A converse theorem for hyperbolic surface spectra and the conformal bootstrap

The conformal bootstrap in physics has recently been adapted to prove remarkably sharp estimates on Laplace eigenvalues and triple correlations of automorphic forms on compact hyperbolic surfaces. These estimates derive from an infinite family of algebraic equations satisfied by this spectral data. The equations encode $G$-equivariance and associativity of multiplication on $\mathrm{\Gamma }\mathrm{\backslash }G$, for $\mathrm{\Gamma }$ a cocompact lattice in $G={\text{PSL}}_2(\mathbf{R})$. The effectiveness of the conformal bootstrap suggests that the equations characterize hyperbolic surface spectra, i.e., that every solution to the equations comes from a compact hyperbolic surface. This paper proves this rigorously with no analytic assumptions on the solution except discreteness of the spectrum. The key intermediate result is an axiomatic characterization of representations of $G$ of the form $L^2(\mathrm{\Gamma }\mathrm{\backslash }G)$.