UNC-MATH-97-1
HOMOLOGICAL REDUCTION OF CONSTRAINED POISSON
ALGEBRAS
JIM STASHEFF
Reduction of a Hamiltonian system with symmetry and/or constraints has a long*
* history.
There are several reduction procedures, all of which agree in "nice" cases [AGJ*
*]. Some have a
geometric emphasis - reducing a (symplectic) space of states [MW], while others*
* are algebraic
- reducing a (Poisson) algebra of observables [SW]. Some start with a momentum *
*map whose
components are constraint functions [GIMMSY]; some start with a gauge (symmetry)
algebra whose generators, regarded as vector fields, correspond via the symplec*
*tic structure
to constraints [D]. The relation between symmetry and constraints is particular*
*ly tight in
the case Dirac calls "first class". The present paper is concerned entirely wit*
*h this first class
case and deals with the reduction of a Poisson algebra via homological methods,*
* although
there is considerable motivation from topology, particularly via the models cen*
*tral to rational
homotopy theory.
Homological methods have become increasingly important in mathematical physic*
*s, espe-
cially field theory, over the last decade. In regard to constrained Hamiltonian*
*s, they came
into focus with Henneaux's Report [H] on the work of Batalin, Fradkin and Vilko*
*visky
[BF,BV 1-3], emphasizing the acyclicity of a certain complex, later identified *
*by Browning
and McMullan as the Koszul complex of a regular ideal of constraints. I was abl*
*e to put the
FBV construction into the context of homological perturbation theory [S1] and, *
*together
with Henneaux et al [FHST], extend the construction to the case of non-regular *
*geometric
constraints of first class. Independently, using a mixture of homological and C*
*1 -patching
techniques, Dubois-Violette extended the construction to regular but not-necess*
*arily-first-
class constraints [D-V].
I am grateful to all of the above for their input and inspiration, whether in*
* their papers
or in conversation. The present version has also profitted from conversations a*
*t the MSRI
Workshop on Symplectic Topology. Finally, I would like to express my thanks to *
*the referee
who has read several versions with extreme care, suggesting extensive improveme*
*nts, both
factual and stylistic. While early revision was in progress, Kimura sent me a c*
*opy of [Ki]
which has also had a significant influence on the present exposition, as has hi*
*s continued
interaction while with me at UNC as an NSF Post-Doc.
__________
Research supported in part by NSF grants DMS-8506637, DMS-9206929, DMS-950487*
*1, a grant
from the Institute for Advanced Study and a Research and Study Leave from the U*
*niversity of
North Carolina-Chapel Hill. Announced in the Bulletin of the American Mathemati*
*cal Society as
"Constrained Poisson algebras and strong homotopy representations" [S2]. This p*
*aper includes the
mathematical version of the physics in [FHST]
1
2 Jim Stasheff
1. Preliminaries
This research touches on questions which it is hoped will be of interest to m*
*athematical
physicists, symplectic and algebraic geometers and homotopy theorists. The tech*
*niques used
here are primarily those of differential commutative algebra and rational homot*
*opy theory.
We write with a dual vision and hopefully a dual audience; for example, the con*
*straints
are functions on a symplectic manifold and the physics literature speaks almost*
* entirely in
terms of the constraints whereas the algebra can be expressed more invariantly *
*in terms of
the ideal generated by the constraints. We work entirely over the reals R as o*
*ur ground
field, although any field of characterisitic 0 would do and the complex numbers*
* C are more
common in certain physical applications. The major Theorem 4.2 is expressed in *
*algebraic
terms, followed by remarks specifically in terms of the constraints themselves.
We begin therefore with a brief (very!) review of the motivating background: *
*a tiny bit
of symplectic geometry, slightly more of Poisson algebra and the essentials of *
*constraint
varieties and their symmetries in the first class case. The reader who desires *
*more extensive
background or a more leisurely exposition may consult a variety of sources list*
*ed in the
bibliography. The relations between the algebra and the motivating geometry are*
* exposed
particularly clearly in [Ki].
1.1. The Hamiltonian Formalism. The motivating physical systems are described as
differential equations of motion or evolution involving smooth functions on a m*
*anifold. The
underlying manifold W is assumed to be symplectic. This means there is a 2-form*
* ! such
that d! = 0 and !dimW 6= 0. Equivalently, ! induces an isomorphism
T W ! T *W:
(With an eye to future applications, we would like to allow W to be infinite di*
*mensional,
in which case the appropriate definition is that the induced map T W ! T *W be *
*one-to-
one.) In local coordinates q1; :::; qn; p1; :::; pn, the form ! looks like dqi^*
* dpi(the summation
convention will be assumed throughout this paper).
From an algebra point of view, the crucial point is two-fold: For any functio*
*n f 2 C1 (W ),
there is a Hamiltonian vector field Xf defined by !(Xf; ) = df. For two functio*
*ns f; g 2
C1 (W ), their Poisson bracket {f; g} 2 C1 (W ) is defined by
{f; g} = !(Xf; Xg) = df(Xg) = -dg(Xf):
This bracket makes C1 (W ) into a Poisson algebra, that is, a commutative algeb*
*ra P (with
product denoted fg) together with a bracket { ; } : P P ! P forming a Lie alge*
*bra such
that {f; } is a derivation of P as a commutative algebra: {f; gh} = {f; g}h + g*
*{f; h}.
A typical Hamiltonian system is one of the form {f; H} = df=dt for fixed H. S*
*ymmetries
of such a system are given by functions g which Poisson commute with H. They fo*
*rm a sub-
Lie algebra of C1 (W ). Symmetries arise also in connection with "constraints".*
* Regarded
as in a dynamical system, solutions can be constrained to lie in a sub-manifold*
* V W
(more generally, V is just a sub-space), hereafter called the constraint locus,*
* also known
Homological Reduction of Constrained Poisson Algebras *
* 3
in the literature as a constraint surface. As in algebraic geometry, we can th*
*ink of V as
the zero set of some functions OEff: W ! R, called constraints. The algebra of *
*functions
C1 -in-the-sense-of-Whitney on V can be identified with C1 (W )=I where I is th*
*e ideal of
functions which vanish on V . If V W is a closed and embedded submanifold, thi*
*s agrees
with the usual notion of smooth functions on V .
Now if W is symplectic (or just given a Poisson bracket on C1 (W )), Dirac ca*
*lls the
constraints first class if I is closed under the Poisson bracket. (If the R-lin*
*ear span of the
OEffis closed under the bracket, physicists say the OEffclose on a Lie algebra;*
* this is a very
nice case, but the more general first class case is where homological technique*
*s are really
important.) When the constraints are first class, we have that the Hamiltonian *
*vector fields
XOEffdetermined by the constraints are tangent to V (where V is smooth) and giv*
*e a foliation
F of V . Similarly, C1 (W )=I is a Lie module over I with respect to the Poisso*
*n bracket.
In symplectic geometry, when V is smooth, it is usually called a coisotropic su*
*bmanifold
(see [W] for generalizations when V is not smooth). For the general case, we wi*
*ll call the
constraint locus coisotropic if the ideal is first class.
In many cases of interest, I does not arise from the Lie algebra of some Lie *
*group of
transformations of W or even V , but the corresponding Hamiltonian vector field*
*s XOEffare
still referred to as (infinitesimal) symmetries. In the nicest case, e.g. whe*
*n the foliation
F is given by a principal G-bundle structure on a smooth V , the algebra C1 (V=*
*F) can be
identified with the I-invariant sub-algebra of C1 (W )=I. In great (if not comp*
*lete) generality,
this I-invariant sub-algebra represents the true observables of the constrained*
* system.
In this context, the "classical BRST construction", at least as developed by *
*Batalin-
Fradkin-Vilkovisky and phrased in terms of constraints, is a homological constr*
*uction for
performing the reduction of the Poisson algebra C1 (W ) of smooth functions on *
*a Poisson
manifold W by the ideal I of functions which vanish on a coisotropic constraint*
* locus. But
the construction produces cohomology in other degrees than zero, which at least*
* in some
cases, admits a geometric interpretation.
Instead of considering just the "observable" functions, one can consider the *
*deRham com-
plex of longitudinal or vertical forms of the foliation F, that is, the complex*
* (V; F) con-
sisting of forms on vertical vector fields, those tangent to the leaves. If we*
* think of F as
an involutive sub-bundle of the tangent bundle to V , then (V; F) consists of s*
*ections of
*F. In adapted local coordinates (x1; :::; xr+s) with (x1; :::; xr) being coord*
*inates on a leaf,
a typical longitudinal form is
fJ(x)dxJ where J = (j1; :::; jq) with 1 j1 < :::jq r; the leaf dime*
*nsion:
The usual exterior derivative of differential forms restricts to determine the *
*vertical exterior
derivative because F is involutive. This complex is familiar in foliation theor*
*y, c.f. [HH].
The classical BRST-BFV construction has, in the nice cases, the same cohomology*
* as this
complex of longitudinal forms.
A major motivating example for the BFV construction was provided by gauge the*
*ory. Here
W is T *A where A is the space of connections for a fixed principal G-bundle G *
*! P ! B.
4 Jim Stasheff
The reduced phase space is T *(A=G) where G is the group of "gauge transformati*
*ons", the
vertical automorphisms of P .
In considering what the physicists [BF],[BV1-3],[FF], [FV],[H],[BM] did in so*
*me special
cases, I recognized a homological "model" for (V; F) in roughly the sense of ra*
*tional homot-
opy theory [Su]. This is the same sense in which the Cartan-Chevalley-Eilenberg*
* complex
[CE] for the cohomology of a Lie algebra g is a "model" for *(G) where G is a c*
*ompact
Lie group with Lie algebra g. The physicists' model is itself crucially a Pois*
*son algebra
extension of a Poisson algebra P and its differential contains a piece which re*
*invented the
Koszul complex for the ideal I. The differential also contains a piece which lo*
*oks like the
Cartan-Chevalley-Eilenberg differential. Generalizations of the Cartan-Chevalle*
*y-Eilenberg
differential as they occur in physics are usually referred to as BRST operators*
*. This honors
seminal work of Becchi, Rouet and Stora [BRS] and, independently, Tyutin [Ty]. *
*Appar-
ently it was the search for such an operator in aid of quantization which motiv*
*ated the work
of Batalin, Fradkin and Vilkovisky.
It was Browning and McMullan [BM] who first identified the Koszul complex wit*
*hin the
construction in the regular case, (Henneaux had already called attention to the*
* relevance of
that acyclicity) leading both Dubois-Violette [D-V] and myself [S1] independent*
*ly to adopt
a more fully homological approach, although with somewhat different emphases. *
*Dubois-
Violette retains some of the symplectic geometry and is able to handle regular *
*general (not
necessarily first class) constraints. On the other hand, by restricting to firs*
*t class constraints,
in joint work with Henneaux et al [FHST], I was able to handle non-regular idea*
*ls in suitable
geometric circumstances.
In the present paper, I start at the level of the purely (Poisson) algebraic *
*structures.
In particular, I adapt the notion of "model" from rational homotopy theory and *
*use the
techniques of homological perturbation theory. Although the treatment of BFV is*
* basis de-
pendent (individual constraints) and nominally finite dimensional, I attempt to*
* work more
invariantly in terms of the ideal generated by the constraints and take care to*
* avoid assump-
tions of finite dimensionality. Although originally invented in the context of *
*quantization,
both BRST cohomology as they described it and the BFV-generalization are mathem*
*atically
interesting in the `classical' setting. The present paper is concerned only wit*
*h the clasical
setting but in the full generality of a first class ideal, in contrast to the p*
*aper of Kostant
and Sternberg [KS] whose main interest is in quantization issues for the case o*
*f an equivari-
ant moment map and hence do not deal with the BFV-generalization nor with homol*
*ogical
perturbation methods.
2. Reduction
We have presented a geometric picture of reduction as referring to W - V ! V*
*=F.
There are a variety (pun intended) of difficulties with this approach. The cons*
*traint locus
V fail to be a submanifold. Even if it is a submanifold, the quotient ^V:= V=F *
*may not be
a manifold, in fact, may not even be Hausdorff. (An intermediate situation of c*
*onsiderable
interest occurs with the quotient V=F being a stratified symplectic space [LS].)
Homological Reduction of Constrained Poisson Algebras *
* 5
When (W; !) is a symplectic manifold with a smooth coisotropic submanifold, o*
*ne of
the nicest cases is called `regular', namely when the quotient V=F is a manifol*
*d and the
projection V ! ^Vis a submersion. This implies further that !|V has constant*
* rank on
T V (so that !|V is a presymplectic form on V ), and F is an involutive distrib*
*ution given
by ker !|V which is fibrating. Then a standard argument, due essentially to E*
*. Cartan
[MW] or [GSte, Thm. 25.2], shows that there exists a unique symplectic form ^!*
*on
V^ satisfying ss*^!= !|V . The reduction of (W; !) is then the symplectic mani*
*fold (V^; ^!)
and the corresponding reduced Poisson algebra is C1 (V^) with the Poisson brack*
*et that is
associated to ^!.
In the "singular" case, when these conditions fail to hold, reduction in the *
*above sense
will not be well defined. Various definitions of reduction are possible, depend*
*ing upon which
aspects of the theory are considered primary. (Of course, each such definition *
*should agree
with regular reduction when both apply.) Below we present two such definitions *
*(following
[AGJ]), although there are undoubtedly others.
The first type of reduction we shall consider is based upon the notion of an *
*"observable".
Following Bergman, we call a function on W an observable iff its Poisson bracke*
*t with
each first class constraint is again a constraint, i.e., h 2 C1 (W ) is an obse*
*rvable if and
only if {h; I} I. Bergman emphasized observables (rather than the points in V *
*which are
states) because observables represent measurable quantities. (The condition {h;*
* I} 0 on
V is a gauge invariance condition.) The set O(V ) of observables forms a subalg*
*ebra of the
associative algebra C1 (W ).
" Dirac reduction" takes two states x; y 2 V to be physically equivalent iff *
*they cannot be
distinguished by observables. This amounts to defining an equivalence relation *
*~ on V by
x ~ y iff h(x) = h(y) for all observables h. The corresponding reduced space is*
* ^V= V= ~.
The observables after reduction are identified with the elements of O(V ) which*
* are fixed
under the adjoint action of I (with respect to Poisson bracket). Since we are d*
*ealing with
first class constraints, these observables inherit a Poisson bracket.
Example: Zero angular momentum in two dimensions.
Here W = T *R2 R2xR2 = {(q; p)} and the angular momentum is q xp = q1p2-q2p1 w*
*ith
constraint set V = {(q; p)|q1p2- q2p1 = 0}. The foliation F is in fact given by*
* the orbits of
the standard circle action on R2 lifted to T *R2. The Dirac reduction can be id*
*entified with
the symplectic orbifold C=Z2.
Sniatycki and Weinstein [SW] have defined an algebraic reduction in the conte*
*xt of group
actions and momentum maps which is guaranteed to produce a reduced Poisson alge*
*bra but
not necessarily a reduced space of states (cf. [W2]). (In contrast, Kostant and*
* Sternberg
use the Marsden-Weinstein reduction [MW].) The S-W (Sniatycki and Weinstein) re*
*duced
Poisson algebra is (C1 (W )=I)G where V = J-1(0) for some equivariant Poisson m*
*ap J :
W ! g* (called a moment map), equivariant with respect to a given G-action on W*
*; g
being the Lie algebra of G. (If G is compact and connected, (C1 (W )=I)G is iso*
*morphic to
6 Jim Stasheff
the Dirac reduction C1 (W )G=IG .) With hindsight, the generalization of S-W re*
*duction to
a general first class constraint ideal I is obvious. The issue of its suitabili*
*ty is not one of
geometry necessarily, but rather one of physics.
The present paper grew out of the realization that the BFV construction could*
* be regarded
as a homological model which in degree zero models the I-invariants of C1 (W )=*
*I. The
whole construction turned out in many cases to be a model for the complex of lo*
*ngitudinal
forms *(V; F). From an algebraic geometric point of view, it is indeed natural *
*to define
the observables on V by restriction of observables on W , that is, to consider *
*the quotient
algebra C1 (W )=I, which corresponds to the algebra of smooth (in-the-sense-of *
*Whitney)
functions on V . In physics, this is expressed by saying two functions on W are*
* weakly equal
(f g) if their difference vanishes on V .
Now let us recast the problem in purely algebraic terms. Consider an arbitrar*
*y Poisson
algebra P with an ideal I which is closed under the Poisson bracket. Reduction*
* is then
achieved by passing to the I-invariant subalgebra of P=I. Note that a class [g]*
* is I-invariant
if {I; g} I, equivalently, if {OE; g} 0 for all constraints OE 2 I. This suba*
*lgebra inherits a
Poisson bracket even though P=I does not: For f; g 2 P and OE 2 I; we have {f +*
* OE; g} =
{f; g} + {OE; g} where {OE; g} need not belong to I, but will if the class of g*
* is I-invariant.
The Poisson algebra of invariants amounts to the quotient NP(I)=I where NP(I)*
* denotes
the normalizer of I in P in the sense of Lie algebras; the ideal I is a Poisson*
* ideal in NP(I).
In this context, the analog of longitudinal forms are the alternating multili*
*near-over-P=I
functions h : I=I2 . . .I=I2 ! P=I which again form a graded commutative algeb*
*ra,
which we denote
AltP=I(I=I2; P=I):
We use I=I2 because the corresponding Hamiltonian vector fields are restricted *
*to V in
providing the foliation F:
The fact that I is a sub-Lie algebra of P but is not a Lie algebra over P (th*
*e bracket
is R-linear but not P -linear) is a significant subtlety. One way to handle thi*
*s is to observe
that I=I2 inherits the structure of what Rinehart called an (R; P=I)-Lie algebr*
*a. This
corresponds to what Herz [Hz] called a quasi-Lie algebra and what Palais [P] ca*
*lled a d-Lie
ring. Since it is Rinehart's paper that establishes the relation to the geometr*
*y and was his
major contribution in a tragically short career, we prefer to refer to the Lie-*
*Rinehart pair
(I=I2; P=I).
Definition 2.1.[R],[P] A Lie-Rinehart pair (L; A) over a ground ring k consists*
* of a com-
mutative k-algebra A and a Lie ring L over k which is a module over A together *
*with an
A-morphism ae : L ! Der A such that
[OE; f ] = (ae(OE)f) + f[OE; ] forOE; 2 L; f 2 A:
Notice this is the condition satisfied by L = I=I2 and A = P=I with ae(OE)f =*
* {OE; f}:
Hence we can consider the Rinehart complex AltP=I(I=I2; P=I) with differential *
*d given
Homological Reduction of Constrained Poisson Algebras *
* 7
by (3.1)
X X
(dh)(OE0; :::; OEq) = (-1)i+jh([OEi; OEj]; :::; ^OEi; :::; ^OEj; :::) + *
* (-1)iae(OEi)h(:::; ^OEi; :::):
i 0, Hi(X) must be represented with ghosts. When this involves only gho*
*sts corre-
sponding directly to constraints (i.e., elements of (s)*) but no ghosts-of-ghos*
*ts, "geomet-
rically" we are looking at longitudinal forms. It is only from the transverse (*
*"gauge-fixed")
point of view that the ghosts inherit their name.
The key to the main theorem comes from the Hamiltonian and BRST formalisms. *
*Let
()* P be given a bigrading (r; s). Assuming P ungraded (see x6 for the graded
or super case), P is already (negatively) graded and this grading is s, calle*
*d the
resolution degree. Then ()* inherits the dual (positive) grading r, called the*
* ghost
degree, adopting the term from the physics literature (where the negative of th*
*e resolution
degree is called the anti-ghost degree). The total degree is the sum r + s of t*
*he ghost degree
and the resolution degrees. Batalin, Fradkin, and Vilkovisky make X into a Pois*
*son algebra
by extending the Poisson bracket on P to one on X by defining
{h; } = h( ) for h 2 *; 2 ;
all other brackets not determined by the derivation property being set equal to*
* zero. This
extended bracket is of total degree zero, but mixed bidegrees.
4.1. The BRST generator. The sought-for differential @ is constructed to be of *
*the form
@ = {Q; } where Q is a formal sum of terms Qn defined by induction (on n). In*
* physics,
Q is referred to as a BRST generator or operator, in keeping with the philosoph*
*y mentioned
in x2 with particular emphasis on the facts that 1) @2 = 0 or equivalently, {Q;*
* Q} = 0 and
2) Q contains a piece corresponding to the Cartan-Chevalley-Eilenberg different*
*ial.
10 Jim Stasheff
The proof of the existence of Q can be handled effectively by the "step-by-st*
*ep obstruction"
methods of homological perturbation theory [G,GM,GSta,GLS,Hu6-9,HK]. We adapt
the details to this case, rather than appealing to the general theory. We make *
*crucial use of
the filtration of X by the form or monomial degree, i.e., (i)* P is the part *
*of X of
form degree i, or equivalently, "form degree i" refers to an i-multilinear grad*
*ed symmetric
function from to P . The filtration is defined by: Fn = FnX is the space of f*
*orms of
degree > n. We use the strict inequality so that this filtration is multiplicat*
*ive with respect
to both parts of the Poisson algebra structure:
FpFq Fp+q+1 Fp+q and{Fp; Fq} Fp+q:
Start with Q0 : ! P as the Koszul-Tate differential ffi restricted to . A*
*s an
element of X, this Q0 is of total degree 1 and form degree 1, but {Q0; } is a*
* sum of two
pieces, of form degree 0 (namely 1 ffi) and of form degree 1. Since the bracke*
*t restricts to
the pairing (by evaluation) of ()* and , the term of form degree 1 includes the*
* adjoint
of ffi taking HomP( P; P ) to itself. The remainder of {Q0; } is given by th*
*e original
bracket (in P ) of the coefficients of Q0 with elements of P .
Since all our objects are at least vector spaces, the model property of P c*
*an be
evidenced by a "contracting homotopy" s : P ! P of degree -1 such that
sffi + ffis = 1 - sswhere ss: P ! P ! P=I ,! P is given by ss composed with*
* an
R-linear splitting P=I ,! P . P
For any element R 2 X, let R2 denote 1=2{R; R}. Now construct Rn = in Qi by
induction so that
{Rn; Rn} 2 Fn+2 and ffi{Rn; Rn} 2 Fn+3:
Define Qn+1 = -s=2{Rn; Rn} = -sR2n.
The following slightly complicated computation shows Rn+1satisfies the induct*
*ive assump-
tion.
Both ffi and s preserve the filtration, and from the way Q0 is defined, {Q0; *
* } - 1 ffi
increases filtration. Start with
R2n+1= (Rn + Q2n+1)2 = R2n- {Rn; sR2n} + (sR2n)2:
The last term (sR2n)2 2 F2n+4since sR2n2 Fn+2 and 2n + 4 n + 4. On the other h*
*and,
{Rn; sR2n} (1 ffi)(sR2n) mod Fn+3;
since Rn = Q0+ Q1+ : :a:nd the {Qi; } for i > 0 increase filtration. Thus
{Rn; sR2n} -(1 sffi)R2n+ R2n mod Fn+3;
so
(4.1) R2n+1 -(1 sffi)R2n+ R2n mod Fn+3
(4.2) 0 mod Fn+3
by the assumption on ffiR2n.
Homological Reduction of Constrained Poisson Algebras *
* 11
Similarly
(4.3) ffiR2n+1 ffiR2n- ffi{Rn; sR2n} + ffi(sR2n)2
(4.4) ffiR2n mod Fn+4:
Now we need to commute ffi with {Rn; }. Since {Rn; } - 1 ffi increases f*
*iltration by
at least one, its square does so by at least two. Thus
{Rn; {Rn; }} - {Rn; 1 ffi} - 1 ffi{Rn; }
applied to sR2nis of filtration at least n + 4. Now the graded Jacobi identity *
*gives
2{Rn; {Rn; }} = {{Rn; Rn}; }
which increases filtration by n + 2, thus
(4.5) ffiR2n+1 ffiR2n+ {Rn; ffisR2n} mod Fn+4
(4.6) ffiR2n+ {Rn; R2n} - {Rn; sffiR2n} mod Fn+4
(4.7) ffiR2n- ffiR2n mod Fn+4
since {Rn; sffiR2n} ffisffiR2n modFn+4 and {x; {x; x}} = 0 for x of any degree*
* (over a field
of characteristic not equal to 3).
Thus we have constructed a differential graded Poisson algebra for any coisot*
*ropic ideal.
Where possible, we will show that we have a model for AltP=I(I=I2; P=I) by the *
*usual
techniques of comparison in homological perturbation theory, namely comparison *
*of spectral
sequences. In one final case, we can do this locally but appeal to a geometric*
* arguement
to patch the local results. After establishing that, we will look at issues inv*
*olving choices
(possibly non-minimal) choices of generators (constraints) for the ideal I.
From the definition of the filtration Fp, the associated graded E0(X) is isom*
*orphic to
()* P . To analyze d0, notice that since s preserves the form degree, Qi+1=
-sR2i2 Fi+2 and hence {Qi; } increases filtration by at least i. As mentionned*
* earlier,
{Qo; } - 1 ffi also increases filtration so d0 is (induced by) the Koszul diff*
*erential ffi. Thus
E1(X) ()* P=I AltR(; P=I);
and E1(X) is concentrated in anti-ghost degree 0; the spectral sequence necessa*
*rily collapses
from E2. To determine H0(X) E0;02, we need only analyze d1 on . For h 2 P=I; c*
*onsider
{Q0; h} : I ! P=I. It is given by {Q0; h}(OE) = {OE; h} for OE 2 I: Thus H0(X) *
*is isomorphic
to the I- invariants of P=I.
When the ideal I is regular, = s and we can analyze d1 on s similarly. For *
*ex-
ample, for h : I ! P=I, consider {Q0; h} : I ^ I ! P=I. It is given by {Q0; h}(*
*OE1; OE2) =
{OE1; h(OE2)} - {OE2; h(OE1)} while {Q1; h}(OE1; OE2) = -1=2{s{Q0; Q0}; h}(OE1;*
* OE2) = -h{OE1; OE2}.
(At this point, one appreciates the facility of non-invariant description in te*
*rms of a gener-
ating set of constraints {OEff} for I and a dual set {jff: I ! P }.)
12 Jim Stasheff
Thus we see d1 (up to sign) looks like the Rinehart generalization of the Car*
*tan-Chevalley-
Eilenberg differential. It is this identification of (E1; d1) which motivates t*
*he name BRST
generator for Q.
Now to make the comparison with the complex of longitudinal forms, since is *
*defined
as a quotient of I, there is the induced chain map
ss : X ! Altk(; P=I) ! Altk(s; P=I) ! AltP(I; P=I) AltP=I(I=I2; P=I)
as described above. In the regular case all maps except ss are isomorphisms. Fo*
*r the con-
strained Hamiltonian setting with which we began, in which P is C1 (W ), we hav*
*e identified
AltP=I(I=I2; P=I) with the longitudinal forms of the foliation F of V and d1 wi*
*th the exterior
derivative "along the leaves".
Theorem 4.2. If I is a regular first class ideal in C1 (W ), the map ss induce*
*s an isomor-
phism H(X) H((V; F)).
When I is not regular, we still have the map but in general lack sufficient i*
*nformation to
conclude an isomorphism in cohomology.
Now the physicists do not work with the ideal explicitly but rather with a se*
*t of constraints,
which is a set (not necessarily minimal) of generators for the ideal. The corre*
*sponding BFV
construction_starts with as the vector space spanned by the constraints, rathe*
*r than with
I=P I. In certain cases, even though the constraints do not form a regular sequ*
*ence, we can
still make the identification of H(X) with H((V; F)).
The redundant case: The set of constraints may be reducible in a trivial way;*
* a proper
subset may consist of a regular sequence of generators. Then we can split as
where is the span of the minimal subset and is spanned by the complementary s*
*ubset.
The Koszul-Tate resolution of P=I splits as the Koszul resolution determined by*
* tensored
with a contractible DCGA. Then Alt(; ) splits similarly and the BRST generator *
*can be
constructed_first_in the part and then extended so the results will be the sam*
*e as when
using = I=P I.
In particular, if the constraints are given by an equivariant_moment map J : *
*W ! g*
where G acts by symplectomorphisms but with kernel H, then I=P I is isomorphic *
*to g=h
but the span of the constraints would be isomorphic to g. Here choose a splitti*
*ng such
that = h and g=h; then proceed as in the redundant case.
In [FHST] and [HT], the setting is specifically that of a symplectic manifold*
* (phase space)
with a constraint submanifold ("surface") and moreover the assumption is made t*
*hat locally
the constraints can be separated into "independent constraint functions" and de*
*pendent
ones which can be expressed as functional linear combinations of the independen*
*t ones with
coefficients which are regular in a neighborhood of the constraint submanifold.*
* Thus locally
we are in the redundant case so identities involving the globally defined BRST *
*generator
and comparisons with the complex of forms along the leaves can be verified loca*
*lly; we again
have H(X) H((V; F)).
Homological Reduction of Constrained Poisson Algebras *
* 13
Finally, the construction of @ and of Q involves a choice of contracting homo*
*topy s and
implicitly of a choice of splitting P=I ,! P . A change in s produces changes i*
*n @ but not
in the homotopy type of (X; @) as DGCA. Moreover the change in s can be realize*
*d by an
automorphism of and the induced one on *. This is an example of what is known *
*as
a canonical transformation, a basic automorphism of any Hamiltonian system.
5. Generalizations: Infinite dimensions and super algebras
If I is regular and finitely generated over P (so is finite dimensional over*
* R), AltP(; P
s) is finitely generated as a P -module and Qn = 0 for sufficiently large n. If*
* I is finitely
generated but not regular, may easily be infinite dimensional, though finite i*
*n each grading,
and so all Qn may be non-zero.
More importantly, there are many examples occurring in physics (field theory)*
* in which
is itself infinite dimensional. That is why we have been careful to emphasize A*
*lt or to take the
dual of rather than (*). Actually both physical and mathematical consideration*
*s (cf.
Gelfand-Fuks cohomology) suggest that the alternating functions might better be*
* restricted
to being continuous in an appropriate topology.
Early in the development of Batalin, Fradkin and Vilkovisky's approach, atten*
*tion was
called to the generalization to a super-Poisson algebra P = P0 P1 with super co*
*nstraints.
This means that P is a GCA (graded by Z=2 = {0; 1}) with a graded bracket { ; *
* }:
(5.1) P0 P0 ! P0
(5.2) P0 P1 ! P1
(5.3) P1 P1 ! P0
with graded anticommutativity, graded Jacobi identity, and graded derivation pr*
*operty
(Leibnitz rule):
{f; gh} = {f; g}h + (-1)|f||g|g{f; h}
where f 2 P|f|; g 2 P|g|.
It has long been known in algebraic topology how to generalize the constructi*
*on of models
such as the Koszul-Tate complex or the Chevalley-Eilenberg complex to the grade*
*d setting,
e.g., is now a graded vector space and s is an isomorphic copy of a regraded d*
*own by 1
so that ffi is still of degree 1. The use of to denote the free graded commuta*
*tive algebra on a
graded vector space means that the only necessary change in our treatment is to*
* specify the
resolution degree as the one implied by the degree on s with ffi being of resol*
*ution degree
1. Notice this is not the same as ignoring the internal grading on s and just c*
*ounting the
algebraic degree. (It is spelled out in [GS1] for example.) From there on, the *
*signs take care
of themselves if we follow the usual conventions, introducing a sign (-1)pqwhen*
*ever a term
of total degree p is pushed past a term of total degree q.
14 Jim Stasheff
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Acknowledgments.The author would like to thank the University of Pennsylvania f*
*or hospitality during his
leave (and many summers) and Lehigh University for a subsequent visiting appoin*
*tment.
Department of Mathematics, University of North Carolina, Chapel Hill, NC 2759*
*9-3250,
USA
E-mail address: jds@math.unc.edu