On Connection between Topological Landau-Ginzburg Gravity and Integrable Systems

We study flows on the space of topological Landau-Ginzburg theories coupled to topological gravity. We argue that flows corresponding to gravitational descendants change the target space from a complex plane to a punctured complex plane and lead to the motion of punctures.It is shown that the evolution of the topological theory due to these flows is given by dispersionless limit of KP hierarchy. We argue that the generating function of correlators in such theories are equal to the logarithm of the tau-function of Generalized Kontsevich Model.Comment: 17 p. late

Two-loop Euler-Heisenberg effective actions from charged open strings

We present the multiloop partition function of open bosonic string theory in the presence of a constant gauge field strength, and discuss its low-energy limit. The result is written in terms of twisted determinants and differentials on higher-genus Riemann surfaces, for which we provide an explicit representation in the Schottky parametrization. In the field theory limit, we recover from the string formula the two-loop Euler-Heisenberg effective action for adjoint scalars minimally coupled to the background gauge field.Comment: 32 pages, 3 eps figures, plain LaTeX. References added, minor changes to the text. Published version, affiliation correcte

On Integrable Systems and Supersymmetric Gauge Theories

The properties of the N=2 SUSY gauge theories underlying the Seiberg-Witten hypothesis are discussed. The main ingredients of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential are presented, the invariant sense of these definitions is illustrated. Recently found exact nonperturbative solutions to N=2 SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to integrable systems are discussed.Comment: LaTeX, 38 pages, no figures; based on the lecture given at INTAS School on Advances in Quantum Field Theory and Statistical Mechanics, Como, Italy, 1996; minor changes, few references adde

Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A

We are interested in the structure of the crystal graph of level $l$ Fock spaces representations of $\mathcal{U}_q (\widehat{\mathfrak{sl}_e})$. Since the work of Shan [26], we know that this graph encodes the modular branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called "canonical" crystal isomorphism turns out to be expressible only in terms of: - Schensted's classic bumping procedure, - the cyclage isomorphism defined in [13], - a new crystal isomorphism, easy to describe, acting on cylindric multipartitions. We explain how this can be seen as an analogue of the bumping algorithm for affine type $A$. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space

M-branes and N=2 Strings

The string field theory of N=(2,1) heterotic strings describes a set of self-dual Yang-Mills fields coupled to self-dual gravity in 2+2 dimensions. We show that the exact classical action for this field theory is a certain complexification of the Green-Schwarz/Dirac-Born-Infeld string action, closely related to the four dimensional Wess-Zumino action describing self-dual gauge fields. This action describes the world-volume of a 2+2d ``M-brane'', which gives rise upon different null reductions to critical strings and membranes. We discuss a number of further properties of N=2 heterotic strings, such as the geometry of null reduction, general features of a covariant formulation, and possible relations to BPS and GKM algebras.Comment: 49 pages, harvmac; 1 figure (uses epsf.tex). References adde