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and j(Wa) <-» 0, and A£a'(X - y) is the s p i n - 1 / 2 Feynman propagator satisfying OXAaa\X
- y ) = Vaa'6(X
-
y)
T h e essential idea used here is that the 0000 b o x is a representation of the
(and K is some uninteresting constant). T h e notion of a reproducing kernel is considerably developed by Fast wood and Ginsberg in [•!]. As well as using the two-variable Penrose transform ll2(P~
x P'~\ 0( —2, —2)) s { = 0 =
§2 The Dot
Product
in the (twistor, dual twistor) ease they consider the corresponding isomorphism in the (twistor, twistor) case. Here the left hand side is II2(P~
x
P+\G(-2,-2)).
We return to this, and to Ginsberg's construction of a twistor kernel for the first order massless scalar <j>'x vertex, a little later. There is another method of getting into / / 2 ( A ' , 0 ( — 2 , — 2)). Take our naive interpretation of the double edge: 1
(Z°Wa)
e II°{P
x P--
A,0(-2,-2))
(where /I = { ( Z ° , W 0 ) <S P x P' : ZaWa = 0 } ) and dot it with an (as yet unidentified) element p €//'((/; 0) (2) where U is a neighbourhood in N of A. It would obviously be preferable if this element p could be identified with s o m e topological object. T h i s was achieved by Ginsberg in [6] who used the following c o m m u t a t i v e diagram of exact sequences
-> I —
1
Hl(N\
±
I • O")
^
H2(N;
ip
—
lr
//2(A',(7;Z)
i -»
1 IIl(U;0) H2(N,U;0) I
Z)
-»
I
lr i in which the rows are induced from the exponential sheaf sequence while the columns are relative e x a c t sequences. Ginsberg started with an e l e m e n t (the generator, in fact) r 6 / / 2 ( A ' , ( / ; Z ) and defined /i = l o g ( p 6 - H ( r ) ) e
H\u-,G).
He then went on to show that the element 1 (Z°W0)2
• I1
(3)
corresponds to the kernel referred to earlier. In fact he managed to achieve all this not just for the case of elementary states (which w e have used in
372
S.A. Huggctt
Coliomology and Twistor
Diagram*
our description above) but for general massless fields in lll(P+\0(— 2)) and lP(P'+; 0(—2)). T h e commutative diagram of exact sequences is the same except that the space N is replaced by P~ x P . Furthermore, Ginsberg used the same commutative diagram to construct a twistor kernel for the first order massless scalar 4>A vertex. Here the underlying space II of the diagram is Pw X Px X Py *
Pz,
where the subscripts refer to the variables used below. form the well-known 'box' (see [9]):
The interior edges
1 Yp X» W-,'/<'< Y6ZS'
W0X°
and the exterior edges comprise the four interacting fields, which we take to be elements of / / W ; < ? ( - 2 ) ) , Jfl(P$-,0(-2)),
/7'(^+;0(-2)),
Hl{P$;Oi-2))-
Ginsberg's kernel is of the form _J
\VUX°
1
I 1
' Y^Z ' YliX»W.,Z-<
'T
(where r, which is an element of an II' group, is obtained by chasing through the commutative diagram). This kernel, being an element of U\Pw
x Px * Py~ x
Px\0{-2,-2,-2,-2))
enables us to assemble the complete diagram using the dot product and evaluate it via Serre duality, as planned. Moreover, the structure of the kernel (in which two of the opposite sides of the box are individually doited into the extra element r) is similar to the structure of the Sparling contour [20] normally used in the integration of the box diagram. (The phrase 'is similar to' can be made precise. There is a relationship between taking a residue on a codimeiision one submanifold defined, say, by s = 0, and dotting with I f s . See [()] or [13] for details). However, a very long-standing problem with the box diagram has been that of crossing symmetry. T h e original hope had been that one twistor diagram (the box diagram) would represent the Feynman vertex for first order massless scalar <*>' scattering, but that there would be three contours for the corresponding twistor integral, one for each of the three channels in the Feynman diagram. One would expect, then, there to be llircc kernels, whereas Ginsberg's procedure only led to one. There is a clue in Ginsberg's kernel, though, as to how one might proceed. Note that two of the edges of the box (1 / YpX'3 and 1 / W y Z y ) arc merely cupped together, not dotted. So the 'rule' that the cohomological version of a twistor diagram is assembled using the dot product has already been broken, which suggests we re-examine it.
§.'f Colioinologlchl
3
Contours
Coliomological
Contours
By way of introduction to the next phase of development of the subject, w e give an alternative description of the dot product. In fact right at the start [7] Eastwood pointed out that the dot product could be thought of as the and cup product followed by the Mayer-Vietoris map: given a € HP(X;S) 0 e HV(Y\T) their cup product is a U 0 <E / / p + , ( A ' n Y\S®T) and the connecting map in the Mayer-Vietoris sequence: d" : W+,'(X
D V; S ® T) -> / / * + « + l ( A' U Y; S ® T)
takes us to the dot product of a and 0: a-0
= d'(o
U 0).
T h i s description of the «lot. product was very useful in a new approach to the cohomology of twistor diagrams initiated by Singer | l l ] . T h e idea (whose roots lie in Penrose [IS]) is almost the opposite of the procedure we have been discussing so far, in that we assemble a given tsvistor diagram by cupping everything together. T h e interior edges are again taken as elements of H° groups (the extra elements such as p or r having been abandoned) and the exterior edges form the fields (assumed now to be elementary states) in various / / ' groups. If there are / of these II 1 elements this new procedure gives an element of / / ' ( I I - T'; Sid) (where T ' is the union of all the subspaces defined by the internal edges, together with the CP1 subspaces on which the / elementary states are singular). There is a mapping from //'(TI — T ' ; Q d ) —*
///+J(ri-T';C)
obtained by using the Dolbeault description of the first group, forgetting the bidegree ( < / , / ) and remembering only the total degree <1 + f . (A Cecil description of this map is given in Penrose [18]). Now we simply use Poincaré duality to tell us that in order to evaluate an element of / / ' + 1 , ( I I — T'; C) we will need a contour in / / / + j ( f l - T': C). This 'coliomological' contour is easy to relate to the traditional ones in //rf(n-T;C), for there is a map ///+,i(II - T ' ; C ) - > //¿(II - T; C)
S.A. Il ugge tt
374
Cohomology
and Twistor Diagram»
given by iterating the Mayer-Vietoris connecting map (in homology) / time«, o n c e for each field. This all works beautifully for the scalar product diagram, and one can show that / / 8 ( I 1 — T'; C ) = C and that the image of the generator of this group under two Mayer-Vietoris maps is the usual physical contour for the scalar product. This tells us that there is only one cohomological contour for the scalar product (as expected) and suggests a method for testing contours to see whether they are cohomological (a point to which we return later). More interesting, though, is the application of this method to the box diagram. Here / = 4 and d — 12, so the cohomological contours are in ffie(Il-T';C).
(4)
A calculation by Singer [19] demonstrated that this space has two generators, one of which does not correspond to any of the three physical channels for ' (because it docs not generalise to positive or negative frequency fields). T h e other generator is the Sparling contour in its cohomological form. (In other words, four Mayer-Vietoris m a p s applied to this generator would yield the Sparling contour). This seems to confirm Ginsberg's work, that is, that there is only one cohomological channel for the box diagram. Indeed, these two approaches (dot product and extra elements on the one hand, and cup product and cohomological contour on the other) were explicitly shown to be equivalent in the scalar product case by Baston using the techniques of cohomology algebra in [3].
4
A General
Procedure
There remained, however, the possibility that there might be other cohomological ways of evaluating the box diagram, corresponding to the other two channels. After all, Hodges had succeeded [8] in constructing contours for all three channels, following a suggestion in [10]. Clearly a method was needed which would exhibit all the cohomological functionals on a given collcction of fields. Some earlier work by Baston provided the starting point. In [2] another interpretation of the dot product is described. Given a complex manifold A' U K, closed subsets F C X and G C Y, and elements A 6 HP(X
— F;
S)
0 € //'(Y
-
T)
and G\
§•/ A Genera/ Procedure
:t75
we can use the connecting maps in the relative cohomology exact sequences Ii"(X-,S)
->
H"(X - F\S)
A
JIpF+l(X\S)
-y
H"+i{X;S)
H"(Y,T)
—
H*{Y-G]T)
A
Hg\Y\T)
—
H"+l(Y,T)
to obtain elements r a and rfi. Then the cup product on relative cohomology U : H?\X;
5 ) x / / ¿ + 1 ( K ; T) -
tlftg2(X
U
Y\S®T)
gives us an element in the image of the connecting map in the following relative cohomology exact sequence: //P+»+I (
a' u V; S ® T) -
IP+"+1(X
u V ; S ® T) -
Y - F
U
+ +2
H" " (X
fl G; 5 ® T) A
u V; 5 ® 7')
and it can be shown that = r-'IroUrg Actually, in order to make use of this new viewpoint we have to be a little more precise (as indeed [2] was). After all, the / interacting fields are given as elements of / / ' groups defined on / different spaces. So we need the cross product in relative cohomology: x : HpF+'(X-,S)®irG+l(Y;T)
- » lipF\^\X
x
Y\S®T).
Here, strictly speaking, 5 ® T should be r. 'XS ® ~yT. As before, r a x r 3 is in the image of the connecting map r in / / p + , + l ( A ' x Y-FxG;
S®T) A / / £ t c " 2 ( A ' X
S ® T ) - H'"i"l+2(X x V; 5®7'),
with Q • ¡3 = r - I ( r a x r/?). We remark that this explains why Ginsberg was able to stop at ¡i in his diagram-chasing. Strictly speaking the diagram chase leads to /-(//), but we then use /£ in a dot product, and for any i> ft • w = r~ (r/i x /•(/). Before we a little more and we let i/, transform on
make use of this description of the dot product we introduce notation. For each i = 1 , . . . , / we let P, stand for P or P', be an open subset of P, of the correct topology for the Penrose / / ' ( i / , ; (9(r,)) to be an isomorphism. We will also need F, = P, -
UI
376
S.A.
i / i i g g e t t — Cohomology
a / n / Twistor
Diagram*
and F = Fx x . . . x F j .
Finally, we denote by L, a projective line contained in /'',, and let A = L\ x . . . x Lj. So, given our / fields we have an element in Hl(UuO(ri))®
U\U,;0(r,))
and our first result [14] is that we lose nothing by choosing to dot all these fields together. In fact the Künneth formula for relative cohomology implies that Hl{Ui;0{rt)) ® ... ® U\Uf; 0(r,)) S //»'(II; O(r)), (5) where r = (T-j,.. . r j ) . Every continuous linear functional on these fields is therefore an element of the compact relative cohomology group H3cÍ J ( í l , U - F\0{-r))
(6)
(but (5) and (6) are not in general dual to each other [16]). We now have to decide how the interior of the diagram picks out some of these f u n c t i o n a l . We regard the interior as a holomorphic kernel h <E // 3 >'<(II - £ ; C ? ( - r ) ) . For example, in the scalar product (spin zero) nw'/ i n
(here and hereafter, we use DWZ
=
W
S
-
L
Z
'
Z
W
-/1;0<"2'-2))
for DIF A DZ, etc.) while in the box
6
" " ' ° <
N
-
2 O ; 0 ( - 2 . - 2 , - 2 , - 2 ) ) .
, 7 )
Usually, q is equal to zero. In these cases li can in principle be calculated by integrating out the interior vertices of the twistor diagram, although it is not always easy to see how to do this in practice. If q is non-zero the determination of h from the diagram is even less clear. Suppose we hail an element o € / / c 0 l / ~ ' ( n - £ , l l - E U F). Then a U h € 1¡V ' { n - S , n - S U F\ C ? ( - r ) ) .
§•/ A (•'<•iier.il I'racedlire
There is a map induced by inclusion i:
(H - E, II - E U /•'; O ( - r ) ) -
//["•'(II, II - F\ 0 ( - r ) )
so i(ct U li) is a functional picked out by the interior of the diagram (i.e. h) as required. But as it stands a does not look very much like a contour. T o overcome this we note first that the embedding of the constant sheaf C into O induces a map / / / - ' ( I I - E, n - E U F\ C ) -
/ / / - ' ( l l - E, n - E U F\ O)
(8)
and second that the groups / / 5 / + , ( r i - E, II - E U F)
and
///"«(11 - E, II - E U F\ C)
are isomorphic. Now all we have to do is insist that a is in the image of the map (8), and then it can be regarded as a contour. When a is a contour we call ¡(a U li) a functional 'associated to' the kernel h, and remark straight away that there are none if F C E, because then
// s ,+,(n - E, n - E u F) = o In this case we refer to the problem as 'ill-posed'. It should be noted that so far in this section our fields are perfectly general. If in fact (he fields are elementary states then F, — I., and F is equal to the closed submanifold A (of real codimension '1 / with orientable normal bundle). Now we can use the T h o m isomorphism / / ; + , ( A - E)
/ / s / + , ( n - E, II - E U A)
to deduce that the contours we seek are in / / y + v ( A — E). Indeed, this result still holds even when the fields are not elementary stales, as long as (II — E, II — E U F)
is homotopic to
(II - E, II - E U A).
To summarise, then: T h e o r e m 4 . 1 / / ( ! ) ) is satisfied
then the Junctionals
ll\Uu0(ri))®...®U\U,,0{rj)) associated
to tin
kernel h £ //3/'(ll
arc given by elements
of the homology
-E;0(-r)) group
///+,(A -
E).
on
(5))
378
S.A. Iluggett
Cohomology
and Twistor Diagram»
Now we consider some applications. For the usual scalar product w e note only that all works well, as usual! Instead let us look at t h e scalar product on P x / } (instead of P x P"). lu this case we can choose E to be the diagonal A in P x P , which defines for each r what Atiyah [1] calls the 'Serre class' k € / / 6 , 2 ( I 1 — A ; 0(r
— 2, r — 2 ) ) .
Our theorem now tolls us that f u n c t i o n a l associated to this kernel are in II,(A - A ) , which again has exactly o n e generator if A ("I A = 0, that is, if the problem is well-posed. It is ill-posed if I.\ and l, 2 intersect, but then the corresponding points in Minkowski space are null-separated, so no functional is expected. T h i s Serre class li was also identified by Eastwood and Ginsberg [4] as the (twistor, twistor) description of their kernel. However, the big question is what happens in the box diagram? Here / = 4, ¿o is as in (7), and in the case of (spin zero) elementary states we are interested in f u n c t i o n a l on
//'(P, - LuO{-2))
® ... ® H\P< - L.\\ 0(-2))
associated to lio- In particular, of course, w e want to know whether functionals exist corresponding to the three channels for the first order massless scalar 4>x vertex. T h e s e functionals will have the property that under a continuous motion of the lines L, into one of the following special configurations exactly o n e survives. (i) L\ = ¿2 and L-s = L.1 (ii) L\ = L. 1 and L 2 = ¿3 (iii) L1 = ¿3 and ¿2 = L.\ In cases (i) and (ii) the problem is ill-posed, so there are no associated functionals. In case (iii) and for the /,, in general position, the problem is well-posed and w e need to study the group //.,(A -
So).
Hut this is exactly the group studied by Sparling [20], It has two generators (for the Li in general position), only one of which can be generalised to positive or negative frequency fields. This functional corresponds to the channel given by the Sparling contour. So our general procedure shows that the box diagram does not have 'crossing s y m m e t r y ' , and explains why the earlier a t t e m p t s at a cohomological understanding of this diagram only led to one functional. T h e contours for the other channels constructed by Hodges remain a bit of a mystery. O n e reasonable guess is that in our scheme they correspond to elements a not in
§5 Traditional
Contours
Regarded as Cohoniological
Functionals
379
the image of the m a p (S). ( T h e s e generalised contours were first considered in [15].) Before we leave our discussion of the applications of this general procedure we should note (omitting the details) that it can be used to show that it is possible to describe all three channels for the first order massless scalar (j>x vertex in twistors. We abandon the 'box' part of the box diagram (i.e. the kernel h0) and choose a new kernel DWXYZ ' ~ [W0X°YpZl>
W0Z°YPX")r
-
This choice of kernel is justified from the space time point "I view in |l l|. where it is also shown that there are three associated f u n d lon.ils, <1111 lm 1... h channel. 5
T r a d i t i o n a l C o n t o u r s R e g a r d e d a s C o h o m o l o g i i nl I
I.
Finally, w e consider the question of deciding which c o n t o u r , .tie C O I M U I I I ' I U | I ical. We take our fields to be elementary states, and choose i c p i e s e u l a l l v c cocycles for them. T h a t is, for each field we choose functions of the form 1
(10)
Z"AaZ'5B,
and o m i t to use the Mayer-Vietoris m a p which would make this into an element of / / ' . Constructing a diagram in this way yields a differential form - A-D,Qd).
H°(X
( 1 1 )
Here X is the rest of the diagram, so that A' = ri - £ U S where
1/-J
S
- (J
imt
t h e Si being (ill pairs) t h e plane , deliiiiur, lepienenl ul lie «ni yiliMi lm I he other / — 1 fields. In other words, in willing • nl nl 1 I I 1 u merely focusing attention on flni one liehl (III) I lie M a v i V l i i u i i pi. in cohomology and homology fii togel.liei in the following way; (w€)
l l ° ( X - A - B,iV)
//'(A
x (^G)
!I
d
(X-A-B\C) i C
In/Mi') x
//,,u(.\
I!.//,() 1
c
S.A. Il ugge tt Cohomology
380
and Twistor Diagram»
T h e traditional contour k evaluates u>, but as far as the field (10) is con ccrned we should be considering d ' u , which means that there should be a contour A such that k = d.X If K passes this test for cach of the fields in the diagram, then it is a coliomolgical functional. Recent work [13] analyses this test in more detail. Consider the two Leray sequences Ilp+i(X
-AO
B)
//„_,(/!-/i) ^
T -
Hp+i(X
HP(X-A)
II - B)
^
//„_,(/!-«) ^
-
T I J
P
( X - A - B )
-
(where O is intersection and 6 is Leray's cobord map). It is shown in [13] that <52n, = d. so that if k = 0.A then (by Leray's Residue T h e o r e m )
which confirms that these contours always involve treating at least o n e of the two singularities in (10) as a Cauchy pole. In fact we believe that they always treat both of these singularities as Cauchy poles (and therefore make an 5 l x Sl contribution to the overall contour for each field) but so far this remains conjectural. G
Concluding
Remarks
A limitation of the general procedure in jj'l (not shared with the work in §5) is that as it stands it only applies to projective twistor diagrams. T h e only result we know of here is that it can be shown from (4) that the non-projcctivc box diagram also lacks three cohomological contours. So the new kernel is still needed. Also, the reader will be aware that almost without c o m m e n t we have restricted our attention to scalar fields. W h a t can be done for other spins? Obviously the kernel in (3) can be modified to give a (twistor, dual twistor) scalar product
Hl(U\\0{n
- 2)) ® //'(£/,; 0(n - 2)) -> C
for any n > —2, but it is not so clear what to do with the other cases. One traditional interpretation [17, 20] of diagram edges of the form
Bibliography
.381
for 11 < —2 lias been to insist that the contour have boundary in the s u b s p a c e Z°Wa = 0. It would be of interest to discover how far the m e t h o d s of §4, 5 generalise to contours with boundary. Indeed it seems likely that these methods will provide a good foundation for future work. I would like to thank all the members of the twistor group, but especially Roger Penrose, for many discussions and constant encouragement. I would also like to thank Andrew Hodges and Michael Singer for our very enjoyable and fruitful collaboration.
References [1] Atiyah, M.P. Green's Functions for self-dual S u p p Studies 7 A , 129-158 (1981)
four-manifolds
Adv. Math.
[2] Baston, R.J. Local coliomology, elementary slates, Twistor Newsletter (Oxford preprint) 2 2 8 1 :j (198G)
and
[.'{] Baston, R.J. Twistor diagram 'magic' and coliomology Newsletter (Oxford Preprint) 2 3 3 1 - 3 5 (1987)
algebra
[4] Eastwood, M.G. & Ginsberg, M . L Duality J. 4 8 1 7 7 - 1 9 « ( 1 9 8 1 )
in twistor
evaluation
Twistor
theory Duke Math
[5] Ginsberg, M.L. A coliomological scalar product construction, in Advances in twistor theory (eds L.P. Ilughston and R.S. Ward). Pitman, San Francisco (1979) [G] Ginsberg, M.L. Scattering theory and the geometry spaces, Trans. Amcr. Math. Soc 2 7 6 , 7 8 9 - 8 1 5 (1983)
of
multi-twist.or
[7] Ginsberg, M.L. fc Huggett, S.A. Slieaf coliomogy and twistor diagrams in Advances in twistor theory (eds L.P. Ilughston and R.S. Ward), P i t m a n , San Francisco (1979) [S] Hodges, A.P. Twistor
diagrams
Physica 1 1 4 A 157- 175 (1982)
[9] Hodges, A.P., Article in this volume. [10] Huggett, S.A., & Penrose R., Thin channel* for tin box diagram Newsletter (Oxford preprint) 10 18-22 (1980)
Twistor
[11] Huggett, S . A . , fc Singer, M.A.. Two philosophies for twistor Twistor Newsletter (Oxford preprint) 2 3 20 30 (1987)
diagrams
[12] Huggett, S.A., k. Singer, M.A., Coliomology and projective twistor grams Twister Newsletter (Oxford preprint) 2 5 33 10 (1987)
dia-
382
S.A. Il ugge tt
Cohomology
[13] Iluggett, S.A., & Singer, M.A., Cohomological ter (Oxford preprint) 2 8 1-9 (19S9)
and Twistor
residues
Diagram»
T w i s l o r Newalot
[14] Iluggett, S.A., & Singer, M.A., Relative cohomology twislor diagrams to appear in Trans. Amer. Math. Soc.
and
projective
[15] Josza, R.O.. Sheaf homology and contour integrals in Advances in twistor theory (eds L.P. Hughston and R.S. Ward), P i t m a n , San Francisco (1979) [16] Laufer, 11.B., On Serre duality and envelopes Math. Soc. 1 2 8 4 1 4 - 4 3 6 (1967)
of holomorphy
Trans. Amer
[17] Penrose, Ii., Twislor Theory: its aims and achievements, in Quantum Gravity, an Oxford Symposium (eds C.J. Isham, II. Penrose & D.W. Sciama), Oxford University Press, Oxford (1975) [18] Penrose, R., On the evaluation of twislor cohomology Newsletter (Oxford preprint) 10 14-15 (1980)
classes
Twislor
[19] Singer, M.A., & lluggett. S.A., On the homology of the box diagram, to appear in Further advances in twistor theory (ed L.J. Mason), Pitman (1990) [20] Sparling, G.A.J., Homology and twistor theory in Quantum Gravity, an Oxford Symposium (eds. C.J. Isham, R. Penrose & D . W . S c i a m a ) , Oxford University Press, Oxford (1975).
Authors' Addresses
T. N. Bailey, Department of Mathematics, University of Edinburgh, J. C. Maxwell Building, T h e King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, U.K. R. J. Baston, Mathematical Institute, 24 29 St. Giles, Oxford, 0 X 1 3LB, U.K. F. E. Burstall, School of Mathematics, University of Bath, Claverton Down, Bath, B A 2 7AY, U.K. C. J. Cutler, Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. E. G. Dunne, Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma, 7407S 0613, U.S.A. M. G. Eastwood, Department of Pure Mathematics, University of Adelaide, G.P.O. Box 49S, Adelaide, 5001, South Australia. J. Fletcher, Mathematical Institute, 21 29 St. Giles, Oxford, 0 X 1 3LB, U.K. J. Fraucndiener, Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. S. G. Gindikin, Molecular Biology Laboratory, Building A, Moscow University, G S P - 2 B Y , Moscow, 119899, U.S.S.R. A. P. Hodges, Mathematical Institute, 2 4 - 2 9 St. Giles, Oxford, 0 X 1 3LB. U.K. S. A. Huggett, Department of Mathematics and Statistics, Polytechnic South West, Plymouth, PL4 8AA, U.K. L. P. Ilughston, Robert Fleming Securities Ltd., 25 Copthall Avenue, London, EC2 7DR, U.K. C. N. Kozameh, Laprida 854, FAMAP, University of Cordoba, Argentina.
C. It. Le Brim, Department of Mathematics, S U N Y , Stony Brook, NY 1179-1 3651, U.S.A. L. J. Mason, Mathematical Institute, 24 29 St. Giles, Oxford, 0 X 1 3LB, U.K. E. T. Newman, Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. It. Penrose, Mathematical Institute, 25-29 St. Giles, Oxford, 0 X 1 3LB, U.K. W. T. Shaw, Intera-ECL, Highland's Farm, Grey's Road, Henley on Thames, RG9 -IPS, U.K. M. A. Singer, Lincoln College, Oxford, 0 X 1 3DR, U.K. I<. P. Tod, Mathematical Institute, 24-29 St. Giles, Oxford, 0 X 1 3LB, U.K. It. S. Ward, Department of Mathematical Sciences, University of Durham, Durham, D i l l 3LE, U.K. N. M. J. Woodhouse, Wadham College, Oxford, OX I 3 P N , U.K.
3S4
LONDON
MATHEMATICAL
LECTURE NOTE f-xiited by
PKOH-SSOR
J. W.
SOCIETY
SERIES
S . CASSULS
Department of Pure Mathematics, and Mathematical 16 Mill Lone. Cambridge. CB2 ISB. England
Statistics
with the assistance: of G. R. Allan (Cambridge) P. M. Cohu (London)
Twistors in Mathematics and Physics F.dited by T. N. Bailey. University of Edinburgh R. J. Baston. University of Oxford Twistor theory has become u diverse subject as it has spread from its origins in theoretical physics to applications in pure mathematics, this unique collection of review articles «.»vers the considerable progress in recent years in a wide range of applications such as relativity, integrahle systems, differential and integral geometry and representation theory . The articles explore the wealth of geometric ideas which provide the unifying themes in twistor theory, from Penrose's quasi-local mass construction in relativity, to the study of eonformally invariant differential operators using recent techniques of representation theory. 'Twistor•> in Maiheniutii.^ and Physics' will therefore be invaluable to research workers and graduate students as a reference and guide to the literature.
ISBN
O - S E l - a ^ f l B - l
9
For n list of hooks available in this series see page I
