TH Journal Club on Strings & QFT

Automorphic Spectra and the Conformal Bootstrap

by Sridip Pal (IAS)

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We point out that the spectral geometry of hyperbolic manifolds provides a remarkably precise model of the modern conformal bootstrap. As an application, we use conformal bootstrap techniques to derive rigorous computer-assisted upper bounds on the lowest positive eigenvalue $\lambda_1(X)$ of theLaplace-Beltrami operator on closed hyperbolic surfaces and 2-orbifolds $X$. In a number of 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 $\lambda_1(X)\approx 3.838887258$. 

I will explain that hyperbolic surface are of the form H\G/K with G=PSL(2,R), K=SO(2) and H being Fuchsian group. For a given hyperbolic surface, one can define a Hilbert space of local operators, transforming under a unitary irrep of a conformal group (PSL(2,R)) and introduce a notion of operator product expanion (OPE). The associativity of this OPE reflects the associativity of function multiplication on the space H\G and leads to the bootstrap equations. Now the functions on H\G can be thought of automorphic forms on the surface H\G/K and I will show that the scaling dimensions of these operators are in fact related to the automorphic spectra in particular the Laplacian eigenvalues on the manifold. Hence the bootstrap equations lead to the bound on the Laplacian eigenvalues.