Operads and Homotopy Algebra

9971434

Voronov

There are two objectives of this project. One is to work on and

eventually prove the following conjecture of Deligne: There exists the

(natural) structure of an algebra over a chain operad of the little

disks operad on the Hochschild complex of any associative algebra.

The other objective is to study holomorphic line bundles over the

moduli space of algebraic curves with a holomorphic disk and prove the

analogue of Borel-Weil-Bott Theorem for this space, considered as

homogeneous space for the Virasoro algebra. The significance of the

project on Deligne's Conjecture consists in studying presumably deep

connections between one complex variable and associative algebras.

The proof of Deligne's Conjecture will also further simplify the proof

of the existence of a deformation quantization of a Poisson manifold,

originally proved by Kontsevich and simplified by Tamarkin. The

completion of the Borel-Weil-Bott part of the project will establish

important relationship between the geometry of the moduli space and

the representation theory of the Virasoro algebra.

The structure of an algebra over a chain operad indicates, usually

explicitly, the presence of a ``homotopy something'' algebra, an

algebra with certain identities, such as associativity, satisfied in

only an approximate way known as ``up to homotopy.'' Such structures

have been used by Stasheff and May in their study of loop spaces,

Beilinson and Ginzburg and Hinich and Schechtman in the study of

deformation theory of algebraic varieties and vector bundles over

them, and by Kontsevich toward knot invariants. Witten and Zwiebach

have effectively used the homotopy Lie structure in string theory.

Deligne's Conjecture may be reformulated as the existence of a certain

string theory associated to every associative algebra. Thus, the

proof of the conjecture will create ground for considerable progress

in studying applications of algebra to geometry and theoretical

physics. The significance of the moduli space part of the project is

in extending the fundamental relationship between the geometry of a

homogeneous space and the representation theory of a semisimple Lie

group to the case of the moduli space of Riemann surfaces and the

Virasoro algebra, respectively. As in the classical case, the

completion of this project would contribute mutually to the two

subjects: the geometry of the moduli space and the representation

theory of the Virasoro algebra. Both subjects are much more

complicated and less studied than those in the classical situation.

The project aims to advance the general understanding of each. In

general, the study of the geometry and topology of moduli spaces has

recently become very important, as moduli spaces play now a

significant role in four-dimensional topology (Donaldson and

Seiberg-Witten invariants), knot theory (Vassiliev invariants),

higher-dimensional topology (Gromov-Witten invariants), and

deformation quantization (Kontsevich's quantization of Poisson

manifolds). Moreover, although moduli spaces have been classical

objects of algebraic geometry (since the seminal work of P. Deligne

and D. Mumford), their topology has not been understood yet and has

presented a challenge for algebraic geometers for the last twenty

years. Thus unraveling the topology of moduli spaces seems to be very

important for progress in both topology and geometry.

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