arXiv subjects

Alexander A. Voronov

Publications and source records attributed to Alexander A. Voronov.

33 records · Page 2Linked to original sources

arXiv2008

Open-closed moduli spaces and related algebraic structures

We set up a Batalin-Vilkovisky Quantum Master Equation for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered J-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory and study the L_\infty and A_\infty algebraic structures associated to it.

math.QA↗

arXiv2007

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 math.AT/0209007. These results are based on the useful notion of a 1/2 PROP introduced by Kontsevich in an e-mail message to the first author.

math.QA↗

arXiv2005

Notes on string topology

This paper is an exposition of the new subject of String Topology. We present an introduction to this exciting new area, as well as a survey of some of the latest developments, and our views about future directions of research. We begin with reviewing the seminal paper of Chas and Sullivan, which started String Topology by introducing a BV-algebra structure on the homology of a loop space of a manifold, then discuss the homotopy theoretic approach to String Topology, using the Thom-Pontrjagin construction, the cacti operad, and fat graphs. We review quantum field theories and indicate how string topology fits into the general picture. Other topics include an open-closed version of string topology, a Morse theoretic interpretation, relation to Gromov-Witten invariants, and "brane'' topology, which deals with sphere spaces. The paper is a joint account of the lecture series given by each of us at the 2003 Summer School on String Topology and Hochschild Homology in Almeria, Spain.

math.GT↗

arXiv2004

Yet another delooping machine

We suggest a new delooping machine, which is based on recognizing an n-fold loop space by a collection of operations acting on it, like the traditional delooping machines of Stasheff, May, Boardman-Vogt, Segal, and Bousfield. Unlike in the traditional delooping machines, which carefully select a nice space of such operations, we consider all natural operations on n-fold loop spaces, resulting in the algebraic theory Map (V_. S^n, V_. S^n). The advantage of this new approach is that the delooping machine is universal in a certain sense, the proof of the recognition principle is more conceptual, works the same way for all values of n, and does not need the test space to be connected.

math.AT↗

arXiv2002

Semi-Infinite Induction and Wakimoto Modules

The purpose of this paper is to suggest the construction and study properties of semi-infinite induction, which relates to semi-infinite cohomology the same way induction relates to homology and coinduction to cohomology. We prove a version of the Shapiro Lemma, relating the semi-infinite cohomology of a module with that of the semi-infinitely induced module. A practical outcome of our construction is a simple construction of Wakimoto modules, highest-weight modules used in double-sided BGG resolutions of irreducible modules.

q-alg↗

arXiv2001

Notes on universal algebra

These are notes of a mini-course given at Dennisfest in June 2001. The goal of these notes is to give a self-contained survey of deformation quantization, operad theory, and graph homology. Some new results related to "String Topology" and cacti are announced in Section 2.7.

math.QA↗

arXiv1998

Cohomology of Conformal Algebras

Conformal algebra is 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, like the Virasoro algebra and loop algebras. In the present paper we develop a cohomology theory of conformal algebras with coefficients in an arbitrary module. It possesses standard properties of cohomology theories; for example, it describes extensions and deformations. We offer explicit computations for most of the important examples.

math.QA↗

arXiv1998

The Swiss-Cheese Operad

We introduce a new operad, which we call the Swiss-cheese operad. It mixes naturally the little disks and the little intervals operads. The Swiss-cheese operad is related to the configuration spaces of points on the upper half-plane and points on the real line, considered by Kontsevich for the sake of deformation quantization. This relation is similar to the relation between the little disks operad and the configuration spaces of points on the plane. The Swiss-cheese operad may also be regarded as a finite-dimensional model of the moduli space of genus-zero Riemann surfaces appearing in the open-closed string theory studied recently by Zwiebach. We describe algebras over the homology of the Swiss-cheese operad.

math.QA↗

arXiv1997

Stability of the Rational Homotopy Type of Moduli Spaces

We show that for g > 2k+2 the k-rational homotopy type of the moduli space M_{g,n} of algebraic curves of genus g with n punctures is independent of g, and the space M_{g,n} is k-formal. This implies the existence of a limiting rational homotopy type of M_{g,n} as g goes to infinity and the formality of it.

alg-geom↗

arXiv1997

Quantizing Poisson Manifolds

This paper extends Kontsevich's ideas on quantizing Poisson manifolds. A new differential is added to the Hodge decomposition of the Hochschild complex, so that it becomes a bicomplex, even more similar to the classical Hodge theory for complex manifolds.

q-alg↗

arXiv1996

Semiquantum Geometry

In this paper we study associative algebras with a Poisson algebra structure on the center acting by derivations on the rest of the algebra. These structures, which we call Poisson fibred algebras, appear in the study of quantum groups at roots of 1 and related algebras, as well as in the representation theory of affine Lie algebras at the critical level. Poisson fibred algebras lead to a generalization of Poisson geometry, which we develop in the paper. We also take up the general study of noncommutative spaces which are close to enough commutative ones so that they contain enough points to have interesting commutative geometry. One of the most striking uses of our noncommutative spaces is the quantum Borel-Weil-Bott Theorem for quantum sl_q (2) at a root of unity, which comes as a calculation of the cohomology of actual sheaves on actual topological spaces.

q-alg↗

arXiv1996

Homotopy Gerstenhaber algebras and topological field theory

We prove that the BRST complex of a topological conformal field theory is a homotopy Gerstenhaber algebra, as conjectured by Lian and Zuckerman in 1992. We also suggest a refinement of the original conjecture for topological vertex operator algebras. We illustrate the usefulness of our main tools, operads and "string vertices" by obtaining new results on Vassiliev invariants of knots and double loop spaces.

q-alg↗

arXiv1995

Homology of moduli spaces of curves and commutative homotopy algebras

We propose an explicit relation between the cohomology of compactified and noncompactified moduli spaces of algebraic curves with punctures. This relationship generalizes one between commutative algebras and Lie algebras proposed by Lazard, Kontsevich and Ginzburg and Kapranov. More explicitly, we show that the cohomology of one moduli space is the graph complex decorated with the cohomology of the other, which in its turn generalizes Getzler's result for the genus zero moduli. We also show that a certain class of algebras over the chain operad of compactified moduli spaces has a natural structure of a commutative homotopy algebra. This may be regarded as a lifting of the dot product of Lian-Zuckerman's BV algebra structure to the cochain level, thus complementing the results of an earlier paper hep-th/9307114 of the authors regarding the BV bracket.

alg-geom↗

arXiv1994

The Cohomology of Algebras over Moduli Spaces

The purpose of this paper is to introduce the cohomology of various algebras over an operad of moduli spaces including the cohomology of conformal field theories (CFT's) and vertex operator algebras (VOA's). This cohomology theory produces a number of invariants of CFT's and VOA's, one of which is the space of their infinitesimal deformations.

hep-th↗

arXiv1994

Topological field theories, string backgrounds and homotopy algebras

String backgrounds are described as purely geometric objects related to moduli spaces of Riemann surfaces, in the spirit of Segal's definition of a conformal field theory. Relations with conformal field theory, topological field theory and topological gravity are studied. For each field theory, an algebraic counterpart, the (homotopy) algebra satisfied by the tree level correlators, is constructed.

hep-th↗