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.
One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d $$ \mathcal{N} $$
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 $$ \mathcal{N} $$
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 $$ \hat{Z} $$
Z
̂
(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.
Abstract: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 make a connection to known mathematical constructions of categorical representations and categorified quantum groups.
By studying the properties of q-series 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 Zinvariants leads to many infinite families of new fermionic formulae for VOA characters.
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