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Gauge theory and wild ramification
"... 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, ..."
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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.
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.
ON THE GEOMETRY OF ISOMONODROMIC DEFORMATIONS
, 804
"... Abstract. This note examines the geometry behind the Hamiltonian structure of isomonodromy deformations of connections on vector bundles over Riemann surfaces. The main point is that one should think of an open set of the moduli of pairs (V,∇) of vector bundles and connections as being obtained by “ ..."
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Abstract. This note examines the geometry behind the Hamiltonian structure of isomonodromy deformations of connections on vector bundles over Riemann surfaces. The main point is that one should think of an open set of the moduli of pairs (V,∇) of vector bundles and connections as being obtained by “twists ” supported over points of a fixed vector bundle V0 with a fixed connection ∇0; this gives two deformations, one, isomonodromic, of (V,∇), and another induced from the isomonodromic deformation of (V0,∇0). The difference between the two will be Hamiltonian. 1.
Two twistor descriptions of the isomonodromy problem
, 2008
"... The connections between Hitchin and Mason’s twistor descriptions of the isomonodromy problem are explored. 1 ..."
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The connections between Hitchin and Mason’s twistor descriptions of the isomonodromy problem are explored. 1
Contents
, 2006
"... Abstract. In this paper we study the Hamiltonian structure of the second Painlevé hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation. The nth element of the hierarchy is a non linear ODE of order 2n in the independent variable z depe ..."
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Abstract. In this paper we study the Hamiltonian structure of the second Painlevé hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation. The nth element of the hierarchy is a non linear ODE of order 2n in the independent variable z depending on n parameters denoted by t1,..., tn−1 and αn. We introduce new canonical coordinates and obtain Hamiltonians for the z and t1,..., tn−1 evolutions. We give explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates.
ASYMPTOTICS FOR GENERAL CONNECTIONS AT INFINITY
, 2003
"... Abstract. For a standard path of connections going to a generic point at infinity in the moduli space MDR of connections on a compact Riemann surface, we show that the Laplace transform of the family of monodromy matrices has an analytic continuation with locally finite branching. In particular the ..."
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Abstract. For a standard path of connections going to a generic point at infinity in the moduli space MDR of connections on a compact Riemann surface, we show that the Laplace transform of the family of monodromy matrices has an analytic continuation with locally finite branching. In particular the convex subset representing the exponential growth rate of the monodromy is a polygon, whose vertices are in a subset of points described explicitly in terms of the spectral curve. Unfortunately we don’t get any information about the size of the singularities of the Laplace transform, which is why we can’t get asymptotic expansions for the monodromy. 1.