Vacua of Massive Hyper-Kahler Sigma Models with Non-Abelian Quotient

TL;DR

The paper develops massive hyper-Kähler sigma models via non-Abelian HK quotients, focusing on ${ m U}(M)$ and ${ m SU}(M)$ gauge groups to obtain non-toric target spaces $T^*G_{N,M}$ and related bundles. It demonstrates that the ${ m U}(M)$ quotient yields a rich discrete vacuum structure with ${}_N C_M$ vacua, enabling various domain-wall configurations, while the ${ m SU}(M)$ quotient lacks discrete vacua and walls. The authors provide complementary off-shell formulations in ${ m N}=1$ superspace and harmonic superspace, derive the scalar potentials and vacuum conditions in the Wess–Zumino gauge, and confirm that the ${}_N C_M$ vacuum counting persists across formulations. They discuss the geometric interpretation of vacua as mutually orthogonal $M$-planes in ${f C}^N$ and outline future directions, including wall dynamics on ${T^*}G_{N,M}$, potential solitons, and couplings to supergravity. Overall, the work reveals a rich non-Abelian quaternionic geometry underlying massive HK sigma models and clarifies when discrete vacua and walls arise due to FI parameters.

Abstract

The Higgs branch of N=2 supersymmetric gauge theories with non-Abelian gauge groups are described by hyper-Kahler (HK) nonlinear sigma models with potential terms. With the non-Abelian HK quotient by U(M) and SU(M) gauge groups, we give the massive HK sigma models that are not toric in the N=1 superfield formalism and the harmonic superspace formalism. The U(M) quotient gives N!/[M! (N-M)!] discrete vacua that may allow various types of domain walls, whereas the SU(M) quotient gives no discrete vacua.

Figures (1)

• Figure 1: The base manifold of $T^*{\bf C}P^1$ and vacua.