Vanishing-radius conjecture for Hodge-integral generating series

Establish whether the formal power series \(f_{j,k}(\mu)\), defined by the Hodge integrals \(\sum_{g\geq\max(j,k)}\mu^{2g-j-k}\int_{\overline{\mathcal M}_{g,1}}\lambda_g\lambda_{g-j}\lambda_{g-k}\psi_1^{j+k-2}\), has radius of convergence zero whenever \(\max(j,k)\geq2\).

Background

The formal series fj,k(μ)f_{j,k}(\mu) control scalar terms in the quantum ILW Hamiltonians after the deep-water specialization. The paper proves that f1,2(μ)f_{1,2}(\mu) has radius of convergence zero using Bernoulli-number estimates, and then proposes the analogous statement for all pairs with max⁡(j,k)≥2\max(j,k)\geq2. This assertion motivates the normalization procedure used in the main conjecture.

References

However, we believe that if $\max(j,k)\ge 2$, then the radius of convergence of the formal power series $f_{j,k}(\mu)$ is zero.

— Quantum hierarchies of the ILW and BO equations and the DR cycles on the moduli spaces of curves  (2610.01841 - Buryak et al., 1 Oct 2026) in Section 2, subsection “Conjecture”