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15
ToledanoLaredo V., Gaudin model with irregular singularities
"... Abstract. We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac–Moody algebra at the critical level, extending the construction of higher ..."
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Cited by 24 (3 self)
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Abstract. We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac–Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from [FFR] to the case of nonhighest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P 1 with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finitedimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.
Flows of CalogeroMoser systems
 Int. Math. Res. Not
"... The CalogeroMoser (or CM) particle system [Ca1, Ca2] and its generalizations appear, in a variety of ways, in integrable systems, nonlinear PDE, representation theory, and string theory. Moreover, the partially completed CM systems—in ..."
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Cited by 7 (3 self)
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The CalogeroMoser (or CM) particle system [Ca1, Ca2] and its generalizations appear, in a variety of ways, in integrable systems, nonlinear PDE, representation theory, and string theory. Moreover, the partially completed CM systems—in
Algebraic and hamiltonian approaches to isostokes deformations
, 2008
"... We study a generalization of the isomonodromic deformation to the case of connections with irregular singularities. We call this generalization Isostokes Deformation. A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points. We study this part of ..."
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Cited by 5 (3 self)
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We study a generalization of the isomonodromic deformation to the case of connections with irregular singularities. We call this generalization Isostokes Deformation. A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points. We study this part of the deformation, giving an algebraic description. Then we show how to use loop groups and hypercohomology to write explicit hamiltonians. We work on an arbitrary complete algebraic
Irregular Wakimoto modules and the Casimir connection
 Selecta Math. (N.S
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The associated map of the nonabelian GaussManin connection
 Central European J. of Math
"... In ordinary Hodge theory for a compact kahler manifold, one can look at the GaussManin connection as the complex structure of the manifold varies. The connection satisfies the Griffiths transversality property, and induces a map on the associated graded spaces of the Hodge filtered cohomology space ..."
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Cited by 1 (0 self)
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In ordinary Hodge theory for a compact kahler manifold, one can look at the GaussManin connection as the complex structure of the manifold varies. The connection satisfies the Griffiths transversality property, and induces a map on the associated graded spaces of the Hodge filtered cohomology spaces of the manifolds. Similarly in nonabelian Hodge theory, where the manifolds are curves and nonabelian cohomology spaces are the moduli spaces of local systems on the curves, the GaussManin connection will be the Isomonodromy deformation, which is a previously known structure on these moduli spaces. One can still define the Hodge filtration and calculate the map induced by the GaussManin connection on the associated graded space. To do this we used deformation theory to express the tangent spaces of the moduli spaces as hypercohomologies of complexes of sheaves over the curves, and write the isomonodromy deformation as a map between such hypercohomology spaces. Under this setting the induced map can be explicitly calculated and is in fact written in an analogous form as the isomonodromy deformation. The induced map turns out to be closely related to another wellknown structure called the Hitchin integrable structure, defined on the moduli spaces that correspond to the associated graded spaces. More specifically it is equal up to a factor of 2 to the quadratic Hitchin map.
Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for NonTrivial Bundles
, 2012
"... Abstract. We describe new families of the Knizhnik–Zamolodchikov–Bernard (KZB) equations related to the WZWtheory corresponding to the adjoint Gbundles of different topological types over complex curves Σg,n of genus g with n marked points. The bundles are defined by their characteristic classes – ..."
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Abstract. We describe new families of the Knizhnik–Zamolodchikov–Bernard (KZB) equations related to the WZWtheory corresponding to the adjoint Gbundles of different topological types over complex curves Σg,n of genus g with n marked points. The bundles are defined by their characteristic classes – elements of H 2 (Σg,n, Z(G)), where Z(G) is a center of the simple complex Lie group G. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness. Key words: integrable system; KZB equation; Hitchin system; characteristic class
WSYMMETRY OF THE ADÈLIC GRASSMANNIAN
, 2008
"... We give a geometric construction of the W1+ ∞ vertex algebra as the infinitesimal form of a factorization structure on an adèlic Grassmannian. This gives a concise interpretation of the higher symmetries and BäcklundDarboux transformations for the KP hierarchy and its multicomponent extensions in ..."
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We give a geometric construction of the W1+ ∞ vertex algebra as the infinitesimal form of a factorization structure on an adèlic Grassmannian. This gives a concise interpretation of the higher symmetries and BäcklundDarboux transformations for the KP hierarchy and its multicomponent extensions in terms of a version of “W1+∞geometry”: the geometry of Dbundles on smooth curves, or equivalently torsionfree sheaves on cuspidal curves.
IRREGULAR WAKIMOTO MODULES AND DMT CONNECTION
, 812
"... Abstract. We study some nonhighest weight modules over an affine Kac– Moody algebra ˆg at noncritical level. Roughly speaking, these modules are noncommutative localizations of some nonhighest weight “vacuum ” modules. Using free field realization, we embed some rings of differential operators i ..."
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Abstract. We study some nonhighest weight modules over an affine Kac– Moody algebra ˆg at noncritical level. Roughly speaking, these modules are noncommutative localizations of some nonhighest weight “vacuum ” modules. Using free field realization, we embed some rings of differential operators in endomorphism rings of our modules. These rings of differential operators act on a localization of the space of coinvariants of any ˆgmodule with respect to a certain level subalgebra. In a particular case this action is identified with the DMT connection. 1.
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, 2005
"... Abstract. We study a generalization of the isomonodromic deformation to the case of connections with irregular singularities. We call this generalization Isostokes Deformation. A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points. We study thi ..."
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Abstract. We study a generalization of the isomonodromic deformation to the case of connections with irregular singularities. We call this generalization Isostokes Deformation. A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points. We study this part of the deformation, giving an algebraic description. Then we show how to use loop groups and hypercohomology to write explicit hamiltonians. We work on an arbitrary complete algebraic curve, the structure group is an arbitrary semisimiple group.