Algebraic Equations Characterizing Hyperbolic Surface Spectra

by Anshul Adve (Princeton University)

Thursday Oct 3, 2024, 2:30 PM → 4:00 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Given a compact hyperbolic surface together with a suitable choice of orthonormal basis of Laplace eigenforms, one can consider two natural spectral invariants: 1) the Laplace spectrum $\Lambda$, and 2) the 3-tensor C_ijk representing pointwise multiplication (as a densely defined map L² x L²  $\to$ L²) in the given basis. Which pairs ($\Lambda$,C) arise this way? Both $\Lambda$ and C are highly transcendental objects. Nevertheless, we will give a concrete and almost completely algebraic answer to this question, by writing down necessary and sufficient conditions in the form of equations satisfied by the Laplace eigenvalues and the C_ijk. This answer was suggested by physicists Kravchuk, Mazac, and Pal, who introduced these equations (in an equivalent form) as a rigorous model for the crossing equations in conformal field theory.

 

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