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We review some unusual facts about the theory of non-relativistic anyons in 2+1 dimensions, and use it as a laboratory to explore how interesting features of non-relativistic field theory correspond to those of many-body quantum mechanics. In particular, we offer an explanation of how Jackiw-Pi vortices arise as the classical limit of certain many-body states in the quantum mechanical theory. Along the way, we make various interesting observations about universal features of the spectrum of anyons subject to different amounts of tuning., Comment: 25 pages + 2 appendices (v2: small corrections, submitted version)

Eight Majorana fermions in $d=1+1$ dimensions enjoy a triality that permutes the representation of the $SO(8)$ global symmetry in which the fermions transform. This triality plays an important role in the quantization of the superstring, and in the analysis of interacting topological insulators and the associated phenomenon of symmetric mass generation. The purpose of these notes is to provide an introduction to the triality and its applications, with careful attention paid to various ${\bf Z}_2$ global and gauge symmetries and their coupling to background spin structures., Comment: 19 pages plus 2 appendices. v2: Typos fixed

In this course, we will discuss dualities in 2+1 dimensions. We begin by briefly reviewing the physics of 1+1d T-duality and bosonization, and discuss flux attachment and the dual photon in 2+1d. Then, we introduce 2+1d particle-vortex duality, bosonization and their associated web of dualities. This includes discussing more elaborate Abelian dualities, including self-dual theories and quiver theories. Next, we move on to non-Abelian physics, discussing QCD in three dimensions and more. This done, we will discuss the relation of these dualities to SUSY dualities, lattice physics, and large N physics; and finally, we present the application of one of the Abelian dualities to quantum Hall physics. These lecture notes are expanded from lectures delivered at XIV Modave Summer School in Mathematical Physics in September 2018., Comment: 136 pages, submitted to PoS

We describe a web of well-known dualities connecting quantum field theories in $d=1+1$ dimensions. The web is constructed by gauging ${\bf Z}_2$ global symmetries and includes a number of perennial favourites such as the Jordan-Wigner transformation, Kramers-Wannier duality, bosonization of a Dirac fermion, and T-duality. There are also less-loved examples, such as non-modular invariant $c=1$ CFTs that depend on a background spin structure., Comment: 29 pages plus appendices. v2: Add references and fix minor typos. v3: More typos, and an extra appendix on the reduction of bosonization from 3d to 2d. Version accepted for publication in SciPost. v4: references added

Quantum Hall matrix models are simple, solvable quantum mechanical systems which capture the physics of certain fractional quantum Hall states. Recently, it was shown that the Hall viscosity can be extracted from the matrix model for Laughlin states. Here we extend this calculation to the matrix models for a class of non-Abelian quantum Hall states. These states, which were previously introduced by Blok and Wen, arise from the conformal blocks of Wess-Zumino-Witten conformal field theory models. We show that the Hall viscosity computed from the matrix model coincides with a result of Read, in which the Hall viscosity is determined in terms of the weights of primary operators of an associated conformal field theory., Comment: v2: 9 pages, references added, to appear in PRB

We study a supersymmetry breaking deformation of the 2d N=(2,2) cigar=Liouville mirror pair, first introduced by Hori and Kapustin. We show that mirror symmetry flows in the infra-red to 2d bosonization, with the theories reducing to massive Thirring and Sine-Gordon respectively. The exact bosonization map emerges at one-loop. We further compactify non-supersymmetric 3d bosonization dualities on a circle and argue that these too flow to 2d bosonization at long distances., Comment: 13 pages. v2: addendum on emergence of Z2 gauge fields upon compactification

We demonstrate explicitly the correspondence between all protected operators in a 2+1 dimensional non-supersymmetric bosonization duality in the non-relativistic limit. Roughly speaking we consider $SU(N)$ Chern-Simons field theory at level $k$ with $N_f$ flavours of fundamental boson, and match its chiral sector to that of a $SU(k)$ theory at level $N$ with $N_f$ fundamental fermions. We present the matching at the level of indices and individual operators, seeing the mechanism of failure for $N_f > N$, and point out that the non-relativistic setting is a particularly friendly setting for studying interesting questions about such dualities., Comment: 30 pages plus references, 2 figures

Yang-Mills instantons are solitonic particles in d=4+1 dimensional gauge theories. We construct and analyse the quantum Hall states that arise when these particles are restricted to the lowest Landau level. We describe the ground state wavefunctions for both Abelian and non-Abelian quantum Hall states. Although our model is purely bosonic, we show that the excitations of this 4d quantum Hall state are governed by the Nekrasov partition function of a certain five dimensional supersymmetric gauge theory with Chern-Simons term. The partition function can also be interpreted as a variant of the Hilbert series of the instanton moduli space, counting holomorphic sections rather than holomorphic functions. It is known that the Hilbert series of the instanton moduli space can be rewritten using mirror symmetry of 3d gauge theories in terms of Coulomb branch variables. We generalise this approach to include the effect of a five dimensional Chern-Simons term. We demonstrate that the resulting Coulomb branch formula coincides with the corresponding Higgs branch Molien integral which, in turn, reproduces the standard formula for the Nekrasov partition function., Comment: 50 pages, 2 figures

We study the spectrum of multiple non-Abelian anyons in a harmonic trap. The system is described by Chern-Simons theory, coupled to either bosonic or fermionic non-relativistic matter, and has an SO(2,1) conformal invariance. We describe a number of special properties of the spectrum, focussing on a class of protected states whose energies are dictated by their angular momentum. We show that the angular momentum of a bound state of non-Abelian anyons is determined by the quadratic Casimirs of their constituents., Comment: 26 pages, 1 figure (plus Feynman diagrams drawn with feynMF). v2: added comments on counting BPS states

Bosonization dualities relate two different Chern-Simons-matter theories, with bosonic matter on one side replaced by fermionic matter on the other. We first describe a more general class of non-Abelian bosonization dualities. We then explore the non-relativistic physics of these theories in the quantum Hall regime. The bosonic theory lies in a condensed phase and admits vortices which are known to form a non-Abelian quantum Hall state. We ask how this same physics arises in the fermionic theory. We find that a condensed boson corresponds to a fully filled Landau level of fermions, while bosonic vortices map to fermionic holes. We confirm that the ground state of the two theories is indeed described by the same quantum Hall wavefunction., Comment: 21 pages. v2: typo corrected. v3: another typo corrected (chemical potential rescaled)

We study a U(N) gauged matrix quantum mechanics which, in the large N limit, is closely related to the chiral WZW conformal field theory. This manifests itself in two ways. First, we construct the left-moving Kac-Moody algebra from matrix degrees of freedom. Secondly, we compute the partition function of the matrix model in terms of Schur and Kostka polynomials and show that, in the large $N$ limit, it coincides with the partition function of the WZW model. This same matrix model was recently shown to describe non-Abelian quantum Hall states and the relationship to the WZW model can be understood in this framework., Comment: 28 pages + appendices. v2: references added. v3: another reference, version to be published in JHEP

We propose a matrix quantum mechanics for a class of non-Abelian quantum Hall states. The model describes electrons which carry an internal SU(p) spin. The ground states of the matrix model include spin-singlet generalisations of the Moore-Read and Read-Rezayi states and, in general, lie in a class previously introduced by Blok and Wen. The effective action for these states is a U(p) Chern-Simons theory. We show how the matrix model can be derived from quantisation of the vortices in this Chern-Simons theory and how the matrix model ground states can be reconstructed as correlation functions in the boundary WZW model., Comment: 35 pages, 2 figures. v2: references added. v3: minor changes, version to be published in Phys. Rev. B

In d=2+1 dimensions, there exist field theories which are non-relativistic and superconformal. These theories describe two species of anyons, whose spins differ by 1/2, interacting in a harmonic trap. We compute the dimensions of chiral primary operators. These operators receive large anomalous dimensions which are related to the unusual angular momentum properties of anyons. Surprisingly, we find that the dimensions of some chiral primary operators violate the unitarity bound and we trace this to the fact that the associated wavefunctions become non-normalisable. We also study BPS non-perturbative states in this theory: these are Jackiw-Pi vortices. We show that these emerge at exactly the point where perturbative operators hit the unitarity bound. To describe the low-energy dynamics of these vortices, we construct a novel type of supersymmetric gauged linear sigma model., Comment: 32 pages + appendix, 3 figures (plus Feynman diagrams drawn with feynMF). v2: references added. v3 typos corrected, version to appear in JHEP

The Omega-deformation is a harmonic trap, penning certain excitations near the origin in a manner consistent with supersymmetry. Here we explore the dynamics of BPS monopoles and vortices in such a trap. We pay particular attention to monopoles in the Higgs phase, when they are confined to a vortex string. Unusually for BPS solitons, the mass of these confined monopoles is quadratic in the topological charges. We compute an index theorem to determine the number of collective coordinates of confined monopoles. Despite being restricted to move on a line, we find that they have a rich dynamics. As the strength of the trap increases, the number of collective coordinates can change, sometimes with constituent monopoles disappearing, sometimes with new ones emerging., Comment: 24 pages + appendices. 5 figures. v2: references added

We introduce a supersymmetric Chern-Simons theory whose low energy physics is that of the fractional quantum Hall effect. The supersymmetry allows us to solve the theory analytically. We quantise the vortices and, by relating their dynamics to a matrix model, show that their ground state wavefunction is in the same universality class as the Laughlin state. We further construct coherent state representations of the excitations of a finite number of vortices. These are quasi-holes. By an explicit computation of the Berry phase, without resorting to a plasma analogy, we show that these excitations have fractional charge and spin., Comment: 31 pages + appendices. 4 Figures. v2: references added and further comments on the chiral edge modes. v3: introduction and abstract rewritten for a condensed matter audience. More references added. Version to be published in Physical Review B

We study the quantum dynamics of N=1 supergravity in four dimensions with a compact spatial circle. Supersymmetry ensures that the perturbative contributions to the Casimir energy on the circle cancel. However, instanton contributions remain. These render supersymmetric compactification on a circle unstable and the background dynamically decompactifies back to four dimensions. The calculation provides a testing ground for some old ideas in Euclidean quantum gravity. In particular, we show that gravitational instantons are associated to a new, infra-red scale which is naturally exponentially suppressed relative to the Planck scale and arises from the logarithmic running of the Gauss-Bonnet term. There are also some interesting technical details, including the non-cancellation of bosonic and fermionic determinants around the background of a self-dual gravitational instanton, despite the existence of supersymmetry., Comment: 48 pages. v2: Added small summary to section 4. v3: Typo corrected

Vacuum solutions admitting a hypersurface-orthogonal repeated principal null direction are an important class of 4d algebraically special spacetimes. We investigate the 5d analogues of such solutions: vacuum spacetimes admitting a hypersurface-orthogonal multiple Weyl aligned null direction (WAND). Such spacetimes fall into 4 families determined by the rank of the 3X3 matrix that defines the expansion and shear of the multiple WAND. The rank 3 and rank 0 cases have been studied previously. We investigate the 2 remaining families. We show how to define coordinates which lead to a considerable simplification of the Einstein equation with cosmological constant. The rank 2 case gives warped product and Kaluza-Klein versions of the 4d Robinson-Trautman solutions as well as some new solutions. The rank 1 case gives product, or analytically continued Schwarzschild, spacetimes., Comment: 25 pages. v2: typos corrected