The progress of TQFT has revealed connections between the algebraic world of tensor categories and the topological world of 3-manifolds, such as Reshetikhin-Turaev and Turaev-Viro theories. Motivated by M-theory in physics, Cho-Gang-Kim recently proposed another relation by outlining a program to construct modular data from certain classes of closed oriented 3-manifolds. In this thesis, we will talk about our mathematical exploration of this program.

We explore how skein theoretic techniques can be applied to the study of quantum

representations of mapping class groups. Of particular interest will be looking into the

asymptotic faithfulness property of quantum representations coming from unimodal versions

of representation categories of quantum groups. We then introduce a combinatorial

property on the graphical calculus of these representation categories which implies asymptotic

faithfulness. We proceed to show that this property is satisfied in some specific cases,

in short we provide support for the conjecture that these quantum representations will

always be asymptotically faithful. This will lead into a discussion of other applications

within low dimensional topology. Finally applications to topological quantum computing

will be given, introducing a potential encoding of qudits making use of these quantum

representations.

We give a construction of Turaev-Viro type (3+1)-TQFT out of a G-crossed braided spherical fusion category for G a finite group. The resulting invariant of 4-manifolds generalizes several known invariants in literature such as the Crane-Yetter invariant and Yetter's invariant from homotopy 2-types. Some concrete examples will be provided to show the calculations. If the category is concentrated only at the sector indexed by the trivial group element, a co-cycle in H^4(G,U(1)) can be introduced to produce another invariant, which reduces to the twisted Dijkgraaf-Witten theory in a special case. It can be shown that with a G-crossed braided spherical fusion category, one can construct a monoidal 2-category with certain extra structure, but these structures do not satisfy all the axioms of a spherical 2-category given by M. Mackaay. Although not proven, it is believed that our invariant is strictly different from other known invariants. It remains to see if the invariant has the power to detect any smooth structures.

We investigate the algebraic theory of symmetry-enriched topological (SET) order in (2+1)D bosonic topological phases of matter and its applications to topological quantum computing. Our goal is twofold: first, to demonstrate how an abstract categorical approach can be applied to understand phenomena in (2+1)D topological phases of matter, and second, to show how ideas from physics can be useful for categorification.

After reviewing modular tensor categories (MTCs) and their role as algebraic theories of anyons in topological phases of matter, we recall their associated quantum representations and their interpretation as quantum gates for a topological quantum computer. Next we recall the characterization of SET order in terms of $G$-crossed braided extensions of MTCs and the mathematical formalism of topological quantum computing (TQC) with anyons and symmetry defects.

We then apply modular tensor category theory to construct algebraic models of symmetry defects in multi-layer (2+1)D topological order with layer-exchange permutation symmetry. Our main result frames a correspondence between bilayer symmetry enriched topological order and monolayer topological order on surfaces with genus, illuminating a connection between quantum symmetry and topological order that first appeared in the work of condensed matter theorists Barkeshi, Ji, and Qian.

Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \ints)$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group $B_3$ with finite image enjoys a congruence subgroup property. In particular, we show that in dimensions two and three, when the projective order of the image of the braid generator $\sigma_1$ is between 2 and 5 the kernel projects onto a congruence subgroup of $\mathrm{PSL}(2,\ints)$ and compute its level. However, for each odd integer $r$ equal to at least 5, we construct a pair of non-congruence subgroups associated with three-dimensional representations. Our techniques use classification results of low dimensional braid group representations and the Fricke-Wohlfarht theorem in number theory, as well as Tim Hsu's work on generating sets for the principal congruence subgroups of $\mathrm{PSL}(2,\ints)$.

Phase transitions can be understood through the formation of Bose condensates. Anyon condensation is similarly an important tool for transitioning between systems modeled by modular tensor categories. The condensation process can be understood as a functor from one modular tensor category to another fusion category with a modular subcategory.

This dissertation focuses on understanding the condensation functor. After reviewing modular tensor categories, we present and comment on the relationship between two descriptions of the resulting category. We then present general results on the modular data of the resulting category and demonstrate how to explicitly compute the new $F$- and $R$-symbols. We finish off with some applications of our work and some speculations about where they might go from here.

Chapter 4 is devoted to implementing the functorial definition of condensation so that it can be carried out by a computer, and a full Mathematica implementation is provided in Supplemental A: Mathematica Code. Supplemental B: $(G_2)_3$ Data provides categorical data for the modular tensor category $(G_2)_3$, which is not otherwise widely available.

We provide a mathematical definition of a low energy scaling limit of a sequence of general non-relativistic quantum theories in any dimension, and apply our formalism to anyonic chains. We formulate Conjecture 2.5.3 on conditions when a unitary rational (1+1)-d conformal field theory would arise as such a limit and verify the conjecture for the Ising minimal model M(4,3) using su(2)_2 anyonic chains. Our work is based on the existence of a Fourier transform relation between Temperley-Lieb generators {e_i} and some finite stage operators of the Virasoro generators {L_m+L_{-m}} and {i(L_m-L_{-m})} for unitary minimal models M(k+2,k+1) (proven for k=2). An earlier attempt [1], called the Koo-Saleur formula, has slower convergence with characteristics that hinder the convergence of algebras of observables, an important contribution of this work. Assuming Conjecture 2.5.3, most of our main results for M(4,3) hold for higher (k >= 3) unitary minimal models M(k+2,k+1) as well. Our approach is supported by extensive numerical simulation and physical proofs in the physics literature. It is also inspired by an eventual application to an efficient simulation of conformal field theories by quantum computers. We approach the definition of the unitary evolution and correlator simulation problems in the same spirit of topological quantum field theory simulation as established by M. Freedman et. al. [2]. Under certain conditions, we present complexity theoretic hardness results on the simulation problems by using the framework of fermionic quantum computation by Bravyi and Kitaev [3].