Connected components of the moduli spaces of Abelian differentials with prescribed singularities

Consider the moduli space of pairs (C,w) where C is a smooth compact complex curve of a given genus and w is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.Comment: 42 pages, 12 figures, LaTe

A remark on the matrix Airy function

An integral representation for matrix Airy function is presentedComment: 4 page

An algebra of deformation quantization for star-exponentials on complex symplectic manifolds

The cotangent bundle $T^*X$ to a complex manifold $X$ is classically endowed with the sheaf of \cor-algebras \W[T^*X] of deformation quantization, where \cor\eqdot \W[\rmptt] is a subfield of \C[[\hbar,\opb{\hbar}]. Here, we construct a new sheaf of \cor-algebras \TW[T^*X] which contains \W[T^*X] as a subalgebra and an extra central parameter $t$. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If $P$ is any section of order zero of \W[T^*X], we show that \exp(t\opb{\hbar} P) is well defined in \TW[T^*X].Comment: Latex file, 24 page

Poisson actions up to homotopy and their quantization

Symmetries of Poisson manifolds are in general quantized just to symmetries up to homotopy of the quantized algebra of functions. It is therefore interesting to study symmetries up to homotopy of Poisson manifolds. We notice that they are equivalent to Poisson principal bundles and describe their quantization to symmetries up to homotopy of the quantized algebras of functions.Comment: 8 page

Quantum Cohomology of a Product

The operation of tensor product of Cohomological Field Theories (or algebras over genus zero moduli operad) introduced in an earlier paper by the authors is described in full detail, and the proof of a theorem on additive relations between strata classes is given. This operation is a version of the Kuenneth formula for quantum cohomology. In addition, rank one CohFT's are studied, and a generalization of Zograf's formula for Weil-Petersson volumes is suggested.Comment: AMSTex file, 30 pages. Figures (hard copies) available from Yu. Manin. Main paper by M. Kontsevich, Yu. Manin, appendix by R. Kaufman

On the variational noncommutative Poisson geometry

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.Comment: Proc. Int. workshop SQS'11 `Supersymmetry and Quantum Symmetries' (July 18-23, 2011; JINR Dubna, Russia), 4 page

The Geometry of the Master Equation and Topological Quantum Field Theory

In Batalin-Vilkovisky formalism a classical mechanical system is specified by means of a solution to the {\sl classical master equation}. Geometrically such a solution can be considered as a $QP$-manifold, i.e. a super\m equipped with an odd vector field $Q$ obeying $\{Q,Q\}=0$ and with $Q$-invariant odd symplectic structure. We study geometry of $QP$-manifolds. In particular, we describe some construction of $QP$-manifolds and prove a classification theorem (under certain conditions). We apply these geometric constructions to obtain in natural way the action functionals of two-dimensional topological sigma-models and to show that the Chern-Simons theory in BV-formalism arises as a sigma-model with target space $\Pi {\cal G}$. (Here ${\cal G}$ stands for a Lie algebra and $\Pi$ denotes parity inversion.)Comment: 29 pages, Plain TeX, minor modifications in English are made by Jim Stasheff, some misprints are corrected, acknowledgements and references adde