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Higgs Bundles, Gauge Theories and Quantum Groups

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Abstract

The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the N-particle sector. This implies the full equivalence between the above gauge theory and the N-particle sub-sector of the quantum theory of the Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine Hecke algebra. We propose a similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra.

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Authors and Affiliations

  1. Institute for Theoretical and Experimental Physics, Moscow, 117259, Russia

    Anton A. Gerasimov

  2. School of Mathematics, Trinity College, Dublin 2, Ireland

    Anton A. Gerasimov & Samson L. Shatashvili

  3. The Hamilton Mathematics Institute TCD, Dublin 2, Ireland

    Anton A. Gerasimov & Samson L. Shatashvili

  4. IHES, 35 route de Chartres, Bures-sur-Yvette, France

    Samson L. Shatashvili

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  1. Anton A. Gerasimov
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  2. Samson L. Shatashvili
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Correspondence to Samson L. Shatashvili.

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Communicated by L. Takhtajan

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Gerasimov, A.A., Shatashvili, S.L. Higgs Bundles, Gauge Theories and Quantum Groups. Commun. Math. Phys. 277, 323–367 (2008). https://doi.org/10.1007/s00220-007-0369-1

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