Stable maps of genus zero to flag spaces

We calculate a generating series for the virtual Euler-Poincar\'e characteristics of the spaces of stable maps of genus zero to flag spaces using the summation over trees technique.Comment: 12 pages, AMSTex. Several annoying misprints and errors in formulas are correcte

Manifolds with multiplication on the tangent sheaf

This is a survey of the current state of the theory of $F$--(super)manifolds $(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here $\circ$ is an \Cal{O}_M--bilinear multiplication on the tangent sheaf \Cal{T}_M, satisfying an integrability condition. $F$--manifolds and compatible flat structures on them furnish a useful weakening of Dubrovin's Frobenius structure which naturally arises in the quantum $K$--theory, theory of extended moduli spaces, and unfolding spaces of singularities.Comment: 16 pages. Talk at the Conference dedicated to the memory of B. Segre, Inst. Mat. Guido Castelnuovo, Rome, June 200

Three constructions of Frobenius manifolds: a comparative study

The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma-models), unfolding spaces of singularities (K. Saito's theory, Landau-Ginzburg models), and the recent Barannikov-Kontsevich construction starting with the Dolbeault complex of a Calabi-Yau manifold and conjecturally producing the B--side of the Mirror Conjecture in arbitrary dimension. Each known construction provides the relevant Frobenius manifold with an extra structure which can be thought of as a version of ``non-linear cohomology''. The comparison of thesestructures sheds some light on the general Mirror Problem: establishing isomorphisms between Frobenius manifolds of different classes. Another theme is the study of tensor products of Frobenius manifolds, corresponding respectively to the K\"unneth formula in Quantum Cohomology, direct sum of singularities in Saito's theory, and presumably, the tensor product of the differential Gerstenhaber-Batalin-Vilkovisky algebras. We extend the initial Gepner's construction of mirrors to the context of Frobenius manifolds and formulate the relevant mathematical conjecture.Comment: 46 pages, AMSTe

Iterated Shimura integrals

In this paper I continue the study of iterated integrals of modular forms and noncommutative modular symbols for $\Gamma \subset SL(2,\bold{Z})$ started in [Ma3]. Main new results involve a description of the iterated Shimura cohomology and the image of the iterated Shimura cocycle class inside it. The concluding section of the paper contains a concise review of the classical modular symbols for SL(2) and a discussion of open problems.Comment: 16 page

Modules and Morita theorem for operads

Associative rings A, B are called Morita equivalent when the categories of left modules over them are equivalent. We call two classical linear operads P, Q Morita equivalent if the categories of algebras over them are equivalent. We transport a part of Morita theory to the operadic context by studying modules over operads. As an application of this philosophy, we consider an operadic version of the sheaf of linear differential operators ona a (super) manifold M and give a comparison theorem between algebras over this sheaf on M and M_{red}. The paper is dedicated to A.N.Tyurin on the occasion of his 60th birthday.Comment: Several revisions and corrections are made in this version. Some topics got a more detailed presentation. 30 pp., no figure

New moduli spaces of pointed curves and pencils of flat connections

It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad $(H_*(\bar{M}_{0,n+1}))$ of the moduli spaces of $n$--pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat connections (or solutions to the Commutativity Equations) and homology of a new series $\bar{L}_n$ of pointed stable curves of genus zero. Whereas $\bar{M}_{0,n+1}$ parametrizes trees of $\bold{P}^1$'s with pairwise distinct nonsingular marked points, $\bar{L}_n$ parametrizes strings of $\bold{P}^1$'s stabilized by marked points of two types. The union of all $\bar{L}_n$'s forms a semigroup rather than operad, and the role of operadic algebras is taken over by the representations of the appropriately twisted homology algebra of this union.Comment: 37 pages, AMSTex. Several typos corrected, a reference added, subsection 3.2.2 revised, subsection 3.2.4 adde