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matrices and monodromy of the quantum cohomology of projective spaces
 Comm. in Math. Physics 207
, 1999
"... 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 singula ..."
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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
Special Functions of the Isomonodromy Type
 Acta Appl. Math
, 2000
"... We discuss relations which exist between analytic functions belonging to the recently introduced class of special functions of the isomonodromy type (SFITs). These relations can be obtained by application of some simple transformations to auxiliary ODEs with respect to a spectral parameter which ass ..."
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Cited by 16 (4 self)
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We discuss relations which exist between analytic functions belonging to the recently introduced class of special functions of the isomonodromy type (SFITs). These relations can be obtained by application of some simple transformations to auxiliary ODEs with respect to a spectral parameter which associated with each SFIT. We consider two applications of rational transformations of the spectral parameter in the theory of SFITs. One of the most striking applications which is considered here is an explicit construction of algebraic solutions of the sixth Painlevé equation. 2000 Mathematics Subject Classification: 34M55, 33E17 1
On a Poisson structure on the space of Stokes matrices
 Internat. Math. Res. Notices 1999
"... 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 ..."
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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
On almost duality for Frobenius manifolds
, 2004
"... We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theo ..."
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Cited by 12 (1 self)
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We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theory of Coxeter groups and Shephard groups, from the Seiberg Witten duality.
Stability conditions and Stokes factors
, 2008
"... 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 a ..."
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Cited by 11 (0 self)
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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.
HYPERKÄHLER MANIFOLDS AND NONABELIAN HODGE THEORY OF (IRREGULAR) CURVES
"... Abstract. Text of talk given at the Institut Henri Poincare ́ January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of arXiv:1111.6228 related to wild character varieties and irregular mappi ..."
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Abstract. Text of talk given at the Institut Henri Poincare ́ January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of arXiv:1111.6228 related to wild character varieties and irregular mapping class groups. 1. Big picture Lets start by recalling the usual picture for nonabelian Hodge theory on curves, due to Hitchin, Donaldson, Corlette and Simpson [29, 22, 21, 49] Fix an integer n and let G = GLn(C). Let Σ be a smooth compact complex algebraic curve. Given this data one may consider the nonabelian cohomology space M = H1(Σ, G). Ignoring stability conditions for the moment (until the next section), this space is naturally a hyperkähler manifold, and there are three viewpoints on it: 1) (Dolbeault) as the moduli space MDol of Higgs bundles, consisting of pairs (E,Φ) with E → Σ a rank n degree zero holomorphic vector bundle and Φ ∈ Γ(End(E) ⊗ Ω1) a Higgs field, 2) (De Rham) as a moduli spaceMDR of connections on rank n holomorphic vector bundles, consisting of pairs (V,∇) with ∇ : V → V ⊗ Ω1 a holomorphic connection, and 3) (Betti) as the space MB = Hom(pi1(Σ), G)/G of conjugacy classes of representation of the fundamental group of Σ. This gives three different algebraic structures on the same underlying space M (since Σ is compact, by GAGA, the holomorphic objects above are in fact algebraic, and have algebraic moduli spaces). MDR and MB are complex analytically isomorphic via the Riemann–Hilbert correspondence, taking a connection to its monodromy representation. MDR and MDol are naturally diffeomorphic as real manifolds via the nonabelian Hodge correspondence, but are not complex analytically isomorphic. Thus there is more than one natural complex structure on M; they form part of the family of complex structures making M into a hyperkähler manifold. To get an idea of this first consider the abelian case n = 1, so G = C∗. Then one finds: 1 ha l0
Matching procedure for the sixth Painlevé equation, the Preprint RIMS1541
, 2006
"... In the context of the isomonodromy deformation method, we present a constructive procedure (a matching procedure) to obtain the critical behavior of Painlevé VI transcendents and solve the connection problem. This procedure yields two and one parameter families of solutions, including logarithmic be ..."
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In the context of the isomonodromy deformation method, we present a constructive procedure (a matching procedure) to obtain the critical behavior of Painlevé VI transcendents and solve the connection problem. This procedure yields two and one parameter families of solutions, including logarithmic behaviors, and three classes of solutions with Taylor expansion at a critical point. 1
On the computation of Stokes multipliers via hyperasymptotics, Surikaisekikenkyusho Kokyuroku 1088
, 1999
"... Abstract. In this paper we explain how the hyperasymptotic expansion of late terms in divergent asymptotic expansions can be used to compute all the Stokes multipliers to arbitrary precision. 1. ..."
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Abstract. In this paper we explain how the hyperasymptotic expansion of late terms in divergent asymptotic expansions can be used to compute all the Stokes multipliers to arbitrary precision. 1.