Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 090, 47 pages      arXiv:2508.10087      https://doi.org/10.3842/SIGMA.2026.090

$c_{\rm eff}$ from Surgery and Modularity

Shimal Harichurn a, Mrunmay Jagadale b, Dmitry Noshchenko c and Davide Passaro b
a) Durban, KwaZulu-Natal, South Africa
b) Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
c) School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, D04 C932, Ireland

Received November 29, 2025, in final form August 23, 2026; Published online September 22, 2026

Abstract
$\widehat{Z}$ invariants, rigorously defined for negative definite plumbed 3-manifolds, are expected - on physical grounds - to exist for every closed, oriented 3-manifold (Gukov et al., 2020). Several prescriptions have been proposed to extend their definition to generic plumbings by reversing the orientation of a negative definite plumbing, thus turning it into a positive definite one (Cheng et al., 2019). Two existing proposals are relevant for this paper: (i) the regularized $+1/r$-surgery conjecture combined with the false-mock modular conjecture (Park, 2021 and Cheng et al., 2024), and (ii) a construction based on resurgence and a false theta function duality (Costin et al., 2023). In this note, we compare these proposals on the class of Brieskorn homology spheres $\overline{\Sigma(s,t,rst\pm1)}$ and find that they are incompatible in general. Our diagnostic is the effective central charge, $c_{\rm eff}$, which governs the asymptotic growth of coefficients of $\widehat{Z}$ (Gukov and Jagadale, 2024). First, we prove that the upper bound on $c_{\rm eff}$ from prescription (i) is governed by the Ramanujan theta function, which regularizes the surgery formula. Second, we develop numerical and modular tools that deliver the lower bounds as well as exact values via mixed mock-modular analysis. Complementing this, we also study $c_{\rm eff}$ for negative definite plumbed 3-manifolds which allow for a better comparison of pairs of 3-manifolds related by orientation reversal. As a result, we find that for some Brieskorn spheres the surgery and false-mock prescriptions violate the expected relation between $c_{\rm eff}$, Chern-Simons invariants and non-abelian flat connections.These findings underscore $c_{\rm eff}$ as a sensitive probe of the ''positive side'' of $\widehat{Z}$-theory.

Key words: effective central charge; GPPV invariants; 3-manifolds; surgery formulae; Chern-Simons theory; $q$-series.

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