Multiplicative Hitchin systems and supersymmetric gauge theory

Abstract

Multiplicative Hitchin systems are analogues of Hitchin's integrable system based on moduli spaces of G-Higgs bundles on a curve C where the Higgs field is group-valued, rather than Lie algebra valued. We discuss the relationship between several occurences of these moduli spaces in geometry and supersymmetric gauge theory, with a particular focus on the case where C = CP 1 with a fixed framing at infinity. In this case we prove that the identification between multiplicative Higgs bundles and periodic monopoles proved by Charbonneau and Hurtubise can be promoted to an equivalence of hyperkähler spaces, and analyze the twistor rotation for the multiplicative Hitchin system. We also discuss quantization of these moduli spaces, yielding the modules for the Yangian Y (g) discovered by Gerasimov, Kharchev, Lebedev and Oblezin.Date: July 12, 2019. 1 Usually Higgs bundles on a curve are defined to be sections of the coadjoint bundle twisted by the canonical bundle. There isn't an obvious replacement for this twist in the multiplicative context, but we'll mostly be interested in the case where C is Calabi-Yau, and therefore this twist is trivial. 1 arXiv:1812.05516v4 [math.AG] 11 Jul 2019 G (CP 1 , D, ω ∨ ) of framed ε-connections on CP 1 . 1.2. Moduli Space of Monopoles. If we focus on the example where a real three-dimensional Riemannian manifold M = C × S 1 splits as the product of a compact Riemann surface and a circle, for the rational case C = C the moduli space of monopoles on M was studied by Cherkis and Kapustin [CK98, CK01, CK02] from the perspective of the Coulomb branch of vacua of 4d N = 2 supersymmetric gauge theory. For general C the moduli space of monopoles on M = C × S 1 was studied by Charbonneau-Hurtubise [CH10], for G R = U(n), and by Smith [Smi16] for general G.These moduli spaces were also studied where C = C with more general boundary conditions than the framing at infinity which we consider. This example has also been studied in the mathematics literature by Foscolo [Fos16, Fos13] and by Mochizuki [Moc17b] -note that considering weaker boundary conditions at infinity means they need to use far more sophisticated analysis in order to work with hyperkähler structures on their moduli spaces than we'll consider in this paper.The connection between periodic monopoles and multiplicative Higgs bundles is provided by the following theorem.Theorem 1.2 (Charbonneau-Hurtubise, Smith). There is an analytic isomorphismbetween the moduli space of (polystable) multiplicative G-Higgs bundles on a compact curve C with singularities at points z 1 , . . . , z k and residues ω ∨ z 1 , . . . , ω ∨ z k and the moduli space of periodic monopoles on C × S 1 with Dirac singularities at (z 1 , 0), . . . , (z k , 0) with charges ω ∨ z 1 , . . . , ω ∨ z kThe morphism H, discussed first by Cherkis and Kapustin in [CK01] and later in [CH10], [Smi16], [NP12] defines the value of the multiplicative Higgs field at z ∈ C to be equal to the holonomy of the monopole connection (complexified by the scalar field) along the fiber circl...

Gauge Theory And Integrability, II

“…This spectral curve is also the spectral curve for the group-valued Hitchin system (also called the multiplicative Hitchin system, see e.g. [22] and references therein). This is because the moduli spaces of classical equations of motion of the four-dimensional Chern-Simons theory define the group-valued Hitchin system on the spectral curve C.…”

A Family of GL_r Multiplicative Higgs Bundles on Rational Base

“…The space of rational multiplicative Higgs fields on X = P 1 with a fixed framing of the gauge bundle at x ∞ forms a Poisson-Lie group [30].…”

“…We believe so and we refer to several geometrical perspectives on the multiplicative case further in the introduction. For the basic definitions see [17,30,33,42,55].…”

“…For simple Lie groups G of the ADE type these moduli spaces appear as Coulomb branches of the moduli space of vacua of the N = 2 supersymmetric ADE quiver gauge theory on R 3 × S 1 [13,14,53,55]. Some constructions from the world of additive Higgs bundles have their versions in the world of multiplicative Higgs bundles [30] leading to difference equations and their monodromy problems [9,58], q-geometric Langlands correspondence [1], q−W algebras [46,54,56].…”

“…The Lax matricesLλ ,x,μ (x;p,q) represent elements of G in a new symplectic leaf Mλ ,x,μ covered by Darboux coordinatesp,q. The symplectic leaves M λ,x,µ arise as moduli spaces of multiplicative Higgs bundles of certain type [30], and like additive Higgs bundles (Hitchin system), the symplectic leaves M λ,x,µ support the structure of an algebraic completely integrable system. In fact, the moduli spaces M λ,x,µ can be also interpreted as moduli spaces of U (r) monopoles on 3-dimensional Riemannian space R 2 × S 1 where R 2 C = P 1 \ {x ∞ }, and consequently [16,17,18,55] as moduli spaces of vacua of certain N = 2 supersymmetric quiver gauge theories on R 3 × S 1 of quiver type A r−1 .…”