arXiv research

Derives localization formulas in Batalin-Vilkovisky formalism.

We explore homotopies in quantum field theory formalism.

The action of Batalin-Vilkovisky Delta-operator on semidensities in an odd symplectic superspace is defined. This is used for the construction of integral invariants on surfaces embedded in an odd symplectic superspace and for more clear interpretation of the Batalin-Vilkovisky formalism geometry.

We give a conceptual formulation of Kontsevich's `dual construction' producing graph cohomology classes from a differential graded Frobenius algebra with an odd scalar product. Our construction -- whilst equivalent to the original one -- is combinatorics-free and is based on the Batalin-Vilkovisky formalism, from which…

In the Batalin-Vilkovisky formalism, gauge conditions are expressed as Lagrangian submanifolds in the space of fields and antifields. We discuss a way of patching together gauge conditions over different parts of the space of fields, and apply this method to extend the light-cone gauge for the superparticle to a conic …

Given a smooth family of massless free fermions parametrized by a base manifold $B$, we show that the (mathematically rigorous) Batalin-Vilkovisky quantization of the observables of this family gives rise to the determinant line bundle for the corresponding family of Dirac operators.

A quantum field theory for Spin(7)-instantons derived from moduli spaces.

The geometry of supermanifolds provided with $Q$-structure (i.e. with odd vector field $Q$ satisfying $\{ Q,Q\} =0$), $P$-structure (odd symplectic structure ) and $S$-structure (volume element) or with various combinations of these structures is studied. The results are applied to the analysis of Batalin-Vilkovisky ap…

This paper analyzes in details the Batalin-Vilkovisky quantization procedure for BF theories on n-dimensional manifolds and describes a suitable superformalism to deal with the master equation and the search of observables. In particular, generalized Wilson loops for BF theories with additional polynomial B-interaction…

In this paper we define Grassmann odd analogues of Jacobi structures on supermanifolds. We then examine their potential use in the Batalin-Vilkovisky formalism of classical gauge theories.

This is a paper about geometry of (iterated) variations. We explain why no sources of divergence are built into the Batalin-Vilkovisky (BV) Laplacian, whence there is no need to postulate any ad hoc conventions such as "$δ(0)=0$" and "$\logδ(0)=0$" within BV-approach to quantisation of gauge systems. Remarkably, the ge…

We consider semidensities on a supermanifold E with an odd symplectic structure. We define a new $Δ$-operator action on semidensities as the proper framework for Batalin-Vilkovisky formalism. We establish relations between semidensities on E and differential forms on Lagrangian surfaces. We apply these results to Batal…

Supergeneralization of $\DC P(N)$ provided by even and odd Kählerian structures from Hamiltonian reduction are construct.Operator $Δ$ which used in Batalin-- Vilkovisky quantization formalism and mechanics which are bi-Hamiltonian under corresponding even and odd Poisson brackets are considered.

Using technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so called quantum master equation satisfying certain boundary conditions are proven to be in 1-1 correspondence with representations of …

This paper shows hypercommutative algebras on Calabi-Yau manifolds are formal.

We analyze geometry of the second order differential operators, having in mind applications to Batalin--Vilkovisky formalism in quantum field theory. As we show, an exhaustive picture can be obtained by considering pencils of differential operators acting on densities of all weights simultaneously. The algebra of densi…

The Schouten bracket (or antibracket) plays a central role in the Poisson formalism and the Batalin-Vilkovisky quantization of gauge systems. There are several (in)equivalent ways to realize this concept on jet spaces. In this paper, we compare the definitions, examining in what ways they agree or disagree and how they…

This is a survey of our program of perturbative quantization of gauge theories on manifolds with boundary compatible with cutting/pasting and with gauge symmetry treated by means of a cohomological resolution (Batalin-Vilkovisky) formalism. We also give two explicit quantum examples -- abelian BF theory and the Poisson…

Let $M$ be a compact oriented $d$-dimensional smooth manifold and $X$ a topological space. Chas and Sullivan \cite{Chas-Sullivan:stringtop} have defined a structure of Batalin-Vilkovisky algebra on $\mathbb{H}_*(LM):=H_{*+d}(LM)$. Getzler \cite{Getzler:BVAlg} has defined a structure of Batalin-Vilkovisky algebra on the…

We study a formulation of the standard Poisson sigma model in which the target space Poisson manifold carries the Hamilton action of some finite dimensional Lie algebra. We show that the structure of the action and the properties of the gauge invariant observables can be understood in terms of the associated target spa…

We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin--Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and di…

Study of Batalin-Vilkovisky algebra on Poisson manifolds with diagonalizable modular symmetry.

A general form for the boundary coupling of a Lie algebroid Poisson sigma model is proposed. The approach involves using the Batalin-Vilkovisky formalism in the AKSZ geometrical version, to write a BRST-invariant coupling for a representation up to homotopy of the target Lie algebroid or its subalgebroids. These consid…

Introduces a new geometric framework for non-perturbative BV-theory.

Let $M$ be a compact oriented $d$-dimensional smooth manifold. Chas and Sullivan have defined a structure of Batalin-Vilkovisky algebra on $\mathbb{H}_*(LM)$. Extending work of Cohen, Jones and Yan, we compute this Batalin-Vilkovisky algebra structure when $M$ is a sphere $S^d$, $d\geq 1$. In particular, we show that $…

This paper reinterprets Khovanov-Sano symmetries using BV formalism.

In 1999 Chas and Sullivan showed that the homology of the free loop space of an oriented manifold admits the structure of a Batalin-Vilkovisky algebra. In this paper we give a direct description of this Batalin-Vilkovisky algebra in the case that the manifold is a compact Lie group G. Our answer is phrased in terms of …

We investigate the perturbative aspects of Rozansky-Witten's 3d $σ$-model using Costello's approach to the Batalin-Vilkovisky (BV) formalism. We show that the BV quantization (in Costello's sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of reg…

The primitive cohomology of Calabi-Yau intersections is described using a twisted de Rham complex.

This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analyses of a large class of TFTs, inspired by the Batalin-Vilkovisky formalism of gauge theories. We begin by considering a simple two-dimensional t…

The string bracket introduced by Chas and Sullivan [math.GT/9911159] is reinterpreted from the point of view of topological field theories in the Batalin-Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern-Simons and BF theories) are considered together with genera…

We extend the notion of a Thomas projective connection (a projective equivalence class of linear connections) for supermanifolds. As a by-product, we arrive at a generalisation of the multidimensional Schwarzian derivative for the super case which was previously unknown. This is combined with our previous construction …

For almost any compact connected Lie group $G$ and any field $\mathbb{F}\_p$, we compute the Batalin-Vilkoviskyalgebra $H^{*+\text{dim }G}(LBG;\mathbb{F}\_p)$ on the loop cohomology of the classifying space introduced byChataur and the second author.In particular, if $p$ is odd or $p=0$, this Batalin-Vilkovisky algebra…

For a Lie-Rinehart algebra (A,L) such that, as an A-module, L is finitely generated and projective of finite constant rank, the relationship between generators of the Gerstenhaber bracket and connections on the highest A-exterior power of L given in an earlier paper arises from the canonical pairing between the exterio…

Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the …

We solve the following problem: to describe in geometric terms all differential operators of the second order with a given principal symbol. Initially the operators act on scalar functions. Operator pencils acting on densities of arbitrary weights appear naturally in the course of study. We show that for the algebra of…

Hitchin's generalized complex geometry has been shown to be relevant in compactifications of superstring theory with fluxes and is expected to lead to a deeper understanding of mirror symmetry. Gualtieri's notion of generalized complex submanifold seems to be a natural candidate for the description of branes in this co…

In 1999 Chas and Sullivan showed that the homology of the free loop space of an oriented manifold admits the structure of a Batalin-Vilkovisky algebra. In this paper we give a complete description of this Batalin-Vilkovisky algebra for complex projective spaces. This builds on a description of the ring structure that i…

Introduces a new operator generating higher Koszul brackets on differential forms.

We present a sigma model field theoretic realization of Hitchin's generalized complex geometry, which recently has been shown to be relevant in compactifications of superstring theory with fluxes. Hitchin sigma model is closely related to the well known Poisson sigma model, of which it has the same field content. The c…

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base…

We construct a differential Gerstenhaber-Batalin-Vilkovisky algebra from Dolbeault complex of any close Kaehler manifold, and a Frobenius manifold structure on Dolbeault cohomology.

The aim of this paper is to define a chain level refinement of the Batalin-Vilkovisky (BV) algebra structure on the homology of the free loop space of a closed, oriented $C^\infty$-manifold. For this purpose, we define a (nonsymmetric) cyclic dg operad which consists of "de Rham chains" of free loops with marked points…

The geometric approach [1312.1262] to iterated variations of local functionals -- e.g., of the (master-)action functional -- resulted in an extension of the deformation quantisation technique to the set-up of Poisson models of field theory [IHES/M/15/13]. It also allowed of a rigorous proof ([1312.1262],[1210.0726]) fo…

A new mathematical approach to general covariance using stacks and Lie algebras.

We define a jet-space analog of the BV-Laplacian, avoiding delta-functions and infinite constants; instead we show that the main properties of the BV-Laplacian and its relation to the Schouten bracket originate from the underlying jet-space geometry.

Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.

Construct algorithms for Frobenius manifolds and residue pairings on Calabi-Yau varieties.