Batalin-Vilkovisky Research Articles

The BV Formalization of Chern—Simons Theory on Deformed Superspace

In this paper we will study non-abelian Chern—Simons theory on a deformed superspace. We will deform the superspace in such a way that it includes the noncommutativity between bosonic and fermionic coordinates. We will first analyse the BRST and the anti-BRST symmetries of the Chern—Simons theory on this deformed superspace. Then we will analyse the extended BRST and the extended anti-BRST symmetries of this theory in the Batalin—Vilkovisky (BV) formalism. Finally, we will express these extended BRST and extended anti-BRST symmetries in extended superspace formalism by introducing new Grassmann coordinates.

Batalin-Vilkovisky Formalism in Perturbative Algebraic Quantum Field Theory

On the basis of a thorough discussion of the Batalin-Vilkovisky formalism for classical field theory presented in our previous publication, we construct in this paper the Batalin-Vilkovisky complex in perturbatively renormalized quantum field theory. The crucial technical ingredient is a proof that the renormalized time-ordered product is equivalent to the pointwise product of classical field theory. The renormalized Batalin-Vilkovisky algebra is then the classical algebra but written in terms of the time-ordered product, together with an operator which replaces the ill defined graded Laplacian of the unrenormalized theory. We identify it with the anomaly term of the anomalous Master Ward Identity of Brennecke and D\"utsch. Contrary to other approaches we do not refer to the path integral formalism and do not need to use regularizations in intermediate steps.

Geometry of differential operators of second order, the algebra of densities, and groupoids

In our previous works, we introduced, for each (super)manifold, a commutative algebra of densities. It is endowed with a natural invariant scalar product. In this paper, we study geometry of differential operators of second order on this algebra. In the more conventional language they correspond to certain operator pencils. We consider the self-adjoint operators and analyze the operator pencils that pass through a given operator acting on densities of a particular weight. There are ‘singular values’ for pencil parameters. They are related with interesting geometric picture. In particular, we obtain operators that depend on certain equivalence classes of connections (instead of connections as such). We study the corresponding groupoids. From this point of view we analyze two examples: the canonical Laplacian on an odd symplectic supermanifold appearing in Batalin–Vilkovisky geometry and the Sturm–Liouville operator on the line, related with classical constructions of projective geometry. We also consider the canonical second order semi-density arising on odd symplectic supermanifolds, which has some similarity with mean curvature of surfaces in Riemannian geometry.

Deformed Super-Yang-Mills in Batalin-Vilkovisky Formalism

In this paper we will analyse a three dimensional super-Yang-Mills theory on a deformed superspace with boundaries. We show that it is possible to obtain an undeformed theory on the boundary if the bulk superspace is deformed by imposing a non-vanishing commutator between bosonic and fermionic coordinates. We will also analyse this theory in the Batalin-Vilkovisky (BV) formalism and show that these results also hold at a quantum level.

BV formulation of higher form gauge theories in a superspace

We discuss the extended BRST and anti-BRST symmetry (including shift symmetry) in the Batalin-Vilkovisky (BV) formulation for two and three form gauge theories. Further we develop the superspace formulation for the BV actions for these theories. We show that the extended BRST invariant BV action for these theories can be written manifestly covariant manner in a superspace with one Grassmann coordinate. On the hand a superspace with two Grassmann coordinates are required for a manifestly covariant formulation of the extended BRST and extended anti-BRST invariant BV actions for higher form gauge theories.

On open-closed extension of boundary string field theory

We investigate a classical open-closed string field theory whose open string sector is given by boundary string field theory. The open-closed interaction is introduced by the overlap of a boundary state with a closed string field. With the help of the Batalin-Vilkovisky formalism, the closed string sector is determined to be the HIKKO closed string field theory. We also discuss the gauge invariance of this theory in both open and closed string sides.

Batalin-Vilkovisky Formalism in the Functional Approach to Classical Field Theory

We develop the Batalin-Vilkovisky formalism for classical field theory on generic globally hyperbolic spacetimes. A crucial aspect of our treatment is the incorporation of the principle of local covariance which amounts to formulate the theory without reference to a distinguished spacetime. In particular, this allows a homological construction of the Poisson algebra of observables in classical gravity. Our methods heavily rely on the differential geometry of configuration spaces of classical fields.

The minimal model for the Batalin–Vilkovisky operad

The purpose of this paper is to explain and to generalize, in a homotopical way, the result of Barannikov–Kontsevich and Manin, which states that the underlying homology groups of some Batalin–Vilkovisky algebras carry a Frobenius manifold structure. To this extent, we first make the minimal model for the operad encoding BV-algebras explicit. Then, we prove a homotopy transfer theorem for the associated notion of homotopy BV-algebra. The final result provides an extension of the action of the homology of the Deligne–Mumford–Knudsen moduli space of genus 0 curves on the homology of some BV-algebras to an action via higher homotopical operations organized by the cohomology of the open moduli space of genus zero curves. Applications in Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry conjecture are given.

SOFT BREAKING OF BRST SYMMETRY AND GAUGE DEPENDENCE

We continue investigation of soft breaking of BRST symmetry in the Batalin–Vilkovisky (BV) formalism beyond regularizations like dimensional ones used in our previous paper [JHEP 1110, 043 (2011)]. We generalize a definition of soft breaking of BRST symmetry valid for general gauge theories and arbitrary gauge fixing. The gauge dependence of generating functionals of Green's functions is investigated. It is proved that such introduction of a soft breaking of BRST symmetry into gauge theories leads to inconsistency of the conventional BV formalism.

NONCOMMUTATIVITY AND NON-ANTICOMMUTATIVITY PERTURBATIVE QUANTUM GRAVITY

In this paper, we will study perturbative quantum gravity on supermanifolds with both noncommutativity and non-anticommutativity of spacetime coordinates. We shall first analyze the BRST and the anti-BRST symmetries of this theory. Then we will also analyze the effect of shifting all the fields of this theory in background field method. We will construct a Lagrangian density which apart from being invariant under the extended BRST transformations is also invariant under on-shell extended anti-BRST transformations. This will be done by using the Batalin–Vilkovisky (BV) formalism. Finally, we will show that the sum of the gauge-fixing term and the ghost term for this theory can be elegantly written down in superspace with a two Grassmann parameter.

Boundary Coupling of Lie Algebroid Poisson Sigma Models and Representations up to Homotopy

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 considerations lead to a conjectural description of topological D-branes on generalized complex manifolds, which includes A-branes and B-branes as special cases.

Open superstring field theory I: gauge fixing, ghost structure, and propagator

The WZW form of open superstring field theory has linearized gauge invariances associated with the BRST operator Q and the zero mode eta_0 of the picture minus-one fermionic superconformal ghost. We discuss gauge fixing of the free theory in a simple class of gauges using the Faddeev-Popov method. We find that the world-sheet ghost number of ghost and antighost string fields ranges over all integers, except one, and at any fixed ghost number, only a finite number of picture numbers appear. We calculate the propagators in a variety of gauges and determine the field-antifield content and the free master action in the Batalin-Vilkovisky formalism. Unlike the case of bosonic string field theory, the resulting master action is not simply related to the original gauge-invariant action by relaxing the constraint on the ghost and picture numbers.

Gauge Fixing of Modified Cubic Open Superstring Field Theory

The gauge-fixing problem in the modified cubic open superstring field theory is discussed in detail for both the Ramond and Neveu-Schwarz (NS) sectors in the Batalin-Vilkovisky (BV) framework. We prove for the first time that the same form of action as the classical gauge-invariant one with the ghost-number constraint on the string field relaxed, gives the master action satisfying the BV master equation. This is achieved by identifying independent component fields by analyzing the kernel structure of the inverse picture-changing operator. The explicit gauge-fixing conditions for the component fields are discussed. In a kind of b0 = 0 gauge, we explicitly obtain an NS propagator that has poles at zeros of the Virasoro operator L0.

Homotopy Batalin–Vilkovisky algebras

This paper provides an explicit cofibrant resolution of the operad encoding Batalin–Vilkovisky algebras. Thus it defines the notion of homotopy Batalin–Vilkovisky algebras with the required homotopy properties. To define this resolution, we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin–Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincaré–Birkhoff–Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad \mathcal{BV} . We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian–Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

Loop homology of spheres and complex projective spaces

Abstract. In his Inventiones paper, Ziller (1977) computed the integral homology as a graded abelian group of the free loop space of compact, globally symmetric spaces of rank 1. Chas and Sullivan (1999) showed that the homology of the free loop space of a compact closed orientable manifold can be equipped with a loop product and a BV-operator making it a Batalin–Vilkovisky algebra. Cohen, Jones and Yan (2004) developed a spectral sequence which converges to the loop homology as a spectral sequence of algebras. They computed the algebra structure of the loop homology of spheres and complex projective spaces by using Ziller's results and the method of Brown–Shih (1959, 1962). In this note we compute the loop homology algebra by using only spectral sequences and the technique of universal examples. We therefore not only obtain Ziller's and Brown–Shih's results in an elementary way, but we also replace the roundabout computations of Cohen, Jones and Yan (2004) making them independent of Ziller's and Brown–Shih's work. Moreover we offer an elementary technique which we expect can easily be generalized and applied to a wider family of spaces, not only the globally symmetric ones.

An algebraic chain model of string topology

A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes non simply-connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology.

Knot invariants and new weight systems from general 3D TFTs

We introduce and study the Wilson loops in general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in the Chern–Simons theory. We study the TFTs within the Batalin–Vilkovisky (BV) and the Alexandrov–Kontsevich–Schwarz–Zaboronsky (AKSZ) framework, and the Ward identities of these theories imply that the expectation value of the Wilson loop is a pairing of two dual constructions of (co)cycles of certain extended graph complex (extended from Kontsevich’s graph complex to accommodate the Wilson loop). We also prove that there is an isomorphism between the same complex and certain extended Chevalley–Eilenberg complex of Hamiltonian vector fields. This isomorphism allows us to generalize the Lie algebra weight system for knots to weight systems associated with any homological vector field and its representations. As an example we construct knot invariants using holomorphic vector bundle over hyperKähler manifolds.

Perturbative quantum gravity on complex spacetime

In this Letter we will study non-anticommutative perturbative quantum gravity on spacetime with a complex metric. After analysing the BRST symmetry of this non-anticommutative perturbative quantum gravity, we will also analyse the effect of shifting all the fields. We will construct a Lagrangian density which is invariant under the original BRST transformations and the shift transformations in the Batalin–Vilkovisky (BV) formalism. Finally, we will show that the sum of the gauge-fixing term and the ghost term for this shift symmetry invariant Lagrangian density can be elegantly written down in superspace with a single Grassmann parameter.

String topology of classifying spaces

Let $G$ be a finite group or a compact connected Lie group and let $BG$ be its classifying space. Let $\mathcal{L}BG:=map(S^1,BG)$ be the free loop space of $BG$ i.e. the space of continuous maps from the circle $S^1$ to $BG$. The purpose of this paper is to study the singular homology $H_*(\mathcal LBG)$ of this loop space. We prove that when taken with coefficients in a field the homology of $\mathcal LBG$ is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology $H^*(\mathcal LBG)$. We also prove an algebraic version of this result by showing that the Hochschild cohomology $HH^*(S_* (G),S_*(G))$ of the singular chains of $G$ is a Batalin-Vilkovisky algebra.

Local BRST cohomologyin(non-)Lagrangian field theory

Some general theorems are established on the local BRST cohomology for not necessarily Lagrangian gauge theories. Particular attention is given to the BRST groups with direct physical interpretation. Among other things, the groups of rigid symmetries and conservation laws are shown to be still connected, though less tightly than in the Lagrangian theory. The connection is provided by the elements of another local BRST cohomology group whose elements are identified with Lagrange structures. This extends the cohomological formulation of the Noether theorem beyond the scope of Lagrangian dynamics. We show that each integrable Lagrange structure gives rise to a Lie bracket in the space of conservation laws, which generalizes the Dickey bracket of conserved currents known in Lagrangian field theory. We study the issues of existence and uniqueness of the local BRST complex associated with a given set of field equations endowed with a compatible Lagrange structure. Contrary to the usual BV formalism, such a complex does not always exist for non-Lagrangian dynamics, and when exists it is by no means unique. The ambiguity and obstructions are controlled by certain cohomology classes, which are all explicitly identified.