BV Structure Research Articles

Poincaré duality for smooth Poisson algebras and BV structure on Poisson cohomology

Similar to the modular vector fields in Poisson geometry, modular derivations are defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson module with the modular derivation, the Poisson cochain complex with values in any Poisson module is proved to be isomorphic to the Poisson chain complex with values in the corresponding twisted Poisson module. Then a version of twisted Poincaré duality is proved between the Poisson homologies and cohomologies. Furthermore, a notion of pseudo-unimodular Poisson structure is defined. It is proved that the Poisson cohomology as a Gerstenhaber algebra admits a Batalin-Vilkovisky operator inherited from some one of its Poisson cochain complex if and only if the Poisson structure is pseudo-unimodular. This generalizes the geometric version due to P. Xu. The modular derivation and Batalin-Vilkovisky operator are also described by using the dual basis of the Kähler differential module.

BV Structure on Hochschild Cohomology of Quantum Exterior Algebra with Two Variables

Let [Formula: see text] over a field [Formula: see text]. We give a clear characterization of the Batalin-Vilkovisky algebraic structure on Hochschild cohomology of [Formula: see text] for any [Formula: see text], and the Gerstenhaber algebraic structure on Hochschild cohomology of [Formula: see text] for [Formula: see text].

A detailed look on actions on Hochschild complexes especially the degree 1 coproduct and actions on loop spaces

We explain our previous results about Hochschild actions (2007/2008) pertaining, in particular, to the coproduct, which appeared in a different form in a work by Goresky and Hingston (2009), and provide a fresh look at the results. We recall the general action, specialize to the aforementioned coproduct and prove that the assumption of commutativity, made for convenience in a previous article (2008), is not needed. We give detailed background material on loop spaces, Hochschild complexes and dualizations, and discuss details and extensions of these techniques which work for all operations given in two previous articles (2007/2008). With respect to loop spaces, we show that the coproduct is well defined modulo constant loops and going one step further that in the case of a graded Gorenstein Frobenius algebra, the coproduct is well defined on the reduced normalized Hochschild complex. We discuss several other aspects such as “time reversal” duality and several homotopies of operations induced by it. This provides a cohomology operation which is a homotopy of the anti-symmetrization of the coproduct. The obstruction again vanishes on the reduced normalized Hochschild complex if the Frobenius algebra is graded Gorenstein. Further structures such as “animation”, the BV structure, a coloring for operations on chains and cochains, and a Gerstenhaber double bracket are briefly treated.

On the BV structure of the Hochschild cohomology of finite group algebras

We give a simple algebraic recipe for calculating the components of the BV operator $\Delta$ on the Hochschild cohomology of a finite group algebra with respect to the centraliser decomposition. We use this to investigate the properties of $\Delta$ and to make some computations for some particular finite groups.

BV structure on the Hochschild cohomology of Sullivan algebras

Abstract Let X be a closed, simply connected manifold of dimension m and LX the space of free loops on X . If (∧ V, d ) is the minimal Sullivan model of X where V is finite dimensional, then there is a Gerstenhaber algebra ( ∧ V ⊗ ∧ s − 1 V # , d 0 ) , where V # is the graded dual of V , and its homology is isomorphic to the loop space homology H * ( L X ) . In this paper we define a BV structure on ( ∧ V ⊗ ∧ s − 1 V # , d 0 ) which extends the Gerstenhaber bracket.

On the Hochschild homology of open Frobenius algebras

We prove that the shifted Hochschild homology HH_*(A,A)[m] of a symmetric open Frobenius algebra A of degree m has a natural coBV-algebra structure which is defined at the chain level. As a consequence HH^*(A,A^\vee)[-m] is a BV algebra. The underlying coalgebra and algebra structure may not be resp. counital and unital. We also introduce a natural BV-algebra structure on HH_*(A,A)[m] which is also defined at the chain level. Hence there is a BV-structure on HH_*(A,A)[m] . Moreover we prove that the product and coproduct on HH_*(A,A)[m] satisfy the Frobenius compatibility condition i.e. HH_*(A,A)[m] is an open Frobenius algebras. If A is commutative, we also introduce a natural BV structure on the shifted relative Hochschild homology \widetilde{HH}_*(A)[m-1] . We conjecture that the product of this BV structure is identical to the Goresky–Hingston [10] product on the cohomology of free loop spaces when A is a commutative cochain algebra model for M .

BV differential on Hochschild cohomology of Frobenius algebras

For a finite-dimensional Frobenius k-algebra R with a Nakayama automorphism ν, we define an algebra HH⁎(R)ν↑. If the order of ν is not divisible by the characteristic of k, this algebra is isomorphic to the Hochschild cohomology algebra of R. We prove that this algebra is a BV algebra. Moreover, we construct a BV differential on it in a canonical way. We use this fact to calculate the Gerstenhaber algebra structure and a BV structure on the Hochschild cohomology algebras of a family of self-injective algebras of tree type Dn.

BV structure on Hochschild cohomology of the group ring of quaternion group of order eight in characteristic two

Let k be an algebraically closed field of characteristic two and let Q8 be the quaternion group of order 8. We determine the Gerstenhaber Lie algebra structure and the Batalin–Vilkovisky structure on the Hochschild cohomology ring of the group algebra kQ8.

CyclicA∞structures and Deligne’s conjecture

First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic $A_\infty$ algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)-operad of CW complexes whose constituent spaces form a homotopy associative version of the Cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyclic $A_\infty$ version of Deligne's conjecture. This chain model contains the minimal operad of Kontsevich and Soibelman as a suboperad and restriction of the action to this suboperad recovers their results in the unframed case. Additionally this proof recovers the work of Kaufmann in the case of a strict Frobenius algebra. We then extend our results to cyclic $A_\infty$ categories, with an eye toward the homotopy BV structure present on the Hochschild cochains of the Fukaya category of a suitable symplectic manifold.

Infinity structure of Poincaré duality spaces

We show that the complex C X of rational simplicial chains on a compact and triangulated Poincare duality space X of dimension d is an A1 coalgebra with1 duality. This is the structure required for an A1 version of the cyclic Deligne conjecture. One corollary is that the shifted Hochschild cohomology HH C d .C X;C X/ of the cochain algebra C X with values in C X has a BV structure. This implies, if X is moreover simply connected, that the shifted homology HC dLX of the free loop space admits a BV structure. An appendix by Dennis Sullivan gives a general local construction of1 structures.

Arc operads and arc algebras

Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identified with open subsets of a compactification due to Penner of a space closely related to Riemann's moduli space. Algebras over these operads are shown to be Batalin-Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy, and it similarly acts on the loop space of any topological space. New operad structures on the circle are classified and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.

A homotopy theoretic realization of string topology

Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In [2] Chas and Sullivan defined a product on the homology H * (LM) of degree -d. They then investigated other structure that this product induces, including a Batalin -Vilkovisky structure, and a Lie algebra structure on the S1 equivariant homology H * S 1 (LM). These algebraic structures, as well as others, came under the general heading of the ”string topology” of M. In this paper we will describe a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We also show that an operad action on the homology of the loop space discovered by Voronov has a homotopy theoretic realization on the level of Thom spectra. This is the ” cactus operad” defined in [6] which is equivalent to operad of framed disks in . This operad action realizes the Chas - Sullivan BV structure on H * (LM). We then describe a cosimplicial model of this ring spectrum, and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology, HH * (C * (M); C * (M)).

Boundary superstring field theory

Using the Batalin–Vilkovisky formalism we provide a detailed analysis of the NS sector of boundary superstring field theory. We construct explicitly the relevant BV structure and derive the master action. Furthermore, we show that this action is exactly equal to the superdisk worldsheet partition function as was recently conjectured.

On a Crystalline Variational Problem, Part II:¶BV Regularity and Structure of Minimizers on Facets

For a nonsmooth positively one-homogeneous convex function φ:ℝ n → [0,+∞[, it is possible to introduce the class ?φ (ℝ n ) of smooth boundaries with respect to φ, to define their φ-mean curvature κφ, and to prove that, for E∈?φ (ℝ n ), κφ∈L ∞(δE) [9]. Based on these results, we continue the analysis on the structure of δE and on the regularity properties of κφ. We prove that a facet F of δE is Lipschitz (up to negligible sets) and that κφ has bounded variation on F. Further properties of the jump set of κφ are inspected: in particular, in three space dimensions, we relate the sublevel sets of κφ on F to the geometry of the Wulff shape ?φ≔{φ≤ 1 }.