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On the Instanton R-matrix

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Abstract

A torus action on a symplectic variety allows one to construct solutions to the quantum Yang-Baxter equations (R-matrices). For a torus action on cotangent bundles over flag varieties the resulting R-matrices are the standard rational solutions of the Yang-Baxter equation, well known in the theory of quantum integrable systems. The torus action on the instanton moduli space leads to more complicated R-matrices, depending additionally on two equivariant parameters t 1 and t 2. In this paper we derive an explicit expression for the R-matrix associated with the instanton moduli space. We study its matrix elements and its Taylor expansion in the powers of the spectral parameter. Certain matrix elements of this R-matrix give a generating function for the characteristic classes of tautological bundles over the Hilbert schemes in terms of the bosonic cut-and-join operators. In particular we rederive from the R-matrix the well known Lehn’s formula for the first Chern class. We explicitly compute the first several coefficients for the power series expansion of the R-matrix in the spectral parameter. These coefficients are represented by simple contour integrals of some symmetrized bosonic fields.

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Authors and Affiliations

  1. Department of Mathematics, Columbia University, New York, NY, USA

    Andrey Smirnov

  2. ITEP, Moscow, Russia

    Andrey Smirnov

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  1. Andrey Smirnov
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Correspondence to Andrey Smirnov.

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Communicated by N. A. Nekrasov

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Smirnov, A. On the Instanton R-matrix. Commun. Math. Phys. 345, 703–740 (2016). https://doi.org/10.1007/s00220-016-2686-8

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