Solitons and dualities in 2+1 dimensions
type: CreativeWork
description: We examine three topics in the physics of three-dimensional systems, paying particular attention to the roles of topological field theory, topological solitons, and duality. First, we consider the dynamics of nonrelativistic Chern--Simons-matter theories in $2+1$ dimensions. The theories we consider support vortex solutions and, at critical coupling, they admit an effective low-temperature description as a topological quantum mechanics on the moduli space of these vortices. Using localisation techniques, we compute the 'expected' dimension of the Hilbert space of this quantum mechanics (that is, the Euler characteristic of the relevant quantum line bundle on the moduli space) in the case of $U(N_c)$ gauge theory with $N_f$ fundamental flavours at arbitrary Chern--Simons level, $\lambda$, compactified on an arbitrary closed Riemann surface. We apply this result to the analysis of the duality between the vortices of these theories and the (composite) fermions that arise in descriptions of strongly correlated electron systems and, in particular, of (nonAbelian, fractional) quantum Hall fluids. In simple cases, where the degeneracies of the fermion fluids are well understood, the results give quantitative evidence for these dualities. For example, when $N_c = N_f = \lambda$ we find strong evidence that the vortices are dual to fermions in the lowest nonAbelian Landau level associated to a background $U(N_f)$ flux. In more complicated examples, the results lead us to quantitative insight into the conjectural dual theory. We find that quantum vortices generally admit a description as composite objects, bound states of (dual) anyons. We comment on potential links with three-dimensional mirror symmetry. We also compute the equivariant expected degeneracy of local Abelian vortices on the $\Omega$-deformed sphere, finding it to be a $q$-analog of the undeformed version. By taking the semiclassical limit, we use our result to compute the volumes of vortex moduli spaces (which …
abstract: We examine three topics in the physics of three-dimensional systems, paying particular attention to the roles of topological field theory, topological solitons, and duality. First, we consider the dynamics of nonrelativistic Chern--Simons-matter theories in $2+1$ dimensions. The theories we consider support vortex solutions and, at critical coupling, they admit an effective low-temperature description as a topological quantum mechanics on the moduli space of these vortices. Using localisation techniques, we compute the 'expected' dimension of the Hilbert space of this quantum mechanics (that is, the Euler characteristic of the relevant quantum line bundle on the moduli space) in the case of $U(N_c)$ gauge theory with $N_f$ fundamental flavours at arbitrary Chern--Simons level, $\lambda$, compactified on an arbitrary closed Riemann surface. We apply this result to the analysis of the duality between the vortices of these theories and the (composite) fermions that arise in descriptions of strongly correlated electron systems and, in particular, of (nonAbelian, fractional) quantum Hall fluids. In simple cases, where the degeneracies of the fermion fluids are well understood, the results give quantitative evidence for these dualities. For example, when $N_c = N_f = \lambda$ we find strong evidence that the vortices are dual to fermions in the lowest nonAbelian Landau level associated to a background $U(N_f)$ flux. In more complicated examples, the results lead us to quantitative insight into the conjectural dual theory. We find that quantum vortices generally admit a description as composite objects, bound states of (dual) anyons. We comment on potential links with three-dimensional mirror symmetry. We also compute the equivariant expected degeneracy of local Abelian vortices on the $\Omega$-deformed sphere, finding it to be a $q$-analog of the undeformed version. By taking the semiclassical limit, we use our result to compute the volumes of vortex moduli spaces (which …
author: Walton, Edward
language: eng
keywords: theoretical physics, solitons, dualities, topological field theory, vortices, quantum Hall effect, mirror symmetry, magnetic Skyrmions, emergent monopoles
license: https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/bea87796-5edf-4fb4-803a-3d28c6c2dc6c/download
license: https://www.rioxx.net/licenses/all-rights-reserved/
date published: 2021-09-11
date modified: 2026-07-22T22:24:16Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.75568