- Email: [email protected]
Equivariant cohomology and cohomological field theories
~1[o]~ ' l [dI i'l,'l[~'~ PROCEEDINGS SUPPLEMENTS ~iI,SEVIER
Nuclear Physics B (Proc. Suppl.) 39B,C (1995) 488490
E q u i v a r i a n t c o h o m o l o g y a n d c o h o m o l o g i c a l field t h e o r i e s R. Stora a aLaboratoire de Physique Th~orique ENSLAPP, URA 14-36 associfie • I'E.N.S. de Lyon, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France and Theory Division, CERN, CH-1211 Geneva 23, SWITZERLAND A general method is given to construct a class of observables in topological field theories.
[W88] E. Witten, Comm. Math. Phys. 117, 353 (1988)
[W88] W i t t e n 1988:yMT4 °p N = 2 Twisted Supersymmetry ---* S(a, ¢, ¢; ...) invariant under
[BS88] L. Baulieu, I.M. Singer, NPB Proc. Supp. 15B, 12-19 (1988)
Q a = ¢ Q ¢ = Da¢ ; f o r m d o Q ¢ = 0 ghost ~
[H89] J.H. Home, Nucl. Phys. B318, 22 (1989) [B88] P. Braam, (1988)
CERN-Strasbourg seminars
a ¢ ¢ 1 1 0 0 1 2
Q2 = 0 on gauge invariant combinations of a, ¢, ¢ in fact: Q2= gauge transformation of parameter
¢
[OSB89] S. Ouvry, R. Stora, P. Van Baal, Phys. Lett. B 220, 1590 (1989)
Q tr ( r ( a ) + ¢ + ¢)2 = - d t r ( F ( a ) + ¢ + ¢)2
[BS91] L. Baulieu, I.M. Singer, CMP 135, 253 (1991)
/
[K93] J. Kalkman, Comm. Math. 447 (1993)
= QA(gauge inv.(a, ¢, ¢; ...) + S(a, ¢, ¢; ...)
Phys. 153,
[C50] H. Cartan, Colloque de Topologie, Bruxelles 1950 CBRM p.15, 56 [BGV92] N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, 1992 , Springer Grundlehren, Vol. 298 [BCI94] C. Becchi, R. Collina, C. Imbimbo, Phys. Lett. B 322, 79 (1994) [STW94] l~. Stora, F. Thuillier, J.C. Wallet, First Carribean Spring School of Math. and Theor. Phys., May 30-June 13 (1993), (1994).
where: F(a) = da q- l[a, a]. Remark: trFAF=
[BS88] Baulieu, Singer 1988
Q
~
s
sa
=
~b - Da w
s¢
=
Da¢ + [¢,w]
s¢
= _
[¢, w]
w : S(a, ¢, ¢ ) =
s2-0
w] + ¢ gauge Faddeev Popov ghost fEtrFAF+sA(a,¢,¢;...)
defect: s has no cohomology ! (Observables ???) [H89] J. H o r n e 1989
0920-5632/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved. SSD1 0920-5632(95)00123-9
R. Stora/Nuclear Physics B (Proc. Suppl.) 39B, C (1995) 488 490
S ( a , ¢ , ¢ ; . . . ) is not only gauge invariant but also does not depend on w Supersymmetric gauge invariance
There is another interpretation, the so-called Weil interpretation [K93] s = 5 + 6w
¢ = 5 a - Daw
r(~) = i(A) + y ( a ) [OSB89] S. Ouvry, R. Stora, P. van Baal (1989) On fields a, ¢, ¢, the action of infinitesimal gauge transformations is given by:
6~ = Is, 2(~)]+ fl(A)w = A
if(A) other = 0
The cohomology of s restricted to 6a and ,Tx invariant objects is non e m p t y and contains Witten's example. This set up has been known since the 50's under the name of "basic cohomology" or "equivariant cohomology".
[K93] J. K a l k m a n (1993) has analyzed OSB in abstract terms and made a very nice algebraic discovery possibly burried in the m a t h e m a t i c a l literature but certainly not widely known. There, s is interpreted as follows: a E .4
:
6 ¢o, ¢
= :
6~o
=
6w¢
=
space of connections on some G-bundle P ( E , G) (with structure group G) differential on .4 generators of a Weil algebra for ~, the gauge group: 1[~,~]+¢
[¢,~]
~(A),t(A), action of G on .4 inherited from the action of ~ on P one may interpret s ¢
= =
6+6w+l(w)-i(¢) 6a
If if(A), E(A) is the action on the Well algebra:
J(h)~ L(h)~ l(~) + L(~)
= = =
~ j(~)¢=0 [~, w] £(~)¢ : [~, ¢] Is, j ( ~ ) ]
489
L(A) = 1(a) + Z(a)
L(a) = Is, i(a)] (compare BS 88-91). This set up generalizes to any situation where .4 is replaced by a space of fields defined on E and by some group acting on .4. Other example: 2d gravity
Y•J• A
-~ M ( E ) --~ DiffE
metrics on I;
One question is: how to find equivariant cohomology classes. One general method which involves equivariant cohomology again goes as follows: find a family of H-bundles over E on which the action of lifts and find a ~ invariant connection F. H characteristic classes of the equivariant curvature of r define equivariant cohomology classes of the basis B of the H-bundle family (e.g. E x .4 in the Yang-Mills case, E x M ( E ) or E x C M ( E ) in the 2d gravity case). (CAd(E) = conformal classes of metrics on E.) Next, " t h e o r e m 3 of Caftan" allows to replace w, the generator of the Weil algebra by a connection ~ for the action of G on the above manifold B, thus yielding basic cohomology classes of B for the action of ~. It follows from general properties of characteristic classes that these cohomology classes do not depend on the various connections used to define t h e m (eg F, ~), hence can be obtained by integrating or averaging out over families of those (e.g., z-~ top in the gauge c a s e F = a i n t h e t ~ r 2 caseP=an extension to the linear frame bundle of the LeviCivita connection). This, we believe, is the reason why such cohomology classes can be expressed as "functional integrals" (to be properly defined). This interpretation allows compact representations of those cohomology classes both in the Y M ~ °v case where it explains the role of F + ¢ + ¢
R. Stora/NuclearPhysicsB (Proc. Suppl.) 39B,C (1995) 48~490
490
(cf. Baulieu, Singer 88-91) and in the Gr~°p case (cf. Becchi, Collina, Imbimbo 94) which remained rather mysterious for some time. It also sheds some light on the N = 2 twisted supersymmetry approach. Introducing the Faddeev Popov charge Q¢~ (not to be confused with the Q in the beginning of the lecture), (and, the corresponding currents, if one wants), one has the following graded commutation relations:
[~, s ( a ) ] + [c(a),L(~')]_ It(a), s(a')]_
= : =
c(a) c([A,a']) s([a, a'])
[s(~), s(~')]+ [O, 4_
: =
o
[Q, S(~)]_ [O, c(~)]_
= =
S(~) 0
Ex.: N = 2 superconformal set up: A E Vir
Q'JJ,
,](.) = G -
s,,~/G +
£(.) = T
This return to the origin requires the introduction of the operation J ( . ) associated to the introduction of the Faddeev Popov ghost w. Conclusion If you are an addict of BRST games, find the correct cohomology first !
Nuclear Physics B (Proc. Suppl.) 39B,C (1995) 488490
E q u i v a r i a n t c o h o m o l o g y a n d c o h o m o l o g i c a l field t h e o r i e s R. Stora a aLaboratoire de Physique Th~orique ENSLAPP, URA 14-36 associfie • I'E.N.S. de Lyon, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France and Theory Division, CERN, CH-1211 Geneva 23, SWITZERLAND A general method is given to construct a class of observables in topological field theories.
[W88] E. Witten, Comm. Math. Phys. 117, 353 (1988)
[W88] W i t t e n 1988:yMT4 °p N = 2 Twisted Supersymmetry ---* S(a, ¢, ¢; ...) invariant under
[BS88] L. Baulieu, I.M. Singer, NPB Proc. Supp. 15B, 12-19 (1988)
Q a = ¢ Q ¢ = Da¢ ; f o r m d o Q ¢ = 0 ghost ~
[H89] J.H. Home, Nucl. Phys. B318, 22 (1989) [B88] P. Braam, (1988)
CERN-Strasbourg seminars
a ¢ ¢ 1 1 0 0 1 2
Q2 = 0 on gauge invariant combinations of a, ¢, ¢ in fact: Q2= gauge transformation of parameter
¢
[OSB89] S. Ouvry, R. Stora, P. Van Baal, Phys. Lett. B 220, 1590 (1989)
Q tr ( r ( a ) + ¢ + ¢)2 = - d t r ( F ( a ) + ¢ + ¢)2
[BS91] L. Baulieu, I.M. Singer, CMP 135, 253 (1991)
/
[K93] J. Kalkman, Comm. Math. 447 (1993)
= QA(gauge inv.(a, ¢, ¢; ...) + S(a, ¢, ¢; ...)
Phys. 153,
[C50] H. Cartan, Colloque de Topologie, Bruxelles 1950 CBRM p.15, 56 [BGV92] N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, 1992 , Springer Grundlehren, Vol. 298 [BCI94] C. Becchi, R. Collina, C. Imbimbo, Phys. Lett. B 322, 79 (1994) [STW94] l~. Stora, F. Thuillier, J.C. Wallet, First Carribean Spring School of Math. and Theor. Phys., May 30-June 13 (1993), (1994).
where: F(a) = da q- l[a, a]. Remark: trFAF=
[BS88] Baulieu, Singer 1988
Q
~
s
sa
=
~b - Da w
s¢
=
Da¢ + [¢,w]
s¢
= _
[¢, w]
w : S(a, ¢, ¢ ) =
s2-0
w] + ¢ gauge Faddeev Popov ghost fEtrFAF+sA(a,¢,¢;...)
defect: s has no cohomology ! (Observables ???) [H89] J. H o r n e 1989
0920-5632/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved. SSD1 0920-5632(95)00123-9
R. Stora/Nuclear Physics B (Proc. Suppl.) 39B, C (1995) 488 490
S ( a , ¢ , ¢ ; . . . ) is not only gauge invariant but also does not depend on w Supersymmetric gauge invariance
There is another interpretation, the so-called Weil interpretation [K93] s = 5 + 6w
¢ = 5 a - Daw
r(~) = i(A) + y ( a ) [OSB89] S. Ouvry, R. Stora, P. van Baal (1989) On fields a, ¢, ¢, the action of infinitesimal gauge transformations is given by:
6~ = Is, 2(~)]+ fl(A)w = A
if(A) other = 0
The cohomology of s restricted to 6a and ,Tx invariant objects is non e m p t y and contains Witten's example. This set up has been known since the 50's under the name of "basic cohomology" or "equivariant cohomology".
[K93] J. K a l k m a n (1993) has analyzed OSB in abstract terms and made a very nice algebraic discovery possibly burried in the m a t h e m a t i c a l literature but certainly not widely known. There, s is interpreted as follows: a E .4
:
6 ¢o, ¢
= :
6~o
=
6w¢
=
space of connections on some G-bundle P ( E , G) (with structure group G) differential on .4 generators of a Weil algebra for ~, the gauge group: 1[~,~]+¢
[¢,~]
~(A),t(A), action of G on .4 inherited from the action of ~ on P one may interpret s ¢
= =
6+6w+l(w)-i(¢) 6a
If if(A), E(A) is the action on the Well algebra:
J(h)~ L(h)~ l(~) + L(~)
= = =
~ j(~)¢=0 [~, w] £(~)¢ : [~, ¢] Is, j ( ~ ) ]
489
L(A) = 1(a) + Z(a)
L(a) = Is, i(a)] (compare BS 88-91). This set up generalizes to any situation where .4 is replaced by a space of fields defined on E and by some group acting on .4. Other example: 2d gravity
Y•J• A
-~ M ( E ) --~ DiffE
metrics on I;
One question is: how to find equivariant cohomology classes. One general method which involves equivariant cohomology again goes as follows: find a family of H-bundles over E on which the action of lifts and find a ~ invariant connection F. H characteristic classes of the equivariant curvature of r define equivariant cohomology classes of the basis B of the H-bundle family (e.g. E x .4 in the Yang-Mills case, E x M ( E ) or E x C M ( E ) in the 2d gravity case). (CAd(E) = conformal classes of metrics on E.) Next, " t h e o r e m 3 of Caftan" allows to replace w, the generator of the Weil algebra by a connection ~ for the action of G on the above manifold B, thus yielding basic cohomology classes of B for the action of ~. It follows from general properties of characteristic classes that these cohomology classes do not depend on the various connections used to define t h e m (eg F, ~), hence can be obtained by integrating or averaging out over families of those (e.g., z-~ top in the gauge c a s e F = a i n t h e t ~ r 2 caseP=an extension to the linear frame bundle of the LeviCivita connection). This, we believe, is the reason why such cohomology classes can be expressed as "functional integrals" (to be properly defined). This interpretation allows compact representations of those cohomology classes both in the Y M ~ °v case where it explains the role of F + ¢ + ¢
R. Stora/NuclearPhysicsB (Proc. Suppl.) 39B,C (1995) 48~490
490
(cf. Baulieu, Singer 88-91) and in the Gr~°p case (cf. Becchi, Collina, Imbimbo 94) which remained rather mysterious for some time. It also sheds some light on the N = 2 twisted supersymmetry approach. Introducing the Faddeev Popov charge Q¢~ (not to be confused with the Q in the beginning of the lecture), (and, the corresponding currents, if one wants), one has the following graded commutation relations:
[~, s ( a ) ] + [c(a),L(~')]_ It(a), s(a')]_
= : =
c(a) c([A,a']) s([a, a'])
[s(~), s(~')]+ [O, 4_
: =
o
[Q, S(~)]_ [O, c(~)]_
= =
S(~) 0
Ex.: N = 2 superconformal set up: A E Vir
Q'JJ,
,](.) = G -
s,,~/G +
£(.) = T
This return to the origin requires the introduction of the operation J ( . ) associated to the introduction of the Faddeev Popov ghost w. Conclusion If you are an addict of BRST games, find the correct cohomology first !