Conformal Bootstrap and Automorphic Spectra

Sridip Pal

Caltech

Wed, Dec. 14th 2022, 14:15

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

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 and conjecture the set of first nontrivial eigenvalues attained in hyperbolic orbifolds. I will particularly focus on this conjecture about the structure of the set of nontrivial eigenvalues attainable on a hyperbolic orbifold and how we go about proving a big part of it. If time permits, I will discuss the connection of this problem with the classic moment problem and some progress about bootstrapping 3D manifolds. This is based on a work (arXiv:2111.12716) with P. Kravchuk and D. Mazac and some ongoing work with James Bonifacio.

Contact : Ioannis
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