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 $U(Nc)$ gauge theory with $Nf$ 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 $Nc=Nf=\lambda$ we find strong evidence that the vortices are dual to fermions in the lowest nonAbelian Landau level associated to a background $U(Nf)$ 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 are closely related to the statistical mechanical partition functions of vortex gases). The volume of the $k$-vortex moduli space is a polynomial in the volume of physical space and, in the local case of $Nc=Nf$, we find a reduction in the degree of this polynomial, from $Nfk$ to $k$, resolving a point of confusion in the literature.

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 $Nf=1$ to pairs consisting of a flat $PSU(Nc+1)$ connection and a holomorphic section of the associated holomorphic $CPNc$-bundle. As well as providing a way to generate exact exotic vortex solutions in nonAbelian gauge theories, this leads to conjectural descriptions of the moduli spaces of exotic vortices and to topology-dependent selection rules for exotic vortex solutions (generalising the fact that Jackiw--Pi vortex solutions on the infinite sphere always have even flux).

Second, we turn our attention to three-dimensional sigma models with $N=4$ supersymmetry. When its target space is a hyperKähler manifold carrying a (so-called) permuting $Sp(1)$ action, such a sigma model admits two topological twists: the A-twist and the B-twist. The B-twist, also known as Rozansky--Witten theory, is well-understood, while the A-twist is a little more mysterious. The two twists are expected to be related by three-dimensional mirror symmetry.

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.