Hyperbolic Manifolds and Conformal Bootstrap

We point out that the spectral geometry of hyperbolic orbifold provides a remarkably precise model of conformal field theory. Given a d-dimensional hyperbolic orbifold, one can construct a Hilbert space of local operators, living in an emergent (d-1) dimension and transforming as unitary representation of the Euclidean conformal group in (d-1) dimensions. The scaling dimensions of these operators are related to the Laplacian eigenvalues on the orbifold. One can further introduce a notion of operator product expansion (OPE) and correlation functions among these operators. The associativity of OPE leads to bootstrap/crossing equations, which can then be used to put rigorous bounds on Laplacian eigenvalues on the orbifold. Specifically, we use conformal bootstrap techniques to derive rigorous computer-assisted upper bounds on the lowest positive eigenvalue $\lambda_1(X)$ of the Laplace-Beltrami operator on closed hyperbolic surfaces and 2-orbifolds $X$. Our bounds are nearly saturated by known surfaces and orbifolds in several notable cases. For instance, our bound on all genus-2 surfaces $X$ is $\lambda_1(X)\leq 3.8388976481$, while the Bolza surface has $\lambda_1(X)\approx 3.838887258$. We use the bounds to identify the set of first nontrivial eigenvalues attained in hyperbolic orbifolds. Including spinors in the game, we produce exclusion plots on the plane of the first nontrivial eigenvalue of the Laplacian and the first nontrivial eigenvalue of the Dirac operator on a spin orbifold.  We identify the orbifolds living on the kink appearing in these exclusion plots. If time permits, I will discuss how a notion of “Euclidean  positivity” emerges out of this framework and some progress toward the application of leveraging “Euclidean positivity” in the conformal bootstrap program.