Mock Modularity, Resurgence, and Dual False Theta Functions
Abstract
The Ẑ invariant of a three-manifold M3, a half-index of the 3d 𝒩=2 theory T[M3], is well understood for negative-definite plumbed three-manifolds. For orientation-reversed manifolds, however, the known formulas fail to produce q-series, and making sense of the operation q → q−1 remains a central open problem. Seifert manifolds, whose Ẑ invariants are built from false theta functions, offer a natural testing ground. We study ``dual false theta functions'', the objects that should replace false theta functions under orientation reversal. Two characterizations of these duals exist in the literature: a modular one, based on mock modularity with a prescribed shadow, and a resurgent one, based on Borel--Mordell integrals and transseries. We prove that the two characterizations are equivalent. We construct a new family of dual false theta functions for values of positive integer p satisfying a Pell-equation condition. Its members are built from Zwegers' indefinite theta functions, have integer coefficients, and have effective central charge ceff ≤ 1. We also show that the mock theta functions of Li and Schwagenscheidt satisfy these characterizations and arise from a natural regularization of the divergent product η3(τ)ψp, a(− τ).
Related
- The Corners of Ramanujan: Partial- and False-Theta Companions to Third-Order Mock Theta Functions
- Congruences modulo powers of 5 and 7 for the crank and rank parity functions and related mock theta-functions
- Resurgent rigidity of mock theta functions: uniqueness and natural boundary crossing
- Resurgence and Modularity in the First Painlevé Equation
- Modularity of higher theta series I: cohomology of the generic fiber
BibTeX
@misc{jagadale2026mock,
title = {{Mock Modularity, Resurgence, and Dual False Theta Functions}},
author = {Mrunmay Jagadale},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00516},
eprint = {2610.00516},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00516},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mock Modularity, Resurgence, and Dual False Theta Functions
AU - Mrunmay Jagadale
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.00516
UR - https://arxiv.org/abs/2610.00516
AB - The Ẑ invariant of a three-manifold M_3, a half-index of the 3d 𝒩=2 theory T[M_3], is well understood for negative-definite plumbed three-manifolds. For orientation-reversed manifolds, however, the known formulas fail to produce q-series, and making sense of the operation q → q^−1 remains a central open problem. Seifert manifolds, whose Ẑ invariants are built from false theta functions, offer a natural testing ground. We study ``dual false theta functions'', the objects that should replace false theta functions under orientation reversal. Two characterizations of these duals exist in the literature: a modular one, based on mock modularity with a prescribed shadow, and a resurgent one, based on Borel--Mordell integrals and transseries. We prove that the two characterizations are equivalent. We construct a new family of dual false theta functions for values of positive integer p satisfying a Pell-equation condition. Its members are built from Zwegers' indefinite theta functions, have integer coefficients, and have effective central charge c_eff ≤ 1. We also show that the mock theta functions of Li and Schwagenscheidt satisfy these characterizations and arise from a natural regularization of the divergent product η^3(τ)ψ_p, a(− τ).
ER -