Supersymmetric Soliton $\sigma$-models from Non-Lorentzian Field Theories

TL;DR

This paper explores how non-Lorentzian supersymmetric field theories, including those related to M-theory branes, can be reduced to supersymmetric sigma models on soliton moduli spaces, revealing new geometric and symmetry structures.

Contribution

It demonstrates the reduction of non-Lorentzian supersymmetric theories to sigma models on BPS soliton moduli spaces, extending the geodesic approximation and explicitly deriving models for instantons.

Findings

Reduced 1+1D sigma model with supersymmetric geodesic approximation.

Derived sigma model on instanton moduli space with extended supersymmetry.

Explicitly constructed the single SU(2) instanton sigma model.

Abstract

Recently, a class of non-Lorentzian supersymmetric Lagrangian field theories was considered, in some cases describing M-theory brane configurations, but more generally found as fixed points of non-Lorentzian RG flows induced upon Lorentzian theories. In this paper, we demonstrate how the dynamics of such theories can be reduced to motion on the supersymmetric moduli space of BPS solitons of the parent theory. We focus first on the $\mathcal{N}=(1,1)$ $\sigma$-model in $(1+1)$-dimensions with potential, where we produce a supersymmetric extension to the standard geodesic approximation for slow kink motion. We then revisit the $(4+1)$-dimensional Yang-Mills-like theory with 24 supercharges describing a null compactification of M5-branes. We show that the theory reduces to a $\sigma$-model on instanton moduli space, extended by couplings to additional fields from the parent theory, and possessing (8+8) super(conformal) symmetries along with 8 further fermionic shift symmetries. We derive this model explicitly for the single $SU(2)$ instanton.