Low-Energy Dynamics of Supersymmetric Solitons

TL;DR

This paper develops a supersymmetric quantum mechanics framework to analyze low-energy soliton dynamics in N=2 sigma models, linking bound states to cohomology classes on the moduli space.

Contribution

It constructs an effective supersymmetric quantum mechanics model for soliton dynamics in N=2 sigma models, connecting bound states to moduli space cohomology.

Findings

Effective action as N=2 supersymmetric quantum mechanics

Bound states correspond to cohomology classes on the moduli space

Quantization reveals zero-energy bound states linked to Dolbeault cohomology

Abstract

In bosonic field theories the low-energy scattering of solitons that saturate Bogomol'nyi-type bounds can be approximated as geodesic motion on the moduli space of static solutions. In this paper we consider the analogous issue within the context of supersymmetric field theories. We focus our study on a class of $N=2$ non-linear sigma models in $d=2+1$ based on an arbitrary K\"ahler target manifold and their associated soliton or ``lump" solutions. Using a collective co-ordinate expansion, we construct an effective action which, upon quantisation, describes the low-energy dynamics of the lumps. The effective action is an $N=2$ supersymmetric quantum mechanics action with the target manifold being the moduli space of static charge $N$ lump solutions of the sigma model. The Hilbert space of states of the effective theory consists of anti-holomorphic forms on the moduli space. The normalisable elements of the dolbeault cohomology classes $H^{(0,p)}$ of the moduli space correspond to zero energy bound states and we argue that such states correpond to bound states in the full quantum field theory of the sigma model.