3d Modularity

TL;DR

This paper explores the underlying modularity structures in 3-manifold topology and 3d $ ext{N}=2$ theories, revealing connections to various advanced mathematical and physical concepts.

Contribution

It provides an explanation for the emergence of diverse modularity-related symmetries in these fields, which were previously not manifest.

Findings

Identification of modular structures in 3-manifold problems and 3d theories

Connection between mock modular forms and physical theories

Explanation for non-manifest modular symmetries

Abstract

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.