Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

3d Modularity

Miranda C. N. Cheng, Sungbong Chun, Francesca Ferrari, Sergei Gukov, Sarah M. Harrison

TL;DR

3d Modularity builds a unifying framework linking topology, 3d N=2 field theories, and number theory through hat{Z}_a(q), a family of q-series functioning as supersymmetric indices, 3-manifold invariants, and quantum modular forms. The authors identify three distinct SL(2,Z) structures—twisted-indices MTCs, abelian S^{(A)} data, and Weil-representation-based S^{(B)} data—tying modular behavior to both abelian and non-abelian flat connections via resurgence and Eichler integrals. A key insight is that hat{Z}_a(q) often realize as characters of logarithmic VOAs, connecting non-semisimple MTCs and log-CFTs to 3-manifold invariants, especially for Seifert and Brieskorn manifolds; hyperbolic cases suggest non-C2-cofinite log-VOAs. The paper provides extensive concrete examples, showing how q-series, their transseries, and A-polynomials encode flat-connection data, while folding by centers and symmetries clarifies modular structure. Together, these results establish a powerful bridge among 3-manifold topology, 3d-3d physics, and number theory, with implications for WRT invariants, resurgence, and log-CFT representation theory.

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.

3d Modularity

TL;DR

3d Modularity builds a unifying framework linking topology, 3d N=2 field theories, and number theory through hat{Z}_a(q), a family of q-series functioning as supersymmetric indices, 3-manifold invariants, and quantum modular forms. The authors identify three distinct SL(2,Z) structures—twisted-indices MTCs, abelian S^{(A)} data, and Weil-representation-based S^{(B)} data—tying modular behavior to both abelian and non-abelian flat connections via resurgence and Eichler integrals. A key insight is that hat{Z}_a(q) often realize as characters of logarithmic VOAs, connecting non-semisimple MTCs and log-CFTs to 3-manifold invariants, especially for Seifert and Brieskorn manifolds; hyperbolic cases suggest non-C2-cofinite log-VOAs. The paper provides extensive concrete examples, showing how q-series, their transseries, and A-polynomials encode flat-connection data, while folding by centers and symmetries clarifies modular structure. Together, these results establish a powerful bridge among 3-manifold topology, 3d-3d physics, and number theory, with implications for WRT invariants, resurgence, and log-CFT representation theory.

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 theories where such structures a priori are not manifest. These modular structures include: mock modular forms, Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs.

Paper Structure

This paper contains 62 sections, 4 theorems, 280 equations, 10 figures, 20 tables.

Table of Contents

  1. Introduction and summary
  2. ... for physicists
  3. ... for topologists
  4. ... for number theorists
  5. From mock to modular, via 3d $\mathcal{N}=2$ theories
  6. The half-index and three-manifolds
  7. Three encounters of modularity
  8. Twisted indices of 3d $\mathcal{N}=2$ theories
  9. $\widehat{Z}_a$ and non-semisimple MTCs
  10. Kazhdan-Lusztig correspondence
  11. The Weil representations
  12. Resurgence and modularity
  13. False theta functions and the asymptotic expansions
  14. Resurgence and Eichler integrals
  15. Flat connections from modularity
  16. ...and 47 more sections

Key Result

Lemma 1

Take $M_3$ to be a plumbed 3-manifold, whose plumbing graph $\mathcal{G}$ is a tree. Denote by $M$ the adjacency matrix of $\mathcal{G}$ and by $M^{-1}$ its inverse. Assume there is only one high-valency vertex and let $v_0$ denote the entry associated to this vertex in the adjacency matrix. Then, i

Figures (10)

  • Figure 1: The different topics involved in this paper.
  • Figure 2: A 3d $\mathcal{N}=2$ theory with a 2d $\mathcal{N}=(0,2)$ boundary condition $\mathcal{B}_a$.
  • Figure 3: A homological block (a.k.a. half-index) counts BPS states of 3d $\mathcal{N}=2$ theory on $(\text{time}) \times (\text{cigar})$.
  • Figure 4: The limit $q \to e^{2\pi i /k}$, with $k \in \mathbb{Z}$, enters many aspects of our story: the Kazhdan-Lusztig correspondence, the relation between $\widehat{Z}_a (M_3)$ and WRT invariants, the relation between mock modular forms and false thetas, etc.
  • Figure 5: From plumbing data to flat connections.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Theorem 4
  • Lemma 5
  • proof
  • Theorem 6
  • proof