2017/12/01 by Sven Balnojan, Balnojan, Sven, Claus Hertling +1 · 1 citation
Mathematics · #15B05 #32S25 #32S35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1712.00388
openalex publication_date 2017/12/01 · openalex created_date 2022/09/27 · openalex updated_date 2026/07/28
Cecotti and Vafa proposed in 1993 a beautiful idea how to associate spectral\nnumbers \α1,...,\αn\∈ mathbb R to real upper triangular n\×\nn matrices S with 1's on the diagonal and eigenvalues of S-1St in the\nunit sphere. Especially, \exp(-2\π i\αj) shall be the eigenvalues of\nS-1St.\n We tried to make their idea rigorous, but we succeeded only partially. This\npaper fixes our results and our conjectures. For certain subfamilies of\nmatrices their idea works marvellously, and there the spectral numbers fit well\nto natural (split) polarized mixed Hodge structures. We formulate precise\nconjectures saying how this should extend to all matrices S as above.\n The idea might become relevant in the context of semiorthogonal\ndecompositions in derived algebraic geometry. Our main interest are the cases\nof Stokes like matrices which are associated to holomorphic functions with\nisolated singularities (Landau-Ginzburg models). Also there we formulate\nprecise conjectures (which overlap with expectations of Cecotti and Vafa). In\nthe case of the chain type singularities, we have positive results.\n We hope that this paper will be useful for further studies of the idea of\nCecotti and Vafa.\n