Entanglement entropy from SU(2) Chern-Simons theory and symmetric webs

A path integral on a link complement of a three-sphere fixes a vector (the "link state") in Chern-Simons theory. The link state can be written in a certain basis with the colored link invariants as its coefficients. We use symmetric webs to systematically compute the colored link invariants, by which we can write down the multi-partite entangled state of any given link. It is still unknown if a product state necessarily implies that the corresponding components are unlinked, and we leave it as a conjecture

Quantum Groups and Integralities in Chern-Simons Theory

In this dissertation, we investigate integralities in Chern-Simons theory. The integralities of interest arise from non-local observables (Wilson lines) in Chern-Simons theory and the partition function itself. In the associated supersymmetric gauge theories (via 3d-3d correspondence), they encode certain BPS spectrum, which are often identified with homological invariants of links and three-manifolds. In this dissertation, we observe that all of them are equipped with non-trivial algebraic structures, such as quantum group actions, modularity, and logarithmic vertex algebras. In the first half of this dissertation, we identify quantum group representations with the dynamics of line operators and their lift to surface operators. In the second half, Chern-Simons partition functions on Seifert manifolds are studied in detail, and its ``hidden'' integralities are identified with quantum modular forms and the characters of logarithmic vertex operator algebra. From the latter, we also observe that quantum group actions control the ``dynamics'' of characters

Junctions of surface operators and categorification of quantum groups

We show how networks of Wilson lines realize quantum groups U_q(sl_m), for arbitrary m, in 3d SU(N) Chern-Simons theory. Lifting this construction to foams of surface operators in 4d theory we find that rich structure of junctions is encoded in combinatorics of planar diagrams. For a particular choice of surface operators we reproduce known mathematical constructions of categorical representations and categorified quantum groups

3d Modularity

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs.Comment: 119 pages, 10 figures and 20 table

3d-3d correspondence for mapping tori

One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d N = 2 SCFT T [M₃] — or, rather, a “collection of SCFTs” as we refer to it in the paper — for all types of 3-manifolds that include, for example, a 3-torus, Brieskorn spheres, and hyperbolic surgeries on knots. The goal of this paper is to overcome this challenge by a more systematic study of 3d-3d correspondence that, first of all, does not rely heavily on any geometric structure on M₃ and, secondly, is not limited to a particular supersymmetric partition function of T [M₃]. In particular, we propose to describe such “collection of SCFTs” in terms of 3d N = 2 gauge theories with “non-linear matter” fields valued in complex group manifolds. As a result, we are able to recover familiar 3-manifold invariants, such as Turaev torsion and WRT invariants, from twisted indices and half-indices of T [M₃], and propose new tools to compute more recent q-series invariants Ẑ (M₃) in the case of manifolds with b₁ > 0. Although we use genus-1 mapping tori as our “case study,” many results and techniques readily apply to more general 3-manifolds, as we illustrate throughout the paper

3d-3d correspondence for mapping tori

One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d $N=2$ SCFT $T[M_3]$ --- or, rather, a "collection of SCFTs" as we refer to it in the paper --- for all types of 3-manifolds that include, for example, a 3-torus, Brieskorn spheres, and hyperbolic surgeries on knots. The goal of this paper is to overcome this challenge by a more systematic study of 3d-3d correspondence that, first of all, does not rely heavily on any geometric structure on $M_3$ and, secondly, is not limited to a particular supersymmetric partition function of $T[M_3]$. In particular, we propose to describe such "collection of SCFTs" in terms of 3d $N=2$ gauge theories with "non-linear matter'' fields valued in complex group manifolds. As a result, we are able to recover familiar 3-manifold invariants, such as Turaev torsion and WRT invariants, from twisted indices and half-indices of $T[M_3]$, and propose new tools to compute more recent $q$-series invariants $\hat Z (M_3)$ in the case of manifolds with $b_1 > 0$. Although we use genus-1 mapping tori as our "case study," many results and techniques readily apply to more general 3-manifolds, as we illustrate throughout the paper.Comment: 53 pages, 8 figure

3-Manifolds and VOA Characters

By studying the properties of $q$-series $\widehat Z$-invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to 3-manifolds with toral boundaries, and to BPS partition functions with line operators. This provides a new physical realization of logarithmic vertex algebras in the framework of the 3d-3d correspondence and opens new avenues for their future study. For example, we illustrate how invoking a knot-quiver correspondence for $\widehat{Z}$-invariants leads to many infinite families of new fermionic formulae for VOA characters.Comment: 85 pages, 3 figures, 6 table

3d modularity

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d N = 2 theories where such structures a priori are not manifest. These modular structures include: mock modular forms, SL(2,ℤ) Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs