Spectral Structure of the Mixed Hessian of the Dispersionless Toda $\tau$-Function

TL;DR

This paper analyzes the spectral properties of the mixed Hessian of the dispersionless Toda tau-function for symmetric conformal maps, revealing a rank-one instability at a critical threshold and extending scalar functions beyond it.

Contribution

It introduces a spectral analysis framework for the Hessian of the dispersionless Toda tau-function, identifying a specific spectral transition and extending scalar functions beyond this point.

Findings

Spectral transition occurs at the analytic threshold 

One eigenvalue diverges logarithmically at the transition

Scalar functions remain regular beyond the spectral transition

Abstract

We study the mixed Hessian of the dispersionless Toda $\tau$-function for the one-harmonic $s$-fold symmetric conformal map $f(w)=rw+aw^{1-s}$. This Hessian is the susceptibility matrix generated by the inverse conformal map. Our spectral statements are formulated for its weighted symmetry-block realizations on a fixed Hilbert space. In that realization, the first spectral transition occurs at the analytic threshold $\zeta_c=(s-1)^{s-1}/s^s$, where the dominant square-root singularity of the inverse map reaches the normalization circle, rather than at the geometric threshold $\zeta_{\mathrm{univ}}=1/(s-1)$, where univalence fails. After symmetry decomposition and weighted realization, each block develops exactly one logarithmically diverging eigenvalue as $\zeta\uparrow\zeta_c$, while the remaining spectrum stays bounded and converges to a compact limit. The instability is therefore rank one in every symmetry sector of the weighted block theory. We then continue the scalar Gram generating functions beyond $\zeta_c$. They are generalized hypergeometric functions on the slit plane $\mathbb{C}\setminus[\zeta_c^2,\infty)$, their branch-point expansion contains the logarithmic term responsible for the divergence, and in the range $1\le p\le s$ they admit Cauchy--Stieltjes and Jacobi-matrix realizations. In particular, the continued scalar quantities remain regular at $\zeta_{\mathrm{univ}}$, so the analytic spectral transition strictly precedes the geometric breakdown of univalence.