Geometry of Batalin-Vilkovisky quantization

Abstract

The present paper is devoted to the study of geometry of Batalin-Vilkovisky quantization procedure. The main mathematical objects under consideration are P-manifolds and SP-manifolds (supermanifolds provided with an odd symplectic structure and, in the case of SP-manifolds, with a volume element). The Batalin-Vilkovisky procedure leads to consideration of integrals of the superharmonic functions over Lagrangian submanifolds. The choice of Lagrangian submanifold can be interpreted as a choice of gauge condition; Batalin and Vilkovisky proved that in some sense their procedure is gauge independent. We prove much more general theorem of the same kind. This theorem leads to a conjecture that one can modify the quantization procedure in such a way as to avoid the use of the notion of Lagrangian submanifold. In the next paper we will show that this is really so at least in the semiclassical approximation. Namely the physical quantities can be expressed as integrals over some set of critical points of solution S to the master equation with the integrand expressed in terms of Reidemeister torsion. This leads to a simplification of quantization procedure and to the possibility to get rigorous results also in the infinite-dimensional case. The present paper contains also a compete classification of P-manifolds and SP-manifolds. The classification is interesting by itself, but in this paper it plays also a role of an important tool in the proof of other results.Comment: 13 page

Graded Chern-Simons field theory and graded topological D-branes

“…is a 'total BPZ form' (total topological metric). The action (2.25) is invariant with respect to infinitesimal gauge transformations of the form: 27) where…”

Linear odd Poisson bracket on Grassmann variables

“…It is a well-known fact that, in contrast with the even Poisson bracket, the nondegenerate odd Poisson bracket has one Grassmann-odd nilpotent differential ∆-operator of the second order, in terms of which the main equation has been formulated in the Batalin-Vilkovisky scheme [5,6,7,8,9,10] for the quantization of gauge theories in the Lagrangian approach. In a formulation of Hamiltonian dynamics by means of the odd Poisson bracket with the help of a Grassmann-odd HamiltonianH (g(H) = 1) [11,12,13,14,15,16,17,18,19,20,21,22] this ∆-operator plays also a very important role being used to distinguish the non-dissipative dynamical systems, for which ∆H = 0, from the dissipative ones [1], for which the Grassmann-odd Hamiltonian satisfies the condition ∆H = 0.…”

The category of Z2n-supermanifolds

“…Of course, a theory of superfields defined on Z n 2 -supermanifolds requires a theory of Z n 2 -superintegration, which is still in process of construction, but σ-models with values in our 'superization' ΠS, for a Z n 2 -graded vector bundle S concentrated in nonzero degrees, can be directly constructed [Sch93]. In particular, a 'higher' Minkowski space can be defined as the color Lie group M associated with the color Lie algebra L = V × ΠS * , where V is the central part corresponding to the vector part of the Minkowski spaceM and the color Lie bracket is determined by a 'color-antisymmetric' morphism Γ : ΠS ⊗ ΠS → V .…”