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

Technical Summary of "ceff from Surgery and Modularity"

Problem Statement
The paper addresses the challenge of defining and understanding $\widehat{Z}$ invariants for general closed, oriented 3-manifolds, specifically focusing on positive definite plumbed manifolds such as Brieskorn homology spheres $\Sigma(s, t, rst \pm 1)$. While $\widehat{Z}$ invariants are rigorously defined for negative definite plumbed manifolds via convergent $q$-series, their extension to the positive definite case remains an open problem. Two primary proposals exist for this extension:

1. The Regularized $+1/r$-Surgery Conjecture: Combining surgery formulae with the false-mock modular conjecture (Park; Cheng et al.), which posits that $\widehat{Z}$ for positive definite manifolds can be expressed via indefinite theta series and mock modular forms.
2. The Resurgence Approach: Based on Borel resummation of the Chern–Simons path integral (Costin et al.), which relates $\widehat{Z}$ to a trans-series expansion involving non-abelian flat connections.

The central question is whether these two approaches yield compatible results, particularly regarding the effective central charge, $c_{\text{eff}}$, which governs the asymptotic growth of the coefficients of the $\widehat{Z}$ series. Physically, $c_{\text{eff}}$ is expected to be related to the Chern–Simons invariants of non-abelian flat connections on the manifold.

Methodology
The authors employ a multi-faceted approach combining numerical analysis, modular form theory, and resurgence techniques to compute and compare $c_{\text{eff}}$ for the class of Brieskorn spheres $\Sigma(s, t, rst \pm 1)$.

1. Upper Bounds via Regularization: The authors analyze the conjectural regularized $+1/r$-surgery formula. They prove that the growth rate of the coefficients (and thus the upper bound of $c_{\text{eff}}$) is governed entirely by the regularization factor (specifically the Ramanujan theta function or the Dedekind eta function inverse) rather than the specific knot data. This establishes a strict upper bound for $c_{\text{eff}}$ dependent only on the surgery slope $r$.
2. Numerical Analysis: Using the first $10^4$ terms of the $q$-series expansion derived from the surgery formula, the authors numerically estimate the lower bounds of $c_{\text{eff}}$. They analyze the singularity structure of the series on the unit circle to determine the dominant singularities, which dictate the asymptotic behavior.
3. Modular Completion and Mock Modularity: For the subclass $\Sigma(s, t, str + 1)$, where the invariant is known to be a mixed mock modular form, the authors construct the modular completion of $\widehat{Z}$. By performing the modular $S$-transformation, they decompose the invariant into holomorphic and non-holomorphic parts ($W(\tau)$, $Z_{G_{c_1}}$, and $Z_{G_{c_2}}$). They perform an asymptotic analysis of these components to derive exact values of $c_{\text{eff}}$.
4. Comparison with Resurgence: The derived values from the modular approach are compared against results obtained from the resurgence framework (specifically the work of Adams et al.), which predicts $c_{\text{eff}}$ based on the trans-series expansion associated with flat connections.

Key Contributions and Results

• Incompatibility of Proposals: The primary result is that the regularized $+1/r$-surgery conjecture and the resurgence-based trans-series expansion are generically incompatible. For many Brieskorn spheres, the two methods yield significantly different values for $c_{\text{eff}}$.
• Violation of the Chern–Simons Relation: The paper demonstrates that for several examples (e.g., $\Sigma(3, 4, 13)$, $\Sigma(3, 4, 11)$), the $c_{\text{eff}}$ computed via the surgery formula does not correspond to any Chern–Simons invariant of a flat connection. Specifically, the computed $c_{\text{eff}}$ often leads to irrational or non-integer parameters $m$ in the relation $c_{\text{eff}} = 24(m^2/4st(rst \pm 1) - l)$, whereas the resurgence framework predicts integer values corresponding to specific flat connections.
• Exact Values via Mock Modularity: For the class $\Sigma(s, t, str + 1)$, the authors provide exact analytical expressions for $c_{\text{eff}}$ using the modular completion of the mixed mock modular form. They find that while the modular analysis agrees with numerical estimates, it frequently produces values that do not fit the expected topological interpretation (i.e., they do not match the Chern–Simons invariants of non-abelian flat connections).
• Singularity Structure: The authors identify that for $r > 1$, the dominant singularity of the $\widehat{Z}$ series is not always at $q=1$ but can shift to roots of unity (e.g., $q = e^{-2\pi i 2/5}$ for $\Sigma(2, 3, 13)$), a feature that must be accounted for in the asymptotic analysis.
• Negative Definite Case: As a complementary result, the authors prove that for negative definite plumbed manifolds, the coefficients of $\widehat{Z}$ are polynomially bounded, implying $c_{\text{eff}} \leq 1$ (after appropriate normalization), providing a coherent baseline for comparison.

Significance
The paper claims that $c_{\text{eff}}$ serves as a sensitive probe for the "positive side" of $\widehat{Z}$-theory. The findings suggest that the current conjectural definitions of $\widehat{Z}$ for positive definite manifolds—whether via surgery formulae or resurgence—may require refinement or that the relationship between these invariants and the Chern–Simons theory of flat connections is more subtle than previously assumed. The incompatibility between the modular and resurgent approaches highlights a gap in the current understanding of the modular structure of $\widehat{Z}$ invariants on the "positive side." The authors conclude that these inconsistencies underscore the need for a more comprehensive definition of $\widehat{Z}$ for general 3-manifolds and suggest that the effective central charge is a crucial diagnostic tool in this pursuit.