838 research outputs found

### 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

### 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

### 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

### 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

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