Geometric Constraints on CFT-Like Data at Genus 0, 1, and 2.

This thesis studies how self-consistency constraints on conformal field theory like observables can be used to extract information about CFT or geometric data. The observables considered are associated with genus-0, genus-1, and genus-2 geometries: the four-punctured sphere, the torus S1 x Sd-1, and the genus-2 manifold (S1 x Sd-1)♯(S1 x Sd-1). Although these observables differ in geometric and analytic complexity, they are united by a common principle: the same observable admits multiple equivalent decompositions, and the requirement that these descriptions agree imposes nontrivial constraints on the underlying spectrum and operator product expansion. The first part develops a hyperbolic bootstrap for compact hyperbolic spin surfaces. Using an analogy between harmonic analysis on hyperbolic manifolds and the conformal bootstrap, we derive rigorous bounds on Laplacian and Dirac spectral gaps by combining representation-theoretic methods with semidefinite programming. The second part studies thermal partition functions with angular twists. We show that, even in the absence of full modular invariance in dimensions greater than two, the high-temperature behavior of these observables is controlled by thermal effective field theory, leading to universal relations among twisted partition functions. The third part studies the genus-2 partition function in conformal field theory. By equating different decompositions of this observable, we derive a genus-2 crossing equation and use it to relate thermal one-point data to heavy operator product coefficients.