Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras

ANKALS

OF PHYSICS

176. 499113

(1987)

Symplectic Reduction, Infinite-Dimensional

BRS Cohomology, Clifford Algebras

and

BERTRAM KOSTANT* AND SHLOMO STERNBERG+

Received

September

22. 1986

This paper gives the mathematical foundations for the BRS quantization procedure. We first discuss the classical linite dimensional BRS procedure and relate it to Marsden-Weinstein reduction. This leads to interesting relations between Lie algebras and Clifford algebras and a novel way of computing Lie algebra cohomology in terms of the spin representation. We then discuss infinite-dimensional Clifford algebras and their spin representations. We find that in the inlinite-dimensional case, the analog of the finite-dimensional construction of Lie algebra cohomology breaks down. the obstruction (anomaly) being the Kac-Peterson class which is the cohomology class associated to the representation of the Lie algebra on spinors which is now only a projective representation. Tensoring by a projective representation of opposite class kills the obstruction and gives rise to a cohomology theory and a quantization procedure. We discuss the gradings and Hermitian structures on the absolute and relative complexes. 1 1987 Academic Press. Inc

CONTENTS. I. Symplectic reduction. 2. Super-Poisson algebras. 3. Clifford algebras. 4. The exterior algebra as a left Clifford module. 5. Lie algebras and Clifford algebras. 6. Reduction and quantization. 7. Infinite dimensional Clifford algebras and their spin representations. 8. Lie algebras and Clifford algebras in the infinite-dimensional case. 9. Reduction and quantization in the infinite dimensional case. IO. The Clifford group and the space of maximal isotropics. I I. Real forms and Hermitian structures. f2. Additional comments.

The purpose of the present paper is to discuss some of the mathematical foundations of the BRS quantization procedure which has recently been developed in the physics literature. Roughly speaking, the BRS procedure gives a method for quantizing a mechanical system reduced by constraints through the introduction of certain “odd variables.” In the finite-dimensional case, if we impose sufficiently severe regularity conditions on the constraints, the BRS method is intimately related to the method of symplectic reduction. Even in the finite dimensional case the BRS method promises to be a useful tool in handling the troublesome singularities that can arise both in the constraint equations themselves and from * Mathematics ’ Mathematics

Department, Department,

MIT, Cambridge, Harvard University,

MA. Cambridge,

MA.

49 0003-4916/87

$7.50

copyrlghi .(‘: 1987 by Academx Press. Inc All rights of reproductmn I” any form reserved

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KOSTANT AND STERNBERG

dividing out by the action of a symmetry group. In the infinite dimensional case (the one that is of physical interest in gauge theories and string theories) new phenomena can arise, principally associated to the fact that infinite-dimensional Clifford algebras do not have unique spin representations and the attendant problem of Wick ordering, giving rise to “anomalies.” The most striking example is Feigin’s computation of the number 26 in conjunction with the Virasoro algebra, which was one of the starting points of this paper. We will begin this paper with a rapid review of the basic facts of symplectic reduction referring the reader to [Gui-St] for details. We then discuss the BRS method for the case of finite-dimensional Marsden-Weinstein reduction. This will lead us to interesting relations between Lie algebras and Clifford algebras and a novel way of computing Lie algebra cohomology in terms of the spin representation. For the infinite-dimensional case we first discuss the spin representations associated to maximal isotropic subspaces. We then discuss the relation between Lie algebras and Clifford algebras and show that the analog of the linite-dimensional construction of Lie algebra cohomology breaks down, the obstruction being essentially the Kac-Peterson class which is the cohomology class associated to the representation of the Lie algebra on spinors which is now only a projective representation. Tensoring by a projective representation with opposite class kills the obstruction and gives rise to a cohomology theory which gives the quantization procedure. We then discuss the pairing between spin modules associated to transverse maximal isotropics. Finally we show how a real fom on the Lie algebra satisfying appropriate conditions induces a Hermitian structure on the spin module. The second author wishes to thank Ezra Getzler for explaining some of the ideas involved in the BRS procedure. Both authors wish to thank David Kazhdan who participated in the initial phases of our current investigations. We also wish to thank J. Bernstein, M. J. Bowick, R. Bott, I. Frenkel, H. Garland, V. Guillemin, V. Kac, Y. Ne’eman, S. G. Rajeev, I. M. Singer, E. Witten, and G. Zuckerman for helpful discussions.

1. SYMPLECTIC REDUCTION The basic idea of symplectic reduction goes back to Lagrange and the “elimination of the nodes” in celestial mechanics. Roughly speaking it says that if we have k integrals of motion that are in involution they can be used to reduce the system of differential equations of motion to a system with 2k fewer variables. In more detail: let f, ,..., fk be k independent functions on phase space which satisfy

where the ct are functions. Then the subset defined by the equations fi= 0, i = l,..., k (if it is a submanifold) is an example of what is called a coisotropic submanifold in the mathematics literature. See [Gui-Stl, p. 1761 for the general

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51

definition of a coisotropic submanifold and for the general theory of symplectic reduction, especially Theorem 25.3 on page 177. In the physics literature such equations as ,f,= 0 are known as first class constraints (according to Dirac’s classification). Every coisotropic submanifold, C, has a “null foliation.” In the example above this will be the k-dimensional family of directions spanned by the infinitesimal motions (vector fields) corresponding to the functions f,, i= l,.... X. Under favorable circumstances, this foliation might be fibrating; that is, there might be a map p: C+ B, where B is a manifold (which then carries a symplectic structure) where the fibers of p, i.e., the sets of the form p l(h), hi B, are exactly the leaves of the null foliation. Now suppose that the -f’s are integrals of motion, that is, that they all satisfy

where H is the Hamiltonian of the system. Then H is constant along the leaves of the null foliation, and so defines a function on B. The corresponding equations on B are known as the reduced equations. The simplest example of this process is the reduction to the center-of-mass motion for a system with translational symmetry: for such a system the three components of the total linear momentum P = (P,, P,, P3) are integrals of motion. If we choose some constant vector d= (d,, n,, d,) and let f, = P, - d, then the submanifold C consists of all points in phase space with total linear momentum tf. Conservation of linear momentum says that if a point in phase space lies on C so will the entire trajectory through that point. The space B is obtained by identifying two points of C if they differ by an overall translation, so B is a phase space for motion “relative to the center of mass.” Once we have solved the reduced system. it is a simple matter to obtain the motion for the original system. There are thus two steps: restriction to the manifold C and projection onto the quotient manifold B. In the example of total linear momentum the functions (zf,all vanish. This is becausethe group of translations is commutative. The generalization to more general symmetry groups is given by the notion of a Hamiltonian group action, see [K-K-S or Gui-91, Sect. 261. For a Hamiltonian group action one is given a symplectic manifold, so that the ring F(M), of all smooth functions on A4 has a Poisson bracket. One is given a Lie group G acting as symmetries of the symplectic structure, and so, in particular of the ring F(M) together with its Poisson bracket structure. Finally, one is given a linear map

where g is the Lie algebra of G and where, for each < ~g, Poisson bracket by the function S(t) is just the infinitesimal automorphism (i.e., derivation) of F(M) given by the Lie algebra element 5. (Also the map 6 should be equivariant for the action of G.) The group theoretical generalization of the total linear momentum is the moment map. The moment map is a map

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where g* denotes the dual space of g, i.e., the space of all linear functions on g. It is defined by

On the left side of this equation Q(m) is to be a linear function on g. The lefthand side denotes its value at the particular element, 5, of g. The equation says that this number is given by the right-hand side, where S(5) is a function on M and the right-hand side is the value of this function at the particular point m. Since, by hypothesis, the function S(5) depends linearly on t, the right-hand side is a linear function of 5 and hence the above equation defines Q(m) as an element of g*. A coadjoint orbit, by definition, is a subset 0 of g* of the form O=G.ci,

cceg*.

That is, 19 is the set of all points that can be obtained from some fixed element cc of g* by applying members of the group, G. The basic theorem of [K-K-S] says that under a certain technical hypothesis (that the map @ intersect the orbit 0 cleanly) the set C = @~ ‘( 0) is a coisotropic submanifold of M. Furthermore, let m be a point of C and /I = Q(m). Let G, denote the isotropy group of /?, that is, the subgroup of G consisting of those elements which keep fi fixed. Then the leaf of the null foliation of C passing through m is just Gs. 1y1,the orbit of m under the connected component of G,. For the case of an abelian group the orbits Cr/are all points and the isotropy group is all of G. This is what happens for the group of translations, for example. A particularly important case is where we take 0 = (0). Thus 0 is a single point, the origin. This is an example of Marsden- Weinstein reduction [M-W]: C=@-‘(O)

(1.1)

and B=C/G

(if G is connected).

(1.2)

Although Marsden-Weinstein reduction is a special case of the reduction associated to coadjoint orbits which is itself only a special case of symplectic reduction, we should point out that many cases of importance in physics are encompassed by the MarsdenWeinstein reduction procedure. Here are some examples: 1. Let the group G act on a manifold N. Let M= T*N, so that M is the “phase space” associated to the “configuration space,” N. The action of G on N induces a Hamiltonian on M whose linear map 6: g + F(M) can be described as follows: each 5 ~g gives a vector field on N, describing the infinitesimal action on N. Any vector field on N defines a function on M= T*N which is linear on each cotangent space. The function S(r) is then the function on T*N defined by the vector field on N associated to 5. Thus C = @- ‘(0) consists of those covectors which

BRS

COHOMOLOGY

53

vanish on tangents to the orbits of G acting on N. In special cases the constraints involved in the definition of C are known as the “Lorentz condition” or the “Lorentz gauge,” see example 2. We thus see that B = C/G can be identified with T*(N/G) under the appropriate regularity conditions for the G action on N (so as to guarantee that N/G is a manifold). Reduction in this case is the passage from the phase space of the original configuration space to the phase space of the space of orbits. 2. In example 1 we could consider the special case where N is a vector space so that T*N = N @ N* and where G is a subspace, W, of N acting by translation. Then C = NO tV”, where IV’ is the subspace of N* consisting of those linear functions which vanish on W. Then B = (N/W) @ v = (N/W) @ (N/W)*. Suppose we take N to consist of one forms on Minkowski space. Any one form B of compact support defines a linear function, fB, on N by 1,(A) = j A . B where the scalar product is the Minkowski scalar product and where the integral is over all of Minkowski space. We can thus think of N* as “generalized” or “distributional” one forms. Let us take W to consist of exact differentials, that is of all one forms of the type &I Then integration by parts shows I, belongs to @ if and only if ri* B=O (or Br, = 0 in tensor language). This is the classical Lorentz condition. 3. A modification of example 2 is to take M to be a symplectic vector space and G = W to be an isotropic subspace (that is, a subspace on which the symplectic form vanishes identically). Again we let W act by translation. Then C= WI. the orthocomplement of W relative to the symplectic form and B= W’/ W. Suppose that M is a complex vector space with a (not necessaily positive definite) Hermitian form. The imaginary part of the Hermitian form is then a symplectic form. If W is a complex subspace, then WI will also be the orthocomplement of W relative to the Hermitian form. For example take M to consist of one forms on Minkowski space which are solutions of the wave equation. More precisely, passing to the Fourier transform, take M to consist of continuous vector valued functions, A, on the light cone which have compact support. So A(k) is a (complex) four vector defined for each X- with li’ = 0. Define the Hermitian form ( , ) on M by (A, B) = j A(k) B(k) rmz(li). where ~Cn(k) is the Lorentz invariant metric on the light cone. Here the pointwise scalar product, .9 is the Lorentz product, so the Hermitian product ( . ) is not positive definite. Let W consist of those functions of the form A(k) =,f‘(k) k. where ,f’ is a scalar function. (So W consists of Fourier transforms of differentials of solutions of the wave equation.) Then W is isotropic, and W ‘/i W inherits a positive definite scalar product. The elements of W’/ W can be identified with solutions of Maxwell’s equation. This is the Gupta-Bleuler construction in embrionic form. 4. Let N be a (complete) Lorentzian manifold. The geodesic flow defines an action of R on M = T*N+, the space of non-zero covectors. Then C is the set of non-zero null covectors and B the space of null geodesics. Of course, in general, B will not be a manifold, but under appropriate homogeneity conditions for the metric it will be. But in most cases, even if B is a manifold, it will not be the cotangent bundle of anything. Thus, starting with a cotangent bundle, the process

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of symplectic reduction can give rise to a symplectic manifold which is not a cotangent bundle. 5. Let N= Q(RdP ‘,l) be the set of all closed loops in RdP ‘.’ (d-dimensional space with a metric of signature d- 1, 1). A curve on N can be thought of as a map of a cylinder into R ‘-‘3’ . The Nambu action associates to any such cylinder its total induced area. The expression for the Nambu action can be written in terms of an integral of a Lagrangian, L, defined on TN. The Lagrangian L defines a Legendre transform, dL: TN + T*N, but this Lagrangian is highly singular, so that its image, dL( TN), is a subvariety, C, of T*N. This C is, in fact, @~ ‘(0) where @ is the moment map for a Hamiltonian action of Diff S’ x Diff S’ on T*N. This is how symplectic reduction enters string theory. We now turn to a general study of (1.1) and (1.2). Any function on C can be extended to be a function on A4 and two functions on M restrict to give the same function on C if their difference vanishes on C. Thus F(C) = F( M)/Z where Z is the ideal consisting of all functions vanishing on C. For every 5 in g the function S(5) vanishes on C by definition, and so belongs to the ideal I. If 0 is a regular value of the moment map then the ideal Z will be generated by these functions, i.e., every element of Z can be written as a linear combination of the S(t) with function coefficients. We write this as

so F(C) = nM)IQw.

h(g).

(1.3)

Now a function on B can be thought of as a function on C which is constant along the leaves of the null foliation. In view of (1.2) this is the same as requiring the function to be invariant under the action of G. Thus F(B) consists of the G-invariant functions on C for the case of MarsdenWeinstein reduction. We write this as F(B) = F(C)‘,

(1.4)

where the superscript G denotes G invariants. The classical use of symplectic reduction is to simplify the equations of motion: a function on B is considered “simpler” than a function on M since it involves fewer variables. But it was shown in [K-K-S] that sometimes the reverse is true. A complicated looking Hamiltonian on B may be the reduced version of a much simpler Hamiltonian on A4. Indeed we solved the Calogero system arising from soliton theory by showing that it was the “reduction” of a much simpler Hamiltonian system whose solution was straightforward. A similar situation can arise in quantization: We might be interested in quantizing a certain class of functions on B as

BRS

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COHOMOLOGY

operators on some Hilbert space. It might be that these functions all come from a class of functions on M which have a natural quantization. We might then be able to develop an analog of reduction at the quantum level and thereby quantize our original functions on B. Indeed, within the framework of group representation theory such a program has been carried out in the series of papers [Gui-St 2 Gui-St 51. Let us now give a homological transcription of ( 1.3) and ( 1.4) using tools+ the Koszul resolution and Lie algebra cohomologyPwhich are familiar to mathematicians. As they may not be lingua franca for physicists we discuss the transcription in more detail than is strictly necessary. First (1.3): Consider the Grassmann (exterior) algebra /ig and form the tensor product ilg@ F(M). It is a (super)commutative superalgebra. (For those who are unfamiliar with the language of superalgebras see the next section for the definitions.) Now define a (super)derivation (which we shall continue to denote by 6) of this superalgebra by defining it on the generators < @ 1, t ~g and 1 @I fi f‘~ F( M) according to the rules ci(<@l)=l@S(i’)

which makes sense since 6( < ) E F(M)

and

The fact that 6 is required &OF(M) and

to be a superderivation

then defines it on all of

Now the square of an odd derivation, such as 6, is an even derivation. (The general statement is that the set of all superderivations form a Lie superalgebra, cf. the next section. For odd elements, the square is just one half the supercommutator of the element with itself.) Thus 6’ is a derivation and so is determined by its action on the generators. But 6’ clearly vanishes on the generators by definition. Hence s*=o. Define H%( AgO F(M)) claim that

to be ker 6/im 6 as a subquotient

H~(/igOF(M))=F(M)/F(M).S(g)=F(C).

of /1“g@ F(M).

We (1.5)

Indeed, all of /i’soF(F(M) lies in ker 6, while im 6 =nOg@F(M).cS(g) F(M) .6(g). Equation ( 1.5) is our homological reformulation of ( 1.3). (For the commutative algebraist, what has happened here is that we have used the commutative algebra structure of F(M) to extend the linear map 6: g -+ F(M) to an algebra homomorphism of S(g) into F(M). This makes F(M) into a S(g)

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module via multiplication. We have then taken the Koszul resolution of this module.) We now turn to (1.4). Let K be any module for the finite-dimensional Lie algebra g. That is, we assume that we are given a representation of g on the vector space K. (The case of interest for us is where K = Ag@F(M) or one of its homology spaces.) Then one defines d:K-+g*@K=Hom(g,

K)

by (dk)(C)

= 5k

(Es,

kEK>

(1.6)

One then extends this to a map d:APg*@K+AP+‘g*@K

d(w@k)=do@k+(-l)Pco

A dk,

w E npg*.

(1.7)

The expression do in (1.7) denotes the usual exterior derivative of o when thought of as a left-invariant p-form on any Lie group G whose Lie algebra is g. (In purely Lie algebra terms d: g* + A’g* is the negative transpose of the map A’g + g given by the Lie bracket. Then d is extended so as to be a superderivation of degree +l of Ag*.) It is easy to check that the d defined by (1.6) and (1.7) satisfies d2 = 0. The space H$( Ag* @ K) = ker d/im d

at

Apg*@K

is usually denoted by HP(K, g) or just by HP(K) and is called the pth cohomology space of g with values on the module K. Notice that at p + 1 = 0 we have im d = 0 and, by (1.6) an element k belongs to ker d if and only if tk = 0 for all 5 E g. In other words, k belongs to ker d if and only if k is (infinitesimally) invariant under g. Thus Ho,( Ag* Q K) = space of g invariants in K.

(1.8)

If 9: K, AK, is a g morphism, i.e., a linear map which commutes with the action of g, then we can extend the map .Y to be the map id @ 4: Ag* OK, -+ Ag* OK,. We will denote this extended map again by .a instead of id 0 9. This extended 9 then commutes with the action of d. Take K, = K2 = Ag@ F(M) and 9 = 6. We thus have two maps, d and 6, of Ag* 0 Ag 0 F(M) into itself: Apg*@Aqg@F(M)~

APg*@A4-‘g@F(M)

d

AP+‘g*$OA”g@F(M).

(1.9)

31

BRS COHOMOLOGY Furthermore,

6” = 0,

d’ = 0,

6d=dh

(1.10)

and HO,(H~(Ag*OAgOF(M)))=F(B). Now the situation (1.9) and (1.10) is familiar what is called a double complex, cf. [Bott-Tu]. useful to consider the “total differential”

(1.11) to topologists. It is an example of Experience has taught that it is

D = d+ ( - 1 )’ ii. Actually, for technical reasons which will become apparent later but have no impact on the homological considerations, it will be useful for us to take D = d + ( - 1 )” 26. Then D: A”g*@

A“g@ F(M)

-+ A P+‘g*OAYgOF(M)+n”g*OA”

‘g@F(M).

Define the “total degree” as p - q. That is, define

P(ng*@ng@F(M))= c APg*@AYg@F(M) P Y=k so that

and D’=O. The operator D is the “classical” BRS operator. Under certain favorable circumstances to be discussed in Section 6 (and which we have been implicitly assuming in our motivating discussion of this section), we have

and so H:‘,(AR*OAgOF(M))=F(B).

(1.131

So far, all we have done is give a mathematically standard, if somewhat complicated, redefinition of the space F(B). Recall our problem: the points of B are the physically interesting states at the classical level. We wish to “quantize” certain

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observables on B, that is, certain elements of F(B), and we know how to “quantize” corresponding elements of F(M). Here is the new idea: Observe that AgO ,4g* = /i(g+g*) has a (split) scalar product given by the evaluation of g* on g. Use this scalar product to construct the Clifford algebra C(g+g*). This induces a superPoisson bracket on n(g + g*). (For a review of the facts about Clifford algebras and super-Poisson algebras and the details of this construction see the next two sections.) By tensor product, /i(g + g*)@ F(M) becomes a super-Poisson algebra. It then turns out that there exists an (odd) element 0 E A(g +g*)@ F(M) such that the Poisson bracket by 0 is precisely D! In other words, adO={O;}=D.

(1.13)

In particular, (ad 0)2 = 0. So now the procedure is: Extend the “known” quantization of elements of F(M) to a quantization of elements of /i(g +g*)@ F(M) as operators on some space, T. Be sure that we have quantized enough of the elements of n(g +g*)@ F(M) so as to include 0 and the “total degree derivation.” Arrange that 0 is “quantized” by an operator Q with Q2 = 0. Also that T is graded:

T=@Tk

with

Q:T"+Tk+'.

Then define the “true space of physical states” to be

Ho,(T) and the elements of F(B) become quantized as operators on this space. For finite-dimensional g this program can be carried out, at least in principle, for any F(M). For example, if we take F(M) = R, corresponding to M =pt., then we have no 6 at all. We just have A(g + g*) with the derivation d= ad 0 singled out. In this special case 0 will actually be an element Q of n3( g + g*). Now the “quantum algebra” whose associated “classical” algebra is n(g+g*) is just the Clifford algebra C(g +g*) which has a unique irreducible module, S, the spin representation. Furthermore there is a canonical way of identifying C(g+g*) with /i(g + g*) as left C(g +g*) modules (as will be explained in Sect. 4). This identification gives us an element Q of C(g +g*) corresponding to 1;2.It turns out that Q2 = 0. Thus Q acting on S is the “quantization” of d. (In fact, if g is unimodular, H,(S) is just the usual Lie algebra cohomology of g, cf. Section 6. So all we have done is give a complicated definition of a well-known object.) However, when g is infinite-dimensional, the Clifford algebra C(g + g*) no longer has a unique irreducible representation. One defines an irreducible representation by the choice of a maximal isotropic subspace. One can no longer get from such space to any other by a finite number of steps. In physics of infinite-dimensional systems one must “fix the vacuum.” The first and best known example of this phenomenon is the Dirac theory of holes where the vacuum is chosen to be that state where all the negative energy single particle electron states are filled-the “Dirac sea.” Then positrons appear as “holes” in this infinite sea.

BRSCOHOMOLOGY

59

Also, in the infinite-dimensional case the form Sz is still defined as an antisymmetric trilinear form, but will not be an element of /l(g +g*), which consists of antisymmetric forms of finite rank. Rather, it will lie in a certain completion of A( g + g*), a completion which also depends on the choice of vacuum. Then the element Q no longer lies in the Clifford algebra but only in a completion, and cannot be chosen in the g-invariant way that works in the finite-dimensional case. (In physics language, this has to do with “renormalization” and “Wick ordering.“) Then Q’ is no longer zero, in general, but gives a cohomology class which is the obstruction to carrying out the above quantization program This is known as an “anomaly.” It is only for certain choices of modules F(M) (and their analogs) that these anomalies will disappear. As we have seen, the BRS operator gives a cohomological description of the reduction process. Quantizing this operator and then passing to the operators on the zeroth cohomology space is the BRS method of quantization. It has some potential benefits in the finite-dimensional case in that it gives a procedure which is applicable even when the reguiarity conditions used above to define the reduced manifold are not satisfied. It also may be a useful tool in studying the singularities of the moment map. But its principal area of application is in the quantization of infinite-dimensional systems. Here the quantization of the “big” space may be unsatisfactory due to the presence of “ghosts.” That is, the space on which the quantization naturally occurs may have a scalar product which is not positive definite. But by passage to a subquotient (which is, after all, what cohomology does) one obtains a positive definite space. This is known as a “no-ghost theorem.“ The prototype of this procedure is the Gupta-Bleuler method of quantizing the electro-magnetic field. cf. [Gu] or [Bl]. Its modern version is the “no ghost theorem” of string theory in 26 dimensions, cf. [Feigin and FrGarZuck]. It is reminiscent of similar theorems in group representation theory [ Raw-Sch- Wo]. Determining its appropriate general setting remains an important subject for investigation.

2. SUPER-POISSON ALGEBRAS These objects were first introduced in [Co-Ne-St] and studied in detail in [Ko]. We briefly recall their definition and main properties. A superalgebra is simply a Z, graded vector space A=A,,+A, together with a bilinear map A x A 4 A which is Z, graded. This means that A,xAu+Ao

A,xA,+A,

A,xA,+A,

A,xA,+A,.

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A is called an associative superalgebra if the multiplication usual sense). A is called supercommutative if (- 1)” bja,

aibi=

where

aiEAi,

bjEAj;

is associative (in the i, j=O,

1.

A derivation of the superalgebra A can be either even or odd: an even derivation isamapd:A,-+A,andd:A,-+A,suchthat d(ab) = (da) b + a(db)

for any pair of elements a and b of A. An odd derivation

maps A, + A, and

A, + A,, and satisfies d(aib) = (dai) b + (- l)‘a,(d),

aiE Aj.

A superalgebra L is called a Lie superalgebra denoted by a bracket) is superanticommutative: [l;, lj] = -( - 1)” [l,, ri]

if the multiplication

(usually

liG L,, liE L,

and if left multiplication is a superderivation (i.e., either an even or an odd derivation depending on the degree of the element):

III,* [lj, ikll = [Cr,, /jlT lkl + (3)”

Clj, li, /,I.

This is the super version of the Jacobi identity In particular, if A = A, + A, is an associative superalgebra we can use its elements to construct a Lie superalgebra by defining the supercommutator of two elements as [ai, a,] = aja, - ( - 1)” a,ai.

(The right-hand side of this equation is the usual commutator unless i and j both equal 1, in which case the right-hand side is what is called the anticommutator in the physics literature.) It is routine to check that the axioms for a Lie superalgebra are satisfied. A fact whose verification is routine, and which we will make use of is the following: Suppose that Q is an odd element, i.e., Q E A 1. Then for any element a, even or odd,

[Q2, al = [Q, [Q, all. We write this as ad Q* = (ad Q)‘, where ad Q denotes the operation

of left supercommutator

bracket by Q.

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A super-Poisson algebra is an object which is both a supercommutative superalgebra and a Lie superalgebra. That is, B is a super-Poisson algebra if it has two kinds of multiplication defined on it: (i)

an associative multiplication B, x B, + B, + , denoted by (b,, b,) + b,b,

which is supercommutative, and (ii)

a Lie bracket multiplication Bj x B, + B, +, denoted by (b,,b,)+

lb,, b,;

called a Poisson bracket and which satisfies the axioms for a Lie superalgebra. Furthermore, the Poisson bracket is a superderivation of the associative multiplication that is, (iii)

(h,,b,b,;

= (b,,b,)

bk+(-l)V$b,,hh)

The functions on classical phase space form a Poisson algebra in which there are no odd elements: B, = (01. In the next section we shall describe an important class of Poisson algebras with odd elements. Recall that if V is a vector space, then its Grassmann algebra n V is a commutative superalgebra. We shall show that putting a scalar product on V endows ,4 V with the structure of a super-Poisson algebra. The tensor product of two super-Poisson algebras is again a super-Poisson algebra. (The usual sign changes occur in the definition of both the associative and the Lie multiplication: (a@b,)(a,@b)=(-l)“rra,@h,h,

with a similar formula for the Poisson bracket of two tensor products.) In particular, the tensor product /1 V@ F(M) becomesa super-Poisson algebra. This is the simplest way of adding “odd variables” to a classical phase space. As we indicated in the last section, we shall take V=g +g* for the case of the BRS procedure. The super-Poisson structure on ii V induced from the scalar product on V is most simply described in terms of the Clifford algebra, which also is needed for quantization, cf. [Co--Ne-St]. We will review the necessary constructions in the next section.

3. CLIFFORD ALGEBRAS In this section we review some of the basic facts about Clifford algebras and their relation to exterior algebras. We refer the reader to the text by Greub [GrJ for more details. In this section V is a vector space and ( , ) is a scalar product on V. No further assumptions are made. In particular, V can be infinite-dimensional and

62

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( , ) can be degenerate. (We shall consider vector spaces as over the reals, but, in fact, very little assumptions will be made about the ground field. In fact, any field of nonzero characteristic will do. In most of the discussion all that is necessary is that the characteristic be different from 2.) A Clifford map is a pair consisting of an associative algebra A with unit and a linear map 4: V -+ A which satisfies

where 1, denotes the multiplicative unit in A and u and u are arbitrary elements of V. The CZifford algebra is the solution to the corresponding universal problem. That is, the Clifford algebra C(V) is an associative algebra with unit together with a Clifford map i: V-t C(V) such that any Clifford map factors through a unique algebra homomorphism from C(V). In other words, given any Clifford map (A, 4) there is a unique algebra homomorphism @: C( V) --+A such that 4 = @0 i:

As usual, if the Clifford algebra exists, it is unique up to isomorphism. To prove that one exists, we need only construct a model for it, and such a model can be given by dividing the tensor algebra

by the two sided ideal J generated by u@u+u@u-2(u,

u) lT(V).

The structure of the Clifford algebra depends crucially on the scalar product. So, strictly speaking we should write C( V, ( , )) for the Clifford algebra associated to the vector space V and the scalar product ( , ). But to avoid typographical unpleasantness we shall write C(V) for the Clifford algebra, where the scalar product ( , ) and the injection i are to be understood. Notice that the algebra r(V) is Z graded. The ideal J is not generated by homogeneous elements and so this Z grading is lost upon passage to the quotient. But J is generated by even elements, so the Z gradation of r(V) induces a Z, grading on C(V): C(V)=C,(V)+C,(V). Thus C(V) is an associative superalgebra. grading on C(V) is as follows:

(3.1)

A more abstract way of seeing the Z,

63

BRS COHOMOLOGY

Let W be a second vector space with scalar product and let C(W) together with i ,,+,:W -+ C( W) be its Clifford algebra. Suppose that I: V + W is an isometry, i.e., that

(k lu),= (4 01, for any two vectors u and v in I/. Then iM.‘, 1: V + C( W) is a Clifford induces a homorphism

map and hence

@,: C( V) --t C( W) of Clifford algebras, by the universal property of the Clifford algebra C( V). If W’ is a third vector space with scalar product and I’: W -+ w’ is an isometry, then the uniqueness property of the homomorphisms @ implies that

as homomorphisms of C( V) + C( w’). In particular, we may take W = w’ = V and 1 and I’ to be invertible isometries, i.e., orthogonal transformations. Let O(V) denote the group of all such orthogonal transformations. Then the above equations say that each IE O(V) induces an automorphism, a,, and that the map O( I’) -+ Aut C( I’),

I --) @,,

is a homomorphism, i.e., a representation of O( V) as automorphisms of the Clifford algebra. In particular, the transformation o -+ -u is an orthogonal transformation for any scalar product, and hence induces an automorphism Q, ~, of C(V) which satisfies (@ _, )’ = identity. Then C,,(V) consists

of those elements, a, which satisfy

while C’,( V) consists

of those elements which satisfy @_,a=

-a.

Clearly every element of C(V) can be written as a sum of an element of C,,(V) and element of C,( V) giving the decomposition (3.1) The fact that @ , is an automorphism of C(V) with (@ _ ,)2 = id implies that this decomposition makes C(V) into a superalgebra. Notice that every element of C,(V) can be written as a sum of products of even numbers of elements of V while elements of C,(V) can be written as sums of products of odd numbers of elements of V. We can use the minimum number of elements required to define a filtration on each piece: That is, we define Ci”( V) to

64

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be that subspace of C,(V) consisting of those elements which can be expressed as sums of products involving at most 2k elements of V. Similarly, define Cfk+ l(V) as the subspace of C,(V) consisting of sums of products involving at most 2k + 1 factors. Thus c~(v)cc~(v)cc~(v)c

...

u G”C v = Co( 0

and CP( V) . cy V) c cp + “( V),

where C”‘(V) denotes either Ci”( V) or C ik+ ‘(V) according to whether m = 2k or m=2k+

1.

Note that P( V)/Crn - 2( V) = A”( If), since the corrections involved in replacing uv by -vu in any product is a scalar, so the resulting product involves two less factors of elements of V. Furthermore, the term of top possible degree in the Clifford product of two elements is just the exterior product of the leading terms. More precisely, we can formulate the situation abstractly as follows: Let C= CO+ C, be an associative superalgebra which is Z filtered in the sense that we have subspaces c+zc;cc~c

...

c; c c; c c; c

u C;k= co, Uly’=C,

..

with cp. PC

cp+q,

where C” stands for Cik or Cfk + ’ according to whether m = 2k or m = 2k + 1. Then define gr2kC = c;“/c;”

-2

and

gr

2k+Ic=cZk+I

c:k-l ,I

The product on C induces a product on grC=@grPC with grPCxgrYC4grP+YC. Thus gr C becomes a Z graded algebra. In particular, gr, C=grO C@gr’C@

...

if we define

.

BRS

COHOMOLOGY

gr, C=gr’

C@gr3

C@ ...

gr C=gr,

C+gr,

C

and

then

becomes an associative

superalgebra.

In the case at hand the assertion

is that

gr C(V) = A( VI, i.e., that the graded algebra corresponding to the Clifford algebra is just the Grassmann algebra. Notice that n(V) is supercommutative. So consider the following situation: C is a Z filtered superalgebra as above whose associated graded algebra B = gr C is supercommutative. We claim that the supercommutator on C induces a super-Poisson structure as follows: for each h, E gr’ C’ choose a representative C,E C’ such that c,jC’ ’ = h,. Given two such elements h, and h, with chosen representatives I’, and c,. the supercommutativity of B implies that CL.,,c,] E C’ + ’ ?. Define

jh,, h,\ = [c,.(.,]/cI+’ -I.

(3.3)

It is easy to seethat this is independent of the choice of representatives (‘, and cm, and that axioms for a super-Poisson algebra are satisfied. In this way the scalar product on V, via the Clifford algebra, induces a Poisson bracket on ,,I( 1’). As an important side remark, we should note that the standard Poisson bracket structure on ordinary 2n-dimensional phase space arises by the same construction: Indeed, let C be the algebra of differential operators in II variables, so that elements of C are finite sums of the form

and where the a, are smooth functions of .Y= (.v, ,..., s,,). Multiplication in C is taken to be composition of differential operators (according to Leibniz’s rule). If we let /aI = TV,+ .. . + a,, then C a,D” is usually called a differential operator of degree at most k if in the sum all the 1~1are dk. We want to consider C as a superalgebra without odd terms, i.e., C= C,, with C, = (0). To be consistent with our preceding formalism, we will give the differential operators of degree at most k the filtration degree 2k, that is,

66

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Then we can identify gr2k C with expressions of the form

That is, gr2k C can be thought of as the space of polynomials in t with coefficients which are smooth functions of x and which are homogeneous of degree k in <. Leibniz’s rule implies that the multiplication on gr C is just the usual multiplication of functions. Thus B = gr C consists of smooth functions of the 2n variables x and r which are polynomials in l. It is immediate to check that the Poisson bracket induced on B from the commutator on C is the usual Poisson bracket on phase space. The relation between C with its commutator bracket and B with its Poisson bracket is an algebraic version of the relation between quantum and classical mechanics. Our analogy shows that this is how we should also view the relation between the Clifford algebra with its supercommutator and the Grassmann algebra with its super-Poisson bracket. Let us now examine the super-Poisson bracket on A V in more detail, looking at elements of low degree. Since 1 commutes with everything in C(V), Poisson bracket by 1 in A V is identically zero. Let us examine Poisson bracket by an element x of V. Poisson bracket by x is a superderivation of A V, so it is enough to compute it on elements y of V, since they generate A Y as an algebra. Now as C;l( I’) = (0) there is no arbitrariness in the choice of representatives and we may identify x and y with their representatives in Ct( V). By the defining relation for the Clifford algebra we have

so

This then extends uniquely as a superderivation.

For example,

{x,uAv}=2(x,u)u-2(x,u)u,

(3.5)

etc. On a general vector space (without a scalar product) each element I of the dual space V* defines a superderivation of A V called the interior product denoted by i(l) and given by i(l)(u A v A . ..)=l(u)u

A ... -I(u)24 A .‘. + . . . .

If I/ has a scalar product, we can use the scalar product to map V+ V*, by sending x into I(x), where Z(x) = (x, v). Thus for a vector space with a scalar product, each XE V defines a superderivation, z(x), of A V given by z(x) = i(f(x)),

BRS

and we can write

67

COHOMOLOGY

(3.4) as {x;)=2l(x):A”V+n”

(3.6)

‘v.

Now let us consider Poisson bracket by an element w of il’k’. It will map A&V-+ A”V. It is a derivation of n V and so is determined by its action on 6’. But this action is given by (3.6) as

Note that this transformation mation, i.e.,

of V into itself is an infinitesimal

orthogonal

transfor-

for all x and y in V. Indeed, it is enough to check this for the case that CO= II A I’ where it follows immediately from (3.5). We thus have a map

sending w into 10, .}, where O( V)fin denotes the Lie algebra of finite rank infmitesimal orthogonal transformations. But any element of O( V)fi, can be exponentiated to give a one-parameter group of orthogonal transformations. Thus each w E A’V’ determines a one-parameter subgroup of O( V):

which

in turn induces a one-parameter

group, @ -I,r,,,,), of C( C’). Thus, if we define

%o)=(ddr)@,

,,,,.,, Ii=,,,

then X(o)) is a derivation of C( V) and so is determined V. But for .YE V we have X(01) r=

by its action on elements of

{o, .Y) = [co, r].

where w is any representative for o in Ci( V). This is because any two choices of representative differ by a scalar which commutes with all of C(V). Thus X(0)

a = [co, u]

for any u in C( V).

Similarly, the one-parameter group A(o, t) induces Y ,4,C,,,,,,of automorphisms of n V, and hence a derivation u(o) = (4df) of ,I V. As Y(o) is determined

a one-parameter

(3.7) group,

PA I,,,.,) 1,-o?

by its action on V, we see that Y(o)a=

(w, fJ;

(3.8)

68

KOSTANT

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for any u E /1 V. We have explained the meaning of Poisson bracket by elements of degree one and two. As we shall see in Section 5, the case of principal interest to us will be Poisson bracket by elements of degree three, which will give the d operator.

4. THE EXTERIOR ALGEBRA

AS A LEFT-CLIFFORD

MODULE

The results of this section are also valid in general-V can be infinite-dimensional and ( , ) arbitrary. But the applications that we have in mind (in the next section) are only valid in the finite-dimensional case. We introduced the operator z(x) = 4 (x, . ] in the preceding section. Let &(x):n”v+n”+‘v be the operation

(all k)

of exterior multiplication E(X)

by X:

w = 9

A w.

It is easy to check directly from the definitions then z(u)

E(L))

+ E(V)

t(u)

that if tl and v are any elements of V =

(u,

v)

Id,

and that I(U) z(v) + z(v) z(u) = 0

and

E(U) E(V) + E(V) E(U) = 0.

From these equations it follows that if we define d(U) = z(u) + E(U)

(4.1)

then d(u) Q(v) + 4(v) 4(u) = 2(4 v) Id. Thus the map 4: I/-+ End(/l V) is a Clifford map. We therefore get a homomorphism 0: C(V) -+ End(n V). In other words, AV is a module for the Clifford algebra C(V). Now ,4 V has a distinguished element, namely 1 E /1’V. So we may define a map I): C(V)+A(V) by $(a) = @(a) 1 By construction, $ is a homomorphism n V= gr C(V) it follows that $ is an isomorphism

UEC(V).

(4.2)

of left C(V) modules. From the fact that of left C(V) modules.

(4.3)

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BRSCOHOMOLOGY

We will need some details about this isomorphism. If we think of C(V) as an associative superalgebra, each u E V defines an odd superderivation, by supercommutator, when though of as an element of Ci( V). On the other hand, if we think of II as an element of il I V, then it defines a superderivation by a Poisson bracket. We claim that these two derivations are related by 9. That is, we claim that (4.4)

~([u,al)=~u,~(Coli=~~(u)II/(~)

for all a in C( V). Indeed, (4.4) is clearly true if a is a scalar or if u is an element of k’. It then can be proved by induction on the filtration degree of u; that is, if (4.4) holds for h then it also holds for a = vh, where I’ is in V, as follows from the definitions. Clearly I/I(U) = II for u E V, where we have abused the language by identifying I’ as a subspace of C( V) and as a subspace of AK Also it follows directly from the definitions that $(ur - vu) = 2(u A 1’) so $ ‘: A’[‘+

C(V) is given by I)

I(24

A

11) =

$(UL)

-

(4.5)

1w).

We can use II/ ‘cl as the o in (3.7) so that 30, as derivations

(4.6)

‘w .I

of C(V). Note that $(UUw)=z4h

Cyclically

= [$

summing

UA

~t’+(U,/~)tt’+(t:,t1.)z4-(z4,M’)L’.

shows that I) ‘:/l”C’+C(V)

is given by ti

‘(141

A

U?

A

u,)=

(;I

c

(w

n)

(4.7)

~,,1)~4,(?)~4,(3),

that is, by total antisymmetrization. The corresponding fact is true in general: the I: A” V + C( V) is given by total antisymmetrization. map ICI 5. LIE ALGEBRAS

AND CLIFFORD ALGEBRAS

In this section we prove some facts about finite-dimensional Lie algebras as motivations for the infinite-dimensional case. Let a be a Lie algebra and suppose that ( . ) is a nondegenerate scalar product on a. Invariance means that for all

II,

D, $1’ E

a.

(5.1)

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For example, if a is a semisimple Lie algebra then the Killing form is nondegenerate and invariant. Here is another class of examples which is important for us. Let g be an arbitrary Lie algebra, and let g* be the dual space of g. Make the vector space g + g* into a Lie algebra as follows: The bracket of two elements of g is the element of g given by the original Lie algebra structure on g. The bracket of two elements of g* is zero. If x l g and f~ g* then define [x, f] l g* by the formula

Cx,flb)=

-f(lX3Yl)

(5.2)

and, of course, [f, x] = -[x, f 1. In other words, we are making g +g* into a Lie algebra by taking the semidirect product of g with g*. Now define the scalar product(,)ong+g*by (x3 Y I= 0, (f,,f2)=0,

X,YEk!v

hEg*Y

(5.3)

and

(x,f)=(S,x)=f(x),

.xEg and

f l g*.

To check that (5.1) holds, observe that both terms on the left of (5.1) vanish if all three of U, U, and w belong to g. Similarly both terms in (5.1) vanish if at least two out of the three elements U, u, and w lie in g*. Thus we need only check (5.1) for the case that one of U, u, and w lies in g* and the other two belong to g. This case follows from (5.2) and (5.3). The set of all finite-dimensional Lie algebras with nondegenerate scalar product has recently been classified by Medina and Revoy [Me-Re]. Basically, what they show is that the most general irreducible such object can be built up from a simple algebra and from an algebra which is a slight generalization of the g +g* constructed above. In any event, suppose that a is a Lie algebra with invariant scalar product ( , ). Define the trilinear form Q by

Q(x, Y,z) = -1(x,

[IY,

21).

(5.4)

The antisymmetry of the Lie bracket together with (5.1) imply that Q is completely antisymmetric in X, y, and z. Now since a is finite-dimensional and ( , ) is nondegenerate, we can identify a with a* and, similarly, we can use the scalar product to identify the space of antisymmetric trilinear forms with A3a. Thus

(These identifications phenomena. )

fail to hold in the infinite-dimensional

case, giving rise to new

71

BRS COHOMOLOGY

Now for any Lie algebra, a, if 1~ a* is a linear function function dl is defined by dl(y, =)=

-l([y,

on a, then the bilinear

21).

The scalar product on a identifies a with a* and hence we can think of d as a map from a to A’a, since a is finite-dimensional. Thus dx(y, z)= Therefore

it follows

-(x,

[y,z])=

[21(x)sZ](y,z).

from (3.6) and the fact that x and Sz are both odd that {Q, x) = {x3 Qj =d.y.

Now both a Poisson bracket by Sz and d are superderivations therefore determined by their action on a. Hence {Q;)

of Aa and are

=d.

(5.5)

We have thus represented the operator d: A”a --f A’+ ‘a as a Poisson RE A’a. Let Q be any element of C:(a) such that gr, where gr, Q means the equivalence

bracket

Q = if&

by

(5.6)

class of Q mod C;(a).

We claim that

Q’ E C,$ a).

(5.7)

Indeed, since Q E C:(a), Q’ E C:(a). But 4 gr, Q’ = R A 52 = 0 since R is odd. Thus Q’ E C:(a). Let 0 = gr, Q’. Then for any .Y in a, 2l(X) cr= (x, IT) = -{a, But 2 grz[Q,

3-I., --gr,CQ",

-~l=gr&CQ. [Q, .x11

x] = {Sz, ,Y) = dx and therefore 4gr,[Q,

[Q,x]]=2{R,d.x)=2d(d,u)=O.

Thus I(X) CJ= 0 for ail x so CJ= gr, Q’ = 0. This proves (5.7). Now consider the element c = gr, Q2 E A’a

(5.8)

We claim that c is a cocycle, i.e., that dc = 0.

(5.9)

Indeed, dc = (Sz, c) and in the computation of {1;2, c> we may choose Q2 as the representative for c and 2Q as the representative for R. Since Q commutes with Q2 we get O=2gr,[Q,Q2]={52,c:=dc,

72

KOSTANT

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proving (5.9). Suppose that we replace Q by a different choice, Q'. Then Q' = Q + u for some u E a and we have (Q + u)~ = Q2 + [Q, u-j + u2 so

srAQ’J2= gr, Q’+ S{Q, ~1, since u2 = +(u, U) 1. We can write this last equation as

c’=c++du.

(5.10)

In other words, the cohomology class, [c], is independent of the choice of the representative, Q. We claim that in the case we are currently considering, where a is finite-dimensional, this cohomology class vanishes. In other words, we claim that we can choose a Q such that

Q’ E C:(a),

(5.11)

i.e., it is a scalar. Indeed, suppose we choose Q=$,-‘52.

(5.12)

Then for any x E a we have

CQ2,xl = [Q, [Q, .x11= t[Q, Ic/~‘kWl = -&WW Q

(5.13)

by (4.4) and (4.6). Recall that X(dx) is the derivation of C(a) induced by the infinitesimal orthogonal transformation of a given by a Poisson bracket by dx. The map $ is defined purely in terms of the scalar product on a and hence commutes with all orthogonal transformations, so X(dx) Q = +I,-‘(

Y(h) 52).

(5.14)

Now for any y, z E a, (2, {dx, y}) = -(z, (y, dx}) = -41(z) z(y) z(x) Q

=2(x, cy,z1)=2(c&.Y1,~) by (5.1). In other words, {dx,. )=2adx

(5.15)

as transformations of a. Since n is defined purely in terms of the Lie bracket and the scalar product, it follows that Y(dx) Sz= 0 and hence that X(dx) Q = 0.

(5.16)

BRS COHoMoLOGY

7.3

This proves (5.11). Equation (5.16) can be formulated as saying that we have made an invariant choice of Q. This will not be possible, in general, in the infinite-dimensional case, and accounts for the fact that there we may have [c] # =O. Suppose that we have made the choice (5.12) so that Q’ is a scalar. We can compute this scalar as follows: Q’= $(Q’) since Q’ is a scalar. But

$(Q’)=

G(Q) ci/(Q) = Q(Q) Q.

Now SzE A3a and the expression for @(Q) involve a sum of products of three terms of the form c(u) + I(U). The only contribution to A”a can come from applying only the J(U) in each such factor to Sz. Thus we see that 4Q’=

-(a,

Q),i,,

(5.17)

where the scalar product is the one induced on A’a from the scalar product on 11. In the important case a = g + g*. where n’(g+g*)=A3g+n2g@g*+g@A2g*+A’g*, the expression (5.4) vanishes unlesstwo out of the three elements X, y, and I belong to g and the other to g*. Thus QEg@ /i’g* (5.18) and hence (G, IZ?),‘,, = 0. Thus Q’=O.

(5.19)

6. REDUCTION AND QUANTIZATION The results of the preceding section have almost carried out the program described at the end of Section 1 for the case where g is finite-dimensional and F(M) = R. That is, we have found an element Q such that (5.5) holds and a Q in C( C’) representing Q such that Q’ = 0. The trouble is that the d occurring in (5.5) does not, at first glance, seemto be the same as the d occurring in Section 1. The d in (5.5) is the differential operator for the semidirect product algebra g +g* on the complex, A(g +g*), in other words for computing the cohomology of g+g* with values in the trivial module. Let us continue to denote this operator by 0. The operator of Section 1 was the operator on Ag* 0 Ag used for computing the cohomology of g with values in Ag. Let us temporarily denote this operator by d. We must show that when we make the superalgebra identification A(g*)OA(g)b

A(g+g*)

the two operators d and d coincide. We shall do this by showing that (i) (ii)

both d and d are superderivations and they agree on generators.

74

KOSTANT

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As to (i), the operator d is a superderivation by definition. So we need that d is a superderivation. For this it is enough to check that d(w) = (dw) a + (- l)dcgw o da,

only

check

oEAg*

which is just (1.7) and that d( 10 ab) = (da) b + ( - 1 )degua(db). Now the left-hand side of this equation, as an element of g* @ /Ig = Hom( g, /Ig), gives, when evaluated on x in g, the value Y(ad x)(ab), where Y(A) is the derivation of the exterior algebra corresponding to the linear transformation, A, of g (and where we take A = ad x). The right-hand side gives (Y(ad x) a) b + uY(ad X) b. (The reason for the plus sign in the second term comes from the multiplication law in ng* @ ng which says that (1 @a) I I@c = C-1) degoZ@ UCwhen IE g*.) Thus our desired identity simply asserts that Y(ad x) is a derivation of ng. This proves (i). As to (ii), we need only check that d and d coincide when evaluated on’elements ZEN* and on elements l ~g. We leave this verification to the reader. Let K be a g module. We can make K into a g + g* module by declaring that all the elements of g* act trivially. It then follows from the preceding discussion and the definition of Lie algebra cohomology with values in a module that on the space ng* 0 ng @ K the coboundary operators for K considered as a g + g* module or Ag@ K considered as a g module coincide.

(6.1)

Now let us consider the situation described in Section 1, where we are given a Hamiltonian action of g on M, in particular, a map 6: g -+ F(M)

such that {s(<),f}

=q,

where u denotes the result of the action of < ~g on f~ F(M). Under the identification of Hom(g, F(M)) with g*@F(M) ( w h ic h is valid for finite-dimensional g), we may regard S as an element of g* 0 F(M) and hence as an element of n(g+g*)@F(M). Now this last space is the tensor product of two super-Poisson algebras and so can be given the structure of a super-Poisson algebra via tensor product. We claim that (6.2)

{&10f}=df {S,<@1}=1@26(~)

for


(6.3)

(6,Ct)=O

for

uEg*.

(6.4)

and

7.5

BRS COHOMOLOGY

Indeed, let 5, ,..., [, be a basis of g and cc’,..., CP be the dual basis of g* SO that

Then

and this element of g* 0 F(M) assigns to each r) E g the element d of F(M). But this is precisely the definition, (1.6), of df, proving (6.2). As to (6.3) the Poisson brackets of the S(<;) with f in F(M) all vanish so that [&i’Olf=~{

a’, 5jOS(5,)=2C

~‘cs,oscir,,=2os(~),

proving (6.3) with a similar proof for (6.4). We can now consider the element a@ 1 in n(g +g*)@F(M), canonical element given by (5.4). In view of (1.7) and (6.1 )-(6.4) ing on generators, that if we define

where fi is the we see, by check-

o=ag1+s

(6.5)

then a Poisson bracket

by 0 is the operator

D of Section 1.

(6.6)

The next step in our program is to “quantize” $ 0. Once again let us first consider the case where F(M) = R so that there is no 6 and 0 =a. The elements of C( g + g* ) constitute the “quantum observables” whose corresponding “classical observables” are the elements of A( g+g*). The scalar product on g +g* has the signature (n, n) (where II = dim g) and hence C(g +g*) is isomorphic, as an algebra, to the algebra of all 2” by 2” matrices. (Cf., for example, [Gr].) Therefore, up to isomorphism, C(g + g*) has a unique irreducible module; call it S. Then C( g + g*) can be identified with End(S), the algebra of all linear transformations of S. If we choose Q according to (5.12) then, by (5.19), Q’=O. Thus the “quantization” of $sZ is the element Q, considered as an operator on S. Let us be more explicit about the construction of S. Let N be any maximal isotropic subspace of g +g*. (That is, the scalar product of any two elements of N is zero, and N is maximal with respect to this property.) Then the left ideal, C(g +g*). N, generated by N is clearly a maximal left ideal and hence S=C(g+g*K’(g+g*). Indeed if we choose some complementary then we can write

N. maximal

C(g+g*)=/iPOAN

isotropic

(6.7) subspace, P, of g + g*

(6.8)

76

KOSTANT

AND

STERNBERG

as a vector space, by the simple device of using the defining relations of the Clifford algebra to move all the factors coming from N to the right in any product (at the expense of introducing lower order terms if necessary). This is known as “Wick ordering.” Then S=AP

(6.9)

is a model for S. For example, we might pick N = g and P =g*

in which case

s = /Ig*.

(6.10)

Let us choose a basis of g and the corresponding dual basis of g* as above. Let cf: be the structure constants of the Lie algebra relative to this basis. Then Q = (4) 1 c;(dd&

+ c&a’+

~&zj).

(6.11)

Indeed it is clear that Q E II/ -‘A3(g + g*), and a direct check shows that [Q, x] = f dx for all x in g + g*. Now the first term on the right of (6.11) in the parenthesis lies in the left ideal generated by g. Also the Clifford algebra identities allow us to replace cr’t,&+ ~,cY’& by 26:~‘. Thus Q=$xc:,a’modC(g+g*).g. NOW the element $ Z c$j of g* is just the linear function which assigns to each 5 ~g the number f tr ad & Let us call this linear function 1, so ~(4) = t tr ad r.

(6.12)

Thus Q=x

mod C(g+g*).g.

(6.13)

Let 1 E /Ig* be the unit element of the exterior algebra so that l=l

c(g+g*) mod Ck+g*Pg

when we identify S = C( g + g*)/C( g + g*) . g with /ig*. Then Q.l=Q

modC(g+g*).g=X.

Any element of S can be written as UI . 1, where w E ng*. Then Q.(o.l)=[Q,o].l+(-l)d”g”~.Q.l = (do).

1+ ( - 1 )dego we x.

By (1.7) we can reformulate this last equation as follows: Let L, denote the one dimensional g module (i.e., L, = R as a vector space) given by 5. Y= ~(5) Y. Then Under the identification of S with /Ig* = /ig* 0 R the operator (2 becomes identified with the coboundary operator for the cohomology of g with values in the one dimensional module L,.

(6.13)

BRS

COHOMOLOGY

77

A Lie algebra g is called unimodulur if x = 0. (Geometrically this means that the corresponding group has a volume form which is both right and left invariant.) Then (6.13) implies that if g is unimodular then under the identification of S with ng* given above, the operator Q becomes identified with the usual coboundary operator for ordinary Lie algebra cohomology.

(6.14)

A number of comments are in order in connection with this result. First, although the ordinary Lie algebra cohomology is a relatively well understood object, the cohomology of a Lie algebra, g, with values in its exterior algebra ng is a much more complicated object. So here the “quantum” object is simpler than its “classical” counterpart. Second, from an abstract point of view, all we have is the operator Q (with Q’=O) and the module S. The identification of S with ,4,e* introduces more structure, in particular, a grading

with p: ivg* --Pnp + “q*.

(6.15)

Let 9 denote the corresponding degree derivation, so 2’ is the scalar operator. muhiplication by p, on nrg*. In terms of the basis and dual basis it is clear that

so that (6.16)

and (6.15) implies that

!I97 Ql = Q.

(6.17)

Let o=gr,Y

so 0=i! c cdAr, and (6.17) implies that do = $2.

(6.18)

Notice that o is just the evaluation form of g* on g but is extended to be an antisymmetric form on g + g* instead of a symmetric form. In contrast to the sym-

78

KOSTANT

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metric extension it is not g+g* invariant. Notice also that a Poisson bracket by o in A(g+g*) z A(g*)@A(g) is just the total degree derivation. So the choice of a “quantization,” 9, of the total degree derivation gives the grading on S regarded as a complex, under the operator Q. Let us now turn to the problem of “quantizing” the operator $0 in the case that F(M) is nontrivial. We assume that we are given a “quantization” of some class of elements of F(M). That is, we are given a vector space, T, and a map from a subspace of F(M) into End(T) which carries a Poisson bracket into a commutator. We assume that this class of functions includes 6(g). So we are given a representation of g on T. Call it z. It is r that is the quantum analog of 6. Now z E Hom( g, End T) N g* 0 End T,

(6.19)

and we can think of g* as lying inside C( g + g*) 2: End S so that rEEndS@End

T%End(S@T).

(6.20)

In terms of our basis and dual basis

z=c @.‘OT(~,).

(6.21)

So now define Q=Q@id+z

(6.22)

as an operator on SO T. The equations Q’ = 0 and [Q, CX]= da imply that Q*=O.

(6.23)

Thus Q is the quantization of 0. Let us close this section with a brief discussion using the language of homological algebra of the relation between the cohomology of the total (classical) complex under the operator D and the functions on the reduced space. We refer the reader back to Section 1 for notations. The results described here will not be used in the rest of the paper and can be omitted without loss of continuity. We are given a double complex so we get a spectral sequence whose E, term is given by Es4 = fJpd(g, fWkOF(W)). Since Ag gives the Koszul resolution by R, we see that

of the trivial S(g) module, which we denote

Ep = W( g, Tor”(g)( R, F(M)).

(6.24)

If sufficiently severe regularity conditions are satisfied, the higher order Tor’s will all vanish and the spectral sequence will collapse so that flD indeed gives the functions on the reduced space as indicated in Section 1. The study of when this happen, in general, seems to be an interesting and nontrivial question.

BRSCOHOMOLOGY 7. INFINITE-DIMENSIONAL

79

CLIFFORD ALGEBRAS AND THEIR SPIN REPRESENTATIONS

Let S be a vector space endowed with the discrete topology. We put the weak topology on End S. So a sequence {A,} of operators converges if, for each s E S, the sequence (An s} eventually stabilizes. Let I/ be a countable-dimensional vector space with a nondegenerate scalar product. We will assume that this scalar product is “split” in the sense that there is a maximal isotropic subspace, N, so that V/N is nonsingularly paired with N. Suppose that we have fixed one such subspace, N. We may choose a countable basis [n,} of N and {p,} of V/N which are dually paired. We may lift the pi to elements of V and inductively arrange, by subtracting off suitable combinations of the n,, that the scalar product of these lifted elements with one another all vanish. We thus get a basis {n,. pi) of V such that (n,, nj) = 0 (Pi?

Pj)=O

(7.1)

(n;, p,) = 6,.

We let P denote the space spanned by the p,‘s so V=N@P

(7.2)

and both N and P are isotropic. Of course, even after having made the choice of N, the basis and the complementary subspace P depend on additional choices. We now consider the Clifford algebra C(V) and the module S = C( V)/C( V) N.

(7.3)

Since C(V) . N is a maximal left ideal in C(V), the algebra C(V) acts irreducibly on S. By the Jacobson-Bourbaki density theorem [Lang] the algebra C(V) is dense in End S. The Z, grading on C(V) induces a Z, grading on S since C(V) N is a homogeneous ideal. Hence End S inherits a Z, grading consistent with the Z, of C( V). In other words, End S = End, S@ End, S, where End, S is the closure of C,(V) and End, S is the closure of C,(V). The filtration on C(V) induces a topological filtration on End S. That is. we may define Endik S = Ci”( V) and

(7.4)

Endfk+’

S= Cfk+‘(V).

Then End, S = U Endik S 54s. 176 i-6

80

KOSTANT

AND

STERNBERG

and End, S= u End:&+iS. k

Now End: S clearly coincides with Ci( V) and consists of all scalar operators. Let us examine End: S. For clarity of exposition let p(a) denote the image of a E C(V) in End S. That is, p(a) denotes the element a when thought of as an operator on S. Our question is: Given a sequence {vi} of elements of P’, when does the sequence {p(vj)} of operators converge in End S? The condition of convergence is that for each s E S the sequence of elements p(vj) s stabilizes. If we take s = 1 + C( V) N, then ~(0,) s = vi + C( I’) N. So the stabilization condition for this choice of s means that there exists some v E I’+ N such that vj+N=v

for all

j % 0.

(7.5)

Now let us take s=JJ+C(V)N=p(p)(l+C(V)N)

for some

p E V.

for all

j$ 0.

Since the Clifford identities say that Vjp

=

-pVj

+

2(py

Vj)

l,

it follows from (7.5) that Plvj) s = -P(P)

v + 2(P, vj) l+ C(N) TV

Thus the sequence p(vj) s will converge if and only if the numerical sequence {(p, u,)} stabilizes for each p E V. We can express this condition as follows: The scalar product induces an injection i: V-+ V*

i(v)(p)

= (p, 01,

where V* denotes the full algebraic dual of V. Furthermore, i(V) is dense in V* under the weak topology. The fact that {(p, vi)} stabilizes for each fixed p E V means that the sequence {i(v,)} converges in V* to some element, call it 1. Let v be any element of V which satisfies v+N=v (see (7.5)). Then (7.5) says that when evaluated on elements of N, we have I(n) = (u, n). Let I$ c V* denote the subspace of linear functions with this property. That is, V$= (1~ V* 1l,,=(v;) Then our convergence condition

for some UE V}.

(7.6)

can be written as (7.5) and

i(vj) converges to 1 E I$.

(7.7)

We have derived (7.7) as a necessary condition for the convergence of p(ui) s, where s is of the form s = p + N. But it is easy to see that, assuming (7.5), (7.7) is

BRS

81

COHOMOLOGY

sufftcient for the eventual stability of ~(IJ,) s for any s E S. Indeed any s is a linear combination of expressions of the form P(P, 1.. P(P,)(l

+ cc v N)

and repeated use of the Clifford identities shows that p(uj) s stabilizes for such an s if (7.7) holds. We have thus defined a map End; S-t Vz

(7.8)

and a moment’s reflection shows that this is an isomorphism. In terms of our basis {n,, p,}, the most general element of V* can be written as a doubly infinite sum 1=x

a,n,+C

h,p,.

The elements of L’E are those with only finitely many nonzero h’s The choice of P gives a linear space isomorphism

pi, A ... Ap,~p;;~~p,+C(V)N.

In terms of this identification,

the element I E VE given by (7.9) acts as P(4 =I

~,WS~

h,dP,).

(7.10)

Although the first sum in (7.10) is infinite, all but a finite number of the interior products will vanish when applied to any fixed element s. So the series converges in the weak topology. Notice that passage to the limit shows that

CP(OtP(U)1= 24oh

IE v;., UEv.

(7.11)

The scalar product on I/ induces a family of subspaces of N of finite codimension-those which are the intersection of annihilator spaces of i(n), clE V. We can use this family to define a (linear) topology on N. In terms of this topology, we can characterize the space Vz as consisting of those linear functions whose restrictions to N are continuous. More generally, let /iF V denote the space of k-linear antisymmetric functions o with the property that for any j vectors U, ,..., vi E V the restriction AN is continuous.

of w( V, ,..., o,, . ,..., ) to

Another way of putting this condition is that for any u1 ,..., 11,in V there is a subspace F belonging to our collection (depending on the D’s) such that ~(0, ,..., u,, n,,,,...,

n,)=O

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KOSTANT

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STERNBERG

if any of the n’s belong to F. For example, for k = 2, the most general element of A**N V can be written as C auni A nj+C

where the finiteness requirements

b,pi A nj+C

copi A pj,

(7.12)

are:

only a finite number of the cii # 0, for each j, only a finite number of the b, # 0, and no condition on the aV. Similarly,

the most general element of Ai* V can be written as

CA~~niAnjAn,+CB~~piAniAn,+CC~~piApiAn,+CDii~piApiAp,,

with the finiteness conditions: only finitely many nonzero D’s, for each fixed k, only finitely many nonzero C,,, for each fixed j and k, only finitely many nonzero B,, no restriction on the A’s. A similar description

holds for all the ni*V.

and

We have the maps

and C“(V) --% End“ S + Endk S/Endke2 S %‘gr, End S. An argument isomorphism

similar

to the one given above shows that these maps induce an /I: gr, ,End S + /1? V.

(7.13)

In contrast to the finite-dimensional case, we no longer have the map $ or its inverse (due to the presence of infinitely many b’s in (7.12), for example). For instance, let us examine the case k = 2 of (7.13) and apply the isomorphism p to the element w given by (7.12). Here /I(o) is a “projective operator,” that is, an element of End; S determined up to a scalar operator. In terms of the basis we have chosen and the representation (7.12) of o we can choose the operator :o: as a representative for /3(o), where :o: is defined by :0:=x

alp

dnj) + 1 b,P(Pi) P(nj) +C

CudPi) P(Pj).

(7.14)

Note that in (7.14) it is essential that in the middle term the p(nj) be written to the right, for otherwise the infinite sum will not converge in the weak topology. This

83

BRS COHOMOLOGY

convention, that all the ~(n)‘s appear to the right of all the p(p)‘s in any product, is known as Wick ordering. The particular expression :w: given by (7.14) depends on the choice of basis, but the projective class p(o) is well defined independently of any choice of basis. Since the scalars commute with all operators, the expression [A, p(o)] E End S makes sensefor any A E End S. Now for y E V the map 1(y): AkV-+ Ak ‘v extends by continuity to a map, which we shall continue to denote by r(y), 10’): Agv+

n”,-‘*v.

Taking A = p( J,) and passing to the limit from (3.6) shows that (7.15)

B[P(?‘). B ‘(WI] = 2d.Y) (‘1.

Similarly, if D E /i N 3*Vthen /Fl(sZ)Eg r3 End S can be thought of as an operator determined up to an operator of the form p(I), where I is an element of P’z,. A choice of basis will give a particular representative, :O:, for b l(Q), and the analog of (7.15) holds. More generally, pk

-I(grk

A EEndk S, ,r~ V.

L([IP(C’),Al))=21(1?)Bk(grkA)),

(7.16)

All the constructions in this section depend on the choice of the maximal isotropic space, N. We should really have written S,V, etc. In contrast to the tinitedimensional case, different choices of N’s may give rise to inequivalent representations of C(V). So in infinite dimensions the choice of N (up to finite-dimensional modification) must be considered as an additional datum. In Section 10 we shall study the Clifford group and its action on the various choices of N.

8.

LIE

ALGEBRAS

AND

CLIFFORD

ALGEBRAS

IN THE

INFINITE-DIMENSIONAL

CASE

Let a be a countable-dimensional Lie algebra with an invariant scalar product ( , ). The form 52 defined by (5.4) is an antisymmetric trilinear form, that is, an element of A3*a. But it will not, in general, be an element of n’a, which consists of trilinear forms of finite rank. In view of the considerations of the preceding section it is reasonable to formulate as an additional hypothesis and as an additional datum the existence and choice of a subspace N c a such that N is maximal isotropic with a/N nonsingularly paired to N and such that Q2En3*a. N

(8.1)

Most of the rest of this section will be devoted to exploring the implications of this hypothesis.

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KOSTANT

AND

STERNBERG

As in Section 5, let us define, for x E a, the antisymmetric formula

WY, 2) = 44

bilinear form dx by the

CY,21).

(8.2)

Since dx = {x, Sz} =22(x) R it follows from (8.1) that for all

dxE AFa

x E a.

(8.3)

Condition (8.1) implies that if we fix ye a and take x and J’ both to lie in N, then there exists a subspace Fc N of finite codimension such that the right-hand side of (8.2) vanishes whenever x E F. Furthermore, F is the orthogonal complement, under ( , ) of some finite-dimensional subspace of V/N, or, what amounts to the same thing, of some subspace M containing N with dim M/N finite. But, in view of the non-singularity of ( , ), this implies that [y, z] E M for all z E N. In other words, dimb,

NM [.v, Nl n N) < co

for any

~1E a.

(8.4)

We can think of (8.4) as saying that N is an “almost ideal” of a in the sense that bracketing by any element of a moves N only finitely many dimensions out of itself. (Kac and Peterson in [Kac-Pet] have studied a more general situation in which we are given a vector space V with scalar product and a subspace N as in the preceding section and a representation of a Lie algebra, b, on V which is orthogonal and which satisfies the analog of (8.4). That is, each element of 6, moves N only finite dimensions out of itself. Our case is where V= a = b and where the representation is the adjoint representation.) In view of (8.3) and (5.16) we can define for

a(x) = tPp’(ddx)

.YE V.

(8.5)

Here a(x) E gr, End S and so it is a projective operator on S. The associative multiplication on End S induces a Poisson bracket on grEndSz:nV=@OnV and a(x) = ${a, x}.

(8.6)

Let Q be any element of End: S with gr, Q = 52. Then the argument

(8.7)

of Section 5 still applies to show that Q’ E End: S, c = gr, Q’

the cohomology

(8.8)

is a cocycle,

class [c] is independent

of the choice of Q.

(8.9)

(8.10)

BRSCOHOMOLOGY

x5

To see (8.8))( 8.10) directly, and to see explicitly what the cocycle c expresses, observe that (7.15) shows that

IP(?t), PI = N(Y1,

(8.1 I )

where x(y) is a representative for a(y) in Ends S. Hence

IP(J’), Q’I = CP(Y),QI Q-QCP(YL PI

(since p( ~1)and Q are both odd )

=cQY)Q-Q~(Y) and therefore

IP(.Y), Cd.vL Q’l =P(-Y)(~Y) Q- Q~Y))+ (4~) Q- Q~Y)) P(.u) = (P(.u)~Y)-~(Y) d-u)) Q+~Y)(P(.~) Q+ QP(-~)) -(p(.u)Q+Qp(.u))cx(~)+Q(p(-~u)a(y)-~(ylp(.~)). By (7.14) and (5.14)

Cp(.y1,4Y )I = -p( [s, .r!] ).

(8.12)

[p(xL [P(Y)> Q21= {a( [x, ~1) - [@XI, r(y)] 1.

(8.13)

Thus we seethat

In other words, z is a projective representation of a, the choice of a representative, Q, determines a choice of representatives, x(x) for the r(s) and the cocycle c is the corresponding cocycle measuring the failure of 5yto be an honest representation. Thus the cohomology class [c] is the usual cohomology class associated to a projective representation. Kac and Peterson introduced (in their more general setting) the projective representation and the corresponding class, cf. [Kac--Pet]. Our observation here is that it is given by Q’. In Section I1 we shall see that under suitable hypotheses the space S has a Hermitian structure. Then a good bit of the freedom in the choice of Q can be eliminated by requiring that Q be skew-adjoint relative to the Hermitian structure on S.

9. REIXI~TION

AND QUANTIZATION

IN THE INFINITE-DIMENSIONAL

CASE

We now want to specialize the results of the preceding section to the intinitedimensional analog of the case a=y+g*. Instead of R*, which denotes the full algebraic dual of g, we shall assumethat we

86

KOSTANT

AND

STERNBERG

are given a countable dimension subspace g# c g* which is nonsingularly paired with g and which is stable under the coadjoint action of g. We then assume that a=g+g#.

(9.1)

We shall also assume that N= (Nng)+

(Nng#).

Thus, if we denote N ng by L, then N=L+L’,

(9.2)

where L c g and Lo denotes the annihilator space of L in g#. An example where all the condition of Section 8 are satisfied is the following: Suppose that g=ogi with dim g, < cc and

[St9gjl

j’

=gi+

Thus we are assuming that g is a Z graded Lie algebra with finite-dimensional graded components. Define g# as (infinite direct sum).

g”=@&+ Applying

the definition

(5.2) we see that

Furthermore, if x, y, and z are homogeneous elements of g or g#, then Q(.lc, y, z) = 0 unless two of the three elements x, y, or z lie in g and the third in g#, with the degrees matching, e.g., unless

k = i + j.

(9.3)

Now take L=

0

gi7

i
(9.4)

so that N= @ gi+ @ g;“. ice j30

(9.5)

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COHOMOLOGY

It is easy to see that with this definition of N, condition (8.1) is satisfied by virtue of (9.3). Under the general hypotheses (9.1) and (9.2) let us choose a basis {tk ) of g is a basis of L. Let c: be the structure constants of g arranged so that (tr}iio relative to this basis so that

Let [a’) denote the dual basis so that {t,, CI’),,~,,~,~ is a basis of N. Relative to the bases we have chosen for g + g* we may take (9.6) Recall that Wick ordering means that all the elements of N are written to the right in any monomial. But also recall that all the c(‘s commute with one another so we may write (9.6) as (9.7) Now the formula

[:a:, x] = :[a, xl:,

a E End S, x E I’

(9.8)

can be proved by induction on the degree of a monomial a in C( I’) and then by passage to the limit for all a in End S. Applied on our case I’= a, taking a = Q, we see that (9.9 and

A direct computation shows that the right-hand side of (8.4) vanishes unless both I and J belong to g. We can thus summarize the results of Section 8 when applied to g + g # as follows: Let L be a subspace of g such that (8.1) holds when N is defined by (9.2). We then get a projective representation of g on the spin space S associated to the Clifford algebra C( g +g# ) and the maximal isotropic space N. In contrast to the finite-dimensional case this need not be equivalent to an actual representation, i.e., the associated cohomology class need not vanish. Indeed, this class is given by the cohomology class of Q’ (when restricted to A’g).

88

KOSTANT

AND

STERNBERG

Since Q* # = 0, in general, the computation of Lie algebra cohomology as given in(6.13) and (6.14) breaks down. (Intuitively this is to be expected as the expression tr(ad x) no longer makes sense, reflecting the “infinite volume” of the “group” G. This interpretation can be justified by the results of Peterson and Kac which show that there is a “renormalized trace” associated to N and an associated “Tate cohomology class” which, according to a formula of [Pet-Kac] is exactly the class of Q*.) But Eq. (6.23) will still hold provided that we take the T in (6.22) to be a projective representation with the opposite cohomology class! Here are the details: A projective representation, r, of g on a vector space T is said to be (locally) L. finite if, for each t E T the vector subspace r(L) t is finite-dimensional. Suppose that r is a locally L finite projective representation whose cohomology class is the negative of the cohomology class [c] = [gr, Q*]. Having made a choice of Q, which fixes the cocycle c, let us choose a representative r E Hom(g, End T) whose corresponding cocycle is --c. We shall assume a slightly stronger condition than local L finiteness. Namely, we shall assume that having fixed Q we can choose the representative r so that for each t E T, the subspace ann,tc3L=(tEL/t(5)t=O}

is a closed subspace of L. Let { ti} be a basis of g chosen as above, and set 7; = d5,). Then the infnite

series lJ=c

(9.11)

c(‘@T,.

converges as an element of End(S@ T). Indeed, for any fixed element s@ t, all but a finite number of the p(cl’) s vanish for i> 0, and all but a finite number of the zjt vanish by our finiteness conditions. We can now compute P=C

Glicxj@t;7i=f

= -$c@id-4

c

(a’a’-dcr’)@T,ti=f

c

dol’@

[Zi,

T,]

1 dor’@r;.

So now if we set

Q,=Q+tT,

(9.12)

Q;=O.

(9.13)

we see that

Also observe that if we write p(t) for p(r) @ id acting on SO T, and if we define Q5) = dt;) 0 id + id 0 r(5),

(9.14)

BRS

89

COHOMOLOGY

then

CQ,t dt)l = d5),

(9.15)

by the definition of ~(5) for the first term on the right of (9.14) and by the defining properties of the Clifford algebra for the second term. Since c( and T have opposite cocycles, it follows that tl, defines an honest representation of g on S@ T. Furthermore, since [Q,, [Q,;]] = [QZ;]. it follows from (9.13) and (9.15) that

CQ,>~,(
for all

4 l g.

(9.16)

Notice that the projective representation of g on S and on T gives rise to a projective representation of g on SO T independent of any choices. The fact that 5 gives rise to the opposite class meant that we are able to choose a representative ar in this projective class which was an honest representation. But the difference between two honest representations in the same class is a linear function on g which must vanish on [g, g]. Hence, if [g, g] = g (i.e., if H’(g) = {O)) there are no such nonzero linear functions. In other words, a, is unique. But then (9.15) implies that Q, is unique. In other words, we have proved if [g. g] = g then the operator Q, is independent choices and is uniquely determined by (9.15).

of all preceding (9.17)

We now consider the issue of the Z gradation on S. That is, we wish to work out the analog of (6.16)-(6.18) for the infinite-dimensional case. So let J E End a by the operator given by J = +id on g and J= -tid on gff. Then define (0 E A’(1 by to(x. J’) = (Js, J’), It follows

directly from the definition

that (0 E A$%l

(9.18)

and hence an element LP E End: S, unique up to a scalar, which under the identification, fi, of gr, End S with At?a. ad Y E End( End S)) is well defined and [y,

p(.U)] = -2p(l(.Y)

projects onto (11 In particular,

OJ)

so

[Isp,o(r)1 = -dir)

for

c;;Eg

and

(9.19)

c=y,p(cc)l= o(2) Since Q is a limit of a sum of monomials term from g it follows from (9.19) that

for involving

I-Y’, Q] = Q.

x[~g#. two terms from g# and one (9.20)

90

KOSTANT

AND

STERNBERG

For any projective representation r, as above, we write dp for L?@ id acting on SO T. Since the expression for T involves only elements of g#, we conclude that

ES, Q,l = Q,.

(9.21)

We now wish to impose enough assumptions to guarantee that ~3’ has a discrete spectrum. Then (9.20) or (9.21) will guarantee that Q, is of degree + 1 relative to the gradation given by the spectrum of L?. The reader can check that all our assumptions will be fulfilled in the case of a graded algebra. We consider the topology on a defined so that a is its own continuous dual. For any subspace b E a we clearly have b G (6’)’ and this space is just the closure of b. Thus b is closed if and only if b = (b’-)I. In particular,

any maximally

isotropic subspace such as N, g, or g# is closed. Thus subspace to L in g, then Lo paired. We now assume the existence of a closed com-

L = Nn g and Lo = Nn g#. If A4 is any complimentary

and A4 are nonsingularly plement, A4, to L so that

g=L@M and such that the elements of Lo induce all the continuous We then have the direct sum decomposition

linear functionals on M.

From these assumptions it follows that Lo and M” are closed,

M” is non-singularly paired to L, and any continuous linear functional on L is induced by elements of MO. Furthermore,

if we set P=M@M’

then a=N@P=L@L’@M@M’.

Under these assumptions and J such that

there is a countable index set K and disjoint K=IuJ,

a basis

(9.22)

subsets I

BRS

91

COHOMOLOGY

of g, a basis

of g# such that

with

Now,

it,iic!

a basis of L

{cx,,itll

a basis of MO

it, 1/EJ

a basis of M

’ \retIJIEJ

a basis of L”.

up to a scalar, 2’ will be Wick .Y =f

ordered relative to this basis so that

c /da,) P(5,)c IEI

c

A<,) ,e,)

IEJ

i

+ ro>

(9.23)

where rO is a scalar. In Section 11 we shall see that in the presence of a Hermitian structure, the scalar r,, will be fixed by the requirement that 9 be skew-adjoint. Now P is a vector space with a trivial scalar product, so C’(P) is isomorphic to AP. But we can regard C(P) as a subalgebra of C(a) and we have the vector space direct sum decomposition C(a)=C(P)@C(a) Therefore

N.

(9.34 )

the map

given by o(u) = u + C(a) N is a bijection. identification

But the direct sum decomposition

(9.25)

P = M @ MO gives the superalgebra

/lP=AM@A(M”) which we can use to put a bigrading

on S. Namely,

S”~Y=,(APM@AY(MO)). Then

we define

92

KOSTANT

Let 1= a( l,,).

AND

STERNBERG

Then we can rewrite (9.24) as (9.26)

Now it follows directly from the definitions

and (9.23) that

23 = t-01.

Hence

ZP(U) 1= cy, P(U)1 1+ YoP(U)1. But now (9.19) and (9.26) imply that .P’=(q-p+r,)id

on

Sp.y.

(9.27)

Thus if we define s, =

@

9.4,

(9.28)

q-p+ro=r

then (9.29)

s= 0 s,, on

L?=r

S,,

(9.30)

and (9.31)

Q: Sr+Sr+, as follows from (9.20). Now for any projective representation of the type we have been considering above, define

of the opposite class

(S@T),=S,@T

(9.32)

so that (9.33)

SO T= 0 (SC3 T),,

dP=r

(9.34)

on (SO T),

and (9.35)

Q,:(S@T),+(SOT),+,.

It now follows from (9.19), (9.20) and (9.15) that

Cp, dOI=

CT, CQmill

= 0.

Thus each of the spaces (SO T), is stable under cc,(g).

(9.36)

93

BRScOHOMOLOC;Y It follows (9.15) that

from (9.35) that we have a gradation

a, induces the trivial

on the cohomology

and from

action of g on each of the H&( S @ T).

(9.37)

So far we have been considering “absolute” cohomology. But it is important to also consider “relative” cohomology : Let h be a Lie subalgebra of g. We say that an element /I of SO T is basic relative to h if

P(OPL=O

(9.3X I

J,(5) p = 0

(9.39)

and

for all i E k. Notice that if p is basic relative to h so is Q,p by (9.15). Notice that (9.15) also implies that if Q,,u =0 and p satisfies (9.38) then it also satisfies (9.39) and hence is basic. In any event, the set of p which are basic relative to 11form a subcomplex. The homology of this subcomplex is called the relative Q, cohomology of g with values in SO T and will be denoted by Ha,( g, h; S@ T). Note that it follows from (9.19) and (9.21) that the subcomplex of h basic forms is stable under 1p. We can therefore consider the gradation induced by Y on the relative cohomology as well. Actually, it will turn out that we may have to choose the constant in (9.23) differently for the relative cohomology in order that Y be adjoint or skew-adjoint on the subcomplex. For example, as we shall see in Section Il. for many interesting algebras the “absolute” cohomology will be graded by half integers while certain relative comohology will be graded by integers.

10. THE CLIFFORD GROUP AND THE SPACE OF MAXIMAL IS~TROPICS We return to the notation and hypotheses of Section 7. We will let L denote the space of all maximally isotropic subspaces of V. Two maximally isotropic subspaces, N and P, are said to be transversal if

If NE L, a basis { rzi) of N is said to be full if there is a transversal P E L to N and a basis ( p, ] of P such that (n,, pj) = 6,. At the beginning of Section 7 we indicated the proof of the fact that every NE L has a full basis. A similar argument establishes the following proposition which characterizes those subspaces which are spanned by part of a full basis. PROPOSITION 10.1. Let NE L and let R he a suhspace two mnditions (III R are equivalent:

qf N. Then the

,follotving

94

KOSTANTAND

STERNBERG

(i) R has a basis which can be extended to a full basis of N. (ii) R is closed and has a closed complement, R h, in N, such that every continuous linear function on R^ is induced by an element of RI. A subspace R of N satisfying these equivalent conditions will be called strongly closed. Notice that if W is a closed subspace of l’ and F is a finite-dimensional subspace, then W+ F is also closed. We can now prove PROPOSITION 10.2. Let NE L. Any jmite-dimensional subspace of N and any closed subspace of N of finite codimension is strongly closed.

Proof If F is a finite-dimensional subspace, any basis of F can be expressed in terms of a finite number, say k, elements of a full basis of N. These expressions can be completed to an invertible k by k matrix which can then be completed by ones along the diagonal to give an invertible matrix which carries the full basis into another full basis passing through F. If R is a closed subspace of finite codimension, any complement, F, to R in N is closed, since it is finite-dimensional. Furthermore, Q.E.D. since R is closed, R’ induces all the linear functions on F. We now define a relation on L by saying that N, -Nz

if

N, n Nz has finite codimension

in N,.

The following is immediate: PROPOSITION 10.3. The relation defined above is an equivalence relation on L. Ij N, w N2 then N,/(N, n N2) and N,/(N, n N2) are nonsingularly paired under ( , ). In particular, they have the same dimension.

Let G = G(V) c C(V) denote the group of invertible elements a E C(V) which satisfy aVa-’ = V. For example, if z E V and (z, z) #O, then z-’ = (l/(z, z)) z and, for any u’ E V zwz - ’ = 2( (z, w))/(z, z)) z - w so conjugation by z preserves V and induces the negative of the reflection through the hyperplane perpendicular to z. 10.4. An element u E C( V) lies in G cf and only if u = z, . . . zk, where the zi E V with (z;, zi) # 0. PROPOSITION

Proof: Any UE C(V) lies in C(F) c C(V), where F is a finite-dimensional subspace. We may enlarge F, if necessary so that ( , ) is nonsingular on F. Then the proposition reduces to the finite-dimensional case, cf. [Greub]. The group G acts on V (by conjugation) as orthogonal transformations (as a consequence of Proposition 10.4, for example). Hence G acts on L. Each a in G sends an NE L into aNa-‘.

95

BRSCOHOMOLOGY THEOREM

10.5.

The orbits of G acting on L are precisely the equivalence classes

qf L under -

First we show that aNa-’ - N. Let P be a transversal null space and 10.4. We can find ‘,n,l,,z and (Pl)!EZ be dual full bases. Write a as in Proposition some finite subset, K, of indices such that all the z, lie in the space spanned by ink, pAJkEK. In particular, aniam ’ =nj if i does not belong to the finite subset K. This proves that aNa--’ - N. Next suppose that N, - Na. Since N, n N, is a strongly closed subspace of finite codimension in N, . we can find a full basis {n,} of N, and a finite subset K of indices such that {nj),gK is a basis of N, n Nz. By Proposition 10.3 we can find elements m, in N, for i E K such that (m,, nj) = 6, and such that the mi extend the Set in ,jleK to a basis of NJ. Let z,=n,+m; for ~EK. Then (:,,=,)=2. (I = ,,, ” -- f,, where K= :i ,,..., i,,). Proof.

Then it is easy to check that ania

and hence that aN,a

+m,

for

iEK,

+n,

for

i$K,

’ = N?, completing

the proof of the theorem.

Q.E.D.

For any NE L consider the maximal left ideal C(V) N and the corresponding irreducible spin representation pN of C(V) on S, = C( I’)/C( I’) N. We let 1,V denote the “vacuum”

in S,, i.e.. l,l

modC(V)N.

For any a E G, C( V) = C(V) a, so that C( I’) Nu- ’ = C( V)(aNa Hence right multiplication

’ ).

by a ’ induces an isomorphism

of left C(V) modules: 7~,(umod C(V) N)=ua-’ The irreducibility

mod C( V)(aNu- ‘).

of the spin modules then immediately

implies

THEOREM 10.6. If NE L and ME L with N - M then the spin representations p N and pM are equivalent and, up to a scalar, the equivalence is implementedby 72,where aEG is such that M==aNa-I.

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KOSTANT

AND

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Let N and P be transversal elements of L and (ni} and {p,} dual full bases. For each finite subset D = {i, < i, < . . < id} of integers let PD=Pl”‘Pd.

Then C(P)-AP=

@ cp,. D

As in the preceding section, since C(V)=C(P)@C(V)N we get a linear isomorphism o: AP + S,,,, where G(U) = p,(n) 1N, which is, in fact, an isomorphism of left C(P) modules. Let S,(D) denote the image of Cp, under cr. Thus (lO.lj S, = 0 S,(D), and it is clear that

IO}

PAni) S,(D) = sN(D _ iij j For of & NE L, ple, it

if if

iq!D, ieD.

(10.2)

any subset E of V, and any left C(V) module JZ, let JZE denote the subspace consisting of those vectors which are annihilated by all elements of E. If the nonzero elements of AN are called the N vacuum states of M. For examfollows from (10.2) that (10.3)

s;=c1,,

so the N vacuum states in SN are the nonzero multiples of 1,. Suppose that N, N N, and we choose K, ni, and mi, iE K as in the proof of Theorem 10.5. We claim that (SN2)N’=CPN~

cn,

Indeed, choose a as in the proof of Theorem a -’ =2-d(n,

“.&,j

lN2.

(10.4)

10.5 so that

+m,)*~~(n,+m,).

(10.5)

Now (SNZ)N’ = 7c,(SE;) = Ca-’ mod C(V) N,. But a-‘32-dnl...n~modC(V)NZ by (10.5). This proves (10.4). Two equivalence classes in L will be called transversal if we can choose representatives in each class which are transversal.

97

BRS CoHoMoLoGY

THEOREM 10.7. Let N, E L and P, E L. Their equivalence classes[N, ] and [P, ] are transversal lf and only if N, + P, is a closed subspaceof V of finite codimension. Furthermore, in this case N, n P, is finite-dimensional and

N, + P, = (N, n P,)‘. Proof: Suppose that the equivalence classes are transversal. This means that we can tind N 5 N, and P - P, such that N 0 P = V. Since N n N, is a closed iinitedimensional subspace of N there exists a finite-dimensional subspace, U. of P such that N n N, = /I’ n N. Thus U’=(NnN,)@P.

Similarly

there is a finite dimensional

subspace, W, of N so that

W’=N@(PnP,).

Then (I!J’+ W)L=lfLn

W’=(NnN,)@(PnP,)

is a closed subspace of finite codimension. But this subspace is contained in N, @P, which therefore also has finite codimension. By the remark preceding Proposition 10.2 it follows that N, 0 P, is also closed. Conversely, suppose that N, + P, is a closed subspace of finite codimension. Then, since N, = Nf and P, = Pf, we have that N, n P, = (N, + P,)’ is finite-dimensional,

and since N, + P, is closed N, f P, = (N, + P,)l’

= (N, n P, )I.

Let n, ,..., n, be a basis of N, n P,. By Propositions 10.1 and 10.2 we can extend it to a full basis {n,} of N, Let P, be some complement to N, and let (pi> be a dual basis to in,:. Let R denote the span of pI,...,pd. Then R’ n (N, n P,) = 10) so (N,+P,)+R=

and, since dim R = codim(N,

V

+ P, ), we have V=(N,+P,)@R.

Let

N=

1 i${l,...d;

Cn, + R.

(10.6)

98

KOSTANT

AND

STERNBERG

Then NE L and N - N, . In fact, N= (Nn N,)@R.

(10.7)

Also N,=(NnN,)@(P,nN,). So N, + P, = P, @ (Nn N,). Thus, by (10.6) and (10.7) NO P, = V. This completes the proof of the theorem. Notice that we only had to modify Ni in its equivalence class to makes it the transversal to P,. We shall say that N, and P, are almost transversal if their equivalence classes are transversal. Let K: C(V) + C(V) be the unique antiautomorphism which is the identity on V. so K(U,

.-uk)=u~...ul

for

USE V.

Let NE L. Let Sz denote the full algebraic dual to S,. We denote the pairing between these two spaces by ( , ). So (s, t) denotes the value of t E S,$ on SES,. Observe that Sz has the structure of a left C(V) module. Indeed, define pj$: C(V) --f End S,?j

If we drop the pN and pX we can write this as (s, ut) = (K(U) s, t).

(10.7)

THEOREM 10.8. Let N, P E L be almost transversal. Then there exists a nontrivial C(V) module homomorphism EN,P. . sp+s;

which is unique up to a nonzero scalar. Furthermore, 0) &N,p is injective and defines a nonsingular pairing ( , > between S, and SN such that = (s, Kc(U)t> for sES,,

tESp anduEC(V).

(10.8)

99

BRS COHOMOLOGY

(ii) (iii)

The image, E~,~(S~), depends only on the equivalence class of P.

Zf N and P are actually transversal,

the pairing can he normalized so that

(IN, 1P) = 1.

(10.9)

Proof. By the proof of Theorem 10.6 we can find some P, equivalent to P such that P, is complementary to N. We claim that

dim(S;)P’

= 1.

(10.10)

Indeed, if r E Sz then r E (S,$)” if and only if p,,,(P,) S,cker

r.

Thus (10.10) is equivalent to the assertion that pN(PI) S, has codimension one in S,. But this follows immediately from (10.2). Let r be a nonzero element of (S*,)“‘. Then the ideal ann r in C( V) is a proper left ideal containing P, , annr=C(V)P,, since C( V) P, is a maximal

left ideal. But ann 1 p, =C(V)

P,.

Hence there is a unique C(V) morphism c,v.p,:

SP,

-+

s;

such that

EN.P,(1p,I= r. and EN.P, is injective. Since any C(V) morphism from S,, to S$ must carry 1 p, into (SE)‘l it follows from (10.10) that E~,~, is uniquely determined up to the nonzero scalar involved in the choice of r. Also, it follows from (10.2) that (l,,,, r) # 0. This implies (iii). We have thus proved the theorem for the case of transversal maximal isotropics. But if P-P, we know by Theorem 10.6 that there exists a C(V) isomorphism, unique up to a nonzero scalar between S, and S,, and this completes the proof of the theorem, in general. Remark. If N and P are almost transversal, Theorem 10.7 implies that there is a unique topology on S, such that sN.JSP) is the space of continuous linear functionals, and that this topology depends only on the class of P.

100

KOSTANTAND STERNBERG

11. REAL FORMS AND HERMITIAN STRUCTURES We return to the hypotheses of Section 9. We assume that our ground field is the complex numbers. But now let us assume that (1) (2)

g = g, + igR is the complexification of a real Lie algebra g,, and where gg = {aEg# 1cc(g,)cR}. Then g# =gR” +igt, (11.1)

a=a,+ia,

is the complexification

of the real Lie algebra a=g,+g,#.

If z=u+iv

we write

so the map sending z to F is a conjugate linear automorphism

of a:

[Z,, Z,] =m

(11.2)

and we also have (z, M’) = (2, M’). In particular, if L is a (complex) then it follows from (11.3) that

(11.3)

subspace of g and Lo its annihilator Lo = (IgO.

space in g# (11.4)

Let us now assume that the subspace L of g satisfies L + L is a closed subspace of finite codimension

of g

(11.5)

of g#.

(11.6)

and Lo + z” is a closed subspace of finite codimension

It follows from (11.6) that L n L s g is finite-dimensional

(11.7)

and from (11.5) that LOnJ?Gg

# is finite-dimensional

(11.8)

101

BRS COHOMOLOGY

Furthermore, (11.9) and (Lo n Lo)’

= L + L.

(11.10)

Now define N by (9.2) so that

We claim that N and I%’are almost complementary.

(11.11)

Indeed ( 11.9) and ( 11.10) say that in a we have (LnL)‘=g@(LO+LO)

(LOnL”)-=(L+L)@g#

and

which implies that (LnL+L”n~“)i=(L+~)@(LO+L”)=N+N.

(11.12)

Thus by ( 11.7) and ( 11.8) N + m is a closed subspace of finite codimension in a, proving (11.11). Using (11.3) we see that the conjugation operation z + 5 on a extends to a conjugate linear involutive automorphism of C(a) which we shall continue to denote by -. Since C(a) N= C(a) @, this conjugation induces a conjugate linear isomorphism s/v ‘S, which se shall also denote by -. For any u E C(a) and s E S we have us = us.

(11.13)

Define a * operator on C(a) according to the formula u* = ti(ti) = K(U).

(11.14)

Then * is a conjugate linear antiautomorphism, i.e., c* ET

for

cECi(a)-C,

(11.15)

)$>*= *

for

\v~Cl(a)-a.

(11.16)

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KOSTANT

AND

STERNBERG

and

(ml)* = u*u*.

(11.17)

Since N and fl are almost transversal there is a (unique up to scalar) nonsingular pairing between S, and S, satisfying (10.8). Use this pairing and the conjugation map from S,,, to Sm to define a nonsingular sesquilinear form ( , ) by the formula b, t) = (s, 0,

s, tES,.

(11.18)

We claim that for any s, t E S, and any u E C(a) we have (us,

t) = (s, u*t).

(11.19)

Indeed, (us, t) = (us, i) = (s, K(U) i)

by (10.8)

= (s, u*t)

by (11.13) and (11.14)

= (s, u*t)

by (11.18).

Note that the uniqueness assertion in Theorem 10.7 implies that up to factor there can be only one sesquilinear form on S, satisfying (11.19). starting from such a form, we can read the preceding argument and backwards to get a bilinear pairing between S, and S,v satisfying (10.8) pairing is unique up to a scalar factor by Theorem 10.7. This implies that there must be some complex number z such that

a scalar Indeed, (11.18) and this

for all u, u E S,, since the left-hand side gives a sesquilinear form satisfying (11.19). A repeated application of this equation shows that ZZ= 1, so z = ezio for some 6. So if we replace ( , ) by e-j”(, ) we can arrange that (UTu) = (4 0).

(11.20)

We shall always make this choice which then fixes ( , ) up to a nonzero real factor. If A E End SN then A may or may not have an adjoint. That is, there may or may not exist an A* E End S, satisfying (Au, u) = (u, A*u) for all u, u E S,. Equation (11.19) asserts that all elements of C( V) have adjoints and that the adjoints are given by the * operator on C(V). If {A;} is a sequence of elements of End SN which have adjoints and if Ai + A and AT -+ B then it follows from the definition of convergence that A* = B.

BRSCOHOMOLOGY

103

THEOREM 11.1. Under the hypotheses of this section it is possible to choose the BRS operator Q so that Q* exists and satisfies

Q* = -Q.

(11.21)

Suppose that the space T has a pseudo-Hermitian scalar product and that the r E Hom(g, End T) used in the construction of Q, sends g, into skew-Hermitian operators. That is, supposethat z([) = -r(c)*.

(11.21)

Then u,ith the choice qf Q giving ( 11.21) we have

Qr* = -Q,

( 11.22)

relatioe to the tensor product pseudo-Hermitian form on S,v @ T. Pro@: According to (11.7) we can modify L without changing the equivalence class of N so as to arrange that L n L = (0). Since L + L is stable under conjugation, we can choose a (finite-dimensional) complement, L,, to L + L in g which is stable under conjugation. We can then choose a basis, u, ,..., qnr of L, consisting of real elements, i.e.,

I‘i; = rf;. We then let it,],

(11.23)

i= -1, -2,..., be a basis of L and define ;-,={,

(11.24)

so that (<, ), i = 1, 2,..., is a basis of L and (11.24) holds for all i # 0. We let {a’ i be the basis of (L,)O c g# dual to the t basis of L + L and {/I,) the dual basis to the t7.s so that the a’s and /Ys give a basis of g” dual to the basis of g given by the I;“s and q’s, We will write the structure constants and bracket relations of g as

Our assumptions guarantee that there are only finitely many nonvanishing f ‘s, g’s, and h’s. The fact that complex conjugation is an automorphism of g, together with the choices (11.23) and (11.24) imply that Cl :-,=c;,

(11.25)

&.i, ~, = d:,,

(11.26)

e:fl,=efp.

(11.27)

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KOSTANT

AND

STERNBERG

Let us use the expression (9.7) for Q relative to the basis we have chosen, taking into account the difference in notation due to the presence of the q’s and the p’s. We can then write 2Q=

c

c;a’aitk+

1

k
c;.tkaiai

k>O

+ 1 d$q,cI’a’+

c

efi,aiPP[k + c

kc0

k>O

e~(kC(i~P

+ a finite sum of elements of C:(a). Now the sum on the first line is skew-Hermitian by virtue of (11.16) ( 11.17), (11.24) (and its dual statement, ti’= a-‘), (11.25) and the fact that the tl’s supercommute with one another. The sum on the second line is skew-Hermitian because of (11.16) (11.17), (11.23), (11.26), the fact that C(‘=C(-~, and the fact that the I]‘S and the U’S supercommute, as do the a’s with one another. Similarly, the sum on the third line is skew-Hermitian using (11.27). Since the elements of C(a) all have adjoints, we see that Q* exists and belongs to End:S,. Also, for X, y, and ZEU, it follows from (11.16) and (11.17) that

C%T~, Q*lll=

-Lx> Cv, C--3 Qlll*,

and by (11.15) (11.2) and (11.3) this means that -Q* can also serve as a possible Q. But then so can $(Q - Q*) which is patently skew-Hermitian. This proves the first assertion of the theorem. The second assertion, that is, Eq. (11.22), follows from (9.12) (11.23) (11.24) and the hypothesis (11.21). Note that it now follows from (9.15) and (11.22) that the representation a, is skew-Hermitian. Let us now show how to choose the constant r. in (9.23) so as to make Y skewHermitian. If we assume that the basis used in (9.23) is of the form (11.23) (11.24), and the corresponding dual elements then (9.23) can be written as .Y=(Y+

-L2y)+dpo

+r,,

where y+ = t 1 PC%) P(L) i-co and % = 4 2 da;) P(B,). Now (Z+ - Y;p*, ) is skew-Hermitian by construction, shows that U,* = --scl, - $ dim Lo. so Y will be skew-adjoint

if we choose r o =$dimL,

in (9.23).

and a direct computation

(11.28)

BRS

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105

Thus the gradation of S and hence of the Q, cohomology will be integers or half integers according to whether dim L, is even or odd. Let us now examine the adjointness properties of Q, and of the degree derivation for the case of the complex of elements which are basic relative to a subalgebra g,,. cf. the discussion at the end of Section 9. We will assume that g, is finite-dimensional and is real, i.e., is stable under the complex conjugation. In fact, we shall assume that the subalgebra g, is the subspace L, in the preceding discussion. Let S’ denote the subspace of S= S, which are basic, relative to the subalgebra L,,. We claim that S’ is totally isotropic for the pseudo-Hermitian product on S, i.e., we claim that (s,, sz)=O for any pair of elements, s,, s, of S”. Indeed, for the present purposes, and also for the needs of the next section, let us describe the scalar product on S in some detail. Recall our situation:

and

so that N=L@L,*@L#

and N and N are almost transversal. As a transversal complement choose

to N we may

P=L@L,@L#

with P=L@L,@L#.

Then S,Y which can be identified

with AP can be written as

s,v= /lL@AL,@flL#.

In computing the scalar product we must use the isomorphism Clifford group which maps

( 11.29 ) coming from the

s,~=AL&lL”@nL#

onto Sp=AL@AL,#@AL#.

This isomorphism is clearly the identity on AL and on AL#. The only nontrivial component of this map will be an isomorphism from n L, to /1L(T. This

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isomorphism must be invariant under the general linear group of Lo and hence the only thing it can be is the map of PLO -+ A-L;,

d=dim

Lo

given by interior product with a real nonzero element in AdL,*. The choice of this element reflects the amount of nonuniqueness in the scalar product. Thus if s1 =x1 QY, 021

and

are two elements of S, which are decomposable their scalar product is given by

s,=x2Qy2Qz*

under the representation

(s,,s2)=(x,,~z)(~z,zl)~(Y,

Am,

(11.29) (11.30)

where ( , ) denotes the evaluation map between dual spaces and where 1 is a nonzero real element of A ‘Lo. Note that if we taken s, = s2 = s = x @ y @ z in (11.30) then (11.31) (w)=l<~,~w(Y AY), so that the index of the Hermitian form is determined by what happens on AL,. Now in the representation S= S, given by (11.29) that we are considering, p(5) for 5 E Lo is given by exterior multiplication by 5. Hence an element s satisfies p(g) s = 0 for all 5 EL, if and only if s lies in AL@ AdL, @ L#. But according to (11.30), the scalar product of any two such elements must vanish. Thus the Hermitian scalar we have been using vanishes identically on the relative complex, so we must consider a modified scalar product when dealing with relative cohomology. It is pretty clear what this modified scalar product should be. Indeed, let a be an element of the Clifford group which satislies P=aNa-‘.

Then for elements which are basic relative to Lo define

In view of the construction of the element a given in the proof of Theorem an alternative description of this new scalar product is as follows: Let

10.5,

Y=D, .BZ...Bd be the element of the Clifford algebra obtained by multiplying the elements of a real basis of L,*, so that y is determined up to a real nonzero scalar. Then

(-Jl, SZ)rel=(Sl, P(Y) 4

(11.32)

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since Yp*t commutes with p(y). Then the element L2&,=.2+

-Yip:

(11.33)

is skew-adjoint with respect to ( , )re, and differs from Y by a scalar on the space of basic elements. Hence on basic elements we still have

[Xc,, Ql = Q.

( 11.34 )

Note that $pre,has integral eigenvalues and that Q advances these eigenvalues by 1. The assumption that L and L, + L are subalgebras implies that ad Sp, has eigenvalues one or zero when applied to each of the monomials entering into the expression for Q (or for Q,) and similarly for -9:. Hence, if we define

Q+=C~+tQl and

(11.35)

Q - = ET, Ql, where we have written

we have

Q=Q++Q

(11.36)

with

C6Pt,Q+l=Q+ and

(11.37)

Also observe that

since P+ and Y commute as they involve different variables. But then by (11.34), (1 l.37), and (11.36) we have

[,sp, +Y~,Q++Q--I=Q++Q

+2C9’+>Q I=Q=Q++Q

and therefore

[sV+.Q I=C~~Q+l=O.

(11.38)

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Thus we have a bigradation with the total Y&, degree being the sum of the Y+ and X degrees, and with Q, advancing the Y+ degree by one and not changing the .E degree and the opposite holding for Q- . Now all of the preceeding discussion holds with Q replaced by Q, in the presence of a representation with opposite class, since the degree operator only affects the spin component. But now (11.36) becomes (11.39)

Qr=Qr+ +Q,-. But now Qt =0 together with the behavior of each summand bidegrees implies that we have a bicomplex, i.e., that

with respect to the

Qf, = CQ,+, Q,- I= QS- = 0.

(11.40)

It is this bicomplex, for the case of the Virasoro algebra, that is used in [Gar-Fra-Zuk] for the computation of the cohomology relative to the circle subalgebra (or more generally when L is graded and L, is the zeroth graded piece). They prove a vanishing theorem for the cohomology using spectral sequences and the ideas of Kostant [Ko2, Ko3] as extended by Kumzar [Ku] to the Kac-Moody situation. We can also examine the adjointness properties of the operator Q with respect to the new scalar product. Let us make the hypothesis (11.41)

(s,, [Q, ~1s~) = 0 for all basic elements S, and s2. For example, ( 11.41) will certainly hold if

Then

(Qslv de, + (~1,QQre, = (s, 7( - QYf rQ, ~1. If d is odd, then y is an odd element and hence the [Q, y] occurring in ( 11.41) is an anticommutator so taking the minus sign in the above equation we see that

(Qs, 9de, = @I9Q&L,

if

d is odd

if

d is even.

while we get (11.43)

Of course, all of the equations we have been considering hold equally well with Q replaced by Q,, since the various degree derivations and modifications of the scalar product involve only the spin component.

I 09

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12. ADDITIONAL

COMMENTS

Due to the length of this paper we thought it best to stop the detailed formal discussion at this point, and leave further theoretical developments and the details of specific computations to later publications. However, a number of general remarks are in order. 1. In our discussion in the first six sections of the relation between symplectic reduction and the BRS operator our starting point was a “bosonic” system with a Lie group of symmetries, i.e., we started with an ordinary symplectic manifold, not a supermanifold. All the “fermions” of the system were “ghosts” introduced by the BRS method. It is reasonable to develop the theory more symmetrically so that it contains bosons and fermions from the very beginning. That is, to start with a super-Poisson manifold with a supergroup or superalgebra of symmetries and consider the reduction process in this more general setting. 2. Our setting in Section 7 and the ensuing sections is purely algebraic. Thus. for example, we consider V=g +g#, where g is just a vector space or a Lie algebra paired to g#. We then consider isotropic subspaces of the form N = L + L” and are interested in orthogonal or infinitesimal orthogonal transformations which “move N a finite dimension outside of itself.” This would not, strictly speaking, include the case of the Dirac sea where we would want g to be a Hilbert space. The point is that with minor modifications one can consider a parallel theory in which k’ is a topological vector space (say g and g# are Hilbert spaces) and “finite rank” is replaced by “Fredholm” or “trace class” as in the theory of cyclic cohomology. 3. In order to be able to apply the methods of Hodge theory in its algebraic setting as developed in [Ko2, Ko3] it is necessary to be have a Laplacian and therefore a “star operator.” Here is a sketch of the construction. The idea is to use the cocycle c = gr, Q2 to construct the analog of the star operator. First, observe that if follows from the cocycle condition that the set of all [ ~g which satisfy c(<, a) = 0 (for all 9) is a subalgebra, call it h. Next observe that if follows from the skew-adjointness of Q that c is skew-adjoint, i.e., that [c(<, ?/)I = -c(f,

ij)

Therefore the subalgebra h is real, i.e., h = h. (For the case of the Virasoro algebra this subalgebra is h = g_ , + g, + g, and is the complexification of sl(2, R) thought of as the Lie algebra of the group of projective transformations of the circle = RP’.) We assume that the subalgebra L, used in the decomposition in Section 11 is /I. Also assume that the restriction of c to L x L (and hence to Lx L) vanishes. Define y:g+g*

Y(5)(V) = 453 vl).

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Notice that ker y = h = L,. Also notice that

r(5) = -r(O. Assume that y maps g into g# and, in fact, gives an isomorphism of L with L# and of e with L#. Finally, assume that h has a real invariant scalar product, so we get a map, E, from h -+ h*. Then y + E: g = (L + I+ h) -+ g# is an isomorphism, so we get an orthogonal transformation 6, of g + g” by defining a=y+song

and

a=(~+&)-’

on g#

Notice that o(N) = P so that (r induces a map from S,,, to S,. Since P is equivalent to N we can follow this by the action of an element of the Clifford group to get a map, c, from S,,, to itself which interchanges the /iL and the AL# components in the decomposition (11.29) by the map induced by cr on the exterior algebra. Thus c plays the role of the “star operator.” The cocycle c defines a Hermitian form on L and on L by c(z,, z2). In many interesting cases this form will be positive definite. This, in turn, induces a Hermitian form on AL and A1 and hence, using 0 on AL” and ,4E# as well. Let us continue to denote these induced Hermitian forms by c. Then (compare with (11.30)) we have (PlY d = c(z19 4 4x19 x2) 4Yl A Y2). 4. For the case of the Virasoro algebra we have been considering two interesting choices for LO, namely LO = g, and LO = g-, + go + g, . This corresponds to two interesting “infinite dimensional Kahler manifolds,” namely, Diff S/S’

and

Diff S’/Sl(2, R).

The first of these has been studied in detail by Bowick and Rajeeb [Bo-Ra] who interpret Diff S/S’ geometrically as consisting of complex structures on the space of loops in d-dimensional Minkowski space. The second of these also has an interesting geometrical interpretation-as the space of all Lorentz metrics on the single sheeted hyperboloid obtained from the standard (Sl(2, R)-invariant) Lorentz metric by a global conformal diffeomorphism. Let us elaborate on this point. A conformal Lorentzian structure on a two dimensional manifold is just two families of curves (the lightlike curves) passing through every point and intersecting transversally at all points. Locally, any two such structures are equivalent-we can locally straighten out the curves to make them lines (meeting at 45”, say). Globally (for example, on a cylinder) there will be many invariants having to do with the number of times a curve from one family will intersect a curve from the other. For example, we can think of the single-sheeted hyperboloid with its Sl(2, R) invariant metric (just the restriction of the Killing form to an adjoint orbit) as a cylinder. Here the null lines (the light like curves) are just the two rulings of the hyperboloid. Each line from one ruling meets all but one line from the other ruling once, and misses one line altogether. On the other hand, we can consider the flat Lorentz

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COHOMOLOGY

Ill

metric on the plane R’.’ whose null lines are just the lines at 45” to the axes. This structure is invariant under translation. In particular, if we identify two points which differ by a unit translation in the x direction, i.e., identify (x, ~1) and (X + 1, y), we obtain a cylinder with a Lorentz structure on it. Let us call this the standard cylinder and denote it by Cy. For the standard cylinder every line from one family meets every line from the other family infinitely often. So the standard cylinder and the single sheeted hyperboloid are not conformally equivalent. In fact, we can find infinitely many open subsets of the standard cylinder each having different intersection properties for the null lines of the restricted metric and hence inequivalent to one another. For closed cylinders, thought of as a manifold with boundary consisting of two circles the situation is even more amusing. Let us assume that the top and bottom circles are each transversals to all null lines. Pick a point on the bottom circle, move along one of the null curves to the top and then come back down to the bottom along the other null curve. This defines a diffeomorphism of the bottom circle. Clearly the conjugacy class of this diffeomorphism is an invariant of the closed conformal Lorentz cylinder. Let us describe the group of conformal automorphisms of the standard cylinder. For this purpose let us first describe the conformal automorphisms of RI,‘: Suppose that we fix one null line from each family, say I, and I,. Of course this really just amounts to choosing their point of intersection which we may call the origin. Then every point on the plane corresponds to a point on each of these lines by following the null line of the opposite family until it intersects each of the chosen lines, and this identification depends only on the conformal structure. So the conformal group of R’,’ is just Diff(l, ) x Diff(I,). If we identify each of these lines with R, we see that Conf(R’,‘)

m Diff(R) x Diff(R),

but this isomorphism depends, of course, on the choices made. For example, let us choose our lines to be the two 45” lines passing through the origin and identified with each other by reflection about the .Y axis and then with R. Then the diagonal subgroup Diff(R) in the above isomorphism has two orbits, a closed orbit consisting of the x axis and an open orbit consisting of the complement to the x axis. The group Z, acting by translations through integers in the .Y direction is the covering group of the cylinder, so Conf(Cy) = Conf(R’.‘)“/Z, where Conf(R’.‘)’ denotes the subgroup of Conf(R’,’ ) which commutes with Z. In view of the above isomorphism this has the same connected component of the identity as Diff S’ x Diff S’ = (Diff R x Diff R)Z/(Z x Z). There is an extra copy of Z in the isometry group of the cylinder generated by the twist map consisting of rotation through 180’ and vertical translation of half a unit. In any event, the diagonal subgroup Diff S’ lies in the conformal group

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of the cylinder. It has as closed orbits the horizontal circles at integer heights and open orbits consisting of the regions between these circles. The subgroup Sl(2, R) c Diff S’ has the same orbits. Hence each of the open orbits is just the single sheeted hyperboloid. In other words, the standard cylinder contains the single sheeted hyperboloid infinitely often, and Diff S’ acts as conformal automorphism of the hyperboloid. Since Sl(2, R) actually preserves the Lorentzian metric, we see Diff S’/S1(2, R) can be thought of as the set of Lorentzian metrics on the hyperboloid transformed from the standard one by conformal automorphisms. REFERENCES

W-PI

[Ba-Vi] [B-R-S]

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lGr1

[Gui-Stl] [Gui-St21 [Gui-St31 [Gui-St41 [Gui-St51

[GUI WI CKa21 [Kac-Pet]

W-01

[K-K-S]

WI W21 CKo31

[Ku1 CLangI CMcl CM-W

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