Joint IAS/PU Groups and Dynamics Seminar

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 $L^2 \times L^2 \to L^2$) 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 conjectured 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.