SO and USp Kähler and hyper-Kähler quotients and lumps

Abstract

We study non-linear σ models whose target spaces are the Higgs phases of supersymmetric SO and U Sp gauge theories by using the Kähler and hyper-Kähler quotient constructions. We obtain the explicit Kähler potentials and develop an expansion formula to make use of the obtained potentials from which we also calculate the curvatures of the manifolds. The 1/2 BPS lumps in the U (1) × SO and U (1) × U Sp Kähler quotients and their effective descriptions are also studied. In this connection, a general relation between moduli spaces of vortices and lumps is discussed. We find a new singular limit of the lumps with non-vanishing sizes in addition to the ordinary small lump singularity. The former is due to the existence of singular submanifolds in the target spaces.

Non-Abelian vortices in SO(N) and USp(N) gauge theories

“…First we note that this theory has a continuous Higgs vacuum which was discussed in detail in ref. [22]. In this paper, we choose to work in a particular point of the vacuum manifold: 5) namely, in the maximally "color-flavor-locked" Higgs phase of the theory.…”

“…They are obtained by the Kähler quotient method and are found for the gauge groups G ′ = SO(N ), USp(N ) in ref. [22] to be: 28) where the G ′ -invariant…”

“…The Kähler metric and its potential on the vacuum moduli space have been obtained in ref. [22]. The distinction between local and semi-local vortices can be made by using the moduli matrix.…”

“…[22] for lumps can be readily applied to vortices. Semi-local vortices always have non-normalizable moduli, which live on the tangent bundle of the moduli space of vacua 11…”

“…Furthermore, vortices were found to provide us with a deep physical intuition about some well-known correspondence between theories in the four and two dimensions [32,33,42]. So far, however, most studies have been limited to the case of U(N ) gauge theories, with a few but notable exceptions [20][21][22].…”