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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 15 (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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Cited by 10 (0 self)
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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
"... on the occasion of his 65th birthday. Abstract. 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 ..."
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Cited by 5 (1 self)
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on the occasion of his 65th birthday. Abstract. 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. 1.
The Klein solution to Painleve’s sixth equation
"... Abstract. We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL2(C) out of triples of generators of threedimensional complex refl ..."
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Cited by 4 (0 self)
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Abstract. We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL2(C) out of triples of generators of threedimensional complex reflection groups. (This involves the Fourier– Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painlevé VI solutions. (In particular this solves a Riemann–Hilbert problem explicitly.) Each step will be illustrated using the complex reflection group associated to Klein’s simple group of order 168. This leads to a new algebraic solution with seven branches. We will also prove that, unlike the algebraic solutions of Dubrovin–Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of SL2(C). Klein’s quartic curve
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 2 (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.
THE KLEIN SOLUTION TO PAINLEVÉ’S SIXTH EQUATION
"... Abstract. We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL2(C) out of triples of generators of threedimensional complex refl ..."
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Abstract. We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL2(C) out of triples of generators of threedimensional complex reflection groups. (This involves the Fourier– Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painlevé VI solutions. (In particular this solves a Riemann–Hilbert problem explicitly.) Each step will be illustrated using the complex reflection group associated to Klein’s simple group of order 168. This leads to a new algebraic solution with seven branches. We will also prove that, unlike the algebraic solutions of Dubrovin–Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of SL2(C). Klein’s quartic curve
FROM KLEIN TO PAINLEVÉ
, 2004
"... Abstract. We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation, generalising that of Dubrovin–Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL2(C) out of triples of gene ..."
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Abstract. We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation, generalising that of Dubrovin–Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL2(C) out of triples of generators of threedimensional complex reflection groups. (This involves the Fourier–Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painlevé VI solutions. (In particular this solves a Riemann–Hilbert problem explicitly.) Each step will be illustrated using the complex reflection group associated to Klein’s simple group of order 168. This leads to a new algebraic solution with seven branches. We will also prove that, unlike the algebraic solutions of Dubrovin–Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of SL2(C). The results of this paper also yield a simple proof of a recent theorem of Inaba– Iwasaki–Saito on the action of Okamoto’s affine D4 symmetry group as well as the correct connection formulae for generic Painlevé VI equations. Klein’s quartic curve
∂tα
, 2003
"... Invariant of the hypergeometric group associated to the quantum cohomology of the projective space. ..."
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Invariant of the hypergeometric group associated to the quantum cohomology of the projective space.