Alexander A. Voronov

Homotopy G -algebras and moduli space operad

This paper emphasizes the ubiquitous role of mod- uli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy G- (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complexX of a Kahler manifold X, thereby yielding the most general type of homotopy G-algebra structure onX. This latter statement is based on a typical construction of supersymmet- ric sigma-model, the construction of Gromov-Witten invariants in Kontsevichs version.

On operad structures of moduli spaces and string theory

We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and Batalin-Vilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory.

Homotopy Gerstenhaber algebras

The purpose of this paper is to complete Getzler-Jones’ proof of Deligne’s Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the B ∞-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: One of them is a B ∞-algebra, another, called a homotopy G-algebra, is a particular case of a B ∞-algebra, the others, a G ∞-algebra, an Ē1-algebra, and a weak G ∞-algebra, arise from the geometry of configuration spaces. Corrections to the paper of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made.

Cohomology of Conformal Algebras

Abstract:The notion of a conformal algebra encodes an axiomatic description of the operator product expansion of chiral fields in conformal field theory. On the other hand, it is an adequate tool for the study of infinite-dimensional Lie algebras satisfying the locality property. The main examples of such Lie algebras are those “based” on the punctured complex plane, such as the Virasoro algebra and loop Lie algebras. In the present paper we develop a cohomology theory of conformal algebras with coefficients in an arbitrary module. It possesses standards properties of cohomology theories; for example, it describes extensions and deformations. We offer explicit computations for the most important examples. To Bertram Kostant on his seventieth birthday

Geometry of superconformal manifolds

The main facts about complex curves are generalized to superconformal manifolds. The results thus obtained are relevant to the fermion string theory and, in particular, they are useful for computation of determinants of super laplacians which enter the string partition function.

Cohomology and deformation of Leibniz pairs

Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebraA together with a Lie algebraL mapped into the derivations ofA. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential.

Semi-infinite homological algebra.

SummaryThe paper provides a homological algebraic foundation for semi-infinite cohomology. It is proved that semi-infinite cohomology of infinite dimensional Lie algebras is a two-sided derived functor of a functor that is intermediate between the functors of invariants and coinvariants. The theory of two-sided derived functors is developed. A family of modules including a module generalizing the universal enveloping algebra appropriate to the setting of two sided derived functors is introduced. A vanishing theorem for such modules is proved.

PROPped-Up Graph Cohomology

We consider graph complexes with a flow and compute their cohomology. More specifically, we prove that for a PROP generated by a Koszul dioperad, the corresponding graph complex gives a minimal model of the PROP. We also give another proof of the existence of a minimal model of the bialgebra PROP from [14]. These results are based on the useful notion of a \(\frac{1}{2}\)PROP introduced by Kontsevich in [9].

On Kontsevich's Hochschild cohomology conjecture

Let an n-algebra mean an algebra over the chain complex of the little n-cubes operad. We give a proof of Kontsevichs conjecture, which states that for a suitable notion of Hochschild cohomology in the category of n-algebras, the Hochschild cohomology complex of an n-algebra is an (n+1)-algebra. This generalizes a conjecture by Deligne for n=1, now proven by several authors.

Elements of supergeometry

This paper is devoted to an exposition of the structure theory of supermanifolds and bundles on them, a description of Serre duality on supermanifolds, investigation of inverse sheaves, definition of characteristic classes and proof of the Grothendieck-Riemann-Roch theorem for supermanifolds.