Odd Scalar Curvature in Anti-Poisson Geometry

Recent works have revealed that the recipe for field-antifield quantization of Lagrangian gauge theories can be considerably relaxed when it comes to choosing a path integral measure \rho if a zero-order term \nu_{\rho} is added to the \Delta operator. The effects of this odd scalar term \nu_{\rho} become relevant at two-loop order. We prove that \nu_{\rho} is essentially the odd scalar curvature of an arbitrary torsion-free connection that is compatible with both the anti-Poisson structure E and the density \rho. This extends a previous result for non-degenerate antisymplectic manifolds to degenerate anti-Poisson manifolds that admit a compatible two-form.


Introduction
The main purpose of this Letter is to report on new geometric insights into the field-antifield formalism. In general, the field-antifield formalism [1,2,3] is a recipe for constructing Feynman rules for Lagrangian field theories with gauge symmetries. The field-antifield formalism is in principle able to handle the most general gauge algebra, i.e. open gauge algebras of reducible type. The input is usually a local relativistic field theory, formulated via a classical action principle in a geometric configuration space. In the field-antifield scheme, the original field variables are extended with various stages of ghosts, antighosts and Lagrange multipliers -all of which are then further extended with corresponding antifields; the gauge symmetries are encoded in a nilpotent Fermionic BRST symmetry [4,5]; and the original action is deformed into a BRST-invariant master action, whose Hessian has the maximal allowed rank. The full quantum master action Here (·, ·) is the antibracket (or anti-Poisson structure), ∆ ρ is the odd Laplacian and ν ρ is an odd scalar, which become relevant in perturbation theory at loop order 0, 1, and 2, respectively. It has only recently been realized that the field-antifield formalism can consistently accommodate a non-zero ν ρ term, thereby providing a more flexible framework for field-antifield quantization [6,7,8].
The classical master equation (1.2) is a generalization of Zinn-Justin's equation [9], which allows to set up consistent renormalization (if the field theory is renormalizable). If the theory is not anomalous at the one-loop level, there will exist a local solution M 1 to the next equation (1.3), and so forth. Although the field-antifield formalism in its basic form is only a formal scheme -i.e. particularly, it assumes that results from finite dimensional analysis are directly applicable to field theory, which has infinitely many degrees of freedom -it has nevertheless been successfully applied to a large variety of physical models. It has mainly been used in a truncated form of the full set of quantum master eqs. are set identically equal to zero. One can for instance mention the AKSZ paradigm [10,11] as a broad example that uses the truncated field-antifield formalism (1.6) to quantize supersymmetric topological field theories [12,13,14,15]. Currently, very few scientific works describe solutions with non-zero M n 's, primarily due to the singular nature of the odd Laplacian ∆ ρ in field theory (again because of the infinitely many degrees of freedom). Nevertheless, it should be fruitful to study generic solutions of the full quantum master equation. See the original paper [1] for an interesting solution with M 1 = 0. Finally, it has in many cases been explicitly checked that the field-antifield formalism produces the same result as the Hamiltonian formulation [16,17,18]. The formalism has also influenced work in closed string field theory [19] and several branches of mathematics. The geometry behind the field-antifield formalism was further clarified in Ref. [20,21,22,23].
In this Letter we shall only explicitly consider the case of finitely many variables. Our main result concerns the odd scalar ν ρ , which is a certain function of the anti-Poisson structure E AB and the density ρ, cf. eq. (6.1) below. It turns out that ν ρ has a geometric interpretation as (minus 1/8 times) the odd scalar curvature R of any connection ∇ that satisfies three conditions; namely that ∇ is 1) anti-Poisson, 2) torsion-free and 3) ρ-compatible. This is a rather robust conclusion as we shall prove in this Letter that it even holds for degenerate antibrackets. (Degenerate anti-Poisson structures appear naturally from for instance the Dirac antibracket construction for antisymplectic second-class constraints [7,21,24,25].) 2 Anti-Poisson structure E AB An anti-Poisson structure is by definition a possibly degenerate (2, 0) tensor field E AB with upper indices that is Grassmann-odd and that satisfies the Jacobi identity In general, an anti-Poisson manifold could have singular points where the rank of E AB jumps, and it is necessary to impose a regularity criterion to proceed. We shall here assume that the anti-Poisson structure E AB admits a compatible two-form field E AB , i.e. that there exists a two-form field E AB with lower indices that is Grassmann-odd and that is compatible with the anti-Poisson structure in the sense that 3) This is a relatively mild requirement, which is always automatically satisfied for a Dirac antibracket on antisymplectic manifolds with antisymplectic second-class constraints [7,21,24,25]. Note that the two-form E AB is neither unique nor necessarily closed. One can define a (1, 1) tensor field as 5) or equivalently, It then follows from either of the compatibility relations (3.3) and (3.4) that P A B is an idempotent The ∆ E Operator An anti-Poisson structure with a compatible two-form field E AB gives rise to a Grassmann-odd, secondorder ∆ E operator that takes semidensities to semidensities. It is defined in arbitrary coordinates as [7] with ρ = 1, and where It is shown in Ref. [7] that the ∆ E operator defined in eq. (4.1) does not depend on the choice of local coordinates, it does not depend on the choice of compatible two-form field E AB , and it does map semidensities into semidensities. Moreover, the Jacobi identity (2.3) precisely ensures that ∆ E is nilpotent Earlier works on the ∆ E operator include Ref. [6,25,26,27,28,29].

The ∆ Operator
Classically, the field-antifield formalism is governed by the anti-Poisson structure E AB , or equivalently, the antibracket Quantum mechanically, the field-antifield recipe instructs one to choose an arbitrary path integral measure ρ, and to use it to build a nilpotent, Grassmann-odd, second-order ∆ operator that takes scalar functions into scalar functions. It is natural to build the ∆ operator by conjugating the ∆ E operator (4.1) with appropriate square roots of the density ρ as follows: In this way the ∆ operator trivially inherits the nilpotency property from the ∆ E operator, In physical applications the nilpotency (5.3) of ∆ is important for the underlying BRST symmetry of the theory. 6 The Odd Scalar ν ρ The odd scalar function ν ρ is defined as where ν (1) , ν (2) , ν (3) , ν (4) , ν (5) are given in eqs. (4.3)-(4.7), and the quantity ν (0) ρ is given as The second-order ∆ operator (5.2) decomposes as where ∆ ρ is the odd Laplacian (4.2). The nilpotency of ∆ implies that The possibility of a non-trivial ν ρ has only recently been observed, cf. Ref. [6,7,8]. In the past, the odd scalar term ν ρ was not present due to a certain compatibility relation between E and ρ, which was unnecessarily imposed, and which (using our new terminology) made ν ρ vanish. In terms of the quantum master equation ∆e

Connection
In the next two Sections 7 and 8 we will briefly state our sign conventions and definitions for the covariant derivative and the curvature in the presence of Fermionic degrees of freedom. A more complete treatment can be found in Ref. [8,30]. Other references include Ref. [31]. Our convention for the left covariant derivative (∇ A X) B of a left vector field X A is [30] ( It is useful to define a reordered Christoffel symbol Γ A BC as A torsion-free connection Γ A BC has the following symmetry in the lower indices: There are in principle two definitions for the divergence divX of a Bosonic vector field X with ε X = 0. The first divergence definition depends on the density ρ while the second definition depends on the connection ∇ The ρ-compatibility condition (7.5) precisely ensures that the two definitions (7.6) and (7.7) coincide, and hence that there is a unique notion of volume [32]. We shall only consider torsion-free connections ∇ that are anti-Poisson and ρ-compatible, i.e. connections that satisfy the above three conditions (7.2), (7.4) and (7.5). Then the odd Laplacian ∆ ρ can be written on a manifestly covariant form

Curvature
The Riemann curvature tensor is (Note that the ordering of indices on the Riemann curvature tensor is slightly non-standard to minimize appearances of sign factors.) The Ricci tensor is

Odd Scalar Curvature
The odd scalar curvature R is defined as the Ricci tensor R AB contracted with the anti-Poisson tensor We now assert that the odd scalar curvature of an arbitrary connection ∇ that is anti-Poisson, torsion-free and ρ-compatible, is equal to (minus eight times) the odd scalar ν ρ . In particular one sees that the odd scalar curvature R carries no information about the connection ∇ used, and it depends only on E and ρ. Equation (9.2) was proven for the non-degenerated case in Ref. [8]. The degenerated case is proven in Appendix A. A Proof of the Main Eq. (9.2) Equation (C.9) in Ref. [8] yields that the odd scalar curvature R can be written as (1) and R I are defined in eqs. (6.2), (4.3) and (A.2), respectively. Since the expression (A.2) below for R I only depends on the torsion-free part of the connection, one does in principle not need the torsion-free condition (7.4) from now on. The heart of the proof consists of the following ten "one-line calculations": so that R III = 1 3 (−ν (3) + ν (4) + 2ν (5) ) . (A.13) Next, R I can be expressed in terms of R III : (2) . (A.14) Inserting eqs. (A.13) and (A.14) into eq. (A.1) yields the main eq. (9.2):