## Solitons and dualities in 2+1 dimensions

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## 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

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

By taking the semiclassical limit, we use our result to compute the volumes of vortex moduli spaces (which are closely related to the statistical mechanical partition functions of vortex gases). The volume of the

We also find new integrability results relating solutions of the exotic vortex equations, which generalise the Jackiw--Pi and Ambjørn--Olesen vortex equations, in theories with

Second, we turn our attention to three-dimensional sigma models with

We construct the A-twist on general three-manifolds and analyse its local and extended operators. We show that compactifying the theory on a circle gives the two-dimensional A-model in the presence of a certain 'defect operator'. We then outline the construction of the 2-category of boundary conditions in the three-dimensional A-twist. In particular, we find that the A-twist induces a monoidal structure on the Fukaya categories of a certain, restricted, class of Kähler manifolds.

Third, we consider magnetic Skyrmions in ferromagnetic materials. We produce a continuum toy model of fractionalised electrons in three spatial dimensions describing magnetic Skyrmions and their creation and destruction via emergent magnetic monopoles. When an external magnetic field is applied, the model has a critical point where confined monopoles are dynamically stabilised and monopole-antimonopole pairs may condense. We find novel 'BPS-like' equations for these configurations. By tuning the model to critical coupling and then deforming it, we find qualitative agreement with the observed phase structure of chiral ferromagnets.

We then consider the critically-coupled model for magnetic Skyrmions on thin films, generalising it to thin films with curved geometry. We find exact Skyrmion solutions on some curved films with symmetry, namely spherical, conical, and cylindrical films. We prove the existence of Skyrmion solutions in the model on general compact films and investigate the geometry of the (resolved) moduli space of solutions.