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In this paper we compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. This problem can be formulated in a “classical ” framework, as the problem of computation of Stokes matrices and monodromy of (systems of) differential equations with regular and irregular singularities. We prove that the Stokes ’ matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups. 1

Abstract: In this paper we study the map associating to a linear differential operator with rational coefficients its monodromy data. The operator is of the form Λ(z) = d V dz − U − z, with one regular and one irregular singularity of Poincaré rank 1, where U is a diagonal and V is a skewsymmetric n × n matrix. We compute the Poisson structure of the corresponding Monodromy Preserving Deformation Equations (MPDE) on the space of the monodromy data. Preprint SISSA 120/98/FM Monodromy preserving deformation equations (MPDE) of linear differential operators with rational coefficients are known since the beginning of the century [Fu, Schl, G]. Particularly, the famous six Painlevé equations are known [G] to be of this type. MPDE were included in the framework of the general theory of integrable systems much later, at the end of 70s [ARS, FN1, JMU]; see also [IN]). Many authors were

It is now twenty years since Jimbo, Miwa and Ueno [23] generalised Schlesinger’s equations (governing isomonodromic deformations of logarithmic connections on vector bundles over the Riemann sphere) to the case of connections with arbitrary order poles. An interesting feature was that new deformation parameters arose: one may vary the ‘irregular

We study D-branes in a nonsupersymmetric orbifold of type C 2 /Γ, perturbed by a tachyon condensate, using a gauged linear sigma model. The RG flow has both higgs and coulomb branches, and each branch supports different branes. The coulomb branch branes account for the “brane drain ” from the higgs branch, but their precise relation to fractional branes has hitherto been unknown. Building on the results of hep-th/0403016 we construct, in detail, the map between fractional branes and the coulomb/higgs branch branes for two examples in the type 0 theory. This map depends on the phase of the tachyon condensate in a surprising and intricate way. In the mirror Landau-Ginzburg picture the dependence on the tachyon phase is manifested by discontinuous changes in the shape of the D-brane. July 20, 2005 1. Introduction and

The JMMS equations are studied using the geometry of the spectral curve of a pair of dual systems. It is shown that the equations can be represented as time-independent Hamiltonian flows on a Jacobian bundle. 1

Abstract. The gauge theory approach to the geometric Langlands program is extended to the case of wild ramification. The new ingredients that are required, relative to the tamely ramified case, are differential operators with irregular singularities, Stokes phenomena, isomonodromic deformation, and, from a physical point of view, new surface operators associated with higher order singularities. 1.

Abstract. Let G be a complex algebraic group and ∇ a meromorphic connection on the trivial G–bundle over P 1 with a pole of order 2 at zero and a pole of order 1 at infinity. We give explicit formulae involving multilogarithms for the map taking the residue of ∇ at zero to the corresponding Stokes factors and for the Taylor series of the inverse map. We show moreover that, when G is the Ringel–Hall group of the category A of modules over a complex, finite–dimensional algebra, this Taylor series coincides with the holomorphic generating function for counting invariants in A recently constructed by D. Joyce [21]. This allows us to interpret Joyce’s construction as one of an isomonodromic family of irregular connections on P 1 parametrised by the space of stability conditions of A. 1.

canonical transformations

The purpose of this paper is to point out and then draw some consequences of the fact that the Poisson Lie group G ∗ dual to G = GLn(C) may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity