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17
P.: Wild nonabelian Hodge theory on curves
 Compos. Math
, 2004
"... Abstract On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. Moduli spaces of these objects are obtained with fixed generic polar parts at each singularity, ..."
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Abstract On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. Moduli spaces of these objects are obtained with fixed generic polar parts at each singularity, which amounts to fixing a coadjoint orbit of the group GL r (C[z]/z n ). We prove that they carry complete hyperKähler metrics.
A RIGID IRREGULAR CONNECTION ON THE PROJECTIVE LINE
, 2009
"... In this paper we construct a connection ∇ on the trivial Gbundle on P 1 for any simple complex algebraic group G, which is regular outside of the points 0 and ∞, has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point ∞, with slop ..."
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In this paper we construct a connection ∇ on the trivial Gbundle on P 1 for any simple complex algebraic group G, which is regular outside of the points 0 and ∞, has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point ∞, with slope 1/h, the reciprocal of the Coxeter number of G. The connection ∇, which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic 0 counterpart of a hypothetical family of ℓadic representations, which should parametrize a specific automorphic representation under the global Langlands correspondence. These ℓadic representations, and their characteristic 0 counterparts, have been constructed in some cases by Deligne and Katz. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. It provides an example of the geometric Langlands correspondence with wild ramification. We compute the de Rham cohomology of our connection with values in a representation V of G, and describe the differential Galois group of ∇ as a subgroup of G.
QUASICOXETER ALGEBRAS, DYNKIN DIAGRAM COHOMOLOGY AND QUANTUM WEYL GROUPS
, 2005
"... ... independently De Concini (unpublished), conjectured that the monodromy of the Casimir connection ∇C introduced in [MTL] is described by Lusztig’s quantum Weyl group operators. This conjecture was proved in [TL1] for all representations of the Lie algebra g = sln and in [TL2] for a number of pai ..."
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... independently De Concini (unpublished), conjectured that the monodromy of the Casimir connection ∇C introduced in [MTL] is described by Lusztig’s quantum Weyl group operators. This conjecture was proved in [TL1] for all representations of the Lie algebra g = sln and in [TL2] for a number of pairs (g, V) including vector and spin representations of classical Lie algebras and the adjoint representation of all complex, simple Lie algebras. The aim of this paper, and of its sequel [TL4] is to prove this conjecture for all g. Our strategy is inspired by Drinfeld’s proof of the equivalence of the monodromy of the Knizhnik–Zamolodchikov equations for g and the R–matrix representations coming from the quantum group U�g. It relies on the use of quasi–Coxeter algebras, which are to the generalised braid group of type g what Drinfeld’s quasitriangular quasibialgebras are to Artin’s braid groups Bn. Using this notion, and the associated deformation cohomology, which we call Dynkin diagram
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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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.
GEOMETRY OF MULTIPLICATIVE PREPROJECTIVE ALGEBRA
, 2007
"... CrawleyBoevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called the multiplicative preprojective algebra. In this paper we study the moduli space of (semi)stable representations of such an algebra (the multiplicative quiver vari ..."
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CrawleyBoevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called the multiplicative preprojective algebra. In this paper we study the moduli space of (semi)stable representations of such an algebra (the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there exists a complex analytic isomorphism between the nilpotent subvariety of the quiver variety and that of the multiplicative quiver variety (which can be extended to a symplectomorphism between these tubular neighborhoods). We also show that when the quiver is starshaped, the multiplicative quiver variety parametrizes Simpson’s (poly)stable filtered local systems on a punctured Riemann sphere with prescribed filtration type, weight and associated graded local
Gbundles, Isomonodromy and Quantum Weyl Groups
 Int. Math. Res. Not
"... 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 deformatio ..."
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Cited by 8 (2 self)
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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
QuasiHamiltonian Geometry of Meromorphic Connections
, 2002
"... For each connected complex reductive group G, we find a family of new examples of complex quasiHamiltonian Gspaces with Gvalued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal Gbundles over a disc, and they generalise the con ..."
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For each connected complex reductive group G, we find a family of new examples of complex quasiHamiltonian Gspaces with Gvalued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal Gbundles over a disc, and they generalise the conjugacy class example of Alekseev–Malkin–Meinrenken (which appears in the simple pole case). Using the ‘fusion product ’ in the theory this gives a finite dimensional construction of the natural symplectic structures on the spaces of monodromy/Stokes data of meromorphic connections over arbitrary genus Riemann surfaces, together with a new proof of the symplectic nature of isomonodromic deformations of such connections.
Poisson groups and differential Galois theory of Schroedinger Equation
 on the circle, Comm. Math. Phys
"... Abstract. We combine the projective geometry approach to Schroedinger equations on the circle and differential Galois theory with the theory of Poisson Lie groups to construct a natural Poisson structure on the space of wave functions (at the zero energy level). Applications to KdVlike nonlinear eq ..."
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Abstract. We combine the projective geometry approach to Schroedinger equations on the circle and differential Galois theory with the theory of Poisson Lie groups to construct a natural Poisson structure on the space of wave functions (at the zero energy level). Applications to KdVlike nonlinear equations are discussed. The same approach is applied to 2 nd order difference operators on a onedimensional lattice, yielding an extension of the lattice Poisson Virasoro algebra.
Symmetries and invariants of twisted quantum algebras and associated Poisson algebras
, 2007
"... ..."
From Klein to Painlevé via Fourier, Laplace and Jimbo
, 2004
"... 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 ..."
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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