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36
Conformal blocks and generalized theta functions
 Comm. Math. Phys
, 1994
"... The aim of this paper is to construct a canonical isomorphism between two vector spaces associated to a Riemann surface X. The first of these spaces is the space of conformal blocks Bc(r) (also called the space of vacua), which plays an important role in conformal field theory. It is defined as foll ..."
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Cited by 141 (8 self)
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The aim of this paper is to construct a canonical isomorphism between two vector spaces associated to a Riemann surface X. The first of these spaces is the space of conformal blocks Bc(r) (also called the space of vacua), which plays an important role in conformal field theory. It is defined as follows: choose a point p ∈ X, and let AX be the
Instantons and affine algebras I: The Hilbert scheme and vertex operators
 Math. Res. Letters
, 1996
"... Abstract. This is the first in a series of papers which describe the action of an affine Lie algebra with central charge n on the moduli space of U(n)instantons on a four manifold X. This generalises work of Nakajima, who considered the case when X is an ALE space. In particular, this should describ ..."
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Cited by 123 (1 self)
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Abstract. This is the first in a series of papers which describe the action of an affine Lie algebra with central charge n on the moduli space of U(n)instantons on a four manifold X. This generalises work of Nakajima, who considered the case when X is an ALE space. In particular, this should describe the combinatorial complexity of the moduli space as beingprecisely that of representation theory, and thus will lead to a description of the Betti numbers of moduli space as dimensions of weight spaces. This Lie algebra acts on the space of conformal blocks (i.e., the cohomology of a determinant line bundle on the moduli space [LMNS]) generalising the “insertion ” and “deletion ” operations of conformal field theory, and indeed on any cohomology theory. In the particular case of U(1)instantons, which is essentially the subject of this present paper, the construction produces the basic representation after FrenkelKac. Then the well known quadratic nature of ch2, ch2 = 1 2 c1 · c1 − c2 becomes precisely the formula for the eigenvalue of the degree operator, i.e. the well known quadratic behaviour of affine Lie algebras.
ON QUANTUM COHOMOLOGY RINGS OF PARTIAL FLAG VARIETIES
 VOL. 98, NO. 3 DUKE MATHEMATICAL JOURNAL
, 1999
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BETTI NUMBERS OF THE MODULI SPACE OF RANK 3 PARABOLIC HIGGS BUNDLES
, 2004
"... Abstract. Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the modul ..."
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Cited by 28 (8 self)
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Abstract. Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles. 1.
Yangians and cohomology rings of Laumon spaces, preprint, arXiv math/0812.4656
"... Abstract. Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yang ..."
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Cited by 18 (4 self)
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Abstract. Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (twoparametric deformation of the universal enveloping algebra of the universal central extension of sln[s ±1, t]) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analogue of the GelfandTsetlin basis. The affine analogue of the GelfandTsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space Mn,d of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center Z of the Yangian of gl n naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on Mn,d is the image of a noncommutative power sum in Z. 1.
FINITE DIFFERENCE QUANTUM TODA LATTICE VIA EQUIVARIANT KTHEORY
, 2005
"... Abstract. We construct the action of the quantum group Uv(sln) by the natural correspondences in the equivariant localized Ktheory of the Laumon based Quasiflags ’ moduli spaces. The resulting module is the universal Verma module. We construct geometrically the Shapovalov scalar product and the Whi ..."
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Cited by 15 (7 self)
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Abstract. We construct the action of the quantum group Uv(sln) by the natural correspondences in the equivariant localized Ktheory of the Laumon based Quasiflags ’ moduli spaces. The resulting module is the universal Verma module. We construct geometrically the Shapovalov scalar product and the Whittaker vectors. It follows that a certain generating function of the characters of the global sections of the structure sheaves of the Laumon moduli spaces satisfies a vdifference analogue of the quantum Toda lattice system, reproving the main theorem of GiventalLee (cf. [7]). Similar constructions are performed for the affine Lie agebra ̂ sln. 1.