Abstract: |
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 a unitary representation of Euclidean conformal group in (d-1) dimensions. The scaling dimension of the operators are related to automorphic spectra and hence to Laplacian eigenvalue 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 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 $X$. In several notable cases, our bounds are nearly saturated by known surfaces and orbifolds. 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 first nontrivial eigenvalue of Laplacian and first nontrivial eigenvalue of Dirac equation. We identify the orbifold living on the kink. If time permits, I will discuss some progress about bootstrapping 3D manifolds and connection with physical CFTs. The first part is based on a work (arXiv:2111.12716) with P. Kravchuk and D. Mazac, the spinorial story is based on ongoing work with D. Simmons-Duffin, E. Gestaeu and Y. Xu and the 3D story is based on one with James Bonifacio, P. Kravchuk and D. Mazac. |