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Q ' L® L* = I, L3 ®Lk = Lj+k. Any line bundle over P„(C) is isomorphic to Lk for some uniquely defined integer k. This integer is related to the first Chern class of L , as will become clear later on. For now however, let us consider TC(P„(C)) and T* (Pn(C)) = A1'0 (Pn(C)), respectively, the complex tangent and cotangent spaces. We have: I®Tc(Pn{C)) /©T*(Pn(C))
= =
L*® • • • © £ * £©•••©/,.
Although this identity does not preserve the holomorphic structure, it is clearly an isomorphism between complex vector bundles.
70 6.2.5
6. Gauge Theories on Riemann Surfaces I: Bundles Principal Bundles
The presentation above makes it crystal clear that a vector bundle is a fiber bundle whose fiber M is a linear vector space, and whose transition functions belong to the general linear group of M. By contrast, a principal bundle P is a fiber bundle whose fiber is actually a Lie group G. The transition functions of P belong to G and act on G by left multiplication. A right action of G on P is the result of exploiting the commutation between right and left multiplication. Related to a principal bundle P are the frame bundle of a vector bundle E and the associated principal bundle. They can be constructed in the following outlined procedure. The fiber Gx of P at X is the set of all frames of the vector space Mx, which is the fiber of E over the point X. With this in hand, we now consider a complex vector space M = C* of dimension k. By definition, the fiber G of the frame bundle P is the collection of k x k non-singular matrices which form the group GL(k;C), where G is the structure group of the vector bundle E. The associated principal bundle has the same transition functions as the vector bundle E. These transition functions are GL (k; C) group elements whose action on the fiber G are essentially through left multiplication. Hence with P, a principal G bundle and p, a representation of G on a finite-dimensional vector space V, one can define the associated vector bundle P xp V by the equivalence relation on P x V: (p, p(g) • v) ~ (p • g,v) for all p G P, v G V,g G G.
6.2.6
Yang-Mills Instanton as Principal Bundle
This is, to date, the most interesting and celebrated example of a principal bundle. The base of this instanton bundle is the compactified Euclidean space-time, i.e. the four-sphere S4, and its fiber is the group SU(2), which we may recall is equivalent to S3. We endow the base S4 with coordinates (6,,ip,r). Similarly, for the fiber we have the coordinates (a,/?, 7). The next step is to split S4 into two pieces, D+ and Z)_ whose boundaries are S3. The intersection of D+ with / ) _ along the equator of S4 is parametrized by
6.2 Vector Bundles
71
the Euler angles (9,faip) of S3. The representation h{9,fa%l>) of SU(2) is
x + iy
t — i\- x , r = r cos§ e ^ 2 ' ^ * )
z + it
= rsinfeW 2 ><*-« "
h =
The fiber coordinates are given in a somewhat similar fashion by SU(2) matrices g (a, f3,7). From there, we have a good picture of what the local bundle patches ought to look like. Explicitly, they are D+ x SU(2), with coordinates (9, far; a+,/? + ,7+) D-. x SU(2), with coordinates (0,faijj,r; a_,/?_,7_). In the intersection area D+ (~l D-, transitions from the SU(2) fiber g (a+., /?+, 7 to g (a_,/3_,7_) can be built using multiplication by the SU(2) matrix h (9, farj)). The result is 5
(«_,/?_, 7 _) = hk(6,,i;)g(a+:f3+,1+).
(6.9)
When k = 1 we have the Hopf fibering of S7 [2, 5], namely: P = S7. This precisely fits the description of the bundle prescribed by the singleinstanton solution of Belavin et al. [8]. More general instanton solutions describe bundles with different values of k. 6.2.7
Dirac's Magnetic Monopole
Dirac's magnetic monopole is in essence a principal U(l) bundle over S2. To explicitly construct such a bundle, we start with a base M = S2 and fiber M' = U{\) = S1. The U{\) coordinate is labelled by e^, while those of the base S2 are (0,), 0 < 9 < n, 0 < < 2x. As with the YangMills instanton, we break S2 into two hemispherical neighborhoods, D- and D+; we also take D+ f) D- to denote the boundary of the equatorial area parametrized by the angle <j>. The local form of the bundle is therefore D+ x U{\)
with coordinates (6, fa e'^)
D_ X f/(l)
with coordinates (9, fa e' 0 -).
72
6. Gauge Theories on Riemann Surfaces I: Bundles
To obtain a principal bundle out of this construction, one has to restrict the transition functions in such a way that they define only U(l) functions of (f> along the intersection D+ (1 JCL . This means literally that one needs to relate the D+ and D- fiber coordinates. This is done by e <*- =
<,«•»*-c*+.
(6.10)
Note that n in equation (6.10) must be an integer for the resulting structure to be a manifold. The fibers ought to fit together perfectly when we complete a full revolution around the equator in >. This is the topological version of the Dirac monopole quantization condition. Let us take a look at various values of n. For n = 0, we have the trivial bundle P = S2 x
S\
For n = 1, P = S3, and we recover the Hopf fibering [2, 5]. This bundle describes a single charged Dirac monopole. Larger values of n correspond to more complicated monopole bundles.
6.3
References
[1] Yang, C. N. and Mills, R. L.: Conservation of Isotopic Gauge Invariance, Phys. Rev. 96 (1954) 191-195. [2] Steenrood, N. : The Topology of Fiber Bundles, Princeton University Press, 1951. [3] Lubkin, E.: Geometric Definition of Gauge Invariance, Ann. Phys. 23 (1963) 233-283. [4] Hermann, R.: Vector Bundles in Mathematical Physics, Vols. I and II, 1970, W. A. Benjamin, Inc. [5] Trautman, A.: Fiber Bundles Associated With Space-Time, on Math. Physics 1 (1970) 29-62.
Reports
[6] Wu, T. T. and Yang, C. N.: Concept of Nointegrable Phase Factors and Global Formulation of Gauge Fields, Phys. Rev. D12 (1975) 3845-3857.
6.3 References
73
[7] Dirac, P. A. M.: Quantized Singularities in the Electromagnetic Field, Proc. Roy. Soc. London A133 (1931) 60-72. [8] Belavin, A., Polyakov, A. M., Schwarz, A. S. and Tyupkin, Y. S.: Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett. B59 (1975) 85-87. [9] 't Hooft, G.: Symmetry Breaking Through the Bell-Jackiw Phys. Rev. Lett. 37 (1976) 8-11.
Anomalies,
[10] Jackiw, R. and Rebbi, C : Vacuum Periodicity in a Yang-Mills Quantum Theory, Phys. Rev. Lett. B37 (476) 172-175. [11] Callan, C. G., Dashen, R. and Gross, D. J.: The Structure of the Gauge Theory Vacuum, Phys. Lett. B63 (1976) 334-340.
Chapter 7 Gauge Theories on Riemann Surfaces II: Connections The physical potential of the theory of fiber bundles lies in connections on bundles. The connection 1-form, for instance, is well known to physicists as a gauge potential; similarly, the Yang-Mills field strength is defined as the curvature associated with the connection. Nowhere is this potential more pronounced than with Chern-Simons-Witten (CSW) theories: in the Hamiltonian approach of CSW theories, an appropiate description of the 3-manifold invariant (see Chapter 1, Section 4) requires the geometric quantization of the space of flat connections on a given Riemann surface [1, 2, 3, 4, 5, 6]. Chapter 8 is devoted to the quantization of CSW theories. For now, however, we aim to focus on the basics of connections on fiber bundles. Apart from its central role in gauge theories, the notion of connection plays an important role in the local differential geometry of fiber bundles. There, a connection defines a covariant derivative which contains a gauge field; connections are important in specifying the way in which a vector bundle E could be parallel-transported along a curve lying in the base manifold M, thus yielding information about holonomy and other related geometrical characteristics in the process. Below, we start our presentation with a description of connections on vector bundles. Connections on principal bundles are treated thereafter.
74
7.1 Connections on Vector Bundles
7.1
75
Connections on Vector Bundles
Physics spells out the need to differentiate sections of a vector bundle. For instance, a charged scalar field in Quantum Electrodynamics (QED) is regarded as a section of a complex line bundle associated with a U(l) bundle, P (M, U{\)) [7]. The differentiation of sections ought to be done covariantly if there is to be a consistent theory with a gauge-invariant action. Lack of such invariance spoils the consistency of gauge theories and gives rise to pathologies called anomalies (Chapters 9, 10, 11, 12 and 13 are devoted to the study of various forms of anomalies in gauge theories in this book). The Levi-Civita connection on a surface in R 3 illustrates this case perfectly. We consider the unit sphere S2 C R 3 as a specific working example. 5 2 is parametrized by the coordinates x{6,<j>) = (sinO cos (f>, sin cf> cos 9,0), with 0 < 0 < TT and 0 < <j> < 2x. The resulting Riemannian metric _ / dgx • dsx dex -d^x\ _ /" 1 — V dg x • dj, x dj,x • d$x ) ~ \ 0
0 \ sin2 6 )
implies ds2 = dO2 + sm26d<}>2. There are two vector fields, namely a,\ — dgx = (cos 6 cos <\>, cos 6 sin <j>, —sin 0) and o-2 = d$x = (—sinOs'mcf), sin 0 cos 0,0), which are tangent to the surface and span the tangent space. Derivatives can be decomposed into tangential components proportional to a,\ and a^, and a normal component h proportional to x. Identifying a,\ and a 2 with the basis d/d6 and d/dfor the tangent space follows mainly from the relations:
df(x) de df(x) dcf>
_ ~
dj_ ai
' dx d]_ dx
76
7. Gauge Theories on Riemann Surfaces II: Connections
where f(x) is a function on R 3 . We now focus on the differentiation to which we referred earlier. We want to differentiate tangential vector fields with respect to the surface. As a first step, we write down the ordinary partial derivatives de(ai)
=
£^(ai)
=
(—sinflcos >, — sin 0 sin ^>, — cos#)
dg(a2) = (—cos0 sin, cos 0 cos , 0) = ——- a 2 sinfl 8^,(02) = (—sin0,cos >, —sin 0 sin>, 0) = —sin2 Ox — cos#sin#a 2 . The next step centers on finding the appropriate intrinsic covariant differentiation V r , which is defined with respect to a tangent vector x. To produce it, we use a rather simple method: one takes the ordinary derivatives and projects them back to the surface. Under this procedure, V x is actually the directional derivative obtained by throwing away the normal component of the ordinary partial derivative. Explicitly: Vai(ai)
=
0
V 01 (a 2 ) V a 2 (a 2 )
= =
V a 2 (ai) = c o t 0 a 2 —cos 6 sin 9 aj.
V is called the Levi-Civita connection on S2. Identifying (0.1,0.2) with (d/00, d/d(f>) gives rise to the formula
and As for the Christoffel symbol, they are given by V 0i (a,) = akT if, or
v 9 ,(^) = rkl3dk, where d\ = d/dO, <92 = d/d(j>. According to previous formulas, T 2 12
=
r 2 2 i = cot 0
r'22
=
— cos 0 sin 0
Tkn
= 0 otherwise.
77
7.1 Connections on Vector Bundles
A few definitions are in order. A curve x(t) C S2 is a geodesic if the acceleration x has only c o m p o n e n t s normal to t h e surface, i.e. V ± ( i ) = 0.
(7.1)
T h e Levi-Civita connection provides a rule for t h e parallel t r a n s p o r t of vectors on a surface. As an illustration, consider x(t) a curve in S2, and let s(t) b e a vector field defined along t h e curve. S is parallel-transported along t h e curve if it satisfies t h e equation V i ( 5 ) = 0. Let x b e t h e geodesic triangle in S2 connecting t h e points ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) a n d ( 0 , 0 , 1 ) . x consists of t h r e e circles:
x(t)
f cos*, s i n f , 0 ) = < 0 , s i n t , —cost { ( - s i n ( f ) , 0, - c o s ( i ) )
t G [0,TT/2] t G [x/2,ir] t G [TT,3TT/2].
Write t h e initial tangent vector 3(0)
= (0,a,/?)
a t ( 1 , 0 , 0 ) . By parallel-transporting s(0) along x(t) connection, we obtain
using t h e Levi-Civita
—asinf, a cos 2, /? t G [0, IT] - a , /3cost /?sin< t G [TT/2, IT] . a cos t — /3, —a sin 2
78
7. Gauge Theories on Riemann Surfaces II: Connections
Below, we provide some properties of the connection V ^ (s) and the total covariant derivative V. The focus is firstly on the connection V x (s). Its linearity is expressed by V * (s + s') = Vx (s) + V * (s'). The linearity in X is Vx+x>(s)
= Vx(a) +
Vx-(s).
The connection often behaves like a first order differential operator, as shown by the formula
VjfW) = s- X(f)
+
(Vx(s))f.
The tensoriality in X follows from
Vfx{s)
=
fVx(s).
Note that f(x) denotes a scalar function, X stands for a vector field and s(x) is a section of E. The linearity of the total covariant derivative V is given by the expression V ( s + s') = V ( s ) + V ( s ' ) . This also behaves as a first order differential operator, as exhibited by the relation
V (sf) = s®df
+ V (s) / .
On the basis of these facts, we can derive a relationship between these two differential operators, namely: (l).V(a) = V a / a r „ ® dx»
(2).Vx(s) where X 6 C°° (T(M))
=,
and V ( s ) G C°° (E ®
T*{M)).
7.2 Curvatures
7.2
79
Curvatures
In the framework of connections, the curvature measures the extent to which parallel transport is path dependent. For instance, when the curvature is trivial, parallel transporting a given path in M results in the identity transformation. (There are some exceptions though, the most notable of which is a path with holes.) For curved manifolds, the story is different since one gets non-trivial results: a parallel-transport around a geodesic triangle on S2 gives a rotation equal to the area of the spherical triangle. In order for us to evaluate the curvature we need to use parallel transport. As an illustration, consider a local coordinate chart (xi,:^? • • •) along with a square path x(t) with vertices. Write the explicit form of the vertices as (0,0,0, • • •), (0, r 1 ' 2 , 0 , • • •), (r 1 / 2 , r 1 ' 2 , 0 , • • •), ( r 1 / 2 , 0 , 0 , • • •); the holonomy matrix r^ (r) is obtained by traversing the path with the vertices above. The curvature matrix in this plane is thus 9H = ^rt](T)\T=0.
(7.2)
Now, let's move our focus to tangent spaces. Here, the curvature is the commutator of the components Dll of the basis for the horizontal subspace oiT{E), that is: [Dll,Dm] the g'j^
= -gijvzi^-r,
(7.3)
can be expressed in terms of Christoffel symbols: i
Ci yi
_i_ p t
pfc
pt
p/c
In the cotangent space approach, the curvature takes on the appearance of a matrix-valued two-form, T'J = dT'j + T'k A r*> = ^T'j^dx11
A dx".
(7.4)
Note that r'j z1 — duj' + F'j A wJ is the covariant differential of the oneform u/ 6 T* (E); although u/ has dzk components, they cancel out in T'J.
80
7. Gauge Theories on Riemann Surfaces II: Connections
It follows from all of this that the curvature measure, in effect, is the extent to which covariant differentiation fails to commute. We now focus on the curvature operator, g (X, Y) (s) = Vx Vy (s) - V y Vx (s) - V[X,Y] (s),
(7.5)
where
Catalogued below are the properties of the curvature operator, with X and Y as vector fields, s(x) a section and f(x) a scalar function. •
Tensoriality g(fX,Y)(s)
•
= g(XJY)(s)
=
fg(X,Y)(s).
Antisymmetry g(X,Y)(s)
•
= g(X,Y)(fs)
=
-g(Y,X)(s).
Multilinearity g(X + X', Y) (s) = g (X, Y) (s) + g (X\ Y) (s).
The total curvature g is a matrix-valued 2-form given by g
= V 2 (s) = V (e3 = ek® Tkj A Pi + ek ® (dT*i zi + ek ® Ykj A dz' = efc ® gkiz\
(s)
The matrix 5 = \\g'j\\ yo
acting on sections s.
c a n De
Pi z' e3 ® dz') z' Tkj A dz') + 0
written as
= - yo 7;—, 7— dx11 A (ix^ 2 \dx^ dx")
(7.6)
7.3 Torsions, Connections, and Tangent Bundles
7.3
81
Torsions, Connections, and Tangent Bundles
The cotangent space formulation V (s) = c-dzi (x) + ti ® F,- z j {x)
(7.7)
of the vector bundle connection V has the advantage that it is independent of the coordinate system {x^} on M. Furthermore, multiple covariant differentiation of an invariant one-form such as p^ dx*1 is also independent of the connection chosen on the cotangent bundle T* (M). In order to differentiate the tensor components z'JM of the covariant derivative of a vector bundle section s (x) = e, z' (x), one needs to specify a connection on T* (M). Torsion is, on the other hand, a property of the connection on the tangent bundle, a crucial ingredient for any study of double covariant derivatives. Consider { P w - } , the Christoffel symbol on the vector bundle E. Let il^nx} denote the Christoffel symbol on the tangent T (M). We then have **';,.;, = dv (0„ z{ + r*'w- z>) + Y\j [dltzi + I V * * ) -l\u(dxz' + Vxjz') for the double covariant derivative of a given section s (x) = e,- z% (x). The sign in front of 7AM„, it should be noted, comes from the requirement of lowering indices in order to obtain the connection on T* (M). The use of the commutator of double covariant differentiation on a section gives rise to % ;f;^i
W}v
j ' i p ^ - r ^
—
(7.8)
We point out the presence of a new tensor in equation (7.8), i.e. the torsion TA J-
— ^A III/
—
/
- -vA pv
I
I'll-
By definition, we regard the torsion operator on T (M) as T(X,Y)
= VXY
- VYX
-
[X,Y],
that is, a vector field with components d T
d \
/ .
\dx»' dx") ~ V""
.
N
0
^""J dxx
82
7. Gauge Theories on Riemann Surfaces II: Connections
Once a metric of the type (X, Y) — g^ xmu y" has been chosen, the LeviCivita connection on the tangent bundle T (M) is uniquely defined because T (X, Y) d(X, Y)
7.4
= 0, torsion-free property = (VX, Y) + (X, V F ) , covariant consistancy of metric.
Connections on Various Bundles
Let E and E', respectively, denote two dual vector bundles with dual frame bases {e;}, and {e"}. The connection V on E' is defined by requiring that the natural inner product between sections s and s' be differentiated according to the following rule: d<s,s'>=
(V(s),s')
+ (a,V'(a')>-
This implies that one can explicitly write V (e t ) and V (e") as follows: V(e ; ) V'(e'J')
= ejP^rfi" = e'Frjdx".
Endowing E with a fiber metric allows one to identify E with E' using a conjugate linear isomorphism. The connection V is said to be Riemannian if V = V , that is,
r„- = -r%„
(7.9)
relative to an orthonormal frame basis. The curvature of a Riemannian connection relative to an orthonormal basis is antisymmetric: g'i = -
(7-10)
The Levi-Civita connection on T (M) is the unique torsion-free Riemannian connection. The question of describing the torsion in terms of the Whitney sum bundle then arises. In order to work it out, let us recall what a Whitney sum bundle E © F is; it is obtained by taking the Whitney sum of the fibers of E and F at each point x £ M. With these descriptions in hand, we proceed to endow E and F, two vector bundles, with connections V and V respectively. The resulting natural connection V © V defined on E © F satisfies (V © V ) (e, © fj) = ek ® 7 \ . dX» © /, ® r" w - d i " .
(7.11)
83
7.4 Connections on Various Bundles
The resulting curvature is just the direct sum of the curvature of E and F, as implied by equation (7.11). For tensor product bundles, the first thing to do is to look at the natural connection V" which takes its value on E © F. This connection is actually given by the formula V " ( s ® s')
= (V1 + 1 ® V ) (s s') = V ( s ) s' + s V ' ( s ' ) .
The curvature of V" reads V " = g ® 1 + 1 ® g'
(7.12)
What about torsion on pullback bundles? Some definitions are in order. We begin with a map / : M —> F and a connection V on the vector bundle E' over F. The resulting natural pullback connection is thus V = / * V with Christoffel symbols which are none other than the of V . More precisely dx'a dx» "
p M • = r"ai ——
The curvature of V is the pullback of the curvature of V : _ 1.
13
fdx^_ dx^
9i»* - 29 3°0 ydxl>
dxV
__ dx^_
dxtf3\
dxV
dxtxj-
Because of their central role in quantization of Chern-Simons-Witten theories, we pause to say a few words about the curvature of projected connections. First, we write the projection as p : F —-> E with E a sub-bundle of M. As before, one thinks of V as a connection on F. The projected connection V on E is V(s)p(V(s)), where as we have seen before, s is a section of F belonging to the sub-bundle E. A peculiar fact is that the curvature V p may be non-trivial, even though the curvature of V may be zero. The Levi-Civita connection on S2 •—• K 3 provides an illustration of this.
84
7.5
7. Gauge Theories on Riemann Surfaces II: Connections
Connections On Principal Bundles
A principal P-bundle is a fiber bundle whose fiber and transition functions both belong to the same matrix group. The gauge potentials of Maxwell's theory of Electromagnetism and Yang-Mills gauge theories are identifiable with connections on principal bundles. The purpose of this present section is to provide a detailled presentation of connections on principal bundles. We shall begin with parallel-transport. Choose a local trivialization with coordinates (x,g) for the principal bundle P, where g € G, G is a matrix group, Q its Lie algebra and g~l dg the Maurer-Cartan form, i.e. a matrix of one-forms belonging to Q. For completeness we note that the Maurer-Cartan equations are dO = }-[8A0]d0i
= | A<*'0fc A 0,; d0' = | [ 0 ' A 0 ' ] .
By a local section of P, we mean a smooth map from a neighborhood U to G. As we have explained earlier, assigning a connection on P provides a rule for the parallel-transport of sections. In view of this, a connection A on P is a Lie algebra valued matrix of 1-forms in T* (M): A(x)
= A\(x)^dx».
(7.13)
Let x (t) denote a curve in M; the section g<j (t) can be defined to be parallel-transported along x if the following differential equation is satisfied: gik + A^^x)
-x'gjk
= 0,
(7.14)
with A^ the connection on the principal bundle P. Equation (7. 14) can be rewritten in a more explicit form, namely:
This concludes the discussion for the parallel-transport case. The focus is now on the tangent space approach. Recall that parallel-transport along a curve parallel-transport along a curve x (t) allows one to compare the fibers of a principal bundle P at various
85
7.5 Connections On Principal Bundles
points of x (t). Drawing on the techniques used for vector bundle connections, we lift curves x (t) in M to curves in P. This procedure requires us to differentiate along a lifted curve. This is actually done by d/dt
M _ | _
= i
~
X
+
^ . ^
A 11 {£))
\dx»
La;
here, we have implicitly used the parallel-transport equation for g^. yields the covariant derivative D, = - ^
- A\{x)La.
This
(7.15)
The curvature is [D„DV\
= -F\„La
(7.16)
where F\u
= dliA\
- dvA\
+ jabcA\Acv.
(7.17)
As for the cotangent bundle approach, we define the connection on P to be a (/-valued one-form u> in T*(P), whose vertical component is the Maurer-Cartan form g~x dg. Locally, we write "•> = 9'1 Ag + g'1 dg, with A{x) =
A\{x){^j)dxr
Notice that A remains invariant under the right action of the group g —+ go while uj transforms tensorially, i.e. w ->
g^uigo.
The resulting curvature is a Lie algebra valued matrix 2-form: il = du> + LJ A w = g~l Fg; with F = dA + AAA
= - F^ 2
^ dx» A dx". 2?
(7.18)
86
7. Gauge Theories on Riemann Surfaces II: Connections
Equation (7.18) satisfies the Bianchi identity dft+u>Afi-fiAw
= 0.
(7.19)
According to previous sections, the transition functions of a principal bundle act on fibers by left multiplication. Suppose we have two overlapping neighborhoods U and U' and a transition function $uu' = $• It then follows that the local fiber coordinates g and g' in U and U' are related by g' =
+ $d*-1.
(7.20)
From there, we show without much difficulty that u = g~l Ag + g'1 dg = g'~l A'g' + g~l dg', which in essence is the statement that w is well defined in T* (P). The transformation (7.19) is referred to as a gauge transformation of A. By contrast, the gauge transformation for M', a submanifold of M is M" =
QM'®-1.
We point out that the curvature 2-form Cl is also well-defined over the manifold i.e. n - g-1 M'g = g'~l M"g'. What are the gauge transformations for pullback bundles? To answer this, we begin by choosing a section g = g (x) by which one can pull back ui and Q to the base space. The procedure in question tells us that A and M' are equivalent to the pullbacks g*ui and g*Cl. The gauge transformations of A and M' are then simply the changes of the sections. Drawing on analogies with gauge field theories, we referred to the gauge group as the structure group G; the choice of G = U(l) [1], for instance, gives the theory of Electromagnetism; G — SU(3) [2], gives the theory of strong interactions or Quantum Chromodynamics; G = E& ® E% [3], the theory of heterotic superstring. In these particular instances, the (pullback) curvature M' gives the strength of the gauge field. Associated vector bundles act to describe matter field contents in the gauge theory.
87
7.5 Connections On Principal Bundles
By taking a particular connection on the U(l) principal bundle over the base space 5 2 and by restricting it to satisfy Maxwell's equations, the resulting physical system corresponds to Dirac's magnetic monopole. A U(l) connection 1-form reads A+ — dtj>+ on D+ A_ + dip- on £>_, where D± are the two hemispheres of S2 . The transition function e'
_ e>ne'0+
implies the gauge transformation A+ = A- + ncf>. Gauge potentials which satisfy Maxwell's equations in R 3 — {0} are of the form A " u O\JA n xdy-ydx A± = -(±cos0)d=. z ± r The curvature is therefore F
= dA± = ^sinOde A d= ^ j (x dy A dz + ydz A dx + zdx A dy).
As for the monopole charge, it corresponds to a negative value of the first Chern number C\ characterizing the bundle:
-Cl = - f c, = ^-F = ~\[ Js*
2TT
2TT7D +
F+ + f
7D_
F_) = n. J
Let us repeat the procedure for the SU(2) Yang-Mills instanton. The fiber is G = SU(2) = S3 with base S4. The metric reads 2 dS
_
dx.dx.
~ (l + r'/a')'
_ dr> + r 2 ( ^ + g\ + a^) _ _
(1+rVa^)
2
3
~ ko
' '
it is obtained by projection from the north or south pole onto R 4 . Using the fact that the overlapping region D+ D D - is equivalent to 5 3 we relate the SU(2) fibers by the transition function 9- = [h(x)]k • g+,
88
7. Gauge Theories on Riemann Surfaces II: Connections ' T ', and A are the SU(2) Pauli matrices.
where k is an integer, h = " The connection 1-form is
+ ff : d9+
u, = I 9+l A9+
° n D+
+
\ gl'A'g-
+ gl'dg-
on D..
Notice that A' can be written as A'(x)
= hk(x)A(x)h-k{xy)
hk(x)dh-k(x).
+
The case k = 1 gives the single instanton solution D+ : A
=
^ - T r2
• h-1 dh -\
The gauge transformed solution is hence £L
: A' h [ ^ y h-1 dh] h'1 + ~~ ~~
hdh'1
l+r^/a2 1+7-2/a-
The field strength in Z)± follows from D_ : F+ = dA + A A A = i\k £ (e° A e* + 1 efc0- e1 A e') ; £)+ : F_
= cM' + / 1 ' A / 1 ' = hF+h'1.
Indeed F is self-dual, that is, *F = F ; consequently, the Bianchi identity implies that the Yang-Mills equation DA * F = d* F + A A *F - +F A A = 0, is indeed well defined. The instanton number k can be written as k
= -C2 = -& = +1.
= -fsi
C2
= -&
fst
TrFAF
[ID, TrF+ A F+ + fD_ Tr F_ A F.]
7.6 References
7.6
89
References
[1] Cowan, E.: Basic Electromagnetism, Academic Press, New York 1968. [2] Yndurain, F. Y.: Quantum Chromodynamics: An Introduction t o the Theory of Quarks and Gluons, Springer-Verlag, 1983 Berlin and New York. [3] Green, M. B., Schwarz, J. H. and Witten, E.: Superstring Theory, Cambridge University Press, 1987 Cambridge and New York.
Chapter 8 Geometric Quantization of Chern-Simons-Witten Theories A classical physical system can be described by the Poisson algebra of functions on the phase space A. The standard Poisson bracket, {f 9}
>
=
^d-q-Tqd-p
(8J)
is associated with Hamiltonian mechanics, in particular with dynamical system of the type:
{f,g} = (xij,
dlxdg_
(8.2)
Quantization is the procedure by which one associates to each theory a Hilbert space Ti of quantum states, and a map m from a subset of the Poisson algebra to the space of symmetric operators on Ti. 90
91 Geometric quantization is essentially a globalization of canonical quantization in which the additional structure needed for quantization is explicitly expressed in geometric terms. According to a classical theorem by Darboux [1], the space of differential A:-forms on the space of smooth real functions satisfies the Jacobi identity: {{/>$},*} + {{9,k}J}
+ {{k,f},g}
= 0
(8.3)
This usually allows one to define a Poisson manifold, together with a Poisson algebra T. In light of this, the Poisson bracket in (8.1) is in reality a bivector field $ that belongs to the space of bivector fields on A. Put differently, {/,
*(df,dg).
Note that \P stands for the mapping of bundles •9: T*{A) -*
T(A),
which is linear on the fibers. We can therefore give a reformulation of the initial Poisson bracket: if, 9} = (df,Vdg); the main novelty here is the angles, which denote the pairing of a form and a vector field. In terms of local coordinates x\, • • •, x-in on A, we have {/,
(8.4)
where dj = -^-. The Jacobi identity, written in terms of the tensor components of tyjk, is of the type £
*>*&tf, m = 0.
(8.5)
UJ.m)
Note that the summation is done only over cyclic permutations. The rank r (*&) = rank 9(x) is in general smaller than the dimension of A. If r ( $ ) is a constant, then we say of the Poisson bracket on A that it is of constant rank. To say that the bracket (8.4) is nondegenerate mean that r ( ^ ) = dim.4. This corresponds to the case for which the dimension of the manifold A is
92
8. Geometric Quantization of Chern-Simons-Witten Theories
necessarily even. A is then referred to as a symplectic manifold [1, 2, 3, 4], and its corresponding closed, nondegenerate 2-form u> u = ]- q (x)-1 J2
dx
> A dxh
(8.6)
is called a symplectic form. The use of symplectic manifolds as phase spaces in physics is well-documented [1, 2, 3], and additionaly [12]. In classical mechanics, the trajectories, .. _
dv OXi
are determined by the values of x and x at t = 0. The Lagrangian C = i (-x2 or
r /
£ = J (px ~
- v(x)\ 1 {T;P2
+ u-
dt "\
dt
encompasses most solutions to the problem of finding the adequate trajectories. The momentum p, an independent variable in these Lagrangians, is equivalent to x for classical orbits. The classical phase space can be thought of as the space of classical solutions of the equations of motions. A classical solution would thus be determined by the initial values x and x. The symplectic manifold A = K 2 " is the phase space with symplectic form u — dp A dx,
(8.7)
which, in view of our above discussion, is equivalent to (8.6). Although u is closed, it is not necessarily exact. For compact manifolds of dimensions 2n > 0 for instance, u would not be exact or the volume form u) A • • • A u> cannot exact and the volume would be zero (which of course is absurd). As a consequence, compact symplectic manifolds must admit non-contractible 2-cycles. A celebrated instance of the lack of this particular property is that of S4: that is why S4 has no symplectic structure whatsoever [1, 2, 3, 4].
8.1 Holomorphic and Symplectic Structures on Bundles
8.1
93
Holomorphic and Symplectic Structures on Bundles
This section is entirely devoted to the study of various aspects of gauge theory on Riemann surfaces which are relevant to the geometric quantization of Chern-Simons-Witten or topological quantum field theories. We begin with a C°° vector bundle C over S 5 , a Riemann surface of genus g. A connection is a differential operator dA: 0 ° ( E 5 ; £ ) -» ft1 ( E , ; £ ) ,
(8.8)
such that dA {fs) = dfs +
fdAs,
where / is a C°° function, s € 0 ° (£<,;£) a section, and fip ( E 3 ; £ ) denotes p-forms on S 3 with values in C. The local form of the connection is dA = d -\- A = d + bdz + cdz, where A is a matrix of 1-forms and b, c are matrix-valued functions. The fundamental invariant of a connection is its curvature, V^. If we extend dA to p-forms, then VA
= d2A: fi° ( £ , ; £ ) -> n2 ( E 5 ; £ ) .
(8.9)
In local terms, (8.9) reads VA(s)
= (d + A)(d + A)(s) = cPs + dAs - Ads + Ads + A2s = (dA + A2)s.
(8.10)
In other words, the curvature is linear over C°°-functions and can be considered as an element of fi2 (E 9 , End C) or, a matrix valued 2-form with respect to (8.10). A gauge transformation is an automorphism of the bundle C. At the local level, a gauge transformation g is a C°°-function with values in GL (rz,C). Gauge transformations act on connections by conjugating the differential operator g~l dAg.
94
8. Geometric Quantization of Chern-Simons-Witten Theories
Again, there is a local interpretation of this formula, namely: g-1 dAg = g~l (d + A)g = dg~x (dg) + g~l Ag.
(8.11)
For the curvature to vanish, a necessary and sufficient condition is for the connection to be flat. The vanishing of the curvature is the integrability condition for the local existence of n-linearly independent solutions of the equation dA (s) = 0. Let (si, • • •, sn) and (s\, • • • , sn) be such two bases of solutions with a
Si = ^
ijdASj.
3
The relation follows that dA Si = dA (53 aijSj) - Yl dciij ® Sj + E aij dA Si = 53 ddij ® Sj
= o, where the a^ are constant matrices. Thus, a flat connection gives rise to a family of constant transition functions for £, which in turn defines a representation of the fundamental group ir\ ( E S ) into GL (n,C) [5]. 8.1.1
Holomorphic Structures on C
A holomorphic structure on £ is a differential operator 4 ' : fl° ( £ , ; £ ) -
O0'1 ( E 8 ; £ )
satisfying the relation
dl{f,s) =d"f ®s + fd'ls, with d"f = -g= dz. In local terms d'l = d" + bdz,
(8.12)
8.1 Holomorphic and Symplectic Structures on Bundles
95
where b is a matrix-valued function. Two holomorphic structures are said to be equivalent if there is a gauge transformation g such that
is a holomorphic isomorphism. A connection on C defines a holomorphic structure providing that we put d"As = (dA s) ' . Consider now a local trivialization of the bundle C endowed with Hermitian structure. We have the form d" = d" + bdz, and a connection dA = d + bdz — b*dz. This connection is compatible with the Hermitian structure since two sections s, t of C satisfy d < s,t>= (dAs,t) + (s, dAt). (8.13) We refer to (8.13) as a unitary connection. Conjugating by a unitary gauge transformation (i.e. one which preserves the Hermitian inner product), takes a unitary connection to a unitary connection. Hence, we draw the conclusion that the space of holomorphic structures on £ and the space of unitary connections are similar objects. Let E 3 x C denote the trivial line bundle. C°° complex line bundles are determined topologically by their first Chern class or degree. To classify holomorphic line bundles, a standard approach is thus to look at bundles of degree zero. On the trivial line bundle E 9 x C, every holomorphic structure is equivalent to the holomorphic structure of a flat unitary connection. These connections, it should be noted, are unique modulo gauge transformations. This implies that one can parametrize holomorphic line bundles of degree zero by unitary equivalence classes of flat connections. As noted earlier, these equivalence classes are in fact classes of constant transition functions. In the abelian case, they correspond to the cohomology group
96
8. Geometric Quantization of Chern-Simons-Witten Theories
a 2<7-dimensional real torus, the Jacobian. This object will play a central role in the geometric quantization of topological quantum field theories below. To endow this torus with a complex structure, the use of Quillen's determinant bundle [6] is required.
8.1.2
Stability of Holomorphic Bundles
A rank two holomorphic bundle C of degree zero is said to be stable whenever every sub-bundle C C C has degree less than zero. This is a natural condition which is unfortunately obscured by its cumbersome derivation using geometric invariant theory. As a general rule, a holomorphic structure on £ originating from a flat, unitary and irreducible connection implies that C is stable. Thus, flat connections give rise to stable holomorphic structures. The theorem which gives to the flat connections their central role is known as the Narasimhan-Seshadri theorem [8]. Roughly, it states that every holomorphic structure on a given vector bundle over S 9 of degree zero which is stable is indeed equivalent to the holomorphic structure of a flat, irreducible unitary connection. The connection is unique modulo unitary gauge transformations. One immediate consequence of the Narasimhan-Seshadri theorem is that the space of equivalence classes of (stable rank n) bundles of degree zero is in one to one correspondence with the equivalence classes of irreducible representations 7TJ (Tig) —> U(n) modulo the action of conjugation by U(n). The resulting space is a Hausdorff space owing to the fact that U(n) is compact. This space plays a crucial role in some forms of geometric quantization of Chern-Simons-Witten theories, as noted by Witten et al. [7] and Atiyah [9].
8.1.3
Symplectic G e o m e t r y a n d G a u g e T r a n s f o r m a t i o n s
A symplectic manifold R 2 n = A is a manifold endowed with a nondegenerate closed 2-form w. Two basic examples of symplectic manifolds are: 1. The cotangent bundle T* (M) of a manifold M [1, 2, 3]; 2. A Kahler manifold M [2, 4]. Symplectic geometry originated from Hamiltonian mechanics. Nowadays it is a powerful tool in shedding light on the relation between flat connections
8.1 Holomorphic and Symplectic Structures on Bundles
97
and stability. A symplectic manifold A admits the action of a Lie group G, which preserves the symplectic form. This can be illustrated as follows. Let v denote a vector field generated by such an action. It leaves u> fixed, meaning that the Lie derivative of u> is zero: Cv (OJ) = d (i (v)u) + i [v) dbj = 0. Since u> is closed, we thus have the relation d {i(v)oj)
= 0,
implying that for H1 (A; K) = 0, there exists a function fiv such that dfiv = i (u)w. There is such a function for each vector field generator of the Lie group G. More precisely, for each element of the Lie algebra Q, the (dual) linear map follows Q - C°° (A), or equivalently fi: A-+
G*.
(8.14)
The map fj. is referred to as a moment map [7, 9] whenever \i commutes with the natural action of G on A and Q*. When G preserves the symplectic form and the metric (i.e. the complex structure in brief) in A, it then follows that Q ® Q yields a complex Lie algebra of holomorphic vector fields. This comes mostly from the action of a complexication GQ of G on the symplectic phase space A [1, 2, 3, 4, 7, 9]. This situation is related to the stability of points in A, providing that they suitably transform under Gc to points on the zero set of the moment map (8.13). Below, we analyze this stability for two finite-dimensional examples. Stable points are essential in the framework of geometric quantization if one is to avoid the occurence of global anomalies. I shall defer to Chapter 9 an explanation of how unstable or degenerate points in symplectic manifolds give rise to global anomalies. Back to the two examples. In the first one, we take the manifold M to be the space of n x re complex matrices with Kahler metric Tr (^4^4*).
98
8. Geometric Quantization of Chern-Simons-Witten Theories
It is required here that G = U(n) act by conjugation. The moment map corresponding to this case is (t(A) = ±i[A,
A*},
(8.15)
where we have identified Q and Q* by the invariant form Tr (AA*). The stable points are those which are conjugate by GL (n,C) to a matrix in the zero set of fi, namely, a normal matrix. Every normal matrix can, in principle, be diagonalized by a unitary transformation, and consequently the stable matrices are the diagonalizable ones. In the second example, we consider M — C P 1 X S2 and set G = S0(3). The moment map is simply the inclusion of the unit sphere in R 3 . For M = C P 1 X C P 1 x C P 1 x C P 1 , the set of ordered quadruple points is C P 1 . The zero set of the moment map consists of those quadruples whose center of mass is at the origin. Thus, the stable points are those which transform under PSL (2,C) = S 0 ( 3 ) c . Our focus is now on the infinite dimensional case. The Kahler manifold M is now taken to be the space of holomorphic structures on the bundle C. This space is essentially an affine space, the difference of two structures being an element 6 £ ft.0'1 (M; E n d £ ) . Tangent vectors are given by 6 € fi0'1 (M; E n d £ ) . We also specify the Kahler metric
i J tr(&*6). JM
As for the group G, it now corresponds to the group of unitary gauge transformations G
w (
(8.16)
with a, b G H 0 ' 1 ( M ; E n d £ ) . The Lie algebra of G consists of the skewHermitian section, $ G fi° ( M ; E n d £ ) for which the action e-'Ve'*,
8.1 Holomorphic and Symplectic Structures on Bundles
99
yields a vector field corresponding to $ : d = day (= s $ + [a,*] locally). Thus, using Stokes's theorem, we find d/i»(d)=
/ JM
tr ( 4 * A d) = /
tr(*
(8.17)
JM
Hence, daa = da + [a, a] locally. However, the curvature V a — da + a2, so V a = da + aa + aa = daa, and consequently, the moment map fi is given by fi(a) = V a G n2(M;End£).
(8.18)
The stability of points allows us to see rather beautifully that the stable holomorphic structures are those which are equivalent to the zero-set of n, i.e. the flat unitary connections. Any consistent geometric quantization of topological quantum field theories has to take this information into account. A powerful observation from the present discusssion is that the Narasimhan-Seshadri theorem established a correspondence between the infinite-dimensional stability of holomorphic bundles with that of the finite dimensional one. This observation plays a central role in the construction of the Donaldson invariants for 4-manifolds. But it is beyond the scope of this book, let alone this chapter, to discuss further this particular subject. 8.1.4
Generalized Gauge Transformations on Symplectic and Holomorphic Structures
The action of the group of gauge transformations on the space of connections leads us, via the moment map, to zero curvature equations, and places them in a natural and general context. We shall use this method to describe more generalized set of equations, and to see, along the way, their relevance in the description of holomorphic structures on M. We start by considering the cotangent bundle T* M of the space of holomorphic structures on the trivial bundle £ = M x C 2 . The tangent vectors to M are
100
8. Geometric Quantization of Chern-Simons-Witten Theories
given by a £ fi0'1 (Af;End£), and so the cotangent vectors are elements of $ € fi0'1 (Af; End C) under the pairing: /
tr ( $ A d ) .
JM
The manifold T* M is symplectic as an infinite-dimensional complex manifold: the full group Qc of complex automorphisms of C acts on it, preserving the symplectic form, and this in turn, yields a moment map. Next, we consider the action of the Lie algebra fi° (M; End C) of Qc on T* M. Since the action of an automorphism g can be written as
O r 1 < s ,) = « ¥ , [ * , * ] ) . (8.19) The following natural symplectic form is given by (8.19) w ((&!,$,), (6 2 ,$ 2 )) - /
tr ( $ , 6 2 - fcafc) .
(8.20)
Consequently, dM
(6,6)
= / M t r ( [ * , * ] & - *d?tf)
= ; M t r ( * ( 4 ' $ + [i,$])), and this gives the moment map /z(6,
(8.21)
The zero set of this complex moment map is the set of holomorphic structures d" and sections $ of E n d £ ® K, which are holomorphic with respect to d". This zero set can be endowed with a complex submanifold of T* M. Let its induced Kahler metric be ||(6, $ ) | | 2 = i [
tr (6*6 + $ $ * )
JM
and consider the moment map for the group Q of unitary automorphisms.
8.2 Geometric Quantization of Chern-Simons-Witten Theories
101
The action on holomorphic structures (or similarly, unitary connections) is the curvature V^. The conjugation action of unitary automorphisms on $ £ fi1,0 (M; End C) corresponds to the finite-dimensional example of U{n) acting on the complex matrices as above. The result is the moment map V n + [*,$*].
(8.22)
In combination with the complex moment map, this gives the natural set of equations d" § = 0
v a + [*,**] = 0,
(8,23)
for a unitary connectiona and section $ (E fi1,0 ( M ; E n d £ ) . Note that equations (8.23) are the self-dual Yang-Mills equations for connection over
8.2
Geometric Quantization of Chern-Simons-Witten Theories
The original discovery by Edward Witten of 3-manifold invariants in 1988 [10] required, in its Hamiltonian version, the geometric quantization of the space of flat connections on a compact surface E 5 . According to Atiyah and Bott [5], the manifold M which corresponds to the space of gauge equivalence classes of fiat G-connections can be regarded as a symplectic quotient of the space of all connections. The canonical symplectic form admits various values of the level k (which, incidentally, correspond to different symplectic structures.) These symplectic manifolds are canonically associated to E 3 . To quantize them requires the choice of a Kahler polarization, and one then needs to prove -for consistency purposes- that the resulting space is independent of that choice. Let us pause a bit to explain what we mean by Kahler polarization. We pick a symplectic manifold, (M, u>) and consider its integral class ^ [w] £ H2 (M; R). Next, we choose a line bundle C with unitary connection whose curvature is to. To choose a complex structure on M such that u is Kahler means that one must choose the (0,1) part of the covariant derivative V of the connection in question in such a way that it gives a holomorphic structure on C. The corresponding projective space is the quantization relative to the polarization. To complete the quantization,
102
8. Geometric Quantization of Chern-Simons-Witten Theories
one shows that the projective space is, in a suitable sense, independent of the choice of polarization. It is worth pointing out the following: as M acquires a complex structure, the Narasimhan-Seshadri theorem transforms it into the moduli space of stable holomorphic vector bundles over S s . We begin with the geometric quantization of 8.2.1
Canonical Spaces
Let A = R 2n denote an affine symplectic space with a symplectic, nondegenerate 2-form, LJ
— uj{j d a,- A
daj,
and coordinatesis viewed as a transformation T —> T*, where T is the tangent space to A and T* its dual. We write u>ij = w _ 1 : T* —> T for the inverse of u>. The constraints are
uj~l dxk
= ukj g|-.
The symplectic form u can be reformulated as n
u = Y, «• A ar
( 8 - 24 )
i=l
To A, we associate a Hilbert space H(-4) or Tix, it is the Hilbert space of all square-integrable functions of ai, - • •, a,in. The coordinates a.{ act on Ti^. The Heisenberg commutation relations [a,i,aj} = -iuij,
(8.25)
yield the Heisenberg algebra. The primary objective of quantizing A is to produce a Hilbert space. In reality though, the quantization procedure for A gives an irreducible unitary Hilbert space of representation of the algebra of (8.25). Roughly speaking, we will construct a representation of the Heisenberg group. This group, it turns out, is an extension by U{\) of the group of affine translations of R 2 n . According to the Stone-von Neumann uniqueness theorem, there is a unique such
8.2 Geometric Quantization of Chern-Simons-Witten Theories
103
representation. The theory furthermore implies that the projective space of HA is independent of the choice of the symplectic coordinates consequence of this fact, we may regard 7i^ as a projective representation of the symplectic group. We will construct the Heisenberg algebra in great detail. For now, however, let us pause to say that there exists another way to construct TH.^ using mostly complex coordinates. As a general rule, this space can be identified with the space of polynomials in a; = a,- + ia,j. In actual fact though, a complex structure on A has to be chosen in such a way that the symplectic form originates from a Hermitian metric. So we begin by picking a unitary line bundle £ over A with a connection V such that the 2form curvature is r 2 = — iu. According to Axelrod, Delia Pietra and Witten whose work appears in [7], such a bundle exists whenever ^ represents an integral cohomology class. Since the isomorphism class H1 (A; U(l)) of £ is trivial, it then follows that £ is unique up to isomorphism. In the process, this verifies the uniqueness which, we may recall, comes as a requirement from the Stone-von Neumann uniqueness theorem. To obtain an irreducible representation of the Lie algebra, we must choose a complex structure c on A; then a second step requires that the symplectic 2-form w be positive and compatible with c, or, put differently, that in c, u> be a (1,1) form. Combining these requirements yields holomorphic linear functions a; which are defined by the formula u> = i Y^ da* A da{. i
Once c is picked, ( £ , V ) acquires a new structure: V has curvature (1,1) with respect to c, so V and c combine to give £ a holomorphic structure. Explicitly, the holomorphicity of £ originates from the condition B2 (VV) 0 ' 2 = u,0'2 = 0, where d is a V 0 , 1 operator on £. We are in a position to give a detailed description of £. To this end, let £Q, endowed with the Hermitian metric, denote the trivial holomorphic line
104
8. Geometric Quantization of Chern-Simons-Witten Theories
bundle on A. Furthermore, let e e C°° {Co). The relations follows |e| 2 b(a)
= e*eexp — 6 = £ once'1.
From results in reference [7], we deduce that the curvature of Co is
-Bdb
=Zdaid*, =
v
—iijj.
'
There is an isomorphism between C and Co which is unique up to projectivity and is a direct consequence of (8.26). What is the Heisenberg algebra to which we alluded earlier? It has a simple expression, [at, a,] = -8ij [ai,aj] = [ai,aj] (8.27) = 0, where, we may recall, Sij is the Kronecker delta (see Chapter 5). The form (8.27) has the following presentation in the Hilbert space "HxP (a,) • e = a, e
For the commutation relations to be well-defined, the following constraints must be taken into consideration:
9 ,
— (aje octi
,
c
d
= bije + oij —e. azi
This representation of the Heisenberg group is irreducible for the simple reason that most holomorphic functions can be expressed in terms of polynomials. We have the Siegel upper half space (which was studied in Chapter 3 and 4), the space of admissible complex structures on A. As such, one can vary c while keeping w positive -and of type (1,1)- while preserving simultaneously the translational invariance of c. We write S for the Siegel upper half space. As we pointed out in Chapters 3 and 4, S is the homogeneous space Sp (2n;M.)/U(n) of complex symmetric n x n matrices with positive
8.2 Geometric Quantization of Chern-Simons-Witten Theories
105
imaginary part. Let Hs denote a bundle of the Hilbert space over S; the symplectic group S p ( 2 u ; E ) acts on Hs- The fiber over c € S is just HcNow, using the Stone-von Neumann uniqueness theorem for the representation of Heisenberg groups, one finds that the identification of all of the fibers of Hs comes from endowing Hs with a natural projectively flat connection. Below we discuss in great detail the nature of this form. The first priority then concerns the expansion of the commutators. They are [V^VJH]
= - i (VuLJjk + V,-jfcUty + VyWjfc + VjyW.-j).
This gives the Lie algebra of Sp (2n; R), £, or more precisely, the algebra of homogeneous quadractic polynomials under Poisson bracket. One would expect £ to act in any representation of the Heisenberg algebra. Notice that, as a group, Sp (2n; R) acts on the Heisenberg algebra by outer automorphism. In doing so, it conjugates the representation Hc of the Heisenberg group to another representation. Sp (2n;R) acts projectively on Hc by the uniqueness of (this) irreducible representation. Consequently, V,j gives the action at the level of £. Note that the V,j are second-order differential operators (the a; act on Hc as zeroth-order differential operators and the or, act on as first order differential operators.) Vector fields generated by the V ; J act transitively on the Siegel upper space S. They define, in turn, a connection on the Hilbert space H and this connection is given by a second-order differential operator which is inherited from the V s: V 1 ' 0 = S1'0 + 1 V (Sc o a,"1 V 1 ' 0 ) .
(8.29)
To extend the quantization over the whole Hilbert space requires the use of the Teichmiiller space T. Essentially, this is done by using the period mapping [7, 9] as defined by equations (8.13-14-17). 8.2.2
T h e Torus
This case corresponds to the geometric quantization of Chern-Simons-Witten theories for the gauge group (7(1). Let us begin by considering an integer lattice A = £ z C Au,
106
8. Geometric Quantization of Chern-Simons-Witten Theories
of maximal dimension 2N, corresponding to the quotient torus T = A/A
~ R " / Z n = A(U(l)).
(8.30)
As was the case previously, our interest lies in the complex quantization case. To this effect, we pick, once again, a complex structure c on A. The torus, as denned in formula (8.30), can then be regarded as an abelian variety, AcThe complex line bundle on A, namely C, with curvature 2iriu>, descends to a holomorphic line bundle on Ac, with first Chern class w. An important issue centers around the fact that the line bundle C is not well-defined, mainly because the torus is not simply connected [5, 7]. There exist however, several ways to get rid of this ambiguity in L. We refer the interested reader to Atiyah's book [9, pages 20-22], to reference [7], and [12]. The action of A commutes with the connection V. Hence, V can be shown to restrict to a connection on the sub-bundle 7{\. This implies that V can be made flat by tensoring ~H\ with a suitable line bundle whose connection lie in S. A short note of caution though: the resulting connection will not necessarily be invariant under the action of the group Sp (2n; Z); this is so because the original connection differs by a central factor. This is the most probable cause of anomalies generated by an analogous problem: the need to rescale the connection 1-form in the infinite dimensional case from the normalization it would otherwise have in finite dimension. Very little is known about these class of anomalies, except some indications that the Atiyah-Patodi-Singer Index theorem may be of potential use in their detection and cancellation, as noted in [7]. 8.2.3
The Symplectic Quotient
The lattice is replaced by a compact Lie group G which acts symplectically on A. There is a lift of the G-action to an action on the line bundle £ , preserving the connection V. In what follows, we shall refocus the original quantization of the affine spaces toward the G-invariant complex structures. Consider the Siegel space SQ to which we associate the Hilbert space bundle H.G whose fiber over c is the G-invariant subspace CHc)a of 7~tc- To obtain a natural projectively flat connection on Tia, we only need restrict our attention to SG and
CHC)G-
8.2 Geometric Quantization of Chern-Simons-Witten Theories
107
As a symplectic manifold, the quotient Ac/Gc is independent of the complex structure c. We write A//G to denotes the symplectic or MarsdenWeinstein quotient of A. It should be noted that A//G acquires a complex structure from its identification with Ac/Gc. The connection (8.29) is once again given by a second-order differential equation. 8.2.4
Non-Abelian Moduli Space of Representations
The method under consideration here can be regarded as a generalization of the previous torus case. Consider a compact, simply connected Lie group with corresponding Lie algebra K. Associated to it is an oriented compact surface without boundary, S p . Let A' be the space of connections on a principal G-bundle, P —• E s ; this is an affine space whose gauge group Q acts on A' by the classical transformation
Write (Tig, G) for the resulting space, i.e. a truly Hausdorff space of dimension 2(n — 1) • dim G. The Lie algebra valued 1-forms a and /? define the skew pairing by {a,/3} * J - ^ ( a A / 3 ) , (8.31) or more generally for G = SU(n): {a,/?} = /
- T r ( a A/3).
(8.32)
Equations (8.31-32) combine to define a natural symplectic structure on A'. The group of gauge transformations Q acts naturally on A' preserving its symplectic structure. The next problem is to find the curvature. This means reformulating the moment map (8.14-15-18) as m : A' -> g. Consequently, the curvature reads m(A')
— QAI.
108
8. .Geometric Quantization of Chern-Simons-Witten Theories
is a Lie algebra valued 2-form. With these ingredients in hand, let us pick a symplectic form w0 on A', an integral class obtained as the curvature of a line bundle over A'. The resulting symplectic quotient is £IA>
M = A'l/g
= {A' : SlA, = 0}/g.
(8.33)
The Narasimhan-Seshadri theorem [8], which was interpreted by Atiyah and Bott [5] as an analogue for the infinite-dimensional affine space of connections, tell us that once a complex structure c is picked on E 0 , the moduli space A4 has a natural identification with the moduli space of holomorphic principal ^-bundles on E 3 . In actual fact though, the family of complex structures on A4 is parametrized by the Teichmiiller space, T . Thus, the quantization result in building a second-order differential operator, that is, projectively flat connections on the bundle £ over T.
8.3
Symplectic Quantization of 2-Dimensional Surfaces
The geometric (or symplectic) quantization of a two-dimensional surface is challenging. Among the obstacles encountered, the lack of polarization (which gives the direction of projection) in the phase space A, and the absence of a primitive 0 for the symplectic form u> are the most noticeable ones. Thus, it is of no surprise that the ordinary rules of geometric quantization fail here. This is best exhibited by the obstruction in defining the cohomology class [fi] G Hl (A\ Z). As shown in previous sections, the need for a (Kahler) polarization and a well-defined cohomology class [y\ £ Hx (A; Z) are ingredients in any consistent quantization of Chern-Simons-Witten theories. In order for us to put these obstacles in a manageable form, we consider the framework in which Stokes's theorem evolves. This theorem suggest, among other things, that integrals of the 2-form u> over E 9 C A with boundary dT,g = M can be regarded as analogues of the integrals of a 1-form 0 over one-dimensional cycles. To properly extend the analogy to a one-dimensional integer-valued class [/*], we need the equivalent of an integer-valued 2-form on the entire symplectic phase space A. Just in the same way as the differential formu> represent the double-cohomology class [u>], this sought integer-valued
8.3 Symplectic Quantization of 2-Dimensional Surfaces
Figure 8.1:
109
110
8. Geometric Quantization of Chern-Simons-Witten Theories
form must represent twice the Chern class 2cx G H2(A;Z). Following Turaev [11], let us define such a form as the index i of E g , i ( E 3 ) . A typical quantization rule can now be written down as
The triangulation of E 5 yields a specific formula for the index of E s : i
( S s) = Yl ispc, + J2 /*/?«>
( 8 - 35 )
where igpa and npa are integers or half-integers; incidentally, a, /3,6 are all indices. Note that the index of summation in (8.35) is defined over all vertices of the triangulation. The following comments are now in order. Firstly, in the case where the Chern class Cj is even (that is, A admits a metaplectic structure ), the index of surfaces usually defines a one-dimensional cohomology class, [/i] £ H1 (M';Z 4 ); also, [/i] (M) = iM (E,), where <9ES = M ' . Secondly, if A posseses a polarization (such as the Kahler one discussed earlier), then the class [fi] is an integer class. Hence, the index in (8.35) is none other then the required two-dimensional analogues of the characteristic class [//]. Let {npa} denotes a cochain on A and cpa a primitive symplectic form: C0a = /
0a — / 00.
Using Stokes's theorem, we find {dc)Spa
+ $60a
= / U>.
In light of this, the quantization rule (8.34) needs to be reconsidered; its new form is 1 x. a T $60a
-
77*«/3a =
a6
+ dSa + <^0a ~ 2?T pS0a;
(8.36)
n I hereps0a denotes an integer-valued 2-cocycle while {d0a} stands for a certain 1-cochain on A. The following relation holds: d
\aiMi + ha ~ F ; =
27rp,M
''
111
8.3 Symplectic Quantization of 2-Dimensional Surfaces Thus, the 1-cocycles with values in U(l) \0a
= exp (i (aPa + -c/)a - ^paj)
,
(8.37)
together with the cohomology class [A] € H1 {M'\ (7(1)) lives on M'. There is, however, a subtlety: the geometry of [A] is not all that transparent from this approach. Furthermore, the contribution of the symplectic form as h —> 0 is mixed with the contribution of the index in equation (8.37). It is with these issues in mind that we proceed to take a closer look at [A]. On the boundary 3 E 3 C M, the exact form for [A] is [A]9E9
= exp f ^J
u - i-iM{Vg)\
•
(8.38)
This result arises by actually writing the integral of the form w in terms of the cocycle $ and the cochain c, that is,
L
E0
<*> = Yl ®Wa + Yl CPa-
Thus,
\ /s,
w
- I «'M ( S P) = E (£ % ~ f ispa) + E (£<*,„ - \ii0a) = —27T J2 a60a + E [PPa + \ Cpa ~ f PPa) •
(8.39)
Combining (8.39) with (8.37) give ex
P (A fzg
w
- {\JM(SS))
= exp (i £ (apa + \c0a - f/*/?«))
= Was,
= n *„„. (8.40) We are now in position to state the following important result which removes the mentioned obstacles to quantizing two-dimensional surfaces: Any two-dimensional surface that admits a quantization rule of the type given by equation (8.34) must have a trivial (or vanishing) [A] E H1 (M; £/(!)).
112
8.4
8. Geometric Quantization of Chern-Simons-Witten Theories
References
[1] Arnold, V. I. and Givental', A. B. : Symplectic G e o m e t r y SpringerVerlag, (1990) Berlin New York. [2] Fomenko, A. T. : Symplectic Geometry, Gordon and Breach, (1988) New York. [3] Weinstein, A. : Symplectic Manifolds and their Lagrangians ifolds, Adv. in Math. 6 (1971) 329-346.
Subman-
[4] Weinstein, A. : Lectures on Symplectic Manifolds, Conf. Board Math. Sciences Regional Conf. Ser. Math. Vol. 29 (1979) Amer. Math. Society, Providence, R. I. [5] Atiyah, M. F. and Bott, R. : The Yang-Mills Equations Over Riemann Surfaces, Trans. Roy. Soc. Lond. A 308 (1982) 523-612. [6] Quillen, D. : Determinants of Cauchy-Riemann Operators Over a Riemann Surface, Funct. Anal, and Applications 19 (1985) 31-34. [7] Axelrod, S., Delia Pietra, S. D. and Witten, E. : Geometric Quantization of Chern-Simons Gauge Theory, J. Diff. Geometry 33 (1991) 787-902. [8] Narasimhan, M. S. and Seshadri, C. S. : Stable and Unitary Vector Bundles on a Compact Riemann Surface, Ann. Math. 82 (1965) 540-567. [9] Atiyah, M. F. : The Geometry and Physics of Knots, Cambridge Univ. Press, 1990 Cambdridge, New York. [10] Witten, E. : Quantum Field Theory and the Jones Comm. Math. Phys. 121 (1989) 351-399.
Polynomial,
[11] Turaev, V. G. : Cocycles for Symplectic First Chern Class and Maslov Indices, Funct. Anal. Applications 18 (1984). [12] Murayama, H.: Explicit Quantization of the Chern-Simons Z. Phys. C-Particles and Fields 48 (1990) 79-88.
Action,
Chapter 9 Deformation Quantization In the present chapter, we will analyze the relation between two seemingly disparate subjects: global anomalies and deformation quantization. The framework for this draws on the work by Baadhio, in reference [1], which asserts that under suitable conditions, global anomalies arise when deformation quantization is performed. We shall learn that in the theory of deformation quantization, there exist essentially two appraoches which may induce the occurrence of global anomalies: the first one has to do with an obstruction to patching a locally deformation quantizable ^-product to a global *-product. Most Poisson manifolds are known to admit regular, or otherwise said, local deformation quantizations. Their generalization, however, represents a substantial problem involving topological obstructions and the like. We will not explicitly discuss the case of global anomalies induced by this technique, the reason being that they have not being investigated at the time of writing. We will, however, address the problem of global anomalies which arise under the degeneracy of the symplectic 2-form ui. In particular, we will study the case in which ui degeneracy extends to the degeneracy of the Poisson structure on the classical phase space, A = R 2n . What one finds is that singularities arise which strongly restrict the possibility of carrying a globalization of local deformations quantization. This is much like the patching approach to globalizing local deformation quantization. We begin with a background review of deformation quantization.
113
114
9.1
9. Deformation Quantization
Aspects of Deformation Quantization
From chapter 8, we have learned that quantization is a procedure consisting of associating a Hilbert space 7i of quantum states to a given classical theory. In view of Dirac's correspondence principle [2], quantization of a classical system is the process of associating Hermitian operators on a Hilbert space to the classical observables. These are functions
/ e C°° (A); where A = M2n, a symplectic manifold, denotes the classical phase space. The operation, up to a constant factor depending on the Planck constant h, associates to the Poisson bracket on A,
if, 9} = (t,
d£ dg_
the commutator of the operators. The resulting operator / associated with / is called the quantization of / . The problem of quantization is then to construct an associative noncommutative algebra of quantum observables satisfying the correpondence principle outlined in [2], and futhermore, to describe its representations by operators in the Hilbert space of quantum states. The issue, as one might guess, is a highly non-trivial one under Dirac's correspondence principles. The reason for this is in fact simple and inescapable: we are asking that the quantization produce similar observables for classical and quantum quantum systems alike. But observables in quantum mechanics, unlike those in classical mechanics, do not commute with one another! The study of deformation (and geometric) quantizations was originally intended to deal with this dichotomy. One mathematical transcription of Dirac's principle is that the obtained representation of the Poisson algebra, C°° ((-4) {, }) be irreducible. A theorem of van Hove [3] shows, however, that the quantization has no such solution. In order to avoid this contradiction, physicists often quantized in two steps: • the pre-quantization which consists of linear representations of the whole algebra C°° {(A) {,}) by operators on a given complex vector space. Then, in a second phase,
9.1 Aspects of Deformation Quantization
115
• the actual quantization, where one restricts the problem to a convenient sub-algebra of C°° ((.4) {,}), and then one represents its irreducibility (in a way that is compatible with the underlying physical theory) on a Hilbert space initially built out of the pre-quantization space. Kostant [4] found, a while ago, that modulo a certain obstruction, the prequantization problem could be solved by means of the space of cross-sections of a complex line bundle, •K
: C -> A ,
the so-called pre-quantization bundle, and, by a well-defined pre-quantization formula. The existence of C is actually related to whether the cohomology class of the symplectic 2-form u> is an integral cohomology class, i.e.
[w] e H2 {A,u). 9.1.1
Star-Product as Deformation Quantization
The theory of deformation quantization was developed a decade ago by Lichnerowicz [5] et al. It originated in the Dirac principle that a quantum system has to reduce to the corresponding classical system as ft —* 0 (ft, = h/2Tt, h — Planck constant). We take ft, to be the numerical value of the Planck constant when it is expressed in a unit of action characteristic of the class of systems under consideration. This present formulation avoids the paradox that we consider the limit ft, —• 0 even though Planck's constant is a fixed physical parameter. The quantization of a classical system, in order to satisfy the correspondence principle, should therefore consist of a deformation of this classical system into a system that depends on a parameter, ft. It is important that we define what we mean by system. According to our presentation of geometric quantization in the previous chapter, from the classical standpoint, the word system refers to the commutative algebra of observables, C°° (A) ® C, e.g. the algebra of smooth functions on the classical space A with ordinary multiplication, and the Poisson bracket {, } on A, induced by the symplectic 2-form 10. The quantization of this system consists of defining an associative, noncommutative deformation of the usual product, into a new operation, the
9. Deformation Quantization
116 •-product ,
/ * 9 (/, 9) G C~ (A) which depends on h, and is such that the commutant, ,,
[f 9]
*
, def U*9
=
~
Th
9*f)
'
will be a deformation of the Lie algebra operation {/,g}. From a purely mathematical standpoint, the goal of deformation quantization is to construct ^-product and interpret the resulting physics. The undeformed product *o is taken to represent the usual pointwise multiplication, so that (A,*o) is the algebra of classical observables. In applying Dirac's general principles [2], the limit lim f(a*fi b — b*rt a) lift\, is equivalent to a given classical Poisson bracket {a, h] on the phase space A. This bracket is a Poisson structure in the sense that it can be shown to satisfy the axioms of a Lie algebra, together with the Leibniz identity, namely, {ab, c} = {a,c}b + a{b,c}. In this context, a, formal deformation D = Do, Di,--- is called a *-product whenever each of the bilinear map Di is indeed a differential operator, annihilating the constant functions when i > 1. These conditions are essential in ensuring the localness of the *product, while simultaneously ensuring that the constant function 1 remains as the unit element. To recapitulate what we have said so far. On the classical phase space C°° {A — R 2 n ) of smooth functions, we have two basic algebraic structures: • the associative product, A X A —• A : (a, b) -+ ab; • a Poisson bracket associated to the symplectic nondegenerate 2-form w, A x A -> A : (a, 6) —• {a,b}. A -^-product is a deformation of the associative structure, which, by antisymmetrization, gives rise to a deformation of the Poisson bracket {a, b}. We call this procedure a deformation quantization. It originated with the need
9.1 Aspects of Deformation Quantization
117
to verify Dirac's correpondence principles [2] between classical and quantum mechanics. Deformation quantization and ^-product are essentially similar objects and from now on, we will use these two terms interchangeably. The symplectic form u, expressed in coordinates (qi, • • • ,qn,Pi, • • • ,Pn), reads UJ
= J2 dqi A dpi, X
and the Poisson structure,
,
,-, _ v-^ (da
db
da
db\
3
is invariant under all diffeomorphisms preserving u, so there is a well-defined Poisson structure on any symplectic manifold. The most readily and extensively studied example of a ^-product is the so-called Moyal-Weyl product [6] on R 2n , with the Poisson structure just described above. The Moyal-Weyl product originates from the composition of operators on C°° (R n ) via Weyl's identification [7] of such operators with functions on R 2 n and was used by Moyal in reference [6] to investigate quantum statistical mechanics. Let V denote a vector space, and consider a, a skew-symmetric bilinear function on V*. The formula {a, b] =
a(da,db)
defines a Poisson structure on v. Associated to the bilinear operator, a, is a unique differential operator: E : C°° (V x V) -» C°° (V x V) with constant coefficients for which {a, b} = A*E(a<8>6). The term ab is the function (y, z) i—• a(y) b(z). Moreover, we think of A* as A* : C ° ( y x F ) - » C°° (V). We can define the Moyal-Weyl product on V by a *n b — A*exp I —— ) (a ® 6). The space C°° (V) [ft] associated with this product is called the Weyl algebra of v.
118 9.1.2
9. Deformation Quantization Local and Global Characters of Deformation Quantization
On a given Poisson manifold, the Leibniz identity implies that the Poisson bracket is given by a skew-symmetric contravariant tensor (or bivector) field a, which is called Poisson tensor. It is given by the formula {a, 6} = a (da,db). A case of interest is the one in which the rank of a (that is, the rank of the matrix function a^ (x) {xk, £/} in local coordinates) is constant. A theorem by Lie [8] is then applied, and roughly speaking, states that the Poisson manifold A is locally isomorphic to a vector space with constant Poisson structure. Thus, Poisson manifolds which are labelled as regular are always locally deformation quantizable [9]. Now, there is a highly non-trivial problem: patching together the local deformations to produce a global ^-product. The issue of giving a global character to the local deformation quantization centered around some forms of obstructions which are especially responsible for the occurrence of quantum pathologies, the so-called global anomalies. The interested reader may want to consult [13] for a thorough investigation of obstructions and global anomalies, particularly insofar as they relate to the context of ten-dimensional physics. Under special circumstances, the patching of local quantizations can be done rather easily. Here is how. It begins with the realization that the Moyal-Weyl product on a vector space V, with constant Poisson structure, is invariant under all the affine automorphisms of V, mostly because an operator with constant coefficients is invariant under such transformations. This, in turn, suggests the possibility of constructing a global quantization of any symplectic manifold A, covered by local isomorphisms, and for which the transition maps are affine. Such a covering, it was revealed in [9], exists whenever the phase space A admits a flat torsionless linear connection (for which the covariant derivative V a is exactly zero). Torsionless Poisson connections play an important role in the analysis of deformation quantization. In particular, the existence of a deformation quantization in the presence of a flat torsionless Poisson connection was first established in Bayen et al.'s landmark papers on deformation quantization [5]. Note that the consistency condition on the connection (e.g. V a = 0) implies the regularity of the Poisson structure. Situations in which this is
9.1 Aspects of Deformation Quantization
119
not the case are those with degenerate Poisson structure (which we shall investigate at length in upcoming sections). At this level, as we shall soon see, quantum pathologies arise and need to be cancelled if the theory is to be unique and consistent. Let us mention another case of interest concerning the possible globalization of local deformation quantizations. This case is actually the most challenging of all, and research activities aimed at solving it are still very much ongoing. When the Poisson manifold A does not admit a flat, torsionless Poisson connection, the authors of [5], and independently Gutt in [10], have pointed out the existence of an obstruction localized in the Hochschild cohomology space H3 (A). More precisely, the obstruction to the existence of the local ^-product, D\, Z?2, • • • lies in H3 (A). However, it was later found that the obstructions could further be pinpointed within the de Rham cohomology class in H3 (A). As a consequence, there is seemingly no obstruction to constructing a deformation quantization when the third Betti number of the phase space A is zero. One can arrive at this conclusion by simply looking at the much smaller Chevalley cohomology space, H^y (A) (taken as a Lie algebra via the Poisson bracket). Deformations of this Lie algebra were initially studied by Vey in reference [11]. But the proof is credited to De Wilde and Lecomte [12] that deformation quantization always exists on any symplectic manifold. The most celebrated consequence of the De Wilde-Lecomte proof is that it renders trivial, in the symplectic case, the class of obstructions (to any generalization of locally deformed quantization) for H^eBham (-^) trivial. For manifolds other than symplectic ones, the question remains and, as we mentioned earlier, research is still ongoing.
9.1.3
Issues W i t h Deformation Quantization
Overall, the purpose of this is chapter is to report on the relationship between deformation quantization and global anomalies. Global anomalies occur whenever large gauge transformations of a classical field theory fail to be symmetries of the corresponding quantum theory. In this case, the anomalies are referred to as global gauge anomalies. Global gravitational anomalies, on the other hand, arise whenever the effective action of a given theory is non-
120
9. Deformation Quantization
invariant under a diffeomorphism group that cannot be smoothly deformed to the identity, that is, to mapping class groups; the later condition is essential to prevent the occurrence of disconnected general coordinate transformations, a sure manifestation of anomalies. Both global gauge and global gravitational anomalies were discovered a decade ago or so by Witten [14]. Since then, the subject has seen a fruitful extension. The case pertinent to the study of global gravitational anomalies in topological quantum field theories [15, 16] has elicited the use of threedimensional mapping class groups and constitutes such an example. There is also the use of some classical invariants of links and knots (such as the Arf invariant ) which have been shown to detect global gravitational anomalies [17]. Finally, although our knowledge of two-dimensional mapping class groups is fairly secure, the recent impetus for studying three-dimensional mapping class groups [18], for which very little was known, originates once again in the newly emerged physical theories in dimension three (including quantum gravity). In carrying out a standard geometric quantization, we are accustomed to the fact that the symplectic manifold, the phase space A — R 2n , ought to be endowed with a nondegenerate 2-form u> (see chapter 8, and additionally, references [1, 3, 4, 20, 32, 33]). We have come to rely on w's nondegeneracy property if we are to achieve a flawless quantization. There are several reasons why physicists choose to work with this particular property. The most obvious one owes much to the fact that a degenerate 2-form w inevitably gives rise to a deformed Poisson bracket; and since deformations of degenerate 2-forms are global functions of a Poisson bracket on A, such a choice renders geometric quantization much less attractive than it would otherwise be. A Poisson bracket which is defined by a degenerate symplectic 2-form ui will be referred to as a deformed or degenerate Poisson bracket. As we shall learn, the degeneracy of w essentially implies the occurrence of singularities in A. Under suitable topological conditions, the study of the degeneracy of <jj can be reduced to that of singularities in the symplectic leaves Oj. A symplectic 2-form u; is said to be degenerate when it is non-zero (or nontrivial), i.e. when it is not exact in the first cohomology class of the symplectic leaf in which it lives.
9.1 Aspects of Deformation Quantization
121
Nondegenerate Poisson brackets do admit small local perturbations whose harmful effects can be put into a manageable form, as they can be eliminated by perturbation theory. We will provide an example of global anomalies arising from deformed Poisson brackets. The most relevant case is an extension of the (singular) topological structures of the leaves fi,. We show in particular that whenever there does not exist a globally smooth homotopy transformation within the Poisson manifold (the collection of all fit- defines the Poisson manifold), global anomalies are bound to arise. This situation is reminiscent of global gravitational anomalies arising due to the presence of disconnected general coordinate transformations [14, 15, 16, 17, 18], for we are dealing with transformation of coordinates around singularities in R n . The relation between quantization and anomalies, it should be noted, has long been a successful endeavor. Schaller and Schwarz, for instance, have worked out anomalies generated by geometric quantization of fermionic fields [22]. The case of stochastic quantization generating anomalies has been studied by Morita and Kaso in [25], and independently in [23] by Morita. Panfil has investigated the general problem of canonical quantization associated to anomalies [24]. Anomalies induced by the proposed quantization approach of Carlip and Kallosh for the Green-Schwarz superstring have surfaced in [26]. Chiral anomalies coming out of stochastic quantization appeared in a recent work by Kim [27]. There is also the very recently published study of chiral anomaliesof WVgravity [28] which are used as a basis to simplify quantization. The emphasis of these works has not been on global anomalies, however. It will become apparent to the reader that we have not specified which type of global anomalies (i.e. global gauge or gravitational) we are dealing with throughout this chapter. There is a two-fold reason for this. While writing this chapter, we have consistently aimed at establishing the general framework in which global anomalies will manifest themselves once it is determined that we have a deformed quantization. This choice has come at the expense of specifying the type of global pathologies we are dealing with. Secondly, in forthcoming chapters (12 and 13 in particular), we will extensively discuss specific types of global anomalies and the quantum field theories they are known to affect.
9. Deformation Quantization
122
9.2
Symplectic Degeneracy of Poisson Brackets
In R 3 , the Poisson bracket takes the form
v, 9} = (e,
d_i dA
(9.1)
with £ G R 3 . The term inside [ ] is a vector product. The 3-form tensor ^
*(0 =
/ o
6
-M
-6
0
&
V6
-6
,
(9.2)
o ;
gives rise to a symplectic form u> — u>a = r s i n # d # A d(6, rji),
where (0, <j>) are spherical angles in R 3 , and the fi; are symplectic leaves. An interesting fact about equation (9.2) is that is it endowed with only one Casimir operator, namely c(£) = |£| 2 . This means that the singular point ^ — 0 corresponds to a point where the rank of $ goes to zero. In other words, £ = 0 is a zero-dimensional singular symplectic leaf, which we write as fi|0 = 0. In R 3 , its corresponding algebra is SU(2). Next, we focus on R 4 . We pick coordinates £o, £i, £21 a n d numbers CQ, Cj, c-i which satisfy the inequalities Co > cx > c2 > 0. The resulting Casimir operator for this bracket is similar to that of R 3 :
co(0 = E ( 6 ) 2 -
(9-3)
1=1
Equation (9.3) is the perfect example of a degenerate Poisson bracket. It can be generalized by the following formula
<*(0 = (£o)2 + £ a • (fc)2-
(9.4)
•=1
The topological consequences of equations (9.3-4) are of the utmost importance in the investigation of global anomalies. According to these, the
9.3 Inducing Global Pathologies
123
symplectic leaves fit- are the joint level surfaces of the Casimir operators Co and c\. Thus, fi = {co = constant; c\ = constant}. These leaves are essentially two-dimensional objects since rank of \P = 2. They also have the following topological structures, to which we shall come back later for a precise study of the manifestations of global anomalies. Q « 5 2 if C! > Co ci > 0 or Co c2 > ci > coc3 > 0 il « T2 = S1 x S1 if CoCi > cx > c0c2 > 0. Because the leaves degenerate and become exactly zero-dimensional at the points where some of these inequalities turn into an equality, these topological structures are of fundamental interest to us.
9.3
Inducing Global Pathologies
Nondegenerate Poisson brackets admit small local perturbations which are not harmful since they can be eliminated by perturbation theory. In the degenerate case however, we have a very different solution: small or infinitesimal perturbation of the bracket (9.1) yields a global change in the topology of the symplectic manifold A. We shall refer to these deformations as global anomalies. Below, we provide an illustration of global anomalies induced by a degenerate Poisson bracket in R 6 . On R 6 = K%f ® Rjf there is a family of linear brackets which depends on a certain parameter e > 0: {Ma,M0} {Xa,X0}
= M, =eM,
(9.5)
The a, (3 and 7 are cyclic permutations of the indices 1,2 and 3; M and X are subsets of R 6 . Altogether, the relations in (9.5) give rise to a bracket on the Lie co-algebra SO(4)* when e > 0. On the other hand, for e = 0, we have a bracket on the co-algebra SO(3)*. The following Casimir operators d =
M-X,
and c2 = |X| 2 + e|M| 2 ,
124
9. Deformation Quantization
generate gives rise to terms which are diffeomorphic to S2 x S2 for e > 0, and diffeomorphic to T* (S2) for e = 0. As a consequence of this, there does not exit a global, smooth homotopic change of variables in K6 which can appropriately transform a non-perturbed or deformed SO(3)* bracket into a perturbed SO(4)* one. This observation is the first manifestation of global anomalies. In quantum field theories, it is often sufficient to know only the first approximation in e if one is to write a pathology-free consistent physical theory. For instance, one may be required to find a Lagrangian transformation of R 6 that maps an SO(3)* bracket into the original and standard Poisson bracket in (9.1). Such a mapping is considered reasonable within 0(e2) accuracy. A transformation of this type often looks like (m,x)
-f (MC,X<)
where v(m,x)=
= (m,x)
+ ev{m,x)
+ 0(e2);
(9.6)
/ (m • x) m — \m\2 x\ (m, ^ j .
Several facts about the point at which the bracket (9.1) degenerates are of interest. Firstly, there are non-singular points in SO(3)* where the rank of (9.1) degenerates. This in turn implies that the generator of the transformation (i.e. the vector field V on R 6 ) possesses a singularity at |a;| — 0, i.e. there are nonregular points in SO(3)* where the rank of the bracket drops. So we are lead in a straightforward way to global anomalies.
9.4
Occurrence and Manifestations of Global Anomalies
To begin with, we choose once again our phase space to be the symplectic manifold A — R 2 "; its corresponding Poisson bracket is of the form {/,
dg >,
(9.7)
where we have used dj = g|-; also, the angle brackets denote the pairing of a form and a vector field. We note that $ belongs to the space of bi-vector
9.4 Occurrence and Manifestations of Global Anomalies
125
fields; incidentally, *P is a mapping of bundles, i.e. * : T*(A)
-
T{A).
A global anomaly corresponds to forms for which the operation * (df, dg) + eV (df, dg)
(9.8)
is a Poisson bracket mod O (e2) corresponding to solutions of the equation [*, $] = 0.
(9.9)
We point out that £
9jkdk9lm = 0,
(9.10)
is the Jacobi identity. This established, our next order of priority is to distinguish between local and global anomalies. Local anomalies have trivial local deformations which are nonetheless solutions of (9.9). Local anomalies will arise whenever we deal with a family of smooth transformations, i -> * = t + *v(t)
+ 0(e2),
(9.11)
in a neighborhood of any point on A. These anomalies typically behave by transforming the initial Poisson bracket (9.1) into the (anomalous) Poisson bracket (9.8). By using a step-by-step approximation with respect to 0(e2), one can show that the effect of the local anomalies is harmless, for they can give a smooth, homotopic, non-anomalous SO(3) transformation into an anomalous SO(4) one. Their local character is preserved as long as they do not generate non-smooth homotopic transformations. Distinguishing between those two classes of anomalies is what we shall refer to as Problem 1. In Problem 2, the issue of interest is to find a way to detect the global trivial deformations in A of equation (9.7). Problem 3 shall focus on determining the numbers of generators that are invariant under the Poisson bracket (9.1). Another focus of interest, which will be covered in Problem 4, is to relate the class of non-locally trivial and non-globally trivial deformations to the
126
9. Deformation Quantization
topology of the symplectic leaf $7. In actuality, we may assume, without loss of generality, that non-local deformations do assume a global pathological character, whereas non-global deformations are simply local anomalies coming from (9.11). However, to clearly differentiate between local and global anomalies in this framework, we need a better way of clarifying these two situations. In Problem 5, we will be concerned with global anomalies arising from the Jacobi identity (9.10). As we shall learn, these anomalies are present whenever the condition (9.9) is violated. To describe the anomalies in this case will require a good look at the class of 3-tensors B for which the condition [¥,$] = B
(9.12)
can be solved with respect to $ . In due course, we will see that the consistency condition (9.12) distinguish between local and global anomalies, thus solving the issues spelled out in Problem 4. Finally, in Problem 6, we aim to find all solutions to the condition (9.12) in order to effectively cancel global anomalies which arise in deformation quantization. There, we will come upon a generalized version of the descent equations, also known as the Wess-Zumino consistency condition.
9.5
^-Product and Anomalies
Consider a dynamical system 4 = (*(<) + €$(<;)) OH (<),
(9.13)
where H stands for the Hamiltonian. This system is relevant to Problems 1-5 for it is a system in which the Poisson structure is perturbed instead of the (usual) Hamiltonian of the theory. The most relevant perturbing term in (9.13) is $ (c), and this perturbation can essentially be approached in two ways. In the first approach, the transformation (9.10) is independent of H; this gives rise to the relations
£ =*(0&(0 Ht
= H + tv{H)
+0(e22 )
+0{e ).
(9.14)
9.5 ^--Product and Anomalies
127
According to equation (9.12), equation (9.13) can be averaged up t o m o d O ( e 2 ) with respect to the Poisson bracket in (9.7). For the quantum states to remain consistent, the averaging method has to respect the Hamiltonian structure of the quantum field theory denned in the symplectic phase space A. As for the second approach, one begins with a thorough description of the field v in (9.10). The condition that the mapping £ — i > <; transforms the Poisson bracket (9.7) into (9.8) (up to modO(e 2 )) is encoded in the formula
| | * ( £ ) ( | | ) =*(<) + **(*) + °^)-
(9-15)
Substituting (9.10) into (9.15) yields the desired formula: $ i j = ^isdsVj
-
^j3dsVi
-
vsds^ij,
(9.16)
which implies the consistency condition
[*,u] = $
(9.17)
To solve Problems 1-5 means finding the global solutions to (9.17). Let us pause to discuss the relationship between anomalies and the deformed Poisson structure. The Poisson fields given by fj. on A (Problem 3) are all solutions of the consistency condition
[<M = 0; while the conformal fields coming out of Problem 4 are solutions of [*,«]• In these two cases, the corresponding quantum field theory is free of global anomalies. Then, depending on how the consistency conditions are handled, anomalies may still arise given that the conditions themselves assume a local character. When this is the case, the induced pathologies are purely local and, as such, they can be put into a rather manageable form using perturbation theory, mostly (9.13). Local anomalies may still manifest themselves even though the theory is shown to be global anomaly-free. For our case, this means the existence of a class of local anomalies induced by homotopic
128
9. Deformation Quantization
transformation. This, for instance, may appear in Lagrangian transformations in R" that map Lie group induced Poisson brackets into deformed ones. The degree of accuracy needed for the counterterm to cancel the deformation is exactly what induces the local or perturbative anomaly. Here is an illustration. Using the transformation (9.10), we define another one, * ( c ) -4 tf (etfe). The transformation £ —• E (£, e), which transforms an initial Poisson bracket < >o to an anomalous one, namely { } £ , originates from the Cauchy problem, i.e. — Z = vt{E),
H|£ = 0 = £.
Here, the vector field vc satisfies the condition (9.17), that is to say [*«,««] = ^ * e .
(9.18)
For the required homotopic transformation to be smooth, the vc solution of (9.18) ought to admit very weak singularities, so weak in fact, that the topology of A is not globally affected. The presence of global anomalies in this case would almost certainly imply the existence of strong singularities, and this, in turn, would violate the smoothness condition spelled out in (9.18). Do such singularities actually exist? The answer is yes. They were initially described by James Eells (circa 1960) [31], and later by Martinet [32]. We point out that even though the Eells singularities predispose to the existence of global anomalies, they do not preclude the manifestation of local anomalies. However, once again, an appropiate use of perturbation theory may put them in a manageable form, primarily by reducing their ill effects on the theory.
9.6
Nondegenerate Symplectic Spaces
In order to better grasp anomalies induced by degenerate Poisson brackets we employ the useful strategy of utilizing nondegenerate Poisson brackets as a comparison tool. Later, we will make a transition to global anomalies.
9.6 Nondegenerate Symplectic Spaces
129
In the nondegenerate case, all points in the symplectic manifold A, are regular, meaning that A is fibered by symplectic leaves ft;. To the symplectic leaves we associate some spaces of A:-forms, Yk(Q), Zk (ft) and Hk (ft), the later being their de Rham cohomology classes. The lack of global anomalies is due to the presence of the following smooth (i.e. non-singular) mappings Yk[Sl]
= C"{A/fl
-»
Yk(il));
zk[n] = e°M/ft -> zk(si)y, Hk[Q] = 0°(A/n -» Hk(0)).
(9.19)
The elements belonging to Yk [ft] and Zk [ft] are all closed forms on the symplectic leaf ft. Thus, ft can be described in terms of smooth coordinates on A/0, as parameters. The set of Casimir operators on A now reads H° [ft] ~ C {A); as for the vector field V {A), it preserves C {A). With these descriptions in hand, we proceed to extract the fc-forms on a given leaf ft. The formula
<* (a) = £
*. ••••* *7,!-. # « A • • •
A
* 7 l «;*.
(9-2°)
with g a tensor, and now £ G ft (instead of £ 6 K 3 in earlier sections.) According to equation (9.20), we have the property <*(g A g') = a(g) A
a(g').
\P assigns to 77 £ Yk [ft] a Ar-tensor ^ (»/). Hence, we can rederive (9.20) in terms of specific fc-forms. That is, -a
WW)
= -ua(*(v)) = V
(9-21)
and it can be shown to preserve the original identity transformation. The 8 operator defines a Poisson mapping S : Mk (A) -> Mk+1
(A)
and moreover, yields the following non-trivial condition, Sg = [*,g].
(9.22)
130
9. Deformation Quantization
Note that Mk denotes the space of antisymmetric tensor forms of degree k. Now, using the Jacobi identity one obtains
[*,*] = 0, which of course implies 62 — 0. In geometrical terms, what we have just found is that the operator 6 is a coboundary operator in the phase space A. We will make appropiate use of this in the next few sections. For now, we aim to discuss the conformal fields in A, which are essential for any study of global anomalies induced by the Jacobi identity, as explained in the preceeding section. Most Hamiltonian versions of field theories have Hamiltonian fields which are encoded in Poisson vector fields. This is most readily shown by the formula r
I* =
Y,
c3Zj + K,
(9.23)
3=s+l
where K is the Hamiltonian field. We say of the Poisson bracket (9.7) that it is homogeneous on A if the symplectic form UIQ is zero on the cohomology class H2 (Ct). When this is the case, U>Q becomes s w
« - Yl 9i "i + dpn
(9.24)
!=1
with the gi belonging to the Casimir operators C (A), while pa G Yk-i [CI]. From this, we deduce that the conformal fields on A are given by 5
/* = E 9i v, + V Pn + it.
(9.25)
!=1
By virtue of its nondegeneracy, the symplectic form u generates a pairing between the tangent and cotangent space of A. A group action Q on the symplectic manifold (.4, u>) is said to be symplectic if , i.e. g*u = w.
(9.26)
This observation is an important ingredient in explaining the lack of global anomalies in the nondegenerate case. DifFeomorphism transformations are
9.7 Cancellation of Global Anomalies
131
obviously smooth ones and so one does not expect the anomalies to arise. Actually, diffeomorphism invariance of quantum theories has been shown to prevent the occurrence of global anomalies, particularly global gravitational anomalies [15, 18].
9.7
Cancellation of Global Anomalies
The goal of this section is to answer Problems 1-5 as spelled out above. A global infinitesimal deformation of the Lagrangian C corresponds to a global deformation of the Poisson bracket at the quantum level. Hence, global anomalies correspond, in essence, to the formula (5$ = AC.
(9.27)
Local anomalies manifest themselves whenever the following condition holds: Sfi = A £ M = $.
(9.28)
The transformation $ gives rise to globally trivial anomalies whenever a solution of (9.28) exists globally on A. The Poisson bracket in (9.7) is said to be anomaly-free (global or otherwise) if each of its locally trivial transformations is globally trivial. (The case in which perturbative local anomalies arise has been treated in Section 9.3.) There is an obstruction for the Poisson bracket to be anomaly-free, and this obstruction lies in H2 (fi). Let us consider the case in which the bracket (9.7) is not anomaly-free. The next question of interest is to find what are the generators for the anomalies. This is especially warranted since we are aware of the obstruction keeping (9.7) from being anomaly-free. Thus, any effort aimed at determining what this obstruction looks like is welcome. A careful investigation tells us that the generators in question belong to Mk=1 (A) and they define closed 2-forms a ( $ - 6u) € Z 2 [fi]. Within the framework laid out by (9.20), global anomalies can be reduced to a manageable size. In essence, this implies that the condition (9.28) is to be reduced to an equation for forms on the space Yk=\ [fl]. When this is
132
9. Deformation Quantization
achieved, one can formulate with great precision the solvability condition for (9.28) in terms of the de Rham cohomology. Here is how. We rederive (9.28) in such a way that its corresponds to d/3 = a ( $ - 6u),
(9.29)
where /? 6 Y\ [0] is the 1-form we are looking for. Thus, solutions to (9.29) mean that the anomalies can be cancelled. Explicitly, by solving (9.29), we have y, = -\S>*(3 + w ; (9.30) in which $* is the dual of $>. The right hand side of equation (9.30) is the needed counterterm to cancel the anomalies. We now shed some light on the complex relation between anomalies and the conformal fields. In the Hamiltonian version of field theories, anomalies are omnipresent, as they often manifest themselves in the $ dki conformal fields. These fields have a global character, as shown by the following equivalence relation: $ dki = V*dhi, (9.31) with hi G yjt (A). Its corresponding symplectic generator is
u = J2 hi Vi.
(9.32)
It is a straighforward exercise to check that (9.31-32) holds in the whole phase space A, but most particularly for simply connected fi. However, for the non-simply connected case, there is an additional condition for w to be well-defined in (9.32). This condition requires the cohomology class of the 1-form a($dki) to be trivial in H1 [ft]. Put differently, in the non-simply connected case, we have an obstruction to defining the anomaly generator. To remove this obstruction, the first cohomology class of the symplectic leaf fi has to vanish! (This is different from the obstruction for the Poisson bracket (9.7) to be free, since this one lies in H2 [fi].) The hi are determined by the set of Casimir operators. The procedure leaves u well-defined in the degenerate case, so there is no danger of losing .4'.s symplectic structure, even in the presence of anomalies.
9.7 Cancellation of Global Anomalies
133
Now that we have elucidated the relationship between conformal fields and global anomalies, we may wonder at the anomaly constraint given by equation (9.29). Observe that the closed 2-form on the right hand side of (9.29) can be written as a ( $ - Su) = a ( $ - £
Vi
A V* dh{) - £
A,-w,-.
(9.33)
This form takes its value in H2 [il] and can be shown to be independent of the choice of the Hamiltonian (i.e. the hi) in (9.31). Several facts arise in light of this finding. One is that, in the presence of global anomalies, we can still manage to preserve the consistency of the vacuum state of the quantum field theory. Adequate manipulations of (9.33) will yield a global anomaly counterterm which will cancel the manifestations of these pathologies. The second consequence which comes to mind, and somewhat reinforces the first, is the observation that, within the appropriate constraints in the Hamiltonian version of the theory, the occurrence of global anomalies induced by deformed Poisson brackets does not consistently destroy the general consistency of the theory. Thus, the real question before us is which of these constraints actually allows us to preserve the general consistency of the theory. According to our previous discussions, there are a handful of potential candidates. We therefore need to rephrase the question: what class of global anomalies are the least likely to destroy the consistency of the theories? To answer this, it is useful to learn more about the anomaly $ . Earlier, we have given a somewhat local character to $ , whose form is given by equations (9.28-29). We said of $ , that it is global if: 1. the cohomology classes of the 1-forms a ($dki) are zero in H1 [fi] (we may recall that this particular condition defines and preserves the symplectic form u> when the leaves fit- are not simply-connected); 2. the quotient class of the 2-form in (9.33) is zero in H2 [ft]. Under condition 1, the relation (9.31) holds true and the generators for the global anomalies simply correspond to u in (9.32). Under condition 2, we can choose the Hamiltonians hi in (9.31) such that the primitive /3 6 F^ =1 [fi] is exact in (9.33). In this case, the global solutions to equation (9.28) take the
134
9. Deformation Quantization
form of equation (9.30). Then, the anomaly generator in (9.32) essentially gives the non-special part of these solutions, that is, the part which has a non-zero projection on the base A/Q. We now focus on the space in which the anomalies live. Consider the following formula to be a specialized form of (9.33) r
A
1 v
= Yl <
A
T
"«' o 53 bi>Vi l
;=i
A v
i-
(9.34)
t.i=i
Since equation (9.34) is antisymmetric, the 6;J are, by definition, equivalent to m (kj) in the space of 2-cocycles. From experience, we know that the m belongs to the space of 1-cocycles, consequently the 6,j belong to the set of Casimir operators on A. The 3-tensor 8 A is a useful tool in determining the closed 3-form a (8 A):
j=\
with Wj = a(6vj)
V
i=l
/
€ 2^=2 [0].
It is a relatively simple exercise to find out under what conditions equation (9.34) corresponds to a certain cocycle. Consider the relation 8 A = 8T, in which T is a tensor. By applying (9.35) to this relation, we have a(8A)
= a (ST).
Owing to some properties of T, this reduces to a (8 A) =
da{T).
a (8 A) is a closed three-form that is exact and well-defined in the formula a [8 A) = dp.
(9.36)
/3 is still defined as in equation (9.29). Using the formula (9.36), we verify that the 2-cocycle is indeed $ = -yp
- A.
(9.37)
135
9.7 Cancellation of Global Anomalies
Recall that the global anomalies are described by $ . Thus, we have arrived at a more descriptive way of writing the anomalies. In fact, there is a more detailed form for (9.37) involving the conformal fields fii, namely r
$ = Sfi + #*// + J2 CHZi
A Z
i-
(9-38)
To obtain this result, we have proceeded as follows. First, using the formula (9.25) for 1-cocycles, we find the representation for the fields /z,-:
( 9 - 39 )
M. = £ <%*> - * * # .
where the Qj belong to the set of Casimir operators, and / ; € Yk- The 6,j, introduced in formulas (9.34-35), are related to the c^ by the formula , ij
_ J Cij for s + 1 < j < r, 1< i < r for 1 < j < s, 1 < i< r.
~~ { 0
As a consequence, a
( N ~ 12 biJvi\
dfj
for 1 < i < r
= | - Ei=i Cji H + df, for 1 < i < r.
Thus, the condition (9.36) can be reduced to the form
dP = E U "i
Ad
fi
- E;.;=. + i en d1} A d1{ + E U + i dn A dfi.
(9.40)
Since the 2-forms a;, are closed, the 3-form on the left hand side of (9.40) is exact. Its primitive 2-form /3 is P
=
Ei=l Ji^i
~ 2 Ei,j=5+1 ciilj
A
li
+ E U + i 7. A dfi.
CQ 4.1 1 (yAi)
The next step is to substitute (9.39) for /i,- into (9.34). This gives A
= \ Ei,i = s + i CijVi A Uj + E,r,J = s + i Cijvt A * * 7 i
- E U «•• A * * # .
(9.42)
136
9. Deformation Quantization
Finally, substituting (9.41) and (9.42) for /?, and A into (9.37), we see that the global anomaly $ manifests itself in equation (9.38). For non-simply connected fi,-, the conclusions are pretty much similar provided that we add to the right hand side of (9.38) the term
V
i=l
/
i=a+l
Note that the 0; are 1-forms satisfying r
J2 W A Oi = dO. i=i
The Schwinger terms arise as an addition to the Poisson bracket in (9.27) and have the familiar look of the anomalies to field theorists. The situation is somewhat different in the case of global anomalies: the last summand in equation (9.38) absorbs local anomalies and thus prevents them from arising under the Schwinger terms.
9.8
Global Anomalies Induced by t h e Jacobi Identity
The purpose of this section is to solve Problems 4-6. The Jacobi identity (9.10) gives rise to global anomalies whenever the condition spelled out by equation (9.9) is violated. In what follows, we make explicit what is meant by global anomalies induced by the Jacobi identity. An antisymmetric 3-form is said to be an anomaly if it satisfies the condition 6 5 = 0,
(9.43)
and if equation (9.12) is solvable. In terms of the 6-function, the condition (9.12) takes on the form 6$ = B. (9.44) What we have done so far is to exchange the global anomaly $ with that of B. Local anomalies, in this curent framework, are just local coboundaries of the operator S of degree three, while global anomalies are obviously global
9.8 Global Anomalies Induced by the Jacobi Identity
137
3-coboundaries. In general, one can expect (9.43-44) to be local anomalies whenever, at each point k £ A, the isotropy condition B{ki, kit kn) = 0 (i,j,n
= 1, •••,!•)
(9.45)
is trivial. The local character of the global anomaly (9.43-44), under the condition (9.45), is actually a rare occurrence. However, it may provide the first known example of a global anomaly degenerating into a local one, and this alone warrants further investigation of this peculiar phenomenon. We are interested in the global solvability of equation (9.44). Our strategy will be to show that (9.44) can be reduced to (9.41) with adequate use of relations of degree less or equal to one, by which we mean forms of the type (6A)(s)
= -6(A(s)).
(9.46)
For an explicit look at A, we refer the reader to equation (9.34). In order to relate (9.44) to the set of 2-forms £?, and vector fields /?,_,, we write the relations Bi = B(ki),
Bn =
Bfakj);
with Hamitonian Bij -V*dhtJ.
(9.47)
Furthermore, the explicit form for the 2-cocycles which are isotropic to Bi is *
= a ( E ; = i Wdhn
A Vj - Bi) - Ei=i
^-W,-.
(9.48)
The Bi and 5,j are cocycles of degrees 2 and 1 respectively. The Casimir operators (hij + hji) aii = 2 ' yield the conformal fields h^ = an + h°i3. (9.49) We now work out the anomalies for the simply connected case, (i.e. the fi; are simply connected.) This means understanding the role played by the
9. Deformation Quantization
138
2-forms a; in equation (9.48). Applying the left and right hand sides of equation (9.44) to the A;,- gives rise to a new formula for the global anomalies 6$i = -Bi,
(9.50)
where we have used the identity $ ; = $ (fct). Obviously, a necessary condition for the solvability of equation (9.44) is that equation (9.50) be solvable. Now the 2-forms a,- come into play: the global solvability of (9.50) corresponds to the assumption that the classes of the a, are zero in H2 [Cl\. On the basis of this, we can choose the h^ from (9.47) in such a way that the <x,j and h®j have representation as in equation (9.49). When this is achieved, the form a,- can be written as a, = dfii,
(9.51)
or equivalently, (9.52) Notice that a° corresponds to the first summand in equation (9.46); these 2-forms are independent of any choice of hij. On the symplectic leaf Q we choose basic 2-cycles K\, • • •, Kn which can be thought of as adjoint to the 2-forms Wi, • • • toy.
L
UJ
— 6{j
Hence, according to (9.52), the condition (9.51) can be reformulated as (9.53) JK
i
j=i
\
i=i
)
One consequence of equations (9.47) and (9.49) is the freedom they give in adding arbitrary operators, particularly those of the ctJ type. Recall that these belong to C (A), the set of Casimir operators on the phase space A. In (9.52), the <7,j could be replaced by a' — a^ — Cij\ the cr,j are fixed by (9.52) while the C;J are arbitrary. This allows one to write a uniquely determined relation: s
J2 <Tij [ui A UJ] = 0.
(9.54)
9.8 Global Anomalies Induced by the Jacobi Identity
139
Later, we shall make appropiate use of this. When the global anomaly B satisfies (9.51), it has a topological defect in the form of an s X s matrix a =
*,• = -** A + E A «»i.
(9-55)
3=1
which, in some ways, is a globalization of (9.30). Our next order of business is to investigate the behavior of the global anomaly (9.43-44) in A. As we have seen in previous sections, the procedural approach will be to reduce (9.44) to an equation of forms using, among other techniques Su(ki) = -Bu (B — SOJ in this case). Recall that u> has the dual role of being a symplectic 2-form on A, and also a generator of anomaly (the most vivid illustration of which is contained in equation (9.32)). Using the equivalence relation u(ki)
= *.-
we deduce that u> can be reconstructed from $,• in (9.55). There is a hitch however, namely, that the hij be antisymmetric. For the global anomaly B to exist, the topological defect, a, must be trivial. When this is the case, the antisymmetric property of the h^ allows us to choose the Hamiltonian in a such a way that the a, are exact. Thus, the explicit form of the anomaly generator ui is
w = - £ > , - A **£,- + - J2 h<Jv>
A v
i-
( 9 - 56 )
As a consequence, the 3-form a (B - 6UJ)
= a(Bi) - E L i ».• A Bi - \ E i i = i (",- A VJ A ty*dhij - E L i w,- A Bit
(9.57)
140
9. Deformation Quantization
corresponds to the global solvability of equation (9.44). We now list the conditions under which the cocycle B is a global anomaly. 1. First, B must satisfy local anomalies condition; 2. The a; in (9.52) ought to be trivial in H2 [O]; 3. The topological defect
— 6u) = dfis-
When these conditions are all satisfied, we obtain the formula for global Poisson bracket deformation: r
/
1 JL
\
(9.58) where the 1-forms /?, were borrowed from equation (9.51). If we consider the Bij as being strictly Hamiltonian, then the conditions (1) to (4) apply to the case of non-simply connected fi;. In the space of 3-forms on A, the global anomaly is given by the formula B
= 8M + V*p + Zt3k=s+i C'jkZi A ZJ A zk + EJJ-.+1 * * ( " « ) « ' A * , -(V*T + E?=i Vi A $* n - \ Y.h=i Vi A Vj A $ * rt]) - ( £ U + 1 Zi A **0,- + Ef=i Zrj=.+i
«,• A
Zi
V,-t>y>
A <S>*0,3) .
A few definitions are in order. M belongs to the space of 2-forms on A, U{j are elements of Z fc=1 [12], T <E Yk=2 [fi], the r, 6 Yk=2 [fi], while 0,-,- £ r fc=1 [ft]. Their explicit forms are revealed by d.Tij = 0 dri = EJ=i «,• A r,, dr
(9.60)
= E;=i w>' A Tj,
and dOu '•j dOi
= 0 = E-=i «; A On.
(9.61)
9.9 Generalized Wess-Zumino Consistency Condition
9.9
141
Generalized Wess-Zumino Consistency Condition
Our primary goal here is to compute the cocycles and coboundaries of the 6 operator: $ : M " ' p + 1 ->
Mq+2'p.
It acts by the formula r
(tiv)h-iP
=E
W
J
A
'fci-.W'
j=l
where the w,- are essentially the same as before, and Mq'p denotes the space of p-forms on fi. We have the condition J2 [Vi A ut] = 0,
(9.62)
l
from which we define the 6 on the direct sum
Mk = 0
M"'\ 8: Mk -> Mk+1.
p+q=k
Using (9.62) gives 62 = 0, dS - Sd = 0. With these descriptions in hand, we can proceed to write a generalized version of the Wess-Zumino consistency condition [33]. The Wess-Zumino consistency condition represents a simple test of anomalies detection. In our context, this condition corresponds to the following linear chain of obstructions which are essentially towers of obstructions in double complexes: dr]o,k dr)i,k <%_ p di]k^i dr]k
—0 = Uj = UJ — UJJ = UJ
A 7/o A j]k-p A rjk-2 A T)k-i.
(9.63)
142
9.10
9. Deformation Quantization
References
[1] Baadhio, R. A.: Global Anomalies Induced by Deformed Quantization, University of California, Berkeley Preprint UCB-PTH-94/20 and LBL-35952, December 1995. Submitted for publication. [2] Dirac, P. A. M.: T h e Principles of Quantum Mechanics, Clarenden Press, Oxford 1930. [3] Weinstein, A.: Lectures on Symplectic Manifolds, Conf. Board Math. Sciences Reg. Conf. Ser. Math. Vol. 29, American Math. Society, 1979 Providence, R. I. [4] Kostant B.: Lectures Notes Mathematics 170 (1970) 87. - Sympo. Mathematics 14 (1974) 139. [5] Bayen, F., Flato, M., Fronsdal, C , Lichnerowicz, A., and Sternheimer, D.: Deformation Theory and Quantization. I. Deformations of Symplectic Structures, Ann. Physics 111 (1978) 61-110. - (Idem): Deformation Theory and Quantization. tions, Ann. Physics 111 (1978) 111-151.
II. Physical Applica-
[6] Moyal, J.: Quantum Mechanics as a Statistical Theory, Proc. Cambridge Phyl. Society 120 (1949) 99-124. [7] Weyl, H.: The Theory of Groups and Quantum Mechanics, Dover (1931) New York. [8] Lie, S.: Theorie der Transformationsgruppen, Leipzig, Teubner 1890. [9] Weinstein, A.: Deformation Quantization, 789, 46 Years 1993-1994. To appear.
Seminaire BOURBAKI
[10] Gutt, S.: Equivalence of Twisted Products on a Symplectic Letters Math. Physics 3 (1979) 495-502.
Manifold,
[11] Vey, J.: Deformation du crochet de Poisson sur une variete symplectique, Comment. Math. Helvetica 50 (1975), 421-454. [12] De Wilde, M. and Lecomte, P.: Formal Deformations of the Poisson Lie Algebra of a Symplectic Manifold and Star Products. Existence, Equivalence, Derivations, in Deformation Theory of Algebra and Structures
9.10 References
143
and Applications. Eds. Hazewinkel, M. and Gerstenhaber, M. Kluwer Academic Publications (1988) 897-960. [13] Witten, E.: Topological Tools in 10-Dimensional Physics, Int. Journal Modern Physics A, vol 1 no. 1 (1986) 37-64. [14] Witten, E.: An SU(2) Anomaly, Physics Letters B117 no. 5 (1982) 324-328. - Global Gravitational Anomalies, Commun. Math. Physics 100 (1985) 197-229. [15] Baadhio, R. A.: Global Gravitational Anomaly-Free Topological Field Theory, Phys. Letters B299 (1993) 37-40. [16] Baadhio, R. A.: Knot Theory, Exotic Spheres and Global Gravitational Anomalies, in Quantum Topology, 78-90, by L. H. KaufFman and R. A. Baadhio, World Scientific, 1993 Singapore, London, New Jersey. [17] Baadhio, R. A. and KaufFman, L. H.: Link Manifolds and Global Gravitational Anomalies, Rev. Math. Physics 5 N.2 (1993) 331-343. [18] Baadhio, R. A.: Mapping Class Groups for D = 2 + 1 Quantum Gravity and Topological Quantum Field Theories, Nuclear Physics B441 Nos. 1-2 (1995) 383-401. [19] See reference [3]. [20] Arnold, V. I. and Givental' A. B.: Symplectic Geometry, SpringerVerlag, 1990 Berlin, New York. [21] Witten, E.: Quantum Field Theory and the Jones Polynomial, Comm. Math. Physics 121 (1989) 351-399. [22] Schaller, P. and Schwarz, G.: Anomalies From Geometric Quantization ofFermionic Fields, J. Math. Physics 31 N. 10 (1990) 2366-2377. [23] Morita, K. and Kaso, H.: Anomalies From Stochastic Quantization and their Path Integral Interpretation, Phys. Reviews D41 N2 (1990) 553560. [24] Morita, K.: Note on the Anomalies From Schotastic Prog. Theor. Physics 81 N6 (1989) 1099-1103.
Quantization,
[25] Panfil, S. L.: Canonical Quantization, the Dirac Problem and the Anomalies, Mod. Phys. Letters A4 N26 (1989) 2561-2167.
144
9. Deformation Quantization
[26] Krammer, U. and Rebhan A.: Anomalous Anomalies in the CarlipKallosh Quantization of the Green-Schwarz Superstring, Phys. Letters B236 N3 (1990) 255-261. [27] Kim, Y. B.: Chiral Anomalies of Higher Derivative Theories from Stochastic Quantization, Mod. Phys. Lett. A7 N30 (1992) 2861-2866. [28] Van Doren, S. and Van Proeyen, A.: Simplification in Lagrangian BV Quantization Exemplified by the Anomalies of Chiral W3 Gravity, Nucl. Physics B411 Nl (1994) 257-306. [29] Eells, J.: Singularities of S m o o t h Maps, Columbia University Mathematics Publication CU-5-ONR-266 (57)-M 1960 New York. [30] Martinet, J.: Singularities of S m o o t h Functions and Maps, London Math. Society Lecture Note Series 58, Cambridge University Press, 1982 Cambridge, New York. [31] Wess, J. and Zumino, B.: Consequences of the Anomalous Identities, Phys. Letters B73 (1971) 95.
Ward
[32] Axelrod, S., Delia Pietra, S. D. and Witten, E.: Geometric Quantization of Chern-Simons Gauge Theory, J. Diff. Geometry 33 (1991) 787-902. [33] Atiyah, M. F.: T h e G e o m e t r y and Physics of Knots, Cambridge University Press, 1990, Cambridge, New York.
Chapter 10 Chiral and Gravitational Anomalies Whenever symmetries of a classical theory are broken by quantum corrections, anomalies are bound to arise. Anomalies originating from global gauge or diffeormorphic (i.e. non-identity component general coordinate) transformations are referred to as global gauge or global gravitational anomalies. This class of pathologies is far more subtle and difficult to evaluate, especially in higher dimensions, than the ones we are about to introduce: chiral and gravitational anomalies. We will dedicate more than one chapter to global anomalies, but before we do so, it is essential that we give some background on the subject. This is the goal of this chapter. The upcoming presentation is self-contained. A celebrated example in the study of classical anomalies (as opposed to global anomalies) is that of the so-called triangle diagram endowed with two vector currents and an axial vector current. The lack of conservation of the axial vector is manifest whenever Bose symmetry and vector current conservation are required. As a consequence, a breakdown of chiral symmetry, most notably in the presence of external gauge fields, is exhibited. Therefore, even though requiring Bose symmetry and vector current conservation may be important for internal consistency reasons, the procedure gives rise to anomalies and challenges the theory's overall consistency. The existence of such anomalies has led to an understanding of 7r° decay, and later, to the
145
146
10. Chiral and Gravitational Anomalies
resolution of the [/(l)-problem in Quantum Chromodynamics (QCD) [1]. Anomalies of this type essentially represent the breakdown of global axial symmetries; their existence in no way jeopardizes unitarity or renormalizability. In contrast, anomalies which arise when chiral currents are coupled to gauge fields are of a more serious nature since theories which are not gauge invariant suffer from nonrenormalizability. In what follows, we shall consider non-abelian gauge theories coupled to fermions in some representations Ra of the gauge group G to which Feynman's rules apply. Anomalies for gauge currents in this class of theories are fairly well known: they are proportional to a purely group-theoretic factor times a given Feynman integral. To get rid of the anomaly, the sum of the group theoretic factor over various representations ought to vanish, as is readily shown by the formula: / = Y, STvRaLRbLRcL RL
- J2 STrRRRbRRR
= 0.
(10.1)
RR
Here, Ri (repectivelyi?fi) stands for the representations of G carried by the left (respectively right) handed fermions; STr denotes the symmetrized trace over the group generators involved. Formula (10.1) has a simple yet farreaching consequence, namely, any given theory will not be gauge-invariant whenever (10.1) is not trivial. This fact is well demonstrated in dimension four where, for instance, the remarkable cancellation of anomalies within the framework of the Glashow-Salam-Weinberg theory is a direct consequence of the vanishing of (10.1). The result arises by writing the known quarks and leptons in terms of left-handed Weyl fermions. It then follows that for each family (there are three of them), we can write down SU(3) x SU(2) X U(l) quantum numbers as (3,2) 1 / 2 © ( l , 2 ) _ 1 / 2 © ( 3 , l ) 1 / 3 © (l,l)x.
(10.2)
Several facts about formula (10.2) are of interest. Firstly, the U{\) hypercharge anomaly cancels because of the contribution of every member of the family. Secondly, it is a source of motivation for understanding the quantum numbers of the quarks and leptons, thus incidentally renewing our interest in grand unified [2] and Kaluza-Klein theories [3]. Theories that allow vector currents to be replaced by the energy-momentum tensor are theories in which the axial vector symmetry is violated in the pres-
147 ence of external gravitational fields. In order to illustrate this, we pick a U(l) gauge field coupled to some V — A currents with external gravitons. We then have a tensor proportional to the trace of K, the generator of U(l). The case Tr K = 0 is of importance given that it is impossible to simultaneously maintain the conservation of the energy-momentum tensor and the U(l) gauge symmetry, as shown in great detail by Alvarez-Gaume and Witten [4]. Therefore, to cancel the anomaly requires that Tr K vanish for the generator of a given U(l) factor in the gauge group G. The quantum numbers in (10.2) do indeed satisfy this consistency condition. As a general rule, Tr K can be shown to vanish whenever the theory's low energy group originates from a bigger, simple or semi-simple unified gauge group. The reason for this is that the trace of any generator always vanishes for a compact, simple group. The standard SU(3) x SU(2) x f/(l) theory to which we alluded earlier is among the class of theories known to satisfy the general rule on the vanishing of Tr K, and it applies to each of the three families. We now focus on anomalies in higher dimensions. In dimensions 2n, chiral fermions coupled to gauge fields have localized anomalies because of the requirement of Bose symmetry for external gauge lines. Furthermore, each vertex contains a factor Ra, and this alone implies that the group theory factor appearing in the anomalies can easily be extracted; explicitly: I Tr = J2 S Tr (R£ • • • RaL"^) - £ RL
S Tr ( 7 $ • • • R%"+1).
(10.3)
RR
Cancelling (10.3) when n > 2 is far more complex than cancelling (10.1). It is therefore of no surprise that the set of anomaly-free chiral representations of fermions is considerably reduced in dimension four. Accordingly, it is only fair to say that anomalies in general restrict the number of potential consistent theories in four dimensions, but even more so in higher dimensions. In dimensions Ak + 2, for instance, the energy-momentum tensor for chiral fermions is anomalous. Alvarez-Gaume and Witten [4] found that theories defined in even dimensions often suffer from anomalies, localized in gauge currents (the so-called gauge anomalies), and in the energy-momentum tensor (i.e. anomalies of a gravitational nature). Additional anomalies are mixed anomalies that correspond to graphs containing gravitons and gauge fields. A theory is said to be consistent when all the anomalies have been cancelled.
148
10. Chiral and Gravitational Anomalies
This leaves us with a very small class of theories that are chiral and anomalyfree in even dimensions. When taking supersymmetry into account [21] there are, to date, only two types of anomaly-free theories: N — 2 chiral supergravity theory in ten dimensions and N — 1 super Yang-Mills theory coupled to N = 1 supergravity, also in dimension ten. The latter has Eg x Eg or SO (32) gauge group and the theory is called the heterotic superstring theory [5, 6]. The anomaly cancellation coupled to the real possibility that this class of theories provides an ultraviolet finite quantum theory of gravity are perhaps the driving forces that sustain our interest in string theory. Physicists working in the field of anomalies have several tools available today in furthering their investigations. Among them is index theory [7] and related applications, exhibiting in the process the ever-increasing interaction between mathematics and physics. The interaction in question began in the mid-seventies with the discovery by Jackiw et al. [8] that the global U(l) axial anomaly could be understood in terms of the Atiyah-Singer Index theorem. The basis for this was the realization that the expectation value of the divergence of the axial current can be related to the spectral asymmetry of the Dirac operator. The authors, in reference [8], demonstrated that the spectral asymmetry is indeed concentrated in the space of zero modes, some of whose properties are determined by index theorem arguments. Independently of obtaining the correct form and normalization of the U(l) anomaly using index theory, the approach provides a powerful way of analyzing some subtleties about the dynamics of fermionic gauge fields. Later, even more powerful forms of the index theorem, namely index theorem for families of elliptic operators [9, 10], were used to provide an understanding of the non-abelian anomaly as well as that of gravitational anomalies [11, 12, 13]. These results were obtained within the Euclidean formulation of field theory. Yet, it is now also possible to understand them from an algebraic [14, 15] or topological [16] point of view in the Hamiltonian formalism. The study of global gauge [17] and gravitational [18] anomalies requires the use of spectral flows and index theory for families of operators and a good deal of algebraic topology, as we shall see in upcoming chapters.
10.1 Spinor Representations of Lorentz Groups
10.1
149
Spinor Representations of Lorentz Groups
Chiral fermions are among the basic ingredients in the study of anomalies, and this alone warrants the present review on spinor representations. To avoid infrared problems we will work on Euclidean space compactified to a sphere. This review is not exhaustive. The interested reader is referred to Cartan's book Theory of Spinors, Dover, New York, 1981 for a detailed study of spinors. We will be concerned primarily with spinor representations of Lorentz groups in Euclidean and Minkowski spaces. Our starting point is the Clifford algebra in dimensions In. A flat metric gkl with signature (t,s) reads: {Tk,T1} = 2gu\ (10.4) k, I = 0, • • •, 2n — 1. A Dirac spinor is a field which, under an infinitesimal S0(t,5) transformation changes as EH +t
r
_ i j r * >r /] = rv--,r*t.
=
( 10-5 )
In even dimensions, f is a matrix which anticommutes with rfc:
f = o r T 1 •••r 2n -\ where we think of a as a phase chosen in such a way that f2 = 1, that is to say a2 = ( - l ) ( ' - ( ) / 2 ; a* = ( - l ) ( - ' ) / 2 a . Notice that {f, Tk} is trivial and so f is free to commute with the generators of the Lorentz group. Using T, we define two projection operators:
P± = | ( i ± r ) , *±
=
p±*-
We say of
150
10. Chiral and Gravitational Anomalies
The Clifford algebra in (10.4) has a unique faithful representation of dimension 2n. Hence Tk, Tk*, —Tk^ are all related owing to the fact that they satisfy the same anticommutation relation (10.4). Next, we need to find a charge conjugate Dirac spinor. To this end we write down the matrix M as
(pk'Y = ME*' M -1 ,
(10.7)
tfc = c # = A T 1 $*.
(10.8)
which gives c
From (10.6) we deduce that $ and $ have identical Lorentz transformation properties, which are:
When c2 = + 1 , the matrix M gives rise to projection operators (1 ± c ) / 2 and to the following Majorana and anti-Majorana fields:
** = (1 + c) y *E =
(1 - c) | .
(10.10)
A question of interest is whether c and T commute. To answer, we begin by writing F in terms of the Efc'; this is readily done by the formula: f = ( - 2 ) " a S 0 1 S 2 3 ••• S 2 "- 2 ' 2 "- 1 ,
(10.11)
which yields cf
= ( - l ) ^ » / 2 f c.
(10.12)
Let us investigate the case in which (s —1)/2 is odd. According to reference [4] c flips the chirality of fermions. However, for (s — t)/2 even, c does not flip the chirality. In dimensions 4k, {c, f } = 0 whereas in dimensions 4k+ 2, c, T = 0. In order to extract the physical importance behind these results, one considers the solutions to the massless Dirac equation in dimension 2k. A free, massless fermion with definite momentum moving in 2k — 1 dimensions possesses a space wave momentum function which is none other than
r°u{P) = r^uip).
(10.13)
10.1 Spinor Representations of Lorentz Groups
151
The helicity operator reads
h = s 12 s 34 • • • E2*-3-2*-2 ~ r 1 r 2 • • • r2fc~2.
(10.14)
Combining (10.12) to (10.13) gives
hu{P) = hr°r°u(p) = r° h r2k~l u (P)
(10.15)
= Tu(p). Formula (10.15) states the well known fact that helicity and chirality are unique manifestations of one another in dimension four. Accordingly, in dimensions 4k, Minkowskian space charge conjugation flips helicity, whereas this does not hold true in 4k + 2 dimensions. Euclidean 4fc-dimensional spinors are similar in several ways to (4k + 2)dimensional Minkowskian spinors, and to 4k + 2 dimensional spinors like the one in 4k dimensional Minkowskian space. Anomaly wise, an interesting fact about dimension 4k + 2 is the triviality of the term c, f , and the consequent imposition of the Weyl-Majorana condition. To impose, for example, a simultaneous Weyl-Majorana condition would require dimension 2 mod 8. Results of this type are important if one is to build minimal on-shell multiplet representations of higher dimensional supersymmetry and supergravity theories. Massless vector fields in dimension ten, for instance, represent eight degrees of freedom - the massless state group in dimension D = 10 is SO (8) - and the state created by a ten-dimensional vector field transforms like a vector under SO (8). To comment further on the simultaneous imposition of the Weyl-Majorana condition, consider a supersymmetric multiplet together with a vector field. Imposing the condition in question requires first finding a fermionic field whose degree of freedom is similar to that of the multiplet. In dimension ten, any given Weyl fermion possesses exactly 24 = 16 degrees of freedom. Thus, imposing both the Majorana and Weyl conditions reduces the number of degrees of freedom to eight. As we have explained, such conditions are necessary for the overall internal consistency of the theory. Anomalies in this case often act to substantially reduce the theory's number of degrees of freedom.
152
10.2
10. Chiral and Gravitational Anomalies
T h e Group Invariant /TT
The evaluation of S Tr RL • • • RR is essential for any thorough analysis of the anomalies. To evaluate this term amounts to computing the quantity Tr r Nn, where AMs a matrix in the representation r of the Lie algebra of interest. It is important to differentiate the cases in which n is even or odd. For even n = 2k, Tr N2k is positive definite; thus, a theory with only left-handed fermions in these dimensions will always be anomalous. A celebrated such instance is the N = 1 super Yang-Mills theory in ten dimensions. The theory is indeed anomalous for any gauge group G [4]. For odd n = 2k+l, we have a purely left-handed anomaly-free representation. A detailed analysis of groups and their representations is often necessary when investigating anomalies. Classically, one works with either of the following representations: (i) the real symmetric (also called simply real) representation in which the group matrices in the representation are equivalent to their complex conjugates. Also noted in this case is the possibility to define a basis in which the Hermitian generators of the representation are purely imaginary; (ii) the real antisymmetric (or pseudo-real) representation. This case is somewhat similar to (i) except that there is no basis in which the Hermitian generators can be chosen as purely imaginary matrices. A readily available example is the two-dimensional representation of SU(2). The Hermitian generators are the Pauli matrices: cr,/2, i = 1,2,3 C =
<7 2
c' = —c.
In these, there is no possible basis in which the <7;'s could be defined as purely imaginary. (iii) the complex representation. In contrast to (i), we encounter complex representations whenever the group matrices in the representation are not equivalent to their complex conjugates. We use these cases to illustrate that Tr N2k+1 vanishes for real or pseu-
10.2 The Group Invariant Iyr
153
doreal representations, i.e. TrN2k+1
= Tr (iV 2 ^ 1 )* = Tr (7Vt)2fc+1
(10.16)
= - T r (c" 1 Nc)2k+1 = 0.
Next, we focus on some given groups and their representations in dimensions 4k. We begin with SO (2A; + 1). This group has two basic representations: a vector representation which is obviously real, and a spinor representation. The later is unique. Moreover, it is equivalent to its complex conjugate. This leaves open the possibility that the SO (2k + 1) spinor representation is either real or pseudoreal. Incidentally, it is worth noticing that all representations of SO (2A; + 1) can be obtained using suitable tensor products of these two representations. Therefore, all representations of SO (2k + 1) are either real or pseudoreal. Hence, a theory whose gauge group is SO (2k + 1) is anomaly-free. Similar conclusions apply to the symplectic group Sp(n), and the exceptional groups G2, -P4, E7, and Eg- The fundamental representations of G2, F4, Ej, and Eg have respective dimensions 7 (real), 26 (pseudoreal), 56 (pseudoreal), 248 (real); all other representations are either real or pseudoreal. Potentially anomalous candidates are SO (2k), SU (A;), and E$. Our previous discussion of spinors in Euclidean space tells us that SO (4k) spinors are not complex, and this leaves SO (4k) anomaly-free. Concerning SO (4k + 2), the situation is far from being safe. For one thing, SO (4k + 2) complex conjugation exchanges the spinor representations, thus indicating the complex character of their nature. As for SU(fc), for different values of k, they suffer from anomalies in virtually all dimensions. E$ is an exceptional group with 78 generators and complex fundamental representations 27, 27. There is an anomaly in the 27 representation which cancels in dimension four [5, 6], but it does not in dimensions 4k, whenever k > 2. One works out the EQ anomaly by embedding E$ in E6 D SO (10) x U ( l ) . The 27 representation now reads 27 = 1(4) + 10(-2) + 16(1);
154
10. Chiral and Gravitational Anomalies
and the anomaly is now computed with respect to SO (10) X U (1).
10.3
The Green-Schwarz Anomaly Cancellation
In this section, we will discuss the Green-Schwarz anomaly cancellation for the Es x Es or SO (32), the heterotic superstring theory [5, 6]. Consider a Lie algebra Q whose maximal subset of commuting generators (the socalled Cartan subalgebra) is 5 , , i = 1, • • •, r; where r is the rank of Q. The eigenvalues of the vector B on the states of the representation B\a. > — a\a > labeled all possible representations. We think of a as a weight vector with r components. The formula Tr K(g) = £
expix
• a,
(10.17)
a
gives the character of g in the representation K, where K is a group matrix. Applying the observation that any group element K can be represented as exp N, one deduces that the computation of equation (10.17) reduces to computing all the traces of the form TriV". Weyl, in reference [19], has provided us with a general formula aimed at computing the form (10.17). Unfortunately, as this formula stands, it has little relevance to the anomaly cancellation. There is, however, a fruitful approach to the computation of the character (10.17). Essentially, the procedure involves computing (10.17) in terms of the character of the fundamental representation of Q. For some representations R, Tr Nn can be written in terms of traces and products of traces in the fundamental representations. Following Green and Schwarz's notations in [5, 20], we denote traces in the fundamental representation by lower case, i.e. tr Nn: TrNn
= e x triV n + e2 triV2 triV n " 2 + •••
(10.18)
The number of independent coefficients e, needed to be taken into account in the analysis of anomaly cancellation thus depends on the number of irreducible traces trNn; these are traces that cannot be factorized into products of lower-order traces. In fact, this number depends on the Casimir operators of the group Q. Group theoretic facts show that the number of independent traces for a simple Lie group is given by the rank of G. For instance, SU(3) enjoys two
10.3 The Green-Schwarz Anomaly Cancellation
155
independent Casimir operators which are related to tr(7V2) and tr(N3). For simple enough representations, one can calculate the coefficients of (10.18) without much trouble. We are interested in computing Tr(JV n ) for the adjoint representation of SO (n) or Sp(n) mainly to illustrate the discussion above. The standard prescription for doing so is to first build the adjoint representation from tensor products of the fundamental representation. In the fundamental representation of SO(rc) or Sp(rc) lives a generic group, call it Qij, whose action on the representation space reads: For SO(n), the adjoint representation is equivalent to the antisymmetric tensor irreducible representation. We obtain as a consequence of this a new transformation, Ui -» g (QikQn ~ QuQik) tkt-
(10-19)
In the symplectic case, the group Sp(n) has an invariant, an antisymmetric bilinear form which has to be taken into account. The adjoint representation that follows from the symmetric second-rank tensor is: ta - • g (QikQn + QuQik) tki-
(10.20)
We are now in a good position to determine the character of the adjoint representation in terms of the character of the fundamental irreducible representation. It is encoded in the formula: TvQ = l- [(trQ) 2 + e t r Q 2 ] ;
(10.21)
where e = +1 for Sp (n) c = - 1 for SO (n). A few comments about (10.21) are in order. Note that Q is equivalent to expTV, and so we could expand both sides of (10.21) in power series in TV in order to extract the desired coefficient e,-. The procedure yields TrJV 2 = (A + 2c) trJV 2 , TrA^4 = (A + 8e) tr N4 + 3 (tr N2)2, TrAf6 =(A + 32e) trA" 5 + 15 tr A^2 tr A^4.
(10.22)
156
10. Chiral and Gravitational Anomalies
What is the analog of formula (10.22) for Eg! The fundamental irreducible representation of E$ decomposes under its subgroup SO (16): 248 = 120 + 128 (+ ). 120 is the adjoint of SO (16), and 128(+) is none other than the positive chirality spinor of SO (16). E& and SO (16) both have rank eight, so the weights of their representations are eight-dimensional vectors. Let e, • tj = 8{j denote the standard basis; the weight of the fundamental irreducible representation of SO (16) are ± e t , i = 1, • • •, 8. Consequently, the weights of the adjoint representation read: ± c,- ± eh i < j , (10.23) to which should be added eight zeros corresponding to the Cartan subalgebra. For the spinor, we have: ta = ± 1
(10.24)
A given matrix N in the adjoint representation of Eg can be written in terms of eight parameters, Xi,--- ,x$. Thus, the character of the 248 representation of E& simply corresponds to the sum of 120 and 128 of SO (16): Tr fi8 Q = Zi<j exp ( ± x ; ± Xj) + £ , exp ( | £|j=i ea xa) = | [(E,- 2cosh*;) 2 - (£,- 2cosh 2Xi)]
(10.25)
+ | [n?=i 2cosh f + n f = i 2 s i n h f ] . The following expansion applies to E$: TriV 4 = ^ ( T r ^ ) 2 , TrAT6 = ^ ( T r i V 2 ) 3 .
(10.26)
The relations (10.22) and (10.26) are crucial ingredients in the GreenSchwarz anomaly cancellation, to which we now dedicate the reminder of this expose. The only groups for which the anomalies cancel are SO (32) and Eg x .Eg [5]. As pointed out earlier, these groups are the internal symmetries required to describe the superstring in ten dimensions. Anomalies of this scale manifest themselves primarily in hexagon diagrams. A general rule for
157
10.3 The Green-Schwarz Anomaly Cancellation
2n dimensions is that the first dangerous diagram is an (n — 1-gon). This is easily understood from the fact that the anomalous divergence of a YangMills gauge current consists of d
. r
~ c « - « » tr (A» F W M • • • F,2n_^n)
.
(10.27)
This algebra in turn is a function of the chiral fermions which go through the loop. Analogous gravitational anomalies can occur, particularly for odd n, in the energy-momentum tensor. The standard N = 2 supergravity theories in ten dimensions are known to be anomaly-free [4]. In one of the theory's incarnations, namely the 2A, this is due to the trivial cancellations between contributions of left-handed and right-handed fermions, whereas in the case of 2B, it is a consequence of highly non trivial cancellations discovered by the authors in reference [4], N — 1 theories present even more difficult challenges. The anomaly cancellation in type / SO(32) superstring theory was discovered by evaluating hexagon loop diagrams in string theory [20]. Consider the effective action £eff of the low-energy expansion of string theory. £eff is defined over fields of massless modes, which we write as $ 0 ; it is obtained by integrating out all the fields associated with massive string modes, the m: e .£ 5 f f (* 0 )
_
£)$me.-£..,l.,(»0.*m)-
(10.28)
£eff can be expanded in a series of operators of increasing dimension. The leading terms correspond to the point-particle theory for N = 1, D = 10 super Yang-Mills theory coupled with supergravity. The higher order terms represent string corrections whose effects are indeed suppressed at low energy E\ovl by powers of E\ovl/m, where m is the characteristic mass scale of the string. Green and Schwarz have found that the theory is known to be anomalous for every Yang-mills group, unless terms not present in the minimal point-particle theory are taken into account [5]. Massless fields consist of a super Yang-Mills multiplet and a supergravity multiplet. The super Yang-Mills multiplet contains vector fields AaM and left-handed Majorana-Weyl spinor fields xi- The index a takes n values corresponding to the generators of the Yang-Mills group G; incidentally, n — dimG. The supergravity multiplet contains a graviton g^, a second-rank antisymmetric tensor field B^, a scalar p, a left-handed Majorana-Weyl gravitino ^ L , and a right-handed Majorana-Weyl spinor A#. These fields
158
10. Chiral and Gravitational Anomalies
are all singlets of the gauge group. The chiral spinors \i Pi an< ^ ^ going through the hexagon (and higher) diagrams give rise to gauge, gravitational, and mixed anomalies. The use of differential forms is an efficient means to investigate the anomalies. To gauge potential, we associate 1-forms: A _ A M A dx u> = w^dx11.
(10.29)
Up, a spin connection, is a 10 x 10 matrix in the fundamental representation of the Lorentz algebra SO (9,1); the Aa are n x n matrices in the adjoint representation of the Yang-Mills algebra, that is, they form the algebra's structure constant. The two-form fields strength reads F R
= \F^dx^ = \R^dx»
A dx" = dA + A2 A dx" = du + w2.
.
. (W M)
-
A few notations are in order. F stands for the Yang-Mills field strength while R denotes the Riemannian curvature tensor. Infinitesimal gauge transformations with parameters a and /3 satisfy the relations: 6A
= da + [A, a] = d/3 + [u, /?],
SUJ
(10.31)
and 6F
= [F,a]
SR
= [R,/3].
(10.32)
It is now useful to introduce the Chern-Simons 3-forms w3y
=
TT(AF
W3L
= tr{uR
|A3)
-
-
fw 3 ),
(10.33)
which have the properties du3Y = TTF duzr, = tvR2,
(10.34)
and SUJSY
=
fiu3L
=
— duj'2Y
~du\L.
(10.35)
10.3 The Green-Schwarz Anomaly Cancellation
159
According to the notations of the authors in [5], the u\L and w\L stand for the two-forms while the superscript 1 indicates that they are linear in the infinitesimal parameters a and /? respectively. Anomalies arising from loops of chiral fermions are easily detected by 12-forms. From this, one extracts the actual 10-forms whose integral is the anomaly. This is done by replacing F —» F + a and R —> R + (3 and by extracting in a second step the terms linear in a or /3. The final result is I 12
oc - £ T r F 6 - ^ T r F 2 (4tri? 4 + 5(trtf 2 ) 2 ) + j-4TrFHvR2 + ( i + ==£§f) (tri? 2 ) 3 2
(10.36)
i
+ (f + ^ ) t r f l t r ^ + ( = g f ) t r # . The big numbers originate from characteristic classes. We will say more about these in the following chapter. For a detailed account of 12-forms characterizing all the gauge, gravitational and mixed anomalies due to x, $, and A loops, the reader is referred to [4]. Meanwhile, Green and Schwarz ask the following question: is it possible for £eff to contain a non-gaugeinvariant local interaction term Cc whose gauge variation SC cancels the anomaly associated with 7i2? They discovered in [5] that this is only possible if 7i2 factorizes into an expression of the form: hi = (tiR2 + ktvF2) Xs.
(10.37)
X& is an 8-form constructed in terms of F and R. The necessary and sufficient condition for this to hold is n = dimG = 496.
(10.38)
When satisfied, the condition gives the following value for k:
and
k = - 1 ,
(10.39)
X8 = ±TrF^-^-0(TvF2)2 2 -^TrF tvR2 + |tri?4 + £(trtf2)2.
(10.40)
Factorizing 1^ in (10.37) tell us that to cancel the anomaly, we need only add the form C ~ / B A X8, (10.41)
160
10. Chiral and Gravitational Anomalies
B is a 2-form congruent to 5M„ dxu A dx" with gauge transformation SB = ulL - ±ulY.
(10.42)
The formulae (10.38) and (10.39) have only two known solutions: SO (32) and Eg x Eg. An exhaustive analysis of the artithmetic of the present finding has been carried out in reference [5] and [20]. We will not repeat them in this chapter, referring interested readers to these two references. Note that additional solutions exit, namely for [[/(l)] and Eg x [£/(l)] . Unfortunately they do not correspond to consistent string theories and this makes them less attractive. Eg x Eg describe, to date, the most promising string theory, the so-called heterotic string theory discovered a decade ago by David Gross and collaborators [6].
10.4
References
[1] Schwinger, J.: Physical Reviews 82 (1951) 664. - Adler, S. L.; Lee, B. W.; Treiman, S. B.; and Zee, A.: Physical Reviews D4 (1971) 3497. - Callan, C ; Dashen, R. and Gross, D. J.: Phys. Letters B63 (1976) 334. [2] Langacker, P.: Physical Rep. 72 (1981) 185. - Zee, A.: in Unity of Forces in the Universe, Vol. I and II, World Scientific, 1982. [3] Salam, A. and Strathdee, J.: Annals of Physics 141 (1982) 316. - Nieuwenhuizen, P.: An Introduction to Simple Supergravity and the KaluzaKlein Program, in Relativity Groups and Topology, II, Ed. R. Stora, North Holland 1985. [4] Alvarez-Gaume, L and Witten, E.: Gravitational Anomalies, Nuclear Physics B234 (1983) 269. [5] Green, M. B. and Schwarz, J. H.: Physics Letters B149 (1984) 117. [6] Gross, D. J.: The Heterotic String, in Unified String Theories, Eds. Gross, D. J. and Green, M. B. 357-399, World Scientific 1986.
10.4 References
161
[7] Atiyah, M. F. and Singer, I. M.: Annals of Mathematics 87 (1968), 485-546. Idem, 93 (1971) 119-139. [8] Jackiw, R.; Nohl, C. and Rebbi, L.: in Particles and Fields, Eds. D. Boch and A. Kamal, Plenum Press 1978. - Nielsen, N. K.; Romer, H. and Schoer, B.: Nuclear Physics B 136 (1978) 478. [9] Atiyah, M. F.; Patodi, V. I. and Singer, I. M.: Mathematical Proceedings Cambridge Philosophical Society 77 (1975) 43. -(Idem), 78 (1975), 405 and 79 (1976) 71. [10] Atiyah, M. F. and Singer, I. M.: in Proc. Sciences USA 81 (1984) 2597.
National Academy of
[11] Alvarez-Gaume, L. and Ginsparg, P.: Nuclear physics B243 (1984) 449. - (Idem) Annals of Physics 161 91985) 423. [12] Alvarez, 0., Singer, I. M. and Zumino, B: Communications Mathematical Physics 96 (1984) 409. [13] Witten, E.: Nuclear Physics B223 (1983) 422. [14] Faddeev, L. D.: Physics Letters B 145 (1984) 81. - Faddeev, L. D. and Shatashvili, S.: Math. Phys. 60 (1984) 206. [15] Zumino, B.: Nuclear Physics B253 91985) 477. [16] Nelson, P. and Alvarez-Gaume, L.: Communications Mathematical Physics 99 (1985) 103. [17] Witten, E.: An SU (2) Anomaly, Phys. Letters B117 (1982) 324. [18] Witten, E.: Global Gravitational Anomalies, Commun. Mathematical Physics 100 (1985) 197. [19] Weyl, H.: Classical Groups, Princeton University Press 1939. [20] Green, M. B. and Schwarz, J. H.: Nuclear Physics B243 (1984) 285. - Green, M.; Schwarz, J. H., and Witten, E.: Superstring Theory, Cambridge University Press, Vol. I and II, New York, 1987. [21] Wess, J. and Bagger, J.: S u p e r s y m m e t r y and Supergravity,
162
10. Chiral and Gravitational Anomalies
Princeton Series in Physics, Princeton University Press, Princeton, NJ 1983.
Chapter 11 Anomalies and the Index Theorem The most suitable topological framework for the study of anomalies is encompassed in the theory of fiber bundles to which we devoted Chapters 6 and 7: anomalies are known to occur whenever the determinant bundle of a suitable operator is nontrivial over the orbit space [1, 2, 3]. Much of this depends on the relative twisting of the line bundle. In turn, such twistings are computed via the family of index theorem, and reflect, on the physical side, the fact that the conservation of anomalous fermionic currents does not survive quantization [4]. The index theorem is essentially a bridge between the analytical properties of differential operators on fiber bundles and the topological properties of the fiber bundles themselves. The most readily available example of such is the Gauss-Bonnet theorem, which relates the number of harmonic forms on a given manifold, M, (i.e. the so-called Betti numbers) to the Euler character given by integrating the Euler form over M. The relevant differential operator here is referred to as the exterior derivative mapping, C°°(A P ) - • C°°(A P + 1 ), and the analytic property to which we alluded is the number of zero-frequency solutions to a generalized Laplace equation. The index theorem is useful in exhibiting general analogs of the GaussBonnet theorem for other differential operators. The index of a given op163
164
11. Anomalies and the Index Theorem
erator is often determined by the number of zero-frequency solutions to a generalized Laplace equation and is expressed in terms of the characteristic classes of the fiber bundles involved. It is, moreover, a powerful tool in the analysis of global gauge and gravitational anomalies. The present chapter is, therefore, a prerequisite for any study of global anomalies in general. We will provide here the general formulation of the index theorem and discuss its relationship with chiral and gravitational anomalies. The use of the index theorem for global gauge and gravitational anomalies will be the topic of upcoming chapters. We shall follow the presentation outlined in reference [5] by Alvarez-Gaume, and in reference [6] by Eguchi and collaborators.
11.1
The Index Theorem
The starting point is a Dirac operator for fermions coupled to external gauge and gravitational fields in even dimensions. To this operator is associated two vector bundles over M, which we write as R+ ® V, and i?_V. The space V carries the representation of the gauge group, while R+ (R~) denotes the space of positive (respectively negative) chirality spinors. A Weyl operator,
D+ =
iflp+,
(whose adjoint is D_ =#> p-.) sends objects localized in R+V into objects in /?_ ® V, thanks to the formula R+ ® V £U
R- ® V;
(11.1)
p is a closed form. The explicit form for D± is A t = H* (d, \^ab
Safc + A,) P±.
(11.2)
The index of D+ is the dimension of the kernel of D+ minus the kernel of D\. = D_: indZ) + = dimker.D + — dimkerZ)_. (11-3) Atiyah and Singer [7] showed that ind D+ is a number which depends only on the topological set up (11.1) and it is given by the integral over M
11.1 The Index Theorem
165
of a particular characteristic class. For instance, for the operator (11.2) the index is
indD+ = JM[ch(F)A(M)]voi,
(11.4)
where
MM) = TT - J / 2 / n . v
'
V sinha; a /2'
and ch(F) = T r e i F / 2 \ ,4 (M), the .4 Dirac genus of M, is a polynomial in the 2-form a:a; it is a finite polynomial since M is finite dimensional. The subscript vol in formula (11.4) means that one has to extract the form whose degree is equal to the dimension of M. There is an expansion of A (M) that makes use of polynomials in the Pontrjagin classes, namely: «.^U^^,
A(M)
ii«.iii*_ij .
= 1 + ( 4 ^ n T r t f 2 + ( ^ [5b (Trfl 2 ) 2 + 4 ^ ] ,W [ i o b (Tri? 2 ) 3 + ^ T r ^ T r / ? 4 + ^ T r / ? 6 ] + (4,r)< ( 4 ^ [ 4 9 ^ ( T ^ 2 ) 4 + loSSo (Tri? 2 ) 2 + Tri?< , ^Tr/i:2Tr^+ ^ ( T r ^ ) 2 + ^ Tri? 8 ] + •
= 1 + & (-bO + Ffe2- &!») _L 1 ^
31 „3
1 11 „ _
1 „ \
+ F l - l M 2 5 f t + 3785P1P2 ~ 55IP3J ;?iP3 + n i o o P 2 - 94I0P4) + • • • ' 2» V 604800^1 ~~ 226800^1 P2 ' 14175?
(11.5) When dimi? - n, we have ch ( F)
,
=
+
-TrF+5^TrF2
+
...
+
- ^
T r ^
r
+
...(11.6)
One can then compute the index theorem by combining equations (11.5) and (11.6). In dimensions four and eight the index is ind£+ indD
+
= jfofM
fiTrF*
+ ^Trtf2) 2
= J^IM g T r ^ ) + | T r F T r i i ! + j^(Tri?2)2+ (j^TVtf)
d =4 2
(11.7) d = 8.
166
11. Anomalies and the Index Theorem
Let us focus now on the operators whose indices are essential in the computation of anomalies. These are obtained by replacing R by some particular vector bundle. In order to illustrate this point, we consider the case of the graviton field R — TM, where TM is the tangent bundle over M. In this instance, A is simply the spin connection taking values on the Lie algebra of S0(2n) in the vector representation, (Tab)cd
= 6\6bd
-
6adSbc.
Thus, we get T r e (fl^T»V^)
=
£2coshxa.
(11.8)
a
The quantization of a spin | field requires adding ghost fields if one is to remove unphysical degrees of freedom. When including the ghost field for a spin j field, the index theorem is = f A(M) (Tre f i / 2 , r - l ) c h ( F ) .
mdifl3/2
(11.9)
The last factor in (11.9) accounts for the possibility that the spin 3/2 field carries extra gauge indices. There is a dimensional dependence of T r c R ' 2 , r , as shown by the term A(M) (Tre f i / 2 x — 1). To order 16, the polynomial in question is A(M)
(Tre^-l)
= - ^ 2 T r R* + ^
[ - | (Tri? 2 ) 2 + | T r R*\
+ jfr h l f c ( T r * 2 ) 3 + i > T r i ? 2 T r ^ - ^ T r t f 5 ] + (^l-^(Tr#2)4 + 4(Tr#2)2Tr^ - J . T r ^ T r t f 5 + £ ( T r ^ ) 2 + ^ T r f l 8 ] + •••
= M*n) + f (M ~ Ift) + ¥(kPi
- TEPW + fsPz)
+ MikoP* - Mpfa ~ T&PH* + §-A ~ WsP*) +
•••
(11.10) In even dimensions, a pair of chiral spinor representations whose selfdual representation is SO (2n) appears along with a number of anomaly-free representations [8]. In light of this, it is important to consider the index for a bi-spinor field $a0- The index theorem for such a field is expressed in
167
11.1 The Index Theorem
terms of the signature theorem [9]. Signatures are computed with respect to Betti numbers. The starting point is to write down harmonic forms in the de Rham complex; they decompose into self-dual and anti-self dual pieces. There is another approach which consists of defining the signature a (M) in terms of the heat kernel expansion:
(H-H)
in this formula, * is the Hodge duality operator and C : Ap —> Ap. (The zeroes of C are called harmonic forms.) Index theory makes use of the fact that the space of harmonic forms is equivalent to that of the closed and co-closed forms. Formula (11.11) is related to the Atiyah-Singer theorem as follows: it can be computed as an index problem for a Dirac-like operator interpolating between R+(R+ + R-) -> R- (R+ R-).
(11.12)
Combining (11.7) with the additivity of the Chern character yields a(M)
= fM ch(R+ = fM(chR+
© R-)A(M) + chi?_) A(M).
K
'
;
The next term to be computed is the character of SO (2n) in the representation R+ © i?_: Trexp — RabZab
= chR+ + c h # _ .
The computation reveals that n x
chR+ + chR_
= Yl 2cosh-^, o=l
^
this allows one to obtain the sought signature, namely,
[
[£(M)]vol;
J>M M
where
C{M) = JJ2
Xa/2
tanh xa/2
(11.14)
168
11. Anomalies and the Index Theorem
C (M) denotes the Hirzebruch /^-polynomial of M, whose expansion is £(M)
=l_I^FITrit:2 + +
(2T)6
+
(2x)8
I
^
T
' -Tk
Jl
-I| 4
(Tri? ) + ^ T r ^ T r / ?
lk( 31104
17010 X 1
2
2 3
Tr^]
5
^Tri?6]
-
Tr2)4-T3k(Tr^)3Tr^
+ _ 3 i _ T r R 2 T r i ? 6 + -&~
[i(Tr^)
XI Jl
(TT4)
~ 64800 \ 2
= 1 + |p! + (-^p
/
2
1
-Z?-TrRs
37800
XI
n
+
+ £p2)
+ fep?- £biP2 + ib 3 ) + ("i^Pl
+
14^*2
-
T4T75P1P2 -
J ^
+ £|P4)
+ •••
(11.15) The Atiyah-Patodi-Singer theorem (APS) deals with compact manifolds with boundaries. The index theorem (11.7), in this case, generalizes to ind#=
/
A ( M ) c h ( F ) - i (77 (0) +h).
(11.16)
The first term in (11.6) is the standard volume term, and the second term the boundary correction; 77(0) is a spectral invariant for the on the boundary of M (we refer the interested reader to Chapter 1 for a thorough definition of 77). It is finite and measures the spectral asymmetry of ^7 2 n _ 1 . The term h, in (11.16), counts the number of zero modes of .D*2"-1) on dM. Applications of the APS index theorem in the study of global anomalies appeared in [10] and [11] and will be discussed further in upcoming chapters.
11.2
Characteristic Classes
11.2.1
The Chern Character
As we have seen earlier, index theory requires the use of characteristic classes. These are actually essential tools in the formulation of the index theorem. The Whitney sum of bundles and the tensor product of bundles are among the most frequently used such tools. The total Chern class works well for the Whitney sum; it is however, unreliable for product bundles. The total Chern class: c(C) = I J ( 1 + xj), £ = vector bundle,
11.2 Characteristic Classes
169
needs to be reformulated in terms of polynomials in the {xi}. One such polynomial is the Chern character, ch(£). It is defined by the invariant polynomial: c h ( a ) = Trexp(-^-a) = V
T Tr(-^-a)*'.
(U-17)
The Chern character of the line bundle C can be expanded in terms of Chern classes: * ch (£) = £ > * < = i=1
11.2.2
Jc^
+a(E)
1 + -{c\
-2c2)(E)
+ •••
(11.18)
=dimof£
Hirzebruch Polynomial and Pontrjagin Classes
The Hirzebruch ^-polynomial
'w-ncsv
(1U9)
is a multiplicative characteristic class, characteristic of the signature index formula. The A polynomial, whose use centers in the computation of the spin index formula, is given by the formula
J
") = nsra-
(1120
'
The total Pontrjagin class (of a real 0 (n) bundle) C with curvature Q, is given by the invariant polynomial p{C) = d e t ( / - i - f l ) = 1 + pi + pa + •-• 2x / is the identity matrix.
(11.21)
When dealing with index theory, it is often convenient to express the Pontrjagin classes of a real bundle in terms of the Chern classes of complex bundles. This procedure employs the relation •^complex
==
*~ Qy *-"/
170
11. Anomalies and the Index Theorem
that is, the complexification of C. Complexification arises naturally as an inclusion of G L ( n , R ) into GL(n,C). Consider a skew-adjoint real matrix, Y. The relation d e t ( / + ^-Y)
= l-Pi(Y)
+
p2(Y).-.
yields Pn(C)
= (-l)'lc2n(£complex).
(11.22)
We can repeat the converse by forgetting the complex structure on C using the so-called forgetful functor. Hence, ^*-Teal J complex — ^
U7 -t*.
C stands for the dual (isomorphic) bundle to £. Using the above formula, we have c(C) = l - c , ( £ ) + c 2 ( £ ) - < * ( £ ) • • • which gives rise to c(-Ccomplex)
=
1 ~ Pi (£real) + ?2 (£real)
~
=c{C)c{C) = [1 + cx {£) + c2(C) + •• •} x [1 - c, (£) + c2 (£) + • • • ] .
(11.23)
Accordingly, Pi (£ rea i)
= (c? - 2c2) (£)
P2(£real)
=
(c^C-lCg
+
2c4)(£).
(11.24)
Anomalies essentially measure the non-triviality of the determinant line bundle of a family of Dirac operators, which is given topologically by the first Chern class. As discussed above, in the case of real bundles, one can derive a local formula for the Chern class of C in terms of characteristic classes. Global anomalies, on the other hand, are detected by Cj (£) and this generalizes the whole issue to the complex case. Anomalies do have another geometrical interpretation, namely, the determinant bundle carries a connection, and the anomaly is represents nothing other than its curvature and holonomy [12, 13, 14].
11.3 The Index Theorem and Anomalies
11.3
!H
T h e Index Theorem and Anomalies
Consider a compact 2fc-dimensional manifold, M2k, to which is associated an infinite dimensional affine space E, the space of all possible gauge fields. Let E/T denote the space of gauge-invariant configurations. The term T stands for the gauge transformations which obey Gauss' law. The existence of a fermion determinant defines a complex line bundle over 5 . The case in which this line bundle is non-trivial is important for a study of the anomalies. In previous chapters, namely Chapters 6 and 7, we have seen that line bundles are classified by their first Chern class (the monopole number in physics terms), which belongs to H2 (M2k; Z). Hence, the 2-form
is a function of H2 ( E / r ; R). The generators of H2 ( E / r ) are 2-cycles deduced from 2-parameter families of gauge fields; the 2-cycle in E / 7 on which we integrate c\ is a 2-sphere. The use of the index theorem for families of elliptic operators is therefore warranted. Roughly speaking, we choose a theory living in M2k, with gauge group G. In a given region of M2k, we write V 6 E for all possible connections. Because of the compactness of M2k, there is a finite number of such possible connections. To each V we associate the Weyl operators D+ (V), JD_ (V). As a consequence, we have an infinite dimensional parameter family of Dirac operators which are parametrized by V. Such a family can be parametrized by E/T owing to the relation D± (g~x Vg) = g'1 D± (V)g. The procedure relies on the elliptic self-adjoint operators Z)_ (V) D+ (V) and D+ (V) Z)_ (V) whose eigenfunctions span the Hilbert spaces H+ (V) and W_ (V) respectively, and a Hilbert bundle over E/T. The term ind Dy is a constant independent of V. But it is also the simplest topological invariant that one can define out of this infinite dimensional family of operators. Next, we consider a finite dimensional subset K of E which projects to some compact set in E/T. On K, we study of the two Hilbert bundles H~x and HR. The eigenvalues of 1_ (p) = D+ (V p ) D_ (V p ) change with respect to p whenever one moves V p in the subset K. The eigenvalues Ao (p)
< Xi (P)
<
• • • < A n (p)
<
• • •
172
11. Anomalies and the Index Theorem
split H+ (V p ) and n~ (V p ) into two pieces, H°± (V p ) and Ti\ (V p ). H\ (V p ) contains the eigenfunctions whose eigenvalues may vanish somewhere on P. For A+ an eigenfunction of L+ (p), A+ ^ 0 for p 6 P , L+{p)<j>+p = A + #
(11.26)
^+(Vp)^p+,
(11.27)
we have that is an eigenfunction of L- (p) with the same eigenvalue, meaning that D+ (V p ) provides an isomorphism between Ti\ (V p ) and Til. (V p ). These observations set the stage for the relation between the anomaly and the index theorem. Note that, by definition, D+ (Vp) : H\ (V p ) -
Til (Vp)
is invertible. We have already established that Ti\ (V p ) and Til. (V p ) are isomorphic. This means that there is no twisting between Til+ (V p ) and Til. (V p ). Therefore, any relative twisting between Ti+ (p), Ti" (p) must originate from Ti°± (V p ). The finite dimensionality of Ti% (V p ) implies that of the vector bundles over P. From now on, let V± denote such bundles. They are easily characterized by their Chern characters, ch(V+) or ch(V_). According to our previous computation of the anomaly, it appears that ch(V + ) or ch(V_) are not sufficient to determine the anomaly. The remainder of the chapter is devoted to illustrating this point. The effective action for a massless Dirac fermion in a 2fc-dimensional Euclidean space is a Gaussian integral: fV 'S] = [ dipdi>exp(- j d2kx^^i e - r '[
$A.
(11.28)
The classical action is invariant under global chiral rotations of the fermions: ,r V> -* ip e'a al,
(11-29)
in part because {f, P) can be shown to be trivial. The effective action in equation (11.28) could be written in terms of the eigenfunctions of i P by
173
11.3 The Index Theorem and Anomalies expanding ip, «/>: ±{x) ifi(x)
= T.k a^k{x) =T.kHt{x)
(11.30)
And so (11.28) now reads:
I d2kx^iipi>
= Y,Xkbka.
J
(11.31)
k
The measures in formula (11.28) are thus TTjc dbda, where a and b are independent Grassmann numbers. A change of variables i/> —> v ^ r a ( i ) , yields a change of the effective action by fdx^ifli,^
I dx~i>iipil> + f d2kXy/ja(x){/\nJg);
(11.33)
where
_ _ Jt = V T T V dx = d2k x yjg. Ward identities, which imply the conservation of the axial current at the quantum level, arise when expanding (11.33) in powers of a(x), taking into account the invariance of T [V] under changes of variables. The observation is due to Fujikawa [15] that the Jacobian factor appearing in the measure thanks to the change of variables in (11.32) is potentially dangerous. Explicitly, he found that the Jacobian is divergent, and thus the computation of the anomaly reduces to computing the change in the measure. For instance, for small a, the measure is described by the formula: 6£ = exp (-2if(dx)a(x) Zl=l ^(x)T^k(x)) = (exp — 2i / (dx) a(x) ,=, (x)). The sum is best regulated by a Gaussian cut-off. allows one to define J
( n34)
Using this information
= 2fdxa(x)rk=lri(x)lTp(x) = 2fdxa(x) Zk=1 i>t (x) r xf>k (x) e~WM = 2 / (dx) a (x) £ (dx) a (x) £ Vjf (x) T eHW/Af 2 i>k (x).
174
11.
Anomalies and the Index Theorem
Taking into account the zero momentum Ward identities (that is, the case in which a(x) is close to being a constant), a factorizes out of the integral. Consequently, J = 2 lim a T r r e ( i p > 2 / M \ (11.35) M2k—*oo
When only contributions from the zero sector are considered, we obtain the relation T r r e x p ( - ( i # > ) 7 M 2 = k+ - fc_ = ind #>) . The relation then follows that f(dx)a{x)
(V„ J5"> = 2 /
J
a(x) \A (M2k) ch (F)\
*/A/
(11.36) vol
(Compare with (11.4)). Axial anomalies arising in the presence of both gauge and gravitational fields follow as a consequence of formula (11.36). In dimension four, for example, we have
t.( V ^-> = (2^/(- T r F i
+
5ITtflI)-
For chiral fermions though, the story is different. effective action cannot be written as
(1L37)
For one thing, the
det D+ = det iftP. The reason simply lies in the ill-defined eigenvalue of D+ = R+ <S> V —• R- ® V. In principle, looking at the perturbative evolution of the fermionic effective action could solve this issue. Demanding gauge invariance of the effective action under infinitesimal gauge transformation would mean, in this particular instance, that the following formula ought to be satisfied: r
[A - D V] - r [A] = J dx V* (x) D, ^ 4 1 .
(11.38)
The consistency condition (11.38) holds by virtue of •I'M
= I d\ dX (exp - / dx A i 0+ A)
M
=(Ir„p+r.A>.
(1139)
11.3 The Index Theorem and Anomalies
175
The much celebrated statement in field theories, that gauge invariance is equivalent to current conservation, is a mere consequence of formulaes (11.3839). According to the material covered in Chapter 10, for complex representations of fermions, the term I \ [A] will generically be anomalous. Hence, the issue before us is to determine whether one can relate the anomaly to more detailed properties of the chiral fermions determinant. Roughly speaking, a theory which does not admit a gauge invariant mass term is potentially anomalous. When such a mass term exists, we could, in principle, regulate the ill-theory by the use of Pauli-Villars fields; this approach is referred to as the Pauli-Villars regularization. Alvarez-Gaume and Ginsparg [2] have shown a rather simple way to regulate Tr [a] in terms of how that could de done. The starting point is the basis for the T-matrices where F is diagonal. Instead of working with the operator i 0 P+, it is useful to now consider the operator D = ir"{dll
+ Allp+).
(n.40)
This is indeed an elliptic operator acting on Dirac rather than Weyl fermions; so we are still left with an eigenvalue problem. Notice, however, that D is not self-adjoint, so its eigenvalues are complex. This means that one has to simultaneouly take into account left and right eigenfunctions: Dk = Ayt 4>k X+D =\kXt (Xk,>i) = Su.
(11.41)
We therefore obtain e-r'fi*J = detD(A).
(11.42)
The anomaly is computed using Fujikawa's method [15]. The action, J dx^Dtjj
(11.43)
is invariant under the transformation
% ^f+Vl*
(11-44)
176
11. Anomalies and the Index Theorem
The expansion of 0 , ip is made possible by the termsk, X£, formula (11.40):
4 = Ef=1 a Pk V> = T.U bXt-
and t/> in (11.45)
The measure reads TT dbda; k
and this changes the effective action in (11.42) in such a way that it is now
k
Moreover, equation (11.41) is easily extracted after a Gaussian integration. The Jacobian factor arising under infinitesimal gauge transformation is 6VT
= f V{D.
5
-^
tD) 2 M2
= /°fc dxTrv{x)Te-( i l
8(x-y).
Its 4-dimensional incarnation is 6V r [A] = j i y / d*xTr v e W = ^
$KVd{AdA
+
dx (A, da AB + \ A, Aa A0) 3
\A ).
The leading term in equation (11.46) is given by
We have reached the conclusion that the term Tr [A] can indeed be defined by detD(A). The next order of priority for us is to demonstrate that the form of the anomaly (11.46) is a consequence of the index theorem (or otherwise, a consequence of the index theorem for families of operators). In order to state the relationship between the 2fc-dimensional anomaly and the 2k + 2 index theorem, we begin by considering a one-parameter family of gauge transformations: g(0,x) = S 1 x S™ -+G , . lii g(0,x) = g(2x,x) = 1. - yj
11.3 The Index Theorem and Anomalies
177
The classification of these maps is given by fl^jt+i G and extends to the cases for which K\ G = 0 and •K2k G = 0. The equivalence transformations in (11.49) yield a one-parameter family of gauge field configurations which reads Ae =g-1(A
+ d)g.
(11.50)
To go from Ae to say, Ae + 6e, is the result of an infinitesimal gauge transformation (with gauge parameter g~l 6gg): Ae+ss
=
As
DAev
= dv +
S6DAe(g^deg)
+
[li i)l
- >
[A\V\.
The anomaly obtained by the infinitesimal transformation (11.49) is
- W M = i ^ .
(U52)
The term eiuj<-A^ is a function mapping S1 —• S2. As such, the anomaly measures the local winding number of e'u(A^)_ To obtain the total winding number, one has to extract the integrated anomaly along the one-parameter family of gauge transformations shown in equation (11.49):
here n stands for the winding number of expiw (A, 6). In view of this, we need to find an appropriate (2k + 2) dimensional Dirac operator whose index equals the winding number (11.51). The integral form for the index, we shall learn, will then generate a term for the anomaly in the form (11.46). The fermion determinant defines a complex line bundle over the 2-sphere whose transition function is the imaginary part of the effective action, expiw (A,0). In turn, the bundles are classified by the winding number of the transition function (11.51). The issue before us, at this point, is to compute a representative of the first Chern class. This is equivalent to computing a particular form of the anomaly. In Ik + 2 dimensions, the Dirac operator takes the form 2k+2
ifl2k+2
_
= i Y, (di + ^ , ) r a .
(11.54)
178
11. Anomalies and the Index Theorem
The use of the index theorem (11.4) yields indi p2k+2
ik+1 r = —————— TrFk+1. fc+1 (27r) (k + 1)! Js* x s "
(11.55)
F is the (2k + 2)-dimensional gauge field strength: F = (dt + de + d)A + A2. Formula (11.53) takes on the final form
x ^ x ^ T r ^ -
1
^ )
2
^ .
[
l bb)
-
The resulting formula
= Sx,^/ = 2-h I
dd
5 1 x S
,TrF
3
1
(11.57) 9
Is* T r s " degd(A dA>
+
e
\A °)
agrees term by term with (11.46). We now pause to discuss the assertion made a while ago that the Chern character ch (V±) of vector bundles on M2k is not relevant in determining the anomaly. In keeping with the spirit of our discussion above, the anomaly is generated by the relative twisting of V>° (t, 0) and V>° (t, 0), the eigenfunctions of p^'9) which are trivial at (#o)- The winding number obtained earlier measures the twisting of ^_ and of course, tjj+ around (to, 0o). Topologically, this amounts to determining the first Chern class of a line bundle. To quantify the relative twisting of V_ and V+ involves the introduction of the difference bundle V+ Q VL. From Chapters 6 and 7, we know how to define the direct sum of vector bundles E © F. The very definition of Chern classes gives ch (E © F) = c h ( £ ) + c h ( F ) and c (E © F) = c(E) • c(F). The index is therefore ind£> = dim (ker£>_) - dim (kerZ>+); (11.58) where ch(indZ)) = /
JM2k
A(B) ch(V).
(11.59)
11.3 The Index Theorem and Anomalies
179
We are recall that A (B) is none other than the Dirac genus originally given by formulae (11.4) and (11.5). Furthermore, B — P x M2k. For gravitational anomalies, the charactersistic polynomial in (11.57) should be interpreted as a polynomial in the Pontrjagin classes. Since Pontrjagin classes are forms of degree divisible by four, the relation follows: ch(ind£>) = c(indi>). The dimension in which it does not vanish is d = ik + 2 (in this case B has dimension 4k) [8]. It is precisely because of this fact that gravitational anomalies exist only in dimensions 4k + 2. In the presence of a gravitational field, a theory describing chiral fermions is consistent when invariance of the effective action under both diffeormorphisms and local Lorentz transformations is achieved. In view of this, for the characteristic polynomial to generate anomalies, a non-trivial index theorem for a two-parameter family of Dirac operators ought to exist. The latter condition is required in part because the anomaly cancellation conditions are the same for diffeomorphism or Lorentz anomalies: one can shift the anomaly from difFeormorphism to local Lorentz transformations simply by adding a local counterterm to the theory's effective action. How does one go about finding a two-parameter family of Dirac operators with a non-trivial family index theorem in dimensions 4k + 2? The answer essentially lies with the characteristic polynomial (11.57) which, as we have seen, gives the anomalies. The discovery, by Atiyah [16], of two-parameter families of two-dimensional Dirac operators with a non-trivial index, certainly provides us an adequate framework from which a detailed analysis of the gravitational anomaly in (11.57) could be carried out. Using Atiyah's result, we see for instance, that for spin 1/2, V = 0, ch(V) = 1 (and A(B) denned as in (11.4)):
'"• - n ^/k-
("-6»>
At this point, we need only expand Ii/2 to order 2k to exhibit the 4fc-form characterizing the gravitational anomaly.
180
11.4
11. Anomalies and the Index Theorem
References
[1] Atiyah, M. F. and Singer, I.: Dirac Operators Coupled to Vector Potentials, Proc. Nat. Academy Sciences USA 81 (1984) 2597. [2] Alvarez-Gaume, L. and Ginsparg, P.: The Topological Meaning of Non-Abelian Anomalies, Nucl. Phys. B243 (1984) 449. [3] Moore, G. and Nelson, P.: The Aetiology of Sigma Model Anomalies, Commun. Math. Physics 100 (1985) 83. [4] Bardeen, W.: Anomalous Phys. Reviews 184 (1969) 1848.
Ward Identities in Spinor Field Theories,
[5] Alvarez-Gaume, L.: An Introduction to Anomalies, in Fundamental Problems of Gauge Field Theory, Eds. Velo, G. and Wightman, A. NATO-ASI Series B, Vol. 141 (1985) 93-206. [6] Eguchi, T.; Gilkey, P. B. and Hanson, A. I.: Gravitation, Gauge Theories and Differential Geometry, Physics Reports 66 No. 6 (1980) 213-393. [7] Atiyah, M. F. and Singer, I.: Annals of Mathematics 87 (1968) 485546. - idem, 93 1 (1971) 119-139. [8] Alvarez-Gaume, L. and Witten, E.: Gravitational Anomalies, Nucl. Physics B 234 (1983) 269. [9] Chern, S. S.: C o m p l e x Manifolds W i t h o u t Potential Theory, Van Nostrand 1967. Milnor, J. and Strasheff, J.: The Theory of Characteristic Classes, Princeton University Press 1974. [10] Alvarez-Gaume, L.; Delia Pietra, S., and Moore, G : Anomalies and Odd Dimensions, Annals of Physics 163 (1985) 288. [11] Niemi, A. and Semenoff, G : Phys. Rev. Letters 55 (1985) 927. [12] Witten, E.: Global Gravitational Anomalies, Commun. Math. Phys. 100 (1985) 197-229. [13] Freed, D. S.: Determinants, Physics 107 (1986) 483-513. [14] Freed, D. S.: Z/k-Manifolds
Torsion and Strings, Commun. Math. and Families of Dirac Operators, Invent.
11.4 References
181
Mathematica 92 (1988) 243-254. [15 Fujikawa, K.: Phys. Rev. Letters 42 (1979) 1195. - idem, 44 (1980) 1733. Phys. Reviews D 21 (1980) 2848. - idem, D 22 (1980) 1499 (E). - idem D 23 (1981) 2262. - idem D 29 (1984) 285. [16] Atiyah, M. F.: The Signature of Fiber Bundles, in Collected Mathematical Papers in Honor of K. Kodaira, Tokyo University Press 1969.
Chapter 12 Global Anomalies We are (finally) in the position to present an exhaustive account of global anomalies. Global anomalies occur when large gauge or diffeomorphim transformations fail to be symmetries of the corresponding quantum theory. There are, to date, two classes of global anomalies: the global gauge anomaly and the global gravitational anomaly. Global gauge anomalies arise whenever a theory's effective action is not invariant under large gauge transformations. The extent to which large takes on a pathological meaning relates to whether the effective action is well-defined under small gauge transformations. These are usually detected by anomalies of the perturbative or chiral type. When the effective action Ceff is well-defined, large gauge transformations of a global character change it in a highly non-trivial way. Global gravitational anomalies, on the other hand, reflect the lack of invariance of the effective action under the group of equivalence classes of diffeomorphisms that cannot be smoothly deformed to the identity, the so-called mapping class groups.
12.1
An SU (2) Global Gauge Anomaly
Elements of homotopy theory are necessary to comprehend this class of anomalies. Primarily, the use of homotopy groups is required. These groups are easily catalogued by Bott's periodicity theorem for i < 2n [1] (with G a 182
183
12.1 An SU(2) Global Gauge Anomaly given gauge group:) TTi(G(n)) = Z for i = odd TTi(G(n)) = 0 fori = even '
, „ > ' '
(
It is a periodicity in the sense that for n = oo, TT; ( ( 7 ( O O ) ) = 7rI+2 (G(oo)) with 7T0(G(oo)) = 0 and Wi(G(oo)) — Z. Global gauge anomalies rely on Bott periodicity and on non-simply connected gauge groups, say G(n) (in the latter, we take account of loops which are not homotopic to the identity map; these are the relevant ones for evaluating the pathologies). The best available illustration of such is Witten's SU (2) global anomaly [2], which we proceed to succinctly analyze below. The problem is essentially easy to formulate conceptually, but, nonetheless requires a good deal of technical knowledge to solve. In dimension four, the number of Weyl doublets is perturbatively anomaly-free. However, as pointed out by Witten in reference [2], the theory is mathematically inconsistent because, according to Bott periodicity, the gauge group of interest, G, is not simply connected and therefore does admit loops which are not homotopic to the identity map. When one encounters such a situation, global anomalies are bound to arise, and thus need to be cancelled if the theory is to be consistent. Let us make this a bit more explicit. Let G = SU (2) be such a group. The formula TT4 (SU(2)) = Z 2 ,
(12.2)
suggests that, in four dimensional Euclidean space, there is a gauge transformation, T (x) —> 1 as |x| —+ oo. For 7rn (SU(n)) = Z, it follows that the transformation in question cannot be continuously deformed to the identity. However, for Z 2 homotopy groups, the gauge transformation wraps twice around SU (2) and thus can be deformed to the identity. Now, in the absence of fermions, the Euclidean path integral is of the form:
J (dAJ exp (-^2
J d*xtTFlu/F'A .
To a given gauge field configuration A^ corresponds a conjugate field,
Al = T-'A^T
-
iT-'d^T,
(12.3)
184
12. Global Anomalies
which makes the same contribution to the functional integral (12.3). Hence, to each contribution originating from the gauge field is to be associated another contribution. Such double counting is harmless in the absence of fermions as it cancels out when computing the vacuum expectation value. We now introduce fermions. The partition function of the Euclidean version of the model of a doublet of massless fermions coupled to an SU(2) gauge field is Z
= f diPLd^t I dA^exp (-fd*x [(l/2g2)trF^ + >^V>L]).
l
j
A,j, is the SU(2) gauge field; ipi a left-handed Weyl fermion doublet; g the gauge coupling constant, and lp = D^j^ is the Dirac operator restricted to act on a Weyl doublet. The fermionic part of the integral (12.4) is ill-defined. It can be formally integrated as the square root of functional integrals over Dirac fermions. As such, it implies the doubling of the fermionic degrees of freedom, meanning we need to consider two Weyl-handed doublets instead of one. Note that the 1/2 representation of SU (2) is pseudoreal. Consequently, a left-handed Weyl doublet can be mapped to a right-handed one. A theory with two left-handed Weyl doublets is therefore equivalent to a vector-like one with a single Dirac doublet. The fermionic functional integral is obtained through det (i $); it is well defined. The formula then follows: J di/>Ldil>texp / 0 L « Pfa
= (deU#)1/2.
(12.5)
At this point we run into a problem: the sign of the square root is ill-defined. It is precisely this ambiguity that defined the global gauge anomaly problem. For a given gauge field, A^, the sign of (detz ]fi) ' can be determined arbitrarily. When certain consistency conditions, such as the Schwinger-Dyson equations, are taken into consideration though, it turns out that the fermion integral (det i ft)1' ought to be defined in such a way that it varies smoothly as A^ is varied. This procedure guarantees the fermion integral to be invariant under infinitesimal gauge transformations, that is to say, the sign does not change abruptly. Such gauge invariance is, however, not necessarily guaranteed under the large, topologically, non-trivial gauge transformation T. In reference [2], it is argued that (det ify)1?2 is indeed odd under T. Here is how.
12.2 Global Gravitational Anomalies
185
Roughly, the lack of invariance under T is best illustrated by the formula dett 0 ( A J 1 / 2 = - (deti # (A^)) 1 / 2 .
(12.6)
Formula (12.6) is the statement that a continuous variation of A^ to AT generates the opposite sign of the square root. How does one solve the issue of the ill-defined sign of the square root? A clever solution, proposed in [2], is to first write down the root in (12.5) as the product of all positive eigenvalues of a Dirac operator, and then continue analytically. As a result of this procedure, the partition function,
Z =
JdA.idetip)1'2
exp(-(l/2 5 2 )/rf 4 xtr^),
V
'>
vanishes due to the contribution of opposite signs from A^ and A . This should come as no surprise since, as we are continuously varying the external field value from A^ to AT, an odd number of these eigenvalues flows through zero, switching their sign in the process. This very phenomenon is related to the existence of an odd number of normalizable zero modes for a welldefined five-dimensional Dirac operator, Ps, in a topologically non-trivial gauge field. It is noted in [2] that Ps, for an SU(2) doublet, is a real, antisymmetric operator. Its eigenvalues either vanish or are imaginary and occur in complex conjugate pairs. A variation of the gauge field AM implies that the number of zero eigenvalues of $$ changes only if a complex conjugate pair of eigenvalues moves to, or away from, the origin. The Atiyah-Singer theorem [3] requires that p$ possess an odd number of zero eigenvalues. This number, modulo two, corresponds to the mod two index of the Dirac operator. Having determined this, it then follows that ( d e t ( j ^ ) ) is odd under the topologically non-trivial gauge transformation T.
12.2
Global Gravitational A n o m a l i e s
Our analysis of global gravitational anomalies shall begin with the N = 2, D = 10 dimensional perturbative anomaly-free chiral supergravity theory. Then, in a second phase, we will address the question of whether the E& X E$ heterotic superstring theory is global anomaly-free. There will be, moreover,
186
12. Global Anomalies
a third focus on the manifestation and cancellation of global gravitational anomalies in some six-dimensional supergravity theories. In the next chapter, we will discuss the relationship between mapping class groups and global gravitational anomalies in dimension three particularly as they relate to topological quantum field theories and Chern-Simons-Witten theories. Any serious study of global gravitational anomalies in ten dimensions, it should be said, requires a specific knowledge of homotopy (i.e. exotic) spheres.
12.2.1
A Case Study of t h e N = 2, D = 10 Supergravity Theory
This theory is known to be chiral anomaly-free [4]. The foremost investigation of global gravitational anomalies in this theory is that of reference [5]. Consider a ten-dimensional manifold, M 1 0 , in which the theory is defined; let the group of diffeomorphisms of this manifold be Diff (M). We are interested in diffeomorphisms which become the identity at infinity, and this suggests we first determine how many components of Diff (M) there are in dimension ten. Finding the number of components of this group is actually a highly non-trivial exercise since, to begin with, there is no general method that can offer a satisfactory answer. So, in order to tackle this problem, one further considers a special class of diffeomorphisms which are different from the identity when restricted to some ten-dimensional ball B10, and become the identity outside. The appropriate use of algebraic topology techniques shows that these diffeomorphisms are the same as diffeomorphisms of S 10 , that is, Diff (B10)
~ Diff (5 1 0 ) .
(12.8)
Formula (12.8) requires a thorough evaluation of TT0 (Diff (S10)), the mapping class group of S10. To detect the gravitational anomaly is now equivalent to asking whether the N = 2, D = 10 supergravity theory considered is invariant under the action of these disconnected diffeomorphism components. The answer to this question depends, in turn, on our evaluation of MilnorKervaire 11-dimensional exotic spheres [6]. Milnor has shown that exotic (n + l)-spheres and topological classes of diffeomorphisms of the (standard) n-sphere are in one to one correspondence. For a twelve-manifold without boundary, M 1 2 , the integral of products of
12.2 Global Gravitational Anomalies
187
Figure 12.1: Sketched here is the topological construction of exotic spheres obtained via surgery on the standard n-sphere.
188
12. Global Anomalies
characteristic classes which are relevant as topological invariants, are: h I2 h
= /M12(TrjR2)3, = / M 1 2 T r / ? 2 -TvR\ = /M^Tri?6.
(12.9)
These quantities are invariant because they are independent of the metric and the connection. When M 1 2 has a non-empty boundary, however, I\, I2 and 73 lose their topological invariance. On the other hand, if, on the boundary of M 1 2 one can solve TT R2 = dH, then there is no obstruction to generalizing some of the quantities in (12.9) to a twelve-manifold with boundary. The generalization takes on the form {i J2
= J M " (TriZ ) - / a M i2_ S ii H (Tr R ) , = fW2 TTR2 • TrR4 - / E „ HTrR*.
From here on, we will take dM12 dimensional exotic sphere.
=
(12 IM ' '
[
E 11 to be the Milnor-Kervaire 11-
Are 7i and I2 invariant under a change of metric on M 1 2 ? The answer to this, as explained in [5], involves some technicalities. The procedure used to obtain generalized topological invariants is as follows. First, we write STr R2 = dA for a variation of Tr R2 under a change of metric. It follows that 6H = A does preserve the requirement that Tr R2 = dH on dM12, and consequently 6 Ii — 612 = 0. But (12.10) is still not well defined via these arguments. For this argument to hold, one needs to establish that I\ and 12 are independent of the choice of H for dH = Tr R?. Let dH = dH' = Tr R2 (i.e. d(H — H') = 0).) In going from H to //"', the quantity I\ changes by Ah
= / E „ (H - H') (TrR 2 ) 2 = / E „ (H - H') • dH • Tr R2 = 0.
y
'
'
Formula (12.11) is essentially the statement that Ix is well-defined. Looking now at I2, its variation is A/2 = /
{H - H')Ti-R4,
(12.12)
JE11
and does not vanish [5]. To demonstrate that I2 is well-defined on the boundary dM12 = E 11 , we consider the term Tr RA = dB. The variation of I2
189
12.2 Global Gravitational Anomalies now becomes: A/2
= fgMia(H - H')dB = - ; E „ d(H - H') • B = 0.
. {
. >
To resume what we have done so far: I\ can be generalized to a topological invariant of M 1 2 with boundary, provided that Tri? 2 — dH on E u . I2 can similarly be generalized if Tr R2 = dH and Trfl 4 = dB on S 1 1 . ^3, on the other hand, does not admit a similar generalization. Our next focus is to concern ourselves with the seven-and eleven-dimensional cases. A closed eight-dimensional manifold, M 8 , has the following expressions for its Pontrjagin numbers: Pi
(^/^|(Tri?2r,
=
P2 = pfr/Af.
(|(Tri?2)2-±Tr^).
(12.14)
According to the discussion above, p 2 has no known generalization for 8manifolds with boundary. The quantity p2, however, admits a generalization to a manifold with boundary, and on the boundary one would expect Tr R2 = dH to be exactly solvable. The index of the Dirac operator for an 8-dimensional spin manifold is of the form mdip
= ^ ( 7 p
2
-4P2).
(12.15)
The signature reads
Combining (12.15-16) yields = — (p2 - 4(7). (12.17) V * 896 KFl ' ' 8 Since M is a compact manifold, it follows that the right hand side of (12.17) is an integer. mdiB
On the basis of formula (12.17), we define the topological invariant
A (E?) =
sie (p*(M8) " 4
(12J8)
12.
190
Global Anomalies
where E 7 is the boundary of the oriented eight-dimensional spin manifold. This invariant depends only on E 7 . To prove this, take two manifolds, M and M', with boundary E. Glue them together while preserving the orientation X = M + (—M'). The resulting manifold has no boundary. Hence, A(M) - A(M') = ind p(X)
= Z.
Thus A (E) depends only on the manifold E. The case in which E is the 7exotic sphere implies that M has signature equal to 8 (i.e. a (M) = 8), and Pi = 0. With these descriptions in hand, one deduces that A(E 7 ) = —^. But A = 0 for the standard sphere S7, therefore E 7 must be an exotic sphere. The connected sum of E 7 generates exactly twenty-seven exotic spheres. Having studied the toy model E 7 , let us move now to the eleven-dimensiona case. In a given closed twelve-manifold M 1 2 , the relevant quantities are: Pi
—
(2TT)6
PlP2 = J^e
Pa
/
I
|(Tri?2)3], |
(Tri? 4 ) - £
(TTR2)
= ^/[-|Tri?
6
(TTR2)3]
,
(12.19)
+ iTri?2Tri?4-i(TrR2)3]
together with the index:
ind
^ = h (-w3
+
s i o ™ - 9i5 P3 )'
(12 20)
-
and the signature:
p? -
^945 r t
—PiP2 + — P a ) .
945ww
945
w
'
(12-21)
V
;
It is best to determine | the Dirac index. This is achieved by a judicious use of equations (12.19), (12.20) and (12.21); the result is: i i n d i ^ = ap\ + pPlP2
- g-^;
(12.22)
where a, and /3 are rational numbers. Witten's computations [5] tell us that (12.22) is always an integer in twelve dimensions. As a consequence, the term ap3 + /?piP2 — 5^5 is an integer for closed twelve-dimensional spin manifolds.
191
12.2 Global Gravitational Anomalies
Now, consider the 11-dimensional exotic sphere, E 1 1 , with Tr R2 = dH, and Tri? 4 = dK. Taking E 11 to be the boundary of the twelve-manifold M12 gives rise to the invariant: A ( £ » ) = aP\ (M 1 2 ) + fap - 2 ( M 1 2 ) - | f ^ l .
(12.23)
As before, modulo one, the invariant (12.23) depends only on the E 11 and not on any given choice of M12. Otherwise, A would be an integer if E n was a standard sphere, which is a contradiction in terms. The example is due to Milnor [6], of an exotic sphere E 11 bounding Af12, having the property that MS") = -
^
(12-24)
(a(Mu) = 8 in this case.) From (12.24) we have a family of 991 exotic spheres via connected sums of the Milnor sphere E 11 . A statement to this effect is encoded in Milnor's observation [3] that exotic 11-spheres bound twelve-manifolds whose signature is divisible by eight. Witten's insight was the realization that a general formula for global gravitational anomalies could be derived in terms of the variation of the effective action that takes mostly into account the Hirzebruch signature and Milnor's observation. Under large diffeomorphism transformations, Witten noticed, the action changes by . a (M") AjLeff
=
^Tl
o
'
o
and the theory is global gravitational anomaly-free when A£ e ff = 0mod27n. We will arrive att this formula in a short while. For now, let us give the flavor of the machinery necessary to carry out the computations in the case of N = 2, D = 10 chiral supergravity theory. We have already shown, namely in Chapters 10 and 11, that this theory is perturbative and chiral anomaly-free [4]. In chapter 11, the anomalies cancellation, it was noted, is function of the Hirzebruch signature a (M) and the index of the Rarita-Schwinger and Dirac operators in twelve dimensions: ^ = ind(R-S) -3indi p.
(12.25)
192
12. Global Anomalies
In ten-dimensional Minkowski space, optimal results are achieved by the inclusion of Weyl-Majorana spinors. The effective action, £eff for such a spinor is half of that C'ef{ for a Weyl spinor. We want to compute the difference
where <JM„ is the metric and IT a diffeormorphism of M. The effective action is written as a functional of the metric. In order to carry out the computation, we choose a one-parameter family of metrics g* which interpolates between g^ and g^u. The anomaly affects only the imaginary part of £eff, and therefore, what is to be computed is simply: AC'eH = zj\tjtlmCef{[gl}.
(12.26)
A series of arguments are then applied [5] to give C'eff = 1*1/(0);
(12.27)
here 7/(0) stands for the spectral asymmetry of the Dirac operator evaluated on the manifold (M X S1)*, a cylinder with base M siding the interval [0,1], whose top and bottom are identified with the twist automorphism TT. Topology-wise, what we have done is to fiber the manifold M over the circle S1 with transition function equal to TT. When taking No Weyl-Majorana fermions into account, the formula (12.27) becomes: A4 ff - NDjr,(Q).
(12.28)
A similar result applies to the Rarita-Schwinger field. It is important though, to make use of the fact that in dimensions n + 1, a Rarita-Schwinger field decomposes to a Rarita-Schwinger field plus a spinor in its n-dimension component. We also take into account the Weyl-Majorana condition. Under this, the change in the effective action is A C r = NRj
ftfl(0)-i?D(0)).
(12.29)
In this formula, NR is taken to represent the number of spin 3/2 WeylMajorana fields while 77 factors in the ghosts contribution. Overall, for a
12.2 Global Gravitational Anomalies
193
theory with No, TVR and Ns chiral Dirac, Rarita-Schwinger, and self-dual tensor fields, there is a general formula for the effective action change under a diffeomorphism 7r, namely A£eff = j
(NDr,D
+ NR(r,R
- TJD) ~ Y113)-
( 1 2 - 3 °)
The inclusion of self-dual tensor fields requires a bit of attention. In particular, the change in the effective action in the interpolation from <7M„ to g* spells out the need for the spectral asymmetry of specified operators. These operators are -kd-forms acting on even forms on (M X S 1 )^. Note that •*- is a Hodge operator: * : C°° (Ap) -» C°° (A"- p ). With the inclusion of self-dual tensor field Ns, the variation in the effective action becomes A£ e f f = - j 7 y ( 0 ) ; (12.31) the minus sign originating from the different statistics with respect to the fermions. Applying the Atiyah-Patodi-Singer Index theorem to (12.30) gives \r,D \ m rjs
= -ind^ + = -md(R-S) = -
fM12A(M"); + i n d t # +fM» fM»L(R).
[K(M12)
- A(M12)};
(12.32)
Some definitions are in order: ind P and indi? are the Dirac and RaritaSchwinger indiceson M12, and a is the signature of M 1 2 . As for the terms A, K, and L, they are polynomials in the curvature tensor, R, whose integral over M12 yields precisely ind P, ind R, and a. K, in particular, is the characteristic polynomial for the index of the Rarita-Schwinger operator. In equation (12.32), we have subtracted the contribution from a spin 1/2 index in f?ft/2 because a Rarita-Schwinger field in dimension twelve decomposes into a Rarita-Schwinger field plus a Dirac field on the boundary dM12 = E 11 . Combining (12.31) and (12.32), one obtains the change in the action under a large diffeomorphism: A£eff
= 27Tt \\rjD (in
r
-2A{M12)\NsL{Ml2)\.
'
\NR{ind(R-S)) (12 331
194
12. Global Anomalies
For the TV = 2 chiral supergravity theory with No = —2, NR = 2, Ns = 1, the integrands (given by M12 curvature) cancel. In twelve-dimensional Euclidean space, charge conjugation does not change the chirality of spinors (see chapter 10 for an explanation); therefore, ind Jf) and ind (R — S) will always be integers. Hence, these terms can be dropped since mod27rz they do not contribute to (12.33). Consequently, (12.33) reduces to the final form A£eff = 2 7 n ^ p .
(12.34)
How does one infer the absence of global gravitational anomalies when the N = 2 supergravity theory is formulated in S10? According to Milnor [6], every exotic 11-sphere E 11 and space (S 1 0 x S1)* ~ ( 5 1 0 x 5 1 ) + E*1 bounds a manifold M12 whose signature is divisible by eight, i.e. a (M 1 2 ) = 0 m o d 8 . Thus, on the basis of these topological observations, A£ e ff = 0mod27ri,
(12.35)
which is precisely the statement that the TV = 2, D — 10 supergravity theory is free of global gravitational anomalies. Equation (12.35) is referred to as Witten's formula for global anomalies and was initially derived in [4]. We willsee in Subsection 12.2.3 how this formula applies to six-dimensional theories as well. 12.2.2
Global Gravitational Anomalies in the Heterotic String Theory
We will follow the definitions and notations of Section 10.3 of Chapter 10 entitled, the Green-Schwarz Anomaly Cancellation. To briefly recall what we have seen so far in this chapter: global anomalies arise whenever the function space over which we are integrating is not simply connected, thereby preventing the effective action from returning to its original value (mod 2-rri), while going through a non-contractible loop in the function space. We will not consider here the heterotic string global world sheet anomaly. For a detailed analysis of this class of anomalies, we refer the interested reader instead to reference [7]. What we will consider though, are the anomalous E$ x E& fields and their overall contribution to the global gravitational anomalies.
12.2 Global Gravitational Anomalies
195
What are the heterotic string fields with anomalous terms? They are listed as (i) the Dirac field, No = 495; (ii) the Rarita-Schwinger field, NB = 1. From the results derived in subsection 12.2.1, a change in the determinant under the large diffeomorphism x in dimension ten is A£ e f f
= iir(^r)D + \VR) = i-K (494ind B 0 + ind B (R)) -iTr JB (i93A(R) + K{R)).
(12.36)
The terms inds p and inds (R) are respectively the Dirac and RaritaSchwinger indices on B, a manifold taken to be B = d(M x S1)*. Recall that in twelve dimensions, ind B J/) and ind B (R) are even. We only want the A£det contribution mod 2iri. The contribution of these terms to (12.36) is trivial and so we may as well drop them. The second point of interest lies in the curvature integral expressed in (12.35): as it stands, it is not a topological invariant since B is endowed with a boundary. In view of this, we rewrite (12.36) as a new formula,
A£
- = -(5? /. iik ( Trfi2 )' + s ? ( * * ) (•***)) • <12-37>
But here too we run into some subtleties. For instance, to produce a topological invariant, we must somehow generalize the terms JB TTR6, JB (Tr R2) and JB (Tr R2) • (TTR4). There is no known generalization for JB TTR6. But this is hardly an obstacle in the anomalies evaluation since Green and Schwarz have worked mainly with No — 495, NR = 1, thereby bypassing any need for the term TTR6. On the other hand, JB (TTR2)3 and fB (TTR2) • (TTR4) admit a generalization (that gives rise to topological invariants), provided that, on (M x S1)*, we solve TTR2 = dH, TTR4 = dK. (H and K are defined as in Subsection 12.2.1). Now, the Es x E8 (or equivalently the SO (32)) string theory has a physical field, H, which needs to obey Tr R2 = dH, and thus, as noted by Witten in [5], this very fact puts a strong requirement that TT R2 = dH indeed be solved on M. Primarily, the procedure to do so amounts to extending H and K to (M x S1)*, a rather difficult endeavor. But this can
196
12. Global Anomalies
be done, and when successfully completed, formula (12.36) generalizes quite smoothly to a topological invariant, provided that
is added to it. Next, consider the purely gravitational term: G
~ (2^F V I w 5 P + 384^
i r K
~
(TTR2)
n o w
576 W 3 ^ l ) •
We point out that P denotes the antisymmetric second rank tensor of the supergravity multiplet, whereas ui^ and LO\ are the Chern-Simons three-and seven-forms. There is a way to extract H, the gauge invariant field strength in terms of P. The relation is H = dP + w£. Under a diffeomorphism transformation evolving from t = 0 to t = I, G in (12.38) changes by
(Tri? 2 ) 2 +
= < # J? dt SMi[^P =
^ / o ^ M ^ m C T r / ?
2
)
-
£-4PTTR<
2
M
^ u t
(12.39) Using the formal 12-form expression [4] for the anomaly (e.g. Chapter 10, Section 10.3 and [8]) and taking into account the regulator factor exhibited in [5], we obtain: A£
=
Z(2TT)6
$ dt fM
+ [T^^Tri?4 +
[^^(Tr/P)
2
^Tri?2^]
= (^/i°^/Mfe-3 L (Tri? 2 ) 2 ] + 3 K ^ #
+ !56d(wM)]-
12.2 Global Gravitational Anomalies
197
Combining (12.39) and (12.40) yields A£reg
= AG + AC = (2 ^' 6 .4 8 f{Mxsi)„
H
• f e W + iM]i
(T> AI\
(
}
where we have used the equivalence relation H = dP + LJ%. By combining equations (12.37) with (12.39) and (12.40) one derives Witten's expression for the change in the effective action in terms of topological invariants, that is, A Ctot
= A £det + A G + A £ r e g = 2»i [ ^ Is (lk ( T r # ) ~ W ) , ^ ' (iki^R3)3 = -2iw • £. [-3p31(B) +
3
+ 5*5 Tr R2 Tr R<) + 3k Tr/?4)] 4plP2(B)},
(the pi are Pontrjagin classes.) When evaluating the anomalies, it is important that the right-hand side of (12.42) be defined only on (M x S'1)7r, and moreover, that it be independent of the choice of B. We are now in position to conclude whether the 0 (32) and/or Eg x Eg superstring theories suffer from global gravitational anomalies when defined in S10. The starting point is the eleven-dimensional spin manifold E 11 , to which we associate the form:
^(M) = lib [ -3p? + 4piP2 l
modL
(12 43)
-
Observe that (S 1 0 x S1)* is the connected sum of E n and (S1 x 5 1 0 ), which we write as (Sw x S% = E u t t ( 5 x x S 1 0 ). Therefore, the relation follows that v((S10
x S1)*)
= /.(E 1 1 ) + niS1 x S10).
But fi ( E u ) = 0 since any 11-exotic sphere bounds a parallelizable manifold [9, 10, 11] . The Pontrjagin classes pi and p 2 then vanish and so p (S1 X S10) is trivial since 5 1 x 5 1 0 bounds D2 x Sw which has p\ = pip2 = 0 as well. D2 is a two-dimensional disc. For the theories in question to be global
12. Global Anomalies
198
gravitational anomaly-free under a large change of diffeomorphism 7r, their effective actions must be invariant under ir. This consistency condition is satisfied whenever the formula H = Omodl
(12.44)
holds. 12.2.3
Global Gravitational Anomalies in S o m e D = 6 Supergravity Theories
We shall analyze here the manifestation of global anomalies generated by disconnected general coordinates transformations. Specifically, we will investigate whether global gravitational anomalies occur in the N = 2, D = 6 as well as the TV = 4, D = 6 supergravity theories. The impetus to study global anomalies in six dimensions lies in superstring theory, which, in order to make contact with 4-dimensional phenomenology, requires the tendimensional theory to break down into product of four-and six-dimensional submanifolds. String theory compactification requires the three-dimensional (complex) submanifold in question to be a Calabi-Yau manifold (see reference [12] for a detailed explanation of this, and additionally, reference [13] for a case study of compactified Calabi-Yau manifolds for the heterotic string theory). On the other hand, supergravity theories are strong candidates for describing low-energy limits of the superstring theory. It is therefore important, for consistency purposes, that supergravity theories be free of global anomalies. As we have discussed throughout this chapter and the preceding ones, there are really two ways to achieve consistency: one is to make sure that the theory we are working with is perturbatively (and chiraly) anomaly-free, and the other complementary option is to cancel any existing global anomalies. Consistency conditions impose such stringent restrictions. This is perhaps why anomalies are seen as good things [17]: they restrict the number of possible consistent theories in any given dimension and are hence a powerful tool in the pursuit of uniqueness. Our approach is as follows. First, we will apply Witten's formula for global anomalies (of Subsection 12.2.1) to the six-dimensional class of theories considered here. The framework in which we will operate draws on the work of Bergshoeff et al. [14], whom, as far as I know, were the first to investigate
12.2 Global Gravitational Anomalies
199
global gravitational anomalies in six-dimensional supergravity theories. A few comments. The reader may wonder why we are not considering here the N = 8, D = 6 supergravity theory. The reason for this is simply that it behave in a vector-like fashion with respect to the gravitational interaction [15], thus making the extraction of information on the anomalies side almost impossible. As for the N = 6, D = 6 theory, when including the field contents, it suffers from perturbative anomalies which cannot be cancelled by the Green-Schwarz mechanism (see Chapter 10, section 10.3 for an exhaustive account of this mechanism.) This very fact therefore rules out the N = 6, D ~ 6 supergravity theory. A. The Case of the TV = 4, D = 6 Supergravity Theory Our initial focus is the perturbative anomaly. The theory has two basic N = 4 chiral multiplets [16] derived from 9nv, %, B$3] B~v, X, $ [ ' J !
supergravity multiplet tensor multiplet.
(12 45) ^ ' '
The gravitino in (12.45) is left-handed. The spinor of the tensor multiplet is a right-handed Majorana-Weyl gravitini. The field strength B+v is self-dual. Perturbative anomalies are characterized by an 8-form polynomial P. We take into account the contributions of a complex left-handed spinor, a complex left-handed Rarita-Schwinger (R-S) spinor and a real self-dual tensor field whose explicit form, taken from reference [4], is / 1 / 2 =A(R) = j ^ [ T r ^ + f (Trtf 2 ) 2 ]; 7 3 / 2 = A (R) [Tr (cos R - 1) + D - 1] = A(R) (Tr cos R - 1) - 2A(R) = ^[245Tr^-^(Tr/?2)2]; h
= - ^
(12.46)
= 5^5 [ 2 8 T r ^ - 1 0 ( T r f t 2 ) 2
In these quantities, A (R) is defined as before, L (R) is the Hirzebruch polynomial, while D stands for the dimension of M, the space-time manifold of interest. According to the formula above, there are four left-handed Majorana-Weyl gravitini, 4A:-right-handed Majorana-Weyl spinors x, an
5)h
(Trtf2)2] .
(12 47)
-
200
12. Global Anomalies
Townsend has shown [16] that for k —- 21, the theory is perturbatively anomaly-free. The Hirzebruch signature derived from formula (12.46), a = -md(R-S)
+ 23ind 0,
(12.48)
hold for any closed eight-dimensional spin manifold. Moreover, we take note of the fact that ind IP = / A{R); \n&(R-S) = /
A(i?)(Trcosi?-l);
a = [ L{R). JM
Next, the focus is on global anomalies. For convenience, let us recall some basic facts which were the basis for our discussion in Subsections 12.2.1 and 12.2.2. Global gravitational anomalies arise whenever the effective action changes under a large difFeomorphism (M) T that is not continuously connected to the identity. We write this change as A£ e ff = £eff(fl£„) - £eff(
VD
+ NR {r,R -
VD)
- Ns T? S /2] .
(12.49)
We have used the Atiyah-Patodi-Singer theorem to find \r\r> = ind jp - fB I1/2 \rjR = ind(R-S) - ind ^ - fB (/ 3 / 2 + /1/2) Vs = o- - fB L(R).
(12.50)
In the N = 4, D = 6 supergravity coupled to 21 tensor multiplets, we have No — —4 x 21, NR — 4 and Ns — 5 — 21. On the basis of this, we combine (12.48) with (12.47) and obtained the result A£eff
= 2iir [-46 ind $ + 2md(R - S) + 2<x] -2*i fB [ - 4 2 / 1 / 2 + 2/3/2 - 167x] .
(12 51)
"
201
12.2 Global Gravitational Anomalies
Observe that the integrand in (12.51) contains terms which are characteristic of perturbative anomalies, as exhibited in formula (12.49), and therefore vanish. Moreover, A£ e ff is an invariant mod 2xi because the terms in the bracket in formula (12.51) are multiples of integers. This establishes the absence of global gravitational anomalies in the N = 4, D = 6 chiral supergravity theory. B . The TV = 2, D = 6 S u p e r g r a v i t y T h e o r y Case This theory contains the following multiplets: g»„, V*, B+v B lv XA, 4> A„ XA Va, (f>a
supergravity tensor Yang-Mills hypermatter.
(12.52)
The $!A and \ A are left-handed, while iia and \ A are right-handed. According to (12.47), to cancel the leading perturbative gravitational anomaly (which is proportional to Tr/? 4 ) requires the consistency condition [14]: dimG = n - 29k + 273 = 0.
(12.53)
We then have to worry about the gravitational anomaly:
For k ^ 9, one cancels (12.54) by the use of the Green-Schwarz mechanism. There is, however, the pure gauge anomaly to be taken into account. It is given by the term Tr A 4 /4! where A is the Yang-Mills curvature corresponding to G. Hence, to actually cancel the (perturbative) anomalies, either Tr AA must vanish or be factorized as r
TvA4
= J2 TTF^TrF",
(12.55)
I
in order for the Green-Schwarz mechanism to work. The resulting mixed anomaly is ^-TIR2TTA2.
(12.56)
202
12. Global Anomalies
The total anomaly polynomial, obtained by combining equations (12.54-5556), reads
Aoiai = E Pa T r F? TTFJ-
(12-57)
1=1
The term (lij denote an (r -+- 1) x (r + 1) symmetric matrix; F ( 0 ) stands for a given Lorentz algebra-valued Riemann curvature 2-form. Under the spell of the Green-Schwarz mechanism, /?,_, takes on the form: Ai = g ( a «Ti +
a
i1i)-
We cancel the anomaly by adding the counterterm - l E i = i a;7jw,j; here, w,j are the Chern-Simons forms, and g^ and H denotes the combination of H+„ and some k H~v tensor fields. In the N — 2 case, the global gravitational anomaly receives two additional contributions, in addition to that of the right hand side of equation (12.51). Setting ND = 2, NR = 2 and Ns = 1 gives A£det
= 2iri[mdR
+ (28fc - 274) ind Z? + (A - 1) | ]
,
.
{
-2xifBZl3ijTrF?TrF?.
Adding the variation of the Green-Schwarz counterterm (12.58), and the Pauli-Villars regulator fields does contribute to (12.59) as follows: A £ f f + A £ r e g = -2-Ki / ZJ2 7.TrF ; 2 . JdB fzi
(12.60)
The total global gravitational anomaly is then easily derived following the presentation given in the above Subsection 12.2.2. Explicitly, A£eff
= A £ d e t + A£'ea + A £ r e g = 2 « [ ( / f c - l ) f + p(B,dB)]
mod 2™.
U^bi]
Note that ji is a topological invariant whose explicit form is given by ?(B,dB)
= / E
/^TrifTrF,2 -
/
Z E
7
203
12.3 References Furthermore, Z is the gauge invariant strength, which we write as Z dH + YJi=\ c
=
i=i
12.3
References
[1] Bott, R. and Tu, L. W.: Differential Forms in Algebraic Topolgy, Springer-Verlag 1982 New York. [2] Witten, E.: An SU(2) Anomaly, Physics Letters B 117 No.5 (1982) 324-328. [3] Atiyah, M. F. and Singer, I. M.: Annals of Mathematics 93 (1971) 119. [4] Alvarez-Gaume, L. and Witten, E.: Gravitational Anomalies, Nuclear Physics B234 (1983) 269-330. [5] Witten, E.: Global Gravitational Anomalies, Commun. Mathematical Physics 100 (1985) 197-229. [6] Milnor, J. W.: D i f f e r e n t i a t e Manifolds Which Are H o m o t o p y Spheres, Princeton University Press 1959 Princeton, NJ. [7] Witten, E.: Global Gravitational Anomalies in String Theory, in Proceedings of S y m p o s i u m in Anomalies, Geometry and Topology, Eds. Bardeen, W. and White, A., World Scientific 1985 p. 61-99. New Jersey, London, Singapore. [8] Alvarez-Gaume, L. and Ginsparg, P.: Nuclear Physics B 243 (1984) 449. - Annals of Physics 161 (1985) 423. [9] Baadhio, R. A.: Global Gravitational Instantons and Their Degree of Symmetry, Journal Mathematical Physics 33 No. 2 (1992) 721-724. - On the Global Gravitational Intantons that Are Homotopy Journal Mathematical Physics 32 No. 10 (1991) 2869-2874.
Spheres,
[10] Baadhio, R. A.: Knot Theory, Exotic Spheres and Global Gravitational Anomalies, in Quantum Topology, by Kauffman, L. H. and Baadhio,
204
12. Global Anomalies
R. A.; p. 78-90, World Scientific 1993. [11] Baadhio, R. A. and Kauffman, L. H.: Link Manifolds and Global Gravitational Anomalies, Reviews in Mathematical Physics Vol. 5, No. 2 (1993) 331-343. [12] Green, M.; Schwarz, J. H. and Witten, E.: Superstring Theory, Cambridge University Press 1987, New York. [13] Baadhio, R. A.: Heterotic Superstring Gauge Residue Via Homogeneous CP4 Topology Change, Commun. Math. (1991) 251-264.
Trivialization Physics 136
[14] Bergshoeff, E.; Kephart, T. W.; Salam, A. and Sezgin, E.: Global Anomalies in Six Dimensions, Modern Physics Letters A Vol. 1 No. 4 (1986) 267-276. [15] Tanni, Y.: Phys. Letters B 145 (1984) 197. [16] Townsend, P. K.: Phys. Letters B 139 (1984) 283. [17] Gross, D.: The Heterotic String, in S y m p o s i u m in Anomalies, G e o m e t r y and Topology, Eds. Bardeen, W. A. and White, R. A., World Scientific, 1985 p. 299-313. New Jersey, Singapore, London.
Chapter 13 Mapping Class Groups and Global Anomalies Throughout this book, we have highlighted the role played by mapping class groups in various physical problems. In this chapter, we will explicitly work out the relationship between global anomalies and mapping class groups in three-dimensional topological quantum field theories, also known as ChernSimons-Witten theories. Our study will draw on the work of Baadhio from reference [14]. We will begin with a presentation for some three-dimensional homeotopy groups of 3-manifolds which have distinct topologies. The basis for this will be Baadhio's work in reference [20]. Then, in a second phase, we study the occurrence of global gravitational anomalies and the role that mapping class groups have in cancelling them. In doing so, we will follow the presentation exhibited in reference [14]. Our knowledge of mapping class groups of dimensions higher than two is very limited. Nonetheless, drawing on the work in reference [20], we will provide a detailed account of 3dimensional homeotopy groups. An essential ingredient of the present study is the Smale-Hatcher conjecture.
13.1
The Smale-Hatcher Conjectures
The Smale conjecture and the Smale-Hatcher conjecture are essential ingredients in our investigation of three-dimensional mapping class groups. We 205
206
13. Mapping Class Groups and Global Anomalies
thus begin by reviewing them. To state the Smale conjecture for the twodimensional case, we begin by choosing a properly-imbedded two-dimensional disc in Mg, a three-manifold of genus g > 2, endowed with at least one boundary component. It is a well known fact [1] that the difFeomorphism group of D2, when restricted to its boundary component dD2, yields a difFeomorphism / , which is the identity; in other words: Diff (£> 2 rel(0£> 2 )) = {/ : D2 -> D2 : f\dD2
= id}.
Smale conjectured in [2] that the mapping class group of D2 is trivial: 7T0Diff+ (D2xe\{dD2))
= 0.
He arrived at this conclusion by studying the inclusion of the orthogonal group O (n) into Diff(S"), the difFeomorphism group of the n-sphere with C°° topology. The conjecture is affirmatively solved whenever the inclusion is shown to be a homotopy equivalence. In actual life, there is an abundance of forms of the Smale conjecture; the best known example is that of Diff(S n ) ~ 0(n + l) x Diff
(Dnrel(dDn)),
for any n. The next in order of priority is a generalization of the Smale conjecture to dimension three. This is referred to as the Smale-Hatcher conjecture. According to this conjecture, for any D3 with one boundary component, its mapping class group is trivial as well: 7r0DifF+ (J)3 rel (dD3))
= 0.
This result follows mostly from Cerf [3], Smale [2], and Hatcher [4]. To complete the present background review, let us point out some useful results by Kirby-Siebenmann and McCullough-Miller [5] which will be of use later: Diff {Mgre\{dMg)) A homeo (M f l rel(0M„)), where a is an inclusion on all 7T; if n < 3. Note that under suitable conditions, this relation equates difFeomorphism groups to homeomorphism groups. As such, they shall prove essential in any investigation of Dehn slides in the study of mapping class groups.
13.2 Mapping Class Groups For 3-Handlebodies
13.2
207
Mapping Class Groups For 3-Handlebodies
This section, which is rather technical, deals with the algebraic topology determination and spresentation of mapping class groups for three-dimensional handlebodies. Specifically, we will be looking at the 3-handlebody with only one boundary component first, then in the later case study that of the 3handlebody with no boundary component at all. The case with one boundary component yields a trivial mapping class group. This renders any evaluation of global anomalies somewhat difficult for reasons that are explained in Subsection 13.4.1. Three-dimensional handlebodies, in which topological quantum field theories live, offer us an additional challenge: the lack of a one-dimensional time component (in contrast to non-proper 3-manifolds which are analyzed next). It would take us beyond the scope of this chapter to analyze them. The interested reader may want to look at the work in reference [20] for a perspective on this, including quantum gravity in its three-dimensional incarnation. 13.2.1
H o m e o t o p y Groups for 3-Handlebodies W i t h One B o u n d ary Component
We begin with a theorem of Baadhio [20]. Theorem 13.1 (Baadhio) Consider a three-dimensional handlebody Hg of genus g < 2. It possesses one boundary component with the identity on the boundary fixed pointwise by ve\dHg. We then have: 7T0 Diff + (HgTeldHg)
= 0.
(13.1)
The proof of Theorem 13.1 will occupy the remainder of this section. We shall rely on involutions within Hg, on the Smale-Hatcher conjecture and on a surgery formula to establish that the mapping class group of Hg with such a topology is indeed trivial. As a first step, notice that the interior of Hg consists of a two-dimensional Riemann surface, S s . Furthermore, there is a homeomorphism, called h, which takes Hg to itself. This homeomorphism becomes the identity on dint Hg — S fl ; by dmt we mean the interior of Hg. A sketch of Hg together with the isotope h is provided in Figure 13.1.
208
13. Mapping Class Groups and Global Anomalies
K'g = 3 =
Figure 13.1: Sketched here is a solid 3-handlebody of genus g = 3 with one boundary component.
13.2 Mapping Class Groups For 3-Handlebodies
209
The isotopy h satisfies the following important properties: h0 — id h\D = id ' where D is a two-dimensional, properly embedded disc in Hg] incidentally, it is depicted in Figure 13.1. We now put forward some useful definitions and notions. By a basic slide homeomorphism within Hg, we shall mean an involution involving h which consists of sliding the disc D in by the trace of h. More generally, any homeomorphism of Hg that uses the disc D and a path c will be referred to as a slide homeomorphism or a slide of the sliding disc D around the sliding path c. When the sliding path c is a well-defined arc however, the slide homeomorphism will be called a Dehn slide. With these descriptions in hand, we can now proceed with the proof of T h e o r e m 13.1. As a first step, we slighty perturb h by a small isotopy to the extent that h(D) involutions intersect transversally. Figure 13.2 provides an illustration of the procedure in question. Another perturbation of h(D) yields circles that are either of the intersection type dD — dh(D) or not. There are far more circles than we need to care about since we are interested only in identity-preserving involutions, that is, in intersections of the type D fl h(D). Therefore, a crucial task is to reduce the number of unwanted circles. To this end, we consider the innermost circles of D D h(D) which are exhibited in Figure 13.3. As h(D) sweeps through D and Hg, it does indeed generate a collection of bold black and regular black discs pictured in Figure 13.4. Their unions are precisely the spheres we are seeking to reduce. A good look at Figure 13.4 tells us that inside the disc there are some balls, B{. To selectively eliminate the unwanted circles, we use a variant of a theorem of Alexander [6, 7]; this theorem asserts that every properlyembedded two-dimensional disc (or sphere for that matter) in Hg is indeed the boundary of a three-dimensional ball 1. Pushing the (bold) discs across the balls in Figure 13.4 allows one to selectively eliminate the corresponding 1 There does not seem to exist a satisfactory reference for Alexander's theorem for the smooth case. It has been curiously neglected in some of the standard books on 3-manifolds. There is however, a rather simple and beautiful proof by Hatcher in unpublished notes. I thank him for bringing this to my attention.
210
13. Mapping Class Groups and Global Anomalies
h(D)
Figure 13.2: Pointwise boundary-fixed preserving homeomorphism involutions on Hg generated by small perturbations of the disc D.
13.2 Mapping Class Groups For 3-Handlebodies
211
h(D)
D
Figure 13.3: Cross view of arcs and circles generated by the intersections D fl h(D) from h(D) involutions.
212
13. Mapping Class Groups and Global Anomalies
D
Figure 13.4: Balls and discs generated by small perturbations of
h(D).
13.2 Mapping Class Groups For 3-Handlebodies
213
h(D)
D
Figure 13.5: Pushing the isotope h(D) to agree with D.
circles made up of D D h(D). eliminate the remaining circles.
We repeat the operation as necessary to
The second step consists of pushing h(D) to agree with D, once it is established that D n h{D) = dD = dh(D). A sketch of this procedure is provided in Figure 13.5. This amounts to a net decrease in the amount of slide homeomorphism involutions. The actual framework that allows us to achieve this result owes much to the Lemma 4.4 by Suzuki in reference [8]. By restricting h(D) to the identity, that is to h\D = id, we can perform a series of surgeries on Hg. We begin by cutting the 3-handlebody along D, as pictured in Figure 13.6.
214
13. Mapping Class Groups and Global Anomalies
CD
Figure 13.6: Initial surgery on Hg along D with induction on the genus g.
13.2 Mapping Class Groups For 3-Handlebodies
215
Figure 13.7: Repeated surgery on Hg with induction on g. This surgery procedure has an induction on the value of g: one keeps cutting Hg along Z), (Figure 13.7). This continuous use of surgery as depicted in Figure 13.7 reduces the Hg genus size to<7 = 3to<7 = 2to<7 = l, and finally to g = 0. The final result is that the handlebody Hg is reduced to D3, as shown in Figure 13.8. It is then just a matter of procedure to apply the Smale-Hatcher conjecture to show that 7r0Diff+ (HgreldHg)
= 7r0DifF+ (l> 3 rel <9£>3) = 0.
This completes the proof of T h e o r e m 13.1.
(13.2)
216
13. Mapping Class Groups and Global Anomalies
H,#,H 2
f~f\ f\\
a:H,Uh(D)H
(D) ' '2
vU w ~~" H,
H,
a
Figure 13.8: Repeated surgeries on Hg have reduced it to a 3-disc, D3. The Smale-Hatcher conjecture is then applied to determine the proper value of its homeotopy group.
13.2 Mapping Class Groups For 3-Handlebodies 13.2.2
217
H o m e o t o p y Groups of 3-Handlebodies W i t h N o Boundary Component
The principal result is contained in the following theorem of Baadhio [20]. T h e o r e m 13.2 (Baadhio) 7r 0 Diff + (# 3 ) = T* C 7r 0 Diff + (£ 9 ,*).
(13.3)
T* is the smallest invariant subgroup of the mapping class group, 7ToDiff+ (E 3 ,*); incidentally, it is a twist group of finite index, generated by canonical twist automorphisms. As for *, it denotes a basepoint in Hg while the group of +-isotopy classes of automorphisms that keeps + fixed is referred to as Diff (E g ,*). The first order of business in proving Theorem 13.2 is to find the canonical twist automorphisms mentioned earlier. This means looking at the automorphism group of 7Ti, where -Ki is the fundamental group of Hg, a free group of rank n with a * base with coordinates xi, • • • ,xn. The automorphisms of the fundamental group give important information about mapping class groups of 3-manifolds including the case of handlebodies. However, finding a presentation for the mapping class group of a 3-handlebody would require additional information about how the Dehn twists around the discs work. The reason lies in the fact that the twists induce the identity automorphism on the fundamental group, and therefore they cannot be detected from the automorphisms of the fundamental group. The automorphism group of Wi is generated by:
A(Xl) A(x2) A{xk) B{xk) C(*i)
C(xk) D{xx) D{xk)
= x2; — Xi] Xk\ = xk+v, = — xi > = Xk] = X\X2\ =
Xk\
(13.4)
218
13. Mapping Class Groups and Global Anomalies
Figure 13.9: Basepoint ^-involutions in Hg that are mapping class preserving.
13.2 Mapping Class Groups For 3-Handlebodies
219
where k = l,---,n and x„+i = x\. Nielsen's original work on automorphisms of surfaces [9] provides us with the right framework to determine the canonical generators of 7T0 Diff+ (E a ,*) = T* on the basis of the A, B, C and D automorphisms in equation (13.4). According to these, when g > 3, F* has six canonical generators; in contrast, when g = 2, there are exactly five canonical generators. In considering the case of interest to us (i.e. g = 3), the action on -K\ (Hg) of four of the six generators is A = B, C and D. The remaining two generators are defined by canonical twist automorphisms. These twists will prove essential in the Proof of Theorem 13.2. It is with this in mind that we aim to determine the twist automorphisms that generate the twist group r*. Firstly, let us clarify what is meant by a twist T(D). homeomorphism satisfying the relation: T(D)
It is essentially a
(re2*'*,*) = (re 2 "< fl+i >).
When restricted to dHg, r (D) is none other than a Dehn twist in 3D. The isotopy class r (D) in the full mapping class group 7r 0 Diff + (E s ,+) can be shown to depend only on the ambient isotopy class of D in Hg. It is precisely the collection of all isotopy classes of twist automorphisms that generates the twist group T*, a normal subgroup of the homeotopy group of H^s interior, E p , with basepoint •. In order to fully understand I \ , we need to address its finiteness. A previous work on the subject by McCullough (on virtually geometrically finite mapping class groups of 3-manifolds) in reference [10] is of interest. The trick is to make use of the compactness and orientability of Hg, and hence to show in the process that T* is contained in the kernel of the topological isomorphism 7r0Diff+ (E s ,*) -> Out (TTJ (Hg)),
(13.5)
or, in the optimal case, is equivalent to that kernel. What are the conditions for the kernel of the homomorphism in equation (13.5) to be finitely generated? The answer to this obviously depends on whether I \ itself is finitely generated. Thus, the whole issue of finiteness amounts to actually seeing whether T* is a geometrically well-defined subgroup (of the mapping class group), and of finite index. Below, we provide some arguments that prove the finiteness of T*.
13. Mapping Class Groups and Global Anomalies
220
Looking at the 3-handlebody Hg (Figure 13.9), there is a fibration
* n (B)
-
T B _ , ( F ) -> ^
( £ ) - • • •
^ ^
The exact sequence of homotopy groups in (13.6) is a consequence of the fibration and restriction p. According to formula (13.6), we have the sequence Diff+ (Hare\dHg) where Diff+ (Hgve\dHg)
-> Diff+ (Hg) A Diff+ (dHg);
(13.7)
= p~l (id). Mapping class group wise, (13.7) reads:
*i DifF+ (dHa) jr 0 Diff+(ff,)
-> 7T0 Diff+ (Hg reldH g ) -> -
7r0Diff+ (dmtHg = S 5 ) .
,
.
{i6
-°>
As a result of equations (13.6-7) we have the exact sequence 0 -> ker(a) - • 7r 0 Diff + (# s ) A [Aut(?n (Hg)) = Out (Aut Hg)\. (13.9) From (13.9), we learn that kera is given by Dehn twists and is not finitely generated. This is in contrast to ITQ DifF+ (Hg). We regard Aut (•K\(Hg)) as acting on the left of 7Ti (Hg) by the action of an injective cyclic group. The term a maps into the inner automorphism, Aut (wi (Hg)); the outer automorphism is a (finitely) generated infinite group. The exact sequence (13.9) yields the final result: TTQ DifF+ [Hgxe\dHg) - • TTQ Diff + (Hg\ ' y0 ' ' *s ' ^ TTO Diff + {dintHg = E 9 ) .
(13.10)
The first term in (13.10) is trivial by virtue of the theorem of Baadhio, namely T h e o r e m 13.1; consequently, the remaining part of T h e o r e m 13.2 will now involve /v, or more precisely, its kernel. It is a straightforward task to show that p^ is injective in (13.9). Thus, as a direct consequence, kera* = {0}. This, it turns out, implies that the mapping class group 7r0Diff+ (Hg) is isomorphic to the twist group T* C 7r0Diff+ ( £ s , + ) . Our proof of T h e o r e m 13.2 parallels, and somewhat strengthens, Waldhausen's work on sufficiently large 3-manifolds with incompressible boundaries (nowadays called Haken 3-manifolds) [11]. Waldhausen was among
13.2 Mapping Class Groups For 3-Handlebodies
221
the first to point out an equivalence relation between homeomorphism and homotopy equivalence. An ongoing problem in geometric topology concerns the classification of such homotopy equivalences, particularly for complicated 3-manifolds such as non-aspherical manifolds and manifolds which are nontrivial connected sums [12, 13].
13.2.3
A Presentation For I \ In Terms of D e h n Twists
When evaluating (13.10), we need to pay attention to homotopic basepoints in the handlebody, as they easily factor out A u t ^ i (Hg)/Out Aut {Hg). On the other hand, homotopic basepoints are generators of xi (Hg) and T*. Determining which ones are relevant is the goal of this subsection. The starting point of the discussion is Figure 13.10. Requiring diffeomorphism extensions of certain Dehn twists in Figure 13.10 is a good method for finding the relevant generators of IV For instance, the Dehn twist Qi doesn't extend to Diff (Hg) since it does not bound a disc. This is not true of a2 which extends along the disc D2, and thus to Diff (Hg). Hence, requiring the infinitely many Dehn twists to bound a disc or equivalently, to extend to Diff (Hg) is a highly non-trivial approach to separate the good Dehn twists from the trivial ones. There are, however additional constraints that need to be satisfied. They have to do with diffeomorphisms of the 3-handlebody that do not induce the identity on 7Ti (Hg). The Dehn twists along the discs /?, induce the identity on K\ (Hg) —> K\ (Hg), because one can always pick a diffeomorphism / : Hg —> Hg that induces an isomorphism on TTI (Hg) -* TTI (Hg). In order for us to present the relevant Dehn twists as generators of I \ , we need to decide which Dehn twists are the relevant ones, that is, we need to determine the ones that are not identityinducing in Wi(Hg), but are nonetheless homotopic to the remaining Dehn twists. The procedural approach is outlined as follows. As depicted in Figure 13.11, we apply a 180° rotational twist along the shaded area of Hg. Our goal here is to generate diffeomorphisms of the 3-handlebody that are not identity inducing. A good look at Figure 13.11 tells us that the half Dehn twist along 7 = T\ is such a diffeomorphism. When endowed with the opposite orientation, A = r^, (A) is equivalent to
222
13. Mapping Class Groups and Global Anomalies
Figure 13.10: Dehn twist involutions in Hg
13.2 Mapping Class Groups For 3-Handlebodies
180° twist
223
0° twist
Order x = 2, x 2 = id.
Figure 13.11: Selection rule by twist automorphisms in order to extract Dehn twists that are generators of T* = 7r0DirF+ (Hg).
224
13. Mapping Class Groups and Global Anomalies
A - 1 in %i (Hg). These are examples of diffeomorphisms that do not induce the identity on ni(Hg), but are nonetheless homotopic to the other Dehn
twists.
13.3
Homeotopy Groups of N o n - P r o p e r 3-Manifolds
We shall provide here a case study of mapping class groups of 3-manifolds which are topologically distinct from the two handlebodies studied above. These are manifold products, and essentially consist of a two-dimensional Riemann surface of genus g cross the time interval. And even though they are constitutionally different from handlebodies, we will see that they have similarities, for they exhibit trivial mapping class groups as well. In reality, the manifolds we are about to investigate are more or less similar. The only difference (perhaps trivial overall but very important for the computation of homeotopy groups) lies in the number of boundary components they have. There were the very first manifolds that entered in Witten's 1989 landmark paper on Chern-Simons-Witten theories [18]. To this day, they remain the most widely studied 3-manifolds for Chern-Simons-Witten theories (including quantum gravity). 13.3.1
Mapping Class Groups for £ g x [0,1] W i t h T w o Boundary Components
According to the following theorem of Baadhio in reference [20], a given 3manifold, E s X [0,1] (g > 2) with two boundary components has a non-trivial mapping class group, namely: T h e o r e m 13.3 (Baadhio) 7r0Diff+ (E a x [0,l]relE„ x {0,1}) 7r0DirT+ ( £ s x Z 2 ) « JTJ Diff+ (E B ).
(13.11)
In Figure 13.12 we provide a sketch of that 3-manifold. To prove Theorem 13.3, we start by picking an element of ir\ Diff+ (E s ). This element is represented by a loop ft of diffeomorphisms (with fo — f\ —
225
13.3 Homeotopy Groups of Non-Proper 3-Manifolds
I = [0,1]
S
g =2X
[0,1]
Figure 13.12: Thickened copy of Eg x [0,1], or equivalently, Eg x [0,1] with two boundary components. identity on Eg); it is defined by Hg (x, t) = ft (x), which in turn comes from the map tf : IT, (Diff Eg) -+ 7r0 (Diff+ ( £ s x [0,1], re\Eg x {0,1})) . Thus, we obtain a single diffeomorphism F of £ s x [0,1]: F E Diff (E s x [0,l]reld 2 ) -» F ( i , 0 =
(ft(x),t);
as shown in Figure 13.12. An essential ingredient in the proof is the realization that any path component of Diff+ (Eg X [0, l ] r e l E s x {0,1}) contains level -preserving elements,
226
13. Mapping Class Groups and Global Anomalies
that is, elements in the image of $ , so \& is surjective. On the basis of these observations, we have the relation 7ToDiff+ (E s x [0,l]reld 2 ) -% TTJ Diff(E s );
(13.12)
which, according to Waldhausen [11], is an isomorphism. On the other hand, from results of the author in reference [14] (see also [15]), it appears that the path component of the identity in Diff (E s ) is contractible if ^r > 1. Combining (13.12) with the contractibility property just mentioned gives:
^Diff^jJ*2 lH\ 13.3.2
.
(13.13)
Mapping Class Groups of E 3 x [0,1] W i t h One Boundary Component
The statement here is that the mapping class group of a 3-manifold £ 3 x [0,1], endowed with one boundary component, is trivial. This statement is best reflected in Theorem 13.4 (see [20]): T h e o r e m 13.4 (Baadhio) 7r0Diff+ ( E , x [ 0 , l ] r e l £ 3 x [0, oo)) « xj Diff+ (£ 3 x Z 2 ) ;
{
'
'
where Z 2 denotes the switching of the two ends of I = [0,1]. Before embarking on the proof of this fourth theorem of Baadhio, let us pause to clarify a topological subtlety: it is slightly incorrect for us to state that E 3 x [0,1] has only one boundary component. In real life, E s x [0,1] always has two boundary components: E 3 x {0} and E 3 x {1}. Hence, when we say of E 3 x [0,1] that it has only one boundary component, we mean that it is diffeomorphic to E 3 x [0, oo). This established, the proof of (13.14) is rather straightforward and draws its base from Cerf's work on diffeomorphism extensions [3]. In particular, it
13.3 Homeotopy Groups of Non-Proper 3-Manifolds
0
227
"X"
Figure 13.13: Sketched in (a): a path component, ft € Diff(E s ) which contains level-preserving elements; in (b): a loop within Diff (£ 9 ) whose homotopic involution (i.e. a 2xt = ft rotation) yields a single isotopic morphism F; in (c): a representation of the involution of F within the annulus A of the 3-manifold, S 5 x [0,1].
228
13. Mapping Class Groups and Global Anomalies
holds true if, and only if, g > 1. We have an exact sequence: 7r0Diff+ (E a x [ 0 , l ] r e l d ( E s x [0,1)) =0ifs>l
7To Diff+ (E s x [0,1])
7r0Diff+
E g x {0,1}
(13.15)
=dl
= (7r 0 Diff+(E 9 )) x (x 0 Diff+(E 3 ) x Z 2 ) . Note that p is injective and acts to restrict the mapping class group to the boundary. E 3 x {0,1} consists of two copies of E s . The image of p is built out of the elements (x,x) £ 7ToDiff+(EJ) x 7ToDiff+(E5) x Z 2 . Isotopic diffeomorphisms of E s , as described by Birman in reference [1] are then chosen to complete the proof of T h e o r e m 13.4.
13.4
Global Anomalies and 3-Homeotopy Groups
The purpose of this section is to demonstrate the absence of global gravitational anomalies for a small class of 3-manifolds in which Chern-SimonsWitten (CSW) theories are defined. The fact that these theories are pathologyfree relies on an intricate relationship between mapping class groups and large difFeomorphism transformations. In effect, we will show that the absence of global anomalies amounts to proving that under large diffeormorphism transformations, the variation of the effective action is continuously connected to the identity, that is, there is no gap or occurrence of disconnected general coordinate transformations. If such gaps existed, one would not expect the variation of the effective action to be invariant under the group of diffeomorphisms that cannot be smoothly deformed to the identity. Rigid deformations of this type are almost exclusively classified (or detected) by homeotopy groups, and this fact alone explains the choice of finding large variations of the effective action which are mapping class group invariant. We will write £csw for the Chern-Simons-Witten effective action. The mapping class group is defined, as above in Section 13.2, with the same notations. For additional references on global anomalies and mapping class groups, we suggest to the interested reader references [16, 17], which are essentially extensions of the work in reference [14].
13.4 Global Anomalies and 3-Homeotopy Groups
229
Consider M " , a manifold of dimension n = 3. In previous sections, when n = 2 + 1, we referred to Mn as a non-proper manifold (a product manifold, e.g. a Riemann surface cross R 1 ), in contrast to n = 3 (handlebody case). To this manifold is associated a compact gauge group, G, and a principal bundle, E. There is a connection on E (e.g. a Lie algebra valued one form, G), which we write as yif, where i is the tangent index to M", and fi is defined over a basis of the Lie algebra Q. Infinitesimal gauge transformations are of the form: Ai -> At - V,e; and e is the generator of G. Another useful ingredient is the covariant derivative, V, V; e = di e - di e + [Ai, e], which gives rise to an equally important form, the curvature 2-form: Fij = [V.-.V,-] - d&j
- djVi + [V.-.V,-].
The Chern-Simons-Witten effective action must be defined as a topological invariant. In so doing, we learn to avoid the standard Yang-Mills Lagrangian, CYM = / ^g gik gil Tr (FtJ Fkl) JMn
since it is metric dependent. A good compromise is to pick the integral of the Chern-Simons form, £csw
= h V
Tr (A A dA + | A A A A A)
= £ / e'>* Tr {{Ai (djAk
- d^)
+ | A{ [A,, Ak}) .
The formula, T) =
d •
TjSI!iv
in reference [18], determines the purely gravitational operator, ^grav Moreover, it is used to write down the phase factor / xd'K \ A = exp ( . — • ?7grav I .
(13.17)
As it stands, equation (13.17) needs to be regularized if general covariance is to apply; furthermore, we point out its metric dependence since two regularizations differs by a local counterterm. This in turn spells out the need
230
13. Mapping Class Groups and Global Anomalies
to find a counterterm which is also metric dependent. In reference [18], it is shown that a multiple of the gravitational Chern-Simons term is that ideal counterterm: /„ = — / T r H i j i - u A w A u ] 3 Air JM« V 3 I
(13.18) '
K
where the subscript g denotes the background metric dependence, and u> stands for the Levi-Civita connection on Mn. Essentially, the gravitational anomaly counterterm is obtained by indirect derivation, often proceeding via conformal field theory. On the other hand, there are, so far, only two possible forms for the CSW counterterm: it can be written in terms of a spin connection or an affmeconnection [18]. In the first form, the anomalous transformations are the SO (2,1) gauge transformations (i.e. transformations of the local Lorentz type), while, in the second form, they are diffeomorphisms. The relation between these two forms is very poorly understood [14, 16, 17, 18]. There is no doubt that any progress on this front will provide us with a better undertanding of the occurrence and manifestation of global anomalies in topological quantum field theories. Let us now review the nature and manifestations of global gravitational anomalies in Chern-Simons-Witten theories . The first symptom is encoded in the ^-invariant. Path integral computations carried out by Witten [18] reveals that, to the lowest order of perturbation, the action for a given CSW theory requires a counterterm proportional to the gravitational CSW action. This is precisely the term we encounter in formula (13.18). Since the gravitational CSW action (13.18) is not invariant under large diffeomorphism transformations, this term represents, or otherwise exhibits, a global gravitational anomaly. The second symptom has to do with the the Wilson-line anomaly. This anomaly appears as a framing dependence [18] because the choice of the framing is mapping class group invariant as well. Finally, there is third manifestation which was the subject of a brief presentation in Chapter 8: the canonical quantization of CSW theories makes succinct the relationship between global anomalies and mapping class groups. Roughly speaking, the quantization exhibits an anomaly that shows up as a dependence of the wave function on the moduli space of (non-proper) three-manifolds. The canonical quantization discussed in Chapter 8 essentially gives a projectively
13.4 Global Anomalies and 3-Homeotopy Groups
231
flat -but not flat by nature- bundle over the moduli space with a curvature given by the anomaly. The mapping class group 7r0 Diff4" (Mn) is then known to have a non-trivial action on this bundle. There is an additional anomaly, namely, the Virasoro global gauge anomaly, in its three-dimensional incarnation. This anomaly shows up when one attempts to centrally extend the mapping class group from dimension two to three using the central charge. This peculiar class of global pathologies was initially analyzed in [19] and, as far as we know, would benefit from additional extensive investigations. We will not discuss them here, referring the interested reader instead to [19]. It should be pointed out that the instances of global anomalies detected by the use of mapping class groups dissscussed here are far from representing the whole story. Several approaches aimed at providing a general framework to detect global anomalies in quantum field theories are in the works. One approach, for instance, targets classical invariants of low-dimensional topology and studies their extension to higher dimensions. In higher dimensions, these knot invariants give rise to exotic spheres. As such, the procedure constitutes a powerful test to detect global anomalies simultanously in low dimensions (e.g. three) and in higher ones (e.g. six, seven, eight, ten, and perhaps beyond eleven). This approach was initiated in [17] by Baadhio and Kauffman, with an additional survey article by Baadhio in [16]. What is still lacking though on the topological side is the equivalent of Witten's formula for global anomalies, which would universally inform us of the presence of global anomalies. Research activities are still ongoing though, and hopefully we will find it. Another approach to the detection and cancellation of global anomalies is that of torsion in mapping class groups. The initial observation that global gravitational anomalies could be understood in terms of torsion is due to Freed [21]. His observation was later applied to the study of the string world sheet global anomalies in [22]. In the mapping class group context, the issue is rendered somewhat manageable by the stability (resp. instability) of the the torsion classes in the homeotopy groups. Torsion stability of mapping class groups could be expressed in terms of the virtual cohomology dimension and the stability range is given by the Harer's stability theorem. Both notions were discussed in Chapter 4 (Section 4.6). An additional tool is perhaps the Farrell-Tate cohomology. This cohomology theory is excellent at detecting some periodicity of the mapping class group. Thus, if somehow combined
232
13. Mapping Class Groups and Global Anomalies
with Harer's stability theorem, there is no doubt that they will provide us with a powerful tool in the analysis of global gravitational anomalies. One can interpret the unstable torsion class of the homeotopy group as being precisely the group of all disconnected general coordinate transformations, although the correct geometric topology picture of this approach remains to be worked out. In the unstable range, the Farrell-Tate cohomology may act as the marker for the anomalies, with respect to diffeomorphism transformations that are continuously connected to the identity. This may provide us with an important quantization tool and a way to easily cancel the anomalies. Details regarding this method will be reported in [23].
13.4.1
Global Gravitational Anomaly-Free Chern-Simons-Witten Theories
The purpose of this subsection is to establish the absence of global anomalies in Chern-Simons-Witten also known as topological quantum field theories. We will follow the presentation of Baadhio in [14]. Let 7ToDiff+ (M n ) denote (non-zero) homeotopy group of a 3-manifold. We ask that (Mn) be either a three-manifold of the type Eff x [0,1] or a handlebody. By E 5 , we mean a two-dimensional Riemannian surface of genus g > 2, while [0,1] labels the associated one-dimensional time component. In order to demonstrate the absence of global anomalies while varying the effective action Ccsw under large diffeomorphism transformations, we need to show that the resulting changes in the effective action are continuously connected to the identity, that is, the collection of values taken by Ccsw while sweeping through the large diffeormorphisms cannot smoothly be deformed to the identity. When such collections are susceptible to non-smooth deformation with respect to the identity, then the theory suffers from disconnected general coordinate transformations, known simply as global gravitational anomalies. The only known diffeormorphism group that cannot be smoothly deformed to the identity (i.e. does not admit general disconnected diffeomorphic transformations) is the mapping class group, or homeotopy group. This is why it is important to establish Ccsw invariance under the mapping class group under those transformations. Not only does the invariance prevent global anomalies from arising, but further, it acts to preserve the consistency and uniqueness of the theory considered. The presence of global anomalies
233
13.4 Global Anomalies and 3-Homeotopy Groups
in this case would almost certainly violate the theory's uniqueness. Once again, it is easy to draw the conclusion reached a decade ago by Witten [24] that global anomalies act to severely limit the number of possible consistent theories in any dimension. A question of interest we are faced with at this point has to do with Theorem 13.1 and T h e o r e m 13.4 which exhibit trivial homeotopy groups. What are the mechanisms in place to check the occurrence and manifestations of global gravitational anomalies when the mapping class group is zero? Currently, there is no known answer to this question, but there is little doubt that if answers are found, they will considerably strengthen our understanding. From the group of orientation preserving diffeormorphisms, Diff+ ( M n ) we extract an element, A, and require that it obey the relation: Ccsw(X)
= Ccsw = Ccsw (id),
(13.19)
for all A; id is the identity. We define ht, a one-parameter family of A, by ht = Xt 0 < t < 1, to keep track of A's variation within Diff+ (Mn). between zero and one, we note that
Incidentally, with values
ht € 7r 0 Diff + (M n ) C Diff+ (M n ). The large diffeomorphism transformation cf> x id : Mn -> Mn,
(13.20)
(e Diff+ ( M n ) ) i nduces the map Mn **idjr*T<*>
Mn
A g.
(13.2i)
Recall that Ai is the 1-form connection on Mn, and ui its gravitational counterterm; Q is the Lie algebra in which the connection takes its values. The term* (A{), is by definition, equivalent to * (u>) and is obtained by pulling back along the formula (13.21).
234
13. Mapping Class Groups and Global Anomalies
Next, we replace A, and w by the variation <j>* (A;) (or similarly, by* (w)). As a consequence, the efFective action in formula (13.16) changes by £ {**(*))
= £ fMn Tr [
dp(Ai)
+ lr(At) A < m - ) A^(A,)] = £ /M« ^ T r [^(A) ( a ^ M t - a^M,-) +
[l6
-^}
lPA,[A3,Ak}}.
These changes apply to the gravitational counterterm (13.18), namely L = — ( Tr ( V w A d*u + - <j>*u A <^>*w A ^ w ) . 4x 7 M " \ 3 /
(13.23)
Now we consider the equivalence relation ir (>) = x (#) for transformations strictly related to the mapping class group . We write it as •K : Diff + (M n ) — 7r 0 DifF + (M n ).
(13.24)
This allows us to verify that under large difFeomorphism transformations, the following relation holds: £ c s w ( r ( A ) ) = 4 (%>))•
(13-25)
Going back to the homeotopic term ht, we consider a set of difFeomorphism transformations a, /?, which obeys (13.24) and for which h0 = 0; /ij = /?. With these values specified, formula (13.25) takes the form £ c s w ( a ( A ) ) = Ig(0(«>))-
(13-26)
To show that equation (13.26) is invariant under 7To DifF (Mn), it suffice to show that it should be invariant under the variation SCcsw ( e ' V ) ~ 6Ig ( e ' ^ + V * ) , 0 < e < 2*.
(13.27)
Recall that ht belongs to the (larger) homeotopy group, the mapping class group 7r0Diff+ ( M n ) , so long as t is bounded by 0 < t < 1. Consequently, ht is none other than ht (e'*,e'e) = ( e ' ' ( f i + V * ) , (13-28)
235
13.5 References and we easily see that mod 2w SCcsv/((Ai)ht)
= SIa{u)
(13.29)
the theory is mapping class group invariant. In other words there is no disconnected general coordinate transformations that show up as we are varying the CSW effective action under large diffeomorphism transformations , and therefore, we conclude that Chern-Simons-Witten theories are global gravitational anomaly-free. In the process, we have recovered Witten's formula for the absence of global anomalies. (We have 6 denoting large transformations in contrast to A in previous sections so as not to confuse it with the 1-form connection, A, defining the CSW effective action.)
13.5
References
[1] Birman, J.: Braids, Links, and Mapping Class Groups, Princeton Annals of Mathematical Studies 82 Princeton University Press, Princeton, NJ 1975. [2] Smale, S.: Diffeomorphisms Society 10 (1959) 621-626.
of the 2-Sphere, Proc.
Amer.
Math.
[3] Cerf, J.: Sur les Diffeomorphismes de la Sphere de Dimension Trois ( r 4 = 0), Lecture Notes in Mathematics Vol 53 Springer-Verlag 1968. [4] Hatcher, A.: A Proof of the Smale Conjecture, Diff(S 3 ) ~ Annals of Mathematics 117 (1983) 553-607.
0(4),
[5] Kirby, R. and Siebenmann, L: Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Princeton Annals of Mathematical Studies 88 (1977). - McCullough, D. and Miller, A.: Homeomorphisms of 3-Manifolds W i t h Compressible Boundary, Memoirs American Mathematical Society, Vol. 61 Number 344 1986. [6] Moise, E.: Geometric Topology in Dimensions 2 and 3, Graduate Texts in Mathematics, Vol. 47 Springer-Verlag, 1977. [7] Alexander, J. W.: An Example of a Simply Connected Surface Bounding a Region Which is Not Simply Connected, Proc. Nat. Acad. Sciences
236
13. Mapping Class Groups and Global Anomalies
USA, Vol. 10 (1924) 6-8. [8] Suzuki, S.: On Homeomorphisms of Z-Dimensional Handlebody, Canadian Journal of Mathematics Vol. 29 (1977) 111-124. [9] Nielsen, J.: Die Isomorphismmengruppe Ann. 91 (1924) 169-209.
der Freien Gruppen, Math.
[10] McCullough, D.: Twist Groups of Compact 3-Manifolds, Topology, Vol. 24 Number 4 (1985) 461-474. [11] Waldhausen, F.: Eint Klasse von 3-Dimensionale Manigfaltigkeiten I, Invent. Math. 3 (1967) 303-333; and II (Idem) 4 (1967) 87-117. - On Irreducible 3-Manifolds 87 (1968) 56-88.
Which Are Sufficiently Large, Ann. Math.
[12] Kalliongis, J. and McCullough, D.: Pacific Journal Mathematics, Vol. 153 Number 1 (1992) 85-117. [13] Swarup, G. A. : Homeomorphisms Vol. 16 (1977) 119-130.
of Compact 3-Manifolds, Topology
[14] Baadhio, R. A.: Global Gravitational Anomaly-Free Topological Field Theory, Physics Letters B299 (1993) 37-40. [15] Earle, C. and Eells, J.: Bulletin American Mathematical Society 73 (1967) 557. [16] Baadhio, R. A.: Knot Theory, Exotic Spheres and Global Gravitational Anomalies, in Quantum Topology, Kauffman, L. H., and Baadhio , R. A. (Eds.). 78-90 (World Scientific) 1993. [17] Baadhio, R. A. and Kauffman, L. H.: Link Manifolds and Global Gravitational Anomalies, Reviews in Mathematical Physics Vol. 5 Number 2 (1993) 331-343. [18] Witten, E.: Quantum Field Theory and the Jones Polynomial, Communications in Mathematical Physics 121 (1989) 351-399. [19] Witten, E.: The Central Charge in Three Dimensions, Phil. Trans. Royal Society London, A-Mathematical and Physical Sciences Series 329 No. 1605 (1989) 349-357. [20] Baadhio, R. A.: Mapping Class Groups for D — 2+1 Quantum Gravity and Topological Quantum Field Theories, Nuclear Physics B441 (1995)
237
13.5 References Nos. 1-2 383-401. [21] Freed, D.: Determinants, Torsions and Strings, Mathematical Physics 107 (1986) 483-513.
Communications
[22] Witten, E.: Global Anomalies in String Theory, in S y m p o s i u m on Anomalies, G e o m e t r y and Topology, Eds. Bardeen, A. and White, A. p. 61-99 (World Scientific) 1985. [23] Baadhio, R. A.: Mapping Class Groups and Global Anomalies, appear.
to
[24] Witten, E.: Global Gravitational Anomalies, Communications Mathematical Physics 100 (1985) 197-229.
Chapter 14 Exotic Spheres by Louis H. Kauffman In Chapter 12, the role played by exotic spheres in the detection and cancellation of global anomalies was extensively analyzed. The purpose of this present chapter is to give a resume (in the signature case) of the mathematical background involving characteristic classes that implies the existence of exotic spheres. To this end, we first review some basic facts about Chern classes, Pontrjagin classes, and the Hirzebruch index theorem. These facts are then marshalled to prove the existence of exotic spheres; in particular, the Milnor seven-sphere, S, and its relatives (see [1] for more information). First, recall the infinite complex projective space CP°° and its interpretations for line bundles and cohomology: Let [X, CP°°] denote the homotopy classes of mappings of a space X to CP 0 0 . Then this homotopy set is isomorphic with the second cohomology group of X: H2 (X) Q* [*,CPco]. This follows from the fact that CPoo is a K (Z, 2), a space whose homotopy groups all vanish except for a Z in dimension two. It follows from the construction of CP°° that [X,CP 2 ] S* C {X), the isomorphism classes of complex line bundles over X. In this case, we have the canonical line bundle A over CP°°, and a map / : X —•* CP°° induces a line bundle / * A over X: 238
239
/*A X
—+
A
-^-+ CP°°.
If i E H2 (CP°°) denotes the generator of the cohomology ring of CP°°, then the first Chern class of /*A, c\ (/*A), is found by taking the pull-back of i via / : ci(f*A) = f*(i)€ H2(X). It is also not hard to see that C (X) — H2 (X) as groups, with tensor product of line bundles corresponding to addition in H2. The first Chern class, Ci, can be interpreted as the self-intersection number of the 0-section of the corresponding bundle. More generally, let E —£-> B be a complex vector bundle. Then, there exist Chern classes c, (E) 6 H2t (B;Z) satisfying the following properties: 14.0.1
Properties of Chern Classes
(0) c, (E) = 0 for i > n = complex fiber dimension of E. c{E) = 1 +Cl(E)
+ c2(E)
+ ••• +
cn(E),
defines the total Chern class. (1) If E and E are complex bundles isomorphic over B, then c(E)
=
c(E).
If E - ^ B and / : B — • B, then f*c(E)
=
c(f*E).
(2) c (E © E) = c(E)c(E), where the product denotes cup product in the cohomology ring of B, and E and E are complex bundles over B. (3) Let A —> CP°° be the canonical line bundle. Then «h (A) = i e H2 (CP°°) as described above. Similarly, if A —> S2 is the canonical line bundle over S2, then ci (A) = g G H2 (S2) is the generator.
240
14. Exotic Spheres
It is known (the splitting principle) that given a complex bundle E —£-• B1 then there exists a mapping / : B — • B such that / * injects the cohomology of B into the cohomology of B and f*E\s& direct sum of line bundles. Thus we can write f E S Lx ffi L2 ffi • • • ffi Ln whence f*c(E)
= c(f*E)
=
=
c(L1)c(L2)--.c(Ln)
n L i (i + cx (/*))• •
In this way, we see that the higher Chern classes can be expressed in terms of elementary symmetric functions of line bundles. • Example Let E = T C P n = the tangent bundle to CP n . Explicitly, CP n =
S^+'/S1
where S1 is the unit complex numbers, S2n+1 = {(z0,zu.-.,zn) If z = {z0,zi,-
G C" +1 ||zo| a + ••• \zn\2 = l } .
• • ,zn) and A G S1 then \z
= \\z0 Azi, • • •, \znj.
{ [ u , v ] } | ||u|| = 1, u • v = 0, (u,v) ~ (Au, Au)}. Here, u,v 6 5
E = 2n+1
and [u, v] denotes the equivalence class of the pair (u, v) under the ^'-action. Let An —y C P n denote the standard line bundle. Then, An = {[u,p}\u
€ S2n+\P
e C, (u,p) ~
(Xu,Xp)}.
Let E' - An © An ffi • • • ffi A„ (n + 1 copies). Then, E' = {[u,v]\{u,v)
G S2n+1 x C " + \ (u,t;) ~ (Au, Xv)}.
Hence, E' D E, and note that E' has the cross section u >~> (u,u). Therefore, E' = T C P „ ffi e, where e —> C P n denotes the trivial bundle in one complex dimension. We conclude that
c ( r C P J = c(E') = (l + Cl(A„))B+1. Letting an = Ci (A„) be the generator of H2 (CP n ), we have the formula c ( r (CP n )) = ( l + a n ) n + 1 .
241 14.0.2
Pontrjagin Classes
If E -£-» B is a real vector bundle, then we get an associated complex vector bundle E — E <8>R C. Note that E and its complex conjugate bundle are isomorphic, i.e. E* = E. This implies that 2c2;+i (E) = 0. We define the ith Pontrjagin class, Pi (E), by the formula: Pi(E)
= (-l)'c2i(£®RC)
H4i(B),
€
and the total Pontrjagin class by the formula P(E)
= 1 + P1(E)
+ ••• +
P[n/2](E),
where [M] denotes the greatest integer in M. It then follows that l(P{E
© £") - P{E)P{E'))
= 0.
The following Lemma (whose proof we omit) is useful. L e m m a 14.1 1.) Let tu be a complex vector bundle. Thenijj^®C (Here WR denotes u> regarded as a real vector bundle.) 2.)
If Pk = Pk (WR), ck = ck (w), then 1 - Px + P2
= OJ®UI*. ± Pn
=
(1 - Ci + C2 - C3 + • • • ± C„) (1 + Cj + • • • + Cn).
• Example. r = r CP", c(r) = (1 + a ) n + 1 . P t = Pfc (TR) . Then 1 - P, + P2 = ( l - a ) " + 1 ( l + a ) " + 1 = ( l - a 2 ) ' 1 + 1 . H e n c e l + P 1 + --- + P n = (1 + a2)n+l . Hence P t (CP") = (U +
1
)a2k.
Now we apply the Pontrjagin classes to study manifolds. Let M4n denote a smooth, compact 4n manifold without boundary. Let M4n € H4n (M; Z) denote the fundamental class of M4n, and suppose that i\ + • • • + ir — n, where 0 < ik < n. Let / denote the sequence i'i,---,i r and define the Pontrjagin number Pi [M4n] by the formula P,[M4n]
=
{pi---Plr,M4n)
14. Exotic Spheres
242
where the brackets denote the evaluation of the product P t l , • • • P{r on the fundamental class. For instance, from our last example, we see that
The following theorem is basic to the relationship of Pontrjagin classes and cobordism. T h e o r e m 14.1 If the smooth manifold M4n is the boundary of a smooth (An+l)-manifold B4n+l, M4n = dB4n+1 then all Pontrjagin numbers Pi (M4n) vanish. Proof. Let fi& denote the fundamental class in H^n+\ (B, M). Then dfiB = (i\r, where d : H4n+i (B,M) —• HAn (M) is the homology boundary mapping. Furthermore, if v £ H4n (M) then (v, dfis) = (Sv, / / B ) , where 6 : H4n(M) -> H4n+l (B, M) is the coboundary map on cohomology. Now, we know that TB\M = rM © e, hence P{ (TB\M) = Pi (TM). It then follows directly from the exact sequence H4n {B) -?-> H4n(M) -?-> H4n+1 {B,M) that 6 (Pi) = 0. Therefore, Pi (M4") = = = =
(Pi, HM) (Pi, due) (S(Pi), M 0.
This completes the proof • Thus we have shown that the C F 2 n are not oriented boundaries. In fact, more is true. We can let Qn denote the oriented cobordism group of an ndimensional smooth manifold. (Two oriented manifolds An and Bn are said to be cobordant if there exists an oriented (n + l)-manifold Cn+1 such that dCn+1 = An U (-Bn), where -Bn denotes Bn with the reverse orientation. A manifold, An, is cobordant to 0 if Bn can be taken to be empty. 0 produces an inverse in cobordism classes since 6 (An x I) = An U (—An) and it is easy to see that the connected sum An U Bn is cobordant to the connected sum Ani(iBn. Thus, An$(-An) is cobordant to 0 . )
243 Cln is a ring with addition the operation of connected sum (j|) and multiplication the cartesian product. It is known that fin is finite for n ^ 0 (mod 4) and that Cl4k
the
= a (jM4k) .
= a (M,4*) a (M 2 4i ) .
Thus a : fi* — • Z is a homomorphism from the cobordism ring to the integers. The Pontrjagin numbers already obey 2) and 3). We need to cook up property 4). For this, we need the concept of a multiplicative sequence: Let flbea commutative ring with unit, 1. Let A* = (A0, A1, A2, • • •) be a graded i?-algebra. Let A* = \ ao + ai + a2 -f • • • |a; £ A11 be the associated formal power series ring. Let Ki (x\,x2, • • •, a;,) be a sequence of polynomials such that each Kn is homogeneous of degree n. Let K\
A* — • Arvia.K(a)
= l + K^ai)
+ K2(aua2)
We say that K is multiplicative if K(ab) = K(a)K(b)
+ #3(01,02,03) + •••. for all a,b £ A*.
244
14. Exotic Spheres
L e m m a 14.1 Given a formal power series f(t) = 1 + A^ -f A2t2 + • • •, there exists a unique multiplicative sequence {K„} such that K(l + t) = f(t). Proof. For uniqueness, let A* = R[t\, t2, ••• ,tn] and a — (1 +
= K(l + <j) K(l + t2) - • • K{\ + tn) = f(h)f(t2)
Thus K(o~\, <72, • • • ,crn) is uniquely determined by f(t). are algebraically independent, this proves uniqueness.
•••
f(tn).
Since iTi,(J2, • • • ,crn
For existence, let / = i\i2 • • • i r be a partition of k and define J2t'it2---t'r
Si (
where this sum means that we sum over all choices of r-subsets, thereby obtaining a symmetric function and hence a polynomial in the elementary symmetric functions <Ji,o~2, • • • ,crn. These polynomials form a basis for the symmetric homogeneous polynomials of degree k in the variables ii, t2, • • •, tn. Thus, letting A/ = Atl At-2 • • • A,r, we can write Kn (°"ir ••,••• crn) = 2 J
A/5/(CTI,
• • • ,crn)
I
where / ranges over all partitions of n. It follows that
5/(a6) = J2
SH(a)Sj(b),
HJ=I
where HJ denotes the partition obtained by juxtaposition. Hence K(ab)
= £/
XIS,(ab)
= Ei
A/ J2HJ=[
= =
HH,J
SH(a)
^H SH{O) \J
K(a)K(b).
Sj(b)
Sj(b)
245 This completes the proof • Now let {Kn (x\ • • • xn)} be a multiplicative sequence of polynomials with rational coefficients. Let M4k be a smooth compact oriented 4A:-manifold. Define the /{"-genus of M4k by the formula K\M4k]
= Kk[M4k] (Kk ( A , • • • , Pk),
[M4k]),
where Pi denotes the ith Pontrjagin class of TM- If 4 j does not divide dim (M), define K [M] = 0. L e m m a 14.1 / / {Kn) is any multiplicative sequence with rational coefficients, then the correspondence M — i » /('[M] defines a ring homomorphism fi* —> Q and hence an algebra homomorphism fi* <8> Q — • QProof. We need only check the behavior on products. M x M' has total Pontrjagin class P x P' modulo elements of order 2. So K ((P x P')) = tf(P) x A-(P') and
{-irm'{K(P),ri(K(P'),S).
Now we can state and prove the Hirzebruch index theorem. T h e o r e m 14.1 ( H i r z e b r u c h ) Let {Lk} be the multiplicative sequence of polynomials corresponding to f{t) = \/i/tanh (y/i). Then a (M4h)
= L[M4k}.
Proof. By the quoted result on Cl^ ® Q, it suffices to check the theorem for Lk[CP3k\. Here P = (1 +a2)2k+1. Since L{1 + a2) = v ^ / t a n h ( V ^ 2 ) , L{P) = (a/ta,nha)2k+1. Hence L[CP 2i ] = (L(P),fi) equals the coefficient of a2* in ( i (1 -f a 2 )) 2 + 1 . We check this coefficient by residues. Let utanh(z) = (ez - e~z) / (e z + e~z) .
246
14. Exotic Spheres
Then du — (1 — u2)dz whence du dz = — ^ — = {l+u2 1 — u*
+ u4 + ---)du. 2/t+l 2Jt+l
~ 2^7* Uanh*/
= 1. Hence, L[CP 2fc+1 ] = 1 = a ( C P 2 i ) . This completes the proof • Here are some useful facts about the series x/i/tanh [y/t) •
v^/tanh (Vt) = 1 + i* - I * ' 3
45
+
- • • • + ( - 1 * - 2 -2fc ^ V
(
+
where B^ is the &f/l Bernoulli number. The first few L-polynomials are:
14.1
L\ L2
= 3 Pi, = i(7P2 -A2),
L3
= gh (62P 3 - 13P 1 P 2 + 2 A 3 ) .
Exotic Spheres
The example that we are about to discuss is not the first example of an exotic differentiable structure on a sphere, but it is diffeomorphic to that example. The first example is due to Milnor [2] and produces a non-standard differentiable structure on a sphere of dimension seven. The example we are about to discuss is due to Brieskorn [3]. The Brieskorn examples arise from studying algebraic varieties associated with polynomials of the form
f(z) = za0° + z? +
...+z*»
where a o , « i , • • • ,fln a r e positive integers and the z,'s are complex variables.
247
14.1 Exotic Spheres Let V(f)
denote the variety of / : V(f)
Let K(f)
= {z G Cn+l\f(z)
= 0}.
denote the intersection of this variety with the unit sphere in C n + 1 :
K(f)
= V(f) n S2n+1 = {zeCn+l\f(z)
= 0 a n d | z | = \z0\2 + ••• + \zn\2 = 1}.
It is not hard to check that V(f) is a manifold away from 0 G V ( / ) and that the intersection of V{f) with S2n+l is transversal. Hence, K2n~1(f) is a smooth manifold of dimension 2rc — 1. Under these conditions, the manifolds K2n~l (/) are sometimes homeomorphic to spheres and sometimes cannot be diffeomorphic to standard spheres. A case in point is K7{f) for / = zza + z\ + z\ + z\ + z\. In general, let E (a 0 , d\, • • •, an) denote K(f) for / = ZQ° + z\x + • • • + z*n. Thus we assert that E (3,5,2,2,2) = E 7 is an exotic sphere. We shall finish this chapter with a number of different points of view on this fact. Here are the facts that we will show: 1) E 7 = E (3,5, 2, 2, 2) is the boundary of a smooth 8-manifold of signature - 8 : E 7 = dNs, a(N8) = - 8 . 2) E 7 is homeomorphic to a 7-dimensional sphere. With these facts in hand, the exoticity of E 7 is proved as follows: An extra fact about the manifold A^8 is that it is connected and has vanishing homology except in dimension four. We can form the topological manifold M 8 = A^8 UE D8 where D8, js a standard 8-ball. If M8 is a smooth manifold, then a(M8) = L[M8]. But H*(M) = 0 for * ^ 4,8. Pi P2
G G
H\M8) H8(M8);
and we have a(M8)
= L2 (M 8 ) = ^ - 8 = -L [p2(M8)
\P2{M8)
- P2(M8)} ;
- P2(M8)} •
14. Exotic Spheres
248
S 2n+1
Figure 14.1:
249
14.1 Exotic Spheres - 8 • 45 = 7 [P2(M8) - 2 3 • 32 • 5 -
7 \P2{M8)
- P?(M8)} - P?(M8)]
; .
Since (P2{MS) - P2(M8)) is an integer and 7 does not divide - 2 3 • 3 2 • 5, we conclude t h a t Ms does not have a differentiable s t r u c t u r e . Since E diffeomorphic t o S7 would allow a differentiable s t r u c t u r e on M 8 , this shows t h a t E is not diffeomorphic t o S7. T h u s E is an exotic sphere. For t h e record, Milnor's original e x a m p l e [2] of an exotic 7-sphere was constructed as follows: For each (h,j) G Z © Z let fhj : S3 — • S 0 ( 4 ) b e defined by t h e equation A > ( u ) • v — ^vu' for v G R 4 . Here we t a k e quaternion multiplication on t h e right. Let £/,- denote t h e 3-sphere b u n d l e over S4 d e t e r m i n e d by t h e m a p fhr T h a t is, with S4 = D\ Uss D4., t h e q u a n t i t y ^ is equivalent t o D4. x S3 over D*. and / ^ provides t h e pasting d a t a for gluing these two trivial bundles to form i ^ . Let Ml denote t h e total space of t h e b u n d l e ^ where h + j = 1 and h — j = k. Milnor shows t h a t Ml is h o m e o m o r p h i c to S7 for all k and t h a t Ml is exotic when k2 ^ 1 (mod 7). T h e a r g u m e n t involves t h e Pontrjagin classes of t h e bundle. Now let us r e t u r n t o t h e Brieskorn manifolds a n d discuss some aspects of their s t r u c t u r e s . Consider f(z) — ZQ° + z\x + • • • + z^" as a m a p p i n g / : C n + 1 —y C. It is easy t o see t h a t / | C " + 1 - V(f) : Cn+1 - V(f) — • C — {0} is a fiber bundle, a n d t h a t by taking t h e restriction to E$ — f~l {S}) -^-» S} where Sj = {z £ C\\z\ = 5} for S small, we also get a fiber bundle a n d t h a t E$(l D2n+2 —> 5 ] gives a fiber b u n d l e with t h e boundary of each fiber diffeomorphic t o K{f). Milnor [9] generalized this fiber bundle s t r u c t u r e t o a b u n d l e > : S2n+l - K(f) — • S\ >{z) = f(z)/\f{z)\. In t h e case of ESD D2n+2 —> S} a n d: S2n+1 - K(f) — • S1 are equivalent bundles by using t h e m a p p i n g (*o,*i, • • • , * „ ) — »
(p1/a°z(hpl^zu--.,p1'a"zn)
for p real (choosing p so t h a t t h e image point is on t h e sphere). In the general case (of / with an isolated singularity at t h e origin) Milnor uses a vector field t o push t h e fibers of Ef, l~l D2n+2 out into t h e sphere. A similar bit of geometric topology lets us see t h a t K [xk + f(z)j fold branched cyclic cover of S
2n+1
branched along K{f).
is a A>
This sets t h e stage
14. Exotic Spheres
250
separate sets of variables) in terms of K(f) C S2n+1 and K(g) C S2m+1, where / = f(z0,- • • ,zn),g = g(zo, • • • ,zm). Here, the idea is as follows. Suppose we are given maps / : D2n+2 — • D2 and g : D2m+2 —>• D2 with singular fiber bundles elsewhere. Then we can form the pull-back Z — {(x,y) G D2^2 x D2™+2\f(x) = g(y)}: Z
-_>
D2m+2
D2n+2
-U
D2.
and dZ <-• d(D2m+2 x D2n+2) = s2(n+m^3. Appropriate analysis shows 2 3 that dZ C S ("+™)+ is equivalent to K{f + g) C 5 2 < n+m )+ 3 . See [4]. For example, in the case of xk + f(z), we have Z
1
—*
D2
1
with g(x) — x2. Here it is easy to see that dZ is the Ar-fold cyclic branched covering of S2n+1 along K(f). Note that this construction gives a canonical embedding in a sphere of two dimensions higher. Thus, if Kn~2 C Sn then we have A'* — • Sn as branched cover, and K" C Sn+2 where K" denotes the a-fold cyclic branched cover of Sn. In this way, we get an inductive definition of the Brieskorn manifolds as iterated branched coverings. E(ao,ai) is a torus link of type (a 0 , fli) in S3. For example S(3,5) C S3 has diagram sketched in Figure 14.2. Our Milnor sphere S(3,5,2,2, 2) C S9 is the result of three 2-fold branched coverings starting from the (3,5) torus knot. #3,5 C S3 —
A'3,5,2 C S5 ^ - tf3,5,2i2 C 5 ? .
These constructions give a clear view of the algebraic topology of the bounding manifolds. We shall only sketch these details here, referring the reader
to [3], [4], and [5]. Given K(f) C S2n+1 we have that K{f) = dN{f) is the fiber of the Milnor fibration alluded above. The intersection form of middle dimension of
251
14.1 Exotic Spheres
E5f"]D 2n+ 2
Figure 14.2:
K3,5
Figure 14.3:
252
14. Exotic Spheres
Given K(f) C 5 2 n + I we have that K(f) = dN(f) is the fiber of the Milnor fibration alluded above. The intersection form of middle dimension of the homology of N(f) is given by 9(f) ± 6(f)T where 9(f) : Hn+l {N{f)) x Hn+i ((f)) —y Z is the Seifert linking pairing obtained by the formula 0(f)(a,b) = lk(a*,b) where Ik denotes the linking number and a* is the cycle in S2n+l — N(f) obtained by pushing a along a positive normal to N(f) into the complement. One finds that 9(f + g) S 9(f) ® 9(g) and consequently it is easy to determine intersection forms for composites. In particular, one has 9 (f + x2) = 9(f). The construction we have discussed generalizes to a tensor product construction for Kn C Sn+2, Lm C Sm+2 (L is a fibered codimension two submanifold of Sm+2 to (K ® L)n+m+1 C Sn+m+3). Thus, we can start 3 with any knot K C S and form [K ® £(2,2,2)] 7 C S9. If 9 is a Seifert pairing for K in S3, then K ® E(2, 2, 2) = &V, N C S9 with the same Seifert pairing. As a result, M has intersection pairing 9 + 9T and hence a (M) = a (9 + 9T) = a(K), the classical signature of the knot. As a consequence, many exotic spheres can be constructed directly in relation to knots and links in S3. The manifolds K ® E(2, 2,2, • • • , 2) (n2's) admit actions of the orthogonal group 0(n) with orbit space D4 and fixed point set K C S3 = dD4. These are called link manifolds and are classified in [6]. We have discussed their relationship with global anomalies in [7]. It is also worth pointing out that the Brieskorn manifolds are tensor products of empty knots [a] : £(a 0 , a\, • • • , a n ) = [a0] ® [ai] <8> • • • , ® K ] where [a] : S1 —> S1, [a}(\) = Aa. The term [a] : S1 — • S1 is a fibration corresponding to the empty knot V> C Sl (the empty set has dimension -1). By looking at the inverse image of a point in 5 1 under [a], we get the fiber consisting of discrete points (a in number), and hence the Seifert pairing of this empty knot. It has the form: / 1 -1
0 1
0\ 0
V o - 1 1 /
253
14.2 References
Finally, we should mention that so far we have only mentioned exotic spheres that are boundaries of parallelizable manifolds. There is a big class of exotic differentiable structures that do not bound in this way. Their properties require homotopy theory for detection. Such very exotic n-spheres are classified by xn+k (Sk)/Im (J), where Kn+k (Sk) denotes a stable homotopy group of the sphere Sk, and Im(7) denotes the image of the J- homomorphism: J : x n (S0(*)) — • x n + , (5*) . See [8] for more information on these matters. Very exotic spheres may have some physical relevance, according to a conjecture by Witten. In reference [10] he postulates that gravitational instantons and/or solitons have the structure of very exotic spheres. Our knowledge of gravitational instantons and solitons is limited, but there is no doubt that deeper knowledge about very exotic spheres should shed light on these relationships. In addition to reference [10] where the role of very exotic spheres is detailed for the case of ten-dimensional supergravity theories, the interested reader may want to consult reference [11], [12], and [13] where the contributions to superstring theory of very exotic spheres are studied.
14.2
References
[1] Milnor, J. W. and Stasheff, J. D.: Characteristic Classes, Annals of Mathematical Studies No. 76, Princeton University Press 1974. [2] Milnor, J. W.: On Manifolds Homeomorphic to the 7-Sphere, Annals of Mathematics 64 (1956) 399-405. [3] Brieskorn, E.: Beispiele zur differential topologic von Invent. Math. 40 (1966) 153-160.
singularitdten,
[4] Kauffman, L. H. and Neumann, W. D.: Products of knots, branched fibrations and sums of singularities, Topology 16 (1977) 369-393. [5] Kauffman, L. H.: On Knots, Annals of Mathematical Studies No. 115, Princeton University Press 1987. [6] Kauffman, L. H.: Link Manifolds, Michigan Mathematics Journal 21 (1974) 33-44.
254
14. Exotic Spheres
[7] Baadhio, R. A. and Kauffman, L. H.: Link Manifolds and Global Gravitational Anomalies, Reviews in Mathematical Physics Vol. 5 No. 2 (1993) 331-343. [8] Kervaire, M. A. and Milnor, J. W.: Groups of Homotopy Spheres I, Annals of Mathematics, Vol. 77 No. 3 (1963) 504-537. [9] Milnor, J. W.: Singular Points of Complex Hypersurfaces, Annals of Math. Study 6 Princeton University Press 1968. [10] Baadhio, R. A. and Lee, P.: On the Global Gravitational Instanton and Soliton that are Homotopy Spheres, Journal of Mathematical Physics Vol.32 No. 10 (1991) 2869-2874. [11] Baadhio, R. A.: Global Gravitational Instantons and their Degrees of Symmetry, Journal of Mathematical Physics Vol. 33 No. 2 (1991) 721-724. [12] Baadhio, R. A.: On Global Gravitational Instantons in Superstring Theory, Journal of Mathematical Physics, Vol. 34 No. 2 (1993) 358-368. [13] Baadhio, R. A.: Vacuum Configuration for Inflationary Journal Mathematical Physics 34 No. 2 (1993), 345-357.
Superstring,
Index 207, 224, 228 4-dimensional phenomenology, 198 7-exotic sphere, 190 8-form, 159, 199 Ee anomaly, 153 J-homomorphism, 253 tf-theory, 30, 45 U(l) anomaly, 146 U(l) axial anomaly, 148 U(l) hypercharge, 146 WVgravity, 121 C°° topology, 206 C°°-function, 93 Z2 homotopy groups, 183 -function, 136 ^-invariant, 3, 5-7, 9, 10, 230 for hyperbolic 3-manifolds, 7 7r° decay, 145 •-product, 113, 116, 117 p-forms, 93, 141 p-torsion class, 45 SU(2) global anomaly, 183
(1,1) current, 56, 59 (1,1) forms, 56 1-cochain, 110 1-cocycles, 111, 134, 135 1-form, 87, 106, 108, 132, 233, 235 10-forms, 159 11-exotic sphere, 197 12-forms, 159 2-cocycle, 110, 134 2-cycles, 56, 138 2-form, 80, 85, 86, 92, 93, 96, 102, 103, 108,113,115, 116,120, 133, 135,139, 160, 165, 171, 202 2-framing of 3-manifold, 9 3-dimensional, 2, 5, 7, 10, 11, 31 handlebodies, 207 homeotopy groups, 205 mapping class groups, 11 physics, 5, 31 topology, 2 universe, 2, 11 3-form, 5, 6, 122, 134-136, 139, 140 3-handlebody, 207, 213, 217, 220, 221 3-manifold, 11, 21, 224, 226, 232 invariants, 2, 10, 101 3-manifolds, 1-3, 5, 8, 10, 11, 205,
abelian group of infinite rank, see Steinberg module action of automorphism, 100 mapping class group, 19, 30, 49 the mapping class group, 50 255
256 unitary automorphisms, 101 admissible weight, 16, 20 affine, 98, 102, 106, 108, 118 connection, 230 Lie algebra, 19 space, 107, 171 symplectic space, 102 translations of R 2n , 102 Alexander polynomial, 7 algebra of classical observables, 116 observables, 115 smooth functions, 115 algebraic construction, 42 group, 36, 37 algebraic topology, 23, 148, 186, 250, 252 ambient isotopy, 219 analysis of deformation quantization, 118 global gauge anomalies, 164 global gravitational anomalies, 164, 232 anomalous transformations, 230 Arf invariant, 120 arithmetic group, 35, 38 Atiyah's canonical framing, 9 Atiyah-Patodi-Singer Index theorem, 106, 193 Atiyah-Singer theorem, 167, 185 automorphism of bundle, 93 axial current, 62, 148, 173 axial symmetries, 146 Bernoulli number, 42, 45, 246 Bers embedding, 51
Index Betti numbers, 39, 40, 163 Bianchi identity, 86, 88 bordism group, 41 Borel-Serre bordification, 36 Bose symmetry, 145, 147 Bott's periodicity theorem, 182 braid description of a link, 7 braid group, 7 braiding matrix, 18 Brieskorn manifolds, 249, 250, 252 Calabi-Yau manifold, 198 cancellation of anomalies, 146 global anomalies, 31, 231 global gravitational anomalies, 39, 186 canonical, 9, 41, 91, 101, 102 embedding, 250 framing, 10 generators, 219 line bundle, 238 quantization, 91, 121, 230 spaces, 3, 102 symplectic form, 101 twist automorphisms, 217, 219 Cartan subalgebra, 154, 156 Cartan-Killing form, 17 Casimir operator, 9, 122 Cauchy problem, 128 center of the mapping class group, 11 central charge, 231 centralizers, 38 chain of obstructions, 141 characteristic classes, 159, 164, 168, 170, 188, 238
Index Chern character, 167, 169, 178 Chern class, 43, 58, 62, 69, 95, 106, 110, 168, 170, 171, 177, 178, 239 Chern-Simons, 5-8, 11, 158, 196, 202, 229 3-form, 6 form, 5, 229 functional, 8 invariant, 3, 5-8, 10, 11 term, 230 Chern-Simons invariant for hyperbolic 3-manifolds, 6 of connections, 8 Chern-Simons-Witten theories, 10, 31, 48, 74, 83, 96, 105, 108, 186, 205, 224, 228, 230, 232, 235 Chevalley cohomology, 119 Chinese lantern relation, 33 chiral £7(1) symmetry, 62 chiral anomalies, 121, 145, 164 chiral currents, 146 chiral fermions, 147, 157, 174, 175, 179 chiral supergravity theory N = 2, 148, 194 N = 2,D = W, 191 N = 4, D = 6 , 201 chiral symmetry, 145 Chow cohomology, 41 ChristofFel symbols, 76, 79, 81, 83 classes of 1-forms, 133 anomalies, 125 arithmetic groups, 38 automorphisms, 217
257 connections, 8 diffeomorphisms, 30, 182, 186 divisors, 58 flat connections, 9, 95 global anomalies, 182 homeomorphisms, 38 hyperbolic 3-manifolds, 6 hyperbolic metrics, 23 irreducible representations, 96 Riemann surfaces, 23, 26 self-diffeomorphisms, 14 transition functions, 95 classical anomalies, 145 field theory, 119 mechanics, 92, 114 modular group, 48 observables, 114 phase space, 92, 113, 114, 116 signature, 252 system, 115 theory, 114, 145 classical invariants of links and knots, 120 Clebsch-Gordan coefficient, 16 Clebsch-Gordan condition for S L ( 2 , C ) , 16 Clebsch-Gordan rule, 18 Clifford algebra, 149, 150 cobordism, 242 cobordism classes, 242 cobordism group, 242 cobordism ring, 243 cobordism theory, 243 cohomology, 31, 42, 44, 45, 56, 95, 133, 240, 242 of moduli space, 23
258 with compact supports, 36 cohomology class, 103, 108, 110, 111, 115, 120, 130, 132 cohomology classes for mapping class group, 39 cohomology ring, 239 commutative ring, 243 compact surface without boundary, 107 compactification locus, 29, 44 compactified Euclidean space-time, 70 complex automorphisms, 100 line bundle, 106 manifold, 49 structure, 56, 96, 97, 101, 103, 106-108, 170 submanifold, 100 complex analysis, 23 complex structure of Teichmuller spaces, 29 on Teichmuller space, 50, 54, 57 complexication, see Lie group conformal blocks, 15, 16 conformal field theory, 15, 16, 230 conformal fields, 127, 130, 132, 133, 135, 137 conformal structures, 24 connection, 9, 16-18, 29, 31, 74, 76, 78, 81-85, 93, 95, 101, 103, 106,107,118,158,170, 188, 233, 235 1-form, 74, 86-88 on Hilbert space, 105 on tangent bundle, 81
Index over R 4 , 101 conservation of axial current, 173 axial vector, 145 energy-momentum tensor, 147 fermion number, 62 fermionic currents, 163 consistency conditions in conformal field theory, 16 construction of 3-manifold invariant, 16 Donaldson invariants, 99 exotic spheres, 187 hyperbolic 3-manifolds, 2 Miller's algebra, 40 torsion, 45 cosine formula, 52, 53, 55 coupling constant, 184 CP-violation, 62 current conservation, 145, 175 currents, 59, 145-147, 163 curvature, 59, 74, 82, 93, 94, 99, 101, 104, 107, 170, 231 2-form, 229 operator, 80 cusped hyperbolic 3-manifolds, 11 cyclic group, 14, 20, 51, 220 cyclic permutations, 91, 123 de Rham cohomology, 119, 129, 132 complex, 10, 167 deformation of associative structure, 116 Lie algebra, 116 Poisson bracket, 116
Index deformation quantization, 113, 116, 118, 119, 126 degeneracy of Poisson structure, 113 symplectic 2-form, 113 degenerate 2-form, 120 Poisson bracket, 120, 122, 123 Poisson structure, 119 symplectic 2-form, 120 degrees of freedom, 151, 166, 184 Dehn slides, 206 surgery, 1, 2 twists, 220, 221, 224 Deligne- Mumford compactification, 29, 49 moduli space, 55, 58 descent equations, 126 detection of global anomalies, 31, 39, 231 global gravitational anomalies, 39 diffeomorphism group, 206 invariance, 131 diffeomorphisms of, 40, 186, 221, 228 3-handlebody, 221 S 1 0 , 186 dimension of fiber, 65 manifold, 91 subgroup of finite index, 37 symmetric space, 37 Teichmuller space, 37 Dirac
259 doublet, 184 equation, 150 fermions, 184 field, 193 genus, 165, 179 index, 190 monopole, 72 operator, 168, 177, 184, 185, 189 spinor, 149, 150 string, 62 Dirac operator for fermions, 164 Dirac's correspondence principle, 114 magnetic monopole, 71, 87 disconnected diffeomorphism, 186 general coordinate transformations, see global gravitational anomalies, 121 Donaldson invariants for 4-manifolds, 99 dual basis, 67 space, 67 duality formula, 50-52, 57 Eells singularities, 128 effective action, 119, 157, 172-174, 176, 177, 179, 182, 191-194, 197, 200, 228, 229, 232, 234, 235 Eilenberg-MacLane space, 41 elliptic operators, 148, 171 energy-momentum tensor, 146, 147, 157
260 equations of motions, 92 Euclidean SU(2) Yang-Mills gauge theory, 62 path integral, 183 space, 149, 153, 172, 183, 194 volume, 29 Euler angles, 71 character, 163 characteristic, 2, 3, 25, 26 form, 163 exotic (n -+ l)-spheres, 186 11-sphere, 194 7-sphere, 249 n-spheres, 253 differentiable structures, 253 sphere, 188, 191, 247, 249 exotic structure, 246 Farrell-Tate cohomology, 231, 232 Fenchel-Nielsen coordinates, 29, 49, 56, 58 twist vector field, 50 fermionic currents, 163 fermions, 147, 152, 157, 159, 175, 183, 184, 193 Feynman integral, 146 Feynman's rules, 146 fibration, 220 field strength, 62, 88, 178, 196, 199 field theories, 130, 132, 175 finite presentation for mapping class group, 31 flat, see connection forgetful functor, 170
Index formula for global anomalies, 138 of Gardiner, 51 Fujikawa's method, 175 fundamental group, 3, 10, 53, 94, 217 fundamental group of 3-manifold, 3 the surface, 53 fusing matrices, 16-18, 21 fusing operators, 15 gauge coupling constant, 184 currents, 146 field, 74, 86, 177, 178, 183-185 field configurations, 177 field strength, 178 group, 86, 107, 146, 152, 153, 158, 164, 171, 183, 229 invariance, 174, 175, 184 invariant mass term, 175 lines, 147 parameter, 177 potential, 62, 74, 158 theories, 61, 74 transformation, 61, 86, 93, 95, 160, 183, 185 variation, 159 Gauss' law, 171 Gauss-Bonnet theorem, 163 Gaussian cut-off, 173 Gaussian integral, 172 generators for, 40, 133 anomalies, 131 global anomalies, 133 geodesic, 26, 50, 51, 54, 55, 79
Index geometric 4-manifolds with boundary, 6 geometric quantization, 74, 91, 97, 108, 115, 120 geometric quantization of canonical spaces, 102 Chern-Simons-Witten theories, 96, 105 fermionic fields, 121 space of flat connections, 101 topological quantum field theories, 93, 96, 99 geometric topology, 221, 249 geometry of symmetric spaces, 54 Riemann surfaces, 61 Teichmuller spaces, 48 Glashow-Salam-Weinberg theory, 146 global rotations of fermions, 172 •-product, 113, 118 deformation, 131 geometry, 64 pathologies, 121 Poisson bracket, 140 quantization, 118 topology, 63 global anomalies, 11, 45, 97, 113, 118,119,121,123-128,130, 131, 133,135,136, 145,164, 168, 182, 183, 194, 198, 200, 205, 207, 228, 230-232 global deformation of Poisson bracket, 131 global gauge anomalies, 119, 164 global gravitational anomalies, 120, 121, 131,145, 185, 186, 194,
261 197-199, 201, 205, 228, 230233 global topology, 64 gluing lemma, 26 graded i?-algebra, 243 Grassmann numbers, 173 gravitational anomalies, 145, 148, 164, 179 Chern-Simons term, 230 counterterm, 234 fields, 147, 164, 174 interaction, 199 operator, 229 gravitini, 199, 200 gravitino, 157, 199 graviton, 157, 166 Green-Schwarz anomaly cancellation, 154, 156 mechanism, 199, 201, 202 group of diffeomorphisms, 186, 228 gauge transformations, 107 homeomorphisms, 25 isotopy classes, 14 Haken 3-manifolds, 220 Hamiltonian, 74, 101, 126, 132, 133, 139, 140, 148 fields, 130 mechanics, 90, 96 Harer's stability theorem, 31, 42, 45, 231, 232 theorem, 31, 39 harmonic Beltrami differentials, 50 forms, 163, 167
262 Hausdorff space, 96, 107 heat kernel, 167 Heegaard decomposition, 21 decomposition of 3-manifolds, 14 gluing, 1 Heisenberg algebra, 102-105 commutation relations, 102 group, 102, 104, 105 Hermitian metric, 103 operators, 114 product, 49, 57 structure, 95 heterotic string theory, 160, 198 heterotic superstring theory E8 x Ea, 185 Hilbert space, 102, 104-106, 114, 115 of quantum states, 90, 114 Hirzebruch ^-polynomial, 168, 169 index theorem, 238, 245 polynomial, 199 signature, 191, 200 signature formula, 6 Hochschild cohomology, 119 Hodge duality operator, 167 Hodge operator, 193 Hodge structures, 45 holomorphic cotangent space, 49 functions, 104 isomorphism, 95 line bundle, 69, 104, 106
Index quadratic differentials, 51, 55 structure, 69, 94-96, 99, 101, 103 tangent space, 50 tangent vectors, 67 holonomy, 14, 17, 74, 77, 79, 170 holonomy of Knizhnik-Zamolodchikov equation, 16 homeomorphism, 1, 21, 24, 25, 30, 31, 207, 209, 210, 213, 219, 221 homeotopy group, 219, 232, 234 homogeneous space, 104 homology, 10, 30, 38, 39, 247, 252 classes, 42 groups, 30, 41 homology of moduli space, 31 homotopic transformation, 128 homotopy class, 24, 25, 50 equivalences, 221 groups, 182, 220 transformation, 121 homotopy spheres, see exotic spheres homotopy theory, 253 Hopf algebra, 42 Hopf fibering, 71, 72 hyperbolic, 50 plane, 48 3-manifolds, 2, 5-7, 10 3-space, 1, 4 3-universes, 10 area, 4, 25, 57 complement, 2 coordinates, 26
263
Index distance, 29 geometry, 23, 24, 49, 52, 54 knot, 2 length, 4 line, 49 metric, 25, 50, 53 mutant manifolds, 11 plane, 4, 26 space, 37 structure, 1-3, 50 surface, 24-26, 54 universes, 2, 10 index of Dirac operator, 189 Rarita-Schwinger operators, 191 surfaces, 110 index theorem, 148, 163-166, 168, 171, 172, 176, 178, 179 index theory, 148, 169 infinite groups, 38 order, 38 infinite-dimensional affine space, 108 complex manifold, 100 infinitesimal deformation, 131 deformation of Lagrangian, 131 gauge transformation, 174, 176, 177 perturbation, 123 transformation, 177 inner automorphism, 220 instanton, 62, 71, 88 integer lattice, 105 integral cohomology, 40, 115
invariant of connection, see curvature 3-manifolds, 8, 10, 11 links, 7 investigation of global anomalies, 122 irreducible connection, 96 lattice, 38 isomorphism, 104, 172 isotopy classes of homeomorphisms, 38 isotropy condition, 137 Jacobi identity, 91, 125, 126, 130, 136 Jacobian, 44, 96, 173, 176 Jones polynomial, 7 Kahler form, 49, 55 Kahler manifold, 98 Kahler metric, 97, 98, 100 Kahler polarization, 101 Kaluza-Klein theories, 146 Kauffman polynomial, 8 Knizhnik-Zamolodchikov equation, 14, 16-18 monodromy equation, 14 knot invariants, 231 Kohno's 3-manifold invariant, 20, 21 Kronecker delta, 52, 104 lack of time component, 207 conservation, 145 global anomalies, 129, 130 invariance, 182, 185
264 polarization, 108 Lagrangian, 8, 92, 124, 128 Lantern relation, 33 Laplace equation, 163, 164 Laplace operator, 6 Laplace-Beltrami operator, 50 large 3-manifolds, 220 diffeomorphim transformations, 182 diffeomorphism, 191, 195, 200, 228, 233 diffeomorphism transformations, 230, 232, 234, 235 gauge transformations, 182 lattice, 36, 38, 105, 106 with generators, 29 Leibniz identity, 116, 118 Levi-Civita connection, 9, 75-77, 82, 83, 230 Lie algebra, 52, 85, 97, 100, 103, 105, 107, 108, 116, 119, 154, 229, 233 Lie bracket formula, 52, 57 Lie co-algebra, 123 Lie group, 8, 70, 97, 106, 107, 128, 154 Lie-Poisson bracket, 90 link manifolds, 252 local anomalies, 125, 136, 140 counterterm, 179 deformation quantization, 113, 118 deformations, 118 isomorphisms, 118 perturbations, 121, 123
Index logarithmic distortion, 26, 48 Lorentz algebra, 158 anomalies, 179 group, 149 transformations, 179 low-energy expansion of string theory, 157 Lyndon-Hochschild-Serre Spectral Sequence, 31 Mobius strip, 63, 66 magnetic monopole, 62 Majorana-Weyl gravitini, 199 Majorana-Weyl gravitino, 157 Majorana-Weyl spinors, 199, 200 manifestation of global anomalies, 124, 198, 230 mapping class group, 14, 15, 20, 21, 23-26, 29-31, 35-38, 40, 48, 52, 206, 207, 217, 219, 220, 226, 228, 230-235 without punctures, 33 mapping class group of 3-manifold, 226 S10, 186 surface with boundary component, 33 surfaces with boundaries, 31 mapping class group's homological stability, 39 marking, 15, 16 Marsden-Weinstein quotient, 107 mass scale of string, 157 massive string modes, 157 massless fermion, 150 massless modes, 157
Index matrix 2-form, 85 matter field, 86 Maurer-Cartan equations, 84 Maurer-Cartan form, 84, 85 Maxwell's equations, 87 Maxwell's theory of Electromagnetism, 61, 84 Mayer-Vietoris spectral sequence, 44, 56 meromorphic sections, 69 metaplectic structure, 110 Miller's polynomial algebra, 40, 42 Milnor-Kervaire exotic sphere, 188 Minkowski space, 192 modular group, 26, 48 moduli space, 23-25, 29, 30, 39, 40, 45, 49, 108, 230 moduli space of conformal structures, 24 stable holomorphic vector bundles, 102 monodromy, 14, 18 monopole, 62, 71, 72, 87 bundles, 72 charge, 87 number, 171 Moore-Seiberg polynomial equations, 17 Moyal-Weyl product, 117, 118 multiloop amplitude computations, 48 multiplicative sequence, 243 mutant, 11 3-manifolds, 11 hyperbolic three-manifolds, 11 Narasimhan-Seshadri theorem, 96,
265 99, 102, 108 non-abelian gauge theory, 61 non-commutative algebra of quantum observables, 114 non-commutative deformation, 115 nondegenerate, 116, 128-130 2-form, 92, 96, 102, 120 observables for classical systems, 114 in quantum mechanics, 114 obstruction to anomaly generator, 132 deformation quantization, 119 generalizing quantities, 188 patching, 113 occurrence of global anomalies, 113, 131, 230 global gravitational anomalies, 205, 233 singularities, 120 operator ordering, 11 orientation of surface, 52 outer automorphism, 105, 220 parallel-transport, 77, 84, 85 along a curve, 84 parallelizable manifold, 197 patching of local quantizations, 118 pathologies, see anomalies, see global anomalies Pauli matrices, 88, 152 Pauli-Villars fields, 175 Pauli-Villars regularization, 175 Pauli-Villars regulator fields, 202 periodicity of
266 mapping class group, 231 perturbation theory, 121, 123, 127, 128 perturbative anomalies, 199, 201 Chern-Simons-Witten theories, 10 perturbative evolution of fermionic effective action, 174 phase factor, 229 phase space, 90, 92, 97, 108, 113, 114,116,118-120, 124, 127, 130, 132, 138 physical system, 87, 90 physical theory, 115, 124 Picard group of the moduli space, 40 Planck constant, 114, 115 Poincare ball, 4 disk, 4 dual, 57 duality, 58 series, 51 point-particle theory, 157 Poisson algebra, 90, 91, 114 bracket, 91, 105, 114-116, 118120, 122, 124, 125, 127, 128, 130-132, 136 connection, 118, 119 fields, 127 manifold, 91, 118, 119, 121 mapping, 129 structure, 116-118, 126 tensor, 118 polynomial algebra, 40-42
Index Pontrjagin classes, 43, 165, 169, 179, 197, 238, 241-243, 249 Pontrjagin number, 7, 241 of 4-rnanifold, 5 pre-quantization, 114, 115 formula, 115 pullback bundles, 83, 86 connection, 83 quadratic form, 243 quantization, 48, 74 condition, 72 rule, 110 quantization of classical system, 115 Chern-Simons-Witten theories, 48, 83, 108 classical system, 114 spin, 166 Quantum Chromodynamics, 86, 146 quantum corrections, 145 Quantum Electrodynamics, 75 quantum field theory, 8, 127 quantum gravity, 31, 120, 207, 224 quantum level, 131 quantum mechanics, 117 quantum numbers, 146, 147 quantum observables, 114 quantum pathologies, 118, 119 quantum states, 90, 114, 127 quantum statistical mechanics, 117 quantum system, 115 quantum theory, 119, 182 quantum theory of gravity, 148 quantum topology, 11, 23, 31 quarks, 146
Index quaternion, 249 Quillen's determinant bundle, 96 random geodesic, 54 rank, see cyclic permutations Rarita-Schwinger field, 192, 193, 195 indices, 193 operators, 191 rational coefficients, 245 cohomology, 23, 40 homology, 9 homology spheres, 10 Ray-Singer torsion, 9 Reshetikhin and Turaev's 3-manifold invariants, 9 Riemann moduli space, 48 Riemann surface, 24, 25, 31, 38, 44, 45, 74, 93 with nodes, 29 Riemannian geometry, 15 ring, 243 Ruberman, see theorem of Satake compactification, 44 satellite knot, 2 Schwinger terms, 136 Schwinger-Dyson equations, 184 Seifert forms, 252 Seifert linking, 252 Seifert pairing, 252 self-dual Yang-Mills equations for connection over R 4 , 101 Serre's list for arithmetic groups, 36 set of
267 2-forms, 137 admissible weights, 16 anomaly-free representations, 147 Casimir operators, 129, 132, 134, 135, 138 chiral representations, 147 diffeomorphism transformations, 234 holomorphic structures, 100 invariants, 11 isolated points, 9 transition functions, 63 sheaf, 43 Siegel moduli space, 44 Siegel upper space, 44, 104, 105 signature defect, 7 signature index formula, 169 signature of 4-manifold, 41 knot, 252 quadratic form, 243 signature theorem, 167 sine-length formula, 52 single-instanton, 71 singular symplectic leaf, 122 singularity, 249 Smale conjecture, 205, 206 Smale-Hatcher conjecture, 205-207, 215 space of 2-cocycles, 134 2-forms, 140 3-forms, 140 p-forms, 141 algebraic curves, 23, 24 complex structures, 104
268 conformal classes, 23 connections, 8, 99, 107 flat connections, 74 gauge fields, 171 holomorphic structures, 95, 98 hyperbolic metrics, 23, 24 linear functionals, 67, 68 tensor forms, 130 unitary connections, 95 zero modes, 148 spectral asymmetry, 148, 168 flows, 148 invariant, 168 spectral asymmetry of Dirac operator, 148, 192 spectrum of, 21 hyperbolic surfaces, 29 spin connection, 230 spin index formula, 169 spin manifold, 190, 197, 200 splitting principle, 240 stability of holomorphic bundles, 99 mapping class groups, 231 points, 97 stable, see holomorphic bundle, see holomorphic structures stable cohomology, 42, 45 stable cohomology of mapping class group, 40 stable points, 98 stationary phase, 9 statistical mechanics, 8 Steinberg module, 36, 37 stochastic quantization, 121 Stokes's theorem, 99, 108, 110
Index Stone-von Neumann uniqueness theorem, 102, 103, 105 string corrections, 157 string theory, 48, 148, 157, 195 string world sheet, 194 global anomalies, 231 subgroup of mapping class group, 30 finite index, 38 mapping class group, 11 supergravity, 151, 157, 186, 198 supergravity multiplet, 157, 196 supergravity theory, 148, 185, 186, 191, 194, 199-201 N = 2, D = 10, 194 TV = 6, D = 6, 199 /V = 8, D = 6, 199 superstring, see superstring theory superstring theory, 148, 154, 157, 198 supersymmetry, 148, 151 surgery on link, 9 switching operator, 18-20 symplectic 2-form, 103, 115, 120, 139 coordinates, 103 form, 92, 97, 98, 100, 102, 103, 108, 110, 111, 117,122,133 generator, 132 group, 103, 105 leaves, 120, 122, 123, 129 manifold, 92, 96, 97, 101, 107, 114, 117-120, 123, 124,129, 130 nondegenerate 2-form, 116 phase space, 97, 108, 127 quotient, 101, 108
Index structure, 92, 107, 132 symplectic geometry, 49, 52, 54, 96 Teichmuller deformations, 48 Teichmuller space, 23-26, 29, 30, 37, 48-52, 54, 105, 108 Teichmuller space, 11 ten-dimensional physics, 118 theorem of Alexander, 209 Baadhio, 207, 217, 220, 224, 226 Hirzebruch, 243 Kodaira, 58 Ruberman, 3 Torelli, 44 van Hove, 114 Mostow, 10 theory of deformation quantization, 115 Electromagnetism, 86 heterotic superstring, 86 quantum groups, 8 strong interactions, 86 three-dimensional ball, 209 mapping class groups, 31, 120, 205 topology, 7 universe, 10 topological defect, 139, 140 invariant, 21, 171, 189, 202, 229 isomorphism, 21, 219 manifold, 247 obstructions, 113 quantum number, 62 structures, 121, 123
269 topological quantum field theories, 93, 120, 186, 205, 207, 230, 232 Torelli group, 30, 36, 38 torsion classes, 45 injiomeotopy groups, 231 torsion in cohomology, 45 mapping class groups, 231 torsion operator, 81 torus knot, 2, 250 torus link, 250 towers of obstructions, 141 transition functions, 63-66, 68-70, 72, 84, 86, 94, 95 translational invariance, 104 triangulation, 110 trivialization of bundle, 95 twist automorphisms, 217, 219 flow, 51 group, 217, 219, 220 parameter, 52 uniform distribution, 54 Uniformization theorem, 25 unitary automorphisms, 100 connection, 95, 96, 101 gauge transformation, 95 Hilbert space of representation, 102 line bundle, 103 transformation, 98 universal bundle, 42, 43 unstable cohomology of
270 mapping class group, 40 unstable torsion class of homeotopy group, 232 vacuum expectation value, 184 vacuum state of quantum field theory, 133 very exotic n-spheres, 253 Virasoro global anomaly, 231 virtual cohomological dimension, 35, 37 duality group, 36, 37 volume of hyperbolic 3-manifold, 4 tetrahedron, 4 von Neumann algebras, 7 Ward identities, 173, 174 wave function on moduli space, 230 Weil-Petersson Kahler form, 29, 50 metric, 49-52, 54-56, 58 volume form, 29 Weil-Petersson geometry, 54, 57 Wess-Zumino consistency condition, 126, 141 Wess-Zumino-Witten model, 15, 16, 20 Weyl doublet, 184 fermions, 146, 175 operators, 171 spinor, 192 Weyl algebra, 117 Weyl-Majorana condition, 151, 192 Weyl-Majorana fermions, 192
Index Weyl-Majorana fields, 192 Weyl-Majorana spinors, 192 Whitney sum, 67, 68, 82, 168 Wigner's 3j-symbols, 18 Wilson-line anomaly, 230 winding number, 177 Witten's formula for global anomalies, 194, 198, 200, 231, 235 global gravitational anomalies, 191 Wolpert's formula, 49 Yang-Baxter equation, 8 Yang-Mills equation, 88 field strength, 74, 158 gauge current, 157 gauge theories, 84 group, 157 instanton, 71, 87 Lagrangian, 229 multiplet, 157 vacua, 62 Yang-Mills algebra, 158 Yang-Mills theory, 61 N = 1,D = 10, 157 N = 1, 148, 152 zero modes, 168, 185