Geometry of solutions of Hitchin equations on ${{\mathbb{R}}^{\mathbf{2}}}$

Abstract

We study smooth SU(2) solutions of the Hitchin equations on R 2 , with the determinant of the complex Higgs field being a polynomial of degree n. When n ≥ 3, there are moduli spaces of solutions, in the sense that the natural L 2 metric is well-defined on a subset of the parameter space. We examine rotationally-symmetric solutions for n = 1 and n = 2, and then focus on the n = 3 case, elucidating the moduli and describing the asymptotic geometry as well as the geometry of two totally-geodesic surfaces. *

Integrable (2k)-Dimensional Hitchin Equations

“…For example, θ(z) = z 3/2 gives an embedding of the 'one-lump' solution on C [21]. Some simple solutions that are not of this embedded type are as follows.…”

Nonabelian probes in holography

“…Such solutions provide limiting configurations for studying the asymptotic regions of the moduli space of solutions to Hitchin's equations on a compact surface [46,47]. Alternatively, one can obtain smooth solutions in the plane if one allows the Higgs fields to diverge polynomially in z [48].…”

Self-dual gravity via Hitchin’s equations

“…We now discuss a solution given by Ward in Ref. [50]. Ward considered smooth SU(2) solutions of the Hitchin's equations on R 2 with boundary conditions involving an integer n, which is the degree of the determinant of the Higgs field Φ.…”