A Family of ${\rm GL}_r$ Multiplicative Higgs Bundles on Rational Base

TL;DR

This paper investigates a specific family of holomorphic symplectic leaves in the Poisson-Lie group related to ${ m GL}_r$, connecting moduli spaces of Higgs bundles, monopoles, and supersymmetric gauge theories, and provides their explicit quantization.

Contribution

It introduces a new class of irregular symplectic leaves at infinity, linking them to gauge theories and quantizes their function algebra via solutions to the quantum Yang-Baxter equation.

Findings

Explicit description of symplectic leaves as fusion products.

Connection to moduli spaces of Higgs bundles and monopoles.

Quantization via solutions to the quantum Yang-Baxter equation.

Abstract

In this paper we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group ${\rm GL}_r(\mathcal{K}_{\mathbb{P}^1_x})$ with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at $\infty \in \mathbb{P}_{1}$. The restriction of the family is that the matrix elements in the defining representation are linear functions of $x$. We study how the symplectic leaves in this family are obtained by the fusion of certain fundamental symplectic leaves. These symplectic leaves arise as minimal examples of (i) moduli spaces of multiplicative Higgs bundles on $\mathbb{P}^{1}$ with prescribed singularities, (ii) moduli spaces of $U(r)$ monopoles on $\mathbb{R}^2 \times S^1$ with Dirac singularities, (iii) Coulomb branches of the moduli space of vacua of 4d $\mathcal{N}=2$ supersymmetric $A_{r-1}$ quiver gauge theories compactified on a circle. While degree 1 symplectic leaves regular at $\infty \in \mathbb{P}^1$ (Coulomb branches of the superconformal quiver gauge theories) are isomorphic to co-adjoint orbits in $\mathfrak{gl}_{r}$ and their Darboux parametrization and quantization is well known, the case irregular at infinity (asymptotically free quiver gauge theories) is novel. We also explicitly quantize the algebra of functions on these moduli spaces by presenting the corresponding solutions to the quantum Yang-Baxter equation valued in Heisenberg algebra (free field realization).