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 homology of the pointed double loop space of $X$, $H_*(Ω^2 X)$. Let $G$ be a topological monoid with a homotopy inverse. Suppose that $G$ acts on $M$. We define a structure of Batalin-Vilkovisky algebra on $H_*(Ω^2BG)\otimes\mathbb{H}_*(M)$ extending the Batalin-Vilkovisky algebra of Getzler on $H_*(Ω^2BG)$. We prove that the morphism of graded algebras $$H_*(Ω^2BG)\otimes\mathbb{H}_*(M)\to\mathbb{H}_*(LM)$$ defined by Felix and Thomas \cite{Felix-Thomas:monsefls}, is in fact a morphism of Batalin-Vilkovisky algebras. In particular, if $G=M$ is a connected compact Lie group, we compute the Batalin-Vilkovisky algebra $\mathbb{H}_*(LG;\mathbb{Q})$.
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. A notion of de Rham chains, which is a certain hybrid of the notions of singular chains and differential forms, is a key ingredient in our construction. Combined with a generalization of cyclic Deligne's conjecture, this dg operad produces a chain model of the free loop space w…
The main result of this paper is to calculate the Batalin-Vilkovisky structure of $HH^*(C^*(\mathbf{K}P^n;R);C^*(\mathbf{K}P^n;R))$ for $ \mathbf{K}=\mathbb{C}$ and $\mathbb{H}$, and $R=\mathbb{Z}$ and any field; and shows that in the special case when $M=\mathbb{C}P^1=S^2$, and $R=\mathbb{Z}$, this structure can not be identified with the BV-structure of $\mathbb{H}_*(LS^2;\mathbb{Z})$ computed by Luc Memichi in \cite{menichi2}. However, the induced Gerstenhaber structures are still identified in this case. Moreover, according to a recent work of Y.Felix and J.Thomas \cite{felix--thomas}, the…
We show that a model of chain complex of the free loop space of a $C^\infty$-manifold, which is proposed in arxiv:1404.0153, admits an action of a certain dg operad. This is a chain level structure under the Chas-Sullivan BV structure on loop space homology. Our dg operad is a variant of the cacti operad, and we introduce combinatorial objects called "decorated cacti" to define it. We also define a chain level Gerstenhaber structure on Hochschild cochains of any differential graded algebra. Applied to the dga of differential forms, this structure is compatible with our chain level structure in…
We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed oriented manifold, and a gravity coalgebra structure on the reduced equivariant homology.
We show that the Connes-Moscovici cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree -2. More generally, we show that a "cyclic operad with multiplication" is a cocyclic module whose cohomology is a Batalin-Vilkovisky algebra and whose cyclic cohomology is a graded Lie algebra of degree -2. This explain why the Hochschild cohomology algebra of a symmetric algebra is a Batalin-Vilkovisky algebra.
We show that a graded commutative algebra A with any square zero odd differential operator is a natural generalization of a Batalin-Vilkovisky algebra. While such an operator of order 2 defines a Gerstenhaber (Lie) algebra structure on A, an operator of an order higher than 2 (Koszul-Akman definition) leads to the structure of a strongly homotopy Lie algebra (L$_\infty$-algebra) on A. This allows us to give a definition of a Batalin-Vilkovisky algebra up to homotopy. We also make a conjecture which is a generalization of the formality theorem of Kontsevich to the Batalin-Vilkovisky algebra lev…
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 $\mathbb{H}_*(LS^2;\mathbb{F}_2)$ and the Hochschild cohomology $HH^{*}(H^*(S^2);H^*(S^2))$ are surprisingly not isomorphic as Batalin-Vilkovisky algebras, although we prove that, as expected, the underlying Gerstenhaber algebras are isomorphic. The proof requires the knowledge of…
Let $M$ be any compact simply-connected $d$-dimensional smooth manifold and let $\mathbb{F}$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$, $HH^*(S^*(M);S^*(M))$, extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjecturated to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$, $H_{*+d}(LM)$ introduced by Chas and Sullivan. We also show that the negative cyclic cohomology $HC^*_-(S^*(M))$ has a Lie bracket. Such Lie bracket is e…
We determine the Batalin-Vilkovisky Lie algebra structure for the integral loop homology of special unitary groups and complex Stiefel manifolds. It is shown to coincide with the Poisson algebra structure associated to a certain odd symplectic form on a super vector space for which loop homology is the super algebra of functions. Over rationals, the loop homology of the above spaces splits into a tensor product of simple BV algebras, and it is shown to contain a super Lie algebra.
Let $A$ be a Koszul Calabi-Yau algebra. We show that there exists an isomorphism of Batalin-Vilkovisky algebras between the Hochschild cohomology ring of $A$ and that of its Koszul dual algebra $A^!$. This confirms (a generalization of) a conjecture of R.~Rouquier.
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 the homology of G, the homology of the space of based loops on G, and the homology suspension. The result is applied to compute the Batalin-Vilkovisky algebra associated to the special orthogonal groups SO(n) with coefficients in the rational numbers and in the integers modulo tw…
Using the machinery of the Batalin-Vilkovisky formalism, we construct cohomology classes on compactifications of the moduli space of Riemann surfaces from the data of a contractible differential graded Frobenius algebra. We describe how evaluating these cohomology classes upon a well-known construction producing homology classes in the moduli space can be expressed in terms of the Feynman diagram expansion of some functional integral. By computing these integrals for specific examples, we are able to demonstrate that this construction produces families of nontrivial classes.
Let X be a topological space. The homology of the iterated loop space $H_*Ω^n X$ is an algebra over the homology of the framed n-disks operad $H_*f\mathcal{D}_n$ \cite{Getzler:BVAlg,Salvatore-Wahl:FrameddoBVa}. We determine completely this $H_*f\mathcal{D}_n$-algebra structure on $H_*(Ω^n X;\mathbb{Q})$. We show that the action of $H_*(SO(n))$ on the iterated loop space $H_*Ω^n X$ is related to the J-homomorphism and that the BV-operator vanishes on spherical classes only in characteristic other than 2.
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 is isomorphicto the Hochschild cohomology $HH^*(H\_*(G),H\_*(G))$. Over $\mathbb{F}\_2$, such isomorphism of Batalin-Vilkovisky algebrasdoes not hold when $G=SO(3)$ or $G=G\_2$.
The goal of this article is to develop BV (Batalin-Vilkovisky) formalism in the $p$-adic Dwork theory. Based on this formalism, we explicitly construct a $p$-adic dGBV algebra (differential Gerstenhaber-Batalin-Vilkovisky algebra) for a smooth projective complete intersection variety $X$ over a finite field, whose cohomology gives the $p$-adic Dwork cohomology of $X$, and its cochain endomorphism (the $p$-adic Dwork Frobenius operator) which encodes the information of the zeta function $X$. As a consequence, we give a modern deformation theoretic interpretation of Dwork's theory of the zeta fu…
We study linear Batalin-Vilkovisky (BV) quantization, which is a derived and shifted version of the Weyl quantization of symplectic vector spaces. Using a variety of homotopical machinery, we implement this construction as a symmetric monoidal functor of $\infty$-categories. We also show that this construction has a number of pleasant properties: It has a natural extension to derived algebraic geometry, it can be fed into the higher Morita category of $E_n$-algebras to produce a "higher BV quantization" functor, and when restricted to formal moduli problems, it behaves like a determinant. Alon…
In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin-Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalised Lie derivative and a generalised (cyclic) cap product that obey a Cartan-Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Ho…
We show under what conditions the complex computing general Ext-groups carries the structure of a cyclic operad such that Ext becomes a Batalin-Vilkovisky algebra. This is achieved by transferring cyclic cohomology theories for the dual of a (left) Hopf algebroid to the complex in question, which asks for the notion of contramodules introduced along with comodules by Eilenberg-Moore half a century ago. Another crucial ingredient is an explicit formula for the inverse of the Hopf-Galois map on the dual, by which we illustrate recent categorical results and answer a long-standing open question. …
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 defind 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 Poincare-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution …
This is a survey of `Cohomological Physics', a phrase that first appeared in the context of anomalies in gauge theory. Differential forms were implicit in physics at least as far back as Gauss (1833) (cf. his electro-magnetic definition of the linking number), and more visibly in Dirac's magnetic monopole (1931). The magnetic charge was given by the first Chern number. Thus were characteristic classes (and by implication the cohomology of Lie algebras and of Lie groups) introduced into physics. The `ghosts' introduced by Fade'ev and Popov were incorporated into what came to be known as BRST co…
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-algebr…
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 is due to Cohen, Jones and Yan. In the course of the proof we establish several new general results. These include a description of how symmetries of a manifold can be used to understand its string topology, and a relationship between characteristic classes and circle actions on s…
Let $M$ be a path-connected closed oriented $d$-dimensional smooth manifold and let ${\Bbbk}$ be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of $M$, $H_{*+d}(LM)$ is a Batalin-Vilkovisky algebra. Let $G$ be a topological group such that $M$ is a classifying space of $G$. Denote by $S_*(G)$ the (normalized) singular chains on $G$. Suppose that $G$ is discrete or path-connected. We show that there is a Van Den Bergh type isomorphism $$ HH^{-p}(S_*(G),S_*(G))\cong HH_{p+d}(S_*(G),S_*(G)). $$ Therefore, the Gerstenhaber algebra $HH^{*}(S_*(G),S_*(G))$ is a …
Using five basic principles we treat Gerstenhaber/Lie brackets, BV operators and Master equations appearing in mathematical and physical contexts in a unified way. The different contexts for this are given by the different types of (Feynman) graphs that underlie the particular situation. Two of the maxims we bring forth are (1) that extending to the non-connected graphs gives a commutative multiplication forming a part of the BV structure and (2) that there is a universal odd twist that unifies and explains seemingly ad hoc choices of signs, and is responsible for the BV operator being a diffe…
We study the homology structure of spaces having an action of the \emph{framed disks} operad. In particular, we compute the relations between Kudo-Araki operations and generalized Batalin-Vilkovisky operators. As an application, we complete the computations of the modulo 2 homology of $Ω^2 S^3$ as a Batalin-Vilkovisky algebra, and give some evidence for a classical conjecture about the modulo 2 Hurewicz homomorphism of the infinite loop space of the sphere spectrum.
Let $H$ be a Hopf algebra with a modular pair in involution $(\Character,1)$. Let $A$ be a (module) algebra over $H$ equipped with a non-degenerated $\Character$-invariant $1$-trace $τ$. We show that Connes-Moscovici characteristic map $φ_τ:HC^*_{(\Character,1)}(H)\rightarrow HC^*_λ(A)$ is a morphism of graded Lie algebras. We also have a morphism $Φ$ of Batalin-Vilkovisky algebras from the cotorsion product of $H$, $\text{Cotor}_H^*({\Bbbk},{\Bbbk})$, to the Hochschild cohomology of $A$, $HH^*(A,A)$. Let $K$ be both a Hopf algebra and a symmetric Frobenius algebra. Suppose that the square of …
