On the Instanton R-matrix

TL;DR

This paper derives an explicit R-matrix for the instanton moduli space, revealing its structure, matrix elements, and connections to characteristic classes, expanding understanding of quantum integrable systems with additional equivariant parameters.

Contribution

It provides the first explicit expression for the instanton R-matrix, linking it to characteristic classes and bosonic operators, extending known rational solutions.

Findings

Explicit formula for the instanton R-matrix.

Connection to characteristic classes of tautological bundles.

Coefficients expressed as contour integrals of bosonic fields.

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, which are 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.