Gauging Lie group symmetry in (2+1)d topological phases

It is well-known that if we gauge a $\mathbb{Z}_n$ symmetry in two dimensions, a dual $\mathbb{Z}_n$ symmetry appears, such that re-gauging this dual $\mathbb{Z}_n$ symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-ano...

We systematically study Lorentz symmetry extensions in quantum field theories (QFTs) and their 't Hooft anomalies via cobordism. The total symmetry $G'$ can be expressed in terms of the extension of Lorentz symmetry $G_L$ by an internal global symmetry $G$ as $1 \to G \to G' \to G_L \to 1$. By enumerating all possible $G_L$ and symmetry extensions, other than the familiar SO/Spin/O/Pin$^{\pm}$ groups, we introduce a new EPin group (in contrast to DPin), and provide natural ph...

It is a long-standing question to extend the definition of 3-dimensional Chern-Simons theory to one which associates values to 1-manifolds with boundary and to 0-manifolds. We provide a solution in case the gauge group is a torus. We also develop from different points of view an associated 4-dimensional invertible topological field theory which encodes the anomaly of Chern-Simons. Finite gauge groups are also revisited, and we describe a theory of "finite path integrals" as a...

We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on $\mathbb{Z}_N$ one-form symmetries. A 3d topological quantum field theory (TQFT) $\mathcal{T}$ with such a symmetry has $N$ special lines that generate it. The braiding of these lines and their spins are characterized by a single integer $p$ modulo $2N$. Surprisingly, if $\gcd(N,p)=1$ the TQFT factorizes $\mathcal{T}=\mathcal{T}'\otime...

This thesis aims at concluding the classification results for topological phases with symmetry in 2+1 dimensions. The main result is that topological phases are classified by a triple of unitary braided fusion categories $\mathcal E\subset\mathcal C\subset\mathcal M$ plus the chiral central charge $c$. Here $\mathcal E$ is a symmetric fusion category, $\mathcal E=\mathrm{Rep}(G)$ for boson systems or $\mathcal E=\mathrm{sRep}(G^f)$ for fermion systems, consisting of the repre...

Topology and generalized symmetries in the $SU(N)/\mathbb{Z}_N$ gauge theory are considered in the continuum and the lattice. Starting from the $SU(N)$ gauge theory with the 't~Hooft twisted boundary condition, we give a simpler explanation of the van~Baal's proof on the fractionality of the topological charge. This description is applicable to both continuum and lattice by using the generalized L\"uscher's construction of topology on the lattice. Thus we can recover the $SU(...

It is well-known that gauging a finite 0-form symmetry in a quantum field theory leads to a dual symmetry generated by topological Wilson line defects. These are described by the representations of the 0-form symmetry group which form a 1-category. We argue that for a d-dimensional quantum field theory the full set of dual symmetries one obtains is in fact much larger and is described by a (d-1)-category, which is formed out of lower-dimensional topological quantum field theo...

We discuss invertible and non-invertible topological condensation defects arising from gauging a discrete higher-form symmetry on a higher codimensional manifold in spacetime, which we define as higher gauging. A $q$-form symmetry is called $p$-gaugeable if it can be gauged on a codimension-$p$ manifold in spacetime. We focus on 1-gaugeable 1-form symmetries in general 2+1d QFT, and gauge them on a surface in spacetime. The universal fusion rules of the resulting invertible a...

We study 't Hooft anomalies for discrete global symmetries in bosonic theories in 2, 3 and 4 dimensions. We show that such anomalies may arise in gauge theories with topological terms in the action, if the total symmetry group is a nontrivial extension of the global symmetry by the gauge symmetry. Sometimes the 't Hooft anomaly for a d-dimensional theory with a global symmetry G can be canceled by anomaly inflow from a (d+1)-dimensional topological gauge theory with gauge gro...

While two-dimensional symmetry-enriched topological phases ($\mathsf{SET}$s) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry $G_s$ on gauge theories (denoted by $\mathsf{GT}$) with gauge group $G_g$. The resulting symmetric gauge theories are dubbed "symmetry-enriched gauge theories" ($\mathsf{SEG}$), which may be served as low-energy effective theories of three-dime...