Homological reduction of constrained Poisson algebras

Abstract

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 symplectic structure to constraints [D]. The relation between symmetry and constraints is particularly tight in the case Dirac calls "first class". The present paper is concerned entirely with 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 central to rational homotopy theory.Homological methods have become increasingly important in mathematical physics, especially field theory, over the last decade. In regard to constrained Hamiltonians, they came into focus with Henneaux's Report [H] on the work of Batalin, Fradkin and Vilkovisky [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 able 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-necessarily-firstclass 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 at 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 improvements, both factual and stylistic. While early revision was in progress, Kimura sent me a copy of [Ki] which has also had a significant influence on the present exposition, as has his continued interaction while with me at UNC as an NSF Post-Doc.

Invariance of the BFV complex

“…The BFV complex was introduced by Batalin, Fradkin and Vilkovisky, with applications to physics in mind [Batalin and Fradkin 1983;Batalin and Vilkovisky 1977]. Later on, Stasheff [1997] gave an interpretation of the BFV complex in terms of homological algebra. The construction we present below is explained with more details in [Schätz 2009a].…”

“…The connection to homological algebra was made explicit in [Stasheff 1997] later on. We focus on the smooth setting; that is, we want to consider arbitrary coisotropic submanifolds of smooth finite-dimensional Poisson manifolds.…”

“…If one leaves the embedding fixed and only changes the connection and the BFV charge, one simply obtains two isomorphic differential graded Poisson algebras; see Theorem 3.4. Note that the dependence on the choice of BFV charge was already well understood; see [Stasheff 1997], for instance. Dependence on the embedding is more subtle.…”

Graded Poisson Algebras

“…For further details on this procedure, and in particular for the construction of D, we refer to [2,13,22,39] and references therein. Observe that some authors refer to this method as BVF [Batalin-Vilkovisky-Fradkin] and reserve the name BRST to the case when the g i 's are the components of an equivariant moment map.…”

Hamiltonian BRST interactions in Abelian theories

“…The reformulation of the Lagrangian BRST symmetry [1,2,3,4,5] on cohomological grounds allowed, among others, the study of consistent interactions that can be introduced among fields with gauge freedom without changing the number of gauge symmetries [6,7,8,9,10] with the help of the deformation of the master equation [11] in the framework of the local BRST cohomology [11,12,13,14,15,16]. This Lagrangian cohomological deformation technique has been successfully applied to many models of interest, like Chern-Simons models, Yang-Mills theories, the Chapline-Manton model, p-forms and chiral p-forms, Einstein's gravity theory, four-and elevendimensional supergravity, or BF models [11], [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32].…”