A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base

Abstract

In this paper, we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group GLᵣ(KP¹ₓ) with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at ∞∈ℙ₁. 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 ℙ¹ with prescribed singularities, (ii) moduli spaces of U(r) monopoles on ℝ²×S¹ with Dirac singularities, (iii) Coulomb branches of the moduli space of vacua of 4d N = 2 supersymmetric Aᵣ₋₁ quiver gauge theories compactified on a circle. While degree 1 symplectic leaves regular at ∞∈ℙ¹ (Coulomb branches of the superconformal quiver gauge theories) are isomorphic to co-adjoint orbits in glᵣ and their Darboux parametrization and quantization are 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 the Heisenberg algebra (free field realization).