Operads and Homotopy Algebra

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9971434Voronov There are two objectives of this project. One is to work on andeventually prove the following conjecture of Deligne: There exists the(natural) structure of an algebra over a chain operad of the littledisks operad on the Hochschild complex of any associative algebra.The other objective is to study holomorphic line bundles over themoduli space of algebraic curves with a holomorphic disk and prove theanalogue of Borel-Weil-Bott Theorem for this space, considered ashomogeneous space for the Virasoro algebra. The significance of theproject on Deligne's Conjecture consists in studying presumably deepconnections between one complex variable and associative algebras.The proof of Deligne's Conjecture will also further simplify the proofof the existence of a deformation quantization of a Poisson manifold,originally proved by Kontsevich and simplified by Tamarkin. Thecompletion of the Borel-Weil-Bott part of the project will establishimportant relationship between the geometry of the moduli space andthe representation theory of the Virasoro algebra. The structure of an algebra over a chain operad indicates, usuallyexplicitly, the presence of a ``homotopy something'' algebra, analgebra with certain identities, such as associativity, satisfied inonly an approximate way known as ``up to homotopy.'' Such structureshave been used by Stasheff and May in their study of loop spaces,Beilinson and Ginzburg and Hinich and Schechtman in the study ofdeformation theory of algebraic varieties and vector bundles overthem, and by Kontsevich toward knot invariants. Witten and Zwiebachhave effectively used the homotopy Lie structure in string theory.Deligne's Conjecture may be reformulated as the existence of a certainstring theory associated to every associative algebra. Thus, theproof of the conjecture will create ground for considerable progressin studying applications of algebra to geometry and theoreticalphysics. The significance of the moduli space part of the project isin extending the fundamental relationship between the geometry of ahomogeneous space and the representation theory of a semisimple Liegroup to the case of the moduli space of Riemann surfaces and theVirasoro algebra, respectively. As in the classical case, thecompletion of this project would contribute mutually to the twosubjects: the geometry of the moduli space and the representationtheory of the Virasoro algebra. Both subjects are much morecomplicated and less studied than those in the classical situation.The project aims to advance the general understanding of each. Ingeneral, the study of the geometry and topology of moduli spaces hasrecently become very important, as moduli spaces play now asignificant role in four-dimensional topology (Donaldson andSeiberg-Witten invariants), knot theory (Vassiliev invariants),higher-dimensional topology (Gromov-Witten invariants), anddeformation quantization (Kontsevich's quantization of Poissonmanifolds). Moreover, although moduli spaces have been classicalobjects of algebraic geometry (since the seminal work of P. Deligneand D. Mumford), their topology has not been understood yet and haspresented a challenge for algebraic geometers for the last twentyyears. Thus unraveling the topology of moduli spaces seems to be veryimportant for progress in both topology and geometry.***