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

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. 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. Two existing proposals are relevant for this paper: (i) the regularized $+1/r$-surgery conjecture combined with the false-mock modular conjecture, and (ii) a construction based on resurgence and a false theta function duality. In this note, we compare these proposals on the class of Brieskorn homology spheres $\Sigma\left(s,t,rst\pm1\right)$ and find that they are incompatible in general. Our diagnostic is the effective central charge, $c_{\text{eff}}$, which governs the asymptotic growth of coefficients of $\widehat{Z}$. First, we prove that the upper bound on $c_{\text{eff}}$ 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 $c_{\text{eff}}$ 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 $c_{\text{eff}}$, Chern-Simons invariants and non-abelian flat connections. These findings underscore $\widehat{Z}$ as a sensitive probe of the "positive side" of $\widehat{Z}$-theory.