω) is strictly negative, zero and strictly positive, respectively. Proposition 25. The following statements hold: 1. (F ) = − (F ) ∪ + (F ). 2. If [ω] = [a, b, c] ∈ + (F ), then: −
a1 a2
(83)
is a well-defined element of the upper-half plane. Proof. Let [ω] ∈ (F ). We show that r(ω) = 0 by proving that the opposite statement leads to a contradiction. Indeed, assume that r(ω) = 0. Since ω ∈ (F ), there exists ωo ∈ such that ω = exp(zN ).ωo = ωo + zN ωo for some z ∈ C. Then: (ω, ω) ¯ = (ωo , ω¯o ) − 2 Im(z) · r(ωo ) = (ωo , ω¯o ) > 0.
(84)
But r(ω) = 0 also implies that {ω, Nω, N ω} ¯ span an isotropic subspace of Lo ⊗ C. Clearly, ω and Nω are independent (otherwise ω ∈ VC , contradicting (84)). Since the
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A. Clingher, J.W. Morgan
largest isotropic subspace in Lo ⊗ C is two-dimensional, it has to happen that N ω¯ is generated by ω and N ω. But that also implies ω ∈ VC , leading to a contradiction. This shows that: (F ) ⊂ − (F ) ∪ + (F ). The reverse inclusion is straightforward. Turning to the second statement, one has r(ω) = 2 Im(a1 a2 ) > 0. The denominator of (83) is therefore non-zero. Moreover, the same formula leads to: a1 r(ω) Im − = a2 2|a2 |2 which assures us that (83) is an element of the upper half-plane.
We are now in position to write explicit formulas for the geometric assignment, described in 2.4, that associates to a nilpotent orbit in B(F ) = (F )/ U(N )C , a pair (H, ψ) consisting of a weight-one Hodge structure H on V and a homomorphism ψ : → J 1 (H). Under the identification V Z2 , provided by the basis {A, B}, the skew-symmetric bilinear form (·, ·)1 is transported to (x, y)1 = x.T y = x1 y2 − x2 y1 . The Hodge structures of weight one on V which are polarized with respect to (·, ·)1 are then indexed by purely imaginary complex numbers τ belonging to the upper half-plane H. Every suchτ induces the polarized weight-one Hodge structure: 0 ⊂ {A + τ B} ⊂ VC
(85)
and the correspondence is one-to-one. Let then [ω] = [a, b, c] be an element in (F ). As described in 2.4, the Hodge structure H associated to the nilpotent orbit of [ω] in B(F ) = (F )/ U(N )C is given by the filtration: 0 ⊂ {[ω]}⊥ ∩ VC ⊂ VC .
(86)
Using the coordinate framework, the middle space in (86) is: {[ω]}⊥ ∩ VC = {(0, y, 0) ∈ VC |a.y = 0}. An identification of the two filtrations (85) and (86) leads one to: τ =−
a1 . a2
(87)
Connecting (87) to Proposition 25 we see that the decomposition B(F ) = (F )/U (N )C = + (F )/U (N )C ∪ − (F )/U (N )C corresponds to the decomposition B(F ) = B + (F ) ∪ B(F ) of Sect. 2.4. The Hodge structure H is polarized with respect to (·, ·)1 if and only if [ω] ∈ + (F ).
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The coordinate framework can also be used to give a straightforward procedure constructing the extension homomorphism:
ψ : → J 1 (H) = VC / {ω}⊥ ∩ VC + V (88) associated to [ω] = [a, b, c]. Ifγ ∈ , choose a lifting γ = (0, β, γ ) ∈ VC⊥ such that γ , ω) = 0. This amounts to choosing β ∈ VC with β.a = γ .c. Clearly, such a β is not ( unique but two different choices always differ by an element in {[ω]}⊥ ∩ VC . Moreover, if one denotes: c z= = z2 − τ z1 , z1 , z2 ∈ R (89) a2 then the homomorphism ψ can be described as assigning: γ −→ ((γ , z1 ), (γ , z2 )) ∈ J 1 (H). The element z ∈ C , defined as in (89), totally controls the homomorphism ψ. We have reached therefore the following conclusion: Theorem 26. The geometric correspondence of Sect. 2.4 which associates to boundary points in B + (F ) pairs (H, ψ) of polarized weight-one Hodge structures on VC and extension homomorphisms ψ : → J 1 (H) induces an identification: B + (F ) = + (F )/ U(N )C H × C .
(90)
Under this identification, the holomorphic C-fibration of (27) is described by the map: a c : + (F ) → H × C , ([a, b, c]) = − 1 ,
. (91) a 2 a2 One immediately verifies in (91) the main features of (27), namely: is an onto holomorphic map. • is invariant under the action of U(N )C on + (F ) and the fibers of coincide • with the orbits of the U(N )C -action. is therefore a holomorphic U(N )C -principal bundle. At this point, recall that one obtains the parabolic cover (80) by further taking the quotient with respect to the action of the parabolic group of integral isometries F+ = P (F )∩ + ( + consists of those isometries preserving the connected components of ). It is important therefore to understand the group F+ and its action on + (F )/ U(N )Z and H × C . 5.2. Description of F+ and its action on B + (F ). The integral isometries F+ can be given a matrix description using the coordinate framework (82). Lemma 27. A transformation in , stabilizing the isotropic sub-lattice V is of the form: m 0 0 Qf , g(m, Q, R, F ) = R m (92) Qt m 0 f where:
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A. Clingher, J.W. Morgan
1. m ∈ GL2 (Z).
−1 . 2. m = mt 3. Q ∈ Hom(, Z2 ), R ∈ End(Z2 ) satisfying R t m + mt R = mt QQt m. 4. f is an isometry of . Moreover g(m, Q, R, f ) ∈ F+ if and only if m ∈ SL2(Z). This gives a matrix characterization for F+ . The group multiplication law goes as follows: g(m1 , Q1 , R1 , d1 ) · g(m2 , Q2 , R2 , d2 ) = g(m1 m2 , Q1 + m 1 Q2 f1t , R1 m2 + m 1 R2 + Q1 f1 Qt2 m2 , f1 f2 ).
(93)
In particular: g(m, Q, R, f )−1 = g(m−1 , −mt Qf, R t , f −1 ). We single out the following special subgroups of F+ : ! (1) U(N )Z = g(I, 0, R, I )|R + R t = 0 Z. (2) S = {g(m, 0, 0, I )|m ∈ SL2 (Z)} SL2 (Z). (3) W = {g(I, 0, 0, f )|f ∈ O()} O(). ! (4) T = g(I, Q, R, I )|R + R t = QQt . It can be verified that:
• U(N )Z ⊂ Z F+ • T F+ . • S ∩ W = {±I }.
• F+ decomposes as a semi-direct product T W ×{±I } S . The parabolic subgroup F+ acts on the total space + (F ) of the holomorphic C∗ -bundle: a1 c + . (94)
: (F ) → H × C , ([a, b, c]) = − , a2 a2 There is a compatible action on H × C which carries an important geometric significance. Recall that a pair (τ, z) ∈ H × C determines a polarized mixed Hodge structure on VC together with a homomorphism ψ : → J 1 (H) given essentially byψ(γ ) = ((γ , z1 ), (γ , z2 )), where z = z2 − τ z1 . As mentioned earlier, the Jacobian J 1 (H) can be regarded as an elliptic curve Eτ = C/Z ⊕ τ Z and, in this setting, the morphism ψ determines a flat G-connection over Eτ (the Lie group Gis E8 ×E8 Z2 if = E8 ⊕E8 lattice and G = Spin(32)/Z2 if = 16 ). Denote by π : H × C → H the projection on the first coordinate. Taking then: Lτ : = {τ } × ⊗ (Z ⊕ τ Z) ⊂ π −1 (τ ) one obtains a family of 32-dimensional lattices, parameterized by τ , moving in the fibration π. Definition 28. Let be the group of holomorphic automorphisms of the fibration π which preserve the lattice family L and cover PSL(2, Z) transformations on H.
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It turns out that two elements (τ, z) and (τ , z ) of H × C determine isomorphic pairs of elliptic curves and flat G-connections if and only if they can be transformed one into another through an isomorphism in . In this sense, the analytic space: ME,G = \ (H × C )
(95)
can be seen as the moduli space of pairs of elliptic curves and flat G-bundles2 . Theorem 29. There is a short exact sequence of groups: {1} → U(N )Z → F+ → → {1} α
(96)
of (94) is α-equivariant. This induces a with respect to which the analytic fibration holomorphic identification: D = F+ \B + (F ) \ (H × C ) = ME,G
(97)
between the Type II boundary divisor D corresponding to F and the moduli space of pairs of elliptic curves and flat G-bundles. Moreover, under (97), the quotient map:
: F+ \+ (F ) → \ (H × C )
(98)
is exactly the parabolic Seifert fibration (28) of Sect. 2.3. Proof. For any ϕ ∈ F+ , one can construct a well-defined automorphism of H × C by taking (τ, z) −→ (ϕ(ω)) ) of (τ, z) in + (F ). We claim that all such transformations where [ω] is a lift (under are elements of . Let (τ, z) ∈ H × C and g(m, Q, R, f ) ∈ F+ defined as in (92). Choose [ω] = −1 (τ, z). It can be assumed that x = (−τ, 1) and z = z2 − z1 τ with z1 , [x, y, z] ∈ z2 ∈ R . If m ∈ SL2 (Z) is has the matrix form: a b cd then
m =
d −c −b a
and the action of g(m, Q, R, f ) is just: " # g(m, Q, R, f ).[ω] = m · x, R · x + m · y + Q · f · z, Qt · m · x + f · z . (g(m, Q, R, f ).[ω]) = (τ , z ) with An easy calculation shows that τ = 2
aτ − b , z = Qt (−τ , 1) + (df (z2 ) − bf (z1 )) + (−cf (z2 ) + af (z1 )) τ . −cτ + d
Again, in the case = 16 , one considers only Spin(32)-liftable connections.
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It is clear then that the transformation: (g(m, Q, R, f ).[ω]) = (τ , z ) (τ, z) →
(99)
covers a PSL(2, Z) transformation on the first factor of H × C corresponding to the matrix action of a −b m = . −c d In addition, one notes that the transformation (99) preserves the lattice family L. It is therefore with a well-defined transformation in . The above assignment induces a group homomorphism α : F+ → . One can see that α (g(m, Q, R, f )) = 1 requires m = I , f = I and Q = 0. This implies that Ker(α) = U(N )Z . Let us check that α is an onto morphism. For this purpose, we single out the following special subgroups of . $
$ +b z • S = ψ ∈ π $ψ(τ, z ⊗ λ) = aτ cτ +d , cτ +d ⊗ λ $ $ • T = ψ ∈ $ψ(τ, z ⊗ λ) = (τ, z ⊗ λ + 1 ⊗ q1 + τ ⊗ q2 ) where (q1 , q2 ) ! ∈⊕ $ $ • W = ψ = id ⊕ ( id ⊗ f ) ∈ $f ∈ O() . The three subgroups S , T and W generate the entire . In addition, note that: $ $ S ∩ W = ψ ∈ $ψ(τ, z ⊗ λ) = (τ, ±z ⊗ λ) = {±I } and if p : → PSL(2, Z) is the projection to PSL(2, Z) then Ker(p) is generated by W and T . One concludes from these facts that is a semi-direct product:
(100) = T W ×{±1} S . The above three subgroups are naturally related through the homomorphism α to the three particular subgroups of F+ described earlier. When restricted to S ⊂ F+ , the morphism α produces an isomorphism S S sending g(m, 0, 0, I ) with a b m= cd to the automorphism in S associated to the matrix: a −b . −c d When restricted to W ⊂ F+ the morphism α produces an isomorphism W W , which sends g(I, 0, 0, f ) to the automorphism induced by f in W . Finally, when restricted to T ⊂ F+ , the morphism α produces a surjective morphism T T with kernel U(N )Z . If Q : → Z2 is given by Q(γ ) = ((γ , q1 ), (γ , q2 )) , q1 , q2 ∈ then α represents the assignment g(I, Q, R, I ) → (q2 , −q1 ).
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All three subgroups, S , T and W are therefore entirely covered by the image of α. Since they generate , the morphism α is surjective. One verifies immediately that the map: a c : + (F ) → H × C , ([a, b, c]) = − 1 ,
(101) a2 a2 is equivariant with respect to α. This leads to the Seifert fibration:
: F+ \+ (F ) → \ (H × C ) = ME,G
(102)
whose fibers are isomorphic to U(N )C / U(N )Z and therefore are copies of C∗ . The identification (97) follows from the arguments above. 5.3. Automorphy factors for the parabolic cover. Let us remark that, based on the above arguments, one obtains a canonical isomorphism D = F+ \B + (F ) = F+ \+ (F )/U (N )C \ (H × C ) = ME,G
(103)
identifying the Type II Mumford divisor D with the moduli space ME,G of pairs of elliptic curves and flat G-bundles. Under this isomorphism, the parabolic cover (80) becomes the induced holomorphic Seifert C∗ - fibration: F+ \+ (F ) → \ (H × C ) .
(104)
Our task in this section is to analyze the holomorphic type of (104). We use the following strategy. The base space of (104) is a complex orbifold \V , where V = H × C . One can describe holomorphic C∗ -fibrations over such a space in terms of equivariant line bundles over the cover V . These equivariant objects are line bundles L → V , where the action of the group on the base is given a lift to the fibers. All holomorphic line bundles over V are trivial and, choosing a trivializing section, one obtains a lift of the action to fibers through a set of automorphy factors (ϕa )a∈ with ϕa ∈ H 0 (V , OV∗ ) satisfying: ϕab (x) = ϕa (b · x)ϕb (x). Such a set determines a 1-cocycle ϕ in Z 1 (, H 0 (VG , OV∗ )). Two automorphy factors provide isomorphic fibrations on ME,G if and only if they determine the same group cohomology class in H 1 (, H 0 (V , OV∗ )). To state this rigorously, there is a canonical map φ entering the following exact sequence: φ
p∗
∗ ) → H 1 (V , OV∗ ) {1}. {1} → H 1 (, H 0 (V , OV∗ )) → H 1 (ME,G , OM E,G
(105)
We are going to write down explicitly a set of automorphy factors for Fibration (104). Since the modular groupis generated by the three subgroups S , W and T it will suffice to describe automorphy factors for elements in those subgroups. The first step towards computing the automorphy factors of (104) is defining a holomorphic trivialization of the covering C-bundle: a1 c +
: (F ) → H × C , ([a, b, c]) = − , . (106) a 2 a2
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Recall that this map provides the arithmetic recipe through which one can obtain out of a given K3 period an elliptic curve Eτ and a morphism ψ : → Eτ which carries the holonomy information of a flat G-connection. Building a trivializing section for (106) amounts then intuitively to finding a way to recover a K3 period out of geometric data given by an elliptic curve endowed with a flat G-connection. Surprisingly, such a method arises in string theory, precisely in the Narain construction (see [24, 25, 16]) of the lattice of momenta related with toroidal compactification of heterotic strings. This construction leads one to consider the following map (see Appendix 6 for details): σn : H × C × C → + (F ), (107) % & 1 (z, z) − (z, z¯ ) τ¯ (z, z) − τ (z, z¯ ) σn (τ, z, u) = exp (u · N ) . (−τ, 1), , ,z . 2 τ¯ − τ τ¯ − τ (108) A brief analysis of the above formula reveals the following: Remark 30. 1. The image of σn indeed lies in the indicated space since for any triplet (τ, z, u), (σn (τ, z, u), σn (τ, z, u)) = 0 and − i(N σn (τ, z, u), σn (τ, z, u)) = Imτ > 0. 2. One has:
σn (τ, z, u), σn (τ, z, u) = Im(u) Im(τ )
and therefore, σn (τ, z, u) is a K3 period for any u ∈ C with strictly positive imaginary part. 3. The map (τ, z) → σn (τ, z, 0)
(109)
makes a smooth section for the line bundle (106). 4. When one factors out the action of U(N )Z , application (108) provides a smooth trivialization for the induced C∗ -bundle: : + (F )/ U(N )Z → H × C .
(110)
The above Narain trivialization has a major drawback! It is not holomorphic. Nevertheless, one can get around this problem and obtain a holomorphic trivialization by perturbing slightly the map (108). Note that the middle term in expression (108) can be rewritten: 1 (z, z) − (z, z¯ ) τ¯ (z, z) − τ (z, z¯ ) , 2 τ¯ − τ τ¯ − τ 1 (z, z) − (z, z¯ ) (z, z) − (z, z¯ ) = , (z, z) + τ 2 τ¯ − τ τ¯ − τ 1 1 (z, z) − (z, z¯ ) = (0, (z, z)) + (1, τ ) 2 2 τ¯ − τ 1 1 (z, z) − (z, z¯ ) = (0, (z, z)) + T (−τ, 1) . 2 2 τ¯ − τ
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Following the above equality, one can see the Narain section (108) as: % & 1 (z, z) − (z, z¯ ) 1 σn (τ, z, 0) = exp N . (−τ, 1), (0, (z, z)) , z ∈ + (F ). 2 τ¯ − τ 2 The second factor in the right-hand side term is holomorphic. This suggests the following perturbation: σ : H × C → + (F ), (111) % & 1 (z, z) − (z, z¯ ) 1 σ (τ, z) = exp − N .σn (τ, z) = (−τ, 1), (0, (z, z)) , z . 2 τ¯ − τ 2 We call this the perturbed Narain map. One can immediately check that: Theorem 31. The perturbed Narain map (111) is a holomorphic section for the line bundle: a c : + (F ) → H × C , ([a, b, c]) = − 1 ,
. (112) a 2 a2 It descends to a holomorphic section for the C∗ -fibration (110), providing therefore a holomorphic trivialization: H × C × C∗ + (F )/ U(N )Z , (τ, z, u) → u · σ (τ, z). The C∗ -action in the right-hand side expression represents the action of U(N )C / U(N )Z upon + (F )/ U(N )Z . We are now in position to compute the automorphy factors of the parabolic cover map:
: F+ \+ (F ) → \ (H × C ) .
(113)
In order to obtain a set of factors, one needs to analyze the variation of the perturbed Narain map σ under the action of the modular group . Lemma 32. Assume (q1 , q2 ) ∈ T . Then: σ (τ, z + q1 + τ q2 ) = eπi(2(q2 ,z)+τ (q2 ,q2 )) · g(I, Q, R, I ).σ (τ, z), (114)
where Q ∈ HomZ , Z2 is given by Q(γ ) = (−(γ , q2 ), (γ , q1 )) and R ∈ End(Z2 ) with R + R t = QQt . Proof. Computing in + (F ), we obtain that σ (τ, z + q1 + τ q2 )amounts to: % 1 (−τ, 1), 0, (z, z) + (q1 , q1 ) + τ 2 (q2 , q2 ) + 2(z, q2 ) + 2τ (z, q2 ) 2 & +2τ (q1 , q2 ) , z + q1 + τ q2 . On the other hand, % & 1 g(I, Q, R, I ).σ (τ, z) = (−τ, 1), R(−τ, 1) + (0, (z, z)) + Qz, z + Qt (−τ, 1) . 2
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But Qz = (−zq2 , zq1 ) and Qt (−τ, 1) = q1 + τ q2 . Moreover, one can see that: 1 1 R(−τ, 1) = −(q1 , q2 ) − τ (q2 , q2 ), (q1 , q1 ) + A(−τ, 1), 2 2 where A ∈ End(Z2 ) skew-symmetric. One obtains then the following equality in + (F ): 1 σ (τ, z + q1 + τ q2 ) = exp (q2 , z) + τ (q2 , q2 ) + α N g(I, Q, R, I ).σ (τ, z) 2 (115) with α ∈ Z. After factoring out the U(N )Z -action, one is led to (114). a b Lemma 33. Assume ∈ SL2 (Z). Then: cd
aτ + b z − π ic(z,z) a −b σ , = e cτ +d · g(m, 0, 0, I ).σ (τ, z), where m = . −c d cτ + d cτ + d (116) Proof. As in the previous lemma, we write the calculations in + (F ). One has: % & aτ + b z z aτ + b 1 (z, z) σ , , = − ,1 , 0, . (117) cτ + d cτ + d cτ + d 2 (cτ + d)2 cτ + d In the same time:
%
& 1 g(m, 0, 0, I ).σ (τ, z) = m(−τ, 1), m (0, (z, z)) , z 2 % & 1 = (−(aτ + b), cτ + d) , (c(z, z), d(z, z)) , z 2 % & aτ + b 1 c(z, z) d(z, z) z = − ,1 , , , . (118) cτ + d 2 cτ + d cτ + d cτ + d
Comparing the two formulas we get the following identity in + (F ): aτ + b z 1 c(z, z) σ , = exp − N g(m, 0, 0, I ).σ (τ, z). cτ + d cτ + d 2 cτ + d Factoring out the U(N )Z -action, one obtains (116).
(119)
We can state then:
Theorem 34. Let ϕg g∈K be the automorphy factors of parabolic cover C∗ -fibration: F+ \+ (F ) → \ (H × C ) associated to the trivialization generated by σ . Then: i(2(q2 ,z)+τ (q2 ,q2 )) for g = (q , q ) ∈ T . 1. ϕg (τ, z) = e−π 1 2
π ic(z,z) a b cτ +d 2. ϕg (τ, z) = e for g = ∈ S . cd 3. ϕg (τ, z) = 1 for g ∈ W .
(120)
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Proof. The first two expressions are direct consequences of Lemmas 32 and 33. The fact that ϕg (τ, z) = 1 for g ∈ W follows from: % & 1 σ (τ, f (z)) = (−τ, 1), (0, (f (z), f (z))) , f (z) 2 % & 1 = (−τ, 1), (0, (z, z)) , f (z) = g(I, 0, 0, f ).σ (τ, z) 2 for anyf ∈ O().
The three subgroups T , S and W generate the entire modular group . Therefore, the above automorphy factors are enough to characterize completely the holomorphic type of fibration (120). 5.4. Theta function interpretation and relation to heterotic string theory. Given the particular automorphy factor expressions computed in the previous section, one can provide for the parabolic cover C∗ -fibration (120) a theta function interpretation. Let H × C be the orbifold cover of ME,G . Since is positive definite, unimodular and even, there is an associated holomorphic theta function (see [19, 23] for details) :
: H × C → C, (τ, z) = eπi(2(z,γ )+τ (γ ,γ )) . (121) γ ∈
The pairing appearing above represents the bilinear complexification of the integral pairing on . The -character function can be written then as a quotient of : B : H × C → C, B (τ, z) =
(τ, z) . η(τ )16
(122)
Here, η is Dedekind’s eta function: η(τ ) = eπiτ/12
∞ '
1 − e2πimτ ,
m=1
which is an automorphic form of weight 1/2 and multiplier system given by a group homomorphism χ : SL2 (Z) → Z/24Z, in the sense that [32]: √ ab ∈ SL2 (Z). η(γ · τ ) = χ (γ ) cτ + dη(τ ) for γ = cd The character terminology for (122) is justified by its role in the representation theory of infinite-dimensional Lie algebras. The function B is the zero-character of the level l = 1 basic highest weight representation of the Kac-Moody algebra associated to G (see [19] for details). According to [19], the character function B obeys the following transformation properties: Proposition 35. Under the action of the modular group , the character function (122) transforms as : B (g · (τ, z)) = ϕgch (τ, z) · B (τ, z). The factors ϕgch , g ∈ can be described as:
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ch • ϕ(q (τ, z) = eπi(−2(q2 z)−τ (q2 q2 )) for (q1 , q2 ) ∈ T . 1 ,q2 ) π ic(zz) ch = e cτ +d for m = a b • ϕm ∈ S . cd ch = 1 for w ∈ W . • ϕw
The holomorphic function B descends therefore to a section of a C-fibration: Z → \ (H × C ) = ME,G
(123)
with automorphy factors ϕgch described above. We call this the character fibration. One can compare then the character fibration (123) with the parabolic cover C∗ fibration:
: F+ \+ (F ) → \ (H × C ) = ME,G
(124)
analyzed in the previous section. A look at Theorem 34 and Proposition 35 is enough to convince us that the two holomorphic fibrations are defined through identical automorphy factors. Therefore: Theorem 36. The parabolic cover fibration
: F+ \+ (F ) → \ (H × C ) = ME,G
(125)
is holomorphically isomorphic to the character fibration (123) with the zero section removed. We conclude the section by placing the outcome of Theorem 36 in connection with the parabolic compactification construction presented in Sect. 2, and comparing the resulting structure to the classical moduli spaces of eight dimensional heterotic string theory. Recall that, up to isomorphism, there exist only two even, positive-definite and unimodular lattices of rank 16. To each choice of one can associate a corresponding Lie group G. For 1 = E8 × E8 one sets G1 = (E8 × E8 ) Z2 . If 2 = 16 then G2 = Spin(32)/Z2 . The moduli space MK3 of K3 surfaces with section admits a partial compactification MK3 obtained by adding two distinct divisors at infinity Di . Each point on Di can be identified with an equivalence class of elliptic Type II stable K3 surfaces with section, in the i -category, and with an isomorphism class of a pair (E, A) consisting of an elliptic curve E and a flat Gi -connection A. The correspondence gives a natural holomorphic isomorphism: Di ME,Gi ,
(126)
where ME,Gi is the 17-dimensional moduli space of pairs of elliptic curves and flat Gi -bundles3 . As explained in 2.3, in each of the two cases, one has the parabolic cover p
Pi → MK3 3
(127)
Again, if G = Spin(32)/Z2 only connections liftable to the Spin(32)-connection are considered.
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modeling the projection F \ → \, where F is the stabilizer in of a rank two isotropic sub-lattice of Lo determining i . Moreover, the space Pi fibers holomorphically over the corresponding divisor: Pi → Di
(128)
with all fibers being copies of C∗ . Theorem 36 shows that, under identification (126), the above fibration is the character fibration of i with the zero-section removed. That allows one to holomorphically identify Pi with the total space of the character C∗ -fibration. Turning our attention to the heterotic side of the duality, it was shown in [6] (see i Theorems 1 and 2 in Sect. 1) that the moduli space MG het of classical vacua for heterotic string theory compactified over the torus is holomorphically isomorphic to the same total space of the character C∗ -fibration corresponding to the lattice i . Corroborating these facts to Theorem 36, one obtains a holomorphic isomorphism of C∗ -fibrations:
Pi
/ MGi
het
Di
(129)
/ ME,Gi
which can be seen as an identification between the parabolic cover space Pi and the classical moduli space of eight-dimensional heterotic string theory with group Gi . But, as described in Sect. 2.3, there exists an open subset Vi ⊂ Pi , which can be seen as a punctured tubular neighborhood of the zero-section in the complex line fibration associated to (128), such that the restriction: p|Vi : Vi → p(Vi )
(130)
is an isomorphism and p(Vi ) is an open punctured tubular neighborhood of the divisor Di in MK3 . This fact allows us to conclude that: Theorem 37 (F-Theory/Heterotic String Duality in Eight Dimensions). There exists a holomorphic isomorphism between an open neighborhood of MK3 near the divisor Di i and an open neighborhood of MG het near the zero-section of the complex line fibration associated to the right fibration in (129). i The open neighborhood of MG het in the above statement corresponds to large volumes of the elliptic curve. Hence, the two regions identified by Theorem 37 are exactly the sectors that physics predicts should closely resemble each other.
6. Appendix: Narain Construction The parameter fields for 8-dimensional heterotic string theory are, after Narain [16, 24], triplets (A, g, B) consisting of a flat G-connection, a flat metric and a constant antisymmetric 2-tensor B, all defined over a two-torus E. The Lie group G involved is either E8 × E8 Z2 or Spin(32)/Z2 . One usually describes a flat torus as a quotient: E = R2 /U
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of the Euclidean space R2 through a rank two lattice U = Ze1 ⊕ Ze2 . In this way, E inherits a flat metric, which in turn generates a volume v ∈ R∗+ and a complex structure parameterized by τ ∈ H. These parameters are obtained as: ( 2 , v = g11 g22 − g12 τ =
g12 + v · i , g11
where gij = ei · ej . A flat G-connection on E is, formally, a morphism A : U → R . The lattice is the coroot lattice of G if G = E8 × E8 Z2 and the lattice of a maximal torus of G if G = Spin(32)/Z2 . As is the standard procedure, one parameterizes holomorphically these flat connections by taking: z = z2 − τ z1 ∈ C , where zi : = A(ei ) ∈ R . The last ingredient, the B-field is seen as a two-form B = b(e1∗ ) ∧ (e2∗ ) with b ∈ R. The B-field holonomy along E is given by ) B = exp (ibv) . exp i E
One considers then the space: R2,18 = R2 ⊕ R2 ⊕ R endowed with the inner product: (x, y, z).(x , y , z ) = x.x − y.y − z.z . The lattice of momenta [16], denoted L(A,g,B) , associated to a heterotic triplet (A, g, B) is obtained as the image of the map: ϕ(A,g,B) : U ⊕ U ∗ ⊕ → R2 ⊕ R2 ⊕ R , 1 1 1 1 ϕ(A,g,B) (w, p, l) = p − bT w − At l − At Aw − w, p 2 2 4 2 1 1 −bT w − At l − At Aw + w, Aw + l . 2 4
(131)
Here T : R2 → R2 is the anti-self adjoint morphism T (x1 , x2 ) = (x2 , −x1 ). One checks that, in the above formulation, the image
L(A,g,B) : = Im ϕ(A,g,B) with the induced inner product forms a lattice of rank 20 embedded in the ambient space R2,18 . The lattice L(A,g,b) is isomorphic to H ⊕ H ⊕ (−). A basis underlying this decomposition is given by:
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1 1 Fi : = ϕ(A,g,B) (−ei , 0, 0) = bT ei + At Aei + ei , bT ei + At Aei − ei , −Aei , 4 4 (132) 1 ∗ 1 ∗ Fi∗ : = ϕ(A,g,B) (0, ei∗ , 0) = (133) e , e ,0 , 2 i 2 i 1 1 L : = ϕ(A,g,B) (0, 0, l) = − At l, − At l, l . (134) 2 2 It satisfies: Fi .Fj = 0, Fi∗ .Fj∗ = 0, Fi .Fj∗ = δij , Fi .L = Fj∗ .L = 0, L.L = −l.l . One is interested in the behavior of the oriented positive 2-plane R2 ⊂ R2,18 with respect to the lattice L(A,g,B) . Let us imagine that the lattice L remains fixed and the oriented R2 is varying inside L ⊗ R parameterized by the heterotic variables. This provides an assignment: { heterotic parameters(A, g, B)} → O(2, 18)/SO(2) × O(18).
(135)
Moreover, the target space in (135) has a natural holomorphic structure. One can equivalently regard positive, oriented, two-planes in L ⊗ R as complex lines ω ⊂ L ⊗ C satisfying ω.ω = 0 and ω.ω > 0. There is then a bijective correspondence: O(2, 18)/SO(2) × O(18) {ω ∈ PLC |ω.ω = 0, ω.ω > 0} and the map (135) can be interpreted as sending triplets of heterotic parameters to the 18-dimensional complex period domain of Sect. 2.1. One can describe explicitly this map. Let (A, g, B) be a heterotic triplet determining (τ, z, v, b) ∈ H × C × R∗+ × R. Then, the complex line ω is generated by βj Fj∗ + γ ω= αi Fi +
(136)
with α1 = −τ, α2 = 1, γ = z, (z, z) − (z, z¯ ) β1 = −2(bv + iv) + , 2(τ¯ − τ ) τ¯ (z, z) − τ (z, z¯ ) β2 = −2τ (bv + iv) + . 2(τ − τ¯ ) Take the decomposition L = H ⊕ H ⊕ (−) with a basis for H ⊕ H given by {F1 , F2 , F1∗ , F2∗ }. The inner product on LC appears as: (a, b, c).(a , b , c ) = (a, b ) + (b, a ) − (c, c ). Let N ∈ End(L) be the nilpotent anti-self adjoint endomorphism N(a, b, c) = (0, T a, 0)
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and let exp(tN ) = I + tN be its exponential. The Narain correspondence between heterotic parameters and period complex lines in appears then as: σn : H × C × H → ,
(137) & % 1 (z, z) − (z, z¯ ) τ¯ (z, z) − τ (z, z¯ ) σn (τ, z, u) = exp (−2u · N ) (−τ, 1), , ,z . 2 τ¯ − τ τ¯ − τ
The complex variable u represents bv + iv ∈ C. It is clear that (137) is not holomorphic. However one can move the non-holomorphic part of (137) to the exponential. Indeed: (z, z) − (z, z¯ ) τ¯ (z, z) − τ (z, z¯ ) =τ + (z.z), τ¯ − τ τ¯ − τ and therefore, one can rewrite: % & (z, z) − (z, z¯ ) 1 −2u + · N (−τ, 1), (0, (z, z)) , z . σn (τ, z, u) = exp 2(τ¯ − τ ) 2 Acknowledgement. The authors would like to thank Robert Friedman for many helpful conversations during the development of this work. The first author would also like to thank Charles Doran for many discussions regarding this work and the Institute for Advanced Study for its hospitality and financial support during the course of the academic year 2002–2003.
References 1. Ash, A., Mumford, D., Rapoport, M., Tai, Y.: Smooth Compactification of Locally Symmetric Varieties. In: Lie Groups: History, Frontiers and Applications, Vol. IV, Brookline, MA: Math. Sci. Press, 1975 2. Aspinwall, P.S., Morrison, D.R.: String Theory on K3 surfaces. In: Mirror symmetry, II, Providence, RI: Amer. Math. Soc., 1997. pp. 703–716 3. Baily, W.L. Jr., Borel,A.: Compactification ofArithmetic Quotients of Bounded Symmetric Domains. Ann. of Math. (2) 84, 442–528 (1966) 4. Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Berlin: Springer-Verlag, 1984 5. Carlson, J.A.: Extensions of Mixed Hodge Structures. In: Journ´ees de G´eometrie Alg´ebrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Alphen aan den Rijn: Sijthoff & Noordhoff, 1980, pp. 107–127 6. Clingher, A.: Heterotic String Data and Theta Functions. http://arxiv.org/abs/math.DG/0110320, 2001 7. Demazure, M., Pinkham, H.C. (eds.): S´eminaire sur les Singularit´es des Surfaces. Volume 777 of Lecture Notes in Mathematics. Berlin: Springer, 1980 8. Freed, D.S.: Dirac Charge Quantization and Generalized Differential Cohomology. In: Surveys in differential geometry, Surv. Differ. Geom., VII, Somerville, MA: International Press, 2000. pp. 129– 194 9. Friedman, R.: Hodge Theory, Degenerations and the Global Torelli Problem. Thesis, Harvard University, 1981 10. Friedman, R.: Global Smoothings of Varieties with Normal Crossings. Ann. of Math. (2) 118(1), 75–114 (1983) 11. Friedman, R.: A New Proof of the Global Torelli Theorem for K3 Surfaces. Ann. of Math. (2) 120(2), 237–269 (1984) 12. Friedman, R.: The Period Map at the Boundary of Moduli. In: Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Princeton, NJ: Princeton Univ. Press, 1984, pp. 183–208 13. Friedman, R., Morgan, J., Witten, E.: Vector Bundles and F theory. Commun. Math. Phys. 187(3), 679–743 (1997) 14. Friedman, R., Morgan, J.W.: Principal Holomorphic Bundles over Elliptic Curves IV: del Pezzo Surfaces, in preparation 15. Friedman, R., Morgan, J.W., Witten, E.: Principal G-bundles over Elliptic Curves. Math. Res. Lett. 5(1–2), 97–118 (1998)
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16. Ginsparg, P.: On Toroidal Compactification of Heterotic Superstrings. Phys. Rev. D (3) 35(2), 648– 654 (1987) 17. Griffiths, P., Schmid, W.: Recent Developments in Hodge Theory: A Discussion of Techniques and Results. In: Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973), Bombay: Oxford Univ. Press, 1975, pp. 31–127 18. Griffiths, P.A.: Periods of Integrals on Algebraic Manifolds: Summary of Main Results and Discussion of Open Problems. Bull. Amer. Math. Soc. 76, 228–296 (1970) 19. Kac, V.G., Peterson, D.H. Infinite-Dimensional Lie Algebras, Theta Functions and Modular Forms. Adv. in Math. 53(2), 125–264 (1984) 20. Kulikov, V.S.: Degenerations of K3 Surfaces and Enriques Surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 41(5), 1008–1042, 1199 (1977) 21. Looijenga, E., Peters, C.: Torelli Theorems for K¨ahlerK3Surfaces. Compositio Math. 42(2), 145–186 (1980/81) 22. Manin, Yu.I.: Cubic Forms:Algebra, Geometry, Arithmetic. Amsterdam: North-Holland Publishing Co., 1986 23. Mumford, D.: Tata Lectures on Theta. I (with the assistance of C. Musili, M. Nori, E. Previato and M. Stillman), Boston, MA: Birkh¨auser Boston Inc., 1983 24. Narain, K.S.: New Heterotic String Theories in Uncompactified Dimensions < 10. Phys. Lett. B 169(1), 41–46 (1986) 25. Narain, K.S., Sarmadi, M.H., Witten, E.: A Note on Toroidal Compactification of Heterotic String Theory. Nucl. Phys. B 279(3–4), 369–379 (1987) 26. Dolgachev, I.V.: Mirror Symmetry for Lattice Polarized K3 Surfaces. J. Math. Sci. 81(3), 2599–2630 (1996) 27. Persson, U., Pinkham, H.: Degeneration of Surfaces with Trivial Canonical Bundle. Ann. of Math. (2) 113(1), 45–66 (1981) ˇ ˇ 28. Pjatecki˘ı-Sapiro, I.I., Safareviˇ c, I.R.: Torelli’s Theorem for Algebraic Surfaces of Type k3. Izv. Akad. Nauk SSSR Ser. Mat. 35, 530–572 (1971) 29. Schmid, W.: Variation of Hodge Structure: The Singularities of the Period Mapping. Invent. Math. 22, 211–319 (1973) 30. Sen, A.: F -Theory and Orientifolds. Nucl. Phys. B 475(3), 562–578 (1996) 31. Cardoso, G., Curio, G., L¨ust, D., Mohaupt, T.: On the duality between the heterotic string and F -theory in 8 dimensions. Phys. Lett. B 389(3), 479–484 (1996) 32. Siegel, C.L.: Advanced Analytic Number Theory. Second edition, Bombay: Tata Institute of Fundamental Research, 1980 33. Siu,Y.T., Trautmann, G.: Deformations of Coherent Analytic Sheaves with Compact Supports. Mem. Amer. Math. Soc. 29(238), iii+155 (1981) 34. Vafa, C.: Evidence for F -theory. Nucl. Phys. B 469(3), 403–415 (1996) 35. Witten, E.: World-Sheet Corrections via D-instantons. J. High Energy Phys. 2, Paper 30, 18 (2000) Communicated by M.R. Douglas
Commun. Math. Phys. 254, 565–580 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1269-2
Communications in
Mathematical Physics
Stable Bundles on Non-K¨ahler Elliptic Surfaces Vasile Brˆınz˘anescu1, , Ruxandra Moraru2, 1 2
Institute of Mathematics “Simion Stoilow”, Romanian Academy, P.O. Box 1-764, 70700 Bucharest, Romania. E-mail:
[email protected] Department of Mathematics and Statistics, Burnside Hall, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 2K6
Received: 2 September 2003 / Accepted: 3 June 2004 Published online: 22 January 2005 – © Springer-Verlag 2005
Abstract: In this paper, we study the moduli spaces Mδ,c2 of stable rank-2 vector bundles on non-K¨ahler elliptic surfaces, thus giving a classification of these bundles; in the case of Hopf and Kodaira surfaces, these moduli spaces admit the structure of an algebraically completely integrable Hamiltonian system. 1. Introduction Vector bundles on elliptic fibrations have been extensively studied over the past fifteen years; in fact, there is by now a well-understood theory for projective elliptic surfaces (see, for example, [D, F1, FMW]). However, not very much is known about the nonK¨ahler case. In this article, we partly remedy this problem by examining the stability properties of holomorphic rank-2 vector bundles on non-K¨ahler elliptic surfaces; their existence and classification are investigated in [BrMo1, BrMo2]. One of the motivations for the study of bundles on non-K¨ahler elliptic fibrations comes from recent developments in superstring theory, where six-dimensional non-K¨ahler manifolds occur in the context of N = 1 supersymmetric heterotic and type II string compactifications with non-vanishing background H-field; in particular, all the non-K¨ahler examples appearing in the physics literature so far are non-K¨ahler principal elliptic fibrations (see [BBDG, CCFLMZ, GP] and the references therein). The techniques developed here and in [BrMo1, BrMo2] can also be used to study holomorphic vector bundles of arbitrary rank on higher dimensional non-K¨ahler elliptic and torus fibrations. A minimal non-K¨ahler elliptic surface X is a Hopf-like surface that admits a holomorphic fibration π : X → B, over a smooth connected compact curve B, whose smooth fibres are isomorphic to a fixed smooth elliptic curve T ; the fibration π can have at most The first author was partially supported by Swiss NSF contract SCOPES 2000-2003, No.7 IP 62615 and by contract CERES 39/2002–2004. Current address: The Fields Institute, 222 College Street, Toronto, Ontario, Canada, M5T 3J1. E-mail:
[email protected].
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a finite number of singular fibres, in which case, they are isogeneous to multiples of T . More precisely, if the surface X does not have multiple fibres, then it is the quotient of a complex surface by an infinite cyclic group (see Example 3.1); multiple fibres can then be introduced by performing a finite number of logarithmic transformations on its relative Jacobian J (X). To study bundles on X, a natural operation is restriction to the smooth fibres of π ; this gives rise to an important invariant, called the spectral curve or cover, which is an effective divisor on J (X) that encodes the holomorphic type of the bundle over each smooth fibre of π. Consider the moduli space Mδ,c2 of stable holomorphic rank-2 vector bundles on X with fixed determinant δ and second Chern class c2 . This moduli space can be identified, via the Kobayashi-Hitchin correspondence, with the moduli space of gauge-equivalence classes of Hermitian-Einstein connections in the fixed differentiable rank-2 vector bundle determined by δ and c2 (see, for example, [Bh, LT]). In particular, if the determinant δ is the trivial line bundle OX , then there is a one-to-one correspondence between MOX ,c2 and the moduli space of SU (2)-instantons, that is, antiselfdual connections. Note that the determinant line bundle δ induces an involution iδ of the relative Jacobian J (X); furthermore, the spectral cover of any bundle in Mδ,c2 is invariant with respect to this involution, thus descending to an effective divisor on the ruled surface Fδ := J (X)/ iδ called the graph of the bundle. We can then define a map G : Mδ,c2 → Div(Fδ ) that associates to each stable vector bundle its graph in Div(Fδ ), called the graph map. In [BH, Mo], the stability properties of vector bundles on Hopf surfaces were studied by analysing the image and the fibres of this map; in particular, it was shown [Mo] that the moduli spaces admit a natural Poisson structure with respect to which the graph map is a Lagrangian fibration whose generic fibre is an abelian variety: the map G admits an algebraically completely integrable system structure. In this paper, we adopt this approach to study stable vector bundles on arbitrary non-K¨ahler surfaces. The article is organised as follows. We begin with a brief review of some existence and classification results for holomorphic vector bundles on non-K¨ahler elliptic surfaces that were proven in [BrMo1, BrMo2]. In the third section, we obtain explicit conditions for the stability of rank-2 vector bundles: we show that unfiltrable bundles are always stable and then classify the destabilising bundles of filtrable bundles. The moduli spaces Mδ,c2 are studied in the last section. We first prove that these spaces are smooth on an open dense subset consisting of vector bundles that are regular on every fibre of π (on a smooth elliptic curve, a bundle of degree zero is said to be regular if its group of automorphisms is of the smallest possible dimension). However, for Hopf and Kodaira surfaces, the moduli spaces are also smooth at points that are not regular; in this case, the moduli are smooth complex manifolds of dimension 4c2 − c12 (δ). Then, we determine the image of the graph map; for simplicity, we focus our presentation on non-K¨ahler elliptic surfaces without multiple fibres, but similar results hold in the multiple fibre case. Furthermore, we give an explicit description of the fibres of the graph map, which follows immediately from the classification results of [BrMo2, Mo] and the stability conditions of the third section; in particular, the generic fibre at a graph G ∈ Div(Fδ ) is isomorphic to a finite number of copies of a Prym variety associated to G. We conclude by noting that for Kodaira surfaces the graph map is also an algebraically completely integrable Hamiltonian system, with respect to a given symplectic structure on Mδ,c2 .
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2. Holomorphic Vector Bundles π
Let X → B be a minimal non-K¨ahler elliptic surface, with B a smooth compact conπ nected curve; it is well-known that X → B is a quasi-bundle over B, that is, all the smooth fibres are pairwise isomorphic and the singular fibres are multiples of elliptic curves [K, Br]. Let T be the general fibre of π, which is an elliptic curve, and denote its dual T ∗ (we fix a non-canonical identification T ∗ := Pic0 (T )); in this case, the relative π Jacobian of X → B is simply p1
J (X) = B × T ∗ → B (see, for example, [K, BPV, Br]) and X is obtained from J (X) by a finite number of logarithmic transformations [K, BPV, BrU]. In addition, if the fibration π has multiple fibres, then one can associate to X a principal T -bundle π : X → B over an m-cyclic covering ε : B → B, where the integer m depends on the multiplicities of the singular fibres; note that the map ε induces natural m-cyclic coverings J (X ) → J (X) and ψ : X → X. To study bundles on X, one of our main tools is restriction to the smooth fibres of the fibration π : X → B. It is important to point out that since X is non-K¨ahler, the restriction of any vector bundle on X to a smooth fibre of π always has trivial first Chern class [BrMo1]. Therefore, a vector bundle E on X is semistable on the generic fibre of π; in fact, its restriction to a fibre π −1 (b) is unstable on at most an isolated set of points b ∈ B; these isolated points are called the jumps of the bundle. Furthermore, there exists a divisor SE in the relative Jacobian of X, called the spectral curve or cover of the bundle, that encodes the isomorphism class of the bundle E over each smooth fibre of π ; for a detailed description of this divisor, we refer the reader to [BrMo1]. We should note that, if the fibration π has multiple fibres, then the spectral cover SE of E is actually defined as the projection in J (X) of the spectral cover Sψ ∗ E ⊂ J (X ) of ψ ∗ E, where ψ : X → X is the m-cyclic covering defined above. 2.1. Line bundles. The spectral cover of a line bundle L on X is a section of J (X) such that the restriction of L to any smooth fibre π −1 (b) of π is isomorphic to the line bundle b of degree zero on T ∼ = π −1 (b). Conversely, given any section of J (X), there exists at least one line bundle on X with spectral cover [BrMo1]. Before giving a classification of line bundles on X, we fix some notation. Suppose that π has a multiple fibre mF over the point b in B; the line bundle associated to the divisor F of X is then such that (OX (F ))m = OX (mF ) = π ∗ OB (b). Let P2 be the subgroup of Pic(X) generated by π ∗ Pic(B) and the OX (Ti ) , where m1 T1 , . . . , mr Tr are the multiple fibres (if any) of X; we then have [BrMo1]: Proposition 2.1. Let be a section of J (X). Then, the set of all line bundles on X with spectral cover is a principal homogeneous space over P2 . 2.2. Rank-2 vector bundles. Consider a rank-2 vector bundle E on X; its discriminant is then defined as c1 (E)2 1 c2 (E) − . (E) := 2 4
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In this case, the spectral curve of E is a divisor SE in J (X) of the form k ∗ + C, {xi } × T SE := i=1
where C is a bisection of J (X) and x1 , · · · , xk are points in B that correspond to the jumps of E. Let δ be the determinant line bundle of E. It then defines the following involution on the relative Jacobian J (X) = B × T ∗ of X: iδ : J (X) → J (X), (b, λ) → (b, δb ⊗ λ−1 ), where δb denotes the restriction of δ to the fibre Tb = π −1 (b). By construction, the spectral curve SE of E is invariant with respect to this involution; in particular, the pair of points lying on the bisection C over b is of the form {λb , δb ⊗ λ−1 b }, where λb and −1 δb ⊗ λb are the subline bundles of E|π −1 (b) . Finally, note that the quotient of J (X) by the involution iδ is a ruled surface Fδ := J (X)/ iδ over B; let η : J (X) → Fδ be the canonical map. The spectral cover SE of E then descends to a divisor on Fδ of the form GE :=
k
fi + A,
i=1
where fi is the fibre of the ruled surface Fδ over the point xi and A is a section of the ruling such that η∗ A = C. 2.2.1. Bundles without jumps We begin with some properties of filtrable bundles without jumps. Let E be a rank-2 vector bundle on X with determinant δ, and spectral cover (1 + 2 ), where 1 and 2 are sections of J (X); there exists a line bundle D on X associated to 1 such that E is given by an extension 0 → D → E → D−1 ⊗ δ → 0.
(2.1)
1 (E) = − (c1 (δ) − 2c1 (D))2 . 8
(2.2)
Consequently,
Given the above considerations, we have the following results. Lemma 2.2. If 1 = 2 , then (E) = 0. Furthermore, the extension (2.1) either splits on every fibre of π or else it splits on at most a finite number of fibres. Proof. Note that c1 (D) = c1 (D−1 ⊗ δ) because 1 = 2 ; referring to (2.2), we then have (E) = 0. Suppose that there exists at least one fibre Tb0 of π over which the extension is non-trivial; therefore, h1 (Tb0 , D−1 ⊗ E) = 1. But if the extension splits over the fibre Tb , then h1 (Tb , D−1 ⊗ E) = 2. The upper semi-continuity of the map b → h1 (Tb , D−1 ⊗ E) thus implies that h1 (Tb , D−1 ⊗ E) = 1 for generic b.
Lemma 2.3. If 1 = 2 , then |1 ∩ 2 | = 4(E). In addition, the extension (2.1) splits globally whenever (E) = 0.
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Proof. Since 1 = 2 , the sheaf π∗ (D−2 ⊗ δ) vanishes and the first direct image sheaf R 1 π∗ (D−2 ⊗ δ) is a skyscraper sheaf supported on the points of 1 ∩ 2 . Therefore, c1 (R 1 π∗ (D−2 ⊗ δ)) = |1 ∩ 2 | and by Grothendieck-Riemann-Roch, 1 |1 ∩ 2 | = − (c1 (δ) − 2c1 (D))2 , 2 which is equal to 4(E) by (2.2). Consequently, if (E) = 0, then 1 ∩ 2 = ∅; in this case, the extension (2.1) splits on every fibre of π and R 1 π∗ (D2 ⊗ δ −1 ) = 0. Hence, the Leray spectral sequence gives H 1 (X, D2 ⊗ δ −1 ) = 0 and the extension splits globally.
We have seen that we can associate to any rank-2 vector bundle on X a bisection in J (X). Conversely, given any bisection of J (X), there exists at least one rank-2 vector bundle on X associated to it; if the bisection is smooth, the bundles that correspond to it are classified as follows (see [BrMo2] for precise statements). Theorem 2.4. Fix a line bundle δ on X and its associated involution iδ of J (X). Let C be a smooth bisection of J (X) that is invariant with respect to this involution; it is then a double cover of B of genus 4(2, c1 , c2 ) + 2g − 1. The set of all rank-2 vector bundles on X with spectral cover C and determinant δ is then parametrised by a finite number of copies of the Prym variety P rym(C/B) associated to the double cover C → B. 2.2.2. Bundles with jumps. Consider a rank-2 vector bundle E on X with determinant δ that has a jump of multiplicity µ over the smooth fibre T = π −1 (x0 ). The restriction of E to the fibre T is then of the form λ⊕(λ∗ ⊗δx0 ), for some λ ∈ Pic−h (T ), h > 0; the integer h is called the height of the jump at T . Moreover, up to a multiple of the identity, there is a unique surjection E|T → λ, which defines a canonical elementary modification of E that ¯ this elementary modification is called allowable [F2]. Therefore, we can we denote E; associate to E a finite sequence {E¯ 1 , E¯ 2 , . . . , E¯ l } of allowable elementary modifications such that E¯ l is the only element of the sequence that does not have a jump at T . Let us now assume that π has a multiple fibre m0 T0 . One can then associate to X an elliptic quasi-bundle π : X → B , over an m0 -cyclic covering ε : B → B, such that T0 := ψ −1 (T0 ) ⊂ X is a smooth fibre of π , where ψ : X → X is the m0 -cyclic covering induced by ε. Given this, we say that E has a jump over T0 if and only if the restriction of ψ ∗ E to the fibre T0 is unstable. Naturally, the height and multiplicity of the jump of E over T0 are defined as the height and multiplicity of the jump of ψ ∗ E over T0 . We can now define the following important invariants. Definition 2.5. Let T be a smooth fibre of π. Suppose that the vector bundle E has a jump over T and consider the corresponding sequence of allowable elementary modifications {E¯ 1 , E¯ 2 , . . . , E¯ l }. The integer l is called the length of the jump at T . The jumping sequence of T is defined as the set of integers {h0 , h1 , . . . , hl−1 }, where h0 = h is the height of E and hi is the height of E¯ i , for 0 < i ≤ l − 1. If the vector bundle E has a jump over a multiple fibre m0 T0 of π , we define the length and jumping sequence of T0 to be the length and jumping sequence of the jump of ψ ∗ E over the smooth fibre T0 = ψ −1 (T0 ) of ψ, where ψ : X → X is the m0 -cyclic covering defined above. Note that if a vector bundle E jumps over a smooth fibre T of π , with multiplicity µ and jumping sequence {h0 , . . . , hl−1 }, then µ = l−1 i=1 hi . For a detailed description of
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jumps, we refer the reader to [Mo, BrMo2]; moreover, the basic properties of elementary modifications can be found, for example, in [F2]. We finish this section by stating the following existence result [BrMo2]: if X does not have multiple fibres and S is a spectral cover in J (X) (that may have vertical components), then one can associate to S at least one rank-2 vector bundle on X. 3. Stable Rank Two Bundles 3.1. Degree and stability. The degree of a vector bundle can be defined on any compact complex manifold M. Let d = dimC M. A theorem of Gauduchon’s [G] states that any hermitian metric on M is conformally equivalent to a metric, called a Gauduchon metric, ¯ d−1 = 0. Suppose that M is endowed with whose associated (1,1) form ω satisfies ∂ ∂ω such a metric and let L be a holomorphic line bundle on M. The degree of L with respect to ω is defined [Bh], up to a constant factor, by deg L := F ∧ ωd−1 , M
where F is the curvature of a hermitian connection on L, compatible with ∂¯L . Any two ¯ ¯ d−1 = 0, the degree is independent such forms F differ by a ∂ ∂-exact form. Since ∂ ∂ω of the choice of connection and is therefore well defined. This notion of degree is an extension of the K¨ahler case. If M is K¨ahler, we get the usual topological degree defined on K¨ahler manifolds; but in general, this degree is not a topological invariant, for it can take values in a continuum (see below). Having defined the degree of holomorphic line bundles, we define the degree of a torsion-free coherent sheaf E on M by deg(E) := deg(det E), where det E is the determinant line bundle of E, and the slope of E by µ(E) := deg(E)/rk(E). The notion of stability then exists for any compact complex manifold: A torsion-free coherent sheaf E on M is stable if and only if for every coherent subsheaf S ⊂ E with 0 < rk(S) < rk(E), we have µ(S) < µ(E). Remark. With this definition of stability, many of the properties from the K¨ahler case hold. In particular, all line bundles are stable; for rank two vector bundles on a surface, it is sufficient to verify stability with respect to line bundles. Finally, if a vector bundle E is stable, then H 0 (M, End(E)) = C. π
Example 3.1. Let X → B be a non-K¨ahler principal elliptic bundle over a curve B of genus g and with fibre T . The surface X is then isomorphic to a quotient of the form X = ∗ /τ , where is a line bundle on B with positive Chern class d, ∗ is the complement of the zero section in the total space of , and τ is the multiplicative cyclic group generated by a fixed complex number τ ∈ C, with |τ | > 1. In this case, the degree of torsion line bundles can be computed explicitly (for details, see [T]). Every line bundle L ∈ Picτ (X)
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∗ decomposes uniquely as L = H ⊗ Lα , for H ∈ ∪d−1 i=0 Pic(B) and α ∈ C . Taking into account this decomposition, the degree of L is given by
deg L = c1 (H ) −
d ln |α|. ln |τ |
In particular, deg(π ∗ H ) = deg H , for all H ∈ Pic(B). We end this example by observing that if X has a multiple fibre m0 T0 , then we have deg(OX (T0 )) = 1/m0 . π
3.2. Stable vector bundles. Let X → B be a non-K¨ahler elliptic surface with multiple fibres of X is then KX = π ∗ KB ⊗ mr1 T1 , . . . , mr T r (if any); the canonical bundle r , giving ω OX (m − 1)T = O (m − 1)Ti as the dualising sheaf i i X/B X i i=1 i=1 of π. Note that deg ωX/B = r − ri=1 1/mi ≥ 0 (see Example 3.1). Fix a rank-2 vector bundle E on X and let δ be its determinant line bundle; there exists a sufficient condition on the spectral cover of E that ensures its stability: Proposition 3.2. Suppose that the spectral cover of E includes an irreducible bisection C of J (X). Then E is stable. Proof. Suppose that there exists a line bundle D on X that maps into E. After possibly tensoring D by the pullback of a suitable line bundle on B, the rank-2 bundle E is then given as an extension 0 → D → E → D−1 ⊗ δ ⊗ IZ → 0,
(3.1)
where Z ⊂ X is a locally complete intersection of codimension 2. In fact, Z is the set of points {x1 , . . . , xk } corresponding to the fibres π −1 (xi ) over which E is unstable. Let 1 and 2 be the sections of J (X) determined by the line bundles D and D−1 ⊗ δ, respectively. The extension (3.1) then implies C = 1 + 2 .
Consequently, the spectral covers of unstable bundles include bisections of the form C = (1 + 2 ), where 1 and 2 are sections of the Jacobian surface. Proposition 3.3. Suppose that the spectral cover of E is given by k ∗ + (1 + 2 ) . {xi } × T i=1
Then, there exist line bundles K1 and K2 on X (corresponding to the sections 1 and 2 , respectively) such that the set of all line bundles that map non-trivially to E is given by
r Kj ⊗ π ∗ H ⊗ OX ai Ti : H ∈ Pic≤0 (B) and ai ≤ 0 . i=1
Also, E is stable if and only if deg K1 and deg K2 are both smaller than deg δ/2. Note that if 1 = 2 , then K1 = K2 . The line bundles K1 and K2 are called the destabilising line bundles of E.
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Proof. Let D be a line bundle that corresponds to the section 1 and suppose that there exists a non-trivial map D → E. We begin by assuming that E is regular on the generic fibre of π. In this case, the direct image sheaf π∗ (D−1 ⊗ E) is a line bundle on B, say L, of positive degree. Set K = D ⊗ (π ∗ L−1 )−1 ; then, K restricts to 1b over the smooth fibres π −1 (b) of π and π∗ (K −1 ⊗ E) ∼ line bun= OB . However, any r dle D on X corresponding to 1 can be written as K ⊗ π ∗ H ⊗ OX a i=1 i Ti , for some H ∈ Pic(B) and integers 0 ≤ a ≤ m − 1. Moreover, one can easily show that i i r π∗ F ⊗ OX = π a T (F), for any locally free sheaf F on X. Consequently, i i ∗ i=1 if D also maps into E, then the line bundle π∗ (D −1 ⊗ E) ∼ = H −1 has a non-trivial −1 section, implying that H ∈ Pic≤0 (B). Note that π∗ (K ⊗E) ∼ = OB for any line bundle r K of the form K ⊗ OX i=1 bi Ti , where 0 ≤ bi ≤ mi − 1. But, the destabilising bundle K1 is the line bundle associated to 1 of maximal degree that maps into E; we r = K ⊗ ωX/B . Clearly, any therefore set K1 = K ⊗ OX (m − 1)T line bundle i i i=1 r corresponding to 1 that maps into E can be written as K1 ⊗ π ∗ H ⊗ OX i=1 ai Ti , for H ∈ Pic≤0 (B) and integers ai ≤ 0. We now assume that E is not regular on the generic fibre of π . The direct image sheaf π∗ (D−1 ⊗ E) is then a rank-2 vector bundle on B, say F; it must have a subline bundle L such that F/L is torsion free. If we set K1 = D ⊗ (π ∗ L−1 )−1 ⊗ ωX/B , then π∗ (K1 −1 ⊗ E) has a nowhere vanishing section and, as above, any line bundle induced by 1 that maps into E is of the required form.
In fact, the destabilising line bundles of filtrable bundles without jumps can be described explicitly as follows: Proposition 3.4. Let E be a holomorphic rank-2 vector bundle on X with invariants det(E) = δ, c2 (E) = c2 , and spectral cover (1 + 2 ), where 1 and 2 are sections of J (X). Let K1 be the destabilising line bundle of E induced by 1 ; there is an extension 0 → K1 → E → K1 −1 ⊗ δ → 0.
(3.2)
(i) If 1 = 2 and the extension is trivial on every fibre of π , then there exists a line bundle H− on B of non-positive degree d0 such that K1 2 = δ ⊗ π ∗ (H− ) ⊗ ωX/B . (ii) If 1 = 2 and the extension splits only on a finite number n ≥ 0 of fibres of π , then K1 2 = δ ⊗ π ∗ (H+ ) ⊗ ωX/B , where H+ is a line bundle of degree n on B that is trivial whenever n = 0. (iii) If 1 = 2 and the extension is non-trivial on a finite number n ≤ 4(E) of fibres, then the second destabilising bundle of E is K2 = K1 −1 ⊗ δ ⊗ π ∗ (H− ) ⊗ ωX/B , where H− is a line bundle of non-positive degree −n on B that is trivial for n = 0. Proof. Let us first assume that the extension (3.2) splits on every fibre of π ; note that if 1 = 2 , then the extension in fact splits globally. If the extension splits globally, then π∗ (K1 ⊗ δ −1 ⊗ E) has a nowhere vanishing global section, implying that the second destabilising bundle of E is K2 = K1 −1 ⊗ δ ⊗ ωX/B . Otherwise, every subline bundle of π∗ (K1 ⊗ δ −1 ⊗ E) has negative degree; let H− be its subline bundle of maximal degree d0 < 0. Then, K1 −1 ⊗ δ ⊗ π ∗ H− ⊗ ωX/B is the destabilising line bundle of E so that it is isomorphic to K1 . This proves (i) and (iii) for n = 0. Next, we suppose that 1 = 2 and that the extension is non-trivial on the generic 2 −1 fibre of π . Note that the restrictionof 1 ⊗ δ is trivial on every fibre of π, implying K r that K1 2 ⊗ δ −1 = π ∗ (H+ ) ⊗ OX a T i=1 i i for a line bundle H+ on B and integers
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0 ≤ ai ≤ mi − 1; tensoring the exact sequence (3.2) by K1 −1 and pushing down to B, we obtain a new exact sequence: 0 → H+ −1 → OB → R 1 π∗ (K1 −1 ⊗ E) → H+ −1 → 0.
(3.3)
Suppose that the extension (3.2) splits over n fibres of π (counting multiplicity). Referring to (3.3), the first direct image sheaf R 1 π∗ (K1 −1 ⊗ E) is then given by the extension 0 → S → R 1 π∗ (K1 −1 ⊗ E) → H+ −1 → 0, where S is a skyscraper sheaf supported on the n points (counting multiplicity) corresponding to these fibres. By Grothendieck-Riemann-Roch, 1 1 deg(R 1 π∗ (K1 −1 ⊗ E)) = − c12 (K1 2 ⊗ δ −1 ) = − c12 (π ∗ (H+ )) = 0. 2 2 The degree of the line bundle H+ is thus clearly, n; H+ = OB if n = 0. Note that by r is the destabilising line bundle of E; a T construction, K1−1 ⊗δ ⊗π ∗ (H+ )⊗OX i i i=1 r however, π∗ K1 ⊗ δ −1 ⊗ π ∗ (H+ )−1 ⊗ OX i=1 bi Ti ⊗ E = OB , for all integers 0 ≤ bi ≤ mi − 1. Therefore, ai = mi − 1 for all i = 1, . . . , r, proving (ii). Finally, let us assume that 1 = 2 . If the extension also splits over m ≤ 4(E) fibres of π (counting multiplicity) corresponding to points in 1 ∩ 2 , then the rank of R 1 π∗ (K1 −1 ⊗ E) jumps at these m points; in fact, the first direct image sheaf is given by the extension 0 → OB → R 1 π∗ (K1 −1 ⊗ E) → R 1 π∗ (K1 −2 ⊗ δ) → 0. Dualising, we get R 1 π∗ (K1 −1 ⊗ E)∗ = H− , for H− ∈ Pic(B). Let n := 4(E) − m; since the skyscraper sheaf R 1 π∗ (K1 −2 ⊗ δ) is supported on 4(E) points (see the proof of Lemma 2.3), the line bundle H− has degree −n. Furthermore, by relative Serre duality, π∗ (K1 ⊗ δ −1 ⊗ E) = R 1 π∗ (K1 −1 ⊗ E)∗ = H− ; therefore, the second destabilising line bundle of E is K2 = K1 −1 ⊗ δ ⊗ π ∗ (H− ) ⊗ ωX/B and we are done.
Recall that, for surfaces X with multiple fibres, the spectral cover of a vector bundle E on X was defined in Sect. 2 in terms of the vector bundle ψ ∗ E on an m-cyclic covering ψ : X → X, where X is an elliptic fibre bundle over an m-cyclic covering B → B. Keeping this in mind, we now state the main result of the section. Theorem 3.5. Consider a filtrable rank-2 vector bundle E on X with determinant δ that has k jumps of lengths l1 , . . . , lk , respectively; furthermore, suppose that j of them occur over multiple fibres mi1 Ti1 , . . . , mij Tij , respectively, for some integer 0 ≤ j ≤ k. We set ν :=
j s=1
ls /mis +
k
lt .
t=j +1
Let K be one of the destabilising bundles of E. There is an extension 0 → ψ ∗ K → ψ ∗ E → ψ ∗ (K ⊗ δ −1 ) → 0, where ψ ∗ E denotes the vector bundle on X obtained by performing successive elementary modifications to eliminate the jumps of ψ ∗ E.
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(i) If 1 = 2 and the extension is trivial on every fibre of π , then E is stable if and only if ν > d0 + deg ωX/B , where d0 is the integer of Proposition 3.4 (i). (ii) Suppose that 1 = 2 and that the extension splits on only a finite number mn of fibres, then E is stable if and only if ν > n + deg ωX/B . (iii) If 1 = 2 and the extension is non-trivial on a finite number mn of fibres of π , then E is stable if and only if deg K ∈ deg δ/2 − ν − n + deg ωX/B , deg δ/2 . Proof. Note that any elementary modification of ψ ∗ E has the same destabilising bundles as ψ ∗ E (which are the pullbacks to X of the destabilising bundles of E). Furthermore, the elementary modification ψ ∗ E has determinant ψ ∗ δ ⊗ OX (−D), where j D := s=1 ls Tis + ( kt=j +1 lt )T . Applying Proposition 3.4 to the bundle ψ ∗ E, we obtain the theorem.
4. Moduli Spaces Let X be a non-K¨ahler elliptic surface and consider a pair (c1 , c2 ) in N S(X) × Z. For a fixed line bundle δ on X with c1 (δ) = c1 , let Mδ,c2 be the moduli space of stable holomorphic rank-2 vector bundles with invariants det(E) = δ and c2 (E) = c2 . We define the following positive rational number:
n n
2 c1 1 µi = c1 . − µi , µ1 , . . . , µr ∈ N S(X), m(2, c1 ) := − max 4 2 1
1
Note that, for any c1 ∈ N S(X), one can choose a line bundle δ on X such that 1 c1 (δ) 2 c1 (δ) ∈ c1 + 2N S(X) and m(2, c1 ) = − ; 2 2
(4.1)
moreover, if there exist line bundles a and δ on X such that δ = a 2 δ , then there is a natural isomorphism between the moduli spaces Mδ,c2 and Mδ ,c2 , defined by E → a ⊗ E. Therefore, if δ is any other line bundle with Chern class in c1 + 2N S(X), it induces a moduli space that is isomorphic to Mδ,c2 . However, the advantage of using such a δ is that its Chern class has maximal self-intersection −8m(2, c1 ). Hence, we restrict our study to moduli spaces Mδ,c2 of stable bundles whose determinant δ satisfies (4.1). 4.1. Existence and dimension. A necessary condition for the existence of holomorphic rank-2 vector bundles is (2, c1 , c2 ) := 1/2 c2 − c12 /4 ≥ 0 [BaL, Br]. Also, a theorem of B˘anic˘a - Le Potier’s [BaL] states that there exists a filtrable holomorphic rank-2 vector bundle with Chern classes c1 and c2 if and only if (2, c1 , c2 ) ≥ m(2, c1 ). Given our choice of line bundle δ, any element E of Mδ,c2 has discriminant 1 (E) = m(2, c1 ) + c2 ≥ 0. 2 Consequently, c2 ≥ −2m(2, c1 ); moreover, if c2 < 0, then E unfiltrable. However, if the vector bundle E is unfiltrable, then its spectral cover contains an irreducible bisection; it is then stable by Proposition 3.2. Therefore, if a rank-2 vector bundle has second Chern class −2m(2, c1 ) ≤ c2 < 0, then it is stable.
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Assume that the moduli space Mδ,c2 is non-empty. Consider one of its elements E; there is a natural splitting of the endomorphism bundle End(E) = OX ⊕ ad(E), where ad(E) is the kernel of the trace map. By deformation theory, the moduli space has expected dimension h1 (X; ad(E)) − h2 (X; ad(E)) at E. Since the vector bundle E is assumed to be stable, we have h0 (X; ad(E)) = 0 and the expected dimension of the moduli space is equal to −χ (E) = 8(2, c1 , c2 ) − 3χ (OX ) = 8(2, c1 , c2 ). 4.2. Smoothness. Let us first assume that X is an elliptic fibre bundle over a curve of genus less than 2. Recall that a vector bundle E on a complex manifold X is said to be good if and only if h2 (X; ad(E)) = 0, or equivalently, if h0 (X; ad(E) ⊗ KX ) = 0 (by Serre duality); furthermore, the moduli space Mδ,c2 is smooth at E if and only if the vector bundle E is good. Given this, one easily proves the following: Proposition 4.1. Let X be a non-K¨ahler elliptic fibre bundle over a curve B of genus less than 2, that is, X is a Hopf surface or a primary Kodaira surface. The moduli spaces Mδ,c2 are then smooth of dimension 8(2, c1 , c2 ). Proof. It is sufficient to prove that every stable bundle on X with Chern classes c1 and c2 is good. In this case, the canonical bundle of the surface is KX = π ∗ (KB ). Since the genus of B is ≤ 1, the canonical bundle is given by OX (−D), where D is an effective divisor. There is an inclusion KX = OX (−D) ⊂ OX , which in turn induces an inclusion on the space of global sections H 0 (X; ad E ⊗ KX ) ⊂ H 0 (X; ad(E)). However, the stability of E implies that h0 (X; ad(E)) = 0 and we are done.
π
For an arbitrary non-K¨ahler elliptic surface X → B, we consider the elements of the moduli space that are regular, that is, vector bundles that are regular on every fibre of π . Note that for such a bundle E, the direct image sheaves π∗ (End(E)) and R 1 π∗ (End(E)) are dual locally free sheaves of rank two; therefore, by Grothendieck-Riemann-Roch, we have c1 (π∗ (End(E))) = 2ch2 (E). Given the natural splitting π∗ (End(E)) = OB ⊕ π∗ (ad(E)), we conclude that deg(π∗ (ad(E))) = 2ch2 (E). The Leray spectral sequence gives us h0 (X; ad(E) ⊗ KX ) = h0 (B; π∗ (ad(E)) ⊗ KB ); hence, if the degree of π∗ (ad(E)) ⊗ KB is negative, we have h0 (X; ad(E) ⊗ KX ) = 0, leading us to the following: Proposition 4.2. Let X be a non-K¨ahler elliptic surface over a base curve B of genus g. Then, if c2 − c12 /2 > g − 1, the moduli space Mδ,c2 is smooth on the open dense subset of regular bundles.
Remark. We can also give a sufficient condition for smoothness of the moduli space at points that do not correspond to regular bundles. Consider a stable vector bundle E that is not regular over the fibres of π lying over the points x1 , . . . , xs in B. In this case, π∗ (End(E)) is again a rank-2 vector bundle, but R 1 π∗ (End(E)) is the sum of a rank 2 vector bundle with a skyscraper sheaf supported on the points x1 , . . . , xs , with multiplicities γ1 , . . . , γs , respectively. Let γ = i γi . Then, one easily verifies that a sufficient condition for smoothness of the moduli space Mδ,c2 at E is given by c12 γ >g−1+ . 2 4 Note that γ depends not only on the spectral cover of E, but also on the geometry of its jumps. c2 −
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4.3. The image of the graph map. Fix any pair (c1 , c2 ) ∈ N S(X) × Z such that (2, c1 , c2 ) ≥ 0 and let δ be a line bundle on X such that m(2, c1 ) = − 21 (c1 (δ)/2)2 . Referring to Sect. 2.2, this line bundle determines an involution iδ of the Jacobian surface and there is an associated ruled surface Fδ := J (X)/ iδ ; the quotient map is denoted η : J (X) → Fδ . Furthermore, to any rank-2 vector bundle E on X with determinant δ and second Chern class c2 , there corresponds a graph in Fδ . It was shown in [BrMo1] that these graphs are elements of linear systems in Fδ of the form |η∗ (B0 ) + bf |, where B0 is the zero section of J (X), b is the pullback to X of a line bundle on B of degree c2 , and f is a fibre of the ruled surface. Let Pδ,c2 be the set of divisors in Fδ of the form ki=1 fi +A, where A is a section and the fi ’s are fibres of the ruled surface, that are numerically equivalent to η∗ (B0 ) + c2 f . We have a well-defined map G : Mδ,c2 −→ Pδ,c2 that associates to each vector bundle its graph, called the graph map. Let us then describe the image of this map; we begin by noting that it is surjective on the open dense subset of graphs in Pδ,c2 that correspond to irreducible bisections in J (X). When considering the remaining graphs, we restrict ourselves, for simplicity, to the case where X has no multiple fibres; however, similar results hold if X does have multiple fibres. π
Proposition 4.3. Let X → B be a non-K¨ahler elliptic fibre bundle. Choose an element c1 ∈ N S(X) such that m(2, c1 ) = 0; in this case, Fδ = B × P1 and the elements of Pδ,c2 are of the form k
({bi } × P1 ) + Gr(F ),
i=1
where b1 , . . . , bk are points in B and Gr(F ) is the graph of a rational map F : B → P1 of degree c2 − k. We then have the following. (i) For c2 = 0, the moduli space Mδ,0 consists of isomorphism classes of bundles of the form L ⊗ π ∗ E , where L is a line bundle on X and E is a stable rank-2 vector bundle on B. (ii) Let S be the set of points λ0 in T ∗ such that the degree of any line bundle on X corresponding to the section B × {λ0 } in J (X) is congruent to deg δ/2 modulo Z. If I is the projection of S onto P1 = T ∗ / iδ , then we denote B × I the set of graphs ¯ b ∈ B and λ¯ ∈ I . ({b} × P1 ) + (B × {λ}) For c2 = 1, the image of the graph map G is Pδ,1 \ (B × I ). (iii) For c2 ≥ 2, the graph map is surjective. Proof. Consider a graph G = ki=1 ({bi } × P1 ) + Gr(F ), where b1 , . . . , bk are points in B and Gr(F ) is the graph of a rational map F : B → P1 of degree c2 − k; we denote C the bisection of J (X) determined by Gr(F ). Referring to Sect. 2, we can construct rank2 vector bundles on X with graph G; therefore, we only have to determine whether or not at least one of them is stable. Let us fix a bundle E with graph G and discuss its stability. Suppose that c2 = 0. Then, (E) = 0 and the map F is constant; moreover, the bisection C is reducible and E is a filtrable bundle without jumps. Set C = (B × {λ1 }) +
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(B × {λ2 }), with λ1 , λ2 ⊂ Pic0 (T ), and let K1 be the destabilising bundle corresponding to λ1 . Referring to Proposition 3.4, if λ1 = λ2 , then the second destabilising bundle of E is K2 = K1 −1 ⊗ δ; therefore, deg K1 + deg K2 = deg δ and at least one of the destabilising bundles has degree greater or equal to deg δ/2. Hence, every bundle with λ1 = λ2 is unstable. Similarly, if λ1 = λ2 and the bundle E is a split extension only on a finite number of fibres of π, then deg K1 ≥ deg δ/2 implying that E is unstable. Finally, if λ1 = λ2 and the bundle E splits on every fibre of π , then there exists a rank-2 vector bundle E on B such that E = K1 ⊗ π ∗ E ; therefore, the bundle E is stable if and only if E is stable, proving (i). Now, assume that c2 ≥ 1. Recall that the vector bundle E may be unstable only if the bisection C is reducible; therefore, suppose that C = 1 + 2 for some sections 1 , 2 ⊂ J (X). If 1 = 2 , then the bundle has at least one jump (otherwise, (E) = 0 (see Lemma 2.2), contradicting the fact that (E) = c2 ≥ 1); in this case, E is always stable by Theorem 3.5. If 1 = 2 , then a stable bundle E can be constructed as follows. Choose a line bundle K corresponding to 1 ; after possibly tensoring K by an element of P2 , one can assume that deg K ∈ (deg δ/2 − k − 2(c2 − k), deg δ/2), unless c2 = 1 and the degree of K is congruent to deg δ/2 modulo Z. Then, consider a regular extension of K −1 ⊗ δ(kT ) by K and perform k elementary modifications (using a line bundle of degree 1 on T ) to introduce the jumps. Note that K is one of the destabilising bundles of E; referring to Theorem 3.5, E is then stable. Finally, if c2 = 1 and 1 = 2 , then a bundle with graph G is stable if and only if the degrees of its destabilising bundles are in the interval (deg δ/2 − 1, deg δ/2); if all bundles corresponding to 1 have degree congruent to deg δ/2 modulo Z, then this is never possible.
π
Proposition 4.4. Let X → B be a non-K¨ahler elliptic fibre bundle. Choose an element c1 ∈ N S(X) such that m(2, c1 ) > 0, so that we may have c2 < 0. Then, the graph map is surjective whenever the moduli spaces are non-empty, except in the following case. Suppose that c2 = 0 and m(2, c1 ) = 1/4. Furthermore, let J be the set of sections A in Pδ,0 such that η∗ A = 1 + 2 is a reducible bisection of J (X) and the degree of any line bundle on X associated to 1 is congruent to deg δ/2 modulo Z. In this case, the image of the graph map is Pδ,0 \J . Proof. Consider a graph G = ki=1 fi + A and set C = η∗ A. We know that there exist bundles corresponding to this graph; let us then discuss the stability of a bundle E that has graph G. If c2 < 0, then (E) < m(2, c1 ): the bisection C is irreducible and the bundle is stable. If c2 = 0, then (E) = m(2, c1 ) > 0. There are now two possibilities. The first is k = 0, implying that A2 < 4m(2, c1 ); therefore, the bisection C = η∗ A is irreducible and the bundle is stable. The second is k = 0 and the bisection is reducible; suppose that C = 1 + 2 , for some sections 1 , 2 ⊂ J (X). Note that 1 = 2 ; otherwise, k = 0 would imply that (E) = 0, which is a contradiction. The vector bundle E is then an extension of K −1 ⊗ δ by K, where K is the destabilising bundle of E corresponding to 1 , that can be assumed to be regular on every fibre of π. Hence, E is stable if and only if deg K ∈ (deg δ/2 − 4m(2, c1 ), deg δ/2), as stated in Theorem 3.5. Clearly, if m(2, c1 ) = 1/4 and the degree of every line bundle corresponding to 1 is congruent to deg δ/2 modulo Z, then E is unstable. Finally, by arguments similar to those used to prove Proposition 4.3, the graph map is surjective whenever c2 ≥ 1.
4.4. Fibre of the graph map. If we consider graphs without vertical components, the description of most fibres of the graph map is then straightforward.
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Proposition 4.5. Let X be a non-K¨ahler elliptic surface over a curve B of genus g. Fix a pair (c1 , c2 ) in N S(X) × Z and let δ be a line bundle on X, with c1 (δ) = c1 , such that Mδ,c2 is non-empty. Consider an element A of Pδ,c2 that does not contain vertical components and let C = η∗ A be the corresponding bisection in J (X). (i) Suppose that C is a smooth bisection of J (X). The fibre of the graph map G at A is then isomorphic to a finite number of copies of the Prym variety P rym(C/B) (see Theorem 2.4). (ii) If the bisection C = 1 +2 is reducible, then the components of G−1 (A) are parametrised by the set of line bundles on X associated to 1 that satisfy the conditions of Theorem 3.5. In particular, the component given by the line bundle K consists of extensions of K −1 ⊗ δ by K that are regular on at least one fibre π −1 (b), where 1,b = 2,b , if deg K is not congruent to deg δ/2 modulo Z, or that are regular on at least two such fibres, otherwise.
For graphs with vertical components, the fibre of the graph map can be described by examining how jumps can be added to vector bundles, that is, by classifying elementary modifications. This is done in detail in [Mo] for vector bundles on Hopf surfaces. For the sake of completion, we briefly state how this translates to bundles on an arbitrary non-K¨ahler elliptic surface X. Let E be a stable rank-2 vector bundle on X with det E = δ, c2 (E) = c2 , and a jump of length l over the smooth fibre T = π −1 (x0 ). This jump can be removed by performing l successive allowable elementary modifications, thus obtaining a bundle with determinant δ(−lT ); note that this procedure is canonical. But, adding a jump to E implies several choices: a jumping sequence {h0 , . . . , hl−1 }, a line bundle N on T for each distinct integer of the jumping sequence, and surjections to N that preserve stability. These choices are parametrised by a fibration that we now describe. Let G be a graph that contains a vertical component over x0 of multiplicity µ and {h0 , . . . , hl−1 } be a jumping sequence such that l−1 i=0 hi = µ. We set G(E) = G and E has a jump ,l E j JGc2,{h = E ∈ Mδ(j T ),c2 of length l at x0 with jumping . 0 ,...,hl−1 } sequence {h0 , . . . , hl−1 } Associating to a bundle E its allowable elementary modification E¯ therefore defines a natural map c2 −h0 ,l ,l+1 −→ E j JG( : E j +1 JGc2,{h ¯ 0 ,h1 ,...,hl } E),{h 1 ,...,hl } ¯ E −→ E.
Proposition 4.6. The fibre of the natural projection at W is given by: (i) AutSL(2,C) (W |T ), if c2 > h0 and h0 = h1 , (ii) Pic−h0 (T ) × AutSL(2,C) (W |T ), if c2 > h0 = 1 and l = 0, (iii) Pic−c2 (T ), if c2 = h0 = 1 and l = 0.
4.5. Integrable systems. A Poisson structure on a surface X is given by a global section π of its anticanonical bundle KX−1 [Bo]. Suppose that X → B is a non-K¨ahler elliptic surface that may have multiple fibres T1 , . . . , Tr of multiplicities m1 , . . . , mr , respectively.
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r The anticanonical bundle of X is then π ∗ KB−1 ⊗ i=1 OX (1 − mi ) , implying that X admits a Poisson structure if the genus of the base curve is smaller or equal to 1 and if π does not have multiple fibres. From now on, we suppose that X is a non-K¨ahler elliptic surface without multiple fibres over a curve B of genus g = 0 or 1, that is, a Hopf surface or a primary Kodaira surface. Let us fix a Poisson structure s ∈ H 0 (X, KX−1 ) on X. A Poisson structure θ = θs ∈ H 0 (M, ⊗2 T M) on the moduli space M := Mc2 ,δ is then defined as follows: for any bundle E ∈ M, θ(E) : TE∗ M × TE∗ M −→ C is the composition ◦
θ (E) : H 1 (X, ad(E) ⊗ KX ) × H 1 (X, ad(E) ⊗ KX ) −→ s
Tr
H 2 (X, End(E) ⊗ KX2 ) −→ H 2 (X, End(E) ⊗ KX ) −→ C, where the first map is the cup-product of two cohomology classes, the second is multiplication by s, and the third is the trace map. If the base curve B is elliptic, the canonical bundle of X is trivial and the Poisson structure s is non-degenerate; in this case, θ has maximal rank everywhere, that is, θ is symplectic. If the base curve is instead rational, the Poisson structure s is now degenerate; we denote its divisor D := (s). Then, at any point E ∈ M, rk θ (E) = 4 dimC M − dim H 0 (D, ad(E|D )). ∼ P1 is given by the divisor x1 + x2 , Suppose that the locally free sheaf OB (2) on B = for some points x1 , x2 ∈ B; then, D = T1 + T2 , where Ti = π −1 (xi ) for i = 1, 2. We now see that the rank of the Poisson structure is generically 4 dimC M − 2 and “drops” at the points of M corresponding to bundles that are not regular over the fibres T1 and T2 (for details, see [Mo]). Referring to Sects. 4.2 and 4.4, the moduli space M has dimension 8(2, c1 , c2 ) and the generic fibres of the graph map G : M −→ Pδ,c2 consist of Prym varieties of dimension 4(2, c1 , c2 )+g −1 (see Proposition 4.5). Also, one can show as in [Mo] that the component functions H1 , . . . , HN of the graph map are in involution with respect to the Poisson structure, that is, {Hi , Hj } = 0 for all i, j . Consequently, the graph map G is an algebraically completely integrable Hamiltonian system. Acknowledgements. The first author would like to express his gratitude to the Max Planck Institute of Mathematics for its hospitality and stimulating atmosphere; part of this paper was prepared during his stay at the Institute. The second author would like to thank Jacques Hurtubise for his generous encouragement and support during the completion of this paper; she would also like to thank Ron Donagi and Tony Pantev for valuable discussions, and the Department of Mathematics at the University of Pennsylvania for their hospitality, during the preparation of part of this article.
References [BaL] [BBDG] [BPV] [Bo] [BH]
B˘anic˘a, C., Le Potier, J.: Sur l’existence des fibr´es vectoriels holomorphes sur les surfaces non-alg´ebriques. J. Reine Angew. Math. 378, 1–31 (1987) Becker, K., Becker, M., Dasgupta,K., Green, P.S.: Compactification of heterotic theory on non-K¨ahler complex manifolds: I. JHEP 0304, 007 (2003) Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Berlin, Heidelberg, New York: Springer-Verlag, 1984 Bottacin, F.: Poisson structures on moduli spaces of sheaves over Poisson surfaces. Invent. Math. 121 421–436 (1995) Braam, P.J., Hurtubise, J.: Instantons on Hopf surfaces and monopoles on solid tori. J. Reine Angew. Math. 400 146–172 (1989)
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Brˆınz˘anescu, V.: Holomorphic vector bundles over compact complex surfaces. Lect. Notes in Math. 1624, Berlin-Heidelberg-New York: Springer, 1996 [BrMo1] Brˆınz˘anescu, V., Moraru, R.: Holomorphic rank-2 vector bundles on non-K¨ahler elliptic surfaces. http://arXiv.org/abs/math.AG/0306191, 2003 [BrMo2] Brˆınz˘anescu, V., Moraru, R.: Twisted Fourier-Mukai transforms and bundles on non-K¨ahler elliptic surfaces. http://arXiv.org/abs/math.AG/0309031, 2003 [BrU] Brˆınz˘anescu, V., Ueno, K.: N´eron-Severi group for torus quasi bundles over curves. In: Moduli of vector bundles (Sanda, 1994; Kyoto, 1994), Lecture Notes in Pure and Appl. Math. 179, New York: Dekker, 1996, pp. 11–32 [Bh] Buchdahl N.P.: Hermitian-Einstein connections and stable vector bundles over compact complex surfaces. Math. Ann. 280, 625–648 (1988) [CCFLMZ] Cardoso, G.L., Curio, G., Dall’Agata, G., L¨ust, D., Manousselis, P., Zoupanos, G.: NonK¨ahler string backgrounds and their five torsion classes. Nucl. Phys B 652, 5–34 (2003) [D] Donagi, R.: Principal bundles on elliptic fibrations. Asian J. Math. 1(2), 214–223 (1997) [F1] Friedman, R.: Rank two vector bundles over regular elliptic surfaces. Invent. Math. 96, 283–332 (1989) [F2] Friedman, R.: Algebraic Surfaces and Holomorphic Vector Bundles. UTX, New YorkBerlin-Heidelberg, Springer, 1998 [FMW] Friedman, R., Morgan, J., Witten, E.: Vector bundles over elliptic fibrations. J. Alg. Geom. 2, 279–401 (1999) [G] Gauduchon, P.: Le th´eor`eme de l’excentricit´e nulle. C. R. A. S. Paris 285, 387–390 (1977) [GP] Goldstein, E., Prokushkin, S.: Geometric model for complex non-K¨ahler manifolds with SU (3) structure. http://arXiv.org/abs/hep-th/0212307, 2002 [K] Kodaira, K.: On the structure of compact complex analytic surfaces I. Am. J. Math. 86, 751–798 (1964) [LT] L¨ubke, M., Teleman, A.: The Kobayashi-Hitchin correspondence. River Edge, NJ: World Scientific Publishing Co., Inc., 1995 [Mo] Moraru, R.: Integrable systems associated to a Hopf surface. Canad. J. Math. 55(3), 609–635 (2003) [T] Teleman, A.: Moduli spaces of stable bundles on non-K¨ahler elliptic fibre bundles over curves. Expo. Math. 16, 193–248 (1998) Communicated by M.R. Douglas
Commun. Math. Phys. 254, 581–601 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1275-4
Communications in
Mathematical Physics
Hamiltonian BRST and Batalin-Vilkovisky Formalisms for Second Quantization of Gauge Theories Glenn Barnich1, , Maxim Grigoriev1,2, 1 2
Physique Th´eorique et Math´ematique and International Solvay Institutes, Universit´e Libre de Bruxelles, Campus Plaine C.P. 231, 1050 Bruxelles, Belgium. E-mail:
[email protected] Tamm Theory Department, Lebedev Physical Institute, Leninsky prospect 53, 119991 Moscow, Russia. E-mail:
[email protected]
Received: 5 November 2003 / Accepted: 19 July 2004 Published online: 22 January 2005 – © Springer-Verlag 2005
Abstract: Gauge theories that have been first quantized using the Hamiltonian BRST operator formalism are described as classical Hamiltonian BRST systems with a BRST ˆ even and with natural ghost and parity degrees for all fields. charge of the form , The associated proper solution of the classical Batalin-Vilkovisky master equation is constructed from first principles. Both of these formulations can be used as starting points for second quantization. In the case of time reparametrization invariant systems, ˆ odd master action is established. the relation to the standard , Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Generalities on BFV and BV Formalisms . . . . . . . . . . . . . . . 2.1 Classical Hamiltonian BRST theory . . . . . . . . . . . . . . . 2.2 Master action for first order gauge theories . . . . . . . . . . . . 2.3 Superfield reformulation . . . . . . . . . . . . . . . . . . . . . 2.4 BRST operator quantization . . . . . . . . . . . . . . . . . . . 3. BFV and BV Formalisms for BRST First Quantized Gauge Systems . 3.1 Geometry of BRST quantum mechanics . . . . . . . . . . . . . 3.2 BRST quantum mechanics as classical BFV system . . . . . . . 3.3 Proper master action for BRST quantum mechanics . . . . . . . 4. Master Action and Time Reparametrization Invariance . . . . . . . . . 4.1 Tensor constructions . . . . . . . . . . . . . . . . . . . . . . . 4.2 Reinterpretation of master action for BRST quantum mechanics 4.3 Time reparametrization invariant systems . . . . . . . . . . . . 4.4 Quantization of trivial pairs and Chern-Simons . . . . . . . . .
Research Associate of the National Fund for Scientific Research (Belgium) Postdoctoral Visitor of the National Fund for Scientific Research (Belgium)
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5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Formulation of Supermanifold in Terms of Complex Coordinates ˆ Appendix B. Dirac Observables and Cohomology of , . . . . . . . . Appendix C. Quantum BRST State Cohomology of Tensor Products . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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597 598 599 599 600
1. Introduction It has been realized in [1–3] (see also [4]) that the action of open bosonic string field theory [5, 6], with free part given by the expectation value of the BRST operator, should be understood as a solution to the classical Batalin-Vilkovisky (BV) master equation. The collection of fields, ghosts, and all the ghosts for ghosts corresponds to the coefficients of the states in negative ghost numbers, while the associated antifields correspond to the coefficients of the states in positive ghost numbers. In the standard gauge field theory context, however, a proper solution to the Batalin-Vilkovisky master equation is obtained from a gauge invariant action, a generating set for its non-trivial gauge symmetries and, if needed, associated reducibility operators [7–11] (see also [12, 13] for reviews). The purpose of this paper is to construct from basic principles the proper solution of the master equation associated to a theory first quantized using the Hamiltonian BRST operator (or "BFV") formalism [14–16] (see also [17]) and to relate it with the free part of the standard master action [1–3]. This involves several steps: 1. the reformulation of BRST quantum mechanics as a classical Hamiltonian BRST system; 2. using the known proper solution of the master equation for Hamiltonian BRST systems; 3. for time reparametrization invariant systems, relating the constructed master action to the standard one by showing that they differ by the quantization of classically trivial pairs. The first two steps are treated in Sect. 3, while Sect. 4 is devoted to the last step. More precisely, for the first step, it has been pointed out by many authors (see e.g. [18, 19] and [20–22] for reviews and further references) that the Hilbert space of quantum mechanics can be understood as a (possibly infinite dimensional) symplectic manifold and that the Schr¨odinger evolution appears as a Hamiltonian flow on this phase space. This point of view provides a useful set-up for second quantization. In order to apply these ideas to gauge systems quantized in the operator formalism according to the Hamiltonian BRST prescription, one also needs to understand in this context the physical ˆ = 0, as well as the identification of BRST closed states up to BRST state condition ψ exact ones. The latter two problems alone have been faced in the context of string field ˆ theory [6, 5, 4, 23], with the somewhat surprising conclusion that the object , is not a BRST charge, but a solution to the master equation. This is due to the fact that the ghost pair associated to the mass shell constraint is quantized in the Schr¨odinger representation (see e.g [24–26]). In our approach, we will start by assuming that the number of independent constraints is even so that there is also no fractionalization of the ghost number. There is no loss of generality in this assumption, since one can always include some Lagrange multipliers among the canonical variables together with the constraints that their momenta should vanish.
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In Subsect. 3.1, we then associate to BRST quantum mechanics a K¨ahler supermanifold. In particular, the even symplectic form of ghost number zero is determined by the imaginary part of the non-degenerate hermitian inner product. In Appendix A, we discuss the geometry of this supermanifold in terms of complex coordinates. In Subsect. 3.2, it is shown that, as for non-gauge systems, time evolution on the supermanifold corresponds to the Hamiltonian flow determined by the "expectation value" of the BRST invariant Hamiltonian H = − 21 , Hˆ , where denotes the "string field". On the supermanifold, the physical state condition then coincides with the constrained surface determined ˆ by the zero locus of the BRST charge = − 21 , . These constraints are first class, and so is H. Furthermore, on the supermanifold, the identification of BRST closed states up to BRST exact ones corresponds to considering Dirac observables, i.e., functions defined on the constraint surface that are annihilated by the Poisson bracket with these constraints. As has been shown in [27], constraints associated to the zero locus of the BRST charge are special in the sense that the cohomology of the BRST charge itself provides directly the correct description of these Dirac observables, without the need to further extend the phase space. In order to make the paper self-contained, a formal proof adapted to the BRST charge is provided in Appendix B. From the point of view of the symplectic supermanifold, BRST quantum mechanics becomes thus a classical Hamiltonian BRST system described by H and . Concerning the second step, the proper solution of the master equation associated to a first order Hamiltonian gauge theory and its relation to the Hamiltonian BRST formalism is well known [28–36]. A convenient "superfield" reformulation [37] of such a master action also exists. These are reviewed in Sect. 2 together with the basic formulas of BRST operator quantization1 . In Subsect. 3.3, the above results are applied to derive the master action S for the classical Hamiltonan BFV system of and H. In Subsect. 4.1, we discuss tensor products of Hamiltonian BRST quantum mechanical systems at the level of the associated classical field theories. For later use, the assumption that the inner product is even is dropped so that the bracket may be either even or odd. In Subsect. 4.2, it is shown that the master action S associated to and ˆ M of the parametrized system: the H can be directly obtained from the BRST charge 1 ˆ M M M , where M is the string field of the master action is given by S = 2 M , parametrized system; the ghost pair of the reparametrization constraint is quantized in the Schr¨odinger representation so that ·, ·M is odd. Finally, to complete the last step, we consider in Subsect. 4.3 the case of systems that are already time reparametrization invariant and are quantized with an odd inner product, originating for instance from the Schr¨odinger representation for the ghosts associated to the mass-shell constraint. The master action S is then shown to differ from the original ˆ st st by two classically trivial pairs, quantized in the Sch¨odinger repSst = 21 st , 2 resentation . More precisely, we show that S corresponds to the tensor product of the system described by Sst with the system described by the Hamiltonian BRST charge aux associated to the trivial pairs. Had these pairs been quantized in the Fock representation instead, we use the results of Subsect. 4.1 to show that S could have been consistently reduced to Sst . In the Schr¨odinger representation, however, the master action S involves two more dimensions than Sst . In Subsect. 4.4, we show that aux is the BRST charge of complex Abelian Chern-Simons theory. Without additional ingredients, the master 1 Except for the conventions related to complex conjugation, we follow closely reference [12], to which we refer for further details. 2 Trivial pairs in string field theory have been used previously in a different context in [38].
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action S can then not be directly reduced to Sst . This is not really surprising since the Fock and the Schr¨odinger quantization are not unitary equivalent. We conclude by giving some additional remarks on the BRST charge aux and the associated master action. 2. Generalities on BFV and BV Formalisms 2.1. Classical Hamiltonian BRST theory. A Hamiltonian approach to gauge theories involves a symplectic manifold M0 with coordinates zA , constraints Ga0 , which we assume for simplicity to be first class and even, Ga0 , Gb0 M = Ca0 b0 c0 Gc0 , and a 0 first class Hamiltonian H0 with H0 , Ga0 M = Va0 b0 Gb0 . The constraints may be 0 a ak−2 reducible, Zaa10 Ga0 = 0, with a tower of reducibility equations Zakk−1 Zak−1 ≈ 0, where ≈ means an equality that holds on the constraint surface. Even though we use a finitedimensional formulation, this section also formally applies to field theories by letting the indices A, a range over both a discrete and a continuous set. In the Hamiltonian BRST approach, the phase space is extended to a symplectic supermanifold M by introducing the ghosts ηak and the ghost momenta Pbk of parity k + 1 with {Pak , ηbk }M = −δabkk . We take these variables to be real. Our convention for complex conjugation involves transposition of variables together with a minus sign when exchanging two odd variables. On the extended phase space, the ghost number of a function A that is homogeneous in ηa and Pb is obtained by taking the extended Poisson bracket { · , · }M with the purely imaginary function i {A, G}M = igh(A) A , (2.1) (k + 1)(ηak Pak − (−1)|ak | Pak ηak ) , G= 2 k
where |ak | = |ηak | + 1. Out of the contraints, one constructs the nilpotent BRST charge of ghost number 1: 1 a ηak Zakk−1 Pak−1 + . . . , (2.2) {, }M = 0. = ηa0 Ga0 + 2 k≥1
Here, . . . denote terms of higher order in ghosts and their momenta. Furthermore, the first class Hamiltonian H0 is extended to the BRST invariant Hamiltonian H of ghost number zero with {H , }M = 0. Physical quantities such as observables are determined by the BRST cohomology of the differential s = {, · }M in the space of functions F (z, η, P) on the extended phase space. Time evolution is generated by the BRST invariant Hamiltonian H according to F˙ = {F , H }M . 2.2. Master action for first order gauge theories. In this subsection, we discuss in some details the proper BV master action for Hamiltonian gauge theories. The reader may wish to skip these details and go directly to the summary, which is the only part that is explicitly needed in the rest of the paper. The information on the symplectic structure, the dynamics and the constraints of the theory is contained in the extended Hamiltonian action, M0 SE [z, λ] = dt (˙zA aA − H0 + λa0 Ga0 ), (2.3) where λa0 are Lagrange multipliers. If the symplectic two-form is defined by
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M0 R M0 ∂ R aA (A+1)(B+1) ∂ aB − (−1) , (2.4) ∂zB ∂zA AB with σ AB σ M0 = δ A . Variathe Poisson bracket is determined by zA , zB M = σM C M0 BC 0 0 tion with respect to all variables zA , λa0 gives as equations of motions both the dynamical equations and the constraints: z˙ A = zA , H0 , Ga0 = 0. (2.5) M0 σAB =−
M0
A generating set of gauge symmetries for this action is given by δ zA = a0 {Ga0 , zA }M0 , δ λ
= ˙
a0
a0
− λ Cb0 c0 c0 b0
(2.6) a0
− Vb0 b0
a0
,
(2.7)
where a0 are gauge parameters. In the field-antifield approach, the functional which contains all the information on the classical action and its gauge algebra is the proper solution S of the classical master equation, 1 (S, S) = 0. (2.8) 2 In the case of the extended Hamiltonian action, the proper solution S is required to start ∗ , λ∗ the like the original action (2.3), to which one couples through the antifields zA a0 gauge transformation (2.6), (2.7) of the fields with the gauge parameters replaced by the ghosts C a0 . One also needs to couple the terms containing the Lagrangian reducibility a operators, (which are determined by the Hamiltonian reducibility operators Zakk−1 ) and introduce associated ghosts for ghosts and their antifields. The antifields can be chosen to be real and are defined to be canonically conjugate to the fields with respect to the antibracket (·, ·). Additional terms in S are then uniquely determined by the master equation (2.8), up to anticanonical transformations in the antibracket. The proper solution S associated to (2.3) can then be shown to be given by M0 S[z, z∗ , λ, λ∗ , η, η∗ ] = dt z˙ A aA + η˙ ak Pak − H ∗ −zA
A
z ,
M
−
[λ
ak
k≥0
Pak , M + ηa∗k ηak , M ] ,
(2.9)
k≥0
where the identifications Pak = −λ∗ak have been made3 . Usually, in order to fix the gauge, one introduces a nonminimal sector, containing antighosts in ghost number −1, their momenta in ghost number 1 and auxiliary fields in ghost number zero. Then, a gauge fixing fermion ϒ in ghost number −1 that does not depend on antifields is chosen. The choice of ϒ is determined by the requirement that there be no more gauge invariance in the dynamics generated by the nonminimal master action obtained after application of the anticanonical transformation generated by ϒ and after setting to zero the transformed antifields. This gauge fixed action can be taken as a starting point for a path integral quantization and the partition function can (formally) be shown to be independent of the choice of ϒ. 3 For later convenience, some signs have been changed in Eqs. (2.3), (2.6) and (2.7) with respect to those of [12]. In (2.9), they imply the change → −.
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For the master action (2.9), it is possible to fix the gauge without introducing a nonminimal sector: indeed, by considering the anticanonical transformation which consists in the exchange of fields and antifields for the sector of the Lagrange multipliers, (λak , λ∗ak ) −→ (−λ∗ak , λak ) ≡ (Pak , P ∗ak ),
(2.10)
the equations of motion are in first order form. The new fields are then the fields zα = (zA , ηak , Pak ) that are naturally associated with the Hamiltonian BRST formalism. The antibracket for two functionals A[z, z∗ ], B[z, z∗ ] is defined by δR A δLB
∗ α ∗ (A, B)[z, z ] = dt − (z ←→ z ) . (2.11) α δzα (t) δzα∗ (t) The solution S of the master equation (2.5) can be rewritten in a compact way as (2.12) S[z, z∗ ] = dt (˙zα aαM − H − zα∗ zα , M ). An additional gauge fixing generated by the fermion ϒ = dt K(z) can then be considered. Its effect is to change the BRST invariant Hamiltonian by a BRST exact term, H → H + {K, }. Note that after putting to zero the antifields zα∗ , the constraint equations (2.5) are no longer imposed as equations of motions since the associated fields have been put to zero. In fact, it turns out to be extremely useful not to put to zero the antifields obtained after the canonical transformation generated by ϒ. They appear as sources that allow to control the Ward identities due to BRST invariance. To lowest order in , it is the antifield dependent BRST cohomology of the differential s = (S, ·) that controls gauge invariance on the quantum level. This cohomology is invariant under canonical transformations and the introduction of a non-minimal sector. Hence, from this point of view, one can forget about gauge fixing and directly discuss the cohomology associated to the master action (2.12). In turn, this cohomology computed in the space of functions in the fields zα , the antifields zα∗ and their (space)time derivatives can be shown to be isomorphic to the Hamiltonian BRST cohomology of the differential s = {, ·}M in the space of functions in zα (and their spatial derivatives). In the space of local functionals, which is the relevant space in the context of renormalization, the relation with the Hamiltonian BRST cohomology is more involved [39]. Summary. From the above construction of the solution (2.12) to the classical master equation, we can learn the following. Suppose that the following data is given: • a (super) phase space with coordinates zα and symplectic 2 form generated by αβ aα (z) with associated Poisson bracket {zα , zβ } = σM (z), • a ghost number grading G on the phase space, • a nilpotent BRST charge in ghost number 1, whose cohomology determines the physically relevant quantities on the phase space, • a BRST invariant Hamiltonian H in ghost number zero determining the time evolution. Then, in the space of functionals in the variables zα (t) “fields” and additional variables zα∗ (t) “antifields” of ghost number −gh(zα ) − 1 equipped with the antibracket given by (2.11), the proper solution of the master equation is given by (2.12). In order to recover the action from which the gauge invariant equations of motion and the constraints follow,
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the interpretation of which variables are the fields and which are the antifields should be reversed for those in negative ghost number before putting the antifields to zero. In fact, this amounts to putting to zero in (2.12) all variables of ghost number different from zero and declare the remaining ghost number zero variables to be dynamical fields. 2.3. Superfield reformulation. A superfield reformulation [37] of the master action (2.12) is achieved by introducing an additional Grassmann odd variable θ of ghost number one. Given an extended phase space M, one associates a space of maps zα = zSα (t, θ ) from the (1|1)-dimensional superspace spanned by t and θ to M. This space is a superextension of the space of field histories zα (t) (maps from t to M). Functionals on can be identified with functionals in the fields and antifields of the previous section. Indeed, one can expand zSα (t, θ ) into components zSα (t, θ ) = zα (t) + θ z∗ β (t)σM (z(t)) , βα
(2.13)
which is consistent with the various ghost number assignments. To every functional A[zS ] one can associate the functional A[z, z∗ ] obtained by using this expansion. Con∗ ] corresponds the functional A[z ] = A[ dθθz , versely, to every functional A[z, z S S dθ zS σ (zS )]. Functionals on are equipped with the odd Poisson bracket δR A δLB αβ (A, B)[zS ] = (−1)|A|+1 dtdθ α σM (zS (t, θ )) β , (2.14) δzS (t, θ ) δzS (t, θ ) with (A, B)[z + θ z∗ σ −1 ] = (A, B)[z, z∗ ]. This Poisson bracket is odd and of ghost number 1. Functional derivatives are defined as δR A δLA = δzα (t, θ ) dtdθ. (2.15) δA = dtdθ δzα (t, θ ) α δz (t, θ ) δzα (t, θ ) The master action S[zS ] corresponding to S[z, z∗ ] given in (2.12) can then be written as
dtdθ DzSα aαM (zS ) − θ H (zS ) − (zS ) , (2.16) S[zS ] = with D = θ ∂t∂ . The superfield reformulation regroups fields and antifields in convenient supermultiplets so that the antibracket is induced by the extended Poisson bracket. 2.4. BRST operator quantization. The BRST operator quantization consists in realizing the functions on the extended phase space as linear operators in a super Hilbert ˆ B] ˆ = i{A, space H together with the correspondence rule [A, B} + O(2 ), where [·, ·] denotes the graded commutator and A, B are phase space functions with associated ˆ B. ˆ linear operators A, ˆ ] ˆ = O(2 ) = [Hˆ , ]. ˆ In the following, These rules imply in particular that 21 [, we assume that we are in the non-anomalous case, where 1 ˆ ] ˆ = 0, [, 2
ˆ = 0, [Hˆ , ]
(2.17)
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ˆ Hˆ are hermitian operators in the inner product ψ, φ, which is non-degenerand , ate but not necessarily positive definite and makes the real classical variables hermitian operators. Furthermore, we take = 1. For a super Hilbert space, ψ, φ = (−1)|ψ||φ| φ, ψ, ˆ = (−1)|A||ψ| Aˆ † ψ, φ, ψ, Aφ
(2.18) (2.19)
ˆ where |φ|, |A| denotes the Grassmann parity of the state, respectively the operator A. The relation to standard Hilbert space with even and odd elements, for which the above formulas do not involve sign factors, is explained for instance in [40]. In what follows, we are not interested in a probabilistic interpretation of the quantum theory, but rather in an associated classical field theory. This is the reason why we are not concerned here with questions related to the normalizability of states or to the infinite dimensionality of the Hilbert space. The ghost number of an operator is obtained by taking the graded commutator (from ˆ We assume that H splits as a sum of eigenthe left) with the antihermitian operator G. ˆ ˆ states of G, H = ⊕p Hp with Gψp = pψp for ψp ∈ Hp . It then follows from the antihermiticity of Gˆ that ψp , φp = 0 only if p + p = 0. This means that the ghost number of the scalar product , is zero. The ghost number p of a state can be shown to be p = p0 + k for some integer k with p0 = 0 or p0 = 21 . The case where p0 = 21 arises if the number of independent constraints is odd. In this case, one can include some of the Lagrange multipliers λa and their momenta ba among the canonical variables, {λa , bb } = δba , together with the new constraint ba ≈ 0. On the level of the classical BRST formalism, this implies adding to the extended phase space the antighosts C¯ a of ghost number −1 and their momenta ρ a of ghost number 1, {ρ a , C¯ b } = −δba . All these variables are chosen to be real. The BRST charge of the system is then modified by the addition of the non-minimal piece nm = ρ a ba . Hence, by adding the cohomologically trivial pairs (λa , ba ), (ρ a , C¯ a ), one can always assume p0 = 0, which is what we do unless otherwise specified. We also assume that the inner product is even, ψ, φ = 0 if ψ and φ are of opposite parity. Physical operators are described by hermitian operators Aˆ such that ˆ ] ˆ = 0, [A,
(2.20)
where two such operators have to be identified if they differ by a BRST exact operator ˆ ]. ˆ Aˆ ∼ Aˆ + [B,
(2.21)
These two equations define the BRST operator cohomology. Similarily, physical states are selected by the condition ˆ = 0. ψ
(2.22)
Furthermore, BRST exact states should be considered as zero, or equivalently, states that differ by BRST exact ones should be identified ˆ . ψ ∼ ψ + χ
(2.23)
These two equations define the BRST state cohomology. Finally, time evolution is governed by the Schr¨odinger equation i
dψ = Hˆ ψ. dt
(2.24)
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3. BFV and BV Formalisms for BRST First Quantized Gauge Systems 3.1. Geometry of BRST quantum mechanics. Let {ea } be a basis over R of the graded Hilbert space H such that the basis vectors are of definite Grassmann parity |a| and ghost number gh(ea ). A general vector can be written as ψ = ea k a , with k a ∈ R. To each ea , one associates a real variable a of parity |ea | = |a| and ghost number −gh(ea ). These variables are coordinates of a supermanifold MH associated to H. The algebra of real valued functions on this supermanifold is denoted by G. Introducing the right module HG = H ⊗R G [41], the "string field" appears as the particular element = ea ⊗R a of this module. In the following, we will drop ⊗R . The string field is even and of total ghost number zero. The sesquilinear form ·, · on H can be extended to elements ψf and φg of HG , with f (), g() ∈ G by the rule ψf, φg = (−1)|f ||φ| ψ, φf g .
(3.1)
ˆ ˆ ) = (Aψ)f . A linear operator Aˆ on H is naturally extended to HG : A(ψf The real and imaginary parts of this inner product, ψ, φ = g(ψ, φ) + iω(ψ, φ)
(3.2)
are respectively graded symmetric and graded skew symmetric, g(ψ, φ) = (−1)|ψ||φ| g(φ, ψ) , ω(ψ, φ) = −(−1)|ψ||φ| ω(φ, ψ).
(3.3) (3.4)
The forms g(·, ·) and ω(·, ·) are extended to HG in the same way as ·, ·. If H is considered as a superspace over real numbers, both g(ψ, φ) and ω(ψ, φ) are R-bilinear forms on H. The complex structure Jˆ is the R-linear operator on H that represents multiplication by i. As a consequence, g(Jˆφ, Jˆψ) = g(φ, ψ) , g(Jˆφ, ψ) = ω(φ, ψ) .
ω(Jˆφ, Jˆψ) = ω(φ, ψ),
(3.5) (3.6)
Furthermore, the operator Jˆ commutes with C-linear operators. Introducing the coefficients ωab = (−1)|a| ω(ea , eb ) and defining ωab through ab ω ωbc = δca , an even graded Poisson bracket on G of ghost number zero is defined by {f , g} =
by
∂ R f ab ∂ L g ω . ∂ψ a ∂ψ b
(3.7)
ˆ one associates a real quadratic function F ˆ () ∈ G To each antihermitian operator A, A FAˆ () =
1 ˆ . , −JˆA 2
(3.8)
ˆ ˆ ). This map is an = − 21 ω(A, Antihermiticity implies that FAˆ () = 21 ω(, A) homomorphism from the super Lie algebra of antihermitian operators to the super Lie algebra of quadratic real functions in G equipped with the Poisson bracket FAˆ , FBˆ = F[A, (3.9) ˆ B] ˆ .
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ˆ The map is compatible with parity and ghost number assignments, gh(FAˆ ) = gh(A), ˆ |FAˆ | = |A|. For hermitian operators, we define A() = F−JˆAˆ . Because of hermiticity 1 1 1 ˆ ˆ ˆ A() = − , A = − g(, A) = − g(A, ). 2 2 2
(3.10)
Furthermore, the properties of Jˆ imply that {A, B} =
1 ˆ B]. ˆ , Jˆ[A, 2
(3.11)
ˆ and the hermitian BRST invariant HamIn particular, for the hermitian BRST charge ˆ iltonian H , Eqs. (2.17) imply 1 {, } = 0, 2
{H, } = 0,
(3.12)
where H, are of total ghost numbers 0 and 1 respectively. 3.2. BRST quantum mechanics as classical BFV system. The Schr¨odinger equation in terms of a can be written as d a = −(JˆHˆ )a = { a , H}, dt
(3.13)
so that time evolution of elements f () ∈ G is determined by the Hamiltonian flow of H, df = {f, H}. dt
(3.14)
On MH , the physical state condition (2.22) defines a submanifold, the constraint surface determined by ˆ ab b ≈ 0.
(3.15)
Because {·, } =
∂R· ˆ a, (−Jˆ) ∂ a
(3.16)
the constraint surface can be identified with the zero locus of the Hamiltonian vector field associated to . Hence, one can take as constraints Ga ≡ { a , } ≈ 0 .
(3.17)
By using the graded Jacobi identity for the Poisson bracket and taking (3.12) into account, these constraints are easily shown to be first class, {Ga , Gb } = {{ a , }, { b , }} = {{{ a , }, b }, } ∂ R {{ a , }, b } c = G ≈ 0. ∂ c Furthermore, since is quadratic in , these constraints are in fact abelian,
(3.18)
Hamiltonian BRST and B-V Formalisms for Second Quantization of Gauge Theories
{Ga , Gb } = 0.
591
(3.19)
On MH , the identification (2.23) of states up to BRST exact ones, corresponds to ˆ a ∂ L ·a . taking functions from G that are annihilated by the distribution generated by b ∂ This distribution is equivalently generated by the adjoint action of the constraints {Ga , ·}. The Hamiltonian H is also first class, ∂ R {H, a } b G . (3.20) ∂ b Hence, from the point of view of MH , BRST quantum mechanics becomes a classical constrained Hamiltonian system. According to the Dirac theory, an observable is a function f () ∈ G such that {f, Ga } ≈ 0. Two such functions should be considered equivalent if they coincide on the constraint surface, f ∼ f +λa Ga . Equivalence classes of observables then form a Poisson algebra with respect to the induced bracket. The classical Hamiltonian BRST approach described in Subsect. 2.1 consists in extending the phase space in order to encode this Poisson algebra in terms of the cohomology of a BRST charge. This will however not be straightforward in the case of the zero locus constraints Ga , because they are reducible due to the nilpotency of , and for the obvious reducibility operators, they are infinitely reducible. In fact, it turns out that for the zero locus constraints Ga , there is actually no need to extend the phase space. Indeed, the Poisson algebra of equivalence classes of observables is isomorphic to the cohomology of the BRST charge itself, equipped with the induced Poisson bracket. This has been shown in [27], where constraint systems originating from the zero locus of a generic Hamiltonian BRST differential have been analyzed. A proof adapted to the particular BRST charge is given in Appendix B. As a side remark, let us note that treating the zero locus of the BRST charge as a constraint surface is analogous to considering the master action S as a classical action; in this case, the zero locus of the BRST differential s = (S, ·) is the stationary surface associated to S (see e.g. [42, 43, 27]). {H, Ga } = {{H, a }, } =
3.3. Proper master action for BRST quantum mechanics. According to Subsect. 2.2, the solution of the master equation associated to the classical Hamiltonian BRST system on the phase space MH described by H and is given by d 1 (3.21) S[, ∗ ] = dt [ ω(, ) − H − {a∗ a , }] 2 dt which can be written as 1 d ˆ ˆ ˜ ∗ , (3.22) ˜ ∗ , + , dt − i, + , Hˆ − S[, ∗ ] = 2 dt ˜ ∗a . As explained in Sect. 2.2, the role of fields and ˜ ∗ = ea ˜ ∗a = ∗ ωba and where b antifields has been exchanged for those fields that are in strictly negative ghost numbers. According to Subsect. 2.3, we now introduce ˜ ∗a (t), Sa (t, θ ) = a (t) + θ
(3.23)
and also the ghost number zero object S = ea Sa (t, θ ). The proper solution (3.22) can then be written as
(3.24)
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S[S ] =
1 2
dtdθ
− iθ S ,
d ˆ S . S + θ S , Hˆ S + S , dt
(3.25)
By construction, it satisfies the master equation with respect to the antibracket (2.13), with zSA (t, θ ) replaced by Sa (t, θ ), δLB δR A ab (A, B)[S ] = (−1)|A|+1 dtdθ . (3.26) ω δSa (t, θ ) δSb (t, θ ) For later purposes, it will be useful to rewrite the master action as 1 d ˆ S . S[S ] = dtdθ S , (−iθ + θ Hˆ + ) 2 dt
(3.27)
4. Master Action and Time Reparametrization Invariance 4.1. Tensor constructions. Given two first quantized BRST systems with super-Hilbert ˆ i and Hamiltonians Hˆ i , the tensor spaces Hi , i = 1, 2, their respective BRST charges product HT = H1 ⊗C H2 is again a super Hilbert space. For later use, we do not assume in this section that the inner products on Hi are even, but we allow them to be of arbitrary parity εi . We also admit the possibility of fractionalization of the ghost number. The Grassmann parity and the ghost number of the state φ1 ⊗ φ2 is naturally |φ1 | + |φ2 | and gh(φ1 ) + gh(φ2 ) respectively. The inner product on HT is determined by φ1 ⊗ φ2 , ψ1 ⊗ ψ2 T = (−1)|ψ1 ||φ2 | φ1 , ψ1 1 φ2 , ψ2 2 .
(4.1)
It is of parity ε1 + ε2 and non-degenerate (for HT considered as a complex space) provided the ones on H1 and H2 are. Linear operators Aˆ i on Hi determine a linear operator Aˆ T on HT by Aˆ T (φ1 ⊗ ψ2 ) = (Aˆ 1 ⊗ 1 + 1 ⊗ Aˆ 2 )(φ1 ⊗ ψ2 ) = (Aˆ 1 φ1 ) ⊗ ψ2 + (−1)|A2 ||φ1 | φ1 ⊗ (Aˆ 2 ψ2 ) .
(4.2)
The various definitions imply that Aˆ †T = (Aˆ 1 ⊗ 1 + 1 ⊗ Aˆ 2 )† = Aˆ †1 ⊗ 1 + 1 ⊗ Aˆ †2 ,
(4.3)
[Aˆ T , Bˆ T ] = [Aˆ 1 , Bˆ 1 ] ⊗ 1 + 1 ⊗ [Aˆ 2 , Bˆ 2 ] .
(4.4)
and
ˆ i and the BRST invariant Hamiltonians Hˆ i deterIn particular, the BRST charges ˆ T, ˆ T ] = 0 and [Hˆ T , ˆ T ] = 0. mine hermitian operators T and Hˆ T such that 21 [ Furthermore, ˆ T , HT ) = H ( ˆ 1 , H1 )⊗C H ( ˆ 2 , H2 ). H (
(4.5)
The formal proof is elementary and given in Appendix C. If {eα , eα¯ } is a basis of H1C (see Appendix A), while {E , E¯ } is a basis of H2C , then {eα ⊗ E , eα¯ ⊗ E¯ } is a basis of HTC . For these basis vectors, one can consider ¯ the complex coordinates α and α¯ for the supermanifold MT associated to the
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superspace HT and also the associated complex valued functions GC T . The string fields can now be defined by ¯
T = (eα ⊗ e ) α + (eα¯ ⊗ E¯ ) α¯ .
(4.6)
The functions 1 ˆ T T T , T = − T , 2
1 HT = − T , Hˆ T T T , 2
(4.7)
satisfy 1 {T , T }T = 0 = {HT , T }T , 2
(4.8)
where { · , · }T denotes a Poisson bracket or antibracket on GC T determined by imaginary part of ·, ·T . Note, however, that when ·, ·T is odd, so is the Poisson bracket {·, ·}T . In this case, it is also called "antibracket" and T is a master action. ˆ 1 , HC ), H ( ˆ 2 , HC ), it follows from If {eθ , eθ¯ }, {E , E¯ } are bases over C of H ( 1 2 Appendix B and Appendix C that the cohomology of {·, T }T is isomorphic to real ¯¯ ˆ 2 , H2 ) is a one dimensional space over functions in θ and θ . In particular, if H ( C with basis vector E such that gh(E) = 0, |E| = 0, then H (T , HT ) H (1 , H1 ). It also follows that the cohomology of {·, T }T in GT is isomorphic to that of {·, 1 }G1 in G1 . Suppose that H2 contains a quartet or a null doublet. Let = (i, ), where i runs over the states of the quartet or the null doublet. Not only do these states not contribute to the cohomology, but they can also be consistently eliminated from T by reducing the string field used in the construction of T to
¯
T = (eα ⊗ E ) α + (eα¯ ⊗ E¯ ) α¯ .
(4.9)
This elimination is algebraic. If the parity of ·, ·T is odd, it corresponds to the elimination of "generalized auxiliary fields" of the master action discussed in [34]. In the case where the parity ·, ·T is even, it is an Hamiltonian analogue of this concept. 4.2. Reinterpretation of master action for BRST quantum mechanics. Consider now the super Hilbert space Ht,θ obtained by quantizing the phase space (t, p0 ), (θ, π ) in the Schr¨odinger representation. The wave functions are ϕ(t, θ) = ϕ0 (t) + θϕ1 (t), while the inner product is given by ϕ, t,θ = dtdθ ϕ(t, ¯ θ)(t, θ) . (4.10) 1 ∂ The ghost number operator is given by Gˆt,θ = θ − so that gh(ϕ0 (t)) = − 21 , ∂θ 2 gh(θ ϕ1 (t)) = 21 and we take |φ0 (t)| = 0, |θ ϕ1 (t)| = 1 so that the inner product is Grassmann odd and of ghost number zero. For the super Hilbert space HM = H⊗C Ht,θ , where H is an even super Hilbert space, states are of the form ea k0a (t) + ea θk1a (t). Real coordinates on the supermanifold MHM associated to HM can be chosen as k0a (t) → a (t), ˜ ∗a (t). The ghost number zero object S introduced in (3.24) can now be k1a (t) → identified with the string field M associated to HM , S ≡ M . The odd inner product on HM is denoted by ·, ·M and extended to HM ⊗ GM , where GM is the algebra of ˜ ∗a (t). The master action (3.27) can now be written as real functions in a (t),
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1 ˆ M M M , M , 2 ˆ = θˆ (pˆ 0 + Hˆ ) + .
S[M ] =
(4.11)
ˆM
(4.12)
In this case, the imaginary part of this inner product determines an odd symplectic structure on MHM . Its inverse is an antibracket, which coincides with (3.26), up to ghost number assignments discussed below. Only when the BRST invariant Hamiltonian Hˆ ˆ and θˆ pˆ 0 . ˆ M is the tensor product of the BRST charges vanishes ˆ From the expression of M , we can deduce how to arrive directly at (4.11): make the original classical Hamiltonian system time reparametrization invariant by including the time t and its conjugate momentum p0 among the canonical variables and adding the first class constraint p0 + H0 ≈ 0. The BRST charge for this system is then given by M , with (θ, π ) the "time reparametrization" ghost pair associated with this new constraint. The quantization of this system then leads directly to HM with its odd inner product and the master action (4.11). From the point of view of HM , the ghost numbers of fields and antifields are half integer and differ from the standard ones by one half. In particular physical fields are those at ghost number 21 ). This originates from the additional ghost number operator ˜ ∗a Gˆt,θ and our convention for the string field ghost number. Indeed, the fields a (t), 1 1 now have ghost numbers 2 −gh(ea ) and − 2 −gh(ea ) instead of −gh(ea ) and −1−gh(ea ). Recall that the latter is the natural assignment from the point of view of the BFV system associated to BRST quantum mechanics, and, as explained in Subsect. 3.3, also leads to the standard ghost number assignments in the associated BV formalism. From the point of view of HM the standard ghost number assignment for the fields, antifields and the antibracket are thus obtained by shifting the ghost number by 21 so that the inner product carries ghost number −1, while the antibracket is of ghost number 1. To summarize, we have thus shown that (4.11) is the proper solution of the master ˆ on H. The antibracket equation for a first quantized BRST system defined by Hˆ and is determined by the inverse of the imaginary part of the inner product ·, ·M defined on HM . Moreover, after shifting, all physical fields are among the fields associated to ghost number zero states while those associated to negative and positive ghost number states are respectively ghost fields and antifields.
4.3. Time reparametrization invariant systems. Suppose now that the BRST invariant Hamiltonian H vanishes, as in time reparametrization invariant systems. Suppose furthermore that the original system has an odd inner product ·, ·st . This is the case for instance for the relativistic particle or for the open bosonic string, where the ghost pair (η, P) associated to the mass-shell constraint p2 + m2 ≈ 0, respectively L0 ≈ 0, is quantized in the Schr¨odinger representation. According to our discussion in Subsect. 2.4, in order to have an even inner product and no fractionalization of the ghost number, the system is extended to include the Lagrange multiplier λ and its momentum b, together ¯ ρ) associated to the constraint b ≈ 0. The BRST charge picks up with the ghost pair (C, ¯ ρ) are both quantized in the Schr¨odinger the additional term bρ and the pairs (λ, b), (C, representation, yielding the odd Hilbert space Hλ,C¯ . Hence, the even Hilbert space H ˆ is of the form Hst ⊗C Hλ,C¯ , where Hst is odd with ˆ the tensor with BRST charge ˆ st and bˆ ρ. product of ˆ
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The master action (4.11) can then be understood as resulting from the original system ˆ st and the associated master described by the odd Hilbert space Hst , the BRST operator action 1 ˆ st st st , Sst = st , (4.13) 2 tensored with the system described by the even Hilbert space Haux = Hλ,C¯ ⊗C Ht,θ with ˆ aux = bˆ ρˆ ⊗ 1 + 1 ⊗ θˆ pˆ 0 and the associated inner product ·, ·aux , the BRST operator BRST charge 1 ˆ aux aux aux . aux = − aux , (4.14) 2 On the classical level, the auxiliary system is described by the pairs ((λ, b), (t, p0 )), ¯ ρ), (θ, π )). The associated BRST the constraints b ≈ 0 ≈ p0 and the ghost pairs ((C, charge aux = ρb + θp0 describes two trivial pairs and its cohomology is generated by a constant. On the classical level, one can thus simply get rid of these pairs. The question then is whether first quantized BRST systems (and the associated classical field theories) that differ by the quantization of classically trivial pairs are equivalent. If the two pairs had been quantized in the Fock instead of the Schr¨odinger representation, then equivalence could have been directly established. Indeed, all the states except for the Fock vacuum | 0 form quartets. Hence, according to the discussion in Subsect. 4.1, the proper master action (4.14) can be consistently reduced to the master action Sst . 4.4. Quantization of trivial pairs and Chern-Simons. When the two pairs are quantized in the Schr¨odinger representation, it is convenient to rename them as σ 1 , p1 , σ 2 , p2 , with [σˆ α , pˆ β ] = iδβα . The associated fermionic ghost pairs are η1 , P1 , η2 , P2 , with β gh(ηα ) = 1, gh(Pα ) = −1, and [Pˆ α , ηˆ β ] = −iδα . Wave functions and inner product are chosen as ¯ φ, ψ = dσ 1 dσ 2 dη1 dP2 φ(σ, η1 , P2 )ψ(σ, η1 , P2 ) = dσ 1 dσ 2 dη1 dP2 hij φ i (σ, η1 , P2 )ψ j (σ, η1 , P2 ) , (4.15) where in the second line we have expressed the hermitian inner product in C in terms of two component real-valued wave functions. The BRST charge and ghost number operators are given by ˆ aux = −iη1
∂ ∂ ∂ − , ∂σ1 ∂σ2 ∂P2
∂ ∂ Gˆ = η1 1 − P 2 2 . ∂η ∂P
(4.16)
In particular, states of the form ψ(x), η1 ψ(x), P2 ψ(x), and η1 P2 ψ(x) are respectively of ghost degrees 0, 1, −1, and 0. The associated string field is = d 2 σ |σ ei i2 (σ ) + η1 P i (σ ) + P2 D i (σ ) + η1 P2 i1 (σ ) , (4.17) where the i2 (σ ), i1 (σ ), P i (σ ), D i (σ ) are the coordinates on the supermanifold associated to the Hilbert space. Their Grassmann parities and ghost numbers are
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|i2 | = |i1 | = 0,
|P i | = |D i | = 1,
gh(i2 ) = gh(i1 ) = 0 ,
gh(P i ) = −1 ,
gh(D i ) = 1 ,
(4.18)
so that gh() = 0, || = 0. From a geometrical point of view, this supermanifold can be understood as the supermanifold of maps from the supermanifold with coordinates σ α , η1 , P2 (configuration space) to C viewed as a 2-dimensional real space. The Poisson bracket corresponding to the symplectic form Im·, · is determined by j D i (σ ), P j (σ ) = ωij δ(σ − σ ) . i2 (σ ), 1 (σ ) = −ωij δ(σ − σ ) , (4.19) Let Jji and ωij denote the complex structure and the symplectic form on C respectively. Integrating out the Grassmann odd variables η1 , P2 , using integrations by parts j and redefining the variables as Ai2 = i2 , Ai1 = −Jji 1 , C i = Jji D j , the BRST charge becomes
aux = − dσ 1 dσ 2 gij Ai2 ∂1 C j − Ai1 ∂2 C j . (4.20) In terms of the new variables, the Poisson bracket is determined by j C i (σ ), P j (σ ) = −g ij δ(σ − σ ) , Ai1 (σ ), A2 (σ ) = −g ij δ(σ − σ ) , (4.21) where g ij = −Jki ωkj and satisfies gij g j k = δki . The adjoint action s = { · , aux } reads sAiα = ∂α C i ,
sC i = 0 ,
sP i = −∂1 Ai2 + ∂2 Ai1 .
(4.22)
From this it follows that the BRST charge aux is the BRST charge of complex Abelian Chern-Simons theory. We conclude by giving some additional remarks on this BRST charge and the associated master action. Remark 1: Superfield formulation. Introducing the Grassmann odd superfields i of ghost number 1, i (σ, η) = C i (σ ) + ηα Aiα (σ ) + η2 η1 P i (σ ), the brackets (4.21) are equivalent to i (σ, η), j (σ , η ) = −g ij δ(σ − σ )δ(η − η ) . The BRST charge (4.20) can then be rewritten as 1 dσ 1 dσ 2 dη1 dη2 gij i (η1 ∂1 + η2 ∂2 )j . aux = − 2
(4.23)
(4.24)
(4.25)
Applying the superfield reformulation to get the master action for a BFV system described by the BRST charge aux and vanishing Hamiltonian, one gets 1 j (4.26) dσ 0 dσ 1 dσ 2 dη0 dη1 dη2 gij iS (η0 ∂0 + η1 ∂1 + η2 ∂2 )S , S= 2 where the time coordinate is denoted by σ 0 , the associated Grassmann odd variable by η0 and iS is the superfield depending on σ µ , ηµ , with µ = 0, 1, 2.
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This coincides with the well-known AKSZ representation [44] of the master action for Abelian Chern Simons theory. The standard formulation can be recovered by iden j tifying ηµ with dσ µ so that the action takes the form S = 21 gij (iS ∧ dS ), where d is the de Rham differential. Remark 2: Coordinate representation for the ghosts. If both ghost pairs are quantized in the coordinate representation, one can arrive directly at (4.25) because the superfield (4.23) then appears as the projection of the string field on σ |. Indeed, the BRST and ghost number operator act on the states as ˆ = −iηα
∂ , ∂σ α
∂ Gˆ = ηα − 1. ∂ηα
(4.27)
In particular, states of the form ψ(σ ), ηα ψα (σ ), and η2 η1 χ (σ ) are respectively of ghost degrees −1, 0, and 1. In order to have a bosonic field theory, we assign Grassmann parity k mod 2 to the states of ghost degree k. This implies defining the inner product on the super Hilbert space by φ, ψ = −i(−1)|ψ| dσ 1 dσ 2 dη1 dη2 φ(σ, η) ψ(σ, η) , (4.28) so that φ, ψ = (−1)|φ||ψ| ψ, φ. The associated string field of total ghost number and Grassmann parity zero is now = d 2 σ |σ ei i (σ, η) , (4.29) ˆ and the corresponding = − 21 , coincides with (4.25). Remark 3: Non Abelian Chern-Simons theory. In [44], the expression for the master action is derived for the Lie algebra of a compact Lie group. Using the same reasoning as above, it can easily be shown that the associated BRST charge can be compactly written as 1 1 =− dσ 1 dσ 2 dη1 dη2 gI J I (η1 ∂1 + η2 ∂2 )J + fI J K I J K ,(4.30) 2 3 where gI J denotes the invariant metric and fI J K = gI L fJLK . Put differently, the BRST charge for Chern-Simons theory has exactly the same form as the AKSZ master action (and therefore the classical action). Only the source supermanifolds are different: for the master action, the superdimension is (3|3), while for the BRST charge it is (2|2). Similar remarks apply for the BRST charge and the master action of the Poisson sigma model [45–47]. 5. Discussion The new feature of the present paper is the shift of emphasis, on the level appropriate for second quantization, from the master action to the BRST charge. Given a gauge system quantized according to the Hamiltonian BRST approach, one can always make ˆ the number of constraints even if necessary. The associated object = − 21 , is then a nilpotent BRST charge with respect to the even Poisson bracket induced by
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the imaginary part of the inner product. Out of this BRST charge, the master action can be constructed according to a standard procedure. In particular, for closed string field theory for instance, is naturally a BRST charge, without the necessity of adding trivial pairs. We plan to discuss this issue in more detail in future work. Acknowledgements. G.B. wants to thank M. Henneaux for suggesting the problem. Useful discussions with I. Batalin, G. Bonelli, L. Houart, A. Semikhatov and I. Tipunin are gratefully acknowledged. The work of GB and MG is supported in part by the “Actions de Recherche Concert´ees” of the “Direction de la Recherche Scientifique-Communaut´e Fran¸caise de Belgique", by a “Pˆole d’Attraction Interuniversitaire” (Belgium), by IISN-Belgium, convention 4.4505.86, by the INTAS grant 00-00262, and by the European Commission RTN program HPRN-CT00131, in which the authors are associated to K. U. Leuven. GB is also supported by Proyectos FONDECYT 1970151 and 7960001 (Chile), while MG is supported by the grants RFBR 02-01-00930, RFBR 02-01-06096 and LSS-1578.2003.2.
Appendix A. Formulation of Supermanifold in Terms of Complex Coordinates The geometrical structures on the supermanifold MH can be conveniently expressed in terms of complex coordinates. We follow [48]. Consider the complexification HC = H⊗R C where H is considered as above as a superspace over R. The complex conjugation of a vector of the form αψ, ψ ∈ H, α ∈ C is defined as αψ ¯ so that the original Hilbert space H is a subspace (over R) of vectors satisfying ψ¯ = ψ. All R-linear operations on H can be be extended to HC by C-linearity. In particular, C H decomposes as HC = H1,0 ⊕ H0,1 with Jˆψ = iψ
∀ψ ∈ H1,0 ,
Jˆφ = −iφ
∀φ ∈ H0,1 .
(A.1)
Complex conjugation defines a real linear isomorphism between H1,0 and H0,1 . Introducing a basis {eα } for H1,0 and {eα¯ } for H0,1 such that eα = eα¯ , the inner product ·, · extended by C bi-linearity is determined by ¯ eα¯ , eβ = (−1)|α| hαβ ¯ ,
eα , eβ¯ = eα , eβ = eα¯ , eβ¯ = 0 .
(A.2)
The graded-symmetric and graded-antisimmetric components g and ω of , are determined by 1 ¯ hαβ (−1)|α| ¯ , 2 1 ¯ ω(eα¯ , eβ ) = (−1)|α| hαβ ¯ , 2i g(eα¯ , eβ ) =
1 ¯ ¯ (−1)|β|+|α||β| hβα ¯ , 2 1 ¯ ¯ ω(eα , eβ¯ ) = − (−1)|β|+|α||β| hβα ¯ , 2i
g(eα , eβ¯ ) =
(A.3) (A.4)
with all other components vanishing. From eα¯ , eβ = eα + eα¯ , eβ + eβ¯ and the fact that eα + eα¯ is a real vector it follows that 1+(1+|α|)(1+|β|) ¯ hαβ hβα ¯ . ¯ = (−1)
(A.5)
By considering H as a space over R, C-linear operators are identified with R linear operators commuting with Jˆ. In the basis eα , eα¯ , this means that the matrix of such an operator Aˆ is block-diagonal with only the diagonal blocks Aαβ and Aαβ¯¯ nonvanishing. The fact that Aˆ is extended from H to HC by C-linearity implies that Aˆ maps real vectors ˆ α +eα¯ ) is again a real vector, which in turn implies that Aα = Aα¯ . to real ones so that A(e β β¯
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Associated to the basis elements eα , eβ¯ , one then introduces variables α , α¯ with | α | = | α¯ | = |α| and gh( α ) = gh( α¯ ) = −gh(eα ) and considers GC , the algebra of complex valued functions in these variables. The complex conjugation in HC naturally determines a complex conjugation in GC through α = α¯ so that the real elements of GC can then be identified with G. The symplectic form ω on HC determines a Poisson bracket in GC determined by ¯
¯
{ α , β } = 2ihα β ,
β γ¯ hαβ = δγα¯¯ . ¯ h
(A.6)
The string field is then given by = eα α + eα¯ α¯ . ˆ Appendix B. Dirac Observables and Cohomology of Ψ, ΩΨ Formally, one can assume that a real basis {ea } ≡ {ei , fm , gm } in the Hilbert space H is chosen such that ˆ i = 0, (−Jˆ)e
ˆ m = gm , (−Jˆ)f
ˆ m = 0. (−Jˆ)g
(B.1)
The associated coordinates of the supermanifold are { i } ≡ { i , ϒ m , m }. On the one hand, the differential {·, } becomes {·, } =
∂R· m ϒ , ∂m
(B.2)
so that H ({·, }) is isomorphic to the algebra of functions in i alone. On the other ˆ hand, the constraints are given by ϒ m ≈ 0. Antihermiticity and nilpotency of −Jˆ implies that the symplectic structure in the basis {ei , fm , gm } becomes ωij ωim 0 ωkj ω ωkn , (B.3) km 0 ωlm 0 with both ωij and ωlm non-degenerate. The inverse has the form ji 0 ω˜ ni ω 0 0 ωnr , ω˜ jp ωmp ω˜ np
(B.4)
with both ωj i and ωmp non-degenerate. This implies that the adjoint action in the Poisson bracket of the constraints ϒ m generate shifts in the m , which are thus coordinates along the gauge orbits. Hence, equivalence classes of Dirac observables also correspond to functions in i alone. Appendix C. Quantum BRST State Cohomology of Tensor Products Let {kα } and {K } be bases over C in H1 and H2 respectively. Then the vectors kα = kα ⊗ K provide a basis (over C) of the tensor product H1 ⊗C H2 . Assume that in the ˆ 1 and ˆ 2 take bases kα = {kθ , fγ , gγ } and K = {K , F , G }, the BRST charges the Jordan form,
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ˆ 1 fm = gm , ˆ 1 gm = 0 , ˆ 1 kθ = 0 , ˆ 2 F M = GM , ˆ 2 GM = 0 . ˆ 2 K = 0 ,
(C.1)
One then can check that the vectors kθ = kθ ⊗ K , fγ = fγ ⊗ F , 0 fθ = kθ ⊗ F ,
1 f˜γ = (gγ ⊗ F − (−1)|fγ | fγ ⊗ G ) , 2 fγ0 = fγ ⊗ K ,
gγ = gγ ⊗ F + (−1) 0 gθ
= (−1)
|kθ |
|fγ |
kθ ⊗ G ,
fγ ⊗ G , gγ0
g˜ γ = (−1)
|gγ |
(C.2)
gγ ⊗ G ,
= gγ ⊗ K
ˆ T . Hence, the cohomology of ˆ T is the linear span over C of form a Jordan basis for ˆ ˆ ˆ 2 , H2 ). kθ = kθ ⊗ K , so that H (T , HT ) = H (1 , H1 )⊗C H ( References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
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25. Dayi, O.F.: Gauge fixing of point particle and open bosonic string field theory actions of Neveu and West. Nuovo Cim. A101, 1 (1989) 26. Dayi, O.F.: A general solution of the BV master equation and BRST field theories. Mod. Phys. Lett. A8, 2087–2098 (1993) 27. Grigoriev, M.A., Semikhatov, A.M., Tipunin, I.Y.: BRST formalism and zero locus reduction. J. Math. Phys. 42, 3315–3333 (2001) 28. Batalin, I., Fradkin, E.: Operatorial quantization of dynamical systems subject to constraints. A further study of the construction. Ann. Inst. Henri Poincar´e (Phys. Theor.) 49, 145–214 (1988) 29. Siegel, W.: Batalin-Vilkovisky from Hamiltonian BRST. Int. J. Mod. Phys. A4, 3951 (1989) 30. Batlle, C., Gomis, J., Par´ıs, J., Roca, J.: Lagrangian and Hamiltonian BRST formalisms. Phys. Lett. B224, 288 (1989) 31. Fisch, J.M.L., Henneaux, M.: Antibracket - antifield formalism for constrained Hamiltonian systems. Phys. Lett. B226, 80 (1989) 32. Henneaux, M.: Elimination of the auxiliary fields in the antifield formalism. Phys. Lett. B238, 299 (1990) 33. Batlle, C., Gomis, J., Par´ıs, J., Roca, J.: Field - antifield formalism and Hamiltonian BRST approach. Nucl. Phys. B329, 139–154 (1990) 34. Dresse, A., Gr´egoire, P., Henneaux, M.: Path integral equivalence between the extended and nonextended Hamiltonian formalisms. Phys. Lett. B245, 192 (1990) 35. Dresse, A., Fisch, J.M.L., Gr´egoire, P., Henneaux, M.: Equivalence of the Hamiltonian and Lagrangian path integrals for gauge theories. Nucl. Phys. B354, 191–217 (1991) 36. Grigorian, G.V., Grigorian, R.P., Tyutin, I.V.: Equivalence of Lagrangian and Hamiltonian BRST quantization. Systems with first-class constraints. Sov. J. Nucl. Phys. 53, 1058–1061 (1991) 37. Grigoriev, M.A., Damgaard, P.H.: Superfield BRST charge and the master action. Phys. Lett. B474, 323–330 (2000) 38. Siegel, W.: Boundary conditions in first quantization. Int. J. Mod. Phys. A6, 3997–4008 (1991) 39. Barnich, G., Henneaux, M.: Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket. J. Math. Phys. 37, 5273–5296 (1996) 40. Deligne, P., Morgan, W.: Quantum Fields and Strings: A Course for Mathematicians, Part I, Chapter-Notes on Supersymmetry. Providence, RI: American Mathematical Society, 1999, pp. 41–98 41. Gaberdiel, M.R., Zwiebach, B.: Tensor constructions of open string theories I: Foundations. Nucl. Phys. B505, 569–624 (1997) 42. Sen, A., Zwiebach, B.: A note on gauge transformations in Batalin-Vilkovisky theory. Phys. Lett. B320, 29–35 (1994) 43. Grigoriev, M.A., Semikhatov, A.M., Tipunin, I.Y.: Gauge symmetries of the master action in the Batalin- Vilkovisky formalism. J. Math. Phys. 40, 1792–1806 (1999) 44. Alexandrov, M., Kontsevich, M., Schwartz, A.: and O. Zaboronsky, The geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. A12, 1405–1430 (1997) 45. Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591–611 (2000) 46. Batalin, I., Marnelius, R.: Generalized Poisson sigma models. Phys. Lett. B512, 225–229 (2001) 47. Cattaneo, A.S., Felder, G.: On the AKSZ formulation of the Poisson sigma model. Lett. Math. Phys. 56, 163–179 (2001) 48. Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Vol. II. New York: John Wiley & Sons, 1996 ed., Orig. published in 1969 Communicated by M.R. Douglas
Commun. Math. Phys. 254, 603–650 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1243-z
Communications in
Mathematical Physics
Quantization of Classical Dynamical r-Matrices with Nonabelian Base Benjamin Enriquez1 , Pavel Etingof2 1 2
IRMA (CNRS), rue Ren´e Descartes, 67084 Strasbourg, France. E-mail:
[email protected] Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. E-mail:
[email protected]
Received: 28 November 2003 / Accepted: 24 June 2004 Published online: 13 January 2005 – © Springer-Verlag 2005
Abstract: We construct some classes of dynamical r-matrices over a nonabelian base, and quantize some of them by constructing dynamical (pseudo)twists in the sense of Xu. This way, we obtain quantizations of r-matrices obtained in earlier work of the second author with Schiffmann and Varchenko. A part of our construction may be viewed as a generalization of the Donin-Mudrov nonabelian fusion construction. We apply these results to the construction of equivariant star-products on Poisson homogeneous spaces, which include some homogeneous spaces introduced by De Concini. Introduction In this paper, we construct generalizations of some classes of classical dynamical rmatrices with nonabelian base, and quantizations of some of them (in the sense of [Xu1]). This way, we obtain quantizations of dynamical r-matrices introduced in [EV1, ES2]. We then apply these results to obtain explicit, equivariant star-products on some homogeneous spaces. In particular, we obtain quantizations of Poisson homogeneous spaces, introduced by De Concini. The classes of r-matrices we consider are the following (a), . . . , (d) (all Lie algebras are assumed to be finite dimensional). (a) Let g = l ⊕ u be a Lie algebra with a nondegenerate splitting (see Sect. 1). g Then the natural map u ⊗ u → l can be “inverted” and yields a solution rl (λ) ∈ 2 l ∧ (g) ⊗ S · (l)[1/D0 ] of the classical dynamical Yang-Baxter equation (CDYBE), g
g
CYB(rl (λ)) − Alt(drl (λ)) = 0. g
rl (λ) is a rational function in λ, homogeneous of degree −1, and is a generalization of the rational classical dynamical r-matrices of [EV1] (here D0 is a certain nonzero homogeneous element in S · (l) and S · (l)[1/D0 ] is the corresponding localization of S · (l)). g rl (λ) also plays a role in “composing” r-matrices:
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Proposition 0.1 (see [EV1], Theorem 3.14 and [FGP], Proposition 1). Let l ⊂ g be an inclusion of Lie algebras. Assume that l has a nondegenerate splitting l = k ⊕ m. Given Z ∈ ∧3 (g)g , let us say that a (l, g, Z)-r-matrix is an l-invariant function ρ : l∗ → ∧2 (g), solution of CYB(ρ(λ)) − Alt(dρ(λ)) = Z. Set σ (λ) := rkl (λ) + ρ|k∗ (λ). Then ρ → σ is a map {(g, l, Z)-r-matrices} → {(g, k, Z)-r-matrices}. Here by a function l∗ → ∧2 (g), we understand an element of ∧2 (g) ⊗ S · (l)[1/], · 1,2 where is a suitable nonzero element of S (l). We set CYB(ρ) := [ρ , ρ 1,3 ] + [ρ 1,2 , ρ 2,3 ] + [ρ 1,3 , ρ 2,3 ], and d(x ⊗ y ⊗ z1 · · · zl ) := li=1 x ⊗ y ⊗ zi ⊗ z1 · · · zˇ i · · · zl . If ξ ∈ g⊗3 is antisymmetric in two tensor factors, we set Alt(ξ ⊗ f ) = (ξ + ξ 2,3,1 + ξ 3,1,2 ) ⊗ f . Proposition 0.1 enables us to construct new r-matrices from known ones. (b) Let (g = l ⊕ u, t ∈ S 2 (g)g ) be a quadratic Lie algebra with a nondegenerate splitting (we do not assume t to be nondegenerate). We may apply Proposition 0.1 to ρ := the Alekseev-Meinrenken r-matrix of g ([AM1]), (l, k, m) := (g, l, u), and obtain: Corollary 0.2 (see also [FGP]). Let c ∈ C, and λ ∈ l∗ , set g ρc (λ) = rl (λ) + c f (c ad(λ∨ )) ⊗ id (t). Then we have CYB(ρc ) − Alt(dρc ) = −π 2 c2 Z, where Z = [t 1,2 , t 2,3 ]. Here we set λ∨ = (λ ⊗ id)(t) and f (x) = −1/x + π cotan(π x). (c) Let (g, t ∈ S 2 (g)g ) be a quadratic Lie algebra, equipped with σ ∈ Aut(g, t). We assume that σ − id is invertible on g/gσ . We set l := gσ , u := Im(σ − id), so g = l ⊕ u, t = tl + tu , tx ∈ S 2 (x) for x = l, u. The following result can be found in [AM2] (see also [S] and [ES2], Theorem A1). Proposition 0.3. Set ∨
e2πicad(λ ) ◦ σ + id ρσ,c (λ) := (cf (cad(λ )) ⊗ id)(tl ) + iπ c( 2πicad(λ∨ ) ⊗ id)(tu ). e ◦ σ − id ∨
Here we set λ∨ = (λ ⊗ id)(tl ) for λ ∈ l∗ . Then ρσ,c is a solution of CYB(ρσ,c ) − Alt(dρσ,c ) = −π 2 c2 Z. Note that if χ : l → C is a character, then χ ∨ is central in l, and if g = l ⊕ u is nondegenerate and tu is nondegenerate, then ρexp(ad(χ ∨ )) (λ) coincides with ρc (λ − χ ), with ρc as in Corollary 0.2. If now l has a nondegenerate splitting l = k ⊕ m, then Proposition 0.1 implies that rkl + (ρσ,c )|k∗ is a (k, g, −4π 2 c2 Z)-r-matrix. (d) Let g = l ⊕ u be a Lie algebra with a splitting. Assume that t ∈ S 2 (g)g decomposes as tl + tu , with tx ∈ S 2 (x) for x = l, u. Let us say that C ∈ End(u) is a Cayley endomorphism if it satisfies the following axioms: C is a l-module endomorphism, and for any x, y ∈ u, we have [C(x), C(y)]u = C([C(x), y]u ) + C([x, C(y)]u ) − [x, y]u .
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Here for x ∈ g, we denote by xu its projection on u parallel to l. If σ is as in (3), then C = C(σ ) := (σ + id)/(σ − id) is a Cayley endomorphism. (Such Cayley endomorphisms are exactly those which do not contain ±1 in their spectrum.) More generally, a limit of C(exp(ad(χ ))), where some eigenvalues of χ tend to ±∞, is a Cayley endomorphism. Proposition 0.4. Assume that (C ⊗id+id⊗C)(tu ) = 0 (when C = C(σ ), this condition means that σ preserves tu ). Set ρC,c (λ) := (cf (cad(λ∨ )) ⊗ id)(tl ) + iπ c
C + itan(π cad(λ∨ )) ⊗ id (tu ). 1 + itan(π cad(λ∨ ))C
Then ρC,c is a (l, g, −π 2 c2 Z)-r-matrix. We define quantizations of solutions of the (modified) CDYBE as solutions of suitable (pseudo)twist equations (Sects. 1.4, 3, and also [EE], Eq. 9). We construct quantizations for the above r-matrices in the following cases. g (a’) Rational r-matrices. We construct quantizations Jl of the rational r-matrices introduced above in the particular case when g is polarized, i.e., u decomposes as a sum of l-submodules u+ ⊕ u− , such that u± are Lie subalgebras of g (Corollary 2.6). We do so by working out a nonabelian generalization of the fusion construction of [EV2] (whose ideas originate from [Fad, AF]). Recently J. Donin and A. Mudrov [DM] (see also [AL]) extended this construction to the case when h is replaced by a Levi subalgebra l ⊂ g; their work relies on Jantzen’s computation of the Shapovalov form for induced modules. To generalize their result, we work directly in (microlocalizations of) universal enveloping algebras. (b’) We quantize the rational-trigonometric r-matrices of Corollary 0.2 in the following situation: g is polarized, and t ∈ S 2 (g)g decomposes as tl +s +s 2,1 , where tl ∈ S 2 (l) and s ∈ u+ ⊗ u− (Theorem 3.2). Our argument is based on nonabelian versions of the g ABRR identities (see [ABRR, EV1, ES2]), which are satisfied by Jl when g is polarized and quadratic, and the use of Drinfeld associators ([Dr2]). When l = g, our construction coincides with the quantization of the Alekseev-Meinrenken r-matrix ([EE]), which is based on renormalizing an associator. (c’) We quantize the r-matrix ρσ,c defined in Proposition 0.3 (Theorem 4.6). We also quantize the r-matrix (ρσ,c )|k∗ + rkl under the assumption that l = k ⊕ m+ ⊕ m− is quadratic polarized (Proposition 4.10). For this, we introduce a compatible differential system, generalizing the Knizhnik-Zamolodchikov (KZ) system, and we adapt Drinfeld’s proof that the KZ associator satisfies the pentagon equations ([Dr2]). In Sect. 4.7, we explain why the quantizations obtained in (b’), (c’) may be interpreted in terms of infinite-dimensional ABRR equations for extended (twisted) loop algebras. In Sect. 6, we apply these results to the construction of equivariant star-products on Poisson homogeneous spaces. In particular, we quantize Poisson homogeneous spaces introduced by De Concini. Remark 0.5. An earlier version of this paper is available at www-math.mit.edu/etingof/ ee.tex; this version is less general but uses a somewhat more intuitive representation theoretic language.
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1. Rational Classical Dynamical r-Matrices In this section, we introduce the notion of a (nondegenerate) Lie algebra with a splitting g g = l ⊕ u. We associate to each such nondegenerate Lie algebra a rational r-matrix rl . We show that in the case of a double inclusion k ⊂ l ⊂ g of Lie algebras, rkl plays a role in a restriction theorem for r-matrices. We introduce the notions of polarized Lie algebras g g and of quantizations of the rational r-matrices rl . As rl is singular at λ = 0 ∈ l∗ , the latter notion involves a microlocalization of U (l) (in the sense of [Spr]). Notation. If A is a Hopf algebra, we use Sweedler’s notation:(x) = x (1) ⊗ x (2) . If x ∈ A, we write x (2) := 1 ⊗ x ⊗ 1 · · · ∈ A⊗n , and if x = i xi ⊗ xi ∈ A⊗2 , we set x 3,2 := i 1⊗xi ⊗xi ⊗1 · · · ∈ A⊗n , x 3,21 := i (xi )(2) ⊗(xi )(1) ⊗xi ⊗1 · · · ∈ A⊗n , etc. 1.1. A family of classical dynamical r-matrices. Let g be a finite dimensional Lie algebra.Assume that we have a decomposition g = l⊕u, where u is an l-invariant complement of l in g; that is, [l, u] ⊂ u. Such a triple (g, l, u) is called a “Lie algebra with a splitting”. We have a linear map l∗ → ∧2 (u)∗ , taking λ ∈ l∗ to ω(λ) : x ∧ y → λ([x, y]). The triple (g, l, u) is called nondegenerate if for a generic λ ∈ l∗ , ω(λ) is nondegenerate. The algebraic translation of this condition is the following: identify ∧2 (u)∗ with a subspace of End(u) using any linear isomorphism u u∗ , then the map λ → det ω(λ) does not vanish identically. This map is a degree d := dim(u) polynomial on l∗ , i.e., an element of S d (l). If (g, l, u) is nondegenerate, then d is even. If E is an even dimensional vector space, denote by ∧2 (E)nondeg the space of nondegenerate tensors in ∧2 (E). Then we have a bijection ∧2 (E ∗ )nondeg → ∧2 (E) nondeg , ω → ω−1 , taking a tensor ω to its image under the inverse of the linear isomorphism E ∗ → E induced by ω. Proposition 1.1 (see [FGP], Proposition 1 and [Xu2], Theorem 2.3). Let (g, l, u) be a nondegenerate Lie algebra with a splitting. Then we have a rational map g
rl : l∗ ⊃ U → ∧2 (u), g
defined by rl (λ) := −ω(λ)−1 . It is homogeneous of total degree −1 in λ. Here U ⊂ l∗ is the l-invariant open subset {λ| det ω(λ) = 0} ⊂ l∗ . g g g Then rl is l-invariant, and is a solution of the CDYBE, i.e., CYB(rl )−Alt(drl ) = 0. g Proof. Set D0 := det ω(λ) ∈ S d (l). Then rl = i ui ⊗ vi ⊗ i belongs to ∧2 (u) ⊗ S · (l)[1/D0 ], and is uniquely determined by the equivalent conditions h(x, ui )h(vi , y) i = −h(x, y) (equality in S · (l)[1/D0 ]) ∀x, y ∈ u, i
or ∀x ∈ g,
h(x, ui ) i ⊗ vi = −1 ⊗ xu
(equality in S · (l)[1/D0 ] ⊗ u).
(1)
i
Here we denote by xu , xl the components of x ∈ g in u, l, and by h : g × g → l the map (x, y) → [x, y]l .
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We define a bilinear map −, − : (g⊗3 ⊗ S · (l)[1/D0 ]) × g⊗3 → S · (l)[1/D0 ] by a ⊗ b ⊗ c ⊗ , x ⊗ y ⊗ z := h(x, a)h(y, b)h(z, c) . This pairing is left-nondegenerate, g g so we will prove that the pairing of CYB(rl ) − Alt(drl ) with x ⊗ y ⊗ z ∈ g⊗3 is zero. g 1,2 g 1,3 g We have [(rl ) , (rl ) ], x ⊗ y ⊗ z = h([yu , zu ], x), therefore CYB(rl ), x ⊗ y ⊗ z = [[yu , zu ], x]l + c.p. = [h(y, z), xl ] + c.p.; the last equality follows from the Jacobi identity and the l-invariance of u (c.p. = cyclicpermutation). On the other hand, differentiating (1), and pairing the resulting identity with z ⊗ y, we get
h(x, ui )h(vi , y)h(ε, z) εi +
i
[h(x, ui ), zl ] i h(vi , y) = 0.
(2)
i
Here we set d( i ) = ε ⊗ εi . g We have drl , x ⊗ y ⊗ z = − i,ε h(x, ui )h(y, vi )h(z, ε) εi , so by (2) this is equal to i h(vi , y)[h(x, ui ), zl ]. i
g
Now (1) implies that this is equal to [h(x, yu ), zl ]. Finally, Alt(drl ), x ⊗ y ⊗ z = [h(x, yu ), zl ] + c.p. = [h(x, y), zl ] + c.p., where the last equality follows from l-invarig g ance of u and the Jacobi identity. Finally, we get CYB(rl ) − Alt(drl ), x ⊗ y ⊗ z = 0, as wanted. Remark 1.2. The nondegeneracy condition means that for a generic λ ∈ l∗ , the tangent space Tλ (Oλ ) of the coadjoint orbit of λ contains u∗ ; this means that a generic element of g∗ is conjugate to an element of l∗ . Remark 1.3. D0 satisfies ad(a)(D0 ) = χ0 (a)D0 , where χ0 : l → C is the character of l defined by χ0 (a) = tr(ad(a)u ), so D0 is l-equivariant.
1.2. Composition of r-matrices. Let us prove Proposition 0.1. Let us first prove that the restriction ρ|k∗ is well-defined. The singular locus {λ ∈ l∗ |(λ) = 0} is l-invariant, so it cannot contain k∗ ; therefore |k∗ is nonzero, and ρ|k∗ ∈ ∧2 (g) ⊗ S · (k)[1/|k∗ ] is well-defined. Both rkl and ρ|k∗ are k-invariant, hence so is σ . Let us write rkl (λ) as i ui (λ) ⊗ ei , where ui (λ) ∈ u ⊗ S · (l)[1/D0 ], and show that ad∗ (ui (λ))(λ) = −εi .
(3)
This equality means that for any x ∈ u, we have i λ([x, ui (λ)])ei = −x, i.e., if j j i ui (λ) = j e ⊗ fi,j (λ), then i,j λ([x, e ])fi,j (λ) ⊗ e = −1 ⊗ x. Taking into account the identification of the function λ → λ(x) with x ∈ S 1 (l), (3) now follows from (1). l We now show that if f ∈ ∧2 (g) ⊗ S · (l)[1/] , then (df )|k∗ − d(f|k∗ ) = −[rk,l (λ)1,3 + rk,l (λ)2,3 , (f|k∗ )1,2 ].
(4)
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d i 1,2 i 3 i dt |t=0 f (λ + tε ) (e ) , where (εi ), (ei ) are dual bases of u∗ and u. According to (3), this l.h.s. is equal to − i ddt f (Ad(etui (λ) )(λ))1,2 (ei )3 , which by invariance of f is the r.h.s. of (4). |t=0 Then we get CYB(σ ) − Alt(dσ ) = CYB(rk,l ) − Alt(drk,l ) + CYB(ρ) − Alt(dρ|k∗ ) + CYB(rk,l , ρ) − Alt((dρ)|k∗ − d(ρ|k∗ )) .
The l.h.s., evaluated at λ ∈ k∗ , is equal to
(Here CYB(a, b) is the bilinear form derived from the quadratic form CYB.) In this equality, the first term is zero by Proposition 1.1, the second term is equal to Z, and the last term is zero by (4). Remark 1.4. In the case where g = l ⊕ u is a nondegenerate Lie algebra with a splitting, Z = 0 and ρ = rl,g , then σ = rk,g . In the polarized case, a quantum analogue of this statement is Proposition 2.15. 1.3. Polarized Lie algebras. We say that the Lie algebra with a splitting (g, l, u) is polarized if we are given a decomposition u = u+ ⊕ u− of u as a sum of two l-submodules, such that u+ and u− are Lie subalgebras of g. We denote by p± the “parabolic” Lie subalgebras p± = l ⊕ u± ⊂ g. Assume that (g, l, u) is nondegenerate and polarized; then dim(u + ) = dim(u− ). In g that case, λ → rl (λ) takes its values in (u+ ⊗ u− ) ⊕ (u− ⊗ u+ ) ∩ ∧2 (u). Therefore g rl = r − (r )2,1 , where r ∈ u+ ⊗ u− ⊗ S · (l)[1/D0 ]. We will call r the “half r-matrix” of (g, l, u+ , u− ). 1.4. Quantization. Let (g, l, u) be a nondegenerate Lie algebra with a splitting. Let D as the microlocalization of in U (l) be a degree ≤ d element with symbol D0 . Define U is independent on the U (l), obtained by inverting D ([Spr]). U (l) embeds into U , and U choice of D up to isomorphism. U is a complete filtered algebra, with associated graded S · (l)[1/D0 ]. . An element of U is represented by a series i∈Z ai D −i , Here is a description of U where ai ∈ U (l) vanishes for −i large enough, and the sequence deg(ai ) − id tends to −∞ as i → ∞. Two such series are equivalent if they differ by a sum i xi , where q(i) q(i) xi has the form k=p(i) αk (i)D −k , k=p(i) αk (i)D q(i)−k = 0 (equality in U (l)) and q(i)
maxk=p(i) (degαk (i) − kd) → −∞ as i → ∞. The degree of f is the minimum of all maxi (deg(ai ) − id) running over all i ai D −i representing f . The product of two elements is
−i ai adα (D)(bj ) D −k . ai D −i bj D −j = α i∈Z
j ∈Z
k∈Z α≥0
. Then if V is a vector space, we set ≤k the degree ≤ k part of U We denote by U V ⊗U = lim← (V ⊗ U )/(V ⊗ U≤k ). The coproduct map of U (l) extends to a map → U (l)⊗ , where the image of D −1 is i≥0 (−1)i (1 ⊗ D −1 )a · · · a(1 ⊗ D −1 ), and U U a = (D) − 1 ⊗ D ∈ U (l) ⊗ U (l)≤d−1 .
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609
g
≤0 , satisU Definition 1.5. A quantization of rl is a l-invariant element J ∈ U (g)⊗2 ⊗ fying the dynamical twist equation J 12,3,4 J 1,2,34 = J 1,23,4 J 2,3,4 , ≤−1 , the reduction j of J − 1 modulo U and the following conditions: J − 1 ∈ U (g)⊗2 ⊗ g ⊗2 ⊗2 · U (g) ⊗U≤−2 (an element of U (g) ⊗ S (l)[1/D0 ]−1 ) satisfies Alt(j ) = rl . Then R := (J 2,1,3 )−1 J 1,2,3 is a solution of the dynamical quantum Yang-Baxter equation R 1,2,4 R 1,3,24 R 2,3,4 = R 2,3,14 R 1,3,4 R 1,2,34 . The PBW star-product on l∗ may be described as follows: l[[]] ⊂ l[[]] is a Lie subalgebra, then the -adic completion of U (l[[]]) ⊂ U (l)[[]] is a flat deformation of S · (l) (which means that as a C[[]]-module, it is isomorphic to S · (l)[[]]), which we · denote by S (l) (it is the quantized formal series algebra associated to the trivial deformation of U (l)). Then U (l)(()) identifies with S · (l) [−1 ]; moreover, this identification −k · takes U (l)≤k [[]] into S (l) . This discussion can be localized. We denote by S · (l)[1/D0 ] the -adic completion (()) generated by l[[]] and (d D)−1 . This is a flat deformation of the subalgebra of U (()) identifies with ≤k [[]] goes of S · (l)[1/D0 ]. Moreover, U S · (l)[1/D0 ] [−1 ], and U −k · into S (l)[1/D0 ] . It follows that J gives rise to an element J (λ) of U (g)⊗2 ⊗ S · (l)[1/D0 ] , with the g 2 expansion J (λ) = 1 + j (λ) + O( ), and Alt(j (λ)) = rl . In Sect. 2, we will quantize the classical dynamical r-matrices arising from nondegenerate polarized Lie algebras. → C(()) are all of the following form: Remark 1.6. The continuous characters χ : U i λ : l → C(()) is a character of l, of the form λ = i≥v λi , with v < 0 and → C(()), D0 (λv ) = 0. Then λ is a character U (l) → C(()), it extends to a character U ≤0 → C[[]]. which restricts to U Remark 1.7. Microlocalization. Springer’s microlocalization ([Spr]) associates to a pair (A, f ), where A is a Z-filtered algebra with gr(A) integral commutative and f ∈ A is nonzero, a complete separated Z-filtered algebra Af , such that gr(Af ) = gr(A)[1/f¯] (here f¯ is the symbol of f , i.e., its nonzero homogeneous component with maximal degree). (Af )≤0 is a subalgebra of Af and contains (Af )≤−1 as an ideal. Af has the following universal property: if B is a Z-filtered, complete separated algebra (i.e., ∩i Bi = {0} and B = lim←i (B/Bi )), and µ : A → B is a morphism of filtered algebras, such that µ(f ) is invertible, then µ extends to a morphism of topological filtered algebras Af → B. Actually, Af depends only on f¯, and when A is graded, Af is the completion of its associated graded. E.g., if A = C[x1 , . . . , xn ] and f ∈ A − {0} is homogeneous, these algebras can be described as follows. Let C(f ) = C ⊂ Cn be the cone defined by the equation f = 0. Then gr(Af ) is the ring on functions on Cn − C. The projective space Pn decomposes as Cn ∪ H , where H is the hyperplane at infinity, and the closure C of C in Pn decomposes as C ∪ C∞ , where C∞ = C ∩ H . Then gr((Af )≤0 ) is the ring of functions on Pn − C, the quotient (Af )≤0 /(Af )≤−1 is the ring of C× -invariant functions on Cn − C. Finally, (Af )≤0 (resp., Af ) is the ring of functions on the formal (resp., formal punctured) neighborhood on H − C∞ in Pn − C. In general, if A is a Z+ -filtered commutative algebra and X = Spec(A), then X has a compactification X = X ∪ X∞ . Here X = Proj(R(A)), where R(A) is the Rees algebra
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of A, and X∞ = Proj(gr(A)). If g ∈ A − {0}, then Ag (resp., (Ag )≤0 ) is the ring of functions on the formal (resp., formal punctured) neighborhood of X∞ − C∞ (g), where ¯ C∞ (g) = V (g) ∩ X∞ , and V (g) ⊂ X is the zero-set of g. C∞ (g) depends only on g, which explains why the same is true about Ag . 1.5. Examples. 1.5.1. Lie algebras with a splitting. (1) An inclusion l ⊂ g of simple Lie algebras with the same Cartan algebra h is called a Borel-de Siebenthal pair ([BS]). Then l has an invariant complement u. If λ ∈ h∗ , the bilinear form g2 → C, (x, y) → λ, [x, y] is nondegenerate for λ generic, and is the sum of two bilinear forms l2 → C and u2 → C, which are therefore nondegenerate. In particular, u2 → C, (x, y) → λ, [x, y] is nondegenerate. So (g, l, u) is a nondegenerate Lie algebra with a splitting. (2) If g is a finite dimensional Lie algebra and r ∈ ∧2 (g) is a nondegenerate triangular r-matrix, then the dual of r is a 2-cocycle on g. Let g = g ⊕ Cc be the corresponding central extension. Then g is a nondegenerate Lie algebra with a splitting with l = Cc, u = g. The corresponding r-matrix is λ → r/λ. (3) We generalize (2) to the case when l is no longer 1-dimensional. Let g be a Lie algebra, let z ⊂ g be a central subalgebra, set g := g/z, and let π : g → g be the canonical projection. Let u ⊂ g be a complement of z. Then ( g, z, u) is a Lie algebra with a g splitting; let us assume it is nondegenerate. Set r := (π ⊗ π ⊗ id)(rl ). Then r satisfies ∗ CYB(r) = 0. In particular, for any λ ∈ l such that D0 (λ) = 0, rλ := (id⊗id⊗λ)(r) is a triangular r-matrix (we identify λ with a character of S · (l)[1/D0 ]). If J is a quantization g ≤0 → C[[]] is a character as in Remark 1.6, then Fχ := (π ⊗π ⊗χ )(J ) of rl , and χ : U g is a solution of the twist equation, quantizing rλ . A quantization of rl was obtained in [Xu2] using Fedosov quantization. 1.5.2. Polarized Lie algebras. (1) If g is a semisimple Lie algebra and l ⊂ g is a Levi subalgebra, then (g, l) gives rise to a nondegenerate polarized Lie algebra, which was g studied in [DM]. Then rl : l∗ → ∧2 (u) is defined by g (ad(λ∨ ))−1 (eα ) ∧ fα , rl (λ) = − α∈+ (g)−+ (l)
g
for λ ∈ l∗ such that ad(λ∨ )|u ∈ End(u) is invertible. rl is also uniquely determined by the requirements that it is an l-equivariant rational function, such that ∀λ ∈ h∗ ,
g
rl (λ) = −
α∈+ (g)−+ (l)
eα ∧ f α . (λ, α)
Here + (g), + (l) are the sets of positive roots of g, l, and x ∧ y = x ⊗ y − y ⊗ x. (2) Let g be a finite dimensional Lie algebra, which can be decomposed (as a vector space) as g = g+ ⊕ g− , where g± ⊂ g are Lie subalgebras. Let r ∈ g+ ⊗ g− be a nondegenerate tensor, such that r := r − ( r)2,1 is a triangular r-matrix (i.e., it satisfies the CYBE). Then we may construct g as above. If we set l = Cc, u± = g± , we get a nondegenerate polarized Lie algebra. (3) Let g = l⊕u+ ⊕u− be a nondegenerate polarized Lie algebra, and A = C[t]/(t n ), then g ⊗ A = (l ⊗ A) ⊕ (u+ ⊗ A) ⊕ (u− ⊗ A) is a nondegenerate polarized Lie algebra.
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1.5.3. Infinite dimensional examples. The definitions of polarized Lie algebras, their r-matrices and quantizations generalize to the case of graded Lie algebras with finite dimensional graded parts. (1) The Virasoro algebra Vir decomposes as l ⊕ u+ ⊕ u− , where l = Cc ⊕ CL0 and u± = ⊕i>0 CL±i . Then (Vir, l, u+ , u− ) is an infinite dimensional polarized Lie algebra. (2) If g is a Kac-Moody Lie algebra and l ⊂ g is a Levi subalgebra, then (g, l) gives rise to an infinite dimensional polarized Lie algebra. Remark 1.8. The proof of Proposition 0.1 shows that if g = l ⊕ u is a Lie algebra with a nondegenerate splitting, U ⊂ g∗ is an invariant open subset and r : U → ∧2 (g) is g g-invariant, such that r|g∗ + rl is a (l, g, Z)-r-matrix, then r is a (g, g, Z)-r-matrix. This leads to the following r-matrix (a quantization of which is unknown). Let g be a semisimple Lie algebra and t ∈ S 2 (g)g be nondegenerate. For ξ ∈ g∗ , set ξ ∨ = (ξ ⊗ id)(t). If h ⊂ g is a Cartan subalgebra, let th be the part of t corresponding to h . Set g∗ss = {ξ ∈ g∗ |ξ ∨ is semisimple}. If x ∈ g is semisimple, let hx = {h ∈ g|[h, x] = 0} be the Cartan subalgebra associated to x. Then the map g∗ss → ∧2 (g), ξ → (ad(ξ )−1 ⊗ id)(t − th(ξ ) ) is a (g, g, 0)-r-matrix. 2. Dynamical Twists in the Polarized Case g
g
In this section, we constuct a dynamical twist Jl quantizing rl . For this, we first construct an element K; it is defined by algebraic requirements, related with the Shapovalov g form. We then construct J = Jl and show that it obeys the dynamical twist equation. We then show that J satisfies nonabelian versions of the ABRR equations. 2.1. Construction of K. Let g = l ⊕ u+ ⊕ u− be a nondegenerate polarized Lie algebra. Denote by H : U (g) → U (l) the Harish-Chandra map, defined as the unique linear map such that H (x+ x0 x− ) = ε(x+ )ε(x− )x0 if x0 ∈ U (l) and x± ∈ U (u± ). Here ε : U (g) → C is the counit map. Let d = dim(u± ) and let D0 ∈ S d (l) be the polynomial taking λ ∈ l∗ to det(λ◦ω)◦i, where ω : u+ ⊗ u− → l is the Lie bracket followed by projection, λ ◦ ω is viewed as a linear map u∗+ → u− and i is a fixed linear isomorphism u− → u∗+ . The relation with the objects introduced in the previous section is d = d/2 and identifies with the microlocalization of U (l) with respect (D0 )2 = D0 . In particular, U to a lift D ∈ U (l)≤d of D0 . , such that if U Theorem 2.1. There exists a unique element K ∈ U (u+ ) ⊗ U (u− ) ⊗ + − we set K = i ei ⊗ ei ⊗ i , then we have ∀x ∈ U (p− ), ∀y ∈ U (p+ ), H (xei+ ) i H (ei− y) = H (xy). (5) i
Equivalently, we have for any x± ∈ U (u± ), x0 ∈ U (l), H (xei+ ) i ⊗ ei− = ε(x+ )x0 ⊗ x− , ei+ ⊗ i H (ei− y) = x+ ⊗ x0 ε(x− ), i
i
where x = x+ x0 x− . K has also K is invariant under the the following properties. ≤−n , U adjoint action of l. K is a sum n≥0 Kn , where Kn ∈ U (u+ )≤n ⊗ U (u− )≤n ⊗
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and the image of Kn in S n (u+ ) ⊗ S n (u− ) ⊗ S · (l)[1/D0 ]−n under the tensor product of ≤−n → U ≤−n /U ≤−n−1 the projection maps U (u± )≤n → U (u± )≤n /U (u± )≤n−1 and U 1 n coincides with n! (r ) , where r := −r ∈ u+ ⊗ u− ⊗ S · (l)[1/D0 ]−1 is the opposite of the “half r-matrix” of (g, l, u+ , u− ) (the index k means the homogeneous part of degree k; the index ≤ k means the part of degree ≤ k; the algebra structure of S · (u+ ) ⊗ S · (u− ) ⊗ S · (l)[1/D0 ] is understood). Equation (5) is a nonabelian version of the condition that K is inverse to the pairing (x, y) → H (xy). Proof. r is a sum i ai ⊗ bi ⊗ Pi (D0 )−1 , with Pi ∈ S d −1 (l). Let P¯i ∈ U (l)≤d −1 be ≤−1 is a lift of a lift of Pi , and let r¯ := i ai ⊗ bi ⊗ P¯i (D )−1 . Then r¯ ∈ u+ ⊗ u− ⊗ U r . Let us set K¯ 0 := exp(ρ). ¯ Lemma 2.2. Set K¯ = i e¯i+ ⊗ e¯i− ⊗ ¯i . If x ∈ U (u− ), set T (x) = i H (x e¯i+ ) ¯i ⊗ e¯i− . ≤0 ⊗ U (u− ), such that if x has degree ≤ n, then Then T is a linear map U (u− ) → U ≤0 ⊗ ≤−1 ⊗ U (u− )≤n−1 + U U (u− ). T (x) − 1 ⊗ x ∈ U
(6)
Proof. The map H is such that if x± ∈ U (u± ) have degrees ≤ n± , then H (x− x+ ) ∈ U (l) has degree ≤ min(n+ , n− ). The bilinear map U (u− ) ⊗ U (u+ ) → U (l), x− ⊗ x+ → H (x− x+ ) therefore induces a collection of bilinear maps S n (u− ) ⊗ S n (u+ ) → S n (l), which turn out to be the symmetric powers of h : u+ ⊗ u− → l. (n) (n) (n) (n) (n) (n) Write r¯ n = i ai ⊗ bi ⊗ i , where ai , bi have degree ≤ n and hi has (n) (n) degree ≤ −n. Then if x has degree ≤ k, H (xai ) i has degree ≤ min(k, n) − n ≤ 0. Moreover, this degree tends to −∞ as n → ∞, so T is well-defined and maps to ≤0 ⊗ U (u− ). U Let us prove (6) when x ∈ u− . We have i h(x ⊗ ai )Pi ⊗ bi = D0 ⊗ x (equality in S d (l) ⊗ u− ), so i H (xai )P¯i ⊗ bi ∈ D ⊗ x + U (l)≤d −1 ⊗ u− . Then i H (xai ) ≤−1 ⊗u− . On the other hand, if n > 1, then H (xa (n) ) (n) ⊗b(n) P¯i (D )−1 ⊗bi ∈ 1⊗x+ U i i i ≤−1 ⊗ U (u− ). has degree ≤ 1 − n ≤ −1. So T (x) − 1 ⊗ x ∈ U (n) (n) Let now x ∈ U (u− ) be of degree k. If n < k, then H (xai ) i has degree ≤ 0, so (n) (n) (n) n i H (xai ) i ⊗ bi ∈ U≤0 ⊗ U (u− )≤n , so the contribution of n
k, then H (xai ) i has degree ≤ k − n ≤ −1. This shows that T (x) − 1 ⊗ x has the required degree properties. ≤0 ⊗ U (u− ), such Lemma 2.3. T extends uniquely to a continuous endomorphism T of U ≤0 and x ∈ U (u− ). T is invertible, and that T( ⊗ x) = ( ⊗ 1)T (x) for any ∈ U T := (T−1 )|1⊗U (u− ) has the same degree properties as T : if x ∈ U (u− ) has degree ≤ n, then ≤0 ⊗ ≤−1 ⊗ U (u− )≤n−1 + U U (u− ). T (x) − 1 ⊗ x ∈ U Proof. Clear.
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End of proof of Theorem 2.1. We now set K := i (e¯i+ ⊗ 1 ⊗ ¯i )(1 ⊗ T (e¯i− )2,1 ). Set K = i ei+ ⊗ ei− ⊗ i . Then if x ∈ U (u− ), we have
H (xei+ ) i ⊗ ei− =
i
(H (x e¯i+ ) ¯i ⊗ 1)T (e¯i− ) i
= (T (x)(1) ⊗ 1)T (T (x)(2) ) = T−1 (T (x)) = x, where we have set T (x) = T (x)(1) ⊗ T (x)(2) . If n ≥ 0, then the reduction of K modThe properties of T then imply the following. ≤0 /U ≤−n−1 lies in U (u+ )≤n ⊗ U (u− )≤n ⊗ (U ≤−n−1 ), and ulo U (u+ ) ⊗ U (u− ) ⊗U n its reduction modulo (U (u+ )⊗U (u− ))<2n ⊗(U≤0 /U≤−n−1 ) lies in S (u+ )⊗S n (u− )⊗ 1 n (r ) . This implies the claim on the decomposition S · (l)[1/D0 ]−n ; it identifies with n! on K. as K, then K := We now prove the uniqueness of K. If K has the same properties K − K = i ai ⊗ bi ⊗ i is such that for any x ∈ U (u− ), i H (xai ) i ⊗ bi = 0. Let (eI− ) be a basis of U (u− ), and set K = I,i ai,I ⊗ eI− ⊗ i,I , then i H (xai ) i,I = 0 for any I . We now prove: is such that i H (xai ) i = 0 for any U Lemma 2.4. If ξ = i ai ⊗ i ∈ U (u+ )⊗ x ∈ U (u− ), then ξ = 0. Proof of Lemma. Set ξ = α ξα , where deg(ξα ) = −α. Let α0 be the largest integer such that ξα0 = 0. We have ξα0 = N s=0 ηs , where ηs ∈ U (u− )≤s ⊗Uα0 −s . Then if x ∈ has degree ≤ min(n, s)+α0 −s. U (u− ) has degree ≤ n, then m◦(H ⊗id) (x⊗1)ηs ∈ U α0 −N , etc. U Pairing ηs with U (u− )≤N , U (u− )≤N−1 , etc., we get ηN ∈ U (u+ )≤N−1 ⊗ Finally ξα0 = 0, and ξ = 0. Therefore K = 0, so K is unique. Then its l-invariance follows from the l-invariance of H . g 2.2. The dynamical twist equation. If K = i ai ⊗ bi ⊗ i , set J = Jl := i ai ⊗ (2) (1) ≤0 . U S(bi )S( i ) ⊗ S( i ). Then J ∈ U (u+ ) ⊗ U (p− ) ⊗ Proposition 2.5. J satisfies the dynamical twist equation J 12,3,4 J 1,2,34 = J 1,23,4 J 2,3,4 .
(7)
This proposition has a representation-theoretic interpretation in terms of intertwiners, analogous to that of the abelian case (see [EV2] or [ES1], Proposition 2.3). Proof. Let us set K = i ai ⊗ bi ⊗ i . Then (7) can be written as follows: (1) (2) (3) (2) (2) (1) (1) ai aj ⊗ ai S(bj )S( j ) ⊗ S(bi )S( i )S( j ) ⊗ S( i )S( j ) i,j
=
i,j
(2)
(3)
(1)
(2)
(2)
(1)
(1)
ai ⊗ S(bi )S( i )aj ⊗ S(bi )S( i )S(bj )S( j ) ⊗ S( i )S( j ).
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Since K is l-invariant, the right-hand side is rewritten as (2) (3) (1) (2) (2) (1) (1) ai ⊗ S(bi )aj S( i ) ⊗ S(bi )S(bj )S( j )S( i ) ⊗ S( j )S( i ). i,j
Now both sides belong to the image of the map → U (u+ ) ⊗ U (g) ⊗ U (p− ) ⊗ , U U U (u+ ) ⊗ U (g) ⊗ U (u− ) ⊗ x ⊗ y ⊗ z ⊗ t → x ⊗ y ⊗ S(z)S(t (2) ) ⊗ S(t (1) ). So we have to prove the equality (1) (2) (2) (1) ai aj ⊗ ai S(bj )S( j ) ⊗ bi ⊗ j i i,j
=
(2)
(2)
(1)
ai ⊗ S(bi )aj S( i ) ⊗ bj bi
(1)
⊗ i j
(8)
i,j
. U in U (u+ ) ⊗ U (g) ⊗ U (u− ) ⊗ The linear map → HomC (U (u− ) ⊗ U (u+ ), U (g)⊗ ), U U U (u+ ) ⊗ U (g) ⊗ U (u− ) ⊗ A ⊗ B ⊗ C ⊗ D → x ⊗ y → BS H (xA)(2) ⊗ H (xA)(1) DH (By) is injective. This map takes the l.h.s. of (8) to (2) (1) (1) α : x ⊗ y → ⊗ (xy )0,i ε((xy (1) )+,i ), y (2) S (xy (1) )−,i S (xy (1) )0,i i
and the r.h.s. of (8) to β : x ⊗ y →
S(x (2) )(x (1) y)+,i ⊗ (x (1) y)0,i ε((x (1) y)−,i ).
i
Here we denote by i x+,i ⊗x0,i ⊗x−,i the image of x ∈ U (g) in U (u+ )⊗U (l)⊗U (u− ) by the inverse of the product map. To prove that α = β, we will prove that the maps (x, y) → (S(y (2) ) ⊗ 1)α(x ⊗ y (1) ) and (x, y) → (S(y (2) ) ⊗ 1)β(x ⊗ y (1) ) coincide. The first map takes (x, y) to (2) (1) ⊗ (xy)0,i ε((xy)+,i ) S((xy)−,i )S (xy)0,i i
and the second map takes (x, y) to S((xy)(2) )((xy)(1) )+,i ⊗ ((xy)(1) )0,i ε(((xy)(1) )−,i ). i
To prove the equality of both maps, it suffices to prove that the maps U (g) → U (g) ⊗ U (l), a → S(a−,i )S((a0,i )(2) ) ⊗ (a0,i )(1) ε(a+,i ) i
Quantization of Classical Dynamical r-Matrices with Nonabelian Base
and
a →
615
S(a (2) )(a (1) )+,i ⊗ (a (1) )0,i ε((a (1) )−,i )
i
coincide. If a0 ∈ U (l) and a± ∈ U (u± ), then the first map takes a+ a0 a− to S(a− )S((a0 )(2) ) ⊗ (a0 )(1) ε(a+ ), and the second map takes a+ a0 a− to S((a− )(2) )S((a0 )(2) )S((a+ )(2) )(a+ )(1) ⊗ (a0 )(1) ε((a− )(1) ), so both maps coincide. Together with the valuation results of Theorem 2.1, and taking into account the change of sign induced by S, Proposition 2.5 implies: g
Corollary 2.6. J is a quantization of rl , in the sense of Sect. 1.4. Example 2.7. If g is the Heisenberg algebra, spanned by x+ , x− , c, with [x+ , x− ] = c, (1) (2) u± = Cx± , l = Cc, then K = exp(−x+ ⊗ x− ⊗ c−1 ), so that J = exp(−x+ x− (c(2) + c(3) )−1 ), i.e., J (λ) = exp(−x+ ⊗ x− (λ + c)−1 ). 2.3. K, singular vectors, and fusion of intertwiners. If u ∈ u− , then x → [u, x] maps U (u+ ) to U (u+ )l ⊂ U (g). Proposition 2.8. If u ∈ u− , then [u(1) , K] − Ku(2) ∈ Im(ϕ), where ϕ : U (u+ ) ⊗ ≤−1 → U (p+ ) ⊗ U (u− ) ⊗ ≤0 is the map taking x+ ⊗ x− ⊗ ⊗ U U x to U (u− ) ⊗ l ⊗ x+ ⊗ x− ⊗ x − x+ ⊗ x− ⊗ x. Proof. Set K = i ai ⊗ bi ⊗ i . Then if x ∈ U (u− ), y ∈ U (u+ ), we have H (x[u, ai ]) i H (bi y) = H ((xu)ai ) i H (bi y), i
i
because H (ξ u) = 0 for any ξ ∈ U (g). Now H ((xu)ai ) i H (bi y) = H (xuy) = H (xai ) i H (bi uy). i
i
− is decomposed as i αi ⊗ βi ⊗ λi , we get So if L = H (xα )λ H (β y) = 0, therefore for any x ∈ U (u ), H (xαi )λi ⊗ βi = 0. i i i − i i ≤0 is such that i H (xα )λ = 0, there U Now if i αi ⊗ λ i ∈ U (u+ ) ⊕ U (u+ )l ⊗ i i ≤−1 , such that i α ⊗ λ = i (α i ) ⊗ U exists i αi ⊗ i ⊗ λ i ∈ U (u+ ) ⊗ l ⊗ i i i λ i − αi ⊗ ( i λ i ). [u(1) , K]
Ku(2)
-module and V is a g-module, the morphism U → If now Y is a topological U U (l)⊗U extending the coproduct of U (l) allows to view Y ⊗V as a U -module. U modules can be constructed as follows: let λ ∈ l∗ be a character such that D0 (λ) = -module, where x ∈ l acts as 0, and let (V , ρV ) be a l-module. Then V (()) is a U ρV (x) + −1 λ(x)idV . Denote by U (g) the microlocalization of U (g) associated with D . Then U (g) is . Let U (p− ) ⊂ U . U U isomorphic to U (u+ ) ⊗ U (u− ) ⊗ (g) be the subalgebra U (u− )⊗ Then any U -module Y may be viewed as a U (p− )-module. We associate to it the U (g) (g) := IndU module Y (Y ). U (p− )
U (g), so if Z is The coproduct of U (g) also extends to a morphism U (g) → U (g)⊗ V is a U aU (g)-module and V is a g-module, then Z ⊗ (g)-module.
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-modules and V is a g-module, and if ξ ∈ HomU (Y, Y ⊗ Proposition 2.9. If Y, Y are U → Y ⊗ V , such that ξ|Y = ξ + higher V ), then there is a U (g)-module morphism ξ : Y (2) (1) degree terms. Set K = i ai ⊗ bi ⊗ i , and set J := i ai ⊗ S(bi )S( i ) ⊗ S( i ). ξ If we write J = i αi ⊗ βi ⊗ λi , then |Y = i (αi ⊗ βi ) ◦ ξ ◦ S(λi ). Proof. The properties of K imply that for any u ∈ u− , (u(1) + u(2) )J = J u(1) + L, ⊗ k ⊗ 1)(e(1) + e(2) − e(3) )(1 ⊗ 1 ⊗ k ), where where L has the form i,α (ki,α α α α i,α i,α (eα )α is a basis of l. When Y = C, this proposition shows how to construct singular vectors in tensor products. We now show that J also controls the fusion of intertwiners. Let Y, Y , Y be -modules and let V , V be g-modules. Let ξ ∈ HomU (Y, Y ⊗ V ) and U , Y ⊗ V ). Then (ξ ⊗ id) ◦ ξ ∈ Hom (Y (V ⊗ V )), and ξ ∈ HomU (Y , Y ⊗ U (g)
(ξ ⊗ id) ◦ ξ =
(id ⊗ αi ⊗ βi ) ◦ (ξ ⊗ id) ◦ ξ ◦ S(λi ),
(9)
i
(V ⊗ V ) of an intertwiner. If A(ξ, ξ ) is where ... denotes the component Y → Y ⊗ the r.h.s. of (9), we even have
(ξ ⊗ id) ◦ ξ = A(ξ,ξ ) . All this follows from the fact that J satisfies the dynamical twist equation. 2.4. Microlocalized Harish-Chandra map. To state the composition formula, we need microlocalized versions of the Harish-Chandra map and the PBW isomorphism, which we now prove. Let a = b ⊕ c+ ⊕ c− be a polarized Lie algebra, let D ∈ S d (a) be a nonzero element, a , U b be the microlocalizations of U (a), U (b) such that D|b∗ ∈ S d (b) is nonzero. Let U ∗ w.r.t. lifts of D, D|b . b as follows: U Define a product on U (c+ ) ⊗ U (c− ) ⊗ (3) −1 ⊗ id) µ = (132) ◦ mU (c+ ) ⊗ mU ⊗ mU (c− ) ◦ (id ⊗ e+ ⊗ id ⊗ e− b ◦ (id ⊗ id ⊗ π ⊗ id ⊗ id) ◦ (132) ⊗ (132) . Here mA is the product map of an algebra A, mA : A⊗3 → A is (mA ⊗ id) ◦ mA , b ⊗ b are the exchange maps defined as the unique continU U (c± ) → U (c± )⊗ e± : U uous extensions of Ub ⊗ U (c± ) f ⊗ x → i xi ⊗ fi ∈ U (c± )⊗Ub , such that f x = i xi fi (identity in the microlocalization of U (b ⊕ c± ) w.r.t. a lift of D|b∗ ), π : U (c− )⊗U (c+ ) → U (c+ )⊗U (b)⊗U (c− ) is the composition of U (c− )⊗U (c+ ) → U (a) with the inverse of U (c+ ) ⊗ U (b) ⊗ U (c− ) → U (a) (both maps are inclusions followed by the product of U (a)). b . The U Lemma 2.10. µ is an associative, continuous product −) ⊗ on U (c+ ) ⊗ U (c b , µ , and U subspace U (c+ ) ⊗ U (c− ) ⊗ U (b) is a subalgebra of U (c+ ) ⊗ U (c− ) ⊗ is isomorphic to (U (a), mU (a) ) under α : x+ ⊗ x− ⊗ f → x+ f x− . (3)
Quantization of Classical Dynamical r-Matrices with Nonabelian Base
617
a → There is a unique morphism of topological algebras P BW : U b , µ , extending the inverse of the isomorphism α. U U (c− ) ⊗
U (c+ ) ⊗
Proof. The associativity of the transport of mU (a) on U (c+ ) ⊗ U (c− ) ⊗ U (a) may be viewed as a consequence of the commutativity of diagrams involving U (c± ) and b , which implies the U (b). These diagrams still commute when U (b) is replaced by U associativity of µ. D 0 of D and D|b∗ in U (a), U (b) in such a way that D ∈ Let us choose lifts D, U (a)≤d , D0 ∈ U (b)≤d and H (D) = D0 , and let us construct an inverse of α(D). Set −D 0 , and define inductively ξn , n ≥ 0 by ξ0 := D (2) −1 −1 −1 ). ξn = −(id ⊗ mU ⊗ id) ◦ (e+ ⊗ id ⊗ e− )(D0 ⊗ ξn−1 ⊗ D 0 b
b is ≤ d − 1, by construction, and the partial degree of The partial degree of ξ0 in U b is ≤ d − 1 − 2nd (because e± has partial degree 0 for the filtration by the ξn in U b -degree; actually its associated graded for this filtration is the identity). Therefore the U −1 + n≥1 ξn1,3,2 converges in U (c+ )⊗U (c− ) ⊗ b , and one shows that U sum 1⊗1⊗ D 0 The construction of PBW then follows from the universal property it is inverse to α(D). of Springer’s microlocalization. := (ε ⊗ ε ⊗ id) ◦ PBW, :U a → U b is a continuous map, then H Remark 2.11. Set H using can be recovered from H extending the Harish-Chandra map H . Moreover, PBW the formula ⊗ π− ) ◦ (l ⊗ id) ◦ r . = (π+ ⊗ H PBW a → U (a)⊗ a , r : U a → U a ⊗ U U (a) are the left- and right-comodHere l : U a under U (a), and π± : U (a) → U (c± ) are the maps U (a) → ule structures of U U (a) ⊗U (b⊕c− ) C → U (c+ ), U (a) → C ⊗U (b⊕c+ ) U (a) → U (c− ), induced by the natural projections and the inverses of the maps x+ → x+ ⊗ 1, x− → 1 ⊗ x− . In is a left U (c+ )-module and right U (c− )-module morphism. particular, PBW b , topologU Remark 2.12. If d ∈ Z, let Xd be the subspace of U (c+ ) ⊗ U (c− ) ⊗ b )≤d−α−β , where α, β ≥ 0. Then ically generated by the U (c+ )α ⊗ U (c− )β ⊗ (U Xd ⊂ Xd+1 , µ(Xd ⊗ Xd ) ⊂ Xd+d , and Xd is contained in the degree ≤ d part of U (c+ ) ⊗ if we set X := ∪d∈Z Xd , X is a topological subU (c− ) ⊗Ub , therefore factors through a algebra of U (c+ ) ⊗ U (c− ) ⊗Ub , µ . One can check that PBW a → X. morphism U 2.5. The composition formula. Assume that g = l ⊕ u+ ⊕ u− and l = k ⊕ m+ ⊕ m− are nondegenerate polarized Lie algebras. Set v± = u± ⊕ m± . Then g = k ⊕ v+ ⊕ v− is a polarized Lie algebra. Lemma 2.13. g = k ⊕ v+ ⊕ v− is nondegenerate. g
g
= dim(m ), d = dim(u ), d = d + d , and D ∈ S dv (k), D ∈ Proof. Let dm ± ± u v m u l k k d S u (l), D l ∈ S dm (k) be the determinants associated to each polarized Lie algebra. Then [m± , u∓ ] ⊂ u∓ , therefore g g D k = (D l )|k∗ D lk .
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Here the map x → x|k∗ is the algebra morphism S · (l) → S · (k), taking x+ x0 x− to ε(x+ )ε(x− )x0 , where x± ∈ S · (m± ) and x0 ∈ S · (k) (it is the associated graded of the Harish-Chandra map, and corresponds to the inclusion k∗ ⊂ l∗ attached to the decomg g position of l). Since D k and D lk are nonzero, so is D l . Let us denote by: l , the microlocalization of U (l) w.r.t. a lift of D g •U l k (resp., U , U ), the microlocalization of U (k) w.r.t. a lift of D g (resp., D l , •U k k k k g (D l )|k∗ ). ⊂ U k , U ⊂ U k of complete filtered Lemma 2.14. 1) We have natural inclusions U k k rings. of degree ≤ 0. l → U (v+ ) ⊗ U (v− ) ⊗ is a continuous map U U 2) PBW k Proof. 1) is clear. 2) follows from Lemma 2.10. g
g
Let us denote by Jkl , Jk and Jl the dynamical twists associated to the polarized Lie algebras l = k ⊕ m+ ⊕ m− , g = k ⊕ v+ ⊕ v− , g = l⊕ u+ ⊕ u− . l → U (v+ ) ⊗ U (p− ) ⊗ , U U We denote by η the linear map U (u+ ) ⊗ U (p− ) ⊗ k (2) (1) taking α ⊗ β ⊗ λ to i αS(λ+,i ) ⊗ βS(λ+,i ) ⊗ ε(λ−,i )λ0,i . Here α ∈ U (u+ ), k . = λ−,i ⊗ λ+,i ⊗ λ0,i , where λ±,i ∈ U (m± ) and λ0,i ∈ U β ∈ U (p− ), and PBW(λ) Here the map PBW is relative to the polarization l = k ⊕ m− ⊕ m+ , so if λ ∈ U (l), we have λ = i λ−,i λ0,i λ+,i . g g k , U Proposition 2.15. We have Jk = η(Jl )Jkl . This is an equality in U (v+ )⊗U (p− ) ⊗ g l where η(Jl ) (resp., Jk ) is viewed as an element of this algebra using the injection k (resp., U ⊂ U k ). ⊂ U U k k g
g
Remark 2.16. This formula allows one to recover Jl uniquely from Jkl , Jk . Indeed, let k → U (u+ )⊗U (p− ) ⊗ k be the map taking u+ λ+ ⊗p− ⊗k U U η : U (v+ )⊗U (p− ) ⊗ to u+ ⊗ p− S(λ+ ) ⊗ k, where u+ ∈ U (u+ ), p− ∈ U (p− ), λ+ ∈ U (m+ ), k ∈ U (k), g then η ◦ η = id ⊗ id ⊗ Hkl . Now Jl can be uniquely recovered from its image by id ⊗ id ⊗ Hkl using its l-invariance, because the map S · (l)l → S · (k), f → f|k∗ is injective (see Remark 1.2). g
g
Remark 2.17. One can prove that the classical limit of η(Jl ) is (rl )|k∗ , so the classical g g limit of Proposition 2.15 is rk = (rl )|k∗ + rkl (see Remark 1.4). Proof. We set Kkl =
, U αi ⊗ βi ⊗ κi ∈ U (m+ ) ⊗ U (m− ) ⊗ k
i
g
Kl =
l , U aj ⊗ bj ⊗ cj+ cj0 cj− ∈ U (u+ ) ⊗ U (u− ) ⊗
j
where
cj±
∈ U (m± ),
cj0
⊂ U k . ∈U k
Quantization of Classical Dynamical r-Matrices with Nonabelian Base
Then Jkl = g
Jl =
(2)
i
619
(1)
αi ⊗ S(βi )S(κi ) ⊗ S(κi ), and −(2)
aj ⊗ S(bj )S(cj
0(2)
)S(cj
+(2)
)S(cj
−(1)
) ⊗ S(cj
0(1)
)S(cj
+(1)
)S(cj
),
j
therefore
g
η(Jl ) =
aj cj+ ⊗ S(bj )S(cj− )S(cj
0(2)
0(1)
) ⊗ S(cj
),
j
and we want to prove that g 0(2) (2) 0(1) (1) Jk = aj cj+ αi ⊗ S(bj )S(cj− )S(cj )S(βi )S(κi ) ⊗ S(cj )S(κi ), i,j
i.e., that g
Kk =
aj cj+ αi ⊗ S(cj
0(2)
0(3) − c j bj
)βi cj
0(1)
⊗ κ i cj
(10)
.
i,j
To prove (10), we will prove that: k , U (a) the r.h.s. of (10) belongs to U (v+ ) ⊗ U (v− ) ⊗ (b) for any x ∈ U (v− ), y ∈ U (v+ ), we have g g 0(1) g 0(2) 0(3) Hk (xaj cj+ αi )κi cj Hk S(cj )βi cj cj− bj y = Hk (xy).
(11)
i,j
g
Here Hk is the Harish-Chandra map U (g) → U (k). Let us now prove (a). We have aj ∈ U (u+ ), cj+ ∈ U (m+ ), αi ∈ U (m+ ), so the first 0(2)
0(3)
factor of the r.h.s. of (10) belongs to U (v+ ). Since [k, m− ] ⊂ m− , S(cj )βi cj ∈ U (m− ); we also have cj− ∈ U (m− ) and bj ∈ U (u− ), therefore the second factor of the and c0(1) ∈ U k , the third factor r.h.s. of (10) belongs to U (v− ). Finally, since κi ∈ U j k k . This proves (a). of the r.h.s. of (10) belongs to U g Let us now prove (b), i.e., identity (11). Since Hk is a left U (k)-module morphism, 0(1)
cj
g
can be inserted in the argument of Hk , so (11) is equivalent to the identity
g
g
g
Hk (xaj cj+ αi )κi Hk (βi cj0 cj− bj y) = Hk (xy).
(12)
i,j
We now prove: Lemma 2.18. If z ∈ U (g) and t ∈ U (l), then g g g g Hk (zt) = Hkl Hl (z)t and Hk (tz) = Hkl tHl (z) . Proof of Lemma. We assume that z = z+ z0 z− , with z± ∈ U (u± ), z0 ∈ U (l). Then g g g g Hk (zt) = ε(z+ )Hk (z0 z− t) = ε(z+ )ε(z− )Hk (z0 t) = Hkl (Hl (z)t). The second identity is proved in the same way.
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It follows that the l.h.s. of (12) is equal to g g Hkl Hl (xaj cj+ )αi κi Hkl βi Hl (cj0 cj− bj y) , i,j
which is equal to
g g Hkl Hl (xaj cj+ )Hl (cj0 cj− bj y) .
(13)
j
g
Now cj+ and cj0 cj− belong to U (l), and Hl is a U (l)-bimodule map, so
g
g
Hl (xaj cj+ )Hl (cj0 cj− bj y) =
j
g
g
g
Hl (xaj )cj+ cj0 cj− Hl (bj y) = Hl (xy).
j
g
g
Therefore l.h.s. of (13) = Hkl (Hl (xy)) = Hk (xy). This proves (b).
2.6. The ABRR equation. We assume now that g = l ⊕ u+ ⊕ u− is a polarized Lie algebra, equipped with t ∈ S 2 (g)g , such that t decomposes as t = tl + s + s 2,1 , where tl ∈ S 2 (l) and s ∈ u+ ⊗ u− . Then tl is l-invariant. We then say that (g, t) is a quadratic polarized Lie algebra. We set s¯ := s 2,1 . Let µ be the Lie bracket, and set γ := − 21 µ(s). Then Lemma 2.19. 1) [γ , l] = 0. 2) [γ , u± ] ⊂ u± . Proof. Let us prove 1). If x ∈ l, we have [s, x (1) +x (2) ] ∈ u+ ⊗u− , [tl , x (1) +x (2) ] ∈ l⊗l, and [¯s , x (1) + x (2) ] ⊂ u− ⊗ u+ . Since the sum of these terms is zero, each of them is zero. Applying µ to [s, x (1) + x (2) ] = 0, we get 1). Let us prove 2). If x ∈ u+ , we have [s, x (1) ] ∈ u+ ⊗u− , [s, x (2) ] ∈ u+ ⊗g, [tl , x (1) ] ∈ u+ ⊗ l, [tl , x (2) ] ∈ l ⊗ u+ , [¯s , x (1) + x (2) ] ∈ g ⊗ u+ . Therefore ([s, x (2) ])u+ ⊗u− = −[s, x (1) ], so [s, x (1) + x (2) ] ∈ u+ ⊗ p+ . Applying µ to this relation, we get [γ , x] ∈ u+ . One proves [γ , u− ] ⊂ u− in the same way. Assume now that g is nondegenerate (as a polarized Lie algebra). − Lemma 2.20. Let us set K = i ai ⊗ bi ⊗ i , s = σ u+ σ ⊗ uσ , tl = λ Iλ ⊗ Iλ , where K is as in Theorem 2.1. Then − u+ ai ⊗ i ⊗ [γ , bi ] + ai ⊗ i Iλ ⊗ [Iλ , bi ] σ ai ⊗ i ⊗ b i uσ = i,σ
i
i,λ
1 − ai ⊗ i ⊗ [Iλ , [Iλ , bi ]]. 2 i,λ
l . U Proof. Let of both sides, then δ 1,3,2 belongs to (U (u+ )⊗U (u− ))⊗ δ be the difference ⊗ δ , it will suffice to prove that for any x, y ∈ U (g), we have Set δ = δ ⊗ δ i i i i i H (xδi )δi H (δi y) = 0.
Quantization of Classical Dynamical r-Matrices with Nonabelian Base
Let x, y ∈ U (g), then
621
− H (xu+ σ ai ) i H (bi uσ y) = H (xm(s)y),
(14)
i,σ
where m is the product map of U (g). Set Cg = 21 m(tg ), Cl = 21 m(tl ). Then Cg = Cl + m(s) + γ , so (14) is equal to H (x(Cg − Cl − γ )y). Since Cg is central, this is H (xyCg )−H (x(Cl +γ )y). Using again Cg = Cl +m(s)+γ and the fact that H (zm(s)) = 0 for any z ∈ U (g), we rewrite (14) as H (xy(Cl + γ )) − H (x(Cl + γ )y), and since H is a right U (l)-module map, this is H (xy)Cl − H (xCl y) + H (x[y, γ ]). Now we have H (xy)Cl = i,λ H (xai ) i Cl H (bi y), since Cl commutes with U (l). Moreover, 1 H (xIλ ai ) i H (bi Iλ y) H (xCl y) = 2 i,λ
1 = H (x([Iλ , ai ] + ai Iλ )) i H (([bi , Iλ ] + Iλ bi )y). 2 i,λ
On the other hand, for any ξ ∈ U (g), we have H ([γ , ξ ]) = [γ , H (ξ )]. Indeed, if ξ = ξ + ξ 0 ξ − , then according to Lemma 2.19, the triangular decomposition of [γ , ξ ] is [γ , ξ + ]ξ 0 ξ − + ξ + [γ , ξ 0 ]ξ − + ξ + ξ 0 [γ , ξ − ], and since ε([γ , ξ ± ]) = 0, we get H ([γ , ξ ]) = ε(ξ + )[γ , ξ 0 ]ε(ξ − ) = [γ , H (ξ )]. Therefore H (x[y, γ ]) = i H (xai ) i H (bi [y, γ ]) = i H (xai ) i H ([γ , bi ]y). Therefore 1 H (xai ) i Cl H (bi y) − H (x([Iλ , ai ] + ai Iλ )) i H (([bi , Iλ ] + Iλ bi )y) (14) = 2 i i,λ + H (xai ) i H ([γ , bi ]y). i
Therefore − u+ ai ⊗ i ⊗ [γ , bi ] σ ai ⊗ i ⊗ b i uσ = i,σ
i
+
1 ai ⊗ [ i , Iλ ]Iλ ⊗ bi + ai ⊗ Iλ i ⊗ [Iλ , bi ] 2 i,λ
+[ai , Iλ ] ⊗ i Iλ ⊗ bi − [ai , Iλ ] ⊗ i ⊗ [Iλ , bi ]. Then we use the l-invariance of K to transform the two last terms. (2) (1) Recall that J = i ai ⊗ S(bi )S( i ) ⊗ S( i ). Corollary 2.21 (The nonabelian ABRR equation). We have 1 s 1,2 J = [−γ (2) + m(tl )(2) + tl2,3 , J ]. 2
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Proof. Uses the facts that tl commutes with (U (l)) ⊂ U (l)⊗2 and that m(tl ) is central in U (l). Remark 2.22. A quadratic polarized Lie algebra g such that l = 0 and t is nondegenerate, is the same as a Manin triple, i.e., as a Lie bialgebra structure on u+ (or u− ). Such a polarized Lie algebra is degenerate (unless g = 0) and does not lead to a classical dynamical r-matrix. Remark 2.23. Corollary 2.21 may be written in a “normally ordered way” 2,3 tl − s 1,2 − m(¯s )(2) J = J tl2,3 − m(¯s )(2) . (15) − Here “normally ordered” means that all expressions involving s = σ u+ σ ⊗ uσ are + − such that uσ appears before uσ if both of them are in the same factor. Remark 2.24. Expression of the r-matrix. Assume that t is nondegenerate. Let t ∨ : g∗ → g be the map λ → (λ ⊗ id)(t). Then t ∨ is an isomorphism and restricts to an isomorphism l∗ → l. So if is a generic element of l, the bilinear form u+ × u− → C, (x, y) → (t ∨ )−1 ( ) [x, y] = ([x, y]) is nondegenerate. By invariance of the scalar product, it follows that for such an , the operators ad( ) ∈ End(u± ) are invertible. If we identify ∧2 (g) with a subspace of End(g) using the scalar product, the r-matrix of Proposition 1.1 is λ → ad(t ∨1 (λ)) P , where P is the projection on u+ ⊕ u− along l and ad(t ∨ (λ)) is viewed as an automorphism of u+ ⊕ u− . The same applies in the case of a Lie algebra with a splitting and a nondegenerate t ∈ S 2 (g)g . 2.7. Multicomponent ABRR equations. Here g is still a quadratic polarized Lie algebra, nondegenerate as a polarized Lie algebra. Proposition 2.25. We have J 12,3,4 (tl2,3 + tl2,4 − s 1,2 − m(¯s )(2) ) = (tl2,3 + tl2,4 − s 1,2 + s 2,3 − m(¯s )(2) )J 12,3,4 (16) and (17) J 1,23,4 (tl3,4 − s 2,3 − m(¯s )(3) ) = (tl3,4 − s 1,3 − s 2,3 − m(¯s )(3) )J 1,23,4 . Proof. Let us prove the first identity. Recall that K = i ai ⊗ bi ⊗ i . The identity follows from (1) (1) (2) (2) − ai ⊗ a i I λ ⊗ I λ i ⊗ b i + ai u + (18) σ ⊗ a i uσ ⊗ i ⊗ b i i,λ
i,σ
(1) + ai
+
(2) + ⊗ a i u− σ uσ
⊗ i ⊗ bi =
(1)
(1)
⊗ I λ ai
(1)
⊗ u+ σ ai
(1)
⊗ I λ ai
ai
(2)
⊗ i Iλ ⊗ b i
i,λ
u+ σ ai
(1)
⊗ u− σ ai
(2)
⊗ i ⊗ bi + ai
i,σ
+ ai
+ ⊗ u− σ uσ a i
(2)
⊗ i ⊗ bi −
i,λ
ai
(2)
(2)
⊗ i ⊗ b i u− σ
⊗ i ⊗ [Iλ , bi ].
Quantization of Classical Dynamical r-Matrices with Nonabelian Base
623
Write the difference of both sides of (18) as i Ai ⊗ Bi ⊗ Ci ⊗ Di . It will suffice to show that for any x ∈ U (u+ ), we have i Ai ⊗ Bi ⊗ Ci H (Di x) = 0. This means that for any x ∈ U (u− ), (2) − (1) + x (1) ⊗ x (2) Iλ ⊗ Iλ + x (1) u+ ⊗ x (2) u− σ ⊗ x uσ ⊗ 1 + x σ uσ ⊗ 1 λ
=
σ
+
σ (1) u+ σx
(2) ⊗ u− σx
+ (2) ⊗ 1 + x (1) ⊗ u− ⊗1 σ uσ x
x (1) ⊗ Iλ x (2) ⊗ Iλ + [Iλ , x](1) ⊗ Iλ [Iλ , x](2) ⊗ 1
λ
+
(2) Aσ,λ (x)(1) ⊗ u+ ⊗ Iλ + σ Aσ,λ (x)
(2) Aσ (x)(1) ⊗ u+ ⊗ 1, (19) σ Aσ (x)
σ
λ,σ
where Aσ , Aλ,σ are the linear endomorphisms of U (u+ ) defined by the condition that u− σ x − (Aσ (x) + λ Aσ,λ (x)Iλ ) belongs to U (g)u− . This identity decomposes into two parts. The first part is (2) x (1) ⊗ x (2) Iλ = x (1) ⊗ Iλ x (2) + Aσ,λ (x)(1) ⊗ u+ (20) σ Aσ,λ (x) , σ
which we proveas follows: since x (1) ⊗ Aσ,λ (x (2) ) = Aσ,λ (x)12 , it suffices to prove of identities in U (u+ ) is equivalent to that [x, Iλ ] = σ u+ σ,λ (x). This collection σ A the identity in U (u+ )l, λ [x, Iλ ]Iλ = λ,σ u+ (x)Iλ . The last identity is proved σ Aσ,λ − as follows: we have [m(t), x] = 0, where m(t) = λ (Iλ )2 + 2 σ u+ σ uσ + 2γ . This gives 2[Iλ , x]Iλ − [[Iλ , x], Iλ ] + 2[γ , x] + 2 u+ Aσ,λ (x)Iλ ∈ U (g)u− σ Aσ (x) + σ
λ
λ
(21) which implies λ [Iλ , x]Iλ + σ,λ u+ σ Aσ,λ (x)Iλ = 0, as wanted. Notice for later use that (21) also implies −[[Iλ , x], Iλ ] + 2[γ , x] + 2 u+ (22) σ Aσ (x) = 0. σ
λ
The second part of (19) is − − + [x (1) ⊗ x (2) , u+ [Iλ , x](1) ⊗ Iλ [Iλ , x](2) σ ⊗ u σ + 1 ⊗ u σ uσ ] = σ
λ
+
(2) Aσ (x)(1) ⊗ u+ σ Aσ (x) .
(23)
σ
Before we prove (23), we prove + [x, u− Iλ [Iλ , x] + u+ σ uσ ] = σ Aσ (x), ∀x ∈ U (u+ ). σ
λ
σ
(24)
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B. Enriquez, P. Etingof
According to (22), this is written as [x,
1 Iλ [Iλ , x] + [Iλ , x]Iλ − [γ , x], 2
+ u− σ uσ ] =
σ
λ
which follows from the fact that m(t) is central. This implies (24). Let us now prove (23). The difference between (23) and (22) applied to the second factor of x (1) ⊗ x (2) is − u+ [Iλ , x (1) ] ⊗ Iλ x (2) [x (1) ⊗ x (2) , σ ⊗ uσ ] = σ
λ
+
(1) (1 ⊗ u+ ⊗ Aσ (x)(2) −x (1) ⊗ Aσ (x (2) )). σ )(Aσ (x) σ
(25) So we should prove (25). Since Aσ (x)12 = (Aσ ⊗ id + id ⊗ Aσ )(x 12 ), (25) is rewritten as − (2) u+ [Iλ , x (1) ] ⊗ Iλ x (2) + A(x (1) ) ⊗ u+ [x (1) ⊗ x (2) , σ ⊗ uσ ] = σx . σ
(26)
σ
λ
− If x ∈ u+ , then [x 1 + x 2 , σ u+ σ ⊗ uσ ] ∈ u+ ⊗ p+ , so if x ∈ U (u+ ), then the l.h.s. of (26) belongs to U (u+ ) ⊗ U (u+ ) ⊕ U (u+ )l . Now − (1) + [x (1) ⊗ x (2) , u+ ⊗ x (2) , Iλ ⊗ Iλ ] − [x (1) ⊗ x (2) , u− σ ⊗ uσ ] = −[x σ ⊗ uσ ] σ
λ
[Iλ , x (1) ] ⊗ Iλ x (2) + x (1) Iλ ⊗ [Iλ , x (2) ] = λ
+
σ
λ (1) u− σx
(2) ⊗ u+ σx
modulo U (g)u− ⊗ U (u+ )
σ
=
[Iλ , x (1) ] ⊗ Iλ x (2) + x (1) Iλ ⊗ [Iλ , x (2) ] λ
+
σ
λ
Aσ,λ (x
(1)
(2) )Iλ + Aσ (x (1) ) ⊗ u+ modulo U (g)u− ⊗ U (u+ ). σx
λ
we get (26). Projecting this identity on U (u+ ) ⊗ U (g) parallel to U (u+−)l ⊗1 U (g), 2 ] = 0 for x ∈ l and Let us now prove the second identity. Using [ σ u+ ⊗u , x +x σ σ the l-invariance of K, we transform this identity into the analogue of (19) with u+ , u− exchanged, which also holds. Corollary 2.21 has a multicomponent version. Namely, let J [n] := J 1,{2,...,n},n+1 · · · J n−2,{n−1,n},n+1 J n−1,n,n+1 (this element corresponds to fusing n intertwiners). 1 1 Here, for example, J 1,{2,...,n},n+1 means that we put the first component of J in component 1, the second in components 2...n (after taking the coproduct n − 2 times), and the third in component n + 1.
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Theorem 2.26 (The multicomponent ABRR equation). For i = 1, . . . , n, the element J [n] satisfies the equations [
n+1
j =i+1
i−1 n 1 i,j tl − γ (i) + m(tl )(i) , J [n] ] = ( s j,i − s i,j )J [n] . 2 j =1
(27)
j =i+1
Proof. We will treat the case n = 3. Then J [3] = J 1,23,4 J 2,3,4 = J 12,3,4 J 1,2,34 . Then (tl3,4 − s 1,3 − s 2,3 − m(¯s )(3) )J [3] = (tl3,4 − s 1,3 − s 2,3 − m(¯s )(3) )J 1,23,4 J 2,3,4 = J 1,23,4 (tl3,4 − s 2,3 − m(¯s )(3) )J 2,3,4 = J 1,23,4 J 2,3,4 (tl3,4 − m(¯s )(3) ) = J [3] (tl3,4 − m(¯s )(3) ), where we have used (17) and (15). This proves (27) when i = 3. Let us treat the case i = 2: (tl2,3 + tl2,4 − s 1,2 + s 2,3 − m(¯s )(2) )J [3] = (tl2,3 + tl2,4 − s 1,2 + s 2,3 − m(¯s )(2) )J 12,3,4 J 1,2,34 = J 12,3,4 (tl2,3 + tl2,4 − s 1,2 − m(¯s )(2) )J 2,3,4 = J 12,3,4 J 1,2,34 (tl2,3 + tl2,4 − m(¯s )(2) ) = J [3] (tl2,3 + tl2,4 − m(¯s )(2) ), where we have used (16) and (15)1,2,34 . This proves (27) when i = 2. In general, (27) for i = 1 is a consequence of (27) for i = 2, . . . , n, and of the land γ -invariances of J , and of [γ , l] = 0. We have already proven the l-invariance of J , and its γ -invariance follows from that of K, which in its turn follows from the identity H ([γ , x]) = [γ , H (x)] for x ∈ U (g). Proposition 2.27 (Compatibility of multicomponent ABRR). Write the multicomponent ABRR equations as ai[n] J [n] = J [n] bi[n] , for i = 2, . . . , n. This is a compatible system, i.e., [ai[n] , aj[n] ] = [bi[n] , bj[n] ] = 0 for any pair i, j ∈ {2, . . . , n}. Proof. The vanishing of these brackets follows from the identities [s 1,2 , s 1,3 ] + [s 1,2 , s 2,3 ] + [s 1,3 , s 2,3 ] = [tl2,3 , s 1,3 ]
(28)
[s 1,2 , m(¯s )(1) + m(¯s )(2) − tl1,2 ] = 0.
(29)
and
To prove (28), one may assume that t is nondegenerate. Both sides of (28) belong to u+ ⊗ g ⊗ u− . Then (28) follows from its pairing with x− ⊗ id ⊗ x+ , with x± ∈ u± . Let us prove (29). Let us project [γ (1) + γ (2) , t] = 0 on u+ ⊗ u− . Lemma 2.19 says that this projection is [γ (1) + γ (2) , s] = 0. (29) then follows from this identity, together with m(¯s ) = 21 (m(t) − m(tl )) + γ , the fact that m(t) is central in U (g), and [s, m(tl )12 ] = 0, which follows from the l-invariance of s.
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3. Dynamical Pseudotwists Associated to a Quadratic Polarized Lie Algebra As we noted in the Introduction, Proposition 0.1 together with [AM1] implies: Lemma 3.1. Let (g = l⊕u, t) be a quadratic Lie algebra with a nondegenerate splitting. S · (l)[1/D0 ] defined by Let c ∈ C, then ρc ∈ ∧2 (g) ⊗ g ρc (λ) := rl (λ) + c f (c adλ∨ ) ⊗ id (t) for λ ∈ l∗ , is a solution of CYB(ρc )−Alt(dρc ) = −π 2 c2 Z. Here we set λ∨ = (λ⊗id)(t) and f (x) = −1/x + π cotan(π x). In this section, we assume that (g = l⊕u+ ⊕u− , t ∈ S 2 (g)g ) is a quadratic polarized Lie algebra, such that g is nondegenerate as a polarized Lie algebra (see Sect. 2.6). Recall that this means that t decomposes as tl + s + s 2,1 , with tl ∈ S 2 (l) and s ∈ u+ ⊗ u− . We will construct a dynamical pseudotwist quantizing ρc in this situation. We fix a formal parameter and a complex parameter c. We set κ = c. If (A, B) is −1 a Lie associator ([Dr2]), we set −1 κ (A, B) = (κA, κB) . An example of an associator is the KZ associator, i.e., the renormalized holonomy from 0 to 1 of the differential A B equation dF dz = ( z + z−1 )F . Theorem 3.2. Set
1,2 2,3 g J¯ = −1 , tl − s 1,2 − m(¯s )(2) Jl . κ t
l [[]] is a solution of the dynamical pseudotwist equation U Then J¯ ∈ U (g)⊗2 ⊗ 1,2 2,3 ¯1,23,4 ¯2,3,4 J , t )J . J¯12,3,4 J¯1,2,34 = −1 κ (t
(30)
Proof. Drinfeld’s algebra T4 is defined by generators τi,j , 1 ≤ i = j ≤ 4, and relations τi,j = τj,i , [τi,j , τk,l ] = 0 if {i, j, k, l} = {1, 2, 3, 4}, and [τi,j + τi,k , τj,k ] = 0 if card{i, j, k} = 3. Then we have the pentagon relation 1,2 2,3 1,2 2,3 1,2 −1 , τ )−1 + τ 1,3 , τ 2,4 + τ 3,4 )−1 ,τ ) κ (τ κ (τ κ (τ 1,3 1,2 2,3 = −1 + τ 2,3 , τ 3,4 )−1 , τ + τ 2,4 ). κ (τ κ (τ
Then we have an algebra morphism T4 → U (g)⊗3 ⊗ U (l), with τ i,j → t i,j for 2,4 → t 2,4 − s 1,2 − s 3,2 − m(¯ i, s )(2) , τ 3,4 → tl3,4 − s 1,3 − s 2,3 − m(¯s )(3) , l j = 4, τ 1≤i<j ≤4 τi,j → 0. Taking the image of the pentagon relation by this morphism, we get an identity l [[]]. Multiply it from the right by the identity (J g )1,23,4 (J g )2,3,4 = U in U (g)⊗2 ⊗ l l g 12,3,4 g 1,2,34 g (Jl ) (Jl ) . Then using the identities of Proposition 2.25 to put (Jl )1,23,4 2,3 , τ 3,4 ) in the l.h.s., and (J g )12,3,4 before the image of before the image of −1 κ (τ l −1 1,2 2,3 2,4 κ (τ , τ + τ ) in the r.h.s., we get the result. Let us study the classical limit of J¯. In Sect. 1.4, we introduced quasi-commutal )≤0 ⊂ tive algebras S · (l) ⊂ S · (l)[1/D0 ] , and inclusions S · (l) ⊂ U (l)[[]], (U · S (l)[1/D0 ] . ¯ is the “formal series” tensor ¯ Then J¯ belongs to A := U (g)⊗2 ⊗ S · (l)[1/D0 ] (here ⊗ product).
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Proposition 3.3 (Classical limit). J¯ − 1 belongs to A, and the reduction of Alt1,2 (J¯ − 1)/ modulo belongs to ∧2 (g) ⊗ S · (l)[1/D0 ]. It coincides with the expansion at the origin of the meromorphic function ρc : l∗ → ∧2 (g), defined in Lemma 3.1. 1,2 , t 2,3 − s 1,2 − Proof. It will be enough to compute the classical limit of X := −1 κ (t l m(¯s )(2) ). We have −1 = exp(φ), where φ is a Lie formal series in A, B. X is the specialization of under A → A0 := ct 1,2 , B → B0 := c(tl2,3 − s 1,2 − m(¯s )(2) ). We have A0 ∈ A and B0 ∈ A, moreover if B0 := ctl2,3 , then B0 = B0 modulo A. Since φ is a Lie series, it is a sum of homogeneous components with partial degree ≥ 1 in A and B, so φ(A0 , B0 ) ∈ A. In particular, X ∈ 1+φ(A0 , B0 )+2 A. Moreover, φ(A0 , B0 ) = φ(A0 , B0 ) = φ1 (A0 , B0 ) modulo 2 A, where φ1 is the part of φ, linear in A. −1 k · Set φ1 = ] is quasi-commutative, k≥1 ck ad(B) (A). Since S (l)[D0 −1 k k+1 ad(B0 ) (A0 ) ∈ A is equal modulo A, to c α,λ1 ,... ,λk eα ⊗ ad(Iλ1 ) · · · ad(Iλk )(eα ) ⊗ (Iλ1 ) · · · (Iλk ) (here t = α eα ⊗ eα ). Its reduction modulo A is the linear map l∗ → g⊗2 , λ → ck+1 ad(1 ⊗ t ∨ (λ))k (t 1,2 ). This map takes values in S 2 (g) if k is even and in ∧2 (g) if k is odd. Only the “even k” part remains after antisymmetrization, and the result follows from the fact that the c2k coincide with the Taylor coefficients of f (see e.g. [EE]).
Let η : U (g)⊗3 → U (g)⊗2 ⊗ U (l) be the linear map associated to g = l ⊕ u+ ⊕ u− (see Sect. 2.5). Proposition 3.4. If P is any noncommutative polynomial in two variables, we have η(P (t 1,2 , t 2,3 )) = P (t 1,2 , tl2,3 − s 1,2 − m(¯s )(2) ). 1,2 , t 2,3 )) = −1 (t 1,2 , t 2,3 − s 1,2 − m(¯ s )(2) ). In particular, η(−1 κ (t κ l
Proof. We have for any x ∈ U (g)⊗3 , η(t 1,2 x) = t 1,2 η(x).
(31)
Let us now show that if x ∈ U (g)⊗3 is g-invariant, then η(t 2,3 x) = (tl2,3 − s 1,2 − m(¯s )(2) )η(x).
(32)
0 + − − ) = 0, we have η(s 2,3 x) = 0. Let us write x = i Ai ⊗ Bi ⊗ λ− i λi λi . Since ε(uσ λi − 0 + 0 + We have tl2,3 x = i,λ Ai ⊗ Iλ Bi ⊗ [Iλ , λ− i,λ Ai ⊗ Iλ Bi ⊗ λi (Iλ λi )λi , i ]λi λi + − so since ε([Iλ , λi ]) = 0, we get +(2) +(1) η(tl2,3 x) = Ai S(λi ) ⊗ Iλ Bi S(λi ) ⊗ ε(λ− )Iλ λ0i = tl2,3 η(x). i,λ
Finally, since x is invariant, we have − 0 + + − 0 + + − s¯ 2,3 x = Ai ⊗ u − σ Bi ⊗ λi λi (λi uσ ) + [Ai , uσ ] ⊗ uσ Bi ⊗ λi λi λi i,σ − 0 + + +Ai ⊗ u− σ [Bi , uσ ] ⊗ λi λi λi ,
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B. Enriquez, P. Etingof
therefore η(¯s 2,3 x) =
+(2)
−Ai S(λi
+(1)
+ ) ⊗ u− σ Bi uσ S(λi
0 ) ⊗ ε(λ− i )λi
i,σ +(2)
−Ai u+ σ S(λi
+(1)
) ⊗ u− σ Bi S(λi
+(2)
+[Ai , u+ σ ]S(λi +(2)
0 ) ⊗ ε(λ− i )λi
+(1)
) ⊗ u− σ Bi S(λi
0 ) ⊗ ε(λ− i )λi
+(1)
+ 0 +Ai S(λi ) ⊗ u− ) ⊗ ε(λ− σ [Bi , uσ ]S(λi i )λi − − + =− (u+ σ ⊗ uσ ⊗ 1 + 1 ⊗ uσ uσ ⊗ 1)η(x). σ
Adding up these results, we get (32). The proposition now follows from (31), (32) and η(1) = 1. 1,2 , t 2,3 )−1 = KZ (t 1,2 , t 2,3 − s 1,2 − m(¯ s )(2) )−1 , Remark 3.5. The relation η KZ κ (t κ l where KZ is the KZ associator, can be derived from the results of Sect. 4.7 (in the untwisted case) together with the composition formula. 1,2 , t 2,3 )J 1,23 J 2,3 , then Remark 3.6. If J ∈ U (g)⊗2 [[]] satisfies J 12,3 J 1,2 = −1 κ (t −1 U (g) := (U (g)[[]], Ad(J ) ◦ 0 ) is a Hopf algebra. l [[]] satisfies the pseudotwist equation (30), and set U Assume that J¯ ∈ U (g)⊗2 ⊗ 1,2 −1 ¯ J := (J ) J , then J satisfies the twist equation in U (g)
J12,3,4 J1,2,34 = J1,23,4 J2,3,4 . (33) Remark 3.7. When g is a semisimple Lie algebra and l ⊂ g is a Cartan subalgebra, c = 1, Z = 41 [t 1,2 , t 2,3 ], we have ρc (λ) = −
1 (λ, α) (eα ∧ fα )coth . 2 2 α∈+
1,2 2,3 1,2 − Here x ∧ y = x ⊗ y − y ⊗ x. Then a quantization of ρc (λ) is J¯ := −1 (t , tl − r g (2) m(¯r ) )Jh (here r is the standard r-matrix of g). Therefore, if J is as in Remark 3.6, then J := (J 1,2 )−1 J¯ satisfies (33). On the other hand, we know from [EV2] another solution J of the same equation, obtained by a quantum analogue of the construction g of Jl . In [EEM], we show that J and J are gauge-related in U (g) (or, which is the same, that J¯ and J 1,2 J are gauge-related in U (g)[[]]).
Remark 3.8. (Expression of the r-matrix.) If t ∈ S 2 (g)g is nondegenerate and we use it for identifying ∧2 (g) with a subspace of End(g), then ρc identifies with λ →
1 P + cf (cad(t ∨ (λ))). adt ∨ (λ)
4. Dynamical Pseudotwists Associated to a Quadratic Lie Algebra with an Automorphism In this section, we quantize the r-matrix ρσ,c of Proposition 0.3, as well as (ρσ,c )|k∗ + rkl , if (l = k⊕m+ ⊕m− , tl ) is a polarized quadratic Lie algebra, nondegenerate as a polarized Lie algebra.
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4.1. Quadratic Lie algebras with an automorphism. Let g be a finite dimensional complex Lie algebra, equipped with t ∈ S 2 (g)g and σ ∈ Aut(g), such that (σ ⊗ σ )(t) = t. We assume that σ − id is invertible on g/gσ . Set l := gσ , u := Im(σ − id). Then g = l ⊕ u is a Lie algebra with a splitting (see Sect. 1.1). Moreover, we have t ∈ S 2 (l) ⊕ S 2 (u). We denote by tl the component of t in S 2 (l). We have tl ∈ S 2 (l)l . Example 4.1. g, l are as in Example 1.5.2 (1), and σ = exp(ad(χ )), where χ is a generic central element of l (see Sect. 2). Example 4.2. g is a simple Lie algebra, σ is an involution of g. Then G/L is a symmetric space for G. Example 4.3. g is a simple Lie algebra, and l ⊂ g is a semisimple Lie subalgebra of the same rank (“a Borel-De Siebenthal pair” [BS]). In this case there is an automorphism (of degree 2, 3 or 5) such that l = gσ . Example 4.4. g is a simple Lie algebra of simply laced type, σ is induced by a Dynkin diagram automorphism (with no fixed edges). Then l = gσ is the Lie algebra corresponding to the quotient diagram. Example 4.5. More generally, in the setting of Example 4.4, one may consider the automorphism σβ = σ ◦ exp(ad(β)), where β is a generic element of hσ . In this case, l = hσ . Let D := {a + ib ∈ C|a ∈ [0, 1[ and b ≥ 0, or a ∈]0, 1] and b < 0}. There is a unique operator log(σ ) in End(g), such that elog(σ ) = σ , and whose eigenvalues belong to 2π iD. 4.2. The main result: definition and properties of κ . Let us define O]0,1[ as the ring of analytic functions on ]0, 1[. We define in the same way OR× −{1} . Set +
X(z) :=
z ⊗ id)(t − tl ) + tl . z−1 z−1
(zlog(σ )/2πi
Then X(z) ∈ g⊗2 ⊗ OR× −{1} . More precisely, the first term of X(z) is a linear combi+ nation of products of powers of log(z) (of degree < dim(g)) with zα /(z − 1), where α is an eigenvalue of log(σ )/2iπ. Let κ be a formal parameter and let κ be the renormalized holonomy from 0 to 1 of the equation z
1 dG = κ X(z)1,2 + tl2,3 + m(tl )(2) G(z), dz 2
(34)
where G ∈ U (g)⊗2 ⊗ U (l) ⊗ O]0,1[ [[κ]]. More precisely, if σ has no strictly positive eigenvalues on g/gσ , there are unique 2,3 1 (2) solutions G0 , G1 of (34), such that G0 (z) = zκ(tl + 2 m(tl ) ) (1 + o(1)) as z → 0, 1,2 G1 (z) = (1 − z)κt (1 + o(1)) as z → 1, and κ ∈ U (g)⊗2 ⊗ U (l)[[κ]] is defined by κ = G1 (z)−1 G0 (z) for any z. If σ has strictly positive eigenvalues, G0 , G1 , κ are defined similarly, replacing [0, 1] by a smooth path γ : [0, 1] → C, such that γ (0) = 0, γ (1) = 1, 0, 1 ∈ / γ (]0, 1[), and the Euclidean scalar product of γ (0) with the eigenvalues of log(σ ) is > 0.
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Theorem 4.6. κ satisfies the pseudotwist equation −1 1,2,3 1,23,4 2,3,4 ((KZ κ κ = κ12,3,4 κ1,2,34 . κ ) )
Proposition 4.7 (Classical limit of κ ). Let c be a complex number, a formal variable, and assume that κ = c. Recall that S · (l) is the -adically complete subalgebra of ¯ U (l)[[]] generated by l. Then κ and (κ − 1)/ belong to U (g)⊗2 ⊗ S · (l) . 2,1 Moreover, the reduction modulo of (κ − κ )/ is the formal function ρσ,c : l∗ → ∧2 (g), such that
e2πic ad(λ ) ◦ σ + id ρσ,c (λ) = c f (c ad(λ )) ⊗ id (tl ) + iπ 2πic ad(λ∨ ) ⊗ id (t − tl ) ; e ◦ σ − id ∨
∨
here for λ ∈ l∗ , λ∨ = (λ ⊗ id)(tl ) ∈ l, and f (x) = − x1 + π cotan(π x). Remark 4.8. This solution of the modified CDYBE was discovered in [AM2], generalizing [ES2], where σ is assumed of finite order. In the case of Example 4.5, this solution was discovered in [S] and quantized using quantum groups in [ESS]. Our quantization is different; it should be related to the quantization of [ESS] by a gauge transformation given by a twisted version of the Kazhdan-Lusztig equivalence between the representation categories of an affine algebra and a quantum group. Remark 4.9. When σ = id, g is semisimple and k is a Cartan subalgebra, (34) is the trigonometric KZ equation, see [EFK, EV3]. Assume now that l is a quadratic polarized Lie algebra, nondegenerate as a polarized Lie algebra. So l = k ⊕ m+ ⊕ m− , and tl = tk + s + s 2,1 , with tk ∈ S 2 (k)k and s ∈ m+ ⊗ m− . We set γ = − 21 µ(s). Let k,l,g be the renormalized holonomy from 0 to 1 of the differential equation 1,2 2,3 1 dG z + tk + m(tk )(2) − γ (2) G(z). (35) = κ X(z) − s dz 2 Then k,l,g ∈ U (g)⊗2 ⊗ U (l)[[κ]]. The map η defined in Sect. 2.5 restricts to η : U (g)⊗2 ⊗ U (l) → U (g)⊗2 ⊗ U (k). g
g
Proposition 4.10. 1) Set l := κ , then η(l ) = k,l,g . g g k [[]] is a dynamical pseudotwist, U 2) Set k := k,l,g Jkl . Then k ∈ U (g)⊗2 ⊗ i.e., it satisfies g
g
g
g
−1 1,2,3 ((KZ (k )1,23,4 (k )2,3,4 = (k )12,3,4 (k )1,2,34 . κ ) )
g ¯ Moreover, if κ = c, k belongs to U (g)⊗2 ⊗ S · (k)[1/Dk ] , and its classical limit is ρ˜c,k,g := rkl (λ) + (ρσ,c )|k∗ (λ). It satisfies the modified CDYBE-Alt(dρ˜c,k,g ) + CYB(ρ˜c,k,g ) = −π 2 c2 Z. Here rkl (λ) is as in Sect. 1 and Dk ∈ S dim(m+ ) (k) is the determinant corresponding to the polarized Lie algebra l = k ⊕ m+ ⊕ m− .
The proofs of Theorem 4.6, Proposition 4.7 and Proposition 4.10 occupy the rest of this section.
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631
g
Remark 4.11. If we assume that gσ = 0, then J := l is a solution of the twist equation J 12,3 J 1,2 = κ (t 1,2 , t 2,3 )J 1,23 J 2,3 , so it gives rise to a quasitriangular Hopf algebra (U (g)[[κ]], m0 , := Ad(J −1 ) ◦ 0 , R := (J −1 )2,1 eκt/2 J ). Its classical limit is the quasitriangular Lie bialgebra structure on g induced by the rmatrix r := ( id+σ id−σ ⊗ id)(t) (r is antisymmetric, and is a solution of the modified CYBE). 4.3. Proof of Theorem 4.6. Consider the system of equations 1 ∂G z = κ X(z)1,2 + X(z/u)3,2 + tl2,4 + m(tl )(2) G, ∂z 2 u
∂G 1 = κ X(u)1,3 + X(u/z)2,3 + tl3,4 + m(tl )(3) G, ∂u 2
(36)
(37)
where the unknown function G(z, u) lies in U (g)⊗2 ⊗ U (l) ⊗ O[[κ]], O is the ring of analytic functions on {(z, u)|0 < z < u < 1}, and G has the form 1 + O(κ). One checks that the system (36,37) is compatible; more generally, the following is true. Let g = l ⊕ u be a Lie algebra with a splitting, equipped with t ∈ S 2 (g)g , such that t = tl + tu , tl ∈ S 2 (l), tu ∈ S 2 (u). Assume that ∈ End(u) commutes with the adjoint action of l on u, and that (a) ( ⊗ id + id ⊗ )(tu ) = tu , (b) [t 1,2 , t 2,3 ] = [tl1,2 , tl2,3 ] + τ1 + τ2 , where ( ¯(1) + ¯(2) + ¯(3) )(τa ) = τa , where ¯ ∈ End(g) is defined by ¯|l = 0, ¯|u = . Then the system (36,37), where X(z) = (z ⊗ id)(tu )/(z − 1) + tl · z/(z − 1), is compatible. The system (36,37) has therefore a solution, unique up to right multiplication by an element of U (g)⊗2 ⊗ U (l)[[κ]] of the form 1 + O(κ). Following [Dr2], we consider five asymptotic zones, corresponding to the parenthesis orders P1 = ((0z)u)1, P2 = (0(zu))1, P3 = 0((zu)1), P4 = (0z)(u1), P5 = 0(z(u1)). Assume for simplicity that σ has no strictly positive eigenvalue. This guarantees that the function zλ , λ in the spectrum of log(σ ), tends to zero as z → 0+ . There exist five solutions of the system (36,37) G1 , . . . , G5 corresponding to these zones. They are uniquely determined by the requirements 2,4
G1 (z, u) = zκ(tl
+ 21 m(tl )(2) ) κ(tl2,3 +tl3,4 + 21 m(tl )(3) )
(1 + g1 (z, u)), u − z κt 2,3 κ t 2,4 +t 3,4 +t 2,3 + 1 m(tl )(2) + 1 m(tl )(3) 2 G2 (z, u) = ( (1 + g2 (z, u)), ) z l l l 2 u u
2,3
G3 (z, u) = (u − z)κt (1 − z)κ(t G4 (z, u) = z
κ(tl2,4 + 21 m(tl )(2) ) 1,3
1,2 +t 1,3 )
(1 + g3 (z, u)),
1,3
(1 − u)κt (1 + g4 (z, u)),
G5 (z, u) = (1 − u)κt (1 − z)κ(t
1,2 +t 2,3 )
(1 + g5 (z, u)),
where g1 (z, u) (resp., g2 , g3 , g4 , g5 ) tends to zero as (u, z/u) (resp., (u, 1 − uz ), (1 − 1−u 2 z, u−z 1−z ), (z, 1 − u), (1 − z, 1−z )) tends to (0, 0) in ]0, 1[ . Here “tends to zero” means
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that these are series in κ with coefficients tensor products of elements of U (g)⊗2 ⊗ U (l) with analytic functions in (z, u) tending to zero in the relevant zone. The expansions of G2 and G3 are based on the identity z
∂G ∂G 1 1 +u = κ X(z)1,2 + X(u)1,3 + tl1,2 + tl1,3 + tl2,3 + m(tl )(2) + m(tl )(3) G. ∂z ∂u 2 2 We have the relations g
g
2,3 1,3 G1 = G2 (l )3,2,4 , G2 = G3 (l )1,23,4 , G3 = G5 KZ , t ), κ (t
g
g
G1 = G4 (l )1,3,24 , G4 = G5 (l )13,2,4 . g
g
g
g
2,3 , t 1,3 )( )1,23,4 ( )2,3,4 = G ( )13,2,4 ( )1,3,24 . Therefore G1 = G5 KZ 5 κ (t l l l l Simplifying by G5 , exchanging factors 2 and 3 and using the antisymmetry relation 3,2,1 = (KZ )−1 , we get the result. (KZ κ ) κ If σ has strictly positive eigenvalues, one applies the same argument after replacing the segment [0, 1] by a smooth path γ : [0, 1] → C, as in the definition of κ . Then if α is any eigenvalue of log(σ ), zα → 0 as z → 0 along γ .
4.4. Proof of Proposition 4.10. Let us prove 1). Similarly to Proposition 3.4, one shows that if P is any noncommutative polynomial in dim(g)+1 variables, and := log(σ )/2π i, then 1 η P (( k ⊗ id)(t)1,2 , k = 0, . . . , dim(g) − 1|tl2,3 + m(tl )(2) ) 2 = P ( k ⊗ id)(t)1,2 , k = 0, . . . , dim(g) − 1|tk2,3 − s 1,2 − m(¯s )(2) + m(tl )(2) 1 = P (( k ⊗ id)(t)1,2 , k = 0, . . . , dim(g) − 1|tk2,3 − s 1,2 + m(tk )(2) − γ (2) ). 2 g
l may be expressed as P (( k ⊗ id)(t)1,2 , k = 0, . . . , dim(g) − 1|tl2,3 + 21 m(tl )(2) ); this implies 1). Let us prove 2). Let us denote by and the renormalized holonomies from 0 to 1 of the differential equations z
1 dG = κ X(z)2,3 − s 1,3 − s 2,3 + tk3,4 + m(tk )(3) − γ (3) G , dz 2
(38)
z
1 dG = κ X(z)1,2 − s 1,2 − s 3,2 + tk2,4 + m(tk )(2) − γ (2) G . dz 2
(39)
We will prove the identities 1,2 2,3 −1 1,23,4 KZ , t ) k,l,g = k12,3,4 κ (t ,l,g
(40)
l 1,23,4 l 12,3,4 (Jkl )1,23,4 = k2,3,4 , (Jkl )12,3,4 = k1,2,34 . ,l,g (Jk ) ,l,g (Jk )
(41)
and
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Then combining (40), (41) and the twist equation (Jkl )1,23,4 (Jkl )2,3,4 = (Jkl )12,3,4 (Jkl )1,2,34 , we get 2). Let us prove (40). Proposition 2.25 implies that if G(z) is a solution of (35) of the −1 form 1 + O(κ), then G (z) := (Jkl )1,23,4 G(z)2,3,4 (Jkl )1,23,4 is a solution of (38) −1 of the form 1 + O(κ), and G (z) := (Jkl )12,3,4 G(z)1,2,34 (Jkl )12,3,4 is a solution of (39) of the form 1 + O(κ). This implies (40). Let us prove (41). We consider the system of equations z
∂G 1 = κ X(z)1,2 + X(z/u)3,2 − s 1,2 − s 3,2 + tk2,4 + m(tk )(2) − γ (2) G, ∂z 2
(42)
u
∂G 1 = κ X(u)1,3 + X(u/z)2,3 − s 1,3 − s 2,3 + tk3,4 + m(tk )(3) − γ (3) G, ∂u 2
(43)
where the unknown function G(z, u) lies in U (g)⊗2 ⊗ U (k) ⊗ O[[κ]] and has the form 1 + O(κ). As before, the system (42,43), supplemented with the condition G = 1 + O(κ), has a solution, unique up to right multiplication by an element of U (g)⊗2 ⊗ U (k)[[κ]] of the form 1 + O(κ). The system (42,43) has unique solutions G1 , . . . , G5 corresponding to the asymptotic zones P1 , . . . , P5 , satisfying 1,2 3,2 2,4 1 (2) (2) 1,3 2,3 2,3 3,4 1 (3) (3) uκ −s −s +tk +tk + 2 m(tk ) −γ (1 + g1 (z, u)), G1 (z, u) = zκ −s −s +tk + 2 m(tk ) −γ G2 (z, u) =
u − z κt 2,3 κ −s 1,2 −s 1,3 +t 2,3 +t 2,4 +t 3,4 + 1 m(t )(2) + 1 m(t )(3) −γ (2) −γ (3) k k 2 2 k k k (1 + g2 (z, u)), z u 2,3
G3 (z, u) = (u − z)κt (1 − z)κ(t G4 (z, u) = zκ
1,2 +t 1,3 )
−s 1,2 −s 3,2 +tk2,4 + 21 m(tk )(2) −γ (2)
1,3
G5 (z, u) = (1 − u)κt (1 − z)κ(t
(1 + g3 (z, u)), 1,3
(1 − u)κt (1 + g4 (z, u)),
1,2 +t 2,3 )
(1 + g5 (z, u)),
where gi (z, u) → 0 in the zone Pi . Then we have 2,3 1,3 , t ), G1 = G2 ( )1,3,2,4 , G2 = G3 (k,l,g )1,23,4 , G3 = G5 KZ κ (t
G1 = G4 ( )1,3,2,4 , G4 = G5 (k,l,g )13,2,4 . As before, this implies (41), and therefore 2).
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1
4.5. Classical limits. Let us prove Proposition 4.7. Set H (z) := G(z)z−κ(tl + 2 m(tl ) ) , 1,2 then κ = limz→1− (1 − z)−κt H (z) . Let us prove that H (z) has the following κ-adic property: it belongs to U (g)⊗2 ⊗ S · (l)⊗ ⊗2 · S (l), and H (z) satisfies the equations O]0,1[ [[κ]]. Set t¯l := κtl , then t¯l ∈ U (g) ⊗ (2)
κ dH = κX(z)1,2 H (z) + [t¯l2,3 + m(tl )(2) , H (z)]. dz 2 The formal expansion of H (z) therefore belongs to U (g)⊗2 ⊗ S · (l)[log(z), za , a ∈ D][[z]][[κ]], and has the form 1 + O(κ) (D is defined in Sect. 4.2). Set H (z) = 1 + κh(z) + O(κ 2 ), then H (0) = 1,
z
h(0) = 0,
z
dh = X(z)1,2 + [t¯l , h(z)]. dz
View h(z) as a formal function l∗ → U (g)⊗2 , then h(0, λ) = 0,
z
Here tl∨ (λ) := (λ ⊗ id)(tl ). Therefore
∂h(z, λ) = X(z)1,2 + [tl∨ (λ)(2) , h(z, λ)]. ∂z
du ∨ (2) Ad (z/u)tl (λ) (X(u)1,2 ) . u 0 ⊗2 · S (l)[[κ]], and it has the form It follows from the form of H (z) that κ ∈ U (g) ⊗ 1 + κψ + O(κ 2 ). We have ∨ (2) ψ(λ) = limz→1− Ad (z/u)tl (λ) h(z, λ) =
z
(ulog(σ )/2πi ⊗ id)(t − tl ) du u + tl − t 1,2 log(1 − z) . u−1 u−1 u Therefore ψ(λ) ∈ g⊗2 ⊗ S · (l). We now use the fact that for Re(x) > 0, one has
z 1 1 1 du lim − log(1 − z) = + − ) = − (x)/ (x). ux−1 ( z→1 1 − u x x + n n 0 ×
n≥1
Using the l-invariance of (ulog(σ )/2πi ⊗ id)(t − tl ) and of tl , and the fact that log(1 − ∨ z)(z−adtl (λ) − id)(t) → 0 as z → 1− , we get log(σ ) ψ(λ) = + adtl∨ (λ) ⊗ id (t − tl ) + 1 + adtl∨ (λ) ⊗ id (tl ). 2πi Then using (log(σ ) ⊗ id + id ⊗ log(σ ))(t − tl ) = 2π i(t − tl ), and the l-invariance of t − tl and tl , we get log(σ ) log(σ ) ψ(λ) − ψ(λ)2,1 = ( + adtl∨ (λ) − 1− 2πi 2π i ∨ − adtl (λ) ⊗ id)(t − tl ) +( 1 + adtl∨ (λ) − 1 − adtl∨ (λ) ⊗ id)(tl ).
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Using the identities (1 − x) − (x) = −πcotan(π x), and (x + 1) − (x) = x1 , we obtain Proposition 4.7. The classical part of Proposition 4.10 now follows from the fact that the classical counterpart of η is the restriction to k∗ ⊂ l∗ . 4.6. Twists by an element of Z(l). One checks that the results of Sect. 4.2 can be generalized as follows. Let (g, l, σ ) be as in Sect. 4.1. Let us denote by Z(l) the center of l and let γ ∈ Z(l). Denote by κ,γ the renormalized holonomy from 0 to 1 of the equation z
dG 1 = κ X(z) + tl2,3 + m(tl )(2) − γ (2) G(z). dz 2
(44)
Then κ,γ satisfies the pseudotwist equation 1,23,4 2,3,4 12,3,4 1,2,34 ((KZ )−1 )1,2,3 κ,γ κ,γ = κ,γ κ,γ ,
and its classical limit is ρσ,c (λ). To prove the first statement, one modifies the system of Eqs.(36,37) by adding −κγ (2) G in the r.h.s. of (36), and −κγ (3) G in the r.h.s. of (37). This is again a compatible system, because of the identity X(z) + X(z−1 )2,1 = tl . Assume in addition that l is quadratic polarized as in the sequel of Sect. 4.2, let κ,l,g (γ ) be the renormalized homolomy from 0 to 1 of (35), modified by the addition of −κγ (2) G in its r.h.s. Then η(κ,γ ) = k,l,g (γ ), and k,l,g (γ )Jll is a dynamical pseudotwist, quantizing ρ˜c,k,l . To prove this, one modifies the system (42,43) as above. 4.7. Relation with twisted loop algebras. Here, we interpret results of Sect. 4.2 in terms of the ABRR equations and the dynamical twist for a twisted loop algebra. More precisely, we show that the compatibility of the systems (36,37) and (42,43) are consequences of the compatibility of multicomponent ABRR equations (Proposition 2.27), and relate G(z) with a dynamical twist. Throughout the section, we assume that g, t, σ, l, k, m± are as in Sects. 4.1, 4.2. We also assume that t ∈ S 2 (g)g is nondegenerate. If s ∈ C× is an eigenvalue of σ , let gs ⊂ g be the generalized eigenspace (we set gs = 0 for other s ∈ C× ). Then t decomposes as a sum s∈C× ts , where ts ∈ gs ⊗ gs −1 . 4.7.1. Twisted loop algebras. Let us say that a function of one variable x is a generalized trigonometric polynomial if it is a linear combination of functions of the form x n eax , n ∈ Z+ , a ∈ C (the sum may involve different a). Let Lσ g be the Lie algebra of g-valued generalized trigonometric polynomials of x satisfying the condition f (x + 1) = σ (f (x)). For notational convenience we will express such functions as multivalued functions of z = e2π ix . We will denote by C[log(z), za , a ∈ C] the ring of generalized trigonometric polynomials. Set e(x) = e2πix , := {α ∈ C|e(α) is an eigenvalue of σ }, C+ = {u + iv|u > 0 or (u = 0 and v ≥ 0)}, C− = −C+ , ± = ∩ C± . If λ, µ ∈ C, we write λ ≤ µ (resp., λ < µ) iff µ − λ ∈ C+ (resp., C+ − {0}).
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d The operator z dz acts on Lσ g, and its eigenvalues belong to . If α ∈ , we denote by (Lσ g)α the corresponding generalized eigenspace. Then Lσ g = ⊕α∈ (Lσ g)α . be expressed as u = (Lσ g)α is the subspace of Lσ g of allα elements u, which can fi , where ai ∈ ge(α) and fi ∈ z C[log(z)]. Then if u = i ai ⊗ i ai ⊗ fi ∈ (Lσ g)α and v = j bj ⊗ gj ∈ (Lσ g)−α , the function i,j (ai , bj )fi gj (z) is in C[log(z)] and is invariant under x → x + 1, and is therefore constant (we denote by (−, −) the pairing on g inverse to t). This defines a nondegenerate pairing (Lσ g)α × (Lσ g)−α → C. We denote by (−, −) the direct sum of these pairings, which is a nondegenerate pairing (Lσ g)2 → C. Let Cg be the endomorphism of g equal to ad(m(t)) (we denote by ad : U (g) → End(g) the algebra morphism extending the Lie algebra morphism g → End(g) induced d by the adjoint action of g). Then Cg ⊗ z dz is a derivation of g ⊗ C[log(z), za , a ∈ C], which restricts to a derivation of Lσ g. d Set ω(x, y) = 21 ((Cg ⊗ z dz )(x), y) for any x, y ∈ Lσ g. Then ω is a cocycle on Lσ g, independent on a rescaling of t, and is a generalization of the critical level cocycle (which corresponds to g simple, σ = id). Define the affine Lie algebra
g¯ = Lσ g = Lσ g ⊕ Ck ⊕ C1 ⊕ Cd ⊕ Cδ, where
[a(z), b(z)] = [a, b](z) + (za (z), b(z))k + ω(a(z), b(z))1,
1 Cg (za (z)), [d, δ] = 0, 2 k and 1 are central. If g is semisimple, g¯ is closely related to a (possibly twisted) affine Kac-Moody algebra. An invariant, nondegenerate symmetric bilinear form is defined on g¯ by the following requirements: its extends the bilinear form on Lσ g, (k, d) = (1, δ) = 1, the other pairings of k, 1, d, δ are zero, and k, 1, d, δ are orthogonal to Lσ g. Define a Lie subalgebra g = Lσ g⊕Ck⊕C1 ⊂ g¯ . We also let ¯l := l⊕Span(k, 1, d, δ). σ We have (Lσ g)0 = g . Set g¯ 0 = ¯l = (Lσ g)0 ⊕ Span(k, 1, d, δ), g¯ ν = (Lσ g)ν if ν ∈ − {0}. Then g¯ = ⊕ν∈ g¯ ν . [d, a(z)] = za (z), [δ, a(z)] =
(Lσ g)−α be the element dual to the 4.8. Critical cocycle. If α ∈ , let sα ∈ (Lσ g)α ⊗ pairing (−, −). Then s0 = tl . Set T = 21 m(s0 ) + α>0 m(¯sα ). Then T belongs to the normal-order completion U (Lσ g) of U (Lσ g). We set 1 γσ := µ(( ⊗ id)(tu )) 2 (where µ denotes the Lie bracket). Proposition 4.12. 1) Denote by Z(l) the center of the Lie algebra l, then γσ belongs to Z(l). 2) The derivation u → [T , u] preserves Lσ g, and we have for u ∈ Lσ g, 1 d [T , u] = − (Cg ⊗ z )(u) + [γσ ⊗ 1, u]. 2 dz
(45)
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Remark 4.13. The critical level cocycle for Lσ g is defined as (u, v) → −([T , u], v); so this cocycle is cohomologous to ω. Remark 4.14. Let g be a simple, simply laced Lie algebra, h be its Coxeter number, + ⊂ h∗ a system of positive roots, and ρ = 21 α∈+ α. Equip h with its scalar product (−, −) such that all roots have length 2. Let hρ ∈ h be the element corresponding to ρ (so [hρ , eα ] = (ρ, α)eα for any root α). Set σ = exp( 2πi h ad(hρ )). Then Lσ g is isomorphic to Lg with the principal gradation. In that case γσ = 0. Indeed, 1 (ρ, α) (ρ, α) 1 γσ = [eα , fα ] + (1 − )[fα , eα ] = ( (ρ, α)hα ) − hρ . 2 h h h α∈+
α∈+
Now if β ∈ h, (γσ , β) = h1 α∈+ (ρ, α)(α, β) − (ρ, β) vanishes because of the iden tity α∈+ α ⊗ α = h(−, −); this identity holds up to scaling by W -invariance, and the contraction of both sides yields 2card(+ ) = h × rank(g), which is Kostant’s identity. Proof of Proposition 4.12. Let us prove 1). γσ clearly belongs to g1 = l. On the other hand, commutes with the adjoint action of l, and tu is l-invariant, so ( ⊗ id)(tu ) is l-invariant. Therefore γσ commutes with l. Let us prove 2). Let us set sα = i eα,i (z) ⊗ e−α,i (z). We first prove: Lemma 4.15. Let λ ∈ and u(z) ∈ (Lσ g)λ . If λ ≥ 0, then 1 [T , u(z)] = − [eα,i (z), [e−α,i (z), u(z)]], 2 α|0≤α<λ
and if λ < 0, then [T , u(z)] =
1 2
[e−α,i (z), [eα,i (z), u(z)]].
α|−λ≤α<0
Proof of Lemma 4.15. Assume that λ ≥ 0, then 1 [T , u(z)] = [e0,i , u(z)]e0,i + e0,i [e0,i , u(z)] 2 i [e−α,i (z), u(z)]eα,i (z) + e−α,i (z)[eα,i (z), u(z)]. + α>0 i
Now if α ∈ , then [e−α,i (z), u(z)] ⊗ eα,i (z) = −eλ−α,i (z) ⊗ [eα−λ,i (z), u(z)].
(46)
So an infinity of cancellations take place, and we get 1 [T , u(z)] = [e0,i , u(z)]e0,i + e0,i [e0,i , u(z)] + [e−α,i (z), u(z)]eα,i (z) 2 0<α≤λ
1 1 [e0,i , u(z)]e0,i + [e−α,i (z), u(z)]eα,i (z) + [e−λ,i (z), u(z)]eλ,i (z). = 2 2 0<α<λ i i 1 [[e0,i , u(z)], e0,i ] + [e−α,i (z), u(z)]eα,i (z), = 2 i
0<α<λ
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where we have used (46) for α = λ. Using again (46) for 0 < α < λ, and using the change of variables α → λ − α, we get the first identity of Lemma 4.15. The second identity is proved in the same way. Lemma 4.16. Let α ∈ be such that 0 ≤ α < 1, and u ∈ ge(α) , then
(
ad(m(tλ ))(u) − ◦ ad(m(t))(u) = −2[γσ , u].
λ|0≤λ<α
Here tλ ∈ ge(λ) ⊗ ge(−λ) is such that
λ|0≤λ<1 tλ
Proof of Lemma 4.16. For λ ∈ , set tλ = have tu = λ|0<λ<1 tλ , so γσ =
1 2
= t.
i tλ,i
⊗ t−λ,i , where t±λ,i ∈ ge(±λ) . We
[ (tλ,i ), t−λ,i ].
λ|0<λ<1 i
Then −2[γσ , u] = −
[ (tλ,i ), [t−λ,i , u]] + [[ (tλ,i ), u], t−λ,i ].
λ|0<λ<1 i
Now using ( ⊗ id + id ⊗ )(tλ ) = tλ , and the change of variables λ → 1 − λ, we get −2[γσ , u] = [(1 − 2 )(tλ,i ), [t−λ,i , u]]. λ|0<λ<1 i
We split this sum as [(1 − 2 )(tλ,i ), [t−λ,i , u]] + λ|0<λ<α
+
[(1 − 2 )(tλ,i ), [t−λ,i , u]]
λ|α<λ<1 i
i
[(1 − 2 )(tα,i ), [t−α,i , u]].
(47)
i
Using the change of variables λ → α − λ, we rewrite the first summand of (47) as S1 =
1 2
[(1 − 2 )(tλ,i ), [t−λ,i , u]] + [(1 − 2 )(tα−λ,i ), [tλ−α,i , u]] .
λ|0<λ<α
Now we have S1 =
1 2
i
i
i tα−λ,i
λ|0<λ<α
⊗ [tλ−α,i , u] =
i [u, t−λ,i ] ⊗ tλ,i .
Therefore
[(1 − 2 )(tλ,i ), [t−λ,i , u]] + (1 − 2 )([u, t−λ,i ]), tλ,i , i
which we rewrite as S1 =
1 2
λ|0<λ<α
[(1 − 2 )(tλ,i ), [t−λ,i , u]] + tλ,i , (1 − 2 )([t−λ,i , u]) . i
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639
Now we have: if t ∈ ge(λ) , v ∈ ge(α−λ) , then [(1 − 2 )(t ), v] + [t , (1 − 2 )(v)] = 2(1 − )([t , v]). Therefore S1 = (1 − ) [tλ,i , [t−λ,i , u]] . 0<λ<α
i
In the same way, we use the change of variables λ → 1 + α − λ and the identity: if α < λ < 1, t ∈ ge(λ) , v ∈ ge(α−λ) , then [(1 − 2 )(t ), v] + [t , (1 − 2 )(v)] = −2 ([t , v]) to prove that the second summand of (47) is S2 = − [tλ,i , [t−λ,i , u]] . α<λ<1 i
Finally (47) is equal to [tλ,i , [t−λ,i , u]] λ|0<λ<α
i
λ|0<λ<1,λ=α
i
−
[tλ,i , [t−λ,i , u]] + [(1 − 2 )(tα,i ), [t−α,i , u]] . (48) i
Now the last sum is (using the invariance of t, then the fact that is an l-module endomorphism, then the invariance of t again) [tα,i , [t−α,i , u]] + [−2 ([u, t0,i ]), t0,i ] = [tα,i , [t−α,i , u]] − 2 [t0,i , [t0,i , u]]. i
i
[t0,i , [t0,i , u]] − [tα,i , [t−α,i , u]] . = (1 − ) i
i
Finally (47) is equal to
[tλ,i , [t−λ,i , u]] − [eλ,i , [e−λ,i , u]] ,
λ|0≤λ<1 i
λ|0≤λ<α
i.e., to the l.h.s. of the identity of Lemma 4.16.
i
End of proof of Proposition 4.12. Denote by u → D1 (u), u → D2 (u) both sides of (45). Then D1 , D2 are derivations of Lσ g, such that ∀u(z) ∈ Lσ g,
1 Di (zu(z)) = zDi (u(z)) − zCg (u(z)). 2
Here we view Lσ g as a module over C[z, z−1 ]. Since ⊕α∈|0≤α<1 (Lσ g)α generates Lσ g as a C[z, z−1 ]-module, it suffices to prove (45) for u(z) ∈ (Lσ g)α , 0 ≤ α < 1. We have then zv (z) = (v(z)) for any v(z) ∈ (Lσ g)α , In particular, Cg (u(z)) ∈ (Lσ g)α , therefore (Cg ⊗ z
d )(u(z)) = ( ◦ Cg ) ⊗ id (u(z)) dz
(in fact, one can show that Cg commutes with ).
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On the other hand, Lemma 4.15 implies that 1 ad(m(tλ ))(u(z)). [T , u(z)] = − 2 λ|0≤λ<α
Finally, we get d 1 Cg ⊗ z (u(z)) 2 dz 1 1 =− ◦ ad(m(tλ )) ⊗ id (u(z)) + ad(m(tλ )) ⊗ id (u(z)) 2 2
[T , u(z)] +
0≤λ<α
0≤λ<1
= [γσ ⊗ 1, u(z)] (by Lemma 4.16), which proves (45).
4.8.1. Infinite dimensional polarized Lie algebras. One checks that the theory of dynamical pseudotwists extends as follows. Let be a subset of C. We assume that Z ⊂ , is stable under the translations by elements of Z, and /Z is finite. We set as before ± = ∩ C± . Let g¯ be a -graded Lie algebra, g¯ = ⊕ν∈ g¯ ν . Here -graded means that [¯gν , g¯ ν ] ⊂ g¯ ν+ν if ν + ν ∈ , and equals 0 otherwise. Set ¯l := g¯ 0 . Assume that ¯l is a polarized Lie algebra ¯l = k¯ ⊕ m+ ⊕ m− . The Lie brackets induce linear maps m+ ⊗ m− → k¯ and g¯ ν ⊗ g¯ −ν → k¯ for ν ∈ + − {0}. We assume that (a) dim(m+ ) = dim(m− ) and dim(¯gν ) = dim(¯g−ν ) for any ν ∈ = dim(m ) and d = dim(¯ + − {0}; we set dm g±ν ), and (b) the corresponding ± ν d d ¯ ¯ m ν determinants D m ∈ S (k) and D ν ∈ S (k) are all nonzero. ¯ These linear maps can therefore be “inverted” and yield ρm ∈ m+ ⊗m− ⊗S · (k)[1/D m ], · ¯ ρν ∈ g¯ ν ⊗ g¯ −ν ⊗ S (k)[1/D ], such that ρ + ρ is a formal solution of the m ν ν∈+ −{0} ν CDYBE. ¯ of D m , D ν . We denote by U the microlocalization of Let Dm , Dν be lifts in U (k) ¯ U (k) w.r.t. all Dm , Dν . Let us set u¯ ± = m± ⊕ (⊕ν∈± −{0} g¯ ν ), then U (¯u± ) are ± -graded algebras with finite dimensional homogeneous parts. As in Sect. 2, we can construct (ν) (ν) (ν) , U ai ⊗ bi ⊗ i ∈ U (¯u+ )ν ⊗ U (¯u− )−ν ⊗ Kν = i
such that for any x ∈ U (¯u− )−ν , y ∈ U (¯u+ )ν , we have (ν) (ν) (ν) H (xai ) i H (bi y) = H (xy). i
¯ corresponding to g¯ = k¯ ⊕ u¯ + ⊕ u¯ − . Here H is the Harish-Chandra map U (¯g) → U (k) (ν) (ν) (ν)(2) (ν)(1) ¯ )⊗S( i ), then Let us set p¯ ± = k⊕ u¯ ± . We also set Jν = i ai ⊗S(bi )S( i . Moreover, J := ν∈ Jν belongs to ⊕ ¯ ν∈+ (U (¯u+ )ν ⊗ U Jν ∈ (U (¯u+ )ν ⊗U (p¯ − )−ν )⊗ + , and satisfies the identity U U (p¯ − )−ν )⊗ J12,3,4 J2,3,4 = J2,3,4 J1,2,34 . Here ⊕ ¯ ν,ν ∈+ (U (¯u+ )ν ⊗ U (¯g)ν −ν ⊗ U (p¯ − )−ν )⊗ ¯ means the direct product. U in ⊕
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4.8.2. The ABRR equation in the infinite dimensional case. Assume that g¯ is equipped ¯ m± = with a nondegenerate invariant pairing of degree 0, −, −, such that k, 2 ¯ m± , m± = 0. Let s ∈ m+ ⊗ m− , tν ∈ g¯ ν ⊗ g¯ −ν and tk¯ ∈ S (k) be dual to this pairing. Then the analogue of the normally-ordered ABRR equation (15) is (s 1,2 + tν1,2 )J = [−m(¯s )(2) − m(t¯ν )(2) + t ¯2,3 , J]. k
ν>0
ν>0
¯ This is an identity in U (¯g)⊗2 ⊗U . g¯ Moreover, the component J0 of J coincides with the twist J ¯ 0 corresponding to the k nondegenerate polarized Lie algebra g¯ 0 = k¯ ⊕ m+ ⊕ m− .
4.8.3. Let us return to the setup of Sect. 4.7.1. Assume that gσ is polarized and nondegenerate, gσ = k ⊕ m+ ⊕ m− . Then we set k¯ = k ⊕ Span(k, 1, d, δ). Then g¯ = ⊕ν∈ g¯ ν , g¯ 0 = k¯ ⊕ m+ ⊕ m− is an example of the situation of Sects. 4.8.1, 4.8.2. ¯ = S · (k) ⊗ S · (Ck ⊕ C1) ⊗ S · (Cd ⊕ Cδ). . We have S · (k) 4.8.4. The algebra U 0 d m Let D m ∈ S (k) be the determinant corresponding to the nondegenerate Lie algebra l = k ⊕ m+ ⊕ m− , then D m = D 0m ⊗ 1 ⊗ 1. Let us now describe D ν when ν ∈ + − {0}. Let us set dg := dim(g). Then D n is an ¯ of the form (nk)dg + polynomial of partial degree element of S dg (k ⊕ Ck) ⊂ S dg (k), < dg in k. So D ν is nonzero. ¯ let U be the corresponding microlocalization. Let Dm , Dν be lifts of D m , D ν in U (k), k⊕Cd⊕Cδ ) be the microlocalization of U (k) (resp., U (k ⊕ Cd ⊕ Cδ)) k (resp., U Let U in U k⊕Cd⊕Cδ ((1/k)). w.r.t. Du . Then the form taken by the D ν allows us to embed U ¯ ¯ the image of K¯ in U (¯g)⊗2 We will still denote by K ⊗ U ((1/k)); actually, K¯ k⊕Cd⊕Cδ k , where ¯ ( U belongs to U (¯u+ )⊗U p− )[1/k] ⊗ p− = (k ⊕ Ck) ⊕ u¯ − . 4.8.5. The affine ABRR equations. We have a morphism π1 : U (¯u+ ) → U (g), taking a(x) ∈ (Lσ g)α to a(0) (recall that α > 0), a ∈ m+ to a, k to 0, 1 to 1. γ If γ ∈ Z(l), we also have a morphism πz : U ( p− ) → U (g)⊗C[za , a ∈ C, log(z)], taking af (x) ∈ (Lσ g)α to a ⊗ f (x) (recall that α < 0), a ∈ m+ to a ⊗ 1, a ∈ l to a − (γ , a) ⊗ 1, k to 0, 1 to 1. −γ k ((1/k))[[za , a ∈ U Set J (z) := (π1 ⊗πz σ ⊗id)(J). Then J (z) belongs to U (g)⊗2 ⊗ a a C+ ]][log(z)]. Here A[[z , a ∈ C+ ]] = A[z , a ∈ D][[z]] (D is defined in Sect. 4.2). ¯ The ABRR equation is an equality in U ( g)⊗2 ⊗Uk ((1/k)). We have −γσ (πz ⊗ π1 )( ν>0 tν ) = −X(z). Moreover, −γσ
(πz
⊗ π1 ⊗ id)([
ν>0 −γσ
so the image of ABRR by πz −k (3) z
1 m(t¯ν )(2) , J]) = [− m(tl )(2) , J (z)], 2
⊗ π1 ⊗ id is
1,2 dJ 1 J (z) + [tk2,3 + m(tk )(2) − γ (2) , J (z)], = X(z) − s dz 2
(49)
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where γ = µ(s). Moreover, the constant coefficient of J (z) is J0 = Jkl . These conditions determine the series J (z) uniquely. 2,3
1
Set κ = −1/k (3) , then G(z) is a solution of (34) iff G(z)z−κ(tk + 2 m(tk ) −γ ) is a solution of (49). 1,2 g g g Let us write Jk (z) = J (z). We then have k = limz→1− (1 − z)−κt Jk (z). The g g g g composition formula implies that Jk (z) = η(Jl (z))Jkl , therefore k = η(l )Jkl . We can now reprove Proposition 4.10, 1) as follows: we have g
g
(2)
(2)
g
η(l ) = k (Jkl )−1 = lim (1 − z)−κt Jk (z)(Jkl )−1 , z→1− 1,2
g
now g(z) := Jk (z)(Jkl )−1 is such that g(0) = 1, and (thanks to the ABRR equation for Jkl ) z
dg 1 = κ(X(z) − s)1,2 g(z) + κ[tk2,3 + m(tk )(2) − γ (2) , g(z)]. dz 2 g
It follows that η(l ) is the renormalized holonomy from 0 to 1 of (35), i.e., k,l,g . The theory of infinite dimensionalABRR equations also underlies the systems (36,37) −γ −γ and (42,43). Indeed, set J (z, u) := (π1 ⊗ πz−1σ ⊗ πu−1σ ⊗ id)(J[2] ), then the multicomponent ABRR implies that J (z, u) satisfies the equations z
u
∂J 1 =κ X(z)1,2 + X(z/u)3,2 − s 1,2 − s 3,2 + tk2,4 + m(tk )(2) − γ (2) J (z, u) 2 ∂z 1 2,3 2,4 (2) (2) − κJ (z, u)(tk + tk + m(tk ) − γ ), 2 ∂J 1 =κ X(u)1,3 + X(u/z)2,3 − s 1,3 − s 2,3 + tk3,4 + m(tk )(3) − γ (3) J (z, u) 2 ∂u 1 3,4 (3) (3) − κJ (z, u)(tk + m(tk ) − γ ), 2 2,3
2,4
1
so that J (z, u) satisfies this system iff it has the form G(z, u)zκ(tk +tk + 2 m(tk ) −γ ) 3,4 1 (3) (3) uκ(tk + 2 m(tk ) −γ ) , where G(z, u) is a solution of (42,43). −γ The compatibility of the systems (36,37) and (42,43) is the image by π1 ⊗ πz−1σ ⊗ (2)
(2)
−γ
πu−1σ ⊗ id of the compatibility relations for the multicomponent ABRR equations (Proposition 2.27). Remark 4.17. If γ is an element of Z(l), and Jγ (z) is the analogue of J (z), where 1,2 −γσ is replaced by γ − γσ , then limz→1− (1 − z)−κt Jγ (z) coincides with κ,γ , as defined in Sect. 4.6. The ABRR arguments of this section can be modified to provide other proofs of the statements of Sect. 4.6.
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5. Cayley r-Matrices Proposition 0.3 follows from Proposition 0.4. Let us prove this proposition. It will be enough to treat the case c = 1. We set ρC := ρC,1 . Set ρl (λ) := (f (ad(λ∨ )) ⊗ id)(tl ),
ρu (λ) := iπ
(C + id) + e−2πiad(λ∨ ) (C − id) (C + id) − e
−2πiad(λ∨ )
(C − id)
⊗ id (tu ).
Then ρC = ρl + ρu . We want to prove that CYB(ρl + ρu ) − Alt(d(ρl + ρu )) = −π 2 Z. We have Z = Zl + Zl,u + Zu , where Zl = [tl1,2 , tl2,3 ], Zl,u = Alt([tl1,2 , tu2,3 ]), Zu = (id ⊗ pu ⊗ id)([tu1,2 , tu2,3 ]), where pu : g → u is the projection on u parallel to l. Applying [AM1] to the quadratic algebra (l, tl ), we already have CYB(ρl )−Alt(dρl ) = −π 2 Zl . It remains to prove that CYB(ρl , ρu ) + CYB(ρu ) − Alt(dρu ) = −π 2 (Zl,u + Zu ). Both sides of this equality belong to ∧3 (g) = ⊕3α=0 ∧α (l) ⊗ ∧3−α (u). Since the equality already holds when projecting it on the components α = 3 and α = 2, it remains to prove its projection on the components α = 0 and α = 1. The projection on the component α = 0 is the equality (2) Alt ◦ pu [ρu1,2 (λ), ρu2,3 (λ)] + π 2 [tu1,2 , tu2,3 ] = 0,
(50)
and the projection on the component α = 1 is (1)
pl ([ρu1,2 (λ), ρu1,3 (λ)]) + [ρl1,23 (λ), ρu2,3 (λ)] − (dρu (λ))2,3,1 + π 2 [tl1,2 , tu2,3 ] = 0. (51) ⊗3 ∨ To prove (50), we apply to it (C + id) − e−2πiad(λ ) (C − id) . We get ⊗2 ∨ ∨ (C + id) − e−2πiad(λ ) (C − id) ⊗ (C + id) + e−2πiad(λ ) (C − id) ⊗3 ∨ + cyclic permutation + (C + id) − e−2πiad(λ ) (C − id) (Zu ) = 0, ∨ i.e., 4 (C + id)⊗3 − (e−2πiad(λ ) (C − id))⊗3 (Zu ) = 0, which follows from (C + id)⊗3 (Zu ) = (C − id)⊗3 (Zu ), which follows from the assumptions on C.
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⊗2 ∨ Let us now prove (51). Let us apply to it id ⊗ (C + id) − e−2πiad(λ ) (C − id) , we get ⊗2 1,2 2,3 ∨ π 2 id ⊗ (C + id) + e−2πiad(λ ) (C − id) ([tl , tu ]) ⊗2 1,2 2,3 ∨ ([tl , tu ]) +π 2 id ⊗ (C + id) − e−2πiad(λ ) (C − id) (2) ∨ ∨ −iπ(e2πiad(λ ) )(2) (C + id) − e−2πiad(λ ) (C − id) (2) ∨ d (C + id) + e−2πiad(λ ) (C − id) (tu2,3 ) (2) ∨ ∨ +iπ(e2πiad(λ ) )(2) (C + id) + e−2πiad(λ ) (C − id) (2) ∨ d (C + id) − e−2πiad(λ ) (C − id) (tu2,3 ) ⊗2 ∨ +iπ id ⊗ (C + id) − e−2πiad(λ ) (C − id) ([ρl1,23 (λ),
C + id + e−2πiad(λ∨ ) (C − id) (2) ∨
C + id − e−2πiad(λ ) (C − id)
(tu2,3 )]) = 0,
(52)
i.e., ∨ ∨ 2π 2 (id − C 2 )(2) ([tl1,2 , tu2,3 ]) − iπ(e2πiad(λ ) )(2) (C 2 − id)(2) d (e−2πiad(λ ) )(2) (tu2,3 ) + last line of (52) = 0. Now last line of (52) ∨ ∨ = iπ id ⊗ C + id + e−2πi ad(λ ) (C − id) ⊗ C + id − e−2πi ad(λ ) (C − id) ([ρl1,3 (λ), tu2,3 ]) ∨ ∨ − iπ id ⊗ C + id − e−2πi ad(λ ) (C − id) ⊗ C + id + e−2πi ad(λ ) (C − id) ([ρl1,2 (λ), tu2,3 ]) ⊗2 1,2 ∨ ([ρl (λ), tu2,3 ]) = −2iπ id ⊗ (C + id)⊗2 − id ⊗ e−2πi ad(λ ) (C − id) ∨ = −2iπ(id − C 2 )(2) 1 − id ⊗ (e−2πi ad(λ ) )⊗2 ([ρl1,2 (λ), tu2,3 ]) = 0,
which follows from the CDYBE identity for the Alekseev-Meinrenken r-matrix for (g, t), restricted to λ ∈ l∗ and projected on l ⊗ ∧2 (u). 6. Quantization of Homogeneous Spaces In this section, we show that the (pseudo)twists constructed in Sects. 2, 3 and 4 enable us to quantize (quasi)Poisson structures on homogeneous spaces. 6.1. Quantization of coadjoint orbits. Let g = l ⊕ u be a Lie algebra with a splitting; we assume that g is nondegenerate. Let G be the formal group with Lie algebra g, and L ⊂ G the subgroup corresponding to l (with suitable restrictions, the following constructions may be extended to other categories, like algebraic or complex Lie groups). Let D0 : l∗ → C be the determinant corresponding to g = l ⊕ u. Thedynamical g r-matrix, rl (λ), enables one to define a Poisson structure on l∗ − D0−1 (0) × G. The
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group L acts on this space by Poisson automorphisms, and the moment map is the prog jection on the first factor. According to P. Xu, a dynamical twist quantizing rl (λ) yields a quantization of this Poisson space. Let us recall how. Let χ ∈ l∗ be a character of l. Then L ⊂ Stab(χ ), where Stab(χ ) := {g ∈ G|Ad∗ (g)(χ ) = χ }. Let Oχ ⊂ g∗ be the coadjoint orbit of χ . Then we have a natural G-map G/L → G/Stab(χ ) = Oχ , taking the class of g to Ad∗ (g)(χ ). Let us further assume that D0 (χ ) = 0. Then we have L = Stab(χ ), so G/L → Oχ g is an isomorphism. The assumptions on χ imply that rl (χ ) ∈ ∧2 (g) is well-defined and g l-invariant. This implies that the bivector R(rl (χ )) on G (R denotes the translation from the right) descends to G/L, and that it equips G/L with a Poisson structure. Moreover, the map G/L → Oχ is Poisson (this is an observation of J.-H. Lu). l , since D0 (χ ) = 0, According to Remark 1.6, we may extend −1 χ to a character of U −1 therefore (id ⊗ id ⊗ χ )(J ) is well-defined; it coincides with J (χ ) as defined in Sect. 1.4, and belongs to U (g)⊗2 [[]]. It satisfies J (χ )12,3 J 1,2 (χ + l (3) ) = J (χ )1,23 J (χ )2,3 . Here J 1,2 (χ + l (3) ) has the usual meaning, and its action on f ⊗ g ⊗ h is the same as that of J (χ)1,2 if h is l-invariant. This relation implies that one can define a G-equivariant star-product on G/L by the formula f ∗ g = m(R(J (χ ))(f ⊗ g)), quantizing the Poisson structure on G/L . By virtue of the results of Sect. 2, these considerations allow us to get equivariant star-products in all nondegenerate polarized cases. In the polarized quadratic case, J (χ ) satisfies the equation (derived from ABRR) 1
s 1,2 J (χ ) = [
2
m(tl ) + −1 tl∨ (χ ) − γ
(2)
, J (χ )].
In the reductive case, this quantization (which yields an explicit equivariant starproduct for all semisimple coadjoint orbits) has been obtained by Alekseev-Lachowska ([AL]) and Donin-Mudrov ([DM]). 6.2. Quantization of Poisson homogeneous spaces. Let g = l ⊕ u be a Lie algebra with a splitting, such that g is nondegenerate. We assume that g is quadratic, i.e., we have t ∈ S 2 (g)g . Let c be a complex number. The dynamical r-matrix ρc (λ) (see Corollary 0.2) can be used to equip U ×G with the structure of a quasi-Poisson homogeneous space under the pair (g, t) ([AKM]). Recall that this means a g-invariant bivector, whose Schouten bracket with itself is given by the action of [t 1,2 , t 2,3 ]. Here U = {λ ∈ l∗ |ad(λ∨ ) has no eigenvalue of the form n/c, n a nonzero integer, and D0 (λ) = 0}. Let χ ∈ l∗ be a character of l, and assume that χ ∈ U . Then G/L is equipped with a quasi-Poisson homogeneous space structure under (g, t), given by the bivector χ = R(ρc (χ )). These quasi-Poisson structures may be viewed as trigonometric versions of the Poisson structures of Sect. 6.1. A quantization J of ρc (λ) gives rise to a quantization of Uformal × G (in the sense of [EE], Sect. 4.5), where Uformal is the intersection of U with the formal neighborhood of 0 ∈ l∗ . Moreover, if J (λ) is regular at χ , then it can be used to construct a quantization of this quasi-Poisson space, according to the formula f ∗ g = m(R(J (χ ))(f ⊗ g)). (Recall that we do not know a quantization of ρc in the nonpolarized case, even if c = 0.)
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Assume now that g is polarized, i.e., u = u+ ⊕u− and t = tl +s +s 2,1 , with tl ∈ S 2 (l), s ∈ u+ ⊗ u− . Then J (λ) has been constructed in Sect. 3, and if we take = KZ , J (λ) is regular on an explicit neighborhood of 0 (see [EE]). This yields a quantization of Uformal × G/L and of (G/L, χ ) for characters χ in this neighborhood. Let ∈ g⊗2 be a quasitriangular structure on g, i.e., + 2,1 = t and CYB() = 0. Let (G, (R − L)()) be the corresponding Poisson-Lie group. We have CYB(π ic( − 2,1 )) = −π 2 c2 Z (equality in ∧3 (g)) and CYB(ρc (χ )) = −π 2 c2 Z (equality in ∧3 (g/l)), where Z = [t 1,2 , t 2,3 ]. Therefore G/L, equipped with the Poisson bivector ,χ := −L(π ic( − 2,1 )) + R(ρc (χ )), is a Poisson homogeneous space under (G, (R − L)()). (Here L stands for left translations.) A quantization of the Poisson homogeneous space (G/L, ,χ ) may be obtained as follows. According to [EK] (in the reductive case, [Dr2, ESS]), there exists a pseudotwist 12,3 1,2 1,2 , t 2,3 )−1 J 1,23 J 2,3 . Then JEK ∈ U (g)⊗2 [[]] quantizing , i.e., JEK JEK = KZ κ (t EK EK the star-product on G/L is defined by the formula −1 f ∗ g = m(R(J (χ ))L(JEK )(f ⊗ g)).
This quantization is equivariant with respect to the quantum group U (g)JEK (U (g) twisted by JEK ). In the case when G, L are reductive, the homogeneous spaces we considered include generic dressing orbits of G, and we get their quantization equivariant under the quantum group Uq (g). A different way of quantizing such Poisson (and quasi-Poisson) homogeneous spaces was proposed in [DGS].
6.3. Quantization of Poisson homogeneous spaces corresponding to an automorphism. Let us assume that (g, t ∈ S 2 (g)g ) is a quadratic Lie algebra, equipped with σ ∈ Aut(g, t). We set l := gσ and assume that σ − id is invertible on g/gσ . As above, the dynamical r-matrix ρσ,c (λ) can be used to equip G/L with a structure of a quasi-Poisson homogeneous space of the group (G, −π 2 c2 Z) (where Z = [t 1,2 , t 2,3 ]). Namely, the quasi-Poisson bivector on G/L is given by the formula = R(ρσ,c (0)). We will set c = 1/(2πi), therefore =R
1 σ + id ( ⊗ id)(tu ) . 2 σ − id
The dynamical pseudotwist κ provides a quantization of this quasi-Poisson structure. Namely, set κ (0) := (id ⊗ id ⊗ ε)(κ ). The non-associative star-product on G/L (which is associative in the representation category of Drinfeld’s quasi-Hopf algebra) is given by the formula f ∗ g = m(R(κ (0))(f ⊗ g)), Let ∈ g⊗2 be a quasitriangular structure on g, i.e., + 2,1 = t and CYB() = 0. 2,1 Let (G, (R−L)()) be the corresponding Poisson-Lie group. Since CYB( −2 ) = Z/4 (in ∧3 (g)) and CYB(ρσ,c (0)) = Z/4 in ∧3 (g/l), we have a Poisson homogeneous space G/L under (G, (R − L)()), with Poisson bivector = −L(
− 2,1 1 σ + id ) + R( ( ⊗ id)(tu )). 2 2 σ − id
(53)
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The above construction yields a star-product quantization of this Poisson homogeneous structure. Namely, the star-product on G is defined by the formula f ∗ g = m(R(κ (0))L(J −1 )(f ⊗ g)), where J is a pseudotwist quantizing (e.g., J = JEK ). As in Sect. 6.2, this quantization is equivariant under the quantum group U (g)J . 6.4. Relation to the De Concini homogeneous spaces. Recall that according to Drinfeld [Dr1], if G is a Poisson-Lie group and L is a subgroup, then Poisson homogeneous space structures on G/L correspond to Lagrangian Lie subalgebras h ⊂ D(g) of the double of g such that g ∩ h = l. C. De Concini explained to us the following construction of Poisson homogeneous spaces. Let g be a factorizable quasitriangular Lie bialgebra. This means that g is a Lie algebra, ∈ g⊗2 is such that CYB() = 0, and t := + 2,1 ∈ S 2 (g)g is nondegenerate. Assume also that σ ∈ Aut(g, t). Then D(g) is isomorphic to g ⊕ g, with bilinear form given by (x1 , x2 ), (y1 , y2 ) = x1 , y1 − x2 , y2 . The graph h of σ is a Lagrangian subalgebra of g ⊕ g, which induces a Poisson homogeneous space structure on G/L = G/Gσ . Theorem 6.1. The construction of Sect. 6.3 yields quantizations of all the De Concini homogeneous spaces, such that σ is invertible on g/gσ . Proof. The Drinfeld subalgebra h ⊂ D(g) corresponding to a Poisson homogeneous space (G/L, ) is defined as h = {(x, ξ ) ∈ g ⊕ g∗ |ξ ∈ l∗ and x = (ξ ⊗ id)((0)) modulo l}, where (0) ∈ ∧2 (g/l) is the value at the origin of . In the case of the Poisson structure (53), (0) is equal to the class of P in ∧2 (g/l), where − 2,1 1 σ + id + (id ⊗ )(tu ), 2 2 σ − id therefore h is the image of the linear map l ⊕ u∗ → g ⊕ g∗ , (x, 0) → (x, x), (0, ξ ) → (x(ξ ), ξ ). Here x(ξ ) = (ξ ⊗ id)(P ). Let us set L(α) := (id ⊗ α)() and R(α) := (α ⊗ id)(), for α ∈ g∗ . If ξ ∈ u∗ , then (ξ ⊗ id)(tu ) = (ξ ⊗ id)(t), therefore (L + R)(ξ ) ∈ u for any ξ ∈ u∗ . If follows that +id x(ξ ) = 21 (R − L)(ξ ) + 21 σσ − id ◦ (L + R)(ξ ). According to [RS], the isomorphism D(g) → g ⊕ g is given by (a, 0) → (a, a) and (0, α) → (−R(α), L(α)). Let us view h as a subalgebra of g ⊕ g using this isomorphism. Then h is the image of the linear map l ⊕ u∗ → g ⊕ g, (x, 0) → (x, x) and id σ (ξ, 0) → x(ξ ) − R(ξ ), x(ξ ) + L(ξ ) = ◦ (L + R)(ξ ), ◦ (L + R)(ξ ) . σ − id σ − id ∗ Since the image of u → u, ξ → (L + R)(ξ ) is exactly u, and σ − id is invertible on u, h is the image of l ⊕ u → g ⊕ g, (x, 0) → (x, x) and (0, y) → (y, σ (y)), i.e., {(z, σ (z))|z ∈ g}, i.e., h is the graph of σ . P =
Remark 6.2. It is useful to describe κ (0) directly, since this is the only information about κ needed in Theorem 6.1: κ (0) is the renormalized holonomy from 0 to 1 of 1,2 + 1 m(t )(2) )G(z). the differential equation dG l 2 dz = κ(X(z)
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6.5. Poisson homogeneous spaces corresponding to a Cayley endomorphism. Assume that g = l ⊕ u is a Lie algebra with a splitting and a factorizable structure, such that t = tl ⊕ tu , tx ∈ S 2 (x) for x = l, u, and C ∈ Endl (u) is a Cayley endomorphism, such that (C ⊗ id + id ⊗ C)(tu ) = 0. Then h ⊂ g ⊕ g defined by h := ldiag ⊕ {(x, y) ∈ u × u|(C + id)(x) = (C − id)(y)} is a Lagrangian subalgebra, generalizing De Concini’s subalgebras (here ldiag = {(x, x)|x ∈ l}). It gives rise to a Poisson homogeneous structure on G/L. A quantization of the dynamical r-matrices of Proposition 0.4 should lead to a quantization of these homogeneous spaces. An example of this situation is: g is semisimple, with Cartan decomposition g = t ⊕ n+ ⊕ n− and the corresponding standard r-matrix ; w is a Weyl group element; l = t, u = n+ ⊕ n− ; C has eigenvalues ±1 on w(n∓ ) ( w (n± ) is independent on the choice of a Tits lift of w , so we denote it w(n± )). Then h = {(h + x+ , h + x− )|h ∈ t, x± ∈ w(n± )}. The corresponding Poisson homogeneous structure on G/T is given by −L() + R(w ⊗2 ()) (if V is a g-module and v ∈ V is a vector of weight 0, then w (v) is independent on the choice of w and is denoted by w(v)). In this case a quantization can be obtained using the formula f ∗ g := m(L(J −1 )R(w ⊗2 (J ))(f ⊗ g)), where J is a pseudotwist quantizing . Another quantization is given by f ∗ g = m (R(Jw )(f ⊗ g)), where f, g ∈ U (g)∗ and m is the product in U (g)∗ , and Jw ∈ U (g)⊗2 is such that Tw⊗2 ◦ ◦ Tw−1 = Ad(Jw ) ◦ , where is the coproduct of the Drinfeld-Jimbo quantum group U (g) and Tw is a Lusztig-Soibelman automorphism corresponding to w. The following fact implies the equivalence of both quantizations. Proposition 6.3. w ⊗2 (J ) and Jw J are gauge-equivalent pseudotwists quantizing w ⊗2 (). Proof. Recall the construction of J : let be a Drinfeld associator, g its specializa ) := (J 2,3 J 1,23 )−1 J 1,2 J 12,3 = tion to (g, t). Then J is a series J (), such that d(J w J ) = g . On the other hand, g . Since Jw satisfies the twist equation, we have d(J ⊗2 ⊗3 ⊗2 d(w (J )) = w (g ) = g . So w (J ) and Jw J are pseudotwists quantizing w⊗2 (). Let us now show that they are gauge-equivalent. Let us still denote by Tw the automorphism of U (g)[[]] obtained by transporting Tw by the isomorphism U (g)[[]] U (g). The reduction mod of Tw is a Tits lift of w, which we denote by w . Then Tw ◦ w −1 is an automorphism of U (g)[[]], whose reduction modulo is the identity, hence is inner (as g is semisimple). Let u ∈ U (g)[[]], u = 1 + O(), be such that Tw ◦ w −1 = Ad(u). Let 0 be the undeformed coproduct of U (g)[[]]. Set J1 = Jw J , J2 = w ⊗2 (J ). We have Ad(J1 ) ◦ 0 = Tw⊗2 ◦ 0 ◦ Tw−1 , Ad(J2 ) ◦ 0 = w ⊗2 ◦ 0 ◦ w −1 , so Ad(J1 ) ◦ 0 = Ad(u)⊗2 ◦ Ad(J2 ) ◦ 0 ◦ Ad(u)−1 , therefore J1 = u⊗2 J2 0 (u)−1 ξ , where ξ ∈ U (g)⊗2 [[]] has the form 1 + O() and is g-invariant. We now prove inductively that ξ = exp(d(η)), where η ∈ (U (g)⊗2 )g [[]] and d(η) = 0 (η) − η ⊗ 1 − 1 ⊗ η. Assume that we have found η1 , . . . , ηn−1 in U (g)g , such that ξ = exp(d(η1 + · · · + n−1 ηn−1 ))(1 + O(n )). Then set u := uexp(−(η1 + · · ·+n−1 ηn−1 )). We get J1 = (u )⊗2 J2 0 (u )−1 ξ , where ξ ∈ (U (g)⊗2 )g [[]] has the form 1 + O(n ). Let ξn ∈ (U (g)⊗2 )g be such that ξ = 1 + n ξn + O(n+1 ), then since 1 ) = d(u ⊗2 J2 0 (u )−1 ), we get d(ξn ) = ξn12,3 − ξn1,23 − ξn2,3 + ξn1,2 = 0. Since the d(J
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cohomology group involved is ∧2 (g) and since ∧2 (g)g = 0, we get ξn = d(ηn ), with ηn ∈ U (g)g . This proves the induction step. Now let η := i≥1 i ηi . We get J1 = (ue−η )⊗2 J2 0 (ue−η )−1 , therefore J1 and J2 are gauge-equivalent. Acknowledgements. The authors thank J. Donin and A. Mudrov for references. P.E. is grateful to C. De Concini for a very useful discussion on Poisson homogeneous spaces related to automorphisms, which was crucial for writing Sect. 6 of this paper. P.E. is indebted to IRMA (Strasbourg) for hospitality. The work of P.E. was partially supported by the NSF grant DMS-9988796.
References [AF] [AKM] [AL] [AM1] [AM2] [ABRR] [BS] [DGS] [DM] [Dr1] [Dr2] [EE] [EEM] [EFK] [EK] [ESS] [ES1] [ES2] [EV1] [EV2] [EV3] [Fad] [FGP] [RS]
Alekseev, A., Faddeev, L.: (T ∗ G)t : a toy model for conformal field theory. Commun. Math. Phys. 141, no. 2, 413–422 (1991) Alekseev, A., Kosmann-Schwarzbach, Y., Meinrenken, E.: Quasi-Poisson manifolds. Canad. J. Math. 54, no. 1, 3–29 (2002) Alekseev, A., Lachowska, A.: Invariant ∗-products on coadjoint orbits and the Shapovalov pairing. http://arxiv.org/abs. /math.QA/0308100, 2003. Alekseev, A., Meinrenken, E.: The non-commutative Weil algebra. Invent. Math. 139, no. 1, 135–172 (2000) Alekseev, A., Meinrenken, E.: Clifford algebras and the classical dynamical Yang-Baxter equation. Math. Res. Lett. 10, no. 2–3, 253–268 (2003) Arnaudon, D., Buffenoir, E., Ragoucy, E., and Roche, P.: Universal solutions of quantum dynamical Yang-Baxter equations. Lett. Math. Phys. 44 no. 3, 201–214 (1998) Borel, A., De Siebenthal, J.: Les sous-groupes ferm´es de rang maximum des groupes de Lie clos. Comment. Math. Helv. 23, 200–221 (1949) Donin, J., Gurevich, D., Shnider, S.: Double quantization on some orbits in the coadjoint representations of simple Lie groups. Commun. Math. Phys. 204, no. 1, 39–60 (1999) Donin, J., Mudrov, A.: Dynamical Yang-Baxter equation and quantum vector bundles. http://arxiv.org/abs/math.QA/0306028, (2003) Drinfeld, V.: On Poisson homogeneous spaces of Poisson-Lie groups. Teoret. Mat. Fiz. 95, no. 2, 226–227 (1993); translation in Theoret. and Math. Phys. 95, no. 2, 524–525 (1993) Drinfeld, V.: Quasi-Hopf algebras. Leningrad Math. J. 1, no. 6, 1419–1457 (1990) Enriquez, B., Etingof, P.: Quantization of Alekseev-Meinrenken dynamical r-matrices (in memory of F.I. Karpelevich). AMS Transl. 210, no. 2, 81–98 (2003) Enriquez, B., Etingof, P., Marshall, I.: Quantization of some Poisson-Lie dynamical r-matrices and Poisson homogeneous spaces. http://arxiv.or/abs/math.QA/0403283, 2004 Etingof, P., Frenkel, I., Kirillov, A.: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations. Providence RI: AMS, 1998 Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras, I, II. Selecta Math. (N.S.) 2, no. 1, 1–41 (1996); 4, no. 2, 213–231 (1998) Etingof, P., Schedler, T., Schiffmann, O.: Explicit quantization of dynamical r-matrices. J. Amer. Math. Soc. 13, no. 3, 595–609 (2000) Etingof, P., Schiffmann, O.: Lectures on the Dynamical Yang-Baxter Equations. in: “Quantum groups and Lie theory (Durham, 1999)”, London Math. Soc. Lecture Note Series, 290, Cambridge: Cambridge Univ. Press, 2001, 89–129 Etingof, P., Schiffmann, O.: On the moduli space of classical dynamical r-matrices. Math. Res. Lett. 8, no. 1–2, 157–170 (2001) Etingof, P., Varchenko, A.: Geometry and classification of solutions of the classical dynamical Yang-Baxter equation. Commun. Math. Phys. 192, no. 1, 77–120 (1998) Etingof, P., Varchenko, A.: Exchange dynamical quantum groups. Commun. Math. Phys. 205, no. 1, 19–52 (1999) Etingof, P., Varchenko, A.: Dynamical Weyl groups and applications. Adv. Math., 167, no. 1, 74–127 (2002) Faddeev, L.: On the exchange matrix of the WZNW model. Commun. Math. Phys. 132, no. 1, 131–138 (1990) Feh´er, L., G´abor, A., Pusztai, P.: On dynamical r-matrices obtained from Dirac reduction and their generalizations to affine Lie algebras. J. Phys. A 34, no. 36, 7335–7348 (2001) Reshetikhin, N., Semenov-Tian-Shansky, M.: Quantum R-matrices and factorization problems. J. Geom. Phys. 5, no. 4, 533–550 (1988)
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Schiffmann, O.: On classification of dynamical r-matrices. Math. Res. Lett. 5, no. 1–2, 13–30 (1998) Springer, T.: Microlocalisation alg´ebrique. In: S´em. alg`ebre Dubreil-Malliavin, Lecture Notes in Mathematics 1146, Berlin Heidelberg: Springer-Verlag, 1983, pp. 299–316 Xu, P.: Triangular dynamical r-matrices and quantization. Adv. Math. 166, no. 1, 1–49 (2002) Xu, P.: Quantum dynamical Yang-Baxter equation over a nonabelian base. Commun. Math. Phys. 226, no. 3, 475–495 (2002)
[Spr] [Xu1] [Xu2]
Communicated by L. Takhtajan
Commun. Math. Phys. 254, 651–658 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1282-5
Communications in
Mathematical Physics
On Some Mean Matrix Inequalites of Dynamical Interest Igor Rivin Department of Mathematics, Temple University, Philadelphia, PA 19122. E-mail: [email protected] Received: 12 December 2003 / Accepted: 20 September 2004 Published online: 22 January 2005 – © Springer-Verlag 2005
Abstract: Let A ∈ SL(n, R). We show that for all n > 2 there exist dimensional strictly positive constants Cn such that log ρ(AX)dX ≥ Cn log A, On
where A denotes the operator norm of A (which equals the largest singular value of A), ρ denotes the spectral radius, and the integral is with respect to the Haar measure on On , normalized to be a probability measure. The same result (with essentially the same proof) holds for the unitary group Un in place of the orthogonal group. The result does not hold in dimension 2. This answers questions asked in [3, 5, 4]. We also discuss what happens when the integral above is taken with respect to measure other than the Haar measure. Introduction A. Wilkinson brought the following question to the author’s attention: Question 1. Let A ∈ SL(n, R), then log ρ(AX)dX ≥ Cn do On
A
Sn
log Buddσn = 0,
for some positive dimensional constant Cn . The above is Question 6.6 in [3]. The question is answered in dimension 2 in the paper [1] – in fact, the authors show that in that case one can take C2 = 1, and replace ≥ by = .
The author is supported by the NSF DMS.
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In the papers [3, 5] the question is also posed as to what happens when the measure on the orthogonal group is not the Haar measure (actually, the question is asked in a slightly different context.) Obviously the result is unchanged if a measure µ is in the same measure class as the Haar measure, but we note that in two dimensions the sign of the inequality in Question 1 can be reversed whenever µ is supported on a proper subset of O(2) = S 1 . The same question with the orthogonal group replaced by the unitary group is answered in the paper [4] (in arbitrary dimension). The dimensional constant is again equal to 1, though the inequality cannot be replaced by an equality. In [4], Dedieu and Shub also conjecture a positive answer to Question 1. The same conjecture is made in [5]. In this note we show Theorem 2. For all n > 2 there exist strictly positive constants Cn such that log ρ(AX)dX ≥ Cn log A,
On
where A denotes the operator norm of A (which equals the largest singular value of A), ρ denotes the spectral radius, and the integral is with respect to the Haar measure on On . The same result (with essentially the same proof) holds for the unitary group Un in place of the orthogonal group. This result easily implies a positive answer to Question 1 in dimensions bigger than 2. Theorem 2 is false in dimension 2. It should also be noted that Theorem 2 immediately implies the following Corollary 3. Let µ be an orthogonally invariant measure on SL(n, R), such that log x ∈ L1 (µ), and n ≥ 3. Then
SL(n,R)
log ρ(X)dµ ≥ Cn
SL(n,R)
log Adµ.
The same statement holds with C replacing R and “unitarily invariant” replacing “orthogonally invariant”. Proof. Integrate both sides of the inequality in Theorem 2.
The plan of the paper is as follows: In Sect. 1 we summarize some necessary facts of linear algebra. In Sect. 2 we outline our approach to the proof of Theorem 2. The actual proof falls naturally into two parts – one part covers the case where A∗ A is far from In and is the content of Sect. 3 and the other part the case where it is close to the identity, and is the content of Sect. 4. This last section requires some simple results on perturbation theory, and these are covered in Sect. 5. It should be remarked that no effort is made to estimate the constant Cn which appears in the statement in Theorem 2 – in order to have a sharp estimate, the methods of the current paper will need to be combined with detailed analysis of the structure of the orthogonal groups, and this will be the subject of another paper. Finally, a remark on notation: the measures on compact groups and spheres are normalized to be probability measures.
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1. Generalities on Matrices First, Definition 4. Let A ∈ Mm×n . Then the nonnegative square roots of the eigenvalues of the (positive semi-definite) matrix At A are called the singular values of A. We will denote individual singular values by σ1 ≥ · · · ≥ σm , and we shall denote the ordered m-tuple (σ1 , . . . , σm ) by (A). In this note we will only work with square matrices (although the results go through without change for non-square matrices), and our use of singular values will be localized to the following two lemmas: Lemma 5. Let A ∈ Mn×n , and let P ∈ O(n). Then (A) = (AP ) = (P A). Proof. Immediate.
Lemma 6. Let A ∈ Mn×n , and let f : R → R be such that f ( Au, Au ) dσn < ∞. I= Sn
Then
I=
Sn
f
n
σi2 (A)u2i
dσn .
i=1
Proof. It is sufficient to note that
Au, Au = ut At Au = (P u)t Diag 2 (A) P u,
where P is an orthogonal matrix (hence an isometry of Sn ). Corollary 7.
n 1 2 do log Budσn = log σi (A)ui dσn . 2 Sn A Sn i=1
Proof. Lemma 5 tells us that the inner integral does not depend on B ∈ A, and Lemma 6 tells us how to evaluate that integral in terms of singular values of A. Theorem-Definition 8 (Gershgorin’s Theorem). Let A ∈ Mn (C). Let ri = |aij |,
(1)
j =i
si =
|aj i |,
(2)
j =i
and let the row- and column- Gershgorin disks (respectively) be: Ri = {z ∈ C | |z − aii | ≤ ri }, Si = {z ∈ C | |z − aii | ≤ si }.
(3) (4)
Let now λ be an eigenvalue of A. Then there exists a k and an l such that λ ∈ Rk and λ ∈ Sl . Proof. See, eg, [2] or [7].
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2. A Lower Bound on Average Spectral Radius In this section we consider the following average: log ρ(AX)dX, A(A) = On
where ρ(M) denotes the spectral radius of the matrix M. The main theorem of the section will be the following estimate: Theorem 9. Let A ∈ SL(n, R), let n ≥ 3, and let σ1 (A) be the largest singular value of A. Then there exists a constant Cn > 0, such that A(A) ≥ Cn log σ1 (A).
(5)
Remark 10. Theorem 9 is false for n = 2 and trivially true for n = 1 (C1 = 1). To prove Theorem 9 we first make a couple of observations. First, Lemma 11. Let A ∈ Mn , let (A) be the singular values of A, and let ρ(X) be the spectral radius of the matrix X. Then f (ρ(AY ))dY = f (ρ (Diag (AY )) dY. On
On
Proof. Form the singular value decomposition of A, that is, write A = P1 Diag ((A))P2 , where P1 , P2 ∈ On . Then, ρ (AY ) = ρ (Diag (σ (A)) P2 Y P1 ) . The statement of the lemma follows immediately.
Lemma 11 tells us that in the statement of Theorem 9, it is enough to consider diagonal matrices A with positive entries. In the rest of this section we will labor under these assumptions. The strategy will be to first demonstrate a bound in any closed set of diagonal matrices not containing the identity matrix In – this will be done in Sect. 3 (particularly Corollary 14 – and then show a bound in some small neighborhood of the identity – this will be the content of Sect. 4. Our measure of the distance from the identity will be the quantity n
L(A) = max | log(σi (A)|. i=1
From n now on, A = Diag(d1 , . . . , dn ), with d1 ≥ d2 ≥ . . . dn > 0. We will assume that i=1 di = 1 when necessary – much of the time we will only use the fact that d1 > 1. 3. Away from Identity Theorem 12. Let A = Diag(d1 , . . . , dn ), with d1 ≥ d2 , . . . , ≥ dn ≥ 0, and let M1 ≤ 1. Let At = A(I + tM). Then, d1 (1 + 2t) ≥ ρ(At ) ≥ d1 (1 − 2nt).
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Proof. Since (At )ij = di (δij + tMij ), it follows that di (1 + t) ≥ |(At )ii | ≥ di (1 − t),
(6)
|ri (At )| ≤ di t.
(7)
while
Let G(At ) =
n
Ri (At ).
i=1
By a simple continuity argument (see, for example, [2] for a similar argument), the perturbation of the eigenvalue d1 of A stays in the connected component of the Gershgorin disk R1 (At ). Since by the above inequalities Eq. (6) and Eq. (7), min
z∈Ri (At )
|z| ≥ di (1 − 2t),
max |z| ≤ di (1 − 2t),
z∈Ri (At )
we see that after time t the connected component of R1 (At ) can only stay between the advertised bounds. Theorem 13. Let A = Diag(d1 , . . . , dn ). For any > 0, there is a constant c > 0, such that A(A) ≥ c (d1 − ). Proof. First, let t0 be so small that log(1 − 2nt0 ) > 1 − . Now let S0 = {x ∈ On and let
| x − I 1 < t0 },
c = Vol(S0 ).
Recalling that Vol On = 1, we see that A(A) ≥ log ρ(AX)dX ≥ c (d1 − ). S0
Corollary 14. Let D be the set of all diagonal matrices with elements d1 ≥ d2 ≥ · · · ≥ dn > 0, and let Dt be the set of those matrices M ∈ D with d1 (M) ≥ t > 1. Then there is a constant Ct > 0 such that A(A) > Ct , log(d1 (A)) for all A ∈ Dt . Proof. Let s = log(t)/2. Then, by Theorem 13, cs log d1 (A) , 2 where the last inequality holds for all A ∈ Dt . Now set Ct = cs /2. A(A) ≥ cs (log(d1 (A) − s) ≥
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4. Close to the Identity In this section we shall prove the following: Theorem 15. Let n ≥ 3. Let A = Diag(exp(d1 t), . . . , exp(dn t)), with d1 ≥ · · · ≥ dn , with max(|d1 |, |dn |) = 1, and n di = 0. i=1
Then there exist constants Cn and n , such that for t < n , A(A) ≥ Cn td1
(8)
for 0 < t ≤ n . Remark 16. The argument which follows can be made to give quite explicit lower bounds for both n and Cn . However, this does not seem useful, since it is quite clear that the bounds will not be close to the truth. It seems plausible that a combination of the methods of this note with a detailed understanding of the structure of the orthogonal group will give a tight result, but we shall not attempt to do this here. It should, on the other hand, be noted that not using any structural property of the orthogonal group means that the arguments in this note work just as well for cosets of unitary groups. To prove Theorem 15 we shall first need the following trivial but crucial observation (already used implicitly in the proof of Theorem 13): Observation 17. Let A ∈ SL(n, R), and let X ∈ On (R). Then ρ(AX) ≥ 1. Proof. det AX = 1.
Now, recall that A(A) =
log ρ(AX)dx. On
By the observation above, it follows that the integrand is everywhere nonnegative, and so to prove a lower bound such as that of Theorem 15 it will be enough to show the inequality (8) with A(A) replaced by log ρ(AX)dx. AS (A) = S⊆On
Let us pick S to be a small neighborhood around a matrix X0 ∈ On . Notice that all eigenvalues of X ∈ On have absolute value 1. We will choose X0 to have a simple eigenvalue λ0 , such that d log |λ0 (Diag(exp(td1 ), . . . , exp(tdn ))X0 )| ≥ C|d1 |, dt
(9)
for some constant C. By analyticity of λ0 (Theorem 18), inequality (9) suffices to show that A(A) ≥ Cn td1 ,
(10)
for t sufficiently small, and by Observation 17 this will suffice to show Theorem 15.
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4.1. Proof of inequality (9). First, we need to find a suitable orthogonal matrix X0 . There will be two cases, depending on the parity of n. The first case is that of odd n. In such a case, we pick X0 to be a rotation by 180 degrees around the axis e1 = (1, 0, . . . , 0), and our λ0 (X0 ) = 1. Its eigenvector is vλ0 = e1 . By Lemma 20, dλ(x) = λ(x)d1 , dx and we are done. The second case is that of n even. In that case, we pick X0 to fix the span of e3 , . . . , en and to rotate the span of e1 and e2 by 90 degrees. In this case, λ0 (X0 ) = i, and the eigenvector of λ0 is 1 vλ0 = √ (1, i, 0, . . . , 0). 2 By Lemma 20, 1 dλ(x) = λ(x) √ (d1 + d2 ). dx 2 If d2 > 0, we are done, since d1 + d2 > d1 . Since n
di = 0,
i=1
it follows that d1 = −
n
di =
i=2
n
|di |.
i=2
Since |dn | ≥ |dn−1 | ≥ |d2 |, it follows that d1 ≥ (n − 1)|d2 |, and so d1 + d2 ≥
n−2 d1 , n−1
so as long as n > 2 we are done. Notice that the argument breaks down when n = 2, and in fact, in that case the result is false. 5. Some Perturbation Theory In this section we will recall some formulas of perturbation theory (an exhaustive treatment can be found in the classic [8]), and derive some estimates needed in the rest of this paper. 2
Theorem 18. Let T be an n × n matrix (thought of as a point in Cn ) and let λ be a simple eigenvalue of T . Then there is a neighborhood of T where λ is a holomorphic function of T . In particular, λ is C ∞ in a neighborhood of T . Proof. This follows from generalities on analyticity of (simple) roots of polynomials as functions of their coefficients. See [8] for some further details.
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Theorem 19. Let λ be a simple eigenvalue of T (0). Then
dT (x) dλ = tr Pλ , dx x=0 dx x=0 where Pλ is the orthogonal projection onto the eigenspace of λ. Proof. See [8, p. 79].
Lemma 20. Let T (x) = Diag(exp(d1 x), . . . , exp(dn x))T , and let λ(x) be a simple eigenvalue of T (x). Let the unit eigenvector of λ be vλ = (v1 , . . . , vn ). Then dλ(x) di |vi |2 . = λ(x) dx n
i=1
In particular, the logarithmic derivative of λ(x) is real. Proof. Use Theorem 19 and the observation that T (x)Pλ(x) = λ(x)Pλ(x) , to write
dλ(x) = tr Diag(d1 , . . . , dn )Pλ(x) . dx Now, note that (Pλ(x) )ij = vi vj . Acknowledgements. The author would like to thank Amie Wilkinson for bringing the question to his attention.
References 1. Avila, A., Bochi, J.: A formula with some applications to the theory of Lyapunov exponents. Israel J. Math. 131, 125–137 (2002) 2. Brualdi, R., Mellendorf, S.: Regions in the complex plane containing the eigenvalues of a matrix. Am. Math. Monthly 10, 975–985 (1994) 3. Burns, K., Pugh, C., Shub, M., Wilkinson, A.: Recent results about stable ergodicity. Proc. Symposia AMS 69, 327–366 (2001) 4. Dedieu, J-P., Shub, M.: On random and mean exponents for unitarily invariant and probability measures on GL(n, C). Technical report, IBM, 2001. http://www.research.ibm.com/people/s/shub/ Dedieu-Shub.pdf, 2001 5. Ledrappier, F., Simo, C., Shub, M., Wilkinson, A.: Random versus deterministic exponents in a rich family of diffeomorphisms. J. Stat. Phys. to appear 6. Hardy, H.G., Littlewood, J., Polya, G.: Inequalities. 2nd Edition, Cambridge: Cambridge University Press, 1988 7. Horn, R., Johnson, C.R.: Matrix Analysis. Corrected reprint, Cambridge: Cambridge University Press, England, 1990 8. Kato, T.: Perturbation Theory for Linear Operators. Reprint of the 1980 edition, Berlin HeidelbergNew York: Springer, 1995 Communicated by P. Sarnak
Commun. Math. Phys. 254, 659–694 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1245-x
Communications in
Mathematical Physics
Freely Generated Vertex Algebras and Non–Linear Lie Conformal Algebras Alberto De Sole1 , Victor G. Kac2 1 2
Department of Mathematics, Harvard University, 1 Oxford st, Cambridge, MA 02138, USA. E-mail: [email protected] Department of Mathematics, MIT 77 Massachusetts Av, Cambridge, MA 02139, USA. E-mail: [email protected]
Received: 12 December 2003 / Accepted: 22 June 2004 Published online: 11 January 2005 – © Springer-Verlag 2005
Abstract: We introduce the notion of a non–linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a unique, up to isomorphism, non–linear Lie conformal superalgebra. This correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras. 1. Introduction After the work of Zamolodchikov [Zam85] it has become clear that the chiral (= vertex) algebra of a conformal field theory gives rise to a Lie algebra with non–linearities in commutation relations. This and the subsequent works in the area clearly demonstrated that the absence of non–linearities, like in the Virasoro algebra, the current algebras and their super analogues, is an exception, rather than a rule. This is closely related to the fact that in the singular parts of the operator product expansions of generating fields, as a rule, not only linear combinations of these fields and their derivatives occur, but also their normally ordered products. In fact, the absence of terms with normally ordered products is equivalent to the absence of non–linearities in the commutation relations. The latter case is encoded in the notion of a Lie conformal superalgebra [Kac96], and a complete classification of finite simple Lie conformal superalgebras was given in [FK02] (completing thereby a sequence of works that began with [RS76] and continued in the conformal algebra framework in [Kac97, DK98, FK02]). A complete classification in the general non–linear case is much harder and is still far away. In the present paper we lay rigorous grounds to the problem by introducing the notion of a non–linear conformal superalgebra. In order to explain the idea, let us define this notion in the quantum mechanical setting. Let g be a vector space, endowed with a gradation g = ⊕j gj , by a discrete subsemigroup
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of the additive semigroup of positive real numbers, and with a linear map [, ] : g ∧ g → T (g), where T (g) is the tensor algebra over g. We write (a) = j if a ∈ gj , and we extend the gradation to T (g) by additivity, i.e. (1) = 0, (A ⊗ B) = (A) + (B) for A, B ∈ T (g), and the bracket [, ] to T (g) ∧ T (g) → T (g) by the Leibniz rule. We write (a) < if (aj ) < for all non-zero homogeneous summands aj of a. Let b, c ∈ g, A, D ∈ T (g) Mj (g) = span A ⊗ (b ⊗ c − c ⊗ b − [b, c]) ⊗ D (A ⊗ b ⊗ c ⊗ D)) ≤ j and let M(g) = ∪j Mj (g). Note that M(g) is the two sided ideal of the tensor algebra generated by elements a ⊗ b − b ⊗ a − [a, b], where a, b ∈ g. The map [, ] is called a non–linear Lie algebra if it satisfies the following two properties (a, b, c are homogeneous elements of g): (grading condition) ([a, b]) < (a) + (b), (Jacobi identity) [a, [b, c]] − [b, [a, c]] − [[a, b], c] ∈ M (g), where < (a) + (b) + (c). Let U (g) = T (g)/M(g). It is not difficult to show by the usual method (see e.g. [Jac62]) that the PBW theorem holds for the associative algebra U (g), i.e. the images of all ordered monomials in an ordered basis of g form a basis of U (g). The purpose of the present paper is to introduce the notion of a non–linear Lie conformal (super)algebra R, which encodes the singular part of general OPE of fields, in order to construct the corresponding universal enveloping vertex algebra U (R), and to establish for U (R) a PBW theorem. The main difficulty here, as compared to the Lie (super)algebra case, comes from the fact that U (R) = T (R)/M(R), where M(R) is not a two–sided ideal of the associative algebra T (R). This makes the proof of the PBW theorem much more difficult. Of course, in the “linear” case these results are well known [Kac96, GMS04, BK03]. Another important result of the paper is a converse theorem, which states that any graded vertex algebra satisfying the PBW theorem is actually the universal enveloping vertex algebra of a non–linear Lie conformal superalgebra. The special case when the vertex algebra is freely generated by a Virasoro field and primary fields of conformal weight 1 and 3/2 was studied in [DS03]. Here is a brief outline of the contents of the paper. After reviewing in Sect. 2 the definition of a Lie conformal algebra and a vertex algebra, we introduce in Sect. 3 the definition of a non–linear Lie conformal algebra R. We also state the main result of the paper, a PBW theorem for the universal enveloping vertex algebra U (R), which is proved in Sects. 4 and 5. In Sect. 6 we state and prove the converse statement, namely to every graded freely generated vertex algebra V we associate a unique, up to isomorphism, non–linear Lie conformal algebra R such that V U (R). The results of this paper have important applications to the classification problem of conformal vertex algebras. Such applications will be discussed in subsequent work. 2. Definitions of Lie Conformal Algebras and Vertex Algebras Definition 2.1. A Lie conformal superalgebra (see [Kac96]) is a Z/2Z–graded C[T ]module R = R0¯ ⊕ R1¯ endowed with a C-linear map R ⊗ R → C[λ] ⊗ R denoted by a ⊗ b → [a λ b] and called λ-bracket, satisfying the following axioms:
Freely Generated Vertex Algebras and Non-linear Lie Conformal Algebras
[T a λ b] = −λ[a λ b],
[a λ T b] = (λ + T )[a λ b]
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(sesquilinearity),
[b λ a] = −p(a, b)[a −λ−T b]
(skewsymmetry),
[a λ [b µ c]] − p(a, b)[b µ [a λ c]] = [[a λ b] λ+µ c]
(Jacobi identity),
for a, b, c ∈ R. Here and further p(a, b) = (−1)p(a)p(b) , where p(a) ∈ Z/2Z is the parity of a, and ⊗ stands for the tensor product of vector spaces over C. In the skewsymmetry relation, [a −λ−T b] means that we have to replace in [a λ b] the indeterminate λ with the operator (−λ − T ), acting from the left. For brevity, we will often drop the prefix super in superalgebra or superspace. For a Lie conformal algebra we can define a C-bilinear product R ⊗ R → R for any n ∈ Z+ = {0, 1, 2, · · · }, denoted by a ⊗ b → a(n) b and given by [a λ b] =
λ(n) a(n) b,
(2.1)
n∈Z+ n
where we are using the notation: λ(n) := λn! . Vertex algebras can be thought of as a special class of Lie conformal superalgebras. Definition 2.2. A vertex algebra is a pair (V , |0 ), where V is a Z2 –graded C[T ]– module, called the space of states, and |0 is an element of V , called the vacuum state, endowed with two parity preserving operations: a λ–bracket V ⊗V → C[λ]⊗V which makes it a Lie conformal superalgebra, and a normally ordered product V ⊗ V → V , denoted by a ⊗ b →: ab :, which makes it a (not necessarily associative nor commutative) unital differential algebra with unity |0 and derivative T . They satisfy the following axioms 1. quasi–associativity
T dλ a [b c] : : (: ab :)c : − : a(: bc :) : = : λ 0 T dλ b [a c] :, + p(a, b) : λ 0
2. skewsymmetry of the normally ordered product : ab : −p(a, b) : ba : =
0
−T
dλ[a λ b],
3. non-commutative Wick formula [a λ : bc :] = : [a λ b]c : +p(a, b) : b[a λ c] : +
0
λ
dµ[[a λ b] µ c].
In quasi–associativity, the first (and similarly the second) integral in the right-hand side should be understood as : (T (n+1) a)(b(n) c) : . n≥0
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An equivalent and more familiar definition of vertex algebras can be given in terms of local fields, [FLM88, FHL93, DL93, Kac96]. A proof of the equivalence between the above definition of a vertex algebra and the one given in [Kac96] can be found in [BK03]. It also follows from [Kac96, Li96] that these definitions are equivalent to the original one of [Bor86]. Remark 2.3. Since |0 is the unit element of the differential algebra (V , T ), we have T |0 = 0. Moreover, by sesquilinearity of the λ–bracket, it is easy to prove that the torsion of the C[T ]–module V is central with respect to the λ–bracket; namely, if P (T )a = 0 for some P (T ) ∈ C[T ]\{0}, then [a λ b] = 0 for all b ∈ V . In particular the vacuum element is central: [a λ |0 ] = [|0 λ a] = 0, ∀a ∈ V . Remark 2.4. Using skewsymmetry (of both the λ–bracket and the normally ordered product) and non commutative Wick formula, one can prove the right Wick formula [BK03]
[: ab : λ c] = : eT ∂λ a [b λ c] : +p(a, b) : eT ∂λ b [a λ c] :
+ p(a, b)
λ dµ[b µ 0
[a λ−µ c]].
For a vertex algebra V we can define nth products for every n ∈ Z in the following way. Since V is a Lie conformal algebra, all nth products with n ≥ 0 are already defined by (2.1). For n ≤ −1 we define nth product as a(n) b = : (T (−n−1) a)b : .
(2.2)
By definition every vertex algebra is a Lie conformal algebra. On the other hand, given a Lie conformal algebra R there is a canonical way to construct a vertex algebra which contains R as a Lie conformal subalgebra. This result is stated in the following Theorem 2.5. Let R be a Lie conformal superalgebra with λ–bracket [a λ b]. Let RLie be R considered as a Lie superalgebra over C with respect to the Lie bracket: 0 dλ[a λ b], a, b ∈ R, [a, b] = −T
and let V = U (RLie ) be its universal enveloping superalgebra. Then there exists a unique structure of a vertex superalgebra on V such that the restriction of the λ–bracket to RLie × RLie coincides with the λ–bracket on R and the restriction of the normally ordered product to RLie × V coincides with the associative product of U (RLie ). The vertex algebra thus obtained is denoted by U (R) and is called the universal enveloping vertex algebra of R. For a proof of this theorem, see [Kac96, GMS04], [BK03, Th 7.12], [Primc99]. In the remainder of this section we introduce some definitions which will be used in the sequel. Definition 2.6. Let R be a (Z/2Z graded) C[T ]–submodule of a vertex algebra V . Choose a basis A = {ai , i ∈ I} of R compatible with the Z/2Z gradation, where I is an ordered set. 1. One says that V is strongly generated by R if the normally ordered products of elements of A span V .
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2. One says that V is freely generated by R if the set of elements i ≤ i2 ≤ · · · ≤ in , B = : ai1 . . . ain : 1 ik < ik+1 if p(aik ) = 1¯ forms a basis of V over C. In this case V is called a freely generated vertex algebra. The normally ordered product of more than two elements is defined by taking products from right to left. In other words, for elements a, b, c, · · · ∈ V , we will denote : abc · · · : = : a(: b(: c · · · :) :) :. Also, by convention, the empty normal ordered product is |0 , and it is included in B. For example, the universal enveloping vertex algebra U (R) of a Lie conformal algebra R is freely generated by R, by Theorem 2.5. From now on, fix an ordered abelian semigroup with a zero element 0, such that for every ∈ there are only finitely many elements ∈ such that < . The most important example is a discrete subset of R+ containing 0 and closed under addition. Definition 2.7. By a –gradation of a vector space U we mean a decomposition U = U [], ∈\{0}
in a direct sum of subspaces labelled by \{0}. If a ∈ U [], we call the degree of a, and we will denote it by (a) or a . By convention 0 ∈ U is of any degree. The associated –filtration of U is defined by letting U [ ]. U = ≤
Let T (U ) be the tensor algebra over U , namely T (U ) = C ⊕ U ⊕ U ⊗2 ⊕ · · · . A –gradation of U extends to a –gradation of T (U ), T (U ) = T (U )[], ∈
by letting (1) = 0, (A ⊗ B) = (A) + (B), ∀A, b ∈ T (U ). The induced –filtration is denoted by T (U ) =
T (U )[ ].
≤
If moreover U is a C[T ]–module, we extend the action of T to T (U ) by Leibniz rule: T (1) = 0, T (A ⊗ B) = T (A) ⊗ B + A ⊗ T (B), ∀A, B ∈ T (U ).
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3. Non–linear Lie Conformal Superalgebras and Corresponding Universal Enveloping Vertex Algebras In the previous section we have seen that, to every Lie conformal algebra R, we can associate a universal enveloping vertex algebra U (R), freely generated by R. In general, though, it is not true that a freely generated vertex algebra V is obtained as the universal enveloping vertex algebra of a finite Lie conformal algebra. In fact the generating set R ⊂ V needs not be closed under the λ–bracket. In this section we will introduce the notion of non–linear Lie conformal superalgebra, which is “categorically equivalent” to that of the freely generated vertex algebra: to every non–linear Lie conformal algebra R we will associate a universal enveloping vertex algebra U (R), freely generated by R. Conversely, let V be a vertex algebra freely generated by a free C[T ]–module R satisfying a “grading condition”; we will show that R is a non–linear Lie conformal algebra and V is isomorphic to the universal enveloping vertex algebra U (R). Definition 3.1. A non–linear conformal superalgebra R is a C[T ]–module which is \{0}–graded by C[T ]–submodules: R = ⊕∈\{0} R[], and is endowed with a λ– bracket, namely a parity preserving C–linear map [ λ ] : R ⊗ R −→ C[λ] ⊗ T (R), such that the following conditions hold 1. sesquilinearity [T a λ b] = −λ[a λ b], [a λ T b] = (λ + T )[a λ b]. 2. grading condition ([a λ b]) < (a) + (b). The grading condition can be expressed in terms of the –filtration of T (R) (see Definition 2.7) in the following way: [R[1 ] λ R[2 ]] ⊂ C[λ] ⊗ T (R),
for some < 1 + 2 .
Note also that T (R)[0] = C and T (R)[δ] = R[δ], where δ is the smallest non–zero element of such that R[δ] = 0. This will be important for induction arguments. The following lemma allows us to introduce the normally ordered product on T (R) and to extend the λ–bracket to the whole tensor algebra T (R), by using quasi–associativity and the left and right Wick formulas. Lemma 3.2. Let R be a non–linear conformal algebra. A) There exist unique linear maps N : T (R) ⊗ T (R) −→ T (R), Lλ : T (R) ⊗ T (R) −→ C[λ] ⊗ T (R),
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such that, for a, b, c, · · · ∈ R, A, B, C, · · · ∈ T (R), the following conditions hold N(1, A) = N (A, 1) = A,
(3.1)
N (a, B) = a ⊗ B,
(3.2)
dλ a , Lλ (B, C))
T
N (a ⊗ B, C) = N (a, N (B, C)) + N (
0
+p(a, b)N (
T
0
dλ B Lλ (a, C)),
(3.3)
Lλ (1, A) = Lλ (A, 1) = 0,
(3.4)
Lλ (a, b) = [a λ b],
(3.5)
Lλ (a, b ⊗ C) = N (Lλ (a, b), C) + p(a, b)N(b, Lλ (a, C))
λ
+ 0
(3.6)
dµ Lµ (Lλ (a, b), C),
Lλ (a ⊗ B, C) = N ( eT ∂λ a , Lλ (B, C)) + p(a, B)N ( eT ∂λ B , Lλ (a, C)) +p(a, B)
λ
dµ Lµ (B, Lλ−µ (a, C)).
0
(3.7)
B) These linear maps satisfy the following grading conditions (A, B ∈ T (R)): (N (A, B)) ≤ (A) + (B), (Lλ (A, B)) < (A) + (B).
(3.8)
Proof. We want to prove, by induction on = (A) + (B), that N (A, B) and Lλ (A, B) exist and are uniquely defined by Eqs. (3.1–3.7) and that they satisfy the grading conditions (3.8). Let us first look at N (A, B). If A ∈ C or A ∈ R, N (A, B) is defined by conditions (3.1) and (3.2). Suppose then A = a ⊗ A , where a ∈ R and A ∈ R ⊕ R ⊗2 ⊕ · · · . Notice that, in this case, (a) < (A), (A ) < (A). By condition (3.3) we have
T
N(A, B) = a ⊗ N (A , B) + N (
0
T
+p(a, A )N (
0
dλ a , Lλ (A , B))
dλ A Lλ (a, B),
and, by the inductive assumption, each term in the right-hand side is uniquely defined and is in T (R). Consider then Lλ (A, B). If either A ∈ C or B ∈ C we have, by (3.4), Lλ (A, B) = 0. Moreover, if A, B ∈ R, Lλ (A, B) is defined by (3.5). Suppose now A = a ∈ R, B = b ⊗ B , with b ∈ R and B ∈ R ⊕ R ⊗2 · · · . In this case we have, by (3.6)
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Lλ (A, B) = N (Lλ (a, b), B ) + p(a, b)N (b, Lλ (a, C)) λ + dµ Lµ (Lλ (a, b), B ), 0
and each term in the right-hand side is well defined by induction, and it lies in T (R), for some < . Finally, consider the case A = a ⊗ A , with a ∈ R and A , B ∈ R ⊕ R ⊗2 ⊕ · · · . In this case we have, by (3.7), Lλ (A, B) = N ( eT ∂λ a , Lλ (A , B)) + p(a, A )N ( eT ∂λ A , Lλ (a, B)) λ +p(a, A ) dµ Lµ (A , Lλ−µ (a, B)), 0
and as before each term in the right-hand side is defined by induction and lies in T (R), for some < . Definition 3.3. Let R be a non–linear conformal algebra. For ∈ , we define the subspace M (R) ⊂ T (R) by A ⊗ (b ⊗ c − p(b, c)c ⊗ b) ⊗ D b, c ∈ R, A, D ∈ T (R), M (R) = spanC . −N ( 0 dλ Lλ (b, c) , D) A⊗b⊗c⊗D ∈ T (R) −T Furthermore, we define M(R) =
M (R).
∈
Remark 3.4. By definition we clearly have M (R) ⊂ M(R) ∩ T (R). But in general M (R) needs not be equal to M(R) ∩ T (R). In particular, if δ is the smallest non–zero element of such that R[δ] = 0, then Mδ (R) = 0, since, for b, c ∈ R, (b) + (c) ≥ 2δ, whereas M(R) ∩ Tδ (R) can be a priori non-zero. Definition 3.5. A non–linear Lie conformal algebra is a non–linear conformal algebra R such that the λ–bracket [ λ ] satisfies the following additional axioms: 1. skewsymmetry [a λ b] = −p(a, b)[b −λ−T a],
∀a, b ∈ R,
2. Jacobi identity Lλ (a, Lµ (b, c)) − p(a, b)Lµ (b, Lλ (a, c))
(3.9)
−Lλ+µ (Lλ (a, b), c) ∈ C[λ, µ] ⊗ M (R), for every a, b, c ∈ R and for some ∈ such that < (a) + (b) + (c). Definition 3.6. A homomorphism of non–linear Lie conformal algebras φ : R → R is a C[T ]–module homomorphism preserving the Z/2Z and gradations such that the induced map T (R) → T (R ) is a homomorphism for the operations N and Lλ modulo M(R ) in the following sense: φ(N (A, B)) − N (φ(A), φ(B)) ∈ M (R ) for ≤ (A) + (B), φ(Lλ (A, B)) − L λ (φ(A), φ(B)) ∈ C[λ] ⊗ M (R ) for < (A) + (B).
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Remark 3.7. Given a λ–bracket on R, consider the map L : T (R) ⊗ T (R) → T (R) given by 0 dλ Lλ (A, B). L(A, B) = −T
It is immediate to show that, if R is a non–linear Lie conformal algebra, then L satisfies skewsymmetry L(a, b) = −p(a, b)L(b, a), and the Jacobi identity L(a, L(b, c)) − p(a, b)L(b, L(a, c)) − L(L(a, b), c) ∈ M (R), for every a, b, c ∈ R and for some ∈ such that < (a) + (b) + (c). Thus we obtain a non–linear Lie superalgebra, discussed in the introduction. Remark 3.8. In the definition of non–linear Lie conformal algebra we could write the skew–symmetry axiom in the weaker form: [a λ b] + p(a, b)[b −λ−T a] ∈ M (R),
for some < (a) + (b). In this way we would get a similar notion of non–linear Lie conformal algebra, which, as it will be clear from the results in Sect. 6, is not more useful than that given by Definition 3.5, from the point of view of classification of vertex algebras. We are now ready to state the main results of this paper. Theorem 3.9. Let R be a non–linear Lie conformal algebra. Consider the vector space U (R) = T (R)/M(R), and denote by π : T (R) → U (R) the quotient map, and by : : the image in U (R) of the tensor product of elements of R: : ab · · · c : = π(a ⊗ b ⊗ · · · ⊗ c),
∀a, b, · · · , c ∈ R.
Let A = {ai , i ∈ I} be an ordered basis of R compatible with the Z/2Z–gradation and the –gradation. We denote by B the image in U (R) of the collection of all ordered monomials, namely : ai1 . . . ain : i1 ≤ i2 ≤ · · · ≤ in , B= ⊂ U (R). and ik < ik+1 if p(aik ) = 1¯ Then 1. B is a basis of the vector space U (R). In particular we have a natural embedding ∼
R → π(R) ⊂ U (R). 2. There is a canonical structure of a vertex algebra on U (R), called the universal enveloping vertex algebra of R, such that the vacuum vector |0 is π(1), the infinitesimal translation operator T : U (R) → U (R) is induced by the action of T on T (R): T (π(A)) = π(T A), ∀A ∈ T (R), the normally ordered product on U (R) is induced by N : : π(A)π(B) : = π(N (A, B)),
∀A, B ∈ T (R),
and the λ–bracket [ λ ] : U (R) ⊗ U (R) → C[λ]U (R) is induced by Lλ : [π(A) λ π(B)] = π(Lλ (A, B)),
∀A, B ∈ T (R).
In other words, U (R) is a vertex algebra freely generated by R.
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The above notion of a homomorphism of non–linear Lie conformal algebras corresponds to that of homomorphism of the corresponding universal enveloping vertex algebras. Namely we have the following. Proposition 3.10. Let φ : R → R be a homomorphism of non-linear Lie conformal algebras. (i) If we extend φ to a homomorphism of the associative tensor algebras T (R) → T (R ), we have φ(M (R)) ⊂ M (R ). (ii) φ induces a homomorphism of the universal enveloping vertex algebras: φ : U (R) → U (R ). Proof. We have 0 φ A ⊗ b ⊗ c ⊗ D − p(b, c)c ⊗ b ⊗ D − N dλLλ (b, c) , D −T = φ(A) ⊗ φ(b) ⊗ φ(c) ⊗ φ(D) − p(b, c)φ(c) ⊗ φ(b) ⊗ (D) −N
−φ(A) ⊗ φ N −φ(A) ⊗ N
0
−T 0
−T
dλ
, φ(D)
dλLλ (b, c) , D − N
−T 0
dλL λ (φ(b), φ(c))
0 −T
dλφ(Lλ (b, c)) , φ(D)
φ(Lλ (b, c)) − L λ (φ(b), φ(c))
, φ(D)
All three terms in the right hand side belong to M (R ) by the assumptions on φ and the results in the next section (see, in particular, Corollary 4.5). The second part of the proposition is obvious. The vertex algebra U (R) has the following universality property: Proposition 3.11. Let φ be a homomorphism from a non–linear Lie conformal algebra R to a vertex algebra V , namely a parity preserving C[T ]- module homomorphism φ : R → V such that, if we extend it to a linear map φ : T (R) → V by letting φ(a1 ⊗ · · · ⊗ as ) =: φ(a1 ) . . . φ(as ):, we have φ(Lλ (a, b)) = [φ(a)λ φ(b)],
∀a, b, ∈ R.
Then: (a) φ(N (A, B)) =: φ(A)φ(B) :, φ(Lλ (A, B)) = [φ(A)λ φ(B)], ∀A, B ∈ T (R). (b) φ extends to a unique vertex algebra homomorphism φ : U (R) → V . Proof. (a) is proved by an easy induction on degree of T (R) and (b) follows from (a). Remark 3.12. Notice that the universal enveloping vertex algebra U (R) is independent of the choice of the grading of R.
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Example 3.13. Suppose Rˆ = R ⊕ C|0 , where C|0 is a torsion submodule, is a Lie conformal algebra. In this case, if we let = Z+ and assign degree = 1 to every element of R, we make R into a non–linear Lie conformal algebra. In this case we have a Lie algebra structure on R defined by the Lie bracket (recall that C|0 is in the center of the λ–bracket) 0 [a, b] = dλ[a λ b], ∀a, b ∈ R. −T
Therefore the space U (R) coincides with the universal enveloping algebra of R (viewed as a Lie algebra with respect to [, ]), and the first part of Theorem 3.9 is the PBW theorem for ordinary Lie superalgebras. Moreover in this case U (R) simply coincides ˆ defined by Theorem with the quotient of the universal enveloping vertex algebra of R, ˆ 2.5, by the ideal generated by |0 − 1. In particular, if R is the Virasoro Lie conformal c 3 algebra [L λ L] = (T + 2λ)L + 12 λ |0 , then U (R) is the universal Virasoro vertex algebra with central charge c. Conversely, suppose R is a non–linear Lie conformal algebra graded by = Z+ and such that every element of R has degree = 1. It follows by the grading condition on [ λ ] that Rˆ = R ⊕ C is a Lie conformal algebra, and therefore Theorem 3.9 holds by the above considerations. Example 3.14. The first example of a “genuinely” non–linear Lie conformal algebra R is Zamolodchikov’s W3 –algebra [Zam85], which, in our language, is W3 = C[T ]L + C[T ]W, where (L) = 2, (W ) = 3, and the λ–brackets are as follows: c [L λ L] = (T + 2λ)L + λ3 , [L λ W ] = (T + 3λ)W, 12 16 c − 10 [W λ W ] = (T + 2λ) (L ⊗ L) + T 2L 22 + 5c 3(22 + 5c) 1 c 5 + λ(T + λ)L + λ . 6 360
(3.10)
We want to prove that this is the unique, up to isomorphisms, non–linear Lie conformal algebra generated by two elements L and W , such that L is a Virasoro element of central charge c and W is an even primary element of conformal weight 3. We assume that [W λ W ] ∈ C[λ]1. We only need to show that [W λ W ] has to be as in (3.10). By simple conformal weight considerations (see Example 6.2) and by skewsymmetry of the λ–bracket, the most general form of [W λ W ] is [W
λ
W ] = (T + 2λ) α(L ⊗ L) + βT W + γ T 2 L + δλ(T + λ)L + λ5 ,
for α, β, γ , δ, ∈ C. We need to find all possible values of these parameters such that the Jacobi identity (3.9) holds. We may assume α = 0, since otherwise W3 ⊕ C|0 is a “linear” Lie conformal algebra, and from their classification [DK98] it follows that [Wλ W ] ∈ C[λ]1, which is not allowed. There are only two non-trivial Jacobi identities
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(3.9), namely the ones involving two or three W elements. We will show how to deal with the first, and we will leave the second to the reader. We need to impose [W
λ
[W
µ
L]] − [W
µ
[W
λ
L]] ≡ [[W
λ
mod C[λ, µ] ⊗ M7 (R). (3.11)
W ] λ+µ L]
Using the axioms of non–linear conformal algebra, we can compute every term of Eq. (3.11). The first term of the left-hand side is (2T + 2λ + 3µ)(T + 2λ) αL ⊗ L + βT W + γ T 2 λ + δλ(T + λ)L + (2T + 2λ + 3µ) λ5 , (3.12) and the second term is obtained by replacing λ by µ. The right-hand side of (3.11) is (λ − µ) 2αL ⊗ T L + 4αT L ⊗ L + 4α(λ + µ)L ⊗ L + 16 αc(T + λ + µ)3 L 1 +α 43 (λ + µ)3 + 25 (λ + µ)2 T + (λ + µ)T 2 L + 40 αc(λ + µ)5 −β(λ + µ)(2T + 3λ + 3µ)W +(γ (λ + µ)2 − δλµ)((T + 2λ + 2µ)L +
(3.13)
c 3 12 (λ + µ) )
.
If d = 0, (3.11) can’t be an exact identity, but only an identity modulo M (R). Indeed in (3.13) we have a term 2αL ⊗ T L, which does not appear in (3.12). But we can replace in (3.13) 1 T L ⊗ L − L ⊗ T L by − T 3 L mod M4 (R). 6 After this substitution, both sides of (3.11) are linear combinations of L ⊗ L, L, W, 1, with coefficients in C[λ, µ, T ]. Hence by Theorem 3.9 all these coefficients are zero. It is then a simple exercise to show that the only solution of (3.11) is given, up to rescaling of W , by choosing the parameters α, β, γ , δ, as in (3.10). After this, one checks that the Jacobi identity involving three W elements is automatic. A large class of non–linear Lie conformal (super)algebras is obtained by quantum Hamiltonian reduction attached to a simple Lie (super)algebra g and a nilpotent orbit in g (see e.g. [FF90, KW04, dBT94]). In particular, the Virasoro algebra and the W3 – algebra are obtained by making use of the principal nilpotent orbit of g = sl2 and sl3 respectively. The next two sections will be devoted to the proof Theorem 3.9. In the next section we will prove some technical results, and in the following section we will complete the proof of Theorem 3.9.
4. Technical Results Let R be a non–linear Lie conformal algebra. For convenience, we introduce the following notations which will be used throughout the following sections (A, B, C · · · ∈
Freely Generated Vertex Algebras and Non-linear Lie Conformal Algebras
671
T (R)): sl(A, B; λ) = Lλ (A, B) + p(A, B)L−λ−T (B, A), sn(A, B, C) = N (A, N (B, C)) − p(A, B)N (B, N (A, C)) 0 −N ( −T dλ Lλ (A, B) , C), λ Wl (A, B, C; λ) = Lλ (A, N (B, C)) − 0 dµ Lµ (Lλ (A, B), C) Wr (A, B, C; λ)
−N (Lλ (A, B), C) − p(A, B)N (B, Lλ (A, C)), λ = Lλ (N (A, B), C) − p(A, B) 0 dµ Lµ (B, Lλ−µ (A, C)) −N ( eT ∂λ A , Lλ (B, C)) − p(A, B)N ( eT ∂λ B , Lλ (A, C)),
Q(A, B, C) = N (N (A, B), C) − N (A, N (B, C)) T −N ( 0 dλ A , Lλ (B, C)) T −p(A, B)N( 0 dλ B , Lλ (A, C)), J(A, B, C; λ, µ) = Lλ (A, Lµ (B, C)) − p(A, B)Lµ (B, Lλ (A, C)) −Lλ+µ (Lλ (A, B), C). Remark 4.1. Using the above notation, we can write in a more concise form all the definitions introduced in the previous section. The maps N : T (R) ⊗ T (R) → T (R) and Lλ : T (R) ⊗ T (R) → C[λ] ⊗ T (R) are defined respectively by the equations Q(a, B, C) = 0,
Wl (a, b, C; λ) = 0,
Wr (a, B, C; λ) = 0,
for all a, b ∈ R and B, C ∈ T (R) . The axioms of non–linear Lie conformal algebras can be written as sl(a, b ; λ) = 0,
J(a, b, c ; λ, µ) ∈ C[λ, µ] ⊗ M (R),
for all a, b, c ∈ R and for some < (a) + (b) + (c). Finally, the subspaces M (R) ⊂ T (R), ∈ , are defined as A, D ∈ T (R), b, c ∈ R, , M (R) = spanC A ⊗ sn(b, c, D) (A) + (b) + (c) + (D) ≤ and, by definition, M(R) = ∪∈ M (R). Lemma 4.2. The endomorphism T : T (R) → T (R) is a derivation of the product N : T (R) ⊗ T (R) → T (R), namely, for A, B ∈ T (R), T N (A, B) = N (T A, B) + N (A, T B).
(4.1)
Furthermore, the λ–bracket Lλ : T (R) ⊗ T (R) → C[λ] ⊗ T (R) satisfies sesquilinearity, namely, for A, B ∈ T (R),
672
A. De Sole, V.G. Kac
Lλ (T A, B) = −λLλ (A, B), Lλ (A, T B) = (λ + T )Lλ (A, B).
(4.2) (4.3)
In particular T is a derivation of Lλ . Proof. We will prove both statements of the lemma by induction on = (A) + (B). If = 0, we have A, B ∈ C, and there is nothing to prove. Suppose then > 0. For A ∈ C, all three Eqs. (4.1), (4.2) and (4.3) are obviously satisfied. We will consider separately two cases: 1. A = a ∈ R, 2. A = a ⊗ A , with a ∈ R, A ∈ R ⊕ R ⊗2 ⊕ · · · ⊂ T (R). Case 1. For A ∈ R, Eq. (4.1) follows by the fact that T is a derivation of the tensor product and that N (a, B) = a ⊗ B. Consider then Eqs. (4.2) and (4.3). For B ∈ C they are trivial, and for B ∈ R they hold by (3.5). Suppose then B = b ⊗ B , with b ∈ R and B ∈ R ⊕ R ⊗2 ⊕ · · · . By (3.6) we have Lλ (T a, B) = N (Lλ (T a, b), B ) + p(a, b)N (b, Lλ (T a, B )) λ + dµ Lµ (Lλ (T a, b), B ) = −λLλ (a, B). 0
In the last identity we used the inductive assumption and the grading conditions (3.8). Similarly we have Lλ (a, T B) = Lλ (a, T b ⊗ B ) + Lλ (a, b ⊗ T B ) = N (Lλ (a, T b), B ) + p(a, b)N (T b, Lλ (a, B )) λ + dµ Lµ (Lλ (a, T b), B ) 0
+N (Lλ (a, b), T B ) + p(a, b)N (b, Lλ (a, T B )) λ + dµ Lµ (Lλ (a, b), T B ) = (λ + T )Lλ (a, B). 0
Case 2. By Eq. (3.3), the inductive assumption and the grading conditions (3.8) we have T
T N(A, B) = T N(a, N (A , B)) + N ( dλ a , Lλ (A , B)) T 0 +p(a, A )N ( dλ A , Lλ (a, B)) = N (T A, B) + N (A, T B), 0
which proves (4.1). Similarly by Eq. (3.7) we prove (4.2) and (4.3): Lλ (T A, B) = Lλ (T a ⊗ A , B) + Lλ (a ⊗ T A , B) = N( eT ∂λ T a , Lλ (A , B)) + p(a, A )N ( eT ∂λ A , Lλ (T a, B)) λ
+p(a, A ) dµ Lµ (A , Lλ−µ (T a, B)) 0 +N ( eT ∂λ a , Lλ (T A , B)) + p(a, A )N ( eT ∂λ T A , Lλ (a, B)) λ +p(a, A ) dµ Lµ (T A , Lλ−µ (a, B)) = −λLλ (A, B), 0
Freely Generated Vertex Algebras and Non-linear Lie Conformal Algebras
and
673
Lλ (A, T B) = N ( eT ∂λ a , Lλ (A , T B)) + p(a, A )N ( eT ∂λ A , Lλ (a, T B)) λ +p(a, A ) dµ Lµ (A , Lλ−µ (a, T B)) = (λ + T )Lλ (A, B). 0
In the above equations we have used the obvious commutation relation eT ∂λ λ = (λ + T )eT ∂λ . This completes the proof of the lemma. As an immediate consequence of the above lemma, we get the following Corollary 4.3. The subspaces M (R) ⊂ T (R) are invariant under the action of T . In particular T M(R) ⊂ M(R). The following lemma follows by a straightforward but quite lengthy computation, which we omit. Lemma 4.4. The following equations hold for all a, b, · · · ∈ R and A, B, · · · ∈ T (R):
Ta +Tb
N(sn(a, b, C), D) = sn(a, b, N(C, D)) +
−Q(
dλ sn(a, b, Lλ (C, D))
0
dλ Lλ (a, b) , C, D)
0
−T
T
−p(a, C)p(b, C)N (
(4.4) T −λ
dλ 0
dµ C , J(a, b, D; λ, µ)),
0
Lλ (a, sn(b, c, D)) = sn(Lλ (a, b), c, D)−p(b, c)sn(Lλ (a, c), b, D)
(4.5)
+ p(a, b)p(a, c)sn(b, c, Lλ (a, D)) λ + dµ Wl (Lλ (a, b), c, D; µ)−p(b, c)Wl (Lλ (a, c), b, D; µ) 0
−W (a, l
−T
+N (
0
dµ sl(Lλ (a, b), c; µ)−p(b, c)sl(Lλ (a, c), b; µ) , D)
λ
−T 0
−N (
dµ Lµ (b, c) , D; λ)
−λ−T
dµ sl(a, Lµ (b, c); λ) , D)
−p(a, b)p(a, c)N(
λ
−T
dµ J(b, c, a; −µ − T , µ − λ) , D)
λ λ + dν dµ Lν (sl(Lλ (a, b), c; µ)−p(b, c)sl(Lλ (a, c), b; µ),D) 0
−
ν
λ
λ
dν 0
ν
dµ Lν (sl(a, Lµ−λ (b, c); λ), D)
−p(a, b)p(a, c)
λ
λ
dν 0
ν
dµ Lν (J(b, c, a; µ − λ, ν − µ), D),
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A. De Sole, V.G. Kac
Lλ (sn(a, b, C), D) = sn( eT ∂λ a , eT ∂λ b , Lλ (C, D)) 0 r dµ Lµ (a, b) , C, D; λ) −W ( −T
0
λ
−p(a, C)p(b, C)
µ
dµ 0
dµ eT ∂λ C , J(a, b, D; µ, λ−µ))
λ+T
−p(a, C)p(b, C)N(
0
dν Lν (C, J(a, b, D; µ−ν, λ−µ)),
TA
Q(N (a, A ), B, C) = N (a, Q(A , B, C)) + p(a, A ) −
0 Ta
−p(a, A )
TA
dλ N(A , Wl (a, B, C; λ))
Ta
dλ N(a, Wr (A , B, C; λ))
+p(a, A )p(a, B) +p(A , B)
TA +TB
Ta +TB
dλ sn(a, B, Lλ (A , C))
TA +TB
+p(a, A )p(a, B)
Ta
Ta −λ
dλ 0
−p(a, A )
dλ Q(A , B, Lλ (a, C))
0
Ta
(4.7)
0
0
−
dλ Q(A , Lλ (a, B), C)
dλ N(a, Wl (A , B, C; λ))
0
+
(4.6)
0 TA
dλ sn(A , B, Lλ (a, C))
TA
dµ N(a, J(A , B, C; λ, µ))
TA −λ
dλ
0
dµ N (A , J(a, B, C; λ, µ)),
0
Wl (a, N (b, B ), C; λ) = p(a, b)N(b, Wl (a, B , C; λ)) − Q(Lλ (a, b), B , C) λ + dµ Wl (Lλ (a, b), B , C; µ) (4.8) 0
+p(b, B )W (a, −
l
T
0 λ
0
dµ B , Lµ (b, C); λ)
dµ Wr (Lλ (a, b), B , C; µ)
+p(a, b)N(
T
dµ b , J(a, B , C; λ, µ))
0
T
+p(a, B )p(b, B )N ( +
0
λ
λ−µ
dµ 0
dµ B , J(a, b, C; λ, µ))
0
dν J(Lλ (a, b), B , C; µ, ν),
Freely Generated Vertex Algebras and Non-linear Lie Conformal Algebras
675
Wl (N (a, A ), B, C; λ) = N ( eT ∂λ a , Wl (A , B, C; λ)) +p(a, A )N ( eT ∂λ A , Wl (a, B, C; λ)) −p(a, A )Q( eT ∂λ A , Lλ (a, B), C) (4.9) +p(A , B)sn( eT ∂λ a , B, Lλ (A , C)) +p(a, A )p(a, B)sn( eT ∂λ A , B, Lλ (a, C)) λ +p(a, A ) dµ Lµ (A , Wl (a, B, C; λ − µ)) 0 λ −p(a, A ) dµ Wr ( eT ∂λ A , Lλ (a, B), C; µ) 0 λ
+p(a, A ) dµ Wl (A , Lλ−µ (a, B), C; µ) 0 λ
+p(a, A )p(a, B) dµ Wl (A , B, Lλ−µ (a, C); µ) λ 0λ−µ +p(a, A ) dµ dν J(A , Lλ−µ (a, B), C; µ, ν), 0
0
(4.10) Wr (N (a, A ), B, C; λ) T ∂λ = N( e a , Wr (A , B, C; λ)) +p(a, A )p(a, B)Q( eT ∂λ A , eT ∂λ B , Lλ (a, C)) +p(A , B)sn( eT ∂λ a , eT ∂λ B , Lλ (A , C)) +p(a, A )p(a, B)sn( eT ∂λ A , eT ∂λ B , Lλ (a, C)) T
r +p(a, A )W ( dµ A , Lµ (a, B), C; λ) 0 λ
−p(a, B)p(A , B) dµ Wl (B, eT ∂λ a , Lλ−µ (A , C); µ) 0 λ
−p(a, A )p(a, B)p(A , B) dµ Wl (B, eT ∂λ A , Lλ−µ (a, C); µ) 0 λ
r +p(a, A )p(a, B) dµ W (A , B, Lλ−µ (a, C); µ) 0 T −N( dµ eT ∂λ a , J(A , B, C; µ, λ − µ)) 0 T
−p(a, A )N ( dµ eT ∂λ A , J(a, B, C; µ, λ − µ)) 0 λ
−p(a, A )p(a, B)p(A , B) dµ N( eT ∂λ sl(B, A ; µ) , Lλ (a, C)) 0 λ
−p(a, B)p(A , B) dµ N( eT ∂λ sl(B, a; µ) , Lλ (A , C)) 0 T λ A
+p(A , B) dµ dν Lν (sl(a, B; µ), Lλ−ν (A , C)) 0 0 Ta λ +p(a, A )p(a, B) dµ dν Lν (sl(A , B; µ), Lλ−ν (a, C)), 0
0
676
A. De Sole, V.G. Kac
J(a, b, N (c, D); λ, µ) = Wl (a, Lµ (b, c), D; λ) − p(a, b)Wl (b, Lλ (a, c), D; µ) −Wl (Lλ (a, b), c, D; λ + µ)
(4.11)
+N (J(a, b, c; λ, µ), D) +p(a, c)p(b, c)N (c, J(a, b, D; λ, µ)) µ + dν J(a, Lµ (b, c), D; λ, ν) 0 λ −p(a, b) dν J(b, Lλ (a, c), D; µ, ν) λ+µ 0 + dν Lν (J(a, b, c; λ, µ), D), 0
J(A, N (b, B ), D; λ, µ) = Wl (A, b, Lµ+Tb (B , D); λ)
+p(b, B )Wl (A, B , Lµ+TB (b, D); λ) −Lλ+µ (Wl (A, b, B ; λ), D)−Wr (Lλ (A, b), B , D; λ+µ) +p(A, b)N (b, J(A, B , D; λ, µ + Tb ))
(4.12)
+p(A, B )p(b, B )N (B , J(A, b, D; λ, µ + TB )) λ + dν J(Lλ (A, b), B , D; ν, λ + µ − ν) 0 µ
+p(b, B ) dν J(A, B , Lµ−ν (b, D); λ, ν) 0 µ +p(A, B )p(b, B ) dν Lν (B , J(A, b, D; λ, µ − ν)), 0
J(N (a, A ), B, C; λ, µ) = −p(a, B)p(A , B)J(B, N (a, A ), C; µ, λ) −Lλ+µ (sl(N (a, A ), B; λ), C),
(4.13)
b)N (b, sn(a, B , C)) (4.14) sn(a, N (b, B ), C) = sn(a, b, N(B , C)) + p(a, T −p(a, b)p(a, B )N ( dλ b , Wl (B , a, C; λ)) 0 0 dλ Q(Lλ (a, b), B , C) − −Ta −Tb −TB T B dλ sn(Lλ (b, a), B , C) +p(a, b) 0 Tb
−p(a, b)p(a, B )N ( dλ b , N (sl(B , a; λ), C)) 0 T +sn(a, dλ b , Lλ (B , C)) 0 T +p(b, B )sn(a, dλ B , Lλ (b, C)) T 0 λ −p(a, b)N( dλ dµ b , Lµ (sl(a, B ; µ − λ), C)), 0
0
Freely Generated Vertex Algebras and Non-linear Lie Conformal Algebras
677
sn(N (a, A ), B, C) = N (a, sn(A , B, C)) + p(A , B)sn(a, B, N (A , C)) T +N ( dλ a , Wl (A , B, C; λ)) (4.15) 0 T +p(a, A )N ( dλ A , Wl (a, B, C; λ)) 0 0 dλ Q( eT ∂λ A , Lλ (a, B), C) −p(a, A ) −Ta −TA −TB 0
dλ Lλ (a, B) , A , C) +p(A , B)sn( −T T
+p(A , B)sn( dλ a , B, Lλ (A , C)) 0 T
+p(a, A )p(a, B)sn( dλ A , B, Lλ (a, C)) 0 0 0
dµ sl(A , dλ Lλ (a, B) ; µ) , C), +p(a, A )N ( −T
−T
sl(a, N (b, B ); λ) = p(a, b)N(b, sl(a, B ; λ)) + sn(Lλ (a, b), B , 1)
−p(a, b)
λ
−T
dµ sl(Lµ−λ (b, a), B ; µ),
sl(N (a, A ), B; λ) = p(A , B)N (sl(a, B; λ + TA ), A ) +N ( eT ∂λ a , sl(A , B; λ))
(4.16) (4.17)
+p(a, B)p(A , B)Wl (B, a, A ; −λ − T ) +p(a, A )sn( eT ∂λ A , Lλ (a, B), 1) λ −p(a, A )p(a, B) dµ sl( eT ∂λ A , L−λ−T (B, a); µ) +p(a, A )
−T
λ
−T
dµ Lµ (A , sl(a, B; λ − µ)).
Corollary 4.5. For every 1 , 2 , 3 ∈ there exists ∈ such that the following inclusions hold N(T1 (R), M2 (R)) ⊂ M1 +2 (R),
(4.18)
N(M1 (R), T2 (R)) ⊂ M1 +2 (R),
(4.19)
Lλ (T1 (R), M2 (R)) ⊂ C[λ] ⊗ M (R), < 1 +2 ,
(4.20)
Lλ (M1 (R), T2 (R)) ⊂ C[λ] ⊗ M (R), < 1 +2 ,
(4.21)
Q(T1 (R), T2 (R), T3 (R)) ⊂ M1 +2 +3 (R),
(4.22)
Wl (T1(R),T2(R),T3(R);λ) ⊂ C[λ]⊗M , < 1+2+3 ,
(4.23)
Wr (T1(R),T2(R),T3(R);λ) ⊂ C[λ]⊗M (R), < 1+2+3 ,
(4.24)
678
A. De Sole, V.G. Kac
J(T1(R),T2(R),T3(R);λ,µ) ⊂ C[λ, µ]⊗M , < 1+2+3 , sn(T1 (R), T2 (R), T3 (R)) ⊂ M1 +2 +3 (R), sl(T1 (R), T2 (R); λ) ⊂ C[λ] ⊗ M , < 1 +2 .
(4.25) (4.26) (4.27)
In particular, M(R) is a two sided ideal with respect to both the normally ordered product N : T (R) ⊗ T (R) → T (R) and the λ–bracket Lλ : T (R) ⊗ T (R) → T (R). Proof. We will prove, by induction on ∈ , that there exists < such that the following conditions hold, for A ∈ TA (R), B ∈ TB (R), C ∈ TC (R), D ∈ TD (R) and E ∈ ME (R): Q(A, B, C) ∈ M (R),
if A + B + C ≤ ,
(4.28)
Wl (A, B, C; λ) ∈ C[λ]M (R),
if A + B + C ≤ ,
(4.29)
Wr (A, B, C; λ) ∈ C[λ]M (R),
if A + B + C ≤ ,
(4.30)
J(A, B, C; λ, µ) ∈ C[λ, µ]M (R),
if A + B + C ≤ ,
(4.31)
N(A, E) ∈ M (R),
if A + E ≤ ,
(4.32)
N(E, D) ∈ M (R),
if D + E ≤ ,
(4.33)
Lλ (A, E) ∈ C[λ] ⊗ M (R),
if A + E ≤ ,
(4.34)
Lλ (E, D) ∈ C[λ] ⊗ M (R),
if D + E ≤ ,
(4.35)
sn(A, B, C) ∈ M (R),
if A + B + C ≤ ,
sl(A, B; λ) ∈ C[λ] ⊗ M (R),
if A + B ≤ .
(4.36) (4.37)
For = 0 there is nothing to prove. Let then > 0 and assume, by induction, that the ¯ < . above conditions hold for all Proof of (4.28). For A ∈ C condition (4.28) is trivially satisfied, and for A ∈ R it holds thanks to Eq. (3.3). Let then A = a ⊗ A , with a ∈ R and A ∈ R ⊕ R ⊗2 ⊕ · · · . In this case Q(A, B, C) is given by Eq. (4.7), and every term in the right-hand side belongs to M (R) by the inductive assumption. Proof of (4.29). For A ∈ C there is nothing to prove. We will consider separately two cases: 1. A = a ∈ R, 2. A = a ⊗ A , with a ∈ R and A ∈ R ⊕ R ⊗2 ⊕ · · · . Case 1. For B ∈ C or B ∈ R condition (4.29) is trivially satisfied, thanks to Eq. (3.6). Let then B = b ⊗ B , with b ∈ R and B ∈ R ⊕ R ⊗2 ⊕ · · · . In this case Wl (A, B, C; λ) is given by Eq. (4.8), and, by the inductive assumption, every term of the right-hand side lies in M (R), for some < .
Freely Generated Vertex Algebras and Non-linear Lie Conformal Algebras
679
Case 2. In this case Wl (A, B, C; λ) is given by Eq. (4.10), and again, by induction, every term of the right-hand side lies in M (R), for some < . Proof of (4.30). For A ∈ C or A ∈ R, condition (4.30) holds trivially, thanks to Eq. (3.7). Suppose then A = a ⊗ A , with a ∈ R and A ∈ R ⊕ R ⊗2 ⊕ · · · . In this case Wr (A, B, C; λ) is given by Eq. (4.10), and every term in the right-hand side lies in M (R), for some < . Proof of (4.31). If either A ∈ C, or B ∈ C, or C ∈ C, J(A, B, C; λ, µ) = 0, so that (4.31) holds. Moreover, if A, B, C ∈ R, condition (4.31) holds by the definition of non–linear Lie conformal algebra. We will consider separately the following three cases: 1. A = a ∈ R, B = b ∈ R, C = c ⊗ D, with c ∈ R and D ∈ R ⊕ R ⊗2 ⊕ · · · , 2. B = b ⊗ B , with b ∈ R and A, B , C ∈ R ⊕ R ⊗2 ⊕ · · · , 3. A = a ⊗ A , with a ∈ R and A , B, C ∈ R ⊕ R ⊗2 ⊕ · · · . In the first case J(A, B, C; λ, µ) is expressed by Eq. (4.11), and, by the inductive assumption, every term in the right-hand side lies in M (R), for some < . In the second case J(A, B, C; λ, µ) is given by Eq. (4.12), and it lies by induction in M (R), for some < . Finally, the third case reduces to the second case, thanks to Eq. (4.13) and the inductive assumption. Proof of (4.32). For A ∈ C condition (4.32) is obvious, and for A ∈ R it holds by definition of M (R). Let then A = a ⊗ B, with a ∈ R and B ∈ R ⊕ R ⊗2 ⊕ · · · . By (3.3) we have T dλ a , Lλ (B, E)) N(A, E) = a ⊗ N (B, C) + N ( 0
T
+p(a, b)N (
0
dλ B Lλ (a, E),
and each term in the right-hand side is in M (R) by the inductive assumption. Proof of (4.33). We will consider separately the following two cases: 1. E = a ⊗ F , with a ∈ R and F ∈ MF , where a + F = E , 2. E = sn(a, b, C), with a, b ∈ R, C ∈ T (R), and a + b + C = E . In the first case we have, by (3.3),
T
N(E, D) = a ⊗ N (F, D)) + N (
0
T
+p(a, F )N(
0
dλ a , Lλ (F, D))
dλ F Lλ (a, D),
and each term in the right-hand side is in M (R) by the inductive assumption. In the second case N(E, D) is expressed by Eq. (4.4), and again each term in the right-hand side lies in M (R). Proof of (4.34). For A ∈ C condition (4.34) is obvious. We will consider separately two cases:
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1. A = a ∈ R, 2. A = a ⊗ B, with a ∈ R and B ∈ R ⊕ R ⊗2 ⊕ · · · . Consider the second case first. We have, by (3.7), Lλ (A, E) = N ( eT ∂λ a , Lλ (B, E)) + p(a, B)N ( eT ∂λ B , Lλ (a, E)) λ +p(a, B) dµ Lµ (B, Lλ−µ (a, E)), 0
and, by induction, each term in the right-hand side lies in M (R), for some < . Suppose then A = a ∈ R. As before, there are two different situations to consider: 2a E = b ⊗ F , with b ∈ R, F ∈ MF and b + F = E , 2b E = sn(b, c, D), with b, c ∈ R, D ∈ T (R), and b + c + D = E . In the first case we have, by (3.6), Lλ (a, b ⊗ F ) = N (Lλ (a, b), F ) + p(a, b)N (b, Lλ (a, F )) λ + dµ Lµ (Lλ (a, b), F ), 0
and each term in the right-hand side lies by induction in M (R), for some < . Finally, if E = sn(b, c, D), Lλ (a, E) is expressed by Eq. (4.5), and again each term in the right-hand side lies in M (R), for some < . Proof of (4.35). Consider separately two cases: 1. E = a ⊗ F , with a ∈ R, F ∈ MF , and a + F = E , 2. E = sn(a, b, C), with a, b ∈ R, C ∈ T (R), and a + b + C = E . In the first case we have, by (3.7), Lλ (E, D) = N ( eT ∂λ a , Lλ (F, D)) + p(a, B)N ( eT ∂λ F , Lλ (a, D)) λ +p(a, B) dµ Lµ (F, Lλ−µ (a, D)), 0
and, by the inductive assumption, each term in the right-hand side is in M (R), for some < . In the second case Lλ (E, D) is given by Eq. (4.6). All the terms of the right-hand side are in M (R), for some < , thanks to the inductive assumption and condition (4.31). Proof of (4.36). For A ∈ C or B ∈ C, condition (4.36) holds trivially. Moreover, for A, B ∈ R, (4.36) holds by definition of M (R). Suppose first A = a ∈ R, B = b ⊗B , with b ∈ R and B ∈ R ⊕ R ⊗2 ⊕ · · · . In this case sn(A, B, C) is given by Eq. (4.14), and it belongs to M (R) by induction. We are left to consider the case A = a ⊗ A , with a ∈ R and A , B ∈ R ⊕ R ⊗2 ⊕ · · · . In this case sn(A, B, C) is given by Eq. (4.15). Every term in the right-hand side lies in M (R) thanks to the inductive assumption and the above result.
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Proof of (4.37). If either A ∈ C, or B ∈ C, or A, B ∈ R, condition (4.37) is trivial. Suppose A = a ∈ R and B = b ⊗ B , with b ∈ R and B ∈ R ⊕ R ⊗2 ⊕ · · · . In this case sl(A, B; λ) is given by Eq. (4.16), and it lies by induction in M (R), for some < . Finally, consider the case A = a ⊗ A , with a ∈ R and A , B ∈ R ⊕ R ⊗2 ⊕ · · · . In this case sl(A, B; λ) is given by Eq. (4.17), and again it lies in M (R), for some < , by the inductive assumption and by condition (4.29). Corollary 4.6. The map L : T (R) ⊗ T (R) → T (R) defined in Remark 3.7 satisfies skewsymmetry: L(A, B) + p(A, B)L(B, A) ∈ M (R), for every A, B ∈ T (R) and for some < (A) + (B), and Jacobi identity, L(A, L(B, C)) − p(A, B)L(B, L(A, C)) − L(L(A, B), C) ∈ M (R), for every A, B, C ∈ T (R) and for some < (A) + (B) + (C). Proof. The corollary follows immediately from Corollary 4.5 and the following identities: L(A, B) = −p(A, B)L(B, A) +
0 −T
dλ sl(A, B; λ),
L(A, L(B, C)) = p(A, B)L(B, L(A, C)) + L(L(A, B), C) 0 0 dλ dµ J(A, B, C; λ, µ). + −T
−λ−T
5. Proof of Theorem 3.9 In this section we will use Corollary 4.5 to prove Theorem 3.9. Notice that by (4.18)– (4.21), the products N : T (R) ⊗ T (R) → T (R) and Lλ : T (R) ⊗ T (R) → C[λ] ⊗ T (R) induce products on the quotient space U (R) = T (R)/M(R), which we denote by : : and [ λ ] respectively. Moreover, by (4.22)–(4.27), the space U (R), with normally ordered product : : and λ–bracket [ λ ], satisfies all the axioms of vertex algebra. This proves the second part of Theorem 3.9. We are left to prove the first part of Theorem 3.9, which provides a PBW basis for U (R), and therefore guarantees that U (R) = 0. Let us denote by B˜ the collection of 1 and all ordered monomials in T (R): B˜ =
i1 ≤ · · · ≤ in , n ∈ Z+ , ai1 ⊗ · · · ⊗ ain . ik < ik+1 if p(aik ) = 1¯
˜ We also denote by B[] the collection of ordered monomials of degree , namely ˜ ˜ ˜ (R). B[] = B∩T (R)[], and by B˜ the corresponding filtration, namely B˜ = B∩T ˜ By definition B = π(B).
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Lemma 5.1. Any element E ∈ T (R) can be decomposed as E = P + M,
(5.1)
where P ∈ spanC B˜ and M ∈ M (R). Equivalently T (R)/M (R) = spanC π(B˜ ),
∀ ∈ ,
and therefore U (R) = spanC B. Proof. It suffices to prove (5.1) for monomials, namely for E = aj1 ⊗ · · · ⊗ ajn ∈ T (R)[], where ∈ and ajk ∈ A. Let us define the number of inversions of E as 1 ≤ p < q ≤ n and d(E) = # (p, q) either ajp is even and jp > jq . or a is odd and j ≥ j jp
p
q
Let d = d(E). We will prove that E decomposes as in (5.1) by induction on the pair ˜ (, d), ordered lexicographically. If d = 0, we have E ∈ B[], so there is nothing to prove. Suppose that d ≥ 1 and let p ∈ {1, . . . , n − 1} be such that jp > jp+1 , or jp = jp+1 for ajp odd. In the first case we have, by definition of M (R), E ≡ p(ajp , ajp+1 )aj1 ⊗ · · · ⊗ ajp+1 ⊗ ajp ⊗ · · · ⊗ ajn 0 +aj1 ⊗ · · · ⊗ N ( dλ[ajp λ ajp+1 ] , · · · ⊗ ajn ) −T
mod M (R).
The first term in the right-hand side has degree and d − 1 inversions, while, by the grading conditions (3.8), the second term lies in T (R), for some < . Therefore, by the inductive assumption, both terms admit a decomposition (5.1). Consider now the second case, namely ajp is odd and jp = jp+1 .. By definition of M (R) we have 0 1 dλ[ajp λ ajp+1 ] , · · · ⊗ ajn ) mod M (R), E ≡ aj1 ⊗ · · · ⊗ N ( 2 −T and again, by the inductive assumption, the right-hand side admits a decomposition (5.1). ˜ Lemma 5.2. Let U˜ be a vector space with basis B: U˜ = CA. A∈B˜
There exists a unique linear map σ : T (R) −→ U˜ such that ˜ 1. σ (A) = A, ∀A ∈ B, 2. M(R) ⊂ ker σ . Proof. We want to prove that there is a unique collection of linear maps σ : T (R) → ˜ for ∈ , such that U,
Freely Generated Vertex Algebras and Non-linear Lie Conformal Algebras
(1) σ0 (1) = 1,
σ T
(R)
683
= σ , if ≤ ,
˜ (2) σ (A) = A, if A ∈ B[], (3) M (R) ⊂ ker σ . This obviously proves the lemma. Indeed for any such sequence we can define the map σ : T (R) → U˜ by σ T (R) = σ , ∀ ∈ , (5.2)
and conversely, given a map σ : T (R) → U˜ satisfying the assumptions of the lemma, we can define such a sequence of maps σ : T (R) → U˜ by Eq. (5.2). The condition σ0 (1) = 1 defines completely σ0 . Notice that, since M0 (R) = 0, σ0 satisfies all the required conditions. Let then > 0, and suppose by induction that σ is uniquely defined and it satisfies all conditions (1)–(3) for every < . Uniqueness of σ . Given a monomial of degree , E = aj1 ⊗ · · · ⊗ ajn ∈ T (R)[], we will show that σ (E) is uniquely defined by induction on the number of inversions d of E, defined above. For d = 0 we have E ∈ B˜ , so it must be σ (E) = E by condition (2). Let then d ≥ 1 and let (jp , jp+1 ) be the “most left inversion”, namely p ∈ {1, . . . , n} is the smallest integer such that either jp > jp+1 or p(ajp ) = 1¯ and jp = jp+1 . By condition (3) and the definition of M (R) we then have, if jp > jp+1 , σ (E) = p(ajp , ajp+1 )σ aj1 ⊗ · · · ⊗ ajp+1 ⊗ ajp ⊗ · · · ⊗ ajn (5.3) +σ aj1 ⊗ · · · ⊗ N (L(ajp , ajp+1 ), · · · ⊗ ajn ) , while, if p(ajp ) = 1¯ and jp = jp+1 , 1 σ (E) = σ aj1 ⊗ · · · ⊗ N (L(ajp , ajp+1 ), · · · ⊗ ajn ) , (5.4) 2 Notice that (aj1 ⊗ · · · ⊗ ajp+1 ⊗ ajp ⊗ · · · ⊗ ajn ) has degree and d − 1 inversions, so the first term in the right-hand side of (5.3) is uniquely defined by the inductive assumption. Moreover, by the grading conditions (3.8), aj1 ⊗ · · · ⊗ N (L(ajp , ajp+1 ), · · · ⊗ ajn ) lies in T (R), for some < , so the second term in the right-hand side of (5.3) and the right-hand side of (5.4) are uniquely defined by condition (1) and the inductive assumption. Existence of σ . The above prescription defines uniquely a linear map σ : T (R) → ˜ which by construction satisfies conditions (1) and (2). We are left to show that conU, dition (3) holds. By definition of M (R), it suffices to prove that, for every monomial of degree , E = aj1 ⊗ · · · ⊗ ajn ∈ T (R)[], and for every q = 1, . . . , n, we have σ aj1 ⊗ · · · ajq−1 ⊗ sn(ajq , ajq+1 , ajq+2 ⊗ · · · ⊗ ajn ) = 0. (5.5) By skewsymmetry of the λ–bracket we can assume, without loss of generality, that ¯ we have 0 dλ[ajq λ ajq ] = 0, so that when jq ≥ jq+1 . Moreover, if p(ajq ) = 0, −T jq = jq+1 Eq. (5.5) is obvious. In other words, we can assume that (jq , jq+1 ) is an “inversion” of E. In particular d = d(E) ≥ 1. We will prove condition (5.5) by induction on d. Let (jp , jp+1 ) be the “most left inversion” of E. For p = q Eq. (5.5) holds by construction. We will consider separately the cases p ≤ q − 2 and p = q − 1.
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Case 1. Assume p ≤ q − 2. For simplicity we rewrite E = A ⊗ c ⊗ b ⊗ D ⊗ f ⊗ e ⊗ H, where A = aj1 ⊗ · · · ⊗ ajp−1 , c = ajp , b = ajp+1 , D = ajp+2 ⊗ · · · ⊗ ajq−1 , f = ajq , e = ajq+1 , H = ajq+2 ⊗ · · · ⊗ ajn . The left-hand side of Eq. (5.5) then takes the form σ A ⊗ c ⊗ b ⊗ D ⊗ f ⊗ e ⊗ H − p(e, f )σ A ⊗ c ⊗ b ⊗ D ⊗ e ⊗ f ⊗ H −σ A ⊗ c ⊗ b ⊗ D ⊗ N (L(e, f ), H ) . (5.6) By definition of σ , the first term of (5.6) can be written as p(b, c)σ (A ⊗ b ⊗ c ⊗ D ⊗ f ⊗ e ⊗ H ) +σ (A ⊗ N (L(c, b), D ⊗ f ⊗ e ⊗ H )).
(5.7)
If e = f , A ⊗ c ⊗ b ⊗ D ⊗ e ⊗ f ⊗ H has degree and d − 1 inversions, so that we can use the inductive assumption to rewrite the second term of (5.6) as −p(e, f )p(b, c)σ (A ⊗ b ⊗ c ⊗ D ⊗ e ⊗ f ⊗ H ) −p(e, f )σ (A ⊗ N (L(c, b), D ⊗ e ⊗ f ⊗ H )).
(5.8)
By the grading conditions (3.8), A ⊗ c ⊗b ⊗ D ⊗ N (L(f, e), H ) ∈ T (R), for some < . We can thus use the fact that σ T (R) = σ and the inductive assumption to rewrite the third term of (5.6) as −p(b, c)σ (A ⊗ b ⊗ c ⊗ D ⊗ N (L(f, e), H ) −σ (A ⊗ N (L(c, b), D ⊗ N (L(f, e), H ))).
(5.9)
Combining (5.7), (5.8) and (5.9) we can rewrite (5.6) as p(b, c)σ (A ⊗ b ⊗ c ⊗ D ⊗ sn(f, e, H )) +σ (A ⊗ N (L(c, b), D ⊗ sn(f, e, H ))).
(5.10)
The first term of (5.10) appears only for b = c, and in this case it is zero by the inductive assumption, since A ⊗ b ⊗ c ⊗ D ⊗ f ⊗ e ⊗ H has degree and d − 1 inversions. Consider the second term of (5.10). By the grading conditions (3.8) and by Corollary 4.5, the argument of σ lies in M (R), for some < . It follows by induction that also the second term of (5.10) is zero. We thus proved, as we wanted, that (5.6) is zero. Case 2. We are left to consider the case p = q − 1. For simplicity we write E = A ⊗ c ⊗ b ⊗ a ⊗ D, where A = aj1 ⊗ · · · ⊗ ajp−1 , c = ajp , b = ajp+1 , a = ajp+2 , D = ajp+3 ⊗ · · · ⊗ ajn . The left-hand side of Eq. (5.5) then takes the form σ (A ⊗ c ⊗ b ⊗ a ⊗ D) − p(a, b)σ (A ⊗ c ⊗ a ⊗ b ⊗ D) −σ (A ⊗ c ⊗ N (L(b, a), D)).
(5.11)
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After some manipulations based on the inductive assumption, similar to the ones used above, we can rewrite (5.11) as σ (A ⊗ N (L(c, b), a ⊗ D)) − p(a, b)p(a, c)a ⊗ N (L(c, b), D) ) (5.12) +σ (A ⊗ p(b, c)b ⊗ N (L(c, a), D)) − p(a, b)N (L(c, a), b ⊗ D) ) +σ (A ⊗ p(a, c)p(b, c)N (L(b, a), c ⊗ D)) − c ⊗ N (L(b, a), D) ). It follows by Corollary 4.5 that there exists < such that A ⊗ a ⊗ N (L(c, b), D) − p(a, b)p(a, c)A ⊗ N (L(c, b), a ⊗ D) ≡ A ⊗ N (L(a, L(c, b)), D)
mod M (R),
A ⊗ b ⊗ N (L(c, a), D) − p(a, b)p(b, c)A ⊗ N (L(c, a), b ⊗ D) ≡ A ⊗ N (L(b, L(c, a)), D)
mod M (R),
A ⊗ c ⊗ N (L(b, a), D) − p(b, c)p(a, c)A ⊗ N (L(b, a), c ⊗ D) ≡ A ⊗ N (L(c, L(b, a)), D)
mod M (R).
We can thus use the inductive assumption to rewrite (5.12) as σ p(b, c)A ⊗ N (L(b, L(c, a)), D) − p(a, b)p(a, c)A ⊗ N (L(a, L(c, b)), D) −A ⊗ N (L(c, L(b, a)), D) . (5.13) By Corollary 4.6, the argument of σ in (5.13) lies in M (R) for some < and therefore (5.13) is zero by the inductive assumption. This concludes the proof of the lemma. Lemma 5.3. If σ is as in Lemma 5.2, the induced map σˆ : U (R) = T (R)/M(R) −→ U˜ is an isomorphism of vector spaces (namely M(R) = ker σ ). Proof. By definition the map σˆ : U (R) → U˜ is surjective. On the other hand, we have a natural map πˆ : U˜ → U (R) which maps every basis element A = ai1 ⊗ · · · ⊗ ain ∈ B˜ to the corresponding πˆ (A) = : ai1 . . . ain : ∈ B ⊂ U (R). By Lemma 5.1 this map is also surjective. The composition map πˆ σˆ U˜ −→ U (R) −→ U˜
is the identity map (by definition of πˆ and σˆ .) This of course implies that both πˆ and σˆ are isomorphisms of vector spaces. The last lemma implies that B is a basis of the space U (R), thus concluding the proof of Theorem 3.9.
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6. Pre–Graded Freely Generated Vertex Algebras and Corresponding Non–Linear Lie Conformal Algebras Let V be a vertex algebra strongly generated by a free C[T ]–module R = C[T ] ⊗ V¯ . Let ¯ A¯ = {a¯ ¯ı , ¯ı ∈ I}, be an ordered basis of V¯ , and extend it to an ordered basis of R, A = {ai , i ∈ I}, where I = I¯ × Z+ and, if i = (¯ı, n), then ai = T n a¯ ¯ı . The corresponding collection of ordered monomials of V (see Definition 2.6) is denoted by B. Recall that V is said to be freely generated by R if B is a basis of V . We will denote by π the natural quotient map π : T (R) −→ V , given by π(a ⊗ b ⊗ · · · ⊗ c) = : ab · · · c :. We will assume that the generating space V¯ is graded by \{0}, and that the basis A¯ of V¯ is compatible with the \{0}–gradation. The –gradation can be extended to R = C[T ] ⊗ V¯ ⊂ V by saying that T has zero degree, and to the whole tensor algebra T (R) by additivity of the tensor product (as defined in Sect. 2). The corresponding –filtration of T (R) naturally induces a –filtration on the vertex algebra V (cf. [Li02]): V = π(T (R)),
∈ .
(6.1)
Definition 6.1. The vertex algebra V , strongly generated by a free C[T ]–module C[T ]⊗ V¯ , where V¯ = ⊕∈\{0} V¯ [], is said to be pre–graded by if the λ–bracket satisfies the grading condition [V¯ [1 ] λ V¯ [2 ]] ⊂ C[λ] ⊗ V ,
for some < 1 + 2 .
Example 6.2. The most important examples of pre–graded vertex algebras are provided by graded vertex algebras. Recall that a vertex algebra V is called graded if there exists a diagonalizable operator L0 on V with discrete non–negative spectrum and 0th eigenspace C|0 , such that for all a ∈ V , [L0 , Y (a, z)] = z∂z Y (a, z) + Y (L0 a, z). The eigenvalue w(a) of an eigenvector a of L0 is called its conformal weight. Let V = ⊕w∈ V (w) be the eigenspace decomposition. Recall that conformal weights satisfy the following rules (see e.g. [Kac96]): w(T a) = w(a) + 1 , w(a(n) b) = w(a) + w(b) − n − 1.
(6.2)
Assume now that V is strongly generated by a free C[T ]–submodule R = C[T ] ⊗ V¯ , where V¯ is invariant under the action of L0 . Consider the decomposition of V¯ in a direct sum of eigenspaces of L0 : V¯ = ⊕w∈\{0} V¯ [], where V¯ [] = V¯ ∩ V (). This is a \{0}–gradation of V¯ , which induces, as explained above, a –gradation on T (R) and hence a –filtration, V , ∈ , on the vertex algebra V . Notice that such –filtration is not induced by the gradation given by the conformal weights. Indeed the filtered space V are preserved by T , while, by (6.2), T increases the conformal weight by 1. On the other hand, we have w≤
V (w) ⊂ V .
(6.3)
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The above \{0}–gradation of V¯ is a pre–gradation of V . Indeed (6.2) and (6.3), imply V (w) ⊂ C[λ] ⊗ V1 +2 −1 , [V¯ [1 ] λ V¯ [2 ]] ⊂ C[λ] ⊗ w≤1 +2 −1
so that the grading condition in Definition 6.1 holds. Remark 6.3. Notice that if R = C[T ] ⊗ V¯ is a non–linear Lie conformal algebra graded by \{0}, then by Theorem 3.9 the universal enveloping vertex algebra U (R) is a pre– graded freely generated vertex algebra. We want to prove a converse statement to Theorem 3.9. Theorem 6.4. Let V be a pre–graded vertex algebra freely generated by a free C[T ]– submodule R = C[T ] ⊗ V¯ . Then R has a structure of the non–linear Lie conformal algebra Lλ : R ⊗ R −→ C[λ] ⊗ T (R), compatible with the Lie conformal algebra structure of V , in the sense that π(Lλ (a, b)) = [a λ b], ∀a, b ∈ R, so that V is canonically isomorphic to the universal enveloping vertex algebra U (R). Any two such structures of a non–linear Lie conformal algebra on R are isomorphic in the sense of Definition 3.6. The proof will be achieved as a result of 8 lemmas (the first two of which can be derived from [Li02]). Lemma 6.5. The vertex algebra structure on V satisfies the following grading conditions: : V1 V2 : ⊂ V1 +2 , [V1
λ
V2 ] ⊂ C[λ] ⊗ V for some < 1 + 2 .
Proof. By definition V is spanned by elements π(A), where A ∈ T (R). We thus want to prove, by induction on = 1 + 2 , that, for A ∈ T1 (R) and B ∈ T2 (R), we have : π(A)π(B) : ∈ V ,
(6.4)
[π(A) λ π(B)] ∈ C[λ] ⊗ V for some < .
(6.5)
If A ∈ C or B ∈ C, both conditions are obvious. Suppose then A, B ∈ R ⊕ R ⊗2 ⊕ · · · . If A ∈ R, we have : π(A)π(B) : = π(A ⊗ B), so that condition (6.4) follows immediately by the definition of the –filtration on V . Let then A = a ⊗ A , with a ∈ R and A , B ∈ R ⊕ R ⊗2 ⊕ · · · . Notice that (a) + (A ) = (A) and (a), (A ) > 0. By quasi–associativity of the normally ordered product we then have T : π(A)π(B) : = : a(: π(A )π(B) :) : + : dλ a [π(A ) λ π(B)] : +p(a, A ) :
0
0
T
dλ π(A ) [a λ π(B)] :,
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and each term in the right-hand side belongs to V by the inductive assumption. We are left to prove condition (6.5). If A, B ∈ R, (6.5) holds by sesquilinearity and by the assumption of the grading condition on V . Suppose now A = a ∈ R and B = b ⊗ B , with b ∈ R and B ∈ R ⊕ R ⊗2 ⊕ · · · . In this case we can use the left Wick formula to get [π(A) λ π(B)] = : [a λ b]π(B ) : +p(a, b) : b[a λ π(B )] : λ + dµ[[a λ b] µ π(B )], 0
and each term in the right-hand side belongs to V , for some < , by induction. Finally, let A = a ⊗ A , where a ∈ R and A , B ∈ R ⊕ R ⊗2 ⊕ · · · . We then have by the right Wick formula [π(A) λ π(B)] = : eT ∂λ a [π(A ) λ π(B)] : +p(a, A ) : eT ∂λ π(A ) [a λ π(B)] : λ
+p(a, A ) dµ[π(A ) µ [a λ−µ π(B)]], 0
and again, by the inductive assumption, each term in the right-hand side belongs to V for some < . As in Sect. 5, we denote by B˜ the collection of ordered monomials in T (R): B˜ =
i1 ≤ · · · ≤ in , n ∈ Z+ , ai1 ⊗ · · · ⊗ ain ⊂ T (R), iq < iq+1 if p(aiq ) = 1¯
so that the basis B of V is the image of B˜ under the quotient map π : B =
i1 ≤ · · · ≤ in , n ∈ Z+ , : ai1 . . . ain : ⊂ V. iq < iq+1 if p(aiq ) = 1¯
The basis B of V induces an embedding ρ : V → T (R), defined by ρ(: ai1 . . . ain :) = ai1 ⊗ · · · ⊗ ain ,
∀ : ai1 . . . ain :∈ B.
Moreover, the basis B induces a primary –gradation on V , V =
V [](1) ,
(6.6)
∈
defined by assigning to every basis element : ai1 . . . ain : ∈ B degree (: ai1 . . . ain :) = (ai1 ) + · · · + (ain ). Lemma 6.6. The –filtration (6.1) of V is induced by the primary –gradation (6.6).
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Proof. We want to prove that for every ∈ , V = V [ ](1) . ≤
By definition of the –filtration of V V = π(T (R)) = spanC : aj1 . . . ajn : (aj1 ) + · · · + (ajn ) ≤ , while, by definition of the primary –gradation of V , we have (aj ) + · · · + (ajn ) ≤ 1 i1 ≤ · · · ≤ in V [ ](1) = spanC . : aj1 . . . ajn : ¯
i < i if p(a ) = 1 ≤ q q+1 iq Therefore we obviously have the inclusion ≤ V [ ](1) ⊂ V . We are left to show that every (non-necessarily ordered) monomial A =: aj1 . . . ajn : can be written as a linear combination of ordered monomials of primary degree ≤ . We will prove this statement by induction on (, d), where d is the number of inversions of A: 1 ≤ p < q ≤ n and either jp > jq d = # (p, q) . or jp = jq and p(ajp ) = 1¯ If d = 0, then A ∈ V [](1) and the statement is trivial. Suppose then d ≥ 1, and let (jq , jq+1 ) be an inversion of A. We consider the case jq > jq+1 . The case jq = jq+1 and p(ajq ) = 1¯ is similar. By skew–symmetry of the normally ordered product, we have A = p(ajq , ajq+1 ) : aj1 · · · ajq+1 ajq · · · ajn : 0 dλ [ajq λ ajq+1 ] ajq+2 · · · ajn : . +aj1 · · · ajq−1
(6.7)
−T
The first term in the right-hand side of (6.7) is a monomial of V with d − 1 inversions. Moreover, by Lemma 6.5, the second term in the right-hand side of (6.7) belongs to V for some < . We can thus use the inductive assumption to conclude, as we wanted, that A ∈ ≤ V [ ](1) . The embedding ρ : V → T (R) introduced above does not commute with the action of T . The main goal in the following will be to replace ρ with another embedding ρT : V → T (R) which commutes with the action of T . We introduce the following collection of elements of T (R): i1 = (¯ı1 , k1 ) ≤ i2 ≤ · · · ≤ in k1 ˜ ⊂ T (R), BT = T (a¯ ¯ı1 ⊗ ai2 ⊗ · · · ⊗ ain ) iq < iq+1 if p(aiq ) = 1¯ and we denote by BT the corresponding image via the quotient map π , namely i1 = (¯ı1 , k1 ) ≤ i2 ≤ · · · ≤ in k1 ⊂ V. BT = T : a¯ ¯ı1 ai2 . . . ain : iq < iq+1 if p(aiq ) = 1¯
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Lemma 6.7. Consider the elements : ai1 . . . ain : ∈ B and T k1 : a¯ ¯ı1 ai2 . . . ain : ∈ BT ordered lexicographically by the indices (, i1 = (¯ı1 , k1 ), i2 = (¯ı2 , k2 ), . . . , in = (¯ın , kn )), where = (ai1 ) + · · · + (ain ) ∈ . (a) Every element T k1 : a¯ ¯ı1 ai2 . . . ain : ∈ BT can be written as T k1 : a¯ ¯ı1 ai2 . . . ain : = : ai1 . . . ain : +M, where M is a linear combination of elements of B smaller than : ai1 . . . ain :. (b) Every element : ai1 . . . ain : ∈ B can be written as : ai1 . . . ain : = T k1 : a¯ ¯ı1 ai2 . . . ain : +N, where N is a linear combination of elements of BT smaller than T k1 : a¯ ¯ı1 ai2 . . . ain :. Proof. (a) Since T is a derivation of the normally ordered product, we have T k1 : a¯ ¯ı1 ai2 . . . ain : = : ai1 . . . ain : k1 + : a(¯ı1 ,l1 ) a(¯ı2 ,k2 +l2 ) · · · a(¯ın ,kn +ln ) : . l1 , · · · , ln l1 ≥ 1, l2 , . . . , ln ≥ 0 l1 + · · · + ln = k1 We just need to prove that every monomial : a(¯ı1 ,l1 ) a(¯ı2 ,k2 +l2 ) · · · a(¯ın ,kn +ln ) : is a linear combination of ordered monomials smaller than : ai1 · · · ain :. In general the monomial : a(¯ı1 ,l1 ) a(¯ı2 ,k2 +l2 ) · · · a(¯ın ,kn +ln ) : is not ordered. But thanks to skew–symmetry of the normally ordered product and to Lemma 6.5, we can rewrite it (up to a sign) as the sum of an ordered monomial : aσ1 · · · aσn : with the same (reordered) indices, and an element R of V , with < . Notice that the reordered monomial : aσ1 · · · aσn : is smaller than : ai1 · · · ain :, since (aσ1 ) + · · · + (aσn ) = and σ1 = (¯ı1 , l1 ) < i1 . Moreover, by Lemma 6.6, R is linear combination of ordered monomials with primary degree < , hence smaller than : ai1 · · · ain :. This proves the first part of the lemma. (b) As before, we can decompose : ai1 . . . ain : = T k1 : a¯ ¯ı1 ai2 . . . ain : k1 − : a(¯ı1 ,l1 ) a(¯ı2 ,k2 +l2 ) · · · a(¯ın ,kn +ln ) :, l1 , · · · , ln l1 ≥ 1, l2 , . . . , ln ≥ 0 l1 + · · · + ln = k1 and we want to prove that every monomial : a(¯ı1 ,l1 ) a(¯ı2 ,k2 +l2 ) · · · a(¯ın ,kn +ln ) : is a linear combination of elements of BT smaller than T k1 : a¯ ¯ı1 ai2 . . . ain :. By the above argument, we can decompose : a(¯ı1 ,l1 ) a(¯ı2 ,k2 +l2 ) · · · a(¯ın ,kn +ln ) : = : aσ1 aσ2 · · · aσn : +R, where : aσ1 aσ2 · · · aσn :∈ B is the ordered monomial obtained by reordering the indices of : a(¯ı1 ,l1 ) a(¯ı2 ,k2 +l2 ) · · · a(¯ın ,kn +ln ) :, and R ∈ V , for some < . Since (¯ı1 , l1 ) < i1 ≤ (iq , kq + lq ), ∀q = 2, . . . , n, it follows by induction on (, i1 , . . . , in ) that : aσ1 · · · aσn : is a linear combination of elements of BT smaller than T k1 : a¯ ¯ı1 ai2 . . . ain :. By Lemma 6.6 R is a linear combination of ordered monomials : aτ1 · · · aτp : ∈ B, with
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(aτ1 ) + · · · + (aτp ) < . We thus conclude, by induction on (, i1 , . . . , in ), that each such ordered monomial is a linear combination of elements of BT smaller than T k1 : a¯ ¯ı1 ai2 . . . ain :. Lemma 6.8. BT is a basis of V . Proof. By Lemma 6.7(b), BT spans V . Suppose by contradiction that there is a relation of linear dependence among elements of BT , and let T k1 : a¯ ¯ı1 ai2 . . . ain : be the largest element of BT (with respect to the ordering defined in Lemma 6.7) with a non zero coefficient c. By Lemma 6.7(a), we can rewrite such a relation as a relation of linear dependence among elements of B, which is not identically zero since the coefficient of : a(¯ı1 ,k1 ) ai2 . . . ain : is c = 0. This clearly contradicts the fact that B is a basis of V . The basis BT of V induces a new embedding ρT : V → T (R), defined by ρT (T k1 : a¯ ¯ı1 ai2 . . . ain :) = T k1 (a¯ ¯ı1 ⊗ ai2 ⊗ · · · ⊗ ain ), for every basis element T k1 : a¯ ¯ı1 ai2 . . . ain : ∈ BT . Notice that ρT commutes with the action of T . Moreover, BT induces a secondary –gradation on V , V =
V [](2) ,
(6.8)
∈
defined by assigning to every basis element T k1 : a¯ ¯ı1 ai2 . . . ain : ∈ BT degree, (T k1 : a¯ ¯ı1 ai2 . . . ain :) = (ai1 ) + · · · + (ain ). Lemma 6.9. (a) The –filtration (6.1) of V is induced by the secondary –gradation (6.8). (b) The embedding ρT preserves the filtration: ρT (V ) ⊂ T (R). Proof. The first part of the lemma is an obvious corollary of Lemma 6.7. The second part follows by the first part and the obvious inclusion ρT (V [](2) ) ⊂ T (R)[], ∀ ∈ . Lemma 6.10. (a) R is a non–linear skew–symmetric conformal algebra with λ–bracket Lλ : R ⊗ R −→ C[λ] ⊗ T (R), given by Lλ (a, b) = ρT ([a λ b]),
∀a, b ∈ R.
(b) Such a λ–bracket is compatible with [ λ ], namely π(Lλ (A, B)) = [π(A) λ π(B)], π(N (A, B)) = : π(A)π(B) :,
∀A, B ∈ T (R).
Here Lλ and N are defined on T (R) thanks to Lemma 3.2.
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Proof. (a) Since ρT commutes with the action of T , it immediately follows that Lλ satisfies sesquilinearity and skew–symmetry. The grading condition for Lλ follows by the assumptions on V and Lemma 6.9(b). Indeed Lλ (R[1 ], R[2 ]) = ρT ([R[1 ] λ R[2 ]]) ⊂ C[λ] ⊗ ρT (V ) ⊂ C[λ] ⊗ T (R), for some < 1 + 2 . (b) Notice that, by definition of π and ρT , we have π ◦ ρT = 1IV . Hence π(Lλ (a, b)) = [a λ b], ∀a, b ∈ R. Part (b) then follows by the definition of Lλ and N on T (R), and by an easy induction argument. Notice that in the proof of Lemma 5.1 we did not use the assumption that R was a non–linear Lie conformal algebra. As an immediate consequence we get the following Lemma 6.11. Every element A ∈ T (R) decomposes as A = B + M, with B ∈ spanC B˜ and M ∈ M (R) (we are using the notation introduced in Sect. 5). Lemma 6.12. The λ–bracket Lλ : R ⊗ R → C[λ] ⊗ T (R) satisfies the Jacobi identity (3.9). Proof. We need to show that for elements a, b, c of R we have J(a, b, c; λ, µ) = Lλ (a, Lλ (b, c)) − p(a, b)Lµ (b, Lλ (a, c)) −Lλ+µ (Lλ (a, b), c) ∈ M (R) for some ∈ such that < (a) + (b) + (c). By the grading condition (3.8) on Lλ , we have J(a, b, c; λ, µ) ∈ T (R), for some < (a) + (b) + (c). Therefore, by Lemma 6.11, we have J(a, b, c; λ, µ) = JB + JM ,
(6.9)
where JB ∈ spanC B˜ and JM ∈ M (R). By Lemma 6.10(b) Lλ is compatible with [ λ ]. Since V is a vertex algebra, we have π(J(a, b, c; λ, µ)) = [a λ [b µ c]] − p(a, b)[b µ [a λ c]] −[[a λ b] λ+µ c] = 0.
(6.10)
Moreover, since M(R) ⊂ Kerπ, we also have π(JM ) = 0.
(6.11)
We thus get, from (6.9),(6.10) and (6.11), that π(JB ) = 0. On the other hand, since V is freely generated over V¯ , we have that π : spanC B˜ → V is an isomorphism of vector spaces. We thus conclude, as we wanted, that JB = 0, hence J(a, b, c; λ, µ) ∈ M (R).
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The above lemmas prove the first part of Theorem 6.4. We are left to prove the second part, namely uniqueness of the structure of non–linear Lie conformal algebra on R. Proof of Theorem 6.4, second part. Suppose Lλ , L λ are two structures of non–linear Lie conformal algebra on R, such that the corresponding enveloping vertex algebras are both isomorphic to V . Let us denote by M (R) (respectively M (R)) the subspace introduced in Definition 3.3 corresponding to Lλ (resp. L λ ). By assumption V T (R)/M (R) T (R)/M (R), hence M (R) = M (R). Let π : T (R) → T (R)/M (R) V be the natural quotient map. Since the vertex algebra structures of V , T (R)/M(R) and T (R)/M (R) are compatible, we must have π (N(A, B)) = π (N (A, B)), with ≤ (A) + (B), π (Lλ (A, B)) = π (L λ (A, B)), with < (A) + (B). It follows that N (A, B) − N (A, B) ∈ M (R), with ≤ (A) + (B), L λ (A, B) − Lλ (A, B) ∈ C[λ] ⊗ M (R), with < (A) + (B), namely the identity map on R gives a non–linear Lie conformal algebra isomorphism between (R, Lλ ) and (R, L λ ) Remark 6.13. Let V be any simple graded vertex algebra strongly generated by a C[T ]– submodule R. Arguments similar to the ones used in the proof of Theorem 6.4 show that the vertex algebra structure of V induces on R a structure of a non–linear conformal algebra. We have to consider separately two cases. 1. If R is a non–linear Lie conformal algebra, then it is not hard to show that V is the simple graded quotient of the universal enveloping vertex algebra U (R) by a unique maximal ideal. In this case we say that the vertex algebra V is non–degenerate. We thus conclude that non–degenerate vertex algebras are classified by non–linear Lie conformal algebras. Their classification will be studied in a subsequent paper. 2. If on the contrary R is a non–linear conformal algebra, but Jacobi identity (3.9) does not hold, we say that the vertex algebra V is degenerate. In this case one can still describe V as a quotient of the universal enveloping vertex algebra U (R) of R. The main difference is that in general U (R) is not freely generated by R, namely the PBW theorem fails. For this reason, the study (and classification) of degenerate vertex algebras is more complicated than in the non–degenerate case. Their structure theory will be developed in subsequent work. Acknowledgement. We would like to thank M. Artin, B. Bakalov, A. D’andrea and P. Etingof for useful discussions. This research was conducted by A. De Sole for the Clay Mathematics Institute. The paper was partially supported by the NSF grant DMS0201017.
References [BK03] [Bor86]
Bakalov, B., Kac, V.: Field algebras. Int. Math. Res. Not. 3, 123–159 (2003) Borcherds, R.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Nat. Acad. Sci. U.S.A. 83(10), 3068–3071 (1986)
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de Boer, J., Tjin, T.: The relation between quantum W algebras and Lie algebras. Commun. Math. Phys. 160(2), 317–332 (1994) [DK98] D’Andrea, A., Kac, V.: Structure theory of finite conformal algebras. Selecta Math. (N.S.) 4(3), 377–418 (1998) [DS03] De Sole, A.: Vertex algebras generated by primary fields of low conformal weight. PhD dissertation, MIT, Department of Mathematics, June 2003 [DL93] Dong, C., Lepowsky, J.: Generalized vertex algberas and relative vertex operators. Progress in Math 112, Boston: Birkhauser, 1993 [FF90] Feigin, B., Frenkel, E.: Quantization of the Drinfel d-Sokolov reduction. Phys. Lett. B 246(1-2), 75–81 (1990) [FK02] Fattori, D., Kac, V.: Classification of finite simple Lie conformal superalgebras. J. Algebra 258(1), 23–59 (2002) [FHL93] Frenkel, I., Huang, Y-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. J. Memoirs Amer. Math. Soc. 104, (1993) [FLM88] Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. Pure and Appl. Math., Vol 134, Boston: Academic Press, 1988 [GMS04] Gorbounov,V., Malikov, F., Schechtman,V.: Gerbes of chiral differential operators. II. Invent. math. 155, 605–680, 2004. [Jac62] Jacobson, N.: Lie algebras. In: Interscience Tracts in Pure and Applied Mathematics, No. 10, New York, London: Interscience Publishers (a division of John Wiley & Sons), 1962 [Kac97] Kac, V.: Superconformal algebras and transitive group actions on quadrics. Commun. Math. Phys. 186(1), 233–252, (1997). Erratum 217, 697–698, 2001. [Kac96] Kac, V.: Vertex algebras for beginners, Volume 10 University Lecture Series. Providence, RI: American Mathematical Society, 1996. Second edition, 1998 [KW04] Kac, V., Wakimoto, M.: Quantum reduction and representation theory of superconformal algebras. Adv. Math. 185, 400–458, 2004. [Li96] Li, H-S.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure and Appl. Alg. 109, 143–195 (1996) [Li02] Li, H-S.: Vertex algebras and vertex Poisson algebras. Commun. Contemp. Math 6, 61–110 (2004) [Primc99] Primc, M.: Vertex algebras generated by Lie algberas. J. Pure and Appl. Alg. 135, 253–293 (1999) [RS76] Ramond, P., Schwarz, J.: Classification of dual model gauge algebras. Phys. Lett. B 64(1), 75–77 (1976) [Zam85] Zamolodchikov, A.: Infinite extra symmetries in two-dimensional conformal quantum field theory. Teoret. Mat. Fiz. 65(3), 347–359 (1985) Communicated by L. Takhtajan
Commun. Math. Phys. 254, 695–717 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1246-9
Communications in
Mathematical Physics
(Non)regularity of Projections of Measures Invariant Under Geodesic Flow Esa J¨arvenp¨aa¨ , Maarit J¨arvenp¨aa¨ , Mika Leikas Department of Mathematics and Statistics, University of Jyv¨askyl¨a, P. O. Box 35, 40014 Finland. E-mail: [email protected]; [email protected]; [email protected] Received: 20 December 2003 / Accepted: 25 June 2004 Published online: 11 January 2005 – © Springer-Verlag 2005
Abstract: We show that, unlike in the 2-dimensional case [LL], the Hausdorff dimension of a measure invariant under the geodesic flow is not necessarily preserved under the projection from the unit tangent bundle onto the base manifold if the base manifold is at least 3-dimensional. In the 2-dimensional case we reprove the preservation theorem due to Ledrappier and Lindenstrauss [LL] using the general projection formalism of Peres and Schlag [PS]. The novelty of our proof is that it illustrates the reason behind the failure of the preservation in the higher dimensional case. Finally, we show that the projected measure has fractional derivatives of order γ for all γ < (α − 2)/2 provided that the invariant measure has finite α-energy for some α > 2 and the base manifold has dimension 2. 1. Introduction Several indications have been brought for and against the importance and relevance of fractality for different observed phenomena. In this context, there are two important aspects related to physical experiments. First of all, the number of degrees of freedom in realistic systems is usually huge, that is, the phase space is high dimensional. On the other hand, the number of measurements which can be reasonably taken in one experiment is relatively small. As a result, one obtains sharp information only on a few variables whilst the remaining ones must be treated in some averaging or effective manner. This may be interpreted by saying that a measurement is a projection which leads to the need to understand the mathematical theory of projections. Indeed, fractal features of projections have recently been the subject of intensive study. These include, for example, projections of SRB-measures of coupled map lattices [BKL, JJ] and those of measures invariant under the geodesic flow [LL]. In the theory of coupled map lattices projections play a crucial rˆole in the very definition of SRB-measures (see [BS, BK1, BK2]). It has turned out that the projectional
MJ and ML acknowledge the support of the Academy of Finland, project #48557.
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properties of dimensions imply that the natural definition of the SRB-measure given by Bunimovich and Sinai [BS] and Bricmont and Kupiainen [BK1, BK2] has to be modified in order to obtain a physically acceptable concept (see [JJ, J]). Dimensional properties of projections of sets and measures have been investigated for decades. The study of the behaviour of Hausdorff dimension under projection-type mappings dates back to the 1950’s when Marstrand [Mar] proved a well-known theorem according to which the Hausdorff dimension of a planar set is preserved under typical orthogonal projections. In [K] Kaufman verified the same result using potential theoretical methods, and in [Mat1] Mattila generalized it to higher dimensions. For measures the analogous principle, discovered by Kaufman [K], Mattila [Mat2], Hu and Taylor [HT], and Falconer and Mattila [FM], can be stated in the following form: Let m and n be integers such that 0 < m < n and let µV be the image of a compactly supported Radon measure µ on Rn under the orthogonal projection onto an m-plane V . Then for almost all m-planes V we have dimH µV = dimH µ provided that dimH µ ≤ m.
(1.1)
On the other hand, for almost all m-planes V , µV Lm provided that dimH µ > m.
(1.2)
(Above dimH is the Hausdorff dimension, Lm is the m-dimensional Lebesgue measure, and the symbol denotes the absolute continuity.) In the case that µ has finite m-energy a substantially stronger form of (1.2) holds: we have for all typical m-planes that µV Lm with Radon–Nikodym derivative in L2 .
(1.3)
Analogies of these results have been investigated for typical smooth mappings in the sense of prevalence and for infinite dimensional spaces in [SY, HK1, and HK2]. In [PS] Peres and Schlag extended (1.1), (1.2), and (1.3) to Sobolev dimensions of measures on compact metric spaces and parametrized families of transversal mappings in an elegant way. Their formalism has turned out to be a powerful tool when considering the uniqueness of SRB-measures of coupled map lattices [JJ]. For the purposes of the present paper, a significant difference between the earlier results and those of [PS] is that Peres and Schlag generalized (1.3) in terms of fractional derivatives by showing that if the original measure has finite (m + ε)-energy, then densities of typical projections onto m-dimensional spaces have fractional derivatives of order ε/2 in L2 . For more detailed information about a variety of related contributions, see [Mat4] and [PS]. In this paper we address the question of studying measures on Riemannian manifolds which are invariant under the geodesic flow. Although they are measures on the unit tangent bundle of the manifold, that is, on (a subset of) the phase space of the system, from the physical point of view it is important to try to describe their properties on the configuration space. After all, in many situations one is interested only in the positions of the particles and not their velocities. This leads to the study of the natural projection from the unit tangent bundle onto the base manifold. (For a discussion of connections to the Besicovitch-Kakeya problem, see [LL].) Even though the above mentioned results (1.1), (1.2), and (1.3) are genuinely “almost all”-results, meaning that they do not provide information about any specified projection, similar methods work for the natural projection from the unit tangent bundle onto the Riemannian surface. This interesting feature was discovered quite recently by Ledrappier and Lindenstrauss in [LL].
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Theorem 1.1 (Ledrappier, Lindenstrauss). Let M be a compact Riemannian surface, let µ be a Radon probability measure on the unit tangent bundle SM, and let : SM → M be the natural projection. Assuming that µ is invariant under the geodesic flow, the following properties hold for the image ∗ µ of µ under : 1. If dimH µ ≤ 2, then dimH ∗ µ = dimH µ. 2. If dimH µ > 2, then ∗ µ L2 . Analogously to (1.3), Ledrappier and Lindenstrauss proved that if µ has finite αenergy for α > 2, then the Radon–Nikodym derivative is a L2 -function. They also addressed the question of whether this could be further generalized in terms of fractional derivatives. In addition to giving a positive answer to this question by employing the techniques from [PS], we consider another issue brought up in [LL] which is the validity of Theorem 1.1 for higher dimensional base manifolds. Quite surprisingly, it appears that the Hausdorff dimension is not necessarily preserved. Recalling the case of (1.1), (1.2), and (1.3), one might first think that the generalization from dimension 2 to higher dimensions is a question of finding correct methods. However, in Sect. 4 we give a new proof for Theorem 1.1 which explains why the preservation fails in higher dimensions. This paper is organized as follows: In Sect. 2 we discuss the general projection formalism of Peres and Schlag [PS] which plays an important rˆole in this work, whereas in Sect. 3 we recall the basic assumptions from [LL] and introduce our setting. The main part of Sect. 4 is devoted to proving that the parametrized family of mappings we are working with is transversal (Proposition 4.1). Then we apply the machinery of [PS] and a result from [JJL] to reprove Theorem 1.1, and explain why this does not work for higher dimensional base manifolds (Remark 4.6). The question concerning the fractional derivatives of the density of the projected measure will be dealt with in Sect. 5. We prove that if the α-energy of µ is finite for some α > 2, then ∗ µ has fractional derivatives of order γ in L2 for all γ < (α − 2)/2 (Theorem 5.1). Finally, in the last section we give examples of higher dimensional manifolds and invariant measures on the unit tangent bundles whose Hausdorff dimensions decrease when projected onto the base manifolds. Remark 4.6 gives a base for constructing such examples. 2. General Projection Formalism of Peres and Schlag In this section we recall the notation and results we need from [PS]. Given γ ≥ 0, let ν2,γ be the Sobolev norm of a finite Borel measure ν on Rn , that is, 1/2 ν2,γ = |ˆν (ξ )|2 |ξ |2γ dLn (ξ ) , where
νˆ (ξ ) =
e−iξ ·x dν(x)
is the Fourier transform of ν. The Sobolev dimension of ν is dimS ν = sup α ∈ R | |ˆν (ξ )|2 (1 + |ξ |)α−n dLd (ξ ) < ∞ . Given α ≥ 0, the α-energy of a finite Borel measure ν on a compact metric space (Y, d) is denoted by Iα (ν), that is,
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d(x, y)−α dν(x)dν(y).
Iα (ν) = Y
Y
For the rest of this section, we restrict our consideration to the one dimensional parameter space. Basic assumptions. Let (Y, d) be a compact metric space, let J ⊂ R be an open interval, and let P : J × Y → R be a continuous function. Assume that for any l = 0, 1, . . . l ≥ 1 such that there is a constant C l |∂tl P (t, y)| ≤ C
(2.1)
for all t ∈ J and y ∈ Y . Here ∂tl is the l th partial derivative with respect to t. For all t ∈ J and x, y ∈ Y with x = y, define Tt (x, y) =
P (t, x) − P (t, y) . d(x, y)
(2.2)
We assume that the following form of transversality holds: there is a constant CT such that for all t ∈ J and for all x, y ∈ Y with x = y the condition |Tt (x, y)| ≤ CT implies that |∂t Tt (x, y)| ≥ CT .
(2.3)
In addition, the function Tt is assumed to be regular in the following sense: for all l = 0, 1, . . . there exists a constant Cl such that |∂tl Tt (x, y)| ≤ Cl
(2.4)
for all t ∈ J and x, y ∈ Y with x = y. In the following theorem from [PS], which serves as a significant tool in Proposition 4.3, we use the notation Pt (·) = P (t, ·). Moreover, we denote by f∗ µ the image of a measure µ on X under a mapping f : X → Z defined as f∗ µ(A) = µ(f −1 (A)) for all A ⊂ Z. Theorem 2.1. Suppose that the assumptions (2.1), (2.3), and (2.4) are satisfied. Let α > 0 and let ν be a finite Borel measure on Y such that Iα (ν) < ∞. Then there is a constant Cγ such that (Pt )∗ ν22,γ dL1 (t) ≤ Cγ Iα (ν) (2.5) J
provided that 0 < 1 + 2γ ≤ α. Moreover, for any σ ∈ (0, min{α, 1}] we have dimH {t ∈ J | dimS (Pt )∗ ν ≤ σ } ≤ 1 + σ − α. Proof. See [PS, Theorem 2.8].
(2.6)
We complete this section by stating a technical lemma which plays an important rˆole in relating our setting to that of [PS].
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Lemma 2.2. For all t ∈ (0, 1), let νt be a compactly supported Radon measure on R. Suppose that µ is a Radon measure on R × (0, 1) such that for all Borel functions g : R × (0, 1) → R, g(x, t) dµ(x, t) = g(x, t) dνt (x) dL1 (t). (2.7) Assume that there is α > 0 such that dimH νt ≥ α for L1 -almost all t ∈ (0, 1). Then dimH µ ≥ 1 + α. Proof. The proof of [JJL, Lemma 3.4] goes through in our setting. One simply needs to replace in the proof of [JJL, Lemma 3.4] the assumption according to which Iα (νt ) < ∞ for all t by the weaker one of Lemma 2.2. 3. Notation In this section, we define a transversal mapping appropriate to the setting of Sect. 2. Our notation is similar to that in [LL]. Assume that M is a smooth compact 2-dimensional Riemannian manifold. Denoting by SM the unit tangent bundle, let µ be a Radon probability measure on SM which is invariant under the geodesic flow, and let : SM → M be the natural projection. Since, in general, the measure µ is too complicated to handle, we have to divide it into small pieces. The fact that µ is invariant under the geodesic flow implies that locally a suitable restriction of µ is roughly of the form ν × L1 , where ν is a measure on a two dimensional square. We will proceed by showing that the projection of this restriction of µ is in a certain sense of the form νt × L1 (see Lemma 3.2) where νt is a projection of ν onto one dimensional space. In this way one obtains a family of projections parametrized by t and this family will turn out to be transversal (see Proposition 4.1). We continue by formalizing this idea. Taking p1 , p2 ∈ M sufficiently close to each other, we denote by γp1 ,p2 the unique shortest geodesic, parametrized by the Riemannian arc length, which connects p1 and p2 , that is, γp1 ,p2 (0) = p1 and γp1 ,p2 (dM (p1 , p2 )) = p2 .
(3.1)
Here dM is the distance induced by the Riemannian metric. Basic assumptions. Let I = [0, 1]. We choose an open set U ⊂ M and a chart : U → R2 with the following properties: (1) I 2 ⊂ (U ). (2) Defining C1 = −1 (I × {0}) and C2 = −1 (I × {1}) and picking any c1 ∈ C1 and c2 ∈ C2 , there exists a unique geodesic γc1 ,c2 connecting c1 and c2 such that its image (γc1 ,c2 (t)) = (x1 (t), x2 (t)) satisfies |x1 (t)| ≤ C|x2 (t)| for some C > 0 for all t ∈ [0, dM (c1 , c2 )]. Thus the tangents of the (images of) geodesics are uniformly bounded away from being horizontal. Further, U is assumed to be so small that geodesics are close to straight lines. (We use scaled normal coordinates around a fixed point m ∈ U with (m) = (1/2, 1/2).)
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(3) Denoting by 1 : [0, t1 ] → M and 2 : [0, t2 ] → M the unique geodesics connecting the left-hand side end points of C1 and C2 , and their right-hand side end points, respectively, we assume that ( 1 ) ⊂ (U ) and ( 2 ) ⊂ (U ). As in [LL], we define a smooth map : I 2 × R → SM as follows: (y1 , y2 , t) = (γp1 ,p2 (t), γp 1 ,p2 (t)),
(3.2)
where p1 = −1 (y1 , 0) and p2 = −1 (y2 , 1) (see Fig. 1). Set D = {(y1 , y2 , t) | (y1 , y2 ) ∈ I 2 , 0 ≤ t ≤ dM (p1 , p2 )}. Then : D → (D) is a diffeomorphism by the uniqueness of geodesics (see (2)). Next we analyze how the preimages of the projection behave on I 2 ⊂ (U ) keeping in mind that we will project the restriction of µ. Any (x1 , x2 ) ∈ I 2 is a projection of some v ∈ SM if there is (an image under of ) a geodesic starting from a ∈ I × {0}, ending at b ∈ I × {1}, and going through (x1 , x2 ). Note that by the uniqueness of geodesics, for each a = (a1 , 0) the corresponding b = (b1 , 1) is unique (if it exists). Thus a pair of points (a1 , b1 ) defines uniquely a point v ∈ SM which is projected onto (x1 , x2 ). Since the pair (a1 , b1 ) contains also the information about the distance dM (a, x), we may suppress the “time” coordinate and define a function a1 → b1 such that all points on the graph of this function are mapped onto (x1 , x2 ) under the projection (see Figs. 2 and 3). Letting x1 vary and keeping x2 fixed, we obtain a family of graphs filling I 2 (see Fig. 4). The fact that all points in the same graph are mapped onto the same point under implies that these graphs define a projection Px2 : I 2 → R associated with . Note that x2 will play the rˆole of the parameter and x1 determines the domain of the associated projection. C2
p2 γp 1 ,p2 (t)
γp1 ,p2 (t)
p1 C1
Fig. 1. The value of ψ(y1 , y2 , t) for some point (y1 , y2 , t) ≈ ( 21 , 21 , 21 )
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(Fx1 ,x2 (y1 ), 1) [0, 1] × {1}
(x1 , x2 )
[0, 1] × {0} Mx1 ,x2
(y1 , 0)
Fig. 2. Definitions of the domain Mx1 ,x2 and the function Fx1 ,x2 x2
1 2
G (Fx1 ,x2 )
0
1
x1
Fig. 3. The graph of Fx1 ,x2 in the situation of Fig. 2
For the purpose of making the above idea rigorous, denote by E the subset of R2 restricted by the curves I × {0}, I × {1}, ( 1 ), and ( 2 ). Given any (x1 , x2 ) ∈ E, let Mx1 ,x2 = {y1 ∈ I | there is y2 ∈ I such that the geodesic γ −1 (y1 ,0), −1 (y2 ,1) goes through −1 (x1 , x2 )}.
(3.3)
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Note that, by (2), for all y1 ∈ Mx1 ,x2 the point y2 ∈ I in (3.3) is unique (provided x2 > 0). Moreover, Mx1 ,x2 = ∅ for all (x1 , x2 ) ∈ E. For all (x1 , x2 ) ∈ E, we define a function Fx1 ,x2 : Mx1 ,x2 → I by Fx1 ,x2 (y1 ) = y2 where y2 is as in (3.3) (see Fig. 2 and 3). (If x2 = 0, we consider the vertical line segment I above y1 recalling that the important object is the graph of Fx1 ,x2 .) Lemma 3.1. The mapping Fx1 ,x2 has the following properties: 1. If (x1 , x2 ), (x˜1 , x2 ) ∈ E such that x˜1 > x1 , we have Fx˜1 ,x2 (y1 ) > Fx1 ,x2 (y1 ) for all y1 ∈ Mx1 ,x2 ∩ Mx˜1 ,x2 . 2. Given (x1 , x2 ), (x˜1 , x2 ) ∈ E with x˜1 → x1 , we have Fx˜1 ,x2 (y1 ) → Fx1 ,x2 (y1 ) for all y1 ∈ Mx1 ,x2 ∩ Mx˜1 ,x2 . 3. For all y1 , y2 ∈ I and x2 ∈ I there exists x1 such that (x1 , x2 ) ∈ E and Fx1 ,x2 (y1 ) = y2 . Proof. The claims follow directly from the definitions by (2).
Now we are ready to define the family of projections associated with . All the points belonging to the same graph G(Fx1 ,x2 ) should be mapped onto the same point. To choose this point, we fix a line Lx2 which is roughly perpendicular to the graphs and define the image of the points in G(Fx1 ,x2 ) to be the intersection point of this graph and the line Lx2 . Since near the corners of I 2 there is no intersection point (see Fig. 4) we have to replace I 2 by a smaller square I2 with the same centre as I 2 . To be more precise, given t ∈ I , let Lt be the line in R2 which goes through (1/2, 1/2) and is orthogonal to the line segment going through the points in ∂(I 2 ) ∩ G(F1/2,t ) (see Fig. 4). (Here the boundary of a set A is denoted by ∂A.) Note that our assumptions guarantee that {(1/2, t) | t ∈ I } ⊂ E, and furthermore, the set ∂(I 2 ) ∩ G(F1/2,t ) contains
y Lt
pt,y
I2 Fig. 4. The foliation of I 2 and the value of Pt for some t ≈ 13
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exactly two points. We may choose I2 ⊂ I 2 such that for all t ∈ I and (y1 , y2 ) ∈ I2 the intersection Lt ∩ G(Fx,t ) is a singleton for x ∈ I with Fx,t (y1 ) = y2 (see Lemma 3.1 (3)). This enables us to define a function P : I × I2 → R by P (t, y) = pt,y ,
(3.4)
where y = (y1 , y2 ) ∈ I2 , pt,y is the unique point in Lt ∩ G(Fx,t ), and the point x is determined by Fx,t (y1 ) = y2 . Here Lt is identified with R such that the origin is at (1/2, 1/2). Later we will use the abbreviation Pt (·) for the map P (t, ·).
Invariant measure under geodesic flow. Similarly as in [LL], we restrict our consid = (D) eration to the normalized restriction measure µ = µ(U˜ )−1 µ|U , where U and = {(y1 , y2 , t) | (y1 , y2 ) ∈ I2 , 0 ≤ t ≤ dM ( −1 (y1 , 0), −1 (y2 , 1))}. D ∩ A) for all A ⊂ SM.) Since µ is invariant under the geodesic (Here µ|U (A) = µ(U µ. We call a measure locally flow, there is a measure ν on I2 such that ∗ (ν × L1 ) = invariant if it is of this form for some ν. Next we will represent the measure ∗ µ in a form which allows us to apply the general projection formalism of Sect. 2. Observe that the preimage of a point (x1 , x2 ) ∈ E whose projection onto I2 is G(Fx1 ,x2 ). Since the distance under ◦◦ is a curve on D is not from (y1 , 0) to (x1 , x2 ) depends on y1 , the “time” coordinate of this preimage on D constant. Hence we have to first rescale “time” and then use the map P . For this purpose, ). We define, for given t ∈ I and ω ∈ ( ◦◦ )−1 {(x, t) | (x, t) ∈ V }, let V = ◦(U B1 (ω1 , ω2 , ω3 ) = (ω1 , ω2 , t).
(3.5)
→ I2 × I is a diffeomorphism since geodesics are not horizontal. Setting Now B1 : D P (ω1 , ω2 , t) = (Pt (ω1 , ω2 ), t) for all (ω1 , ω2 , t) ∈ I2 × I , we find for all (x, t) ∈ V a ◦ B1 (( ◦ ◦ )−1 {(x, t)}) = (x, unique point x˜ ∈ R such that P ˜ t). Defining ˜ t) B2 (x, t) = (x, and using the fact that B1 (( ◦ ◦ )−1 {(x, t)}) = {(y1 , y2 , t) | y2 = Fx,t (y1 )}, we get a diffeomorphism B2 : V → B2 (V ). Lemma 3.2. The following properties hold: ◦ B1 )∗ (ν × L1 ). (1) ( ◦ )∗ µ = (B2−1 ◦ P (2) For all non-negative Borel functions f : R2 → R, ∗ (ν × L1 ))(x, t) = f (x, t)d((Pt )∗ ν)(x) dL1 (t). f (x, t)d(P
(3.6)
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(3) For all non-negative Borel functions g : R3 → R, g d((B1 )∗ (ν × L1 )) = g| det DB1−1 | d(ν × L1 ), where det DB1−1 is the determinant of the derivative of B1−1 . Furthermore, there is a constant C > 0 such that C −1 ≤ | det DB1−1 | ≤ C. (4) There exists a constant C > 0 such that for all Borel sets A ⊂ R2 , 1 ◦ B1 )∗ (ν × L1 )(A) ≤ C P ∗ (ν × L1 )(A). P∗ (ν × L1 )(A) ≤ (P C (5) There is a constant C > 0 such that C −1 ≤ | det DB2−1 | ≤ C. Proof. Clearly, (1) follows from the definitions, and (2) is a straightforward consequence of Fubini’s theorem. Noting that B1 can be written in the form B1 (x1 , x2 , t) = (x1 , x2 , b(x1 , x2 , t)), Fubini’s theorem gives the equality in (3). Our basic assumption (2) guarantees the existence of a constant C such that C −1 ≤ | det D(B1−1 )| ≤ C concluding the proof of (3). Finally, applying (3) gives (4), and (5) follows similarly as (3). 4. Transversality and Preservation of Hausdorff Dimension in Two Dimensional Manifolds In this section we discuss connections between [LL] and [PS]. In particular, we give a new proof of Theorem 1.1 which explains why the corresponding result fails if the dimension of the base manifold is more than 2 (see Remark 4.6). The machinery developed in this section leads us to prove in Sect. 5 that the Radon–Nikodym derivative d∗ µ has fractional derivatives in the Sobolev sense. An essential step is to prove that d L2 the function Tt , defined as in (2.2) in terms of the function P given in (3.4), has the crucial property of being transversal. Proposition 4.1. Let P be as in (3.4). Then (2.1) is satisfied. Furthermore, defining for all t ∈ I and x = y ∈ I2 , Tt (x, y) =
P (t, x) − P (t, y) , |x − y|
properties (2.3) and (2.4) hold. Proof. Observing that (2.1) and (2.4) follow directly from the definitions, it suffices to prove that the transversality condition (2.3) is satisfied. The idea of the proof of transversality is most easily explained if we assume that the manifold is a flat torus. Then the geodesics are straight lines and the graphs G(Fx,t ) are parallel straight lines with slopes given by the equation tan α = (1 − t)/t (see Fig. 5 and (4.1) where now a = b). Moreover, P (t, ·) is an orthogonal projection and Tt (x, y) reduces to P (t, v), where v = (x − y)/|x − y|. By (4.2), the change of the parameter
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(x, t)
b y1 Fig. 5. The notation for determining the slope of the graph Fx,t at a point (y1 , y2 )
t means a comparable change of the slope of Lt . Therefore the proof of transversality reduces to the case where Lt is spanned by (cos t, sin t), v = (0, 1), and t = 0. Then ∂t P (t, v)|t=0 = ∂t sin t|t=0 = 1. To check that this simple idea works also in the general case involves several careful estimates which we make below. Given (x, t), let α be the slope of the graph of Fx,t at a point (y1 , y2 ). Using the notation introduced in Fig. 5, one may deduce the formula tan α =
(1 − t) sin2 b(x, t) t sin2 a(x, t)
(4.1)
from elementary geometrical arguments. Note that the basic assumption (2) in Sect. 3 guarantees that both the angles a and b are bounded away from 0 and π and are close to each other. Combining this with Eq. (4.1), in turn, implies the existence of a positive constant C1 such that dα (4.2) ≥ C1 dt for all t. Letting ε > 0, consider x = y such that |P (t, x) − P (t, y)| ≤ ε|x − y|.
(4.3)
We will show that, choosing ε small enough, we have for all small h, |P (t + h, x) − P (t + h, y) − (P (t, x) − P (t, y))| ≥ ε|x − y|h. This clearly gives the transversality condition (2.3).
(4.4)
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Note that our assumptions guarantee the existence of a constant CF (independent of x, y, and t) such that max {|z1 − z2 | | z1 ∈ G(Fx,t ) ∩ K, z2 ∈ G(Fy,t ) ∩ K}
KLt
≤ CF min {|z1 − z2 | | z1 ∈ G(Fx,t ) ∩ K, z2 ∈ G(Fy,t ) ∩ K},
(4.5)
KLt
where both the maximum and the minimum are taken over all lines K that are parallel to Lt (denoted by the symbol K Lt ). Using the notation shown in Fig. 6, we have |x − a| ≤ εCF |x − y|, |a − b| ≥ C2 |x − y| h, |c − d| − |e − f | ≤ C3 ε|x − y| h,
(4.6)
where both C2 and C3 are constants that do not depend on x, y, and t. In fact, the first inequality in (4.6) is a consequence of (4.5) and (4.3). Choosing ε < 1/(2CF ), the second inequality follows from the first one and the fact that there is a constant C such that |a − b| ≥ C|a − y| h (see (4.2)). For the last one, observe first that, since the geodesics are close to lines in V and depend smoothly on the initial data, there is a constant C (independent of x, y, and t) such that |c − d| − |e − f | ≤ C |w1 − w2 | − |w3 − w4 |, (4.7) Lt+h
Lt
Kt g f
d
c
e b a
x
y
Fig. 6. Above the line Kt goes through x and is parallel line to Lt , {a} = Kt ∩ G (F·,t ), {b} = Kt ∩ G (F·,t+h ), c = P (t, x), d = P (t, y), e = P (t + h, x), f = P (t + h, a), and g = P (t + h, y)
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where w1 , w2 , w3 , and w4 are as in Fig. 7. Using the fact that the closer to each other the geodesics are, the more they look like parallel curves in V , we get |w1 − w2 | − |w3 − w4 | ≤ C|w 1 − w2 | h ≤ C|x − a| h. and C are constants that are independent of x, y, and t.) This, in turn, combined (Here C with (4.7) and the first inequality in (4.6), completes the proof of the last inequality of (4.6). Finally, after noting that for small h we have |f − g| ≥ (1/(2CF ))|a − b| by (4.5), we deduce from (4.6) |c − d| − |e − g| = |f − g| − |c − d| − |e − f | ≥ C3 ε|x − y| h for ε < min{1/(2CF ), C2 /(4CF C3 )}. Hence (4.4) follows.
As a corollary of Proposition 4.1, one obtains quite easily a new proof for Theorem 1.1. This is achieved by means of Proposition 4.3. Recall that the Hausdorff dimension of a finite Borel measure µ on a Riemannian manifold X is defined using lower local dimensions, dimloc , as follows: dimH µ = µ- ess inf dimloc µ(x), x∈X
where log µ(B(x, r)) . log r
dimloc µ(x) = lim inf r→0
x2
t +h t
a2
w3 w1
e1 c1 f1 d1
e2
c2 f2 d2
w4 w2
x1
a1
Fig. 7. The setting for the proof of the last inequality in (4.6). The notation corresponds to Fig. 6 in a natural way
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Here B(x, r) is the open ball with centre at x and radius r > 0. The following equality relates Hausdorff dimension of measures to that of sets: dimH µ = inf{dimH A | A is a Borel set with µ(A) > 0} (see [F, Proposition 10.2]). Remark 4.2. It follows from [Mat4, Proposition 5.1] and [Mat3, Theorem 8.7] that dimH µ ≥ dimS µ provided that dimS µ < dim X. Proposition 4.3. With the notation introduced in Sect. 3, we have: 1. Assuming that dimH ν ≤ 1, we have dimH (Pt )∗ ν = dimH ν for L1 -almost all t ∈ (0, 1). 2. Assuming that dimH ν > 1, we have (Pt )∗ ν L1 for L1 -almost all t ∈ (0, 1). Proof. To verify (1), let β < dimH ν. Defining νi = ν|Ai for all i = 1, 2, . . . , where Ai = {x ∈ R2 | ν(B(x, r)) ≤ ir β for all r > 0}, one easily checks that Iα (νi ) < ∞ for all α < β, and νi (B) → ν(B) for all B ⊂ R2 . Given σ < α, we get from inequality (2.6) in Theorem 2.1 using Remark 4.2 that for L1 -almost all t ∈ (0, 1), dimH (Pt )∗ νi ≥ dimS (Pt )∗ νi > σ
(4.8)
for all i. This, in turn, implies that dimH (Pt )∗ ν ≥ σ for L1 -almost all t ∈ (0, 1). Finally, taking a sequence σj → dimH ν, gives (1), since Pt does not increase dimension as a Lipschitz function. For (2), we consider 1 < β < dimH ν and proceed as above to find a sequence (νi ) of measures with Iβ (νi ) < ∞ such that νi (B) → ν(B) for all B ⊂ R2 . Now inequality (2.5) in Theorem 2.1 implies that for L1 -almost all t ∈ (0, 1) one has ((Pt )∗ νi )∧ ∈ L2 for all i, and therefore (Pt )∗ νi L1 for all i. This gives (2). We continue by explaining how Theorem 1.1 follows from Proposition 4.3. For this purpose we need two intermediate steps: Corollary 4.4. Using the same notation as in Sect. 3, we have: ∗ (ν × L1 ) = dimH 1. If dimH µ ≤ 2, then dimH P µ. 2. If dimH µ > 2, then P∗ (ν × L1 ) L2 . µ = dimH ν + 1 (see [H] or [Mat3, Theorem 8.10]). To prove (1), Proof. Note that dimH Proposition 4.3 (1) gives dimH (Pt )∗ ν = dimH ν for L1 -almost all t ∈ R. From Lemma ∗ (ν × L1 ) ≥ dimH ν + 1 = dimH 2.2 and Lemma 3.2 (2), we deduce that dimH P µ. The is a Lipschitz mapping yields (1). fact that P For (2), let A ⊂ R2 be a Borel set with L2 (A) = 0. Setting At = {x ∈ R | (x, t) ∈ A} for all t ∈ R, and using Fubini’s theorem and Proposition 4.3 (2), we get (Pt )∗ ν(At ) = 0 for L1 -almost all t ∈ R. Combining this with Lemma 3.2 (2) concludes the proof. Corollary 4.5. Using the notation given in Sect. 3, we have: 1. If dimH µ ≤ 2, then dimH ( ◦ )∗ µ = dimH µ. 2. If dimH µ > 2, then ( ◦ )∗ µ L2 .
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Proof. Corollary 4.4, Lemma 3.2 (4), and the fact that B2−1 is a bi-Lipschitz mapping (see ◦ B1 )∗ (ν × L1 ) = dimH Lemma 3.2 (5)) combine to give the equality dimH (B2−1 ◦ P µ −1 provided that dimH µ ≤ 2, and furthermore, (B2 ◦ P ◦ B1 )∗ (ν × L1 ) L2 under the assumption dimH µ > 2. This in turn gives the claim by Lemma 3.2 (1). Since is bi-Lipschitz mapping, Theorem 1.1 follows immediately from Corollary 4.5 by representing the original measure µ as a finite sum of measures µi having the same properties as the measure µ above. Remark 4.6. In Sect. 6 we construct examples which show that Theorem 1.1 fails for higher dimensional base manifolds. The reason for the failure, which may be deduced from the above methods, is as follows: The local invariance produces a parametrized family of projections onto (n − 1)-dimensional planes in 2(n − 1)-dimensional space. The parameter is given by the time coordinate, and therefore the family is one dimensional. Since the dimension of the space of (n − 1)-planes in 2(n − 1) dimensional space is greater than 1, if n ≥ 3, the transversality condition cannot hold. 5. Fractional Derivatives In this section we answer the question concerning the fractional derivatives of the density of the projected measure ∗ µ addressed in [LL]. The main theorem of this section is as follows: Theorem 5.1. Let M be a compact smooth Riemannian surface and let : SM → M be the natural projection from the unit tangent bundle SM onto the base manifold M. Assume that µ is a Radon probability measure on SM such that µ is invariant under the geodesic flow and Iα (µ) < ∞ for some α > 2. Then for all γ < (α − 2)/2 the projected measure ∗ µ has fractional derivatives of order γ in L2 , that is, ∗ µ2,γ < ∞. Below the proof of Theorem 5.1 is divided into a sequence of lemmas. Observe that Theorem 2.1 combined with Proposition 4.1 implies the existence of fractional derivatives for almost all horizontal slices of ∗ µ, which are, in fact, diffeomorphic images of the measures (Pt )∗ ν. However, since this approach does not give the desired result for the measure ∗ µ, we modify the methods of [PS] in a more effective way. The idea of the proof is roughly as follows: In order to estimate Sobolev norms, we will first use the Littlewood–Paley decomposition to separate different frequencies (see Lemma 5.5). When estimating the Sobolev norm of the projection of µ with an appropriate energy of ν, one is essentially forced to deduce that the measure of parameters t for which |Pt (q) − Pt (q )| is small is less than some power of |q − q | (see (5.5)). This estimate will be divided into several steps (see Lemmas 5.3, 5.4, and 5.6), where we will use effectively two properties of ψ given by the Littlewood–Paley decomposition. First of all, ψ decays faster than any power guaranteeing the desired behaviour of the integral over domains where the argument of ψ is not too small. Secondly, after using the first property several times, we are reduced to a domain where the argument is small. Then we will use the fact that ψ has vanishing moments of all orders, and so the integral over this domain may be calculated over its complement. Finally, the fast decay of ψ will be applied again. Using the same notation as in the previous sections, we begin with a small technical lemma.
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Lemma 5.2. Let α > 1. Assume that µ = F∗ (ν × L1 |K ), where K ⊂ R is a compact set and F is a diffeomorphism such that C −1 ≤ | det DF | ≤ C for some C > 0. Then Iα (µ) < ∞ ⇐⇒ Iα−1 (ν) < ∞. Proof. The claim follows from straightforward calculations.
The next lemma shows that for fixed q = q ∈ I2 the mapping a → Ta (q, q ) is small only in neighbourhoods of finitely many zeroes. Lemma 5.3. For any q = q ∈ I2 there exist a1 , . . . , aN ∈ I such that {a ∈ I | |Ta (q, q )| ≤ d} ⊂
N
B(ai , CT−1 d)
i=1
for all d < CT . Moreover, the mapping a → Ta (q, q ) is a diffeomorphism on B(ai , C1−1 CT ) for all i = 1, . . . , N, and N ≤ C1 /CT + 2. (Here CT is as in (2.3) and C1 as in (2.4).) Proof. Let a1 , . . . , aN−2 be the zeroes of the function a → Ta (q, q ), and let aN−1 = 0 and aN = 1. Then all the claims follow from (2.3) and (2.4). We continue by defining mappings Hq,q and by studying their basic properties which will be needed in the proof of Lemma 5.6. Lemma 5.4. Given q = q ∈ I2 , let r = |q − q |. Define Hq,q : I 2 → R2 by Hq,q (a, b) = (Ta (q, q ) + r −1 (Pa (q ) − Pb (q )), r −1 (a − b)). Let a1 , . . . , aN ∈ I be as in Lemma 5.3. For any i = 1, . . . , N, set
Oi = {(a, b) ∈ B(ai , C1−1 CT ) ∩ (0, 1) ×(0, 1) | |Ta (q, q )| < CT and 2 )−1 CT r}, |a − b| < (2C 2 , CT , and C1 are as in (2.1), (2.3), and (2.4), respectively. Then the restricwhere C tion of Hq,q to the set Oi is a diffeomorphism onto Hq,q (Oi ). Furthermore, there are constants c and c(l) for all l ∈ N which are independent of q and q such that −1 −1 η −1 η DHq,q
< c, |∂ Hq,q | < c(|η|), and |∂ det DHq,q | < c(|η|) η
(5.1)
η
for all indices η = (η1 , η2 ) ∈ N2 . Here |η| = η1 + η2 and ∂ η = ∂a 1 ∂b 2 . Proof. By (2.1) and (2.3) we have for all (a, b) ∈ Oi , | det DHq,q (a, b)| = r −1 |∂a Ta (q, q ) − r −1 (∂b Pb (q ) − ∂a Pa (q ))| ≥ (2r)−1 CT .
(5.2)
For the first claim it is therefore sufficient to show that the restriction of Hq,q to Oi is an injection. This, in turn, follows from two easy observations: If (a, b), (a , b ) ∈ Oi with a − b = a − b , then clearly Hq,q (a, b) = Hq,q (a , b ). On the other hand, Hq,q is strictly monotone on the line segments {(a, b) ∈ Oi | b − a = d}, where d ∈ R, since |∂a Ta (q, q ) − r −1 (∂a Pa+d (q ) − ∂a Pa (q ))| ≥ 2−1 CT .
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For (5.1) note that −1 DHq,q
(y)
−r −1 r −1 ∂b Pb (q ) = −r −1 ∂a Ta (q, q ) + r −1 ∂a Pa (q ) −1 −1 A, =: (det DHq,q (Hq,q
(y))) −1 −1 (det DHq,q (Hq,q
(y)))
−1 where (a, b) = Hq,q
(y). Combining this with inequality (5.2), (2.1), and (2.4), gives
−1 DHq,q
< c. Using similar arguments and the fact that for all l ∈ N there exists a constant C(l) such that |∂ η Aij | < r −1 C(|η|) for all η and i, j , the second claim in (5.1) follows by induction. Finally, the last estimate is a consequence of the previous one.
In the following lemma which is from [PS] we denote by S(Rn ) the Schwartz space of smooth functions such that all of their derivatives decay faster than any power. Lemma 5.5. There exists ψ ∈ S(Rn ) such that ψˆ > 0, spt ψˆ ⊂ {ξ ∈ Rn | 1 ≤ |ξ | ≤ 4}, ˆ −j and ∞ j =−∞ ψ(2 ξ ) = 1 for all ξ = 0. Furthermore, for any finite Radon measure ν n on R and any γ ∈ R there exists a constant C such that ∞ 1 2 2j γ 2 (ψ2−j ∗ ν)(x)dν(x) ≤ Cν22,γ , ν2,γ ≤ n C R j =−∞ where ψ2−j (x) = 2j n ψ(2j x). (Above ∗ is the convolution.) Proof. See [PS, Lemma 4.1].
Next we prove a lemma which is a modification of [PS, Lemma 7.10] tailored for our purposes. Lemma 5.6. Assume that ρ is a smooth non-negative real valued function which is supported inside the open unit square (0, 1)2 . Let ψ be as in Lemma 5.5. Then for all q, q ∈ I2 with q = q , j ∈ Z, and k ∈ N \ {0} we have ρ(a, b)ψ(2j (Pa (q) − Pb (q ), a − b)) dL1 (a) dL1 (b) R R
≤ C min{(1 + 2j |q − q |)−k , (1 + 2j )−1 },
where the constant C does not depend on q, q , and j . Proof. Observing that it is enough to study positive integers j , and using the fast decay of ψ, we have ρ(a, b)ψ(2j (Pa (q) − Pb (q )), 2j (a − b)) dL1 (a) R ≤ c2−j + c (2j t)−2 dL1 (t) ≤ C(1 + 2j )−1 . t>2−j
As indicated by the above calculation, the difficult part to handle is the domain where a is roughly equal to b and |Pa (q) − Pb (q )| is much smaller than |q − q |. We will first estimate the “easy” parts using the fast decay of ψ, and finally, we will use the
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fact that ψ has vanishing moments of all orders to replace the “difficult” domain by its complement. For the other upper bound, fix k, j ∈ N such that k ≥ 1. Setting r = |q − q |, we may assume that 2j r > 1. Let φ : R2 → R be a smooth function such that 0 ≤ φ ≤ 1, φ ≡ 1 on [−1, 1]2 , and φ ≡ 0 on R2 \ [−2, 2]2 . Letting Hq,q : I 2 → R2 be as in Lemma 5.4, one obtains ρ(a, b)ψ(2j (Pa (q) − Pb (q ), a − b)) dL1 (a) dL1 (b) R R = ρ(a, b)ψ(2j rHq,q (a, b))φ(CT−1 Hq,q (a, b)) dL1 (a) dL1 (b) R R + ρ(a, b)ψ(2j rHq,q (a, b))(1 − φ(CT−1 Hq,q (a, b))) dL1 (a) dL1 (b) R R
=: A1 + A2 . Since the integrand of A2 is non-zero only if |Hq,q | > CT , the fact that the support spt ρ of ρ is inside (0, 1)2 and ψ ∈ S(R2 ) implies |A2 | ≤ ρ(a, b)(CT 2j r)−k dL1 (a) dL1 (b) ≤ C(1 + 2j r)−k . R R
We continue by estimating A1 . Picking a1 , . . . , aN as in Lemma 5.3, we find d2 , d3 < min{CT , C1−1 CT } such that
{a ∈ (0, 1) | |Ta (q, q )| ≤ d3 } ⊂
N
B(ai , d2 /2)
(5.3)
i=1
and N
B(ai , d2 ) ∩ (0, 1) ⊂ {a ∈ (0, 1) | |Ta (q, q )| ≤ CT /4}.
(5.4)
i=1
2 )−1 CT , (4C 1 )−1 CT }. For all i = 0, . . . , N, there exists a smooth Let d1 < min{(2C function χi : R → [0, 1] with the following properties: (1) spt χ0 ⊂ B(0, d1 ). (2) spt χi ⊂ B(ai , d2 ) for all i = 1, . . . , N. (3) Letting Oi be as in Lemma 5.4, we have χ0 (r −1 (a − b))χi (a) = 0 for all i = 1, . . . , N and (a, b) ∈ (0, 1)2 \ Oi . 1 )−1 d3 we have (4) For all (a, b) ∈ spt ρ with |Ta (q, q )| ≤ d3 and r −1 |a − b| ≤ (8C N
χ0 (r −1 (a − b))χi (a) = 1.
i=1
(5) For all l ∈ N there is a constant cl such that sup ∂ l χi ∞ ≤ cl .
0≤i≤N
(Non)regularity of Projections of Measures
713
(Note that above property (3) follows from (1), (2), and (5.4), and (5.3) makes the choice of property (4) possible.) Combining (2.1), (5.4), and properties (2) and (3) leads to χ0 (r −1 (a − b))χi (a) = χ0 (r −1 (a − b))χi (a)φ(CT−1 Hq,q (a, b)) for all (a, b) ∈ R2 and i = 1, . . . , N. (This follows from the fact that φ(CT−1 Hq,q (a, b)) = 1 if the left-hand side in the above equality is non-zero.) Therefore A1 =
N i=1
R R
+
R R
ρ(a, b)χ0 (r −1 (a − b))χi (a)ψ(2j rHq,q (a, b)) dL1 (a)dL1 (b)
N ρ(a, b) 1 − χ0 (r −1 (a − b))χi (a) ψ(2j rHq,q (a, b)) i=1
× φ(CT−1 Hq,q (a, b)) dL1 (a) dL1 (b) =:
N
Di + D.
i=1
From (4) we deduce that on the support of the integrand of D we have |Hq,q (a, b)| ≥ 1 )−1 d3 , and so, similarly as before, we get (8C |D| ≤ C(1 + 2j r)−k . Since N ≤ C1 /CT + 2 by Lemma 5.3, it suffices to show that each Di has an upper bound of the desired form. Fixing 1 ≤ i ≤ N and applying (3) and Lemma 5.4 gives Di = ρ(a, b)χ0 (r −1 (a − b))χi (a)ψ(2j rHq,q (a, b)) dL2 (a, b) O i −1 −1 j = ρ(Hq,q
(u, v))χ0 (v)χi ((Hq,q )1 (u, v))ψ(2 r(u, v)) Hq,q (Oi )
−1 2 × | det(DHq,q
(u, v))| dL (u, v),
−1 −1 where (Hq,q
)1 (u, v) is the first coordinate of Hq,q (u, v). Since the integrand of Di is zero outside Oi by (3), we may modify Hq,q in such a way that it becomes a diffeomorphism on R2 , and all the bounds given in Lemma 5.4 remain unchanged. Defining for all (u, v) ∈ R2 , −1 −1 −1 G(u, v) := ρ(Hq,q
(u, v))χ0 (v)χi ((Hq,q )1 (u, v))| det(DHq,q (u, v))|,
and choosing 0 < ε < 1 such that (k + 2)(1 − ε) > k, we rewrite Di as j 2 Di = G(y)ψ(2 ry) dL (y) + G(y)ψ(2j ry) dL2 (y) |y|<(2j r)ε−1
|y|>(2j r)ε−1
=: J1 + J2 .
From (5.1) we obtain that |J2 | ≤ c
t>(2j r)ε−1
(2j rt)− ε −1 t dL1 (t) ≤ C(1 + 2j r)−k−1 ,
and therefore, it remains to estimate J1 .
k
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E. J¨arvenp¨aa¨ , M. J¨arvenp¨aa¨ , M. Leikas
ˆ Note that ψ has vanishing moments of all orders since ∂ η ψ(0) = 0 for all η. Using the Taylor expansion for the function G, we calculate
J1 = −
|η|
+
|y|>(2j r)ε−1
j ε−1 |η|=k |y|<(2 r)
=: −
(η!)−1 ∂ η G(0)y η ψ(2j ry) dL2 (y) (η!)−1 ∂ η G(t (y)y)ψ(2j ry)y η dL2 (y)
Kη + K.
|η|
η
Here y η = y1 1 y2 2 , η! = η1 !η2 !, and t (y) ∈ [0, 1]. Finally, |K| ≤ c
|y|<(2j r)ε−1
sup ∂ η G∞ |y|k dL2 (y)
|η|=k
≤ c sup ∂ η G∞ (2j r)−(1−ε)(k+2) ≤ C sup ∂ η G∞ (1 + 2j r)−k |η|=k
|η|=k
and |Kη | ≤ c∂ η G∞
|y||η| |2j ry|−|η|−1− ε dL2 (y) k
|y|>(2j r)ε−1
≤ C∂ η G∞ (1 + 2j r)−k−1 . Thus the claim follows from Lemma 5.4.
As an immediate consequence of Lemma 5.6 we obtain the following result. Corollary 5.7. Let ρ and ψ be as in Lemma 5.6, and let q, q ∈ I2 with q = q . Then for any k, n ∈ N \ {0} we have ρ(a, b)ψ(2j (Pa (q) − Pb (q ), a − b)) dL1 (a) dL1 (b) R R
≤ C(1 + 2j |q − q |)− n (1 + 2j )− k
n−1 n
,
where C does not depend on q, q or j . Proof of Theorem 5.1. Assume that µ is a Radon probability measure on SM such that µ is invariant under the geodesic flow and Iα (µ) < ∞ for α > 2. Let γ < (α − 2)/2. ∗ (ν × ρL1 ) By Lemma 3.2 we may restrict our consideration to the measures µδ = P where δ > 0 and ρ is a smooth function such that spt ρ ⊂ (0, 1) and ρ(t) = 1 for all δ < t < 1 − δ. Letting n, k ∈ N such that α > 2 + 2γ + 1/n and k > n(1 + 2γ + 1/n), and using Lemma 5.5 and Corollary 5.7 for positive j , we have
(Non)regularity of Projections of Measures
R2
| µδ (ξ )|2 |ξ |2γ dL2 (ξ ) ≤ C
≤C
22j γ +2j
∞
=C
∞
2
I2
≤C
I2
I2 j =−∞ I2
R2
(ψ2−j ∗ µδ )(x) dµδ (x)
ψ(2j (x − y))
∗ (ν × ρL1 )(x)d P ∗ (ν × ρL1 )(y) × dP ρ(a)ρ(b)ψ(2j (Pa (q) − Pb (q ), a − b))
(5.5)
× dν(q)dL1 (a)dν(q )dL1 (b)
∞
22j γ
I2 ×R I2 ×R
j =−∞
≤C
2j γ +2j
∞ j =−∞
R2 R2
j =−∞
715
22j γ +2j (1 + 2j )−
n−1 n
(1 + 2j |q − q |)− n dν(q)dν(q ) k
|q − q |−(1+2γ +1/n) dν(q)dν(q ) = CI1+2γ +1/n (ν).
Here the last inequality follows by picking a positive integer j0 such that 2−j0 −1 ≤ r < 2−j0 and by dividing the sum into 3 parts: j < 0, 0 ≤ j ≤ j0 , and j > j0 . Using the choice of n and applying Lemma 5.2 gives the claim. 6. Non-Preservation of Hausdorff Dimension in Higher Dimensional Manifolds In this section we construct examples of (locally) invariant measures whose Hausdorff dimensions decrease under the projection onto the base manifold. Because of Remark 4.6 the following setting is natural for such examples. Example 6.1. For any n ≥ 3 there exist an n-dimensional compact smooth Riemannian manifold M and a measure µ on the unit tangent bundle SM such that it is locally invariant and its Hausdorff dimension decreases under the projection : SM → M. In fact, let M be the flat n-dimensional torus [−1, 2]n and let I n = [0, 1]n ⊂ M. Using the notation of Sect. 3, we set C1 = I n−1 × {0} and C2 = I n−1 × {1}, and define a diffeomorphism : D → (D) by
(t)), (x, y, t) = (γp,q (t), γp,q
where p = (x1 , . . . , xn−1 , 0) ∈ C1 , q = (y1 , . . . , yn−1 , 1) ∈ C2 , γp,q is the unique shortest geodesic parametrized by the Riemannian arc length which connects p and q, and D = {(x, y, t) | x, y ∈ I n−1 , 0 ≤ t ≤ dM (p, q)}. Taking any measure ν such that spt ν ⊂ {(x, y) ∈ I n−1 × I n−1 | xn−1 = yn−1 = 0}
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and defining µ = ∗ (ν × L1 ), we have dimH ∗ µ ≤ n − 1 since ∗ µ is supported by the (n − 1)-dimensional plane {(m1 , . . . , mn ) ∈ I n | mn−1 = 0}. Furthermore, µ is locally invariant, and dimH µ = dimH ν + 1. Choosing ν such that dimH ν > n − 2 gives dimH ∗ µ < dimH µ. Remark 6.2. (a) Example 6.1 is easily modified to verify the existence of a globally invariant measure whose Hausdorff dimension is not preserved when projected onto the base manifold. To see this, take ν = L2(n−2) and replace I n by M in Example 6.1. Then it is a straightforward calculation to show that dimH ( ◦ ψ)∗ (L2(n−2) × L1 ) = n − 1. Clearly, ψ∗ (L2(n−2) × L1 ) = L2n−3 is globally invariant under the geodesic flow. (b) In the case n = 3 Example 6.1 may be reduced to the 2-dimensional case. Therefore we may apply the results of Sect. 4 to deduce that dimH µ, if dimH ν ≤ 1 dimH ∗ µ = 2, if dimH ν > 1. Acknowledgement. We thank the referee for valuable comments clarifying the exposition.
References [BKL] [BK1] [BK2] [BS] [F] [FM] [H] [HT] [HK1] [HK2] [J] [JJ] [JJL] [K] [LL] [Mar] [Mat1] [Mat2] [Mat3] [Mat4]
Bonetto, F., Kupiainen, A., Lebowitz, J.L.: Absolute continuity of projected SRB measures of coupled Arnold cat map lattices. To appear in Ergodic Theory Dynam. Systems, http://arxiv.org/abs/nlin/0310009, 2003 Bricmont, J., Kupiainen, A.: Coupled analytic maps. Nonlinearity 8, 379–396 (1995) Bricmont, J., Kupiainen, A.: High temperature expansions and dynamical systems. Commun. Math. Phys. 178, 703–732 (1996) Bunimovich, L.A., Sinai, Ya. G.: Spacetime chaos in coupled map lattices. Nonlinearity 1, 491–516 (1988) Falconer, K.: Techniques in Fractal Geometry. Chichester: John Wiley & Sons, Ltd., 1997 Falconer, K., Mattila, P.: The packing dimension of projections and sections of measures. Math. Proc. Cambridge Philos. Soc. 119, 695–713 (1996) Haase, H.: On the dimension of product measures. Mathematika 37, 316–323 (1990) Hu, X., Taylor, J.: Fractal properties of products and projections of measures in Rn . Math. Proc. Cambridge Philos. Soc. 115, 527–544 (1994) Hunt, B.R., Kaloshin, Yu., V.: How projections affect the dimension spectrum of fractal measures?. Nonlinearity 10, 1031–1046 (1997) Hunt, B.R., Kaloshin, Yu., V.: Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. Nonlinearity 12, 1263–1275 (1999) J¨arvenp¨aa¨ , E.: SRB-measures for coupled map lattices. To appear in Lecture Notes in Phys., Springer J¨arvenp¨aa¨ , E., J¨arvenp¨aa¨ , M.: On the definition of SRB-measures for coupled map lattices. Commun. Math. Phys. 220, 1–12 (2001) J¨arvenp¨aa¨ , E., J¨arvenp¨aa¨ , M., Llorente, M.: Local dimensions of sliced measures and stability of packing dimensions of sections of sets. Adv. Math. 183, 127–154 (2004) Kaufman, R.: On Hausdorff dimension of projections. Mathematika 15, 153–155 (1968) Ledrappier, F., Lindenstrauss, E.: On the projections of measures invariant under the geodesic flow. Int. Math. Res. Not. 9, 511–526 (2003) Marstrand, M.: Some fundamental geometrical properties of plane sets of fractional dimension. Proc. London Math. Soc. 4, 257–302 (1954) Mattila, P.: Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. Math. 1, 227–244 (1975) Mattila, P.: Orthogonal projections, Riesz capacities and Minkowski content. Indiana Univ. Math. J. 39, 185–198 (1990) Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability. Cambridge: Cambridge University Press, 1995 Mattila, P.: Hausdorff dimension, projections, and Fourier transform. Publ. Mat. 48, 3–48 (2004)
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Peres, Y., Schlag, W.: Smoothness of projections, Bernoulli convolutions, and the dimensions of exceptions. Duke Math. J. 102, 193–251 (2000) Sauer, T. D., Yorke, J. A.: Are dimensions of a set and its image equal for typical smooth functions?. Ergodic Theory Dynam. Systems 17, 941–956 (1997)
Communicated by A. Kupiainen
Commun. Math. Phys. 254, 719–760 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1247-8
Communications in
Mathematical Physics
Dynamical Yang-Baxter Equation and Quantum Vector Bundles J. Donin† , A. Mudrov Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel Received: 22 December 2003 / Accepted: 24 May 2004 Published online: 13 January 2005 – © Springer-Verlag 2005
Abstract: We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices, dynamical twists, etc. In this context, we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits. We build a dynamical twist for any pair of a reductive Lie algebra and its Levi subalgebra. Using this twist, we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive Lie groups. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Dynamical r-Matrix and Compatible Star Product . . . . . . . . . . . 2.1 Classical dynamical Yang-Baxter equation . . . . . . . . . . . . 2.2 Quantum dynamical Yang-Baxter equation over an abelian base . 2.3 Compatible star product . . . . . . . . . . . . . . . . . . . . . . 3. Generalizations of Dynamical Yang-Baxter Equations . . . . . . . . . 3.1 Base algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Dynamical associative algebras . . . . . . . . . . . . . . . . . . 3.3 Infinitesimal analogs of base algebras and dynamical associative algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Poisson-Lie manifolds. . . . . . . . . . . . . . . . . . . 3.2.2 Poisson base algebras and Poisson base manifolds. . . . 3.2.3 Poisson dynamical algebras. . . . . . . . . . . . . . . .
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The research is supported in part by the Israel Academy of Sciences grant no. 8007/99-03, the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center “Group Theoretic Methods in the study of Algebraic Varieties” of the Israel Science foundation, and by Russian Foundation for Basic Research grant no. 03-01-00593. † Deceased January 2004
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7. 8.
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3.4 Quantization of Poisson base algebras and Poisson dynamical algebras 3.5 Dynamical Yang-Baxter equations . . . . . . . . . . . . . . . . . . . Dynamical Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Base algebra in a monoidal category . . . . . . . . . . . . . . . . . . 4.2 Dynamical categories over base algebras . . . . . . . . . . . . . . . . 4.3 Morphisms of base algebras . . . . . . . . . . . . . . . . . . . . . . ¯ H∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Category M ¯ H∗ . . . . . . . . . . . . . . . ¯ H∗ and M 4.5 Comparison of categories M H 4.6 Dynamical extension of a monoidal category over a module category . ¯ H;L and M ¯ H∗ . . . . 4.7 Comparison of categories O¯ B and BO¯ with M 4.8 Dynamical associative algebras . . . . . . . . . . . . . . . . . . . . . Categorical Approach to Quantum DYBE . . . . . . . . . . . . . . . . . . 5.1 Dynamical twisting cocycles . . . . . . . . . . . . . . . . . . . . . . 5.2 Quantum dynamical R-matrix . . . . . . . . . . . . . . . . . . . . . 5.2.1 Dynamical Yang-Baxter equation. . . . . . . . . . . . . . . . 5.2.2 Dynamical (pre-) braiding. . . . . . . . . . . . . . . . . . . . A Construction of Dynamical Twisting Cocycles . . . . . . . . . . . . . . . 6.1 Associative operations on morphisms and twists . . . . . . . . . . . . 6.2 Dynamical adjoint functors . . . . . . . . . . . . . . . . . . . . . . . 6.3 Generalized Verma modules . . . . . . . . . . . . . . . . . . . . . . 6.4 Dynamical twist via generalized Verma modules . . . . . . . . . . . . Dynamical Associative Algebras and Quantum Vector Bundles . . . . . . . 7.1 Classical vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Quantum vector bundles . . . . . . . . . . . . . . . . . . . . . . . . Vector Bundles on Semisimple Coadjoint Orbits . . . . . . . . . . . . . . . 8.1 Dynamical quantization of the function algebra on a group . . . . . . 8.2 Deformation quantization of the Kirillov brackets and vector bundles on coadjoint orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The quantum group case . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The quantum dynamical Yang-Baxter equation (DYBE) appeared in the mathematical physics literature, [GN, AF, Fad, F, ABB], in connection with integrable models of conformal field theories. The classical DYBE was first considered in [BDFh], rediscovered in [F], and systematically studied by Etingof, Schiffmann, and Varchenko in [EV1, ES2, S]. For a guide in the DYBE theory and an extended bibliography the reader is referred to the lecture course [ES1]. The theory of the DYBE over the Cartan subalgebra in a simple Lie algebra has been developed in detail. Classical dynamical r-matrices were classified in [EV1] and their explicit quantization built in [EV2, EV3, ESS]. Concerning the classical DYBE over an arbitrary (non-commutative) base, much is known about classification of its solutions and there are numerous explicit examples, [AM, ES2, Fh, S, Xu2]. At the same time, there is no generally accepted definition of quantum DYBE over a non-commutative Lie algebra or, say, over an arbitrary Hopf algebra. A generalization of the quantum DYBE for several particular cases was proposed in [Xu2] and [EE1]. Such a generalization was motivated by a relation between DYBE and the star product, [Xu1, Xu2]. An open question is an interpretation of the quantum DYBE of [Xu2] and [EE1] from a categorical point of view.
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Another interesting question is a relation of DYBE to the equivariant quantization. It was observed by Lu, [Lu1], that the list of classical r-matrices over the Cartan subalgebra of a simple Lie algebra is in intriguing correspondence with the list of Poisson-Lie structures on its maximal coadjoint orbits. However, the precise relation between quantum dynamical R-matrices and the equivariant quantization has not been established. The purpose of the present paper is to develop a theory of DYBE over an arbitrary Hopf algebra and relate it to equivariant quantization of vector bundles. Firstly, we generalize the classical dynamical Yang-Baxter equation for any Lie bialgebra h extending the concept of base manifold, which is the dual space h∗ in the standard approach. Secondly, we build dynamical extensions of monoidal categories and define the quantum dynamical R-matrix over an arbitrary base. Our third result is a construction of dynamical twist for Levi subalgebras in a reductive Lie algebra. Finally, we introduce a notion of dynamical associative algebras as algebras in dynamical categories and relate them to equivariant quantization of vector bundles. As an application, we construct an equivariant star product quantization of vector bundles (including function algebras) on semisimple coadjoint orbits of reductive Lie groups. ¯ of every It turns out that there is a general procedure of “dynamical extension”, O, monoidal category O over a base B, which is an O-module category. This new category has the same objects as O but more morphisms. The objects are considered as functors from B to B by the tensor product action. Morphisms in O¯ are natural transformations between these functors. This category admits a tensor product making it a monoidal category with O being a subcategory. One can consider the standard notions as alge¯ In terms of the original category O, they bras, twists, and R-matrices relative to O. satisfy “shifted” axioms, like shifted associativity, shifted cocycle condition, shifted or dynamical Yang-Baxter equation. The construction of dynamical extension admits various formulations. One of them uses the so-called base algebras, which are commutative algebras in the Yetter-Drinfeld categories. From the algebraic point of view, a Yetter-Drinfeld category is a category of modules over the double D(H) of a Hopf algebra H. In the quasi-classical limit, the base algebras are function algebras on the so-called Poisson base manifolds. A Poisson base manifold L is endowed with an action of the double D(h) of the Lie bialgebra h, the classical analog of H. The Poisson structure on L is induced by the canonical r-matrix of the double. The category of H-modules can be dynamically extended over the dual Hopf algebra H∗ . This approach is convenient for definition of dynamical associative algebras. A dynamical associative algebra is equipped with an equivariant family of binary operations (multiplications) depending on elements of H∗ . This family satisfies a “shifted” associativity condition. We show that the dynamical associative algebras give vector bundles on quantum spaces. In this paper we consider vector bundles on coadjoint orbits. In the classical situation, the function algebra on a homogeneous space is a subalgebra in the function algebra on the group. In general, the quantized function algebra on a homogeneous space cannot be realized as a subalgebra in a quantized function algebra on the group. For example, in the case of semisimple coadjoint orbits, such a realization exists only for symmetric or bisymmetric orbits, [DGS1, DM1]. Nevertheless, a quantization of the function algebra on the group as a dynamical associative algebra contains quantum orbits as (associative) subalgebras. Moreover, a dynamical quantization on the group quantizes the algebra of sections of homogeneous vector bundles on orbits. Such quantizations are parameterized by group-like elements of H∗ .
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A way of constructing (quantum) dynamical R-matrices and dynamical associative algebras is by twists in dynamical categories. We build such twists for Levi subalgebras in simple Lie algebras, using generalized Verma modules. This gives a construction of star product on the semisimple orbits. The paper is organized as follows. In Sect. 2 we recall basic definitions concerning DYBE and the compatible star product of [Xu2]. Section 3 presents generalizations of DYBE using the concepts of base algebras and base manifolds. Section 4 is devoted to various formulations of dynamical categories; therein we study dynamical associative algebras. In Sect. 5 we study objects that are interesting for applications: dynamical twists and dynamical R-matrices. We consider various types of dynamical categories and give expressions of dynamical twists and R-matrices in terms of the original category. In Sect. 6 we suggest a method of constructing dynamical twists. The method is based on a notion of dynamical adjoint functors. We build such functors using generalized Verma modules corresponding to Levi subalgebras in the (quantum) universal enveloping algebra of simple Lie algebras. In Sect. 7 we study relations between quantization of vector bundles and dynamical associative algebras in a purely algebraic setting. In Sect. 8 we give a detailed consideration to the dynamical associative algebra which is a quantized function algebra on a simple Lie group G. We relate this algebra to quantum vector bundles on coadjoint semisimple orbits of G. Note that the equivariant star product on function algebras on coadjoint orbits was also constructed in the papers [AL] and [KMST] which appeared after the first version of this article. Our method of building dynamical twists is developed for a more general case in [EE2].
2. Dynamical r-Matrix and Compatible Star Product 2.1. Classical dynamical Yang-Baxter equation. In this section we recall basic definitions concerning the dynamical Yang-Baxter equation. Let g be a Lie algebra and h its Lie subalgebra. The dual space h∗ is considered as an h-module with respect to the coadjoint action. Let {hi } ⊂ h be a basis and {λi } ⊂ h∗ its dual. Definition 2.1 ([F, EV1]). A classical dynamical r-matrix over the base h is an h-equivariant meromorphic function r : h∗ → g ⊗ g satisfying 1. the normal condition: the sum r(λ) + r21 (λ) is g-invariant, 2. the classical dynamical Yang-Baxter equation (DYBE): ∂r23 i
∂λi
(1)
hi −
∂r13 (2) ∂r12 (3) h + h = [r12 , r13 ] + [r13 , r23 ] + [r12 , r23 ]. ∂λi i ∂λi i
(1)
A constant dynamical r-matrix is a solution to the ordinary Yang-Baxter equation. It follows that the sum r(λ) + r21 (λ) does not depend on λ, [ES2]. If it is identically zero, the r-matrix is called triangular.
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2.2. Quantum dynamical Yang-Baxter equation over an abelian base. Suppose that h is a commutative Lie algebra and V is a semisimple h-module. Given a family (λ), λ ∈ h∗ , of linear operators on V ⊗3 , let us denote by (λ + th(1) ) the family of operators on V ⊗3 acting by v1 ⊗ v2 ⊗ v3 → (λ + t wt(v1 ))(v1 ⊗ v2 ⊗ v3 ), where wt(v) stands for the weight of v ∈ V with respect to h and t is a formal parameter. The operators (λ + th(i) ), i = 2, 3, are defined similarly. Definition 2.2. Let h be a commutative Lie subalgebra of a Lie algebra g. Let R(λ) be an h-equivariant meromorphic function h∗ → U(g)⊗2 (we consider h∗ equipped with the coadjoint and U(g) with adjoint action of h). Then R(λ) is called (universal) quantum dynamical R-matrix if it satisfies the quantum dynamical Yang-Baxter equation (QDYBE) R12 (λ)R13 (λ + th(2) )R23 (λ) = R23 (λ + th(1) )R13 (λ)R12 (λ + th(3) ).
(2)
Assuming R(λ) = 1 ⊗ 1 + t r(λ) + O(t 2 ), the element r(λ) satisfies Eq. (1), i.e. Eq. (1) is the quasi-classical limit of Eq. (2). In this case R(λ) is called quantization of r(λ). The problem of quantizing classical DYBE has been solved for g a complex semisimple Lie algebra and h its reductive commutative subalgebra, [ESS]. As to the case of general h, there is no generally accepted concept of what should be taken as the quantum DYBE. In the next subsection we render a construction of [Xu2] suggesting a version of quantum DYBE as a quantization ansatz for triangular dynamical r-matrices. This will be the starting point for our study. 2.3. Compatible star product. Let g be a complex Lie algebra and G the corresponding connected Lie group. Let h be a Lie subalgebra in g. Denote by ξ the left invariant vector field on G induced by ξ ∈ g via the right regular action. Let πh∗ denote the Poisson-Lie bracket on h∗ . Theorem 2.3 ([Xu2]). A smooth function r : h∗ → ∧2 g is a triangular dynamical r-matrix if and only if the bivector field πh∗ +
∂ ∧ h i + r (λ) ∂λi
(3)
i
is a Poisson structure on h∗ × G. Thus, the bivector field r (λ) on G is a “part” of a special Poisson bracket on a bigger space, h∗ × G. Xu proposed to look at a star product on h∗ × G of special form, as a quantization of (3). Let ht := h[[t]] be the Lie algebra over C[[t]] with the Lie bracket [x, y]t := t[x, y] for x, y ∈ h. The universal enveloping algebra U(ht ) can be considered as a deformation quantization of the polynomial algebra on h∗ . It is known that this quantization can be presented as a star product on h∗ by the PBW map S(h)[[t]] → U(ht ), where elements of the symmetric algebra S(h) are identified with polynomial functions on h∗ . We call this star product the PBW star product. Definition 2.4 ([Xu2]). A star product ∗t on h∗ × G is called compatible if 1. when restricted to C ∞ (h∗ ), it coincides with the PBW star product;
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2. for f ∈ C ∞ (G) and g ∈ C ∞ (h∗ ), (f ∗t g)(λ, x) := f (x)g(λ), (g ∗t f )(λ, x) ∞ k ∂ k g(λ) t := hi . . . h ik f (x); k! ∂λi1 . . . ∂λik 1
(4)
k=0
3. for f, g ∈ C ∞ (G), (f ∗t g)(λ, x) := F(λ)(f, g)(x),
(5)
where F(λ) is a smooth function F : h∗ → U(g) ⊗ U(g)[[h]] such that F = 1 ⊗ 1 + t 2 2 r(λ) + O(t ). For this star product to be associative, F should satisfy a certain condition called the shifted cocycle condition. Also, Xu proposed a generalization of the quantum DYBE (2) for an arbitrary Lie algebra h in the form R12 (λ) ∗t R13 λ + th(2) ∗t R23 (λ) = R23 λ+th(1) ∗t R13 ∗t R12 λ+th(3) , (6) where R is an equivariant function h∗ → U(g) ⊗ U(g), and the subscripts mark the tensor components in U ⊗3 (g). Notation f (λ + th) for f ∈ C ∞ (h∗ ) means f (λ + th) :=
∞ k t ∂ k f (λ) hi . . . hik . k! ∂λi1 . . . ∂λik 1
(7)
k=0
Here {hi } ⊂ h and {λi } ⊂ h∗ are dual bases; the superscript of h(i) , i = 1, 2, 3, in (6) means that h is embedded in the i th component of U ⊗3 (g). The compatible star product of [Xu2] is defined on smooth functions on h∗ × G. When restricted to polynomial functions on h∗ , it gives the multiplication in the universal enveloping algebra U(h). Formula (4) expresses the product of elements from U(h) and C ∞ (G) through the comultiplication in U(h) and the action of U(h) on C ∞ (G). It seems natural to replace U(h) with an arbitrary Hopf algebra H and C ∞ (G) with a left H-module A. However, the bidifferential operator F(λ) in (5) may be a meromorphic or even a formal function in λ ∈ h∗ . This requires to consider appropriate extensions of U(h), which may no longer be Hopf algebras. On the other hand, there is a class of admissible algebras which are close, in a sense, to the Hopf ones. Those are commutative algebras in the so-called Yetter-Drinfeld category of H-modules and H-comodules, which are, roughly speaking, modules over the double of H. We will define a dynamical extension of the monoidal category of H-modules over an admissible algebra, where the notions of compatible star products, dynamical Yang-Baxter equations, etc., acquire a natural algebraic formulation. Depending on a particular choice of admissible algebra, we come to different quasi-classical limits of quantum dynamical objects. Also, it appears useful (and often technically simpler) to consider a “dual” version of the dynamical extension, for example, a dynamical extension of the monoidal category of H∗ -comodules. In this way we obtain a “linearization” of the theory; in particular, smooth or meromorphic functions on h∗ become linear functions on U(h)∗ . Moreover, it will be useful to introduce the notion of dynamical extension of an arbitrary monoidal category, defined without involving any Hopf algebra. Below we present all the formulations.
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3. Generalizations of Dynamical Yang-Baxter Equations 3.1. Base algebras. In this subsection we define two objects of our primary concern: a base algebra L and a dynamical associative algebra over L. By k we mean a commutative ring over a field of zero characteristic. The reader may think of it as C or C[[t]], the ring of formal series in t. Given a Hopf algebra H over k we denote the multiplication, comultiplication, counit, and antipode by m, , ε, and γ . We use the standard Sweedler notation for the comultiplication in Hopf algebras: (x) = x (1) ⊗ x (2) . In the same fashion we denote the H-coaction on a right comodule A: δ(a) = a [0] ⊗ a (1) , where the square brackets label the A-component and the parentheses mark that belonging to H. The Hopf algebra with the opposite multiplication will be denoted by Hop while with the opposite comultiplication by Hop . The Hopf algebra H is considered as a left module over itself with respect to the adjoint action x ⊗ a → x (1) aγ (x (2) );
(8)
then the multiplication in H is equivariant. It is a standard fact that for any left H-module A the map H ⊗ A → A ⊗ H, h ⊗ a → h(1) a ⊗ h(2) , is H-equivariant. Recall that an algebra and H-module A is called a module algebra if the multiplication in A is H-equivariant. An algebra and H-comodule A is called a comodule algebra if the coaction A → H ⊗ A is a homomorphism of algebras. Definition 3.1 (Base algebras). A left H-module and left H-comodule algebra L is called base algebra over H if the coaction δ : L → H ⊗ L satisfies the condition (1) (1) (2) (1) [2] = x (1) (1) ⊗ x (2) [2] (9) x x ⊗ x for all x ∈ H and ∈ L, and the condition (1) 1 2 = 1 2 [2] 1 ,
(10)
for all 1 , 2 ∈ L. The coaction δ defines a permutation τA : L ⊗ A → A ⊗ L with every H-module A: τA ( ⊗ a) := (1) a ⊗ [2] ,
⊗ a ∈ L ⊗ A.
(11)
Condition (9) ensures that this permutation is H-equivariant. Condition (10) means that the multiplication in L is τL -commutative. Remark 3.2. A base algebra is a commutative algebra in the braided category of Yetter-Drinfeld modules. From the purely algebraic point of view, Yetter-Drinfeld modules are modules over the double Hopf algebra D(H). The left H-coaction induces a ∗ -action. Together with the H-action, the H∗ -action gives a D(H)-action. In our left Hop op theory, an H-base algebra plays the same role as the U(h)-module algebra of functions on h∗ in the theory of DYBE over a commutative base. One can also introduce the dual notion of a base coalgebra as a comodule over D(H). We will use H∗ , a dual to the Hopf algebra H, as an example of such a base coalgebra. Example 3.3. The algebra H itself is a base algebra over H with respect to the left adjoint action and the coproduct considered as the left regular H-coaction. Conditions (9) and (10) are checked directly.
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Example 3.4. Suppose that H is the tensor product of two Hopf algebras, H = H0 ⊗H1 . Then both H0 and H1 are natural base algebras over H. The H-action on Hi is the adjoint action restricted to Hi . The H-coaction on Hi is the coproduct coaction considered as a map with values in Hi ⊗ Hi ⊂ H ⊗ Hi . Example 3.5 (PBW star product). Consider the algebra C ∞ (h∗ )[[t]] from Definition 2.4 equipped with the PBW star product. It is obviously a left U(h)-module algebra, and formula (7) defines a coaction C ∞ (h∗ )[[t]] → U(h) ⊗ C ∞ (h∗ )[[t]] (the completed tensor product). It is straightforward to check that C ∞ (h∗ )[[t]] is a base algebra over U(h). The algebra C ∞ (h∗ )[[t]] is an extension of U(ht ), which is realized as the subalgebra in U(h)[[t]] generated by th. The algebra U(ht ) is a Hopf one, hence it is a base algebra over itself. At the same time, it is a base algebra over U(h)[[t]]. Indeed, it is invariant under the adjoint U(h)-action, and it is a left U(h)-comodule under the map (ϕt ⊗id)◦, where is the coproduct in U(ht ) and ϕt the natural embedding of U(ht ) in U(h)[[t]]. Proposition 3.6. Suppose that H is a quasitriangular Hopf algebra, with the universal R-matrix R. Let L be a quasi-commutative H-module algebra, i.e. obeying (R2 2 )(R1 1 ) = 1 ⊗ 2 for all 1 , 2 ∈ L. Then L is an H-base algebra, with the left H-coaction δ( ) := R2 ⊗ R1 ,
∈ L.
(12)
Proof. The condition (10) is satisfied by construction. The equality ( ⊗ id)(R) = R13 R23 implies that the map (12) is an algebra homomorphism. The map (12) makes L a left H-comodule, because of (id ⊗ )(R) = R13 R12 . The condition (9) holds by virtue of R(h) = op (h)R for every h ∈ H. Corollary 3.7. Within the hypothesis of Proposition 3.6, suppose that R ∈ H ⊗ K ⊂ H ⊗ H, where K is a Hopf subalgebra in H. Then L is endowed with a structure of the K-base algebra, with the K-coaction (12). Proof. The H-coaction (12) is, in fact, an K-coaction. Now the statement immediately follows from Proposition 3.6. Remark that an R-commutative algebra L is commutative with respect to the element R−1 21 , which is also a universal R-matrix for H. Thus L has two H-base algebra structures, and they are different in general. In particular, an H-base algebra has two different D(H)-base algebra structures. Example 3.8 (The FRT algebras). The FRT-dual Hopf algebra H∗ , [FRT], of a quasitriangular Hopf algebra H is a quasi-commutative H ⊗ Hop -algebra. Therefore it has two structures of H ⊗ Hop -base algebras. Example 3.9 (Reflection equation algebras). Recall that a twist of a Hopf algebra H ˜ is a Hopf algebra with the same multiplication and the new comultiplication (x) := F −1 (x)F; the element F called a twisting cocycle satisfies certain conditions, see [Dr3]. For every quasitriangular Hopf algebra H with the R-matrix R, there is a twist, R
H ⊗ H, of its tensor square, [RS]. It is obtained by applying the twisting cocycle R23 ∈ (H⊗H)⊗(H⊗H) to the comultiplication in H ⊗ H. The twisted tensor square is a quasitriangular Hopf algebra with the R-matrix R
R
− + + ⊗ ⊗ R := R− 14 R13 R24 R23 ∈ (H H) ⊗ (H H),
(13)
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⊗ where R+ := R and R− := R−1 21 . Recall that H is a Hopf subalgebra in H H through the embedding : H → H ⊗ H. Observe that the R-matrix (13) can be presented as + − + R = (R− 1 ⊗ R1 ) ⊗ (R2 )(R2 ). In other words, its right tensor component belongs R
to (H) ⊂ H ⊗ H. Applying the argument from Corollary 3.7 to K = (H ), we come to the following proposition. R
Proposition 3.10. A quasi-commutative H ⊗ H-module algebra is a base algebra over H. The reflection equation algebra associated with a finite dimensional representation R
of H, [KSkl, KS], is a quasi-commutative H ⊗ H-algebra, [DM3]. As a corollary of Proposition 3.10, we obtain that the reflection equation algebra is an H-base algebra. More examples of base algebras are obtained by quantizing Poisson base algebras (see Subsect. 3.2.2), according to Theorem 3.23. 3.2. Dynamical associative algebras. Let L be a base algebra over a Hopf algebra H. Definition 3.11. A left H-module A is called a dynamical associative algebra over the base algebra L if it is equipped with an H-equivariant bilinear map : A⊗A → A⊗L such that the following diagram is commutative: id⊗τA ⊗id A⊗L⊗A - A⊗A⊗L - A⊗L⊗L
⊗id
id⊗m -
A⊗L
6
id⊗
A⊗A⊗A - A⊗A⊗L
⊗id
A⊗L⊗L
id⊗m -
(14)
A⊗L
Here m stands for the multiplication in L and the permutation τA is defined by (11). An example of dynamical associative algebra is the function algebra on a group G twisted by the dynamical twist from [Xu2]. It defines the compatible star product in the sense of Definition 2.4; it turns out that the multiplication in a dynamical associative algebra over an arbitrary base can be extended to an ordinary associative multiplication in a bigger algebra, according to the following proposition. Proposition 3.12. Let A be a left H-module equipped with an equivariant map : A ⊗ A → A ⊗ L. Then A is a dynamical associative algebra with respect to if and only if the operation τA
⊗m
m
(A ⊗ L) ⊗ (A ⊗ L) −→ A ⊗ A ⊗ L ⊗ L −→ A ⊗ L ⊗ L −→ A ⊗ L makes A ⊗ L an associative H-module algebra, denoted further by A L. Proof. The proof can be conducted by a straightforward verification. Below we give another proof using our categorical approach to dynamical associative algebras, see Example 4.21.
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3.3 Infinitesimal analogs of base algebras and dynamical associative algebras. In the present subsection, we introduce quasi-classical analogs of base algebras and dynamical associative algebras. 3.2.1. Poisson-Lie manifolds. Let us recall some basic facts about Poisson-Lie manifolds. Throughout the text an g-manifold means a manifold equipped with a left g-action on functions. This corresponds to a right action on the manifold of a Lie group G relative to g. Let g be a Lie bialgebra, i.e. a Lie algebra equipped with a cobracket map µ : g → ∧2 g. The cobracket defines on the dual space g∗ a Lie algebra structure compatible with the Lie algebra structure on g in the sense of [Dr1]. Recall from [Dr2] that µ induces a Poisson structure on the Lie group G such that the multiplication map G × G → G is a Poisson map (the manifold G × G is equipped with the standard Poisson structure of Cartesian product of two Poisson manifolds). A right G-manifold P is called a PoissonLie manifold if the action P × G → P is Poisson. The right G-action on P induces a left action of the universal enveloping algebra U(g) on the function algebra A(P ). For an element x ∈ U(g), let x P (or simply x , if P is clear from the context) denote the corresponding differential operator on P . For the bidifferential operator on P generated by a bivector field π , we use the notation π(a, b) := (m ◦ π )(a ⊗ b), a, b ∈ A(P ), where m is the multiplication in A(P ). The following fact is well known and can be checked directly. Proposition 3.13. Let g be a Lie bialgebra with cobracket µ, G the corresponding connected simply connected Poisson-Lie group, and P a right G-manifold equipped with a Poisson bracket π . Then P is a Poisson-Lie G-manifold if and only if for any x ∈ g and a, b ∈ A(P ) −−→ x π(a, b) − π( x a, b) − π(a, x b) = µ(x)(a, b).
(15)
Any Lie bialgebra structure on g can be quantized to a C[[t]]-Hopf algebra Ut (g) (quantum group), see [EK]. If At (P ) is a Ut (g)-equivariant quantization of A(P ), then the quasi-classical limit of At (P ) gives a Poisson-Lie bracket on P . An important particular case of Lie bialgebras is a coboundary one, with the cobracket µ(x) := [x ⊗ 1 + 1 ⊗ x, r], where the element r ∈ ∧2 g satisfies the modified classical Yang-Baxter equation [[r, r]] := [r12 , r13 ] + [r13 , r23 ] + [r12 , r23 ] = ϕ ∈ ∧3 (g)g .
(16)
Formula (15) then reads [ x ⊗ 1 + 1 ⊗ x , π − r ] = 0.
(17)
In other words, a Poisson-Lie bracket differs from r by an invariant bivector f := π − r such that [[f, f ]] is equal to −ϕ from (16). Here the operation f → [[f, f ]] is defined by (16) for the Lie algebra of vector fields; this operation is proportional to the Schouten bracket. Note that the Poisson-Lie bracket on a Poisson-Lie g-manifold P is the infinitesimal object for the Ut (g)-equivariant quantization of the function algebra on P , where Ut (g) is the corresponding quantum group. Such brackets are classified in [DGS1, Kar, D2, DO] for homogeneous manifolds G/H , where G is a simple Lie group and H its reductive Lie subgroup of maximal rank.
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3.2.2. Poisson base algebras and Poisson base manifolds. Let D(h) denote the double of a Lie bialgebra h, [Dr1]. As a linear space, D(h) is the direct sum h + h∗op , where h∗ is the dual Lie algebra. The double D(h) is endowed with a non-degenerate symmetric bilinear form induced by the natural pairing between h and h∗op . There is a unique extension of the Lie algebra structure from h and h∗op to a Lie algebra on D(h) such that this form is ad-invariant. The double is a coboundary Lie bialgebra with the r-matrix r := i ηi ∧ hi = 21 i (ηi ⊗ hi − hi ⊗ ηi ), where {hi } is a basis in h and {ηi } is the dual basis in h∗op . The canonical element θ := 21 i (ηi ⊗ hi + hi ⊗ ηi ) is ad-invariant. The pair (r, θ ) makes D(h) a quasitriangular Lie bialgebra. Definition 3.14. A commutative D(h)-algebra L0 is called a Poisson base algebra over h, or simply an h-base algebra, if θ induces the zero bidifferential operator on L0 . When a Poisson base algebra L0 over h appears as the function algebra on a manifold L, i.e. L0 := A(L), we call L a Poisson base manifold over h, or simply an h-base manifold. Proposition 3.15. An h-base manifold L is a Poisson-Lie D(h)-manifold with respect to the bracket := (18) η i ∧ h i , i
which is automatically equal to the bivector field i η i ⊗ h i . Proof. The element i ηi ∧ hi ∈ ∧2 D(h) satisfies the modified Yang-Baxter equation (16) with ϕ := [θ12 , θ23 ]. Since θ yields the zero bivector field on L, the three-vector field induced by [θ12 , θ23 ] is zero, too. This implies the following two assertions. Firstly, the bivector defines a Poisson structure on L. Secondly, for any D(h)-invariant Poisson bracket f the bracket f + hence , is automatically a Poisson-Lie one. The following are examples of Poisson base manifolds. According to Theorem 3.23 below, they can be quantized to Ut (h)-base algebras, where Ut (h) is the quantized universal enveloping algebra of h. Example 3.16 (Group spaces H ∗ and H ). Let H be the Lie subgroup in the double D(H ) corresponding to the Lie subalgebra h ⊂ D(h). We will show that the left coset space H \D(H ) is an h-base manifold. Note that the manifold H \D(H ) is locally isomorphic to the Lie group H ∗ corresponding to the Lie algebra h∗ . The algebra of functions on H \D(H ) is realized as a subalgebra of functions f ∈ A D(H ) obeying f (hx) = f (x) for h ∈ H . This subalgebra is invariant under the right regular action of D(H ) on itself. The element θ is D(h)-invariant, hence the bivector θ l,l −θ r,r , where the superscripts l,r denote the left- and right-invariant field extensions, gives the zero operator on A D(H ) . Therefore the bivector θ l,l gives the zero operator on the left H -invariant functions, where it equals θ l,l − θ r,r . Thus, H \D(H ) and therefore H ∗ are Poisson h-base manifolds. In this example, the Poisson bracket on H ∗ is the Drinfeld-Sklyanin bracket on D(H ) projected to H ∗ . Similarly to H \D(H ), one can consider the coset space H ∗ \D(H ), which is locally isomorphic to the group space H . So H is a Poisson h-base manifold as well. By the function algebra A(P ) we understand, depending on a particular type of the manifold P , the algebra of polynomial, analytical, meromorphic, or smooth functions.
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Example 3.17 (Coset spaces K\H ). Let us generalize Example 3.16. Suppose that k is a Lie sub-bialgebra in h. Then the linear sum k + h∗op is a Lie sub-bialgebra in D(h). Let K be the Lie subgroup in H corresponding to k. Using the same arguments as in Example 3.16, one can prove that the coset space K\H is a Poisson h-base manifold. Indeed, let K · H ∗ denote the connected subgroup in D(H ) whose Lie algebra is k + h∗op . The coset space (K · H ∗ )\D(H ) is locally isomorphic to K\H as a smooth manifold. Consider the functions on the group D(H ) that are invariant under K · H ∗ as functions on (K · H ∗ )\D(H ). The rest of the construction is exactly the same as in the previous example. Namely, one can check that the projection of the Drinfeld-Sklyanin bracket from D(H ) makes K\H a Poisson h-base manifold. It follows that the quotient spaces of the standard Drinfeld-Jimbo simple Poisson-Lie group H by the Levi and parabolic subgroups are Poisson h-base manifolds. Obviously, the same construction works for the dual Lie bialgebra h∗op and its subbialgebras; the corresponding coset spaces will be h-base manifolds. Example 3.18 (Group H , the quasitriangular case). Suppose that h is a quasitriangular Lie bialgebra, i.e. h is endowed with an r-matrix r and a symmetric invariant element ω ∈ h⊗h such that r satisfies (16) with ϕ := [ω12 , ω23 ]. We can treat r and ω as linear maps from h∗op to h via pairing with the first tensor factor. Consider the Lie group H corresponding to h as a right H -manifold via the action x → y −1 xy, x, y ∈ H . This action generates the action of h on the function algebra A(H ) by vector fields h := hl − hr , h ∈ h. Here the superscripts l, r stand for the left- and right- H -invariant vector fields generated, respectively, by the right and the left regular actions of H on itself. The group H is also a right h∗op -manifold. Namely, the element η ∈ h∗op acts on functions from A(H ) by the vector field η := r(η)l − r(η)r + ω(η)l + ω(η)r . We have 2θ = (r l,l − r r,l − r l,r + r r,r ) + (ωl,l − ωr,l + ωl,r − ωr,r ) −(r l,l − r l,r − r r,l + r r,r ) + (ωl,l − ωl,r + ωr,l − ωr,r ) = 2(ωl,l − ωr,r ),
(19)
which vanishes on functions, because ω is invariant. These actions of h and h∗op define an action of the double D(h), thus the group space H is an h-base manifold. In this example, is the reflection equation Poisson bracket, [Sem]. Quantization of this bracket is an RE algebra, cf. Example 3.9. 3.2.3. Poisson dynamical algebras. In this subsection, we define a Poisson dynamical bracket as an infinitesimal object for the deformation quantization of a commutative algebra A to a dynamical associative algebra, in the sense of Definition 3.11. We assume A := A(P ), a function algebra on a manifold P . Given a linear space X, by Alt we denote a linear endomorphism of X⊗3 acting by Alt : x1 ⊗ x2 ⊗ x3 → x1 ⊗ x2 ⊗ x3 − x2 ⊗ x1 ⊗ x3 + x2 ⊗ x3 ⊗ x1 ,
xi ∈ X.
Definition 3.19. Let h be a Lie bialgebra with the cobracket µ, L an h-base manifold, and P an h-manifold. Let T (P ) denote the tangent bundle to P . A function π : L → ∧2 T (P ) is called a Poisson dynamical bracket on P (or on A(P )) over the base manifold L (or over the Poisson h-base algebra A(L)) if
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1. for any h ∈ h and a, b ∈ A(P ) h L π(λ)(a, b) + h P π(λ)(a, b) − π(λ)(h P a, b) − π(λ)(a, h P b) −−→ = µ(h)P (a, b),
(20)
2. π satisfies the equation
Alt h i P ⊗ η i L π(λ) = [[π(λ), π(λ)]].
(21)
i
In this definition the expression π(a, b) is a function on P × L. The vector fields h L −−→ and h P are generated by the actions of h on L and P , respectively; µ(h)P is a bivector i is induced by 2 field induced on P by the Lie cobracket µ(h) ∈ ∧ h. The vector field η L ∗ the actions of hop on L (recall that L is a D(h)-manifold). When P is endowed with a Poisson dynamical bracket over a base L, we say that A(P ) is a Poisson dynamical algebra. While a Poisson base manifold is a classical analog of a base algebra, a Poisson dynamical algebra is a classical analog of dynamical associative algebra. The Poisson dynamical bracket may be viewed as a map π : A(P ) ∧ A(P ) → A(P ) ⊗ A(L). The following proposition is a generalization of Theorem 2.3. Proposition 3.20. Let h be a Lie bialgebra with cobracket µ, L an h-base and P an h-manifold. A function π : L → ∧2 T (P ) is a Poisson dynamical bracket on P over the base L if and only if the bivector i
η i L ∧ h i L + 2
η i L ∧ h i P + π
(22)
i
is a Poisson-Lie bracket on the h-manifold P × L. Proof. This statement is proven by a direct computation. It can be considered as an infinitesimal analog of Theorem 3.12. If the base manifold L has h-stable points, then a Poisson dynamical bracket on P can be restricted to the coset space P /H , where it becomes an ordinary Poisson bracket. This is formalized by the following proposition. Proposition 3.21. Let λ0 ∈ L be a stable point under the action of h. Then π(λ0 ) restricts to a Poisson bracket on the subalgebra of h-invariants in A(P ). Proof. By the equivariance condition (20), the function π(λ0 )(f, g) is h-invariant when f, g ∈ A(P ) are h-invariant. The Schouten bracket of π(λ0 ) with itself vanishes on h-invariant elements from A(P ), as follows from (21). Proposition 3.21 gives rise to a quantization method for the class of Poisson structures coming from Poisson dynamical structures. This method is developed in Sect. 7 and uses dynamical associative algebras, which are quantizations of Poisson dynamical algebras.
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3.4. Quantization of Poisson base algebras and Poisson dynamical algebras. Let h be a Lie bialgebra and Ut (h) the corresponding quantization of U(h). Suppose L0 is a Poisson h-base algebra. By Proposition 3.15, L0 is endowed with a Poisson bracket induced by the tensor i ηi ⊗ hi ∈ D(h)⊗2 , where {hi } ⊂ h and {ηi } ⊂ h∗op are dual bases. Definition 3.22. A quantization of the Poisson h-base algebra L0 is a base algebra Lt over Ut (h) that is a Ut (h)-equivariant deformation quantization of L0 with the multiplication a ∗t b = ab + O(t), a ∗t b − b ∗t a = t ( ηi a)(h i b) + O(t 2 ) (23) i
and the coaction δ : Lt → Ut (h) ⊗ Lt , δ(a) = 1 ⊗ a + t
hi ⊗ ( ηi a) + O(t 2 ),
(24)
i
where a, b ∈ Lt . When L0 = A(L), the function algebra on an h-base manifold, one may require in the definition that Lt is a star product. Then D(h) acts on L0 by vector fields. Theorem 3.23. Any Poisson base algebra can be quantized. Proof. Let θ = 21 i (hi ⊗ ηi + ηi ⊗ hi ) ∈ D(h) ⊗ D(h) be the canonical symmetric invariant of the double Lie algebra D(h). Consider the quasi-Hopf algebra U D(h) [[t]] with the R-matrix etθ and the associator t , which is expressed through tθ12 and tθ23 , [Dr3]. Since θ vanishes on L0 , so do etθ and t . Therefore L0 [[t]] is a commutative algebra not only in the classical monoidal category of U D(h) [[t]]-modules, but also in the category with the associator t , i.e. L0 is etθ -commutative and t -associative. to [EK], there exists Jt converting the quasi-Hopf algebra a twist According U D(h) [[t]] into a Hopf one, Ut D(h) . This Hopf algebra contains the quantized enveloping algebras Ut (h) and Ut (h∗op ) as Hopf subalgebras. The Hopf algebra Ut D(h) tθ ∗ is quasitriangular, with the universal R-matrix Rt = (Jt )−1 21 e Jt lying in Ut (hop ) ⊗ Ut (h) ⊂ Ut D(h) ⊗ Ut D(h) . Applying the twist Jt to the L0 [[t]], we obtain a quasi-commutative alge algebra bra Lt in the category of Ut D(h) -modules. We introduce on Lt a structure of the Ut (h)-comodule algebra by setting δ( ) := (Rt )2 ⊗ (Rt )1 , ∈ Lt . (25) Together with the Ut (h)-action restricted from Ut D(h) , the coaction (25) makes Lt a U algebra. This follows from Corollary 3.7, where one should set H = t (h)-base Ut D(h) and K = Ut (h). Definition 3.24. Let L be an h-base manifold. Let P be an h-manifold and π a Poisson dynamical bracket on P over L. A quantization of Poisson dynamical h-algebra A(P ) is a pair Lt , At (P ) , where a) Lt is a quantization of the Poisson base algebra A(L) in the sense of Definition 3.22 and b) At (P ) is a flat C[[t]]-module and a dynamical associative Ut (h)-algebra over Lt such that At (P )/tAt (P ) = A(P ) and a b − b a = tπ(a, b) + O(t 2 ).
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Conjecture 3.25. Any Poisson dynamical algebra can be quantized. In Subsect. 7.2, we develop a method of quantizing vector bundles on the coset space P /H , using dynamical associative algebras. By duality, the construction of Sect. 7.2 can be formulated in terms of base algebras rather than coalgebras. Namely, let A be a dynamical associative algebra over an H-base algebra L and let χ be an H-invari
id⊗χ
ant character of L. Then the composition map A ⊗ A −→ A ⊗ L −→ A yields an associative multiplication on the subspace of H-invariant elements of A. Thus invariant characters of base algebras are important for our approach to quantization (see also [DM1]). In the deformation situation, the infinitesimal analogs of Ut (h)-invariant characters of the base algebra Lt are h-stable points on the h-base manifold L. By Proposition 3.21, each h-stable point defines a Poisson structure on P /H . It is natural to quantize this Poisson structure by the corresponding invariant character applying it to the dynamical associative quantization of A(P ). The question is whether every h-stable point can be quantized to a Ut (h)-invariant character of Lt . The answer to this question is affirmative. Proposition 3.26. Let Lt be the quantization of the function algebra on a base manifold L built in Theorem 3.23. Then every h-stable point λ0 on L defines an Ut (h)-invariant character of Lt by χ λ0 (f ) = f (λ0 ) for f ∈ Lt . Proof. As follows from the explicit form of the twist Jt constructed in [EK], it reduces to 1 ⊗ 1 at every h-stable point λ0 ∈ L. It follows from the proof of Theorem 3.23 that the star product in Lt satisfies (f ∗ g)(λ0 ) = (f g)(λ0 ) = f (λ0 )g(λ0 ) for any pair of functions f, g ∈ A(L).
3.5. Dynamical Yang-Baxter equations. In this subsection we give definitions of the classical and quantum dynamical Yang-Baxter equations over an arbitrary base algebra. Definition 3.27. Let g be a Lie bialgebra and h ⊂ g its sub-bialgebra; let µ denote the Lie cobracket on h. Let L be a Poisson h-base manifold. A function r¯ : L → g ⊗ g is called a classical dynamical r-matrix over base L if 1. for any h ∈ h h L r¯ (λ) + [h ⊗ 1 + 1 ⊗ h, r¯ (λ)] = µ(h), 2. the sum r¯ (λ) + r¯21 (λ) is g-invariant, 3. r¯ satisfies the equation Alt hi ⊗ η i L r¯ (λ) = [[¯r (λ), r¯ (λ)]].
(26)
(27)
i
We call (27) the classical dynamical Yang-Baxter equation over the base L. The skew part of r¯ satisfies the “modified” version of the dynamical Yang-Baxter equation with non-zero right-hand side being an invariant element from ∧3 g. We call it skew dynamical r-matrix. Condition (26) means quasi-equivariance of r¯ (λ) with respect to the action of h. In fact, the symmetric part ω¯ = 21 (¯r + r¯21 ) is constant on every D(h)-orbit in L, i.e. x L ω¯ = 0 for any x ∈ D(h).
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Consider the opposite Lie bialgebra hop equipped with the opposite bracket and the same cobracket. Endowed with the opposite bracket, the manifold L becomes a Poisson base manifold for hop . The Cartesian product L × L is a Poisson base manifold with respect to the Lie bialgebra h ⊕ hop . Let G be the Lie group corresponding to g. The following proposition characterizes the dynamical r-matrices. Proposition 3.28. A function r¯ : L → g ∧ g is a skew dynamical r-matrix over the Poisson h-base manifold L if and only if M := G × L × L is equipped with the h ⊕ hop Poisson Lie structure such that the projection M → L × L is a Poisson map and i (28) ( ηL f ) hi l + hi r (a), {f, a} := i
{a, b} := r¯ l,l (λ ) − r¯ r,r (λ ) (a, b)
(29)
for f ∈ A(L × L), a, b ∈ A(G), and (λ , λ ) ∈ L × L.
Proof. Straightforward.
Suppose that g is a quasitriangular Lie bialgebra with an r-matrix rg ∈ g ⊗ g. Let ωg = 21 (rg + rg21 ) denote the symmetric part of rg . Assume that L is D(h)-transitive, i.e. D(h)-invariants in A(L) are scalars. Proposition 3.29. A function r¯ : L → g ⊗ g subject to 21 (¯r + r¯ ) = ωg ∈ g ⊗ g is a dynamical r-matrix if and only if M := L × G is equipped with an H -invariant Poisson structure, such that the projection M → L is a Poisson map, and i ( ηL f ) hi l a , {a, b} := (¯r l,l − rgr,r )(a, b) (30) {f, a} := i
for f ∈ A(L), a, b ∈ A(G), Proof. Straightforward.
Proposition 3.28 implies that the bivector field r¯ l,l (λ ) − r¯ r,r (λ ) makes A(G) a Poisson dynamical algebra over the h ⊕ hop -base manifold L × L. By Proposition 3.29, the bivector field r¯ l,l (λ) − rgr,r makes A(G) a Poisson dynamical algebra over the h-base manifold L. Proposition 3.30. Let r¯ : L → g ⊗ g be a classical dynamical r-matrix on an h-base manifold L. Let rg ∈ g ⊗ g be a constant r-matrix whose symmetric part coincides with the symmetric part of r¯ . Suppose that λ0 ∈ L is an h-stable point. Then the bivector field r¯ l,l (λ0 ) − rgr,r yields a g-Poisson-Lie structure on the coset space G/H . Proof. Applying Proposition 3.21 to the Poisson dynamical bracket r¯ l,l (λ) − rgr,r on G, we obtain a Poisson structure on the subalgebra in A(G) that consists of invariants under the action of h by the left-invariant vector fields. This algebra is canonically identified with the algebra of functions on the coset space G/H . Obviously, this Poisson structure is a Poisson-Lie one, with respect to the right g-action on A(G/H ) induced by the left G-action on G/H . Remark 3.31. Let h = g be quasitriangular, with the classical r-matrix rg whose symmetric part is equal to the symmetric part of r¯ . Then r¯ (λ) − rg is the dynamical r-matrix of [FhMrsh].
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We complete this subsection with a definition of the quantum dynamical Yang-Baxter equation over an arbitrary base algebra. This definition naturally follows from our categorical point of view presented in Sect. 5. The quantum DYBE will be defined for any triple (U, H, L), where H is a Hopf subalgebra in a Hopf algebra U and L is an H-base algebra. ¯ =R ¯1 ⊗ R ¯2 ⊗ R ¯ 3 ∈ U ⊗ U ⊗ L is called a universal Definition 3.32. An element R quantum dynamical R-matrix of U over the H-base algebra L if it satisfies the equivariance condition ¯ 1 ⊗ h(1) R ¯ 2 ⊗ h(3) R ¯3 = R ¯ 1 h(1) ⊗ R ¯ 2 h(2) ⊗ R ¯ 3, h(2) R
h ∈ H,
(31)
and the quantum dynamical Yang-Baxter equation ¯ 12 R
(2) ¯
¯ 23 = R13 R
(1) ¯
¯ 13 R23 R
(3) ¯
R12 ,
(32)
in U ⊗ U ⊗ U ⊗ L. ¯ means the following. Applying the coaction to the L-compoHere the notation (i)R ¯ := R ¯1 ⊗ R ¯2 ⊗ R ¯ (1) ⊗ R ¯ [2] . The other two are nent of R, we get the element (3)R 3 3 ¯ := R ¯1 ⊗ R ¯ (1) ⊗ R ¯2 ⊗ R ¯ [2] and obtained from this by permutations, namely (2)R 3 3 (1)R ¯ := R ¯ (1) ⊗ R ¯1 ⊗R ¯2 ⊗R ¯ [2] . 3 3 Equation (32) specializes to (6) for H = U(h), U = U(g), and L being the extension of U(ht ) to the PBW star product on functions on h∗ , cf. Example 3.5. Also, Eq. (32) coincides with the conventional dynamicalYang-Baxter equation (2) for h a commutative Lie subalgebra in g and L being the algebra of functions on h∗ , [EV2]. Suppose that U and H are quantizations of the universal enveloping algebras U(g) and U(h) and L is a quantization of the function algebra on a Poisson h-base manifold L. Suppose that the universal dynamical R-matrix has the form R = 1⊗1⊗1+t r¯ +O(t 2 ). Then r¯ is a function on L with values in g ⊗ g. It satisfies Eq. (26) and (27), which are the consequences of Eqs. (31) and (32). Remark 3.33. The definitions of the classical and quantum dynamical R-matrix given above admit further generalization. The reader is referred to [DM5], where the classical dynamical r-matrices are studied in connection with Lie bialgebroids. The present definitions are conditioned by our specific approach confined to the strict monoidal categories (i.e. with trivial associator). If one considers general monoidal categories, as in Example 5.5, Eq. (32) would involve an associator. In the quasi-classical limit, it will give the dynamical r-matrix of the Alekseev-Meinrenken type, [AM]. 4. Dynamical Categories 4.1. Base algebra in a monoidal category. A dynamical associative algebra A from Definition 3.11 may serve as a model for further generalizations. It turns out that there is a monoidal category where A is an associative algebra. Such categories can be built for all Hopf algebras and they include dynamical categories of Etingof-Varchenko introduced for commutative cocommutative Hopf algebras in [EV3]. Such notions as dynamical twist and the dynamical Yang-Baxter equation can be naturally formulated and generalized within the dynamical categories, which are the subject of our further study.
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ˆ ⊗) be a monoidal category. We will work, for simplicity, with only strict Let (O, monoidal categories, i.e. having the trivial associator; all the constructions can be carˆ be the center of O, ˆ see ried over to the general case in a straightforward way. Let Z(O) [Kas]. The center is a braided monoidal category consisting of pairs (A, τ ), where A is an object of Oˆ and τ the collection of permutations τX : A ⊗ X → X ⊗ A, satisfying natural conditions. ˆ Definition 4.1. A base algebra in the category Oˆ is commutative algebra from a Z(O). In other words, a base algebra is an algebra in Oˆ and a collection of morphisms ˆ such that the following diagrams are τA ∈ HomOˆ (L ⊗ A, A ⊗ L) and A ∈ Ob O, commutative: L⊗B idL ⊗ψ
? L⊗A
τB
- B ⊗L ψ⊗idL
(33)
? - A⊗L
τA
τA⊗B - A⊗B ⊗L L⊗A⊗B HH j τB τA H A⊗L⊗B
(34)
τA τA L⊗L⊗A - L⊗A⊗L - A⊗L⊗L mL
mL
? L⊗A
? - A⊗L
τA
L⊗L
τL
@
mL @ R
L
(35)
- L⊗L mL
(36)
ˆ ψ ∈ Hom ˆ (B, A). for all A, B ∈ Ob O, O Example 4.2. The unit object 1Oˆ is the simplest example of a base algebra. The algebra structure and permutation are defined by the canonical isomorphisms 1Oˆ ⊗ A A ˆ A ⊗ 1 ˆ for all A ∈ Ob O. O
Example 4.3. When the category Oˆ is braided with braiding σ , any commutative algebra L in this category has two natural base algebra structures, with respect to the τ = σ and τ = σ −1 . Example 4.4. Let H be a Hopf algebra and Oˆ the monoidal category of left H-modules. ˆ Any H-base algebra in the sense of Definition 3.1 is a base algebra in the category O. Indeed, for a left H-module A we define the permutation τA : L ⊗ A → A ⊗ L by ⊗ a → (1) a ⊗ [2] ,
a ∈ A, ∈ L.
(37)
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The permutation (37) is H-equivariant, as follows from (9), hence the condition (33) is satisfied. Conditions (34) and (35) hold because L is an H-comodule algebra. Equation (36) follows from (10). Example 4.5. Let Oˆ be the category of semisimple modules over a commutative finite dimensional Lie algebra h. Take L to be the algebra of functions on h∗ , which is a trivial h-module. Let A be a semisimple The permutation τA between L and A is h-module. defined by f (x) ⊗ a → a ⊗ f x + α(a) , where f ∈ L, a ∈ A, and α(a) is the weight of a. ˆ ⊗) be a monoidal category and 4.2. Dynamical categories over base algebras. Let (O, ˆ ˆ let us construct a O be a monoidal subcategory in O. Given a base algebra (L, τ ) in O, new monoidal category O¯ L . Objects in O¯ L are the same as in O. For two objects A and ˆ B in O¯ L , morphisms HomO¯ L (A, B) are O-morphisms HomOˆ (A, B ⊗ L). Since the algebra L is unital, every morphism φ ∈ HomO (A, B) naturally becomes a morphism φ
from HomO¯ L (A, B) through the composition A −→ B ⊗ 1Oˆ → B ⊗ L. ψ
φ
The composition of two morphisms A −→ B and B −→ C in O¯ L is defined as the composition ψ
φ
mL
A −→ B ⊗ L −→ C ⊗ L ⊗ L −→ C ⊗ L,
(38)
ˆ where the rightmost arrow is the multiplication in L. It is easy to see that the comin O, position is associative. The identity morphism idA for A ∈ Ob OL is the composition A → A ⊗ 1Oˆ → A ⊗ L, where the first arrow is the canonical isomorphism and the second one is the natural inclusion 1Oˆ → L via the unit of L. Thus O¯ L is a category. ¯ in O¯ L setting it on objects as in O; on the Let us introduce a monoidal structure ⊗ morphisms it is defined by the composition φ⊗ψ
τD
mL
A ⊗ C −→ B ⊗ L ⊗ D ⊗ L −→ B ⊗ D ⊗ L ⊗ L −→ B ⊗ D ⊗ L,
(39)
for φ ∈ HomO¯ L (A, B) and ψ ∈ HomO¯ L (C, D). ¯ defined by (39) makes O¯ L a monoidal category. Proposition 4.6. The tensor product ⊗ ¯ Let us prove associaProof. The unit object 1O is obviously the neutral element for ⊗. ¯ Using compatibility (35) of τ with the multiplication mL and associativity tivity of ⊗. (34) we find that the diagram L⊗A⊗L⊗B
τA
- A⊗L⊗L⊗B
mL
- A⊗L⊗B
τB
? L⊗A⊗B ⊗L
τB τA⊗B
- A⊗B ⊗L⊗L
? - A⊗B ⊗L
mL
is commutative for all A, B ∈ Ob O. From this one can readily deduce associativity of ¯ ⊗.
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¯ It is equivalent to the four conditions: Now we will prove functoriality of ⊗. ¯ ◦(id⊗ψ) ¯ ¯ ◦¯ ψ), (id⊗φ)¯ = id⊗(φ ¯ ¯ ¯ (φ ⊗id)¯◦(id⊗ψ) = φ ⊗ψ, ¯ ◦(ψ ⊗id) ¯ ¯ (φ ⊗id)¯ = (φ ◦¯ ψ)⊗id, ¯ ◦(φ ⊗id) ¯ ¯ (id⊗ψ)¯ = φ ⊗ψ
(40) (41) (42) (43)
¯ B = (idA ⊗ τB ) ◦ (φ ⊗ idB ) and for any pair of morphisms φ, ψ. Observe that φ ⊗id φ ¯ = idB ⊗ φ for any morphism A −→ A and any object B. This immediately idB ⊗φ leads to (40) and (41). Condition (42) follows from (35). Let us prove condition (43) assuming φ ∈ HomO¯ L (A, A ) and ψ ∈ HomO¯ L (B, B ). It suffices to show that the following diagram is commutative (the identity maps are suppressed): A⊗B
φ
τB
- A ⊗ L ⊗ B
HH ψ H φ⊗ψ H j H ? A ⊗ L ⊗ B ⊗ L HH
- A ⊗ B ⊗ L ψ
τB ⊗L
τB HH
? - A ⊗ B ⊗ L ⊗ L * τL
H j A ⊗ B ⊗ L ⊗ L HH
(44)
mL
mLHH
H j ? A ⊗ B ⊗ L
Commutativity of the rectangle follows from (33); the two lower triangles are commutative by virtue of (34 ) and (36). ¯ The category OL naturally includes O as a monoidal subcategory. We call O¯ L the dynamical extension of O over the base algebra L. ˆ then the category O¯ L is Example 4.7. The simplest example is L = 1Oˆ and O = O; canonically isomorphic to O. Example 4.8. Let H be a Hopf algebra and Oˆ the category of left H-modules. As was mentioned in Example 4.4, any H-base algebra, including H itself, is a base algebra in ˆ Let MH be the subcategory of locally finite H-modules (a module is called locally O. finite if every one of its elements lies in a finite dimensional submodule). Its dynamical ¯ H;L , or simply M ¯ H for L = H. extension over a base algebra L is denoted further by M 4.3. Morphisms of base algebras. By a morphism of base algebras (L1 , τ 1 ) → (L2 , τ 2 ) f ˆ in a category Oˆ we mean a morphism of O-algebras L1 −→ L2 such that the diagram L1 ⊗ A f ⊗idA
? L2 ⊗ A
ˆ is commutative for all A ∈ Ob O.
τA1
- A ⊗ L1 idA ⊗f
? - A ⊗ L2
τA2
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Example 4.9. Let H be a Hopf algebra and Oˆ the category of left H-modules. A homomorphism of two H-base algebras can be defined as a homomorphism of H-algebras ˆ cf. Example 4.4. and H-comodules. Then it is a morphism of base algebras in O, Example 4.10. Any invariant character χ of L defines a homomorphism of base algebras L → H by the formula → (1) χ [2] . Indeed, this is an algebra and comodule map because L is an H-comodule algebra. This map is equivariant for invariant χ , by virtue of (9). Recall that a functor from one monoidal category to another is called strong monoidal if it is unital (relates the units) and commutes with tensor products. We conclude this subsection with an obvious proposition. Proposition 4.11. A morphism of base algebras (L1 , τ 1 ) → (L2 , τ 2 ) induces a strong monoidal functor O¯ L1 → O¯ L2 . ∗
¯ H . The dynamical extension of a monoidal category can be defined 4.4. Category M using a notion of base coalgebra instead of base algebra. We will present such a formulation for the case when the monoidal category Oˆ is a category of H-modules and the base coalgebra is a restricted dual to H. Let H∗ denote the Hopf algebra formed by matrix elements of finite dimensional representations of H (we assume that the supply of such elements is big enough to induce a non-degenerate pairing between H∗ and H). We equip H∗ with the structure of a left H-module with respect to the action x ⊗ λ → x (2) λ γ (x (1) ),
x ∈ H,
λ ∈ H∗ ,
(45)
expressed through the coregular left and right actions, and , of H on H∗ . Let Oˆ be the category of left H-modules. We can consider the category of locally ˆ since every right H∗ -comodule is a finite right H∗ -comodules as a subcategory in O, ∗ natural left H-module. We denote this category by MH . The following statement introduces a permutation between H∗ and other ∗ H -comodules. ∗
Proposition 4.12. For any A ∈ Ob MH the map τ A : H∗ ⊗ A → A ⊗ H∗ defined as τ A (λ ⊗ a) := a [0] ⊗ λa (1)
(46)
is an isomorphism of H-modules. Proof. First of all observe that τ A is invertible and its inverse is (τ A )−1 (a ⊗ λ) = λγ −1 (a (1) ) ⊗ a [0] ,
λ ∈ H∗ , a ∈ A.
Further, for all x, y ∈ H we have τ A x (λ ⊗ a) , id ⊗ y = τ A x (1) λ ⊗ a [0] , id ⊗ ya (1) , x (2) = a [0] ⊗ (x (1) λ)a (1) , ya (2) , x (2) = a [0] ⊗ x (1) λ, y (1) a (1) , y (2) x (2) .
(47)
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On the other hand, x τ A (λ ⊗ a), id ⊗ y = a [0] ⊗ a (1) , x (1) λa (2) , γ (x (2) )yx (3) = a [0] ⊗ a (1) , x (1) λ, γ (x (3) )y (1) x (4) a (2) , γ (x (2) )y (2) x (5) = a [0] ⊗ a (1) , y (2) x (3) λ, γ (x (1) )y (1) x (2)
(48)
for all x, y ∈ H, λ ∈ H∗ , and a ∈ A. The resulting expression in (48) is easily brought to (47). ¯ H∗ , of the category MH∗ . The objects in Let us define the dynamical extension, M ∗ ¯ H are locally finite right H∗ - comodules. The set of morphisms Hom ¯ H∗ (A, B) M M consists of H-equivariant maps from H∗ ⊗ A to B. The composition φ ◦¯ ψ of morphisms φ ∈ Hom(A, A ) and φ ∈ Hom(A , A ) is defined as the composition map H∗ ⊗ A
⊗idA
- H∗ ⊗ H∗ ⊗ A
idH∗ ⊗ψ
- H ∗ ⊗ A
φ
- A
(49)
This operation is apparently associative and ε ⊗ idA is the identity in HomM ¯ H∗ (A, A); here ε is the counit in H∗ . ¯ H∗ . We put the tensor product of Now we introduce a monoidal structure on M ∗ ∗ ¯ H as in MH . The tensor product φ ⊗ψ ¯ of φ ∈ HomM objects from M ¯ H∗ (A, A ) and ) is defined as the composition ∗ ψ ∈ HomM (B, B ¯H τA
φ⊗ψ
H∗ ⊗ A ⊗ B −→ H∗ ⊗ H∗ ⊗ A ⊗ B −→ H∗ ⊗ A ⊗ H∗ ⊗ B −→ A ⊗ B . (50) ∗
¯ H a monodial category. ¯ makes M One can check that, indeed, the operation ⊗ ¯ H∗ and M ¯ H∗ . Since MH∗ is a subcategory in the 4.5. Comparison of categories M H ¯ H∗ over the base category of H-modules, it can be extended to the dynamical category M H algebra L = H along the lines of Subsect. 4.2. Our next goal is to compare the categories ¯ H∗ and M ¯ H∗ . Since they have the same supply of objects, we will study relations M H between their morphisms. Introduce a pairing between H∗ and H by the formula (h, x) := γ −1 (h), x,
(51)
where ., . is the canonical Hopf pairing H∗ ⊗ H → k. It is H-invariant, since H is considered as the adjoint H-module (8) and H∗ is an H-module by (45). ∗
Lemma 4.13. For any right H∗ -comodule A ∈ MH the diagram H∗ ⊗ A ⊗ H
A τ-
A ⊗ H∗ ⊗ H
τA−1
? H∗ ⊗ H ⊗ A is commutative. Proof. Straightforward.
(.,.)
(.,.)
? - A
(52)
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To any equivariant map φ : A → B ⊗ H we put into correspondence an equivariant map φ : H∗ ⊗ A → B being the composition φ
τB
(.,.)
H∗ ⊗ A −→ H∗ ⊗ B ⊗ H −→ B ⊗ H∗ ⊗ H −→ B.
(53)
H∗
¯ ¯ H∗ Clearly, this correspondence induces a natural embedding Hom M H → Hom M . Note that this embedding is not an isomorphism, in general. ¯ H∗ → Hom M ¯ H∗ , φ → φ , given by Proposition 4.14. The correspondence Hom M H ∗ ∗ ¯H →M ¯H . (53), induces a strong monoidal functor M H The proof of this proposition uses the diagram technique, the properties of permutations {τA } and {τ A }, and relies on Lemma 4.13. The details are left to the reader. 4.6. Dynamical extension of a monoidal category over a module category. Let O be a monoidal category and B its left module category, see [O]. For example, B is a monoidal category and O its monoidal subcategory. We denote the tensor product in O and action of O on B by the same symbol ⊗. For simplicity, all monoidal categories are assumed to be strict (with trivial associativity); the same is assumed for actions on module categories. Let us define a dynamical extension, O¯ B , of O over B in the following way. The collection of objects in O¯ B coincides with that of O. An object A of O¯ B is treated as A a functor from B to B, namely X −→ A ⊗ X for all X ∈ Ob B. Morphisms of O¯ B are natural transformations of the functors. Namely, φ ∈ HomO¯ B (A, B) is a collection {φX } of morphisms φX ∈ HomB (A ⊗ X, B ⊗ X} such that φX ◦ (idA ⊗ ξ ) = (idB ⊗ ξ ) ◦ φX
(54)
for any ξ ∈ HomB The composition of morphisms in O¯ B is “pointwise”, (φ ◦¯ ψ)X = φX ◦ ψX . Obviously, the condition (54) holds for ◦¯ . Clearly, O¯ B defined in this way is a category. It includes O as a subcategory. Indeed, any morphism φ from O gives rise to the family {φ ⊗ idX }, which is a morphism in O¯ B . (X , X).
Proposition 4.15. O¯ B is a monoidal category with respect to the tensor product on the objects as in Ob O and defined on the morphisms by ¯ X := (idC ⊗ ψX ) ◦ (φB⊗X ) = (φD⊗X ) ◦ (idA ⊗ ψX ), (φ ⊗ψ)
(55)
for φ ∈ HomO¯ B (A, C), and ψ ∈ HomO¯ B (B, D). ¯ X } defines a morphism of functors, A⊗B → Proof. Let us check that the family {(φ ⊗ψ) C ⊗ D. First of all, observe that condition (54) is satisfied. We will show that operation (55) is functorial. Take {αX } ∈ HomO¯ B (A , A) and {βX } ∈ HomO¯ B (B , B). We have ¯ ◦¯ β): for (φ ◦¯ α)⊗(ψ idC ⊗ (ψX ◦ βX ) ◦ (φB ⊗X ◦ αB ⊗X ) = (idC ⊗ ψX ) ◦ (idC ⊗ βX ) ◦ φB ⊗X ◦ αB ⊗X = (idC ⊗ ψX ) ◦ φB⊗X ◦ (idC ⊗ βX ) ◦ αB ⊗X ¯ X ¯ X ◦ (α ⊗β) = (φ ⊗ψ) for all X ∈ Ob B. In transition to the middle line we used the condition (54), in order to permute the morphisms idC ⊗ βX and φB ⊗X . To prove associativity, we
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J. Donin, A. Mudrov
take ζ ∈ HomO¯ B (A, U ), φ ∈ HomO¯ B (B, V ), ψ ∈ HomO¯ B (C, W ) and find that ¯ ⊗ψ) ¯ ¯ ⊗ψ ¯ ζ ⊗(φ and (ζ ⊗φ) are equal to the same composition map X X ζB⊗C⊗X
φC⊗X
ψX
A ⊗ B ⊗ C ⊗ X −→ U ⊗ B ⊗ C ⊗ X −→ U ⊗ V ⊗ C ⊗ X −→ U ⊗ V ⊗ W ⊗ X. This completes the proof.
Note that O is a monoidal subcategory in O¯ B . Definition 4.16. The category O¯ B is called a dynamical extension of O over B. ¯ of a monoiRemark 4.17. Similarly to O¯ B , one can define a dynamical extension, BO, dal category O over its right module category B. Thus, the set HomBO¯ (A, B) is formed by families {Xψ} from HomB (X ⊗ A → X ⊗ B) subject to the natural condition analogous to (54). The composition ◦¯ is defined as the composition of functor morphisms, similarly to the O¯ B case. Formula (55) for tensor products of morphisms is changed to ¯ X(φ ⊗ψ)
:= (Xφ ⊗ idD ) ◦ X⊗Aψ, φ ∈ HomBO¯ (A, C),
ψ ∈ HomBO¯ (B, D). (56)
¯ H;L and M ¯ H∗ . Let L be a base 4.7. Comparison of categories O¯ B and BO¯ with M algebra over a Hopf algebra H. Let B be the category of left L-modules, and O the category MH of locally finite left H-modules. Then B is a left O-module category. The tensor product of A ∈ Ob O and X ∈ Ob B is an L-module by (a ⊗ x) = (1) a ⊗ [2] x,
∈ L, a ∈ A, x ∈ X,
(57)
where denotes the action of L and the action of H. ¯ H;L of MH over the base algebra L as in Consider the dynamical extension M Example 4.8. Let ψ be a morphism from HomM ¯ H;L (A, B). Consider the family of maps ψX : A ⊗ X → B ⊗ X, X ∈ B, defined by the composition idB ⊗
ψ⊗idX
A ⊗ X −→ B ⊗ L ⊗ X −→ B ⊗ X.
(58)
The maps ψX defined by (58) are L-equivariant, due to quasi-commutativity of L. The following proposition is immediate. Proposition 4.18. The correspondence of morphisms ψ → {ψX } induces a strong ¯ H;L → O¯ B identical on objects. monoidal functor M ∗
Now take O to be the category MH of right locally finite H∗ -comodules also considered as left H-modules. Put B the category of locally finite H-modules. ∗
¯ H → BO. ¯ Proposition 4.19. There exists a strong monoidal functor M ¯ H∗ have the same Proof. We will give a sketch of the proof. Categories BO¯ and M collection of objects, and the functor in question is set to be identical on objects. Let ¯ H∗ . For every us define it on morphisms. Let f : H∗ ⊗ A → B be a morphism in M finite dimensional H-module X there is a natural map X∗ ⊗ X → H∗ , where X ∗ ⊗ X is considered as the (left) dual module to the space of right endomorphisms (over k) of X. Hence, f defines a collection of H-equivariant maps X∗ ⊗ X ⊗ A → B, or, equivalently, a collection {fX } of H-equivariant maps X ⊗ A → X ⊗ B. This family extends
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to all locally finite H-modules X. Thus we have built an embedding of morphisms ¯ f → {fX }. It remains to check that the above correspon¯ H∗ → Hom BO, Hom M dence is functorial and respects the composition and the tensor product of morphisms. We leave the details to the reader. Remark 4.20. The functor from Proposition 4.19 is an isomorphism when H∗ decomposes into the direct sum of X∗ ⊗ X, where X runs over simple H-modules. 4.8. Dynamical associative algebras. Dynamical associative algebra as an algebra in a monoidal (dynamical) category is defined in the standard way. Below we give examples ¯ H;L . ¯ H∗ and M of dynamical associative algebras in the categories M ¯ H;L ). Let us consider the dynamical extenExample 4.21 (Dynamical algebras in M ¯ H;L of the category MH over a H-base algebra L. An algebra A in M ¯ H;L is sion M an object equipped with a morphism A ⊗ A → A obeying the associativity axiom. In terms of MH , this is equivalent to Definition 3.11. Namely, the multiplication in A is an H-equivariant map : A ⊗ A → A ⊗ L, which is shifted associative in the sense of (14). Now let us prove Proposition 3.12. This is a corollary of the following general fact. Let (C, ⊗, 1C ) be a monoidal category whose objects are vector spaces over k and morphisms are linear maps. Suppose there is an object A ∈ Ob C, a morphism ι : 1C → A, and an
operation HomC (X, A) ⊗k HomC (Y, A) −→ HomC (X ⊗ Y, A) for all X, Y ∈ Ob C. We say that is a) natural if (φ ◦ α) (ψ ◦ β) = (φ ψ) ◦ (α ⊗ β), b) associative if (φ ψ) ϑ = φ (ψ ϑ), and c) unital if φ (ι ◦ χ ) = φ ⊗ χ , (ι ◦ χ ) φ = χ ⊗ φ for all morphisms χ with target in 1C . The multiplication m in A and the operation are ¯ H;L . related by m = idA idA , φ ψ = m ◦ (φ ⊗ ψ). Now let A be an algebra in M ¯ H;L gives the unit map k → A ⊗ L in MH . The The unit morphism ι : k → A in M multiplication in A defines a natural associative unital operation on morphisms from ¯ H;L with target in A. Hence it defines a natural associative unital operation on Hom M morphisms from Hom MH with target in A ⊗ L. ¯ H∗ ). Let us describe dynamical associative Example 4.22 (Dynamical algebras in M ∗ ¯ H∗ is an H¯ H . The multiplication in an algebra A ∈ Ob M algebras in the category M ∗ equivariant map : H∗ ⊗ A ⊗ A → A. Associativity, in terms of MH , is formalized by the following commutative diagram:
H∗ ⊗ A ⊗ A ⊗ A −→ H∗ ⊗ H∗ ⊗ A ⊗ A ⊗ A −→ H∗ ⊗ A ⊗ A −→ A τA ↓ . (59) H∗ ⊗ A ⊗ H∗ ⊗ A ⊗ A −→ H∗ ⊗ A ⊗ A −→ A This diagram is a “partial dualization” of the diagram (14). The algebra A is unital if there is an element 1 ∈ A such that (λ, 1, a) = (λ, a, 1) = ε(λ)a, for all a ∈ A, λ ∈ H∗ . λ The map defines a family of bilinear operations ∗ depending on elements λ ∈ H∗ . λ
In terms of ∗ , the “shifted” associativity (59) reads (summation implicit) λ(2)
λ(1)
λ(1)
(a ∗ b) ∗ c = a [0] ∗ (b
λ(2) a (1)
∗
c).
(60)
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J. Donin, A. Mudrov
I. Kantor proposed to consider the multiplication map H∗ ⊗ A ⊗ A −→ A as a ternary operation λ ⊗ a ⊗ b → (λab) which is associative in the sense (1) (2) λ (λ ab)c = λ(1) a (λ(2) bc) . Here a ⊗ λ = τ A (λ ⊗ a), the permutation (46). 5. Categorical Approach to Quantum DYBE 5.1. Dynamical twisting cocycles. In this subsection we study transformations of dynamϒ ical categories. Recall that a functor C˜ −→ C between two monoidal categories is called ˜ monoidal if there is a functor isomorphism F between ϒ(A) ⊗ ϒ(B) and ϒ(A⊗B). This implies a family of isomorphisms, FA,B
˜ ϒ(A) ⊗ ϒ(B) −→ ϒ(A⊗B) fulfilling the cocycle conditions (for simplicity, we assume the trivial associator) FA⊗B,C (61) ◦ FA,B ⊗ idC = FA,B ⊗C ˜ ˜ ◦ idA ⊗ FB,C , FA,1 = idA = F1,A , (62) where 1 is the unit of C. We are mostly interested in the situation when Ob C˜ = Ob C and ϒ is identical on objects. Suppose that F is a cocycle in C, i.e. a family of invertible morphisms FA,B ∈ AutC (A ⊗ B) fulfilling the conditions (61) and (62). Then it is possible to define a new monoidal structure on C. It is the same on objects and defined by ˜ := F ◦ (φ ⊗ ψ) ◦ F −1 φ ⊗ψ
(63)
on morphisms. This new monoidal category C˜ coincides with the old one if F respects morphisms of C, i.e. φ ⊗ ψ = F ◦ (φ ⊗ ψ) ◦ F −1
(64)
for all f, g ∈ Hom C. Remark 5.1. One can define the category C˜ using an arbitrary family FA,B ∈ AutC (A, B) of morphisms, which is not necessarily a cocycle. Then C˜ will not be strictly monoidal, −1 −1 FAB,C , which satisfies the but rather with the associator A,B,C = FA,BC FB,C FA,B ˜ The identity functor C˜ → C yields an isomorphism of monoidal pentagon identity in C. categories. Definition 5.2 (Dynamical twist). Let O¯ be a dynamical extension of a monoidal category O. Dynamical twist is a cocycle in O¯ that respects morphisms from O. A dynamical twist is identical on O, therefore O remains a subcategory in the twisted ˜¯ category O. One of the applications of twist is transformation of algebras. Any cocycle F in a ˜ category C makes a C-algebra with the multiplication m into a C-algebra, with the multiplication m ◦ F −1 . Let us apply this to the specific situation of dynamical twist and ¯ build an O-algebra out of O-algebra. ¯ Let A be an algebra in O with mulProposition 5.3. Let F be a dynamical twist in O. tiplication m. Then the multiplication m ◦ F makes A a dynamical associative algebra, ¯ i.e. an algebra in O.
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Proof. It follows from (63) that dynamical twist preserves O as a monoidal subcategory ˜¯ Therefore A turns out to be an algebra in O ˜¯ as well. The family {F −1 } is a cocycle in O. A,B ˜¯ the corresponding twist of O ˜¯ gives O. ¯ Applying this inverse twist to the algebra in O; ¯ A we obtain an O-algebra with the multiplication m ◦ F . Below we specialize the cocycle equations (61) and (62) for various types of dynamical categories. Example 5.4 (Dynamical twist in OB ). Let us express a cocycle in dynamical category O¯ B in terms of O and B. A cocycle in O¯ B is a collection (FV ,W )X from Aut B (V ⊗ W ⊗ X), V , W ∈ Ob O, X ∈ Ob B, satisfying conditions (FV ⊗W,U )X ◦ (FV ,W )U ⊗X = (FV ,W ⊗U )X ◦ (FW,U ), (FV ,1O¯ )X = idV ⊗X = (F1O¯ ,V )X .
(65) (66)
Example 5.5 (Drinfeld associator as a twist in O O). Let g be a complex simple Lie algebra. In [EE1], Enriquez and Etingof proposed a quantization of the Alekseev-Meinrenken dynamical r-matrix [AM] using the Drinfeld associator ∈ U ⊗3 (g)[[t]]. This ¯ where O is the catquantization can be interpreted as a twist in the category O O, egory of free C[[t]]-modules of finite rank with U(g)[[t]]-action. Indeed, let us put X(FA,B ) := X,A,B . Then the pentagon identity on takes the form A,B,C ◦ X(FA⊗B,C ) ◦ X(FA,B ) = X(FA,B⊗C ) ◦ X⊗A(FB,C ). The twisted dynamical category is not strictly monoidal, cf. Remark 5.1. It is equipped with the associator {A,B,C }. Example 5.6 (Dynamical twist in O¯ L ). Consider a cocycle in O¯ L , the dynamical extension of a category O over a base algebra (L, τ ). In terms of O, condition (61) reads mL⊗L ◦ FV ⊗W,U ◦ (idV ⊗W ⊗ τU ) ◦ FV ,W = mL⊗L ◦ FV ,W ⊗U ◦ FW,U ,
(67)
where FV ,W ∈ HomO (V ⊗ W, V ⊗ W ⊗ L) (the id-automorphisms are dropped from the formulas). ∗
¯ H ). Let us specialize the notion of cocycle for the Example 5.7 (Dynamical twist in M f ∗ ∗ ¯ H . A morphism H∗ ⊗ A → category M B in the category MH can be thought of as a λ ∗ family of maps f : A → B parameterized by elements λ ∈ H . Let λ be a family of ∗ linear operators on the tensor product ⊗m l=1 Vl of H -comodules Vl , l = 1, . . . , m. By Viλ , or simply by iλ , we denote the family of linear operators on ⊗m V defined by l=1 l (1)
i λ
(v1 ⊗ . . . ⊗ vm ) := λvi (v1 ⊗ . . . ⊗ vi[0] ⊗ . . . ⊗ vm ), (1)
where vi[0] ⊗ vi denotes the right H∗ -coaction δ(vi ) (as always, the summation is implicit). The collection of morphisms FVλ,W ∈ HomMH∗ (H∗ ⊗ V ⊗ W, V ⊗ W ) ¯ H∗ if and only if satisfies condition (61) and (62) in M (1)
(2)
(1)
(2)
λ , FVλ ⊗W,U FVλ,W = FVλ,W ⊗U VFW,U
FVλ,k Here λ(1) ⊗ λ(2) stands for H∗ (λ).
= idV =
λ Fk,V .
(68) (69)
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Example 5.8 (Universal cocycle). Assume that H is a Hopf subalgebra of another Hopf algebra, U. Consider the category MU as a subcategory of the category MH . Let L be a base algebra over H. Suppose there is an invertible element F¯ = F¯ 1 ⊗ F¯ 2 ⊗ F¯ 3 ∈ U ⊗ U ⊗ L that satisfies the condition h(1) F¯ 1 ⊗ h(2) F¯ 2 ⊗ h(3) F¯ 3 = F¯ 1 h(1) ⊗ F¯ 2 h(2) ⊗ F¯ 3
(70)
for all h ∈ H, and the conditions ¯ F¯ 23 ), ¯ (3)F¯ 12 = (id ⊗ )(F)( ( ⊗ id)(F) ¯ = 1 ⊗ 1 ⊗ 1 = (id ⊗ ε ⊗ id)(F), ¯ (ε ⊗ id ⊗ id)(F)
(71) (72)
¯ where where Equation (71) is in U ⊗ U ⊗ U ⊗ L. Here the notation (3)F¯ means δ(F), δ is the coaction L → H ⊗ L; the H-component is embedded to the third tensor fac¯ U ;L , namely FV ,W := tor in U ⊗ U ⊗ U ⊗ L. The element F¯ defines a cocycle in M ρV (F¯ 1 ) ⊗ ρW (F¯ 2 ) ⊗ F¯ 3 for U-modules V and W . This cocycle clearly respects morphisms in MU , hence it is a dynamical twist. The element F¯ may be called a universal dynamical twist, by the analogy with the universal R-matrix. Equation (71) leads to the shifted cocycle condition of [Xu2] for H being a universal enveloping algebra. 5.2. Quantum dynamical R-matrix. 5.2.1. Dynamical Yang-Baxter equation. Let us consider the Yang-Baxter equation in dynamical categories. Let C be a braided monoidal category with braiding σ . The braidσA,B ing is a collection, {σA,B }, of morphisms A ⊗ B −→ B ⊗ A for A, B ∈ Ob C obeying conditions σA,B ◦ σA,C ◦ σB,C = σB,C ◦ σA,C ◦ σA,B , σA⊗B,C = σA,C ◦ σB,C , σC,A⊗B = σC,B ◦ σC,A
(73) (74)
and respecting morphisms, i.e. (f ⊗ g) ◦ σ = σ ◦ (g ⊗ f ) for all f, g ∈ Hom C (in fact, (73) follows from (74) and functoriality of σ ). Condition (73) is called the Yang-Baxter equation, conditions (74) are, in fact, the hexagon identities. If σ fulfills (73) and (74) but is not functorial (does not respect morphisms), we call it pre-braiding. This is the case when σ is a braiding in a subcategory C of C such that Ob C = Ob C, e.g., when C is a dynamical extension of a C . Then C has more morphisms than C , and they are not respected by σ , in general. Given a pre-braiding σ in C, it is possible to restrict it to a braiding in a subcategory Cσ defined as follows. The objects in Cσ are those of C. A morphism f ∈ HomC (A, B) is a morphism in HomCσ (A, B) if and only if σB,C ◦ (f ⊗ idC ) = (idC ⊗ f ) ◦ σA,C ,
(f ⊗ idC ) ◦ σC,A = σC,B ◦ (idC ⊗ f ) (75)
for all C ∈ Ob C. Proposition 5.9. Cσ is a braided category with braiding σ . For instance, the dynamical extension O¯ L of a braided category (O, σ ) over a commutative algebra L in O (cf. Example 4.3) is braided if and only if σA,L ◦ σL,A = idL⊗A for all A ∈ Ob O, e.g. when O is a symmetric category.
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Proof. It follows from (73) and (74) that σ lies in Cσ . Condition (74) guarantees that Cσ is a monoidal category. Therefore, σ is a pre-braiding in Cσ and respects morphisms in it by construction; hence σ is a braiding in Cσ . Proposition 5.10. Let σ be a pre-braiding in C and let F be a cocycle in C respecting morphisms from Cσ . Then the family −1 σ¯ A,B := FB,A ◦ σA,B ◦ FA,B
(76)
satisfies the Yang-Baxter equation (73). Proof. Define A,B,C := FA⊗B,C ◦ (FA,B ⊗ idC ) = FA,B⊗C ◦ (idA ⊗ FB,C ) for all A, B, C ∈ Ob C. Since F respects morphisms from Cσ , we have −1 A,C,B ◦ (idA ⊗ σB,C ) ◦ −1 A,B,C = idA ⊗ σ¯ B,C and B,A,C ◦(σA,B ⊗idC )◦A,B,C = σ¯ A,B ⊗idC for all A, B, C. Multiplying Eq. (73) by −1 C,B,A from the left and by A,B,C from the right, we prove the statement. Applied to dynamical twists, Proposition 5.10 yields the following corollary. Corollary 5.11. Let O be a braided category with the braiding σ . Let O¯ be a dynamical ¯ The collection of morphisms (76) for extension of O and F a dynamical twist in O. ¯ A, B ∈ O¯ satisfies the Yang-Baxter equation in O. ˜ However, In general, a twist destroys the hexagon identities in the twisted category C. ˜ it yields a pre-braiding in an equivalent category to C, which is constructed in Subsect. ¯ H,L , the dynamical exten5.2.2. There is another way to fix the situation when C = M sion of the category of H-modules over a base algebra L. There exists a realization ¯ H,L as a category of modules over a certain bialgebroid, [DM5]. A dynamical of M twist gives rise to a bialgebroid twist, which transforms the braiding in the category of modules over the bialgebroid. We call a solution of (73) in a dynamical category a dynamical R-matrix. Below we specialize this definition of dynamical R-matrix to various types of dynamical categories. Example 5.12 (Dynamical R-matrix in O¯ B ). The dynamical R-matrix in the category O¯ B is defined by (σA,B )X ◦ (σA,C )B⊗X ◦ (σB,C )X = (σB,C )A⊗X ◦ (σA,C )X ◦ (σA,B )C⊗X ,
(77)
where σ is a collection of invertible morphisms (σA,B )X ∈ AutB (A ⊗ B ⊗ X). Example 5.13 (Dynamical R-matrix in O¯ L ). Consider the category O¯ L , a dynamical extension of a monoidal category O over a base algebra (L, τ ), cf. Subsect. 4.2. Let m be the multiplication in the algebra L and m3 denote the three-fold product m◦(m⊗idL ). In terms of O and (L, τ ), Eq. (73) reads m3 ◦ σA,B ◦ τB ◦ σA,C ◦ σB,C = m3 ◦ τA ◦ σB,C ◦ σA,C ◦ τC ◦ σA,B , where σA,B ∈ HomOˆ (A ⊗ B, A ⊗ B ⊗ L).
(78)
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¯ H ). We use the notation of Example 5.7. A Example 5.14 (Dynamical R-matrix in M ∗ λ ¯ H∗ if and only if collection of morphisms {σA,B } from Hom MH fulfills Eq. (73) in M A λ(1) σB,C
(2)
λ σA,C
C λ(3) σA,B
(1)
λ = σA,B
B λ(2) σA,C
(3)
λ σB,C .
(79)
Example 5.15 (Universal dynamical R-matrix). Consider the situation of Example 5.8 assuming that H is a Hopf subalgebra in another Hopf algebra, U, and L is a H-base algebra. Definition 3.32 of Subsect. 3.5 introduces a universal quantum dynamical R-matrix of U over the base L. For any pair V and W of U-modules considered as modules over ¯ H, it gives σV ,W := PV ,W RV ,W , where P is the usual flip and RV ,W = (ρV ⊗ ρW )(R) is the image of R¯ in End(V ) ⊗ End(W ) ⊗ L. Proposition 5.16. Suppose the Hopf algebra U is quasitriangular and let R be its universal R-matrix. Let L be a base algebra over H ⊂ U and F¯ ∈ U ⊗ U ⊗ L a universal ¯ := F¯ −1 RF¯ is a universal dynamical R-matrix. dynamical twist. Then the element R 21 Proof. This statement can be checked directly. Another way to verify it is to consider representations of U. Then the statement follows from Proposition (76). 5.2.2. Dynamical (pre-) braiding. Let C be a monoidal category. Let F be a cocycle in C and C˜ be the twisted category defined in Subsect. 5.1. Suppose σ is a pre-braiding ˜ We are in C. As was mentioned above, the hexagon identities (74) are destroyed in C. going to construct an equivalent monoidal category F (C) where the twist of σ will be a pre-braiding. We consider formal sequences (words) A := (A1 , A2 , . . . , An ), n > 0, of objects from C. For two words A and B, let A • B denote the concatenation (A1 , A2 , . . . , An , B1 , B2 , . . . , Bm ). Let α(A) denote the tensor product A1 ⊗. . .⊗An ∈ Ob C. By induction on the length of words, let us introduce an isomorphism A of α(A) ∈ Ob C setting A := idA for A = A ∈ Ob C and A•B := Fα(A),α(B) (A ⊗ B ).
(80)
One can check, using the cocycle condition (61), that A does not depend on a particular partition of A into two concatenated words. Using the family {A }, define a f
transformation F (f ) of morphisms α(A) → α(B) in C setting F (f ) := B f A −1 .
(81)
Let us construct the category F (C). The objects of F (C) are finite formal sequences of objects from C. The set of morphisms HomF (C ) (A, B) consists of F (f ), where f is a morphism from HomC α(A), α(B) . We define the tensor product of objects A and B of F (C) as the concatenation A • B. The empty word plays the role of the unit object. Let us define the tensor product of morphisms in F (C). Let F (f ) : A → A and F (g) : B → B be two morphisms. Then we put F (f ) • F (g) := F (f ⊗ g) : A • B → A • B .
(82)
The category F (C) is equivalent to C. Indeed, the correspondence A → α(A), F (f ) → f gives a strong monoidal functor α : F (C) → C. Consider also the functor β : C →
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F (C) defined on objects by β(A) = A, the word of length n = 1, and on morphisms by β(f ) = f . This functor is monoidal. Indeed, one can interpret A as a morphism in F (C), namely, −1 A = F (idA1 ⊗ . . . ⊗ idAn ) ∈ Hom F (C ) (A1 • . . . • An , A1 ⊗ . . . ⊗ An ). −1 (A,B)
So we obtain the transformation of the tensor products β(A) • β(B) −→ β(A ⊗ B). The functors α and β give the equivalence of categories C and F (C). Proposition 5.17. Let σ be a pre-braiding σA,B := F (σα(A),α(B) ) is a pre-braiding in F (C).
in
C.
Then
Proof. Apply F to Eq. (73) and (74) and use the definition (82).
the
collection
For example, let us specialize σA,B for A = A and B = B. In this case, we have A,B = FA,B . Applying formula (82), we obtain σA,B = FB,A σA,B F −1 A,B for A = A and B = B. 6. A Construction of Dynamical Twisting Cocycles 6.1. Associative operations on morphisms and twists. Let C be a monoidal category and C a subcategory in C. We are going to show that cocycles in C (see Subsect. 5.1) are in one-to-one correspondence with natural associative operations on morphisms HomC (A, V ), where A ∈ Ob C and V ∈ Ob C . First of all observe that a cocycle F in C defines such an operation by the formula φ ψ := F ◦ (φ ⊗ ψ). The converse is also true. Lemma 6.1. Suppose there is an associative operation
HomC (A, V ) ⊗ HomC (B, W ) −→ HomC (A ⊗ B, V ⊗ W ) for all A, B ∈ Ob C and V , W ∈ Ob C that is natural with respect to its C-arguments: (φ ◦ α) (ψ ◦ β) = (φ ψ) ◦ (α ⊗ β),
(83)
whenever φ ∈ HomC (A, V ), ψ ∈ HomC (B, W ), α, β ∈ Hom C. Suppose it is unital, i.e. φ χ = φ ⊗ χ, χ φ = χ ⊗ φ for any morphism φ and any χ ∈ HomC (B, 1C ). Then the family FV ,W := idV idW ∈ EndC (V ⊗ W ) C.
This cocycle respects morphisms from a subcategory is a cocycle in only if the operation is natural with respect to C -arguments, i.e.
(84) C
in
C
if and
(ζ ◦ φ) (η ◦ ψ) = (ζ ⊗ η) ◦ (φ ψ) whenever φ ∈ HomC (A, V ), ψ ∈ HomC (B, W ), ζ, η ∈ Hom C . Proof. By the definition (84), the expression FU ⊗V ,W ◦ (FU,V ⊗ idW ) is equal to (85) idU ⊗V idW ◦ (idU idV ) ⊗ idW = idU idV idW . Here we have used condition (83). Similarly, the expression FU,V ⊗W ◦ (idU ⊗ FV ,W ) is brought to the right-hand side of (85). Thus FV ,W satisfies the cocycle condition.
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6.2. Dynamical adjoint functors. In this subsection we formulate the notion of dynamical adjoint functor, which appears to be very useful in constructing dynamical twists. Let O be a monoidal category and O its monoidal subcategory; the embedding functor O → O is denoted by R. Let B and B be right module categories over O and O , respectively. M
Definition 6.2. A functor B −→ B is called dynamical adjoint to R if there is an isomorphism of the following three-functors from B × B × O to the category of linear spaces: Y ×X×V → HomB Y, X⊗R(V ) Y ×X×V → HomB M(Y ), M(X)⊗V . (86) Given a pair of dynamical adjoint functors, we introduce an operation on mor¯ ¯ phisms from Ob BO to its subcategory Ob BO in the following way. A pair {Xφ} ∈ HomBO¯ A, R(V ) and {Xψ} ∈ HomBO¯ B, R(W ) defines a family of B -morphisms M(X ⊗ A ⊗ B) → M(X) ⊗ V ⊗ W , for all X ∈ B, via the composition ˜ (X⊗A)ψ
˜ Xφ⊗id W
M(X ⊗ A ⊗ B) −→ M(X ⊗ A) ⊗ W −→ M(X) ⊗ V ⊗ W.
(87)
By the tilde we denote the image of a morphism from Hom BO¯ under the correspondence (86). By condition (86), the composition (87) yields a morphism, X(φ ψ)
X ⊗ A ⊗ B −→ X ⊗ R(V ⊗ W ),
(88)
in the category B. Functoriality with respect to the first argument in (86) implies that ¯ the family {X(φ ψ)} is in fact an BO-morphism. The associativity of the operation follows from the associativity of composition of morphisms in the category B . Thus we obtain the following result. Proposition 6.3. A pair of dynamical adjoint functors defines, by formula (88), an associative operation φ ⊗ψ → φ ψ that satisfies the conditions of Lemma 6.1 for C = BO¯ and C = BO¯ . It is O -functorial and thus yields a dynamical twist of O . In the next subsection, using Lemma 6.1 and Proposition 6.3, we construct a dynamical cocycle in the category of g-modules considered as a subcategory of l-modules, where l is an arbitrary Levi subalgebra in g. 6.3. Generalized Verma modules. Let g be a complex reductive Lie algebra with the Cartan subalgebra h and g = n− ⊕ h ⊕ n+ a polarization with respect to h. We fix a Levi subalgebra l, which is, by definition, the centralizer of an element in h. The algebra l is reductive, so it is decomposed into the direct sum of its center and the semisimple part, l = c ⊕ l0 , where l0 = [l, l]. Also, there exists a decomposition + g = n− l ⊕ l ⊕ nl ,
(89)
± ± ± where n± l are subalgebras in n . Let p denote the parabolic subalgebras l ⊕ nl . Let X be a finite dimensional semisimple representation of l. We consider X as a left U(l)-module. Being extended by the trivial action of n+ l , this representation can be considered as a left U(p+ )-module. We denote by MX the generalized Verma module
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MX := U(g) ⊗U (p+ ) X. It is a left U(g)-module, and the natural map U(n− l ) ⊗C X → U(g) ⊗U (p+ ) X is an isomorphism of vector spaces. Let us consider the dual representation X ∗ as a left U(l)-module with the action (uϕ)(x) = ϕ(γ (u)x),
(90)
where ϕ ∈ X ∗ , x ∈ X, u ∈ l, and γ denotes the antipode in U(g). Analogously to MX , we define the generalized Verma module MX−∗ := U(g) ⊗U (p− ) X ∗ naturally isomorphic ∗ as a vector space to U(n+ l ) ⊗C X . There exists the following equivariant pairing between MX−∗ and MX. Let u1 ⊗ ϕ ∈ − ∗ U(n+ l ) ⊗C X , u2 ⊗ x ∈ U(nl ) ⊗C X. We put u1 ⊗ ϕ, u2 ⊗ x = ϕ s(γ (u1 )u2 )x , where s is the projection U(g) → U(l) along the direct sum decomposition + U(g) = U(l) ⊕ (n− l U(g) + U(g)nl ).
It is obvious that this pairing defines the U(g)-equivariant map MX−∗ → MX∗ ,
(91)
where MX∗ denotes the restricted dual U(g)-module to MX , which is defined as follows. It is clear that MX = ⊕µ MX [µ], where MX [µ] is the finite dimensional subspace of weight µ ∈ h∗ . We put MX∗ := ⊕µ (MX [µ])∗ with the U(g)-action similar to (90). It is known that map (91) is an isomorphism for representations X satisfying conditions of Proposition 6.4 below. Since U(l) = U(l0 ) ⊗ U(c), where l0 is the semisimple part of l and c its center, a U(l)-module X is irreducible if and only if it can be presented as the tensor product of two representations: X = X0 ⊗ C λ .
(92)
Here X0 is an irreducible representation of l0 , and Cλ is a one dimensional representation of c defined by a character λ ∈ c∗ ; both X0 and Cλ are lifted to U(l)-modules in the natural way. It is clear that representation (92) is unique. We call the element λ from (92) the character of X. Let αi , i = 1, . . . , dim c, be the simple roots with respect to h that are not roots of l, and e±αi the corresponding root vectors such that (eαi , e−αi ) = 1 for the Killing form (., .) in g. Put hi := [eαi , e−αi ], i = 1, . . . , dim c. Denote by Y the union of hyperplanes in c∗ consisting of λ ∈ c∗ having at least one coordinate λ(hi ) integer. Proposition 6.4 ([J]). Let X be a semisimple representation of l. If the characters of its irreducible components do not belong to Y, then the map (91) is an isomorphism. We call an l-module X generic if it satisfies this proposition. 6.4. Dynamical twist via generalized Verma modules. In this subsection we construct a dynamical cocycle for the case when the Hopf algebra H is a (quantum) universal enveloping algebra of a Levi subalgebra l in a reductive Lie algebra g. Our method is a generalization to noncommutative and non-cocommutative Hopf algebras of the construction of Etingof and Varchenko, [EV3]. For simplicity we consider only classical universal enveloping algebras U = U(g), H = U(l). The construction carries over to the quantum groups in a straightforward way. Recall that MU (l) and MU (g) denote the categories of locally finite semisimple modules over U(l) and U(g), respectively.
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Lemma 6.5. For all Y ∈ MU (l) , V ∈ MU (g) , and generic X ∈ MU (l) , Homg (MY , MX ⊗ V ) Homl (Y, X ⊗ V ).
(93)
Proof. Since, by Proposition 6.4, the module MX∗ is isomorphic to MX−∗ for generic X, we have Homg (MY , MX ⊗ V ) Homg (MY ⊗ MX∗ , V ) Homg (MY ⊗ MX−∗ , V ), (94) g
where MX∗ is the restricted dual to MX . Since MY ⊗MX−∗ Indl (Y ⊗X∗ ) as a g-module, we can apply the Frobenius reciprocity and obtain Homg (MY ⊗ MX−∗ , V ) Homl (Y ⊗ X ∗ , V ) Homl (Y, X ⊗ V ). Combining (94) and (95) we prove the lemma.
(95)
Set, in terms of Definition 6.2, O to be the full subcategory in MU (l) of modules whose characters belong to the weight lattice of g relative to h. This category contains MU (g) as a subcategory, which we put to be O . Let B be the full subcategory in MU (l) of modules whose characters do not belong to Y; it is a module category over O. Let B be the category of all U(g)-modules. Put R : O → O to be the restriction functor making an U(g)-module a module over U(l). We define the adjoint functor M as follows. For X ∈ ObMU (l) we put M(X) = MX , the generalized Verma module corresponding to X. It is clear that any morphism X → Y of U(l)-modules naturally corresponds to a morphism MX → MY in the category B . M
Corollary 6.6. The functor X → MX is dynamical adjoint to the restriction functor R
MU (g) → O. Proof. All we have to check is that correspondence (93) is natural with respect to Y , V , and generic X. This holds because the Frobenius reciprocity gives a natural isomorphism between adjoint functors for generic X. ∗
Let us consider the category MU (g) of locally finite semisimple right U ∗ (g)-com∗ odules. Note that MU (g) is naturally isomorphic to the category MU (g) of locally finite semisimple left U(g)-modules and hence to a subcategory of locally finite semisimple ∗ ¯ U ∗ (l) the full left U(l)-modules. We call the dynamical extension of MU (g) within M ∗ (g) ∗ (l) U U ¯ subcategory in M whose objects belong to M . Let us consider in more detail the structure of U ∗ (l). First of all, U ∗ (l) can be interpreted as the algebra of polynomial functions on the connected simply connected Lie group Hˆ corresponding to the Lie algebra l. That is, U ∗ (l) is generated over C by matrix elements of all finite dimensional semisimple representations of l. The group Hˆ is presented as the Cartesian product Hˆ 0 × c of the semisimple subgroup Hˆ 0 and c viewed as an abelian group. It is well known that U ∗ (l) = ⊕V End∗C (V ), where V runs over the irreducible U(l)modules. Each irreducible representation of l has the form V = V0 ⊗ Cµ , where V0 is a module over the semisimple part l0 of l and Cµ is a one dimensional representation of V0 the matrix elements the center c. Let eµ : c → C× be the matrix element of Cµ and eij V0 µ of V0 . The elements {eij e } form a basis in the vector space U ∗ (l). We call an element ∗ λ ∈ U (l) generic if the decomposition of λ via this basis contains no eµ , for µ ∈ Y.
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Theorem 6.7. Let l be a Levi subalgebra in a reductive Lie algebra g, U(l) the corresponding Hopf subalgebra in the universal enveloping algebra U(g). There exists an U(l)-equivariant map F¯ : U ∗ (l) → U(g) ⊗ U(g) ¯ such that for generic λ ∈ U ∗ (l) the family FVλ,W := (ρV ⊗ ρW ) F(λ) , V, W ∈ U ∗ (g) , is a dynamical twist (68–69) in the dynamical extension of the category Ob M ∗ ¯ U ∗ (l) . MU (g) within M Proof. By Corollary 6.6 and Proposition 6.3, there exists a dynamical twist in the cate¯ which is a collection of morphisms X(FV ,W ) ∈ Endl (X⊗V ⊗W ), where V and gory BO, W are g-modules and X is a generic l-module. Using the natural filtration in generalized ¯ U ∗ (l) . The Verma modules, one can prove that morphisms {X(FV ,W )} are invertible in M ∗ morphisms X(FV ,W ) define a collection of l-equivariant maps X ⊗X → Endk (V ⊗W ), which gives rise to a collection of maps X∗ ⊗ X → U(g) ⊗ U(g) for generic X, since the dynamical twist is natural with respect to the arguments V and W . This collection determines an l-equivariant map F¯ : U ∗ (l) → U(g) ⊗ U(g) defined for generic of elements U ∗ (l). Indeed, by Remark 4.20 the dynamical category BO¯ is isomorphic to ¯ U ∗ (l) . Under this isomorphism, the dynamical twist {X(FV ,W )} goes over to the map M ¯ λ → F(λ) for λ ∈ X∗ ⊗X and X generic, which reduces to the twisting cocycle (68–69) in representations. 7. Dynamical Associative Algebras and Quantum Vector Bundles 7.1. Classical vector bundles. Let H be a Lie group and P be a principal H -bundle. Denote by A = A(P ) the algebra of functions on P . Let V be a finite dimensional left H -module. An associated vector bundle V (M) on M = P /H with the fiber V is defined as the coset space (P × V )/H by the action (p, v) → (ph, h−1 v), (p, v) ∈ (P × V ), h ∈ H . The global sections of V (M) are identified with the space H A(P ) ⊗ V HomH V ∗ , A(P ) . Let us denote by AV the space of global sections of V (M). When V = k, the trivial H -module, the space Ak is canonically identified with the subalgebra in A of H -invariant functions; in other words, Ak = A(M). The tensor product of vector bundles corresponds to the tensor product of sections, which is induced by multiplication in A: given sV ∈ AV and sW ∈ AW the section sW ⊗ sV ∈ AV ⊗W is (sW ⊗ sV )(w ⊗ v) := sW (w)sV (v),
w ⊗ v ∈ W ∗ ⊗ V ∗ (V ⊗ W )∗ .
In particular, the tensor product of sections makes the space AV a two-sided module over Ak . 7.2. Quantum vector bundles. Fix a Hopf algebra H over the ground ring k and con¯ H∗ , cf. Example 4.22. We are sider a dynamical associative algebra A in the category M going to introduce associated vector bundles over the “non-commutative coset space” corresponding to the action of H on A. Definition 7.1. Let V be a right H-module. The associated vector bundle AV with fiber V is the space of all H-equivariant maps (sections) sV : V ∗ → A.
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Observe that the restriction of the dynamical multiplication in A to a group-like eleλ ment λ of H∗ defines a bilinear operation ∗ : A ⊗ A → A, which is H-equivariant, since group-like elements are invariant in H∗ under the coadjoint action (45). Now we can define a product of sections. Let V and W be two H-modules. Take sV : V ∗ → A and sW : W ∗ → A to be sections of AV and AW . Fix a group-like element λ ∈ H∗ . The λ map sW ∗ sV : (V ⊗ W )∗ W ∗ ⊗ V ∗ → A, λ
λ
(sW ∗ sV )(w ⊗ v) := sW (w) ∗ sV (v),
w ⊗ v ∈ W ∗ ⊗ V ∗,
(96)
is a section of the bundle AV ⊗W . The subspace of H-invariants Ak ⊂ A is obviously λ closed under ∗ for every group-like element λ ∈ H∗ . Theorem 7.2. For any group-like element λ ∈ H∗ and any finite dimensional H-modλ ule V the multiplication ∗ provides Ak with the structure of an associative algebra, Akλ , and makes the space AV a left Akλ -module. If V = kα , i.e. is the 1-dimensional representation of H defined by the character α, then the line bundle AV is also a right λ Akλα −1 -module with respect to ∗ . For any a ∈ Akλ , sV ∈ AV , and sW ∈ AW λ
λ
λ
λ
a ∗ (sV ∗ sW ) = (a ∗ sV ) ∗ sW .
(97)
Proof. Sections of the line bundle Akα may be treated as elements a ∈ A such that h a = α −1 (h)a for all h ∈ H (the inverse is understood in the sense of the algebra H∗ ). For a ∈ Akα and b, c ∈ A, the formula (60) turns into λ
λ
λ
λα −1
(a ∗ b) ∗ c = a ∗ (b ∗ c)
(98)
under the assumption that λ ∈ H∗ is group-like. Setting α = 1 (the unit of H∗ ) in λ (98), we find that ∗ is associative when restricted to Ak and makes it an associative algebra, Akλ . Also we see that A is a left Akλ -module. This induces the structure of a left Akλ -module on AV for every H-module V , by formula (98). Assuming b, c ∈ Ak in (98) we obtain a right Akλα −1 -module structure on the space Akα . 8. Vector Bundles on Semisimple Coadjoint Orbits The problem of equivariant quantization of function algebras on semisimple coadjoint orbits of simple Lie groups was studied, e.g., in [DGS1, DolJ, DM1, DM2, DM4, DS]. Quantization of vector bundles on semisimple orbits as modules over the quantized function algebras was considered in [D1, GLS]. In this section we use dynamical associative algebras for quantization of the entire “algebra” of sections of all vector bundles on semisimple coadjoint orbits. 8.1. Dynamical quantization of the function algebra on a group. Let g be a simple Lie algebra and G the corresponding connected Lie group. We will apply the previous considerations to the problem of equivariant quantization of vector bundles on semisimple orbits in g∗ with respect to the coadjoint action of G. Denote by A(G) the algebra of polynomial functions on G. The group G acts on itself by the left and the right regular actions. These actions induce two left commuting actions of U(g) on A(G) via the differential operators ρ1 (x) and ρ2 (x), x ∈ U(g), respectively. Here ρ1 (x) (ρ2 (x)) is the
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differential operator on G that is the right (left) invariant extension of γ (x) (x), where γ denotes the antipode in U(g). Given a representation π : G → End(V ) assign to each f ∈ End(V )∗ the function f ◦ π on G. Identifying End(V )∗ with End(V ) = V ⊗ V ∗ via the trace pairing, these assignments give the well known isomorphism ⊕E E ⊗ E ∗ → A(G),
(99)
where E runs over all irreducible representations of G. Then the ρ1 -action can be treated as an action on the E-factor while the ρ2 -action – on the E ∗ -factor of each term E ⊗ E ∗ in the direct sum (99). As a manifold, a semisimple orbit is the quotient M = G/H , where H is a Levi subgroup with the Lie algebra l ⊂ g. Recall the decomposition l = l0 ⊕ c, where l0 is the semisimple part and c the center of l. The manifold G may be considered as a principal H -bundle on M. Any equivariant vector bundle on M is associated to the bundle G via a representation V of H . We denote this vector bundle by V (M). It has the vector space V as the fiber. l The global sections of the bundle V (M) are identified with the space A(G) ⊗ V = Homl V ∗ , A(G) , where A(G) is considered as a U(g)-module with respect to the ρ2 action. Since M is an affine variety, one can identify the vector bundle V (M) with its global sections and consider it as a U(g)-module with respect to the ρ1 -action. In particular, suppose V = Cλ is the one dimensional representation of l induced by the character λ ∈ c∗ . Let us consider λ as an element of h∗ via the embedding c∗ ⊂ h∗ along the decomposition h = c⊕c⊥ . Due to isomorphism (99), the assignment Homl C∗λ , A(G) ϕ → ϕ(1) ∈ A(G) gives a natural isomorphism of U(g) ⊗ U(l)-modules Homl C∗λ , A(G) → A(G)[−λ],
(100)
where A(G)[−λ] is a subspace of l0 -invariant elements of A(G) of weight −λ with respect to the ρ2 -action. It is obvious that A(G)[−λ] is a U(g)-module with respect to the ρ1 -action and it is naturally isomorphic to ⊕E E ⊗ E ∗ [−λ], where E ∗ [−λ] denotes the subspace of l0 -invariant elements of E ∗ of weight −λ. It is clear that the isomorphism (100) is actually non-zero only if λ is an integer weight. In this case the map (100) identifies A(G)[−λ] with the space of global sections of the line bundle Cλ (M). In particular, the function algebra on M is naturally isomorphic to the U(g)-module algebra ⊕E E ⊗ E ∗ [0] ⊂ A(G). Applying the dynamical twist F¯ constructed in Theorem 6.7 to the U(g)-module algebra A(G) with respect to the ρ2 -action, we obtain a dynamical associative U(l)-algebra ∗ in the category O¯ U (l) . This algebra is equal to A(G) as a U(g)-module (with respect to ρ1 -action) and has the family of multiplications parameterized by generic λ ∈ U ∗ (l) and defined as m ¯ λ = m ◦ F¯ λ , where m is the original multiplication in the algebra A(G). Applying Theorem 7.2 to this dynamical associative algebra, we obtain a quantization of vector bundles on G/H . Obviously, this quantization is equivariant with respect to the ρ1 -action of U(g). ¯ Let us consider the dynamical twist F(λ) restricted to U ∗ (c). Applied to the suball 0 gebra A(G) ⊂ A(G) of l0 -invariant functions on G with respect to the ρ2 -action, this restriction makes A(G)l0 a dynamical associative U(l)-algebra over the base U ∗ (c). As a U(g)-module, it is formed by sections of all linear bundles on M. Let us describe this dynamical algebra in more detail.
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Let Mλ = U(g)⊗U (p+ ) Cλ be theVerma module corresponding to the one dimensional representation of U(l) associated to λ ∈ c∗ . Let Hom0 (Mλ , Mµ ) denote the subspace of locally finite elements in HomC (Mλ , Mµ ) with respect to the adjoint action of U(g). In fact, Hom0 (Mλ , Mµ ) is not zero only when µ − λ is an integer weight. Recall that for a U(g)-module E, we denote by E[λ], λ ∈ c∗ , the subspace of l0 -invariant elements of weight λ. Proposition 8.1. Let V be a finite dimensional representation of U(g). Then, there is a natural morphism of U(g)-modules: V ⊗ V ∗ [λ − µ] → Hom0 (Mλ , Mµ ), λ, µ ∈ c∗ , µ is generic. When V is irreducible, this morphism is embedding. These embeddings give rise to the natural isomorphism jλ,µ : ⊕E E ⊗ E ∗ [λ − µ] → Hom0 (Mλ , Mµ )
(101)
of U(g)-modules, where E runs over all finite dimensional irreducible representations of U(g). Proof. It is enough to prove the first part of the proposition and show that the multiplicity of V in Hom0 (Mλ , Mµ ) is equal to dim V ∗ [λ − µ]. Applying the Frobenius reciprocity, one proves that for generic µ ∈ c∗ the space HomU (g) (Mλ , Mµ ⊗ V ∗ ) is naturally isomorphic to V ∗ [λ−µ]; the proof is the same as in [ES1]. But HomU (g) (Mλ , Mµ ⊗V ∗ ) ∼ = HomU (g) V , Hom0 (Mλ , Mµ ) , which proves the proposition. Compositions Mν → Mµ → Mλ generate the map Hom0 (Mµ , Mλ ) ⊗ Hom0 (Mν , Mµ ) → Hom0 Mν , Mλ , λ, µ, ν ∈ c∗ . Due to isomorphisms (101) and (99), this map defines the morphism of U(g) ⊗ U(l)-modules A(G)[µ − λ] ⊗ A(G)[ν − µ] → A(G)[ν − λ].
(102)
Since A(G)[β] = 0 unless β is a positive integer weight, this morphism is defined for generic λ ∈ c∗ , i.e. for λ ∈ Y, where Y is from Proposition 6.4. Indeed, if λ ∈ Y, then also µ, ν ∈ Y when the differences µ − λ and ν − µ are integer weights. Fixing a generic λ in (102) and varying µ and ν, we obtain from (102) a morphism A(G)l0 ⊗ A(G)l0 → A(G)l0 .
(103)
These morphisms form a family of multiplications parameterized by elements eλ ∈ U ∗ (c) (or λ ∈ c∗ ) for generic λ. Since the elements eλ form a basis of U ∗ (c) over C, this family extends by linearity to all generic elements of U ∗ (c). One can check that this family makes A(G)l0 a dynamical associative U(l)-algebra over the base U ∗ (c). Comparing the construction of this multiplication and the construction of twist from Theorem 6.7, we come to the following. Proposition 8.2. For generic λ ∈ c∗ , the dynamical associative multiplication (103) ¯ has the form m ¯ λ = m ◦ F(λ), where F¯ is the dynamical twist over the base U ∗ (l) from Theorem 6.7.
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8.2. Deformation quantization of the Kirillov brackets and vector bundles on coadjoint orbits. Dynamical twist from Theorem 6.7 applied to A(G) gives a dynamical algebra, which, by Theorem 7.2, defines quantization of vector bundles on M = G/H as left modules over the quantized algebra of functions on M. This quantization of vector bundles is obviously g-equivariant. Restricted to the function algebra on M, it gives quantization of the Kirillov brackets in the following way. Let t be an independent variable. Denote by gt the Lie algebra over C[[t]] with bracket [x, y]t := t[x, y] for x, y ∈ g, where [., .] is the original bracket in g. Then there is an algebra morphism ϕt : U(gt ) → U(g)[[t]] induced by the correspondence x → tx for x ∈ g. As was shown in [DGS2], the equivariant quantization of the Kirillov Poisson bracket corresponding to the semisimple orbit passing through λ ∈ c∗ ⊂ g∗ is identified with the image of U(gt ) by the composition map U(gt ) → U(g)[[t]] → Hom0 (Mλ/t , Mλ/t ), where the first arrow is ϕt and the second one is the representaion map. Using this fact, one can show that the multiplication m ¯ λ,t := m ¯ λ/t from Proposition 8.2 being restricted to A(M) = A(G)[0] gives a U(g)-equivariant deformation quantization of the Kirillov Poisson bracket on M realized as a coadjoint orbit passing through λ. Since the multiplication m ¯ λ,t depends, in fact, on λ/t, we do not need λ to be generic in the deformation quantization. Note that also for any formal path λ(t) = λ0 + tλ1 + . . . ∈ c∗ [[t]] the multiplication m ¯ λ(t),t gives a U(g)-equivariant deformation quantization on the orbit passing through λ0 , with the appropriate Kirillov bracket. Remark 8.3. Any equivariant deformation quantization of the Kirillov bracket on the orbit can be obtained in this way, and different paths in c∗ give non-equivalent quantizations, [D1]. ¯ For λ ∈ c∗ , consider m ¯ λ = m ◦ F(λ), where F¯ is the dynamical twist from Theorem 6.7 and m the classical multiplication, as a map A(G)⊗2 → A(G). Recall that c∗ can be naturally identified with a subspace in g∗ . Let us call an element λ0 ∈ c∗ regular if its stabilizer in g (under the coadjoint action) is the Levi subalgebra l. In a similar way, one can prove Proposition 8.4. For any regular λ0 ∈ c∗ and for any formal path λ(t) = λ0 + tλ1 + . . . ∈ c∗ [[t]], the multiplication m ¯ λ(t),t := m ¯ λ(t)/t gives a U(g) ⊗ U(l)-equivariant map A(G)⊗2 → A(G)[[t]] that coincides modulo t with the original multiplication in A(G). Now, we fix regular λ0 ∈ c∗ and consider M = G/H as the semisimple orbit passing through λ0 . Recall that the space of global sections of the vector bundle V (M) l corresponding to an H -representation V is identified with the space A(G) ⊗ V = ∗ Homl V , A(G) . We identify V (M) with its global sections. Applying Theorem 7.2 to the dynamical algebra obtained from A(G) by the dynamical twist from Theorem 6.7 and using Proposition 8.4, we obtain Theorem 8.5. Let λ0 be a regular element from c∗ and λ(t) = λ0 + tλ1 + . . . a formal path in c∗ . Then the dynamical multiplication m ¯ λ(t)/t defines a U(g)-equivariant λ multiplication ∗ on the global sections of equivariant vector bundles on M. This multiplication is a deformation of the usual tensor product of the sections and satisfies the following properties: λ(t) 1) Restricted to A(M), the operation ∗ defines a deformation quantization Aλ(t) (M) of the function algebra A(M) corresponding to the Kirillov bracket on the orbit passing through λ0 ;
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2) Let sV and sW be global section of vector bundles V (M) and W (M), and a is a function on M. Then λ λ λ λ a ∗ (sV ∗ sW ) = (a ∗ sV ) ∗ sW . In particular, any vector bundle V (M) is a left Aλ(t) (M)-module with respect to the λ(t) action map a ⊗ s → a ∗ s, where a ∈ Aλ(t) (M) and s ∈ V (M). 3) The line bundle Cα (M), where α ∈ c∗ is a positive integer weight, is also a right λ(t) module over the algebra Aλ(t)−tα (M) with respect to the action map s ⊗ b → s ∗ b, where s ∈ Cα (M), b ∈ Aλ(t)−tα (M). 8.3. The quantum group case. Let Uq (g) be the Drinfeld-Jimbo quantum group corresponding to g and Uq (l) be considered as its quantum subgroup corresponding to the Levi subalgebra l ⊂ g. Let Aq (G) denote the dual algebra to Uq (g) consisting of matrix elements of finite dimensional representations of Uq (g). The algebra Aq (G) is a quantization of the classical algebra A(G), it is equivariant under the left and the right regular actions of Uq (g), which we replace by two left actions, ρ1 and ρ2 , as above. Let F¯ q be the dynamical twist constructed in Theorem 6.7 with the help of generalized Verma modules over Uq (g). Applying this twist to the algebra Aq (G), we obtain a ¯ Uq∗ (l) . This algebra is equal to Aq (G) dynamical algebra A(G), m ¯ q,λ in the category M as a Uq (g)-module (with respect to ρ1 -action) and has the family of multiplications m ¯ q,λ parameterized by generic λ ∈ Uq∗ (l). They are defined by m ¯ q,λ := mq ◦ F¯ q,λ , where mq is the original multiplication in Aq (G). It is obvious that Aq,λ (G) is a Uq (g)-module algebra with respect to the ρ1 -action. One can show that replacing simultaneously λ by λ/t and q by q t we obtain the family of multiplications m ¯ q t ,λ,t := m ¯ q t ,λ/t that gives a Uq t (g)-equivariant deformation quantization of A(M). Also, there exists a q-analog of Theorem 8.5 which reduces to Theorem 8.5 when q = 1. Acknowledgement. We are grateful to J. Bernstein, V. Ostapenko, and S. Shnider for stimulating discussions within the “Quantum groups” seminar at the Department of Mathematics, Bar Ilan University. We appreciate useful remarks by M. Gorelik, V. Hinich, and A. Joseph during a talk at the Weizmann Institute. Our special thanks to P. Etingof for his comments on various aspects of the subject.
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Communicated by L. Takhtajan
