Conjectures on spectral numbers for upper triangular matrices and for singularities

TL;DR

This paper investigates spectral numbers associated with upper triangular matrices and singularities, clarifies previous conjectures, and proposes extensions, with positive results for chain type singularities and potential applications in algebraic geometry.

Contribution

The paper rigorously analyzes Cecotti and Vafa's conjectures on spectral numbers, fixing previous results and proposing precise conjectures for all matrices, especially in singularity and algebraic geometry contexts.

Findings

Spectral numbers match well with polarized mixed Hodge structures in certain cases.

Positive results obtained for chain type singularities.

Formulated precise conjectures for general matrices and singularities.

Abstract

Cecotti and Vafa proposed in 1993 a beautiful idea how to associate spectral numbers $\alpha_1,...,\alpha_n\in{\mathbb R}$ to real upper triangular $n\times n$ matrices $S$ with 1's on the diagonal and eigenvalues of $S^{-1}S^t$ in the unit sphere. Especially, $\exp(-2\pi i\alpha_j)$ shall be the eigenvalues of $S^{-1}S^t$. We tried to make their idea rigorous, but we succeeded only partially. This paper fixes our results and our conjectures. For certain subfamilies of matrices their idea works marvellously, and there the spectral numbers fit well to natural (split) polarized mixed Hodge structures. We formulate precise conjectures saying how this should extend to all matrices $S$ as above. The idea might become relevant in the context of semiorthogonal decompositions in derived algebraic geometry. Our main interest are the cases of Stokes like matrices which are associated to holomorphic functions with isolated singularities (Landau-Ginzburg models). Also there we formulate precise conjectures (which overlap with expectations of Cecotti and Vafa). In the case of the chain type singularities, we have positive results. We hope that this paper will be useful for further studies of the idea of Cecotti and Vafa.