GQFI-UGent-WST Seminar: Sridip Pal "Automorphic Spectra and the Conformal Bootstrap" скачать в хорошем качестве

GQFI-UGent-WST Seminar: Sridip Pal "Automorphic Spectra and the Conformal Bootstrap"

In this talk, Sridip Pal (IAS) points out that the spectral geometry of hyperbolic manifolds provides a remarkably precise model of the modern conformal bootstrap. As an application, Sridip uses 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 a number of notable cases, the 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 $\lambda_1(X)\approx 3.838887258$. Sridip explains that hyperbolic surface are of the form \Gamma\G/K with G=PSL(2,R), K=SO(2) /{\pm I} and \Gamma being Fuchsian group. For a given hyperbolic surface, one can define a Hilbert space of local operators, transforming under unitary irreps of a conformal group (PSL(2,R)) and introduce a notion of operator product expansion (OPE). The associativity of this OPE reflects the associativity of function multiplication on the space \Gamma\G and leads to the bootstrap equations. Now the functions on \Gamma\G can be thought of automorphic forms on the surface \Gamma\G/K and Sridip shows that the scaling dimensions of these operators are in fact related to the automorphic spectra in particular the Laplacian eigenvalues on the surface. Hence the bootstrap equations imply bound on the Laplacian eigenvalues. Finally, Sridip makes some remarks about how his methods can be generalized to higher-dimensional hyperbolic manifolds (or orbifolds) and to yield stronger bounds in the two-dimensional case. This is based on a work (arXiv:2111.12716) with P. Kravchuk and D. Mazac. The Virtual Seminar Series “GQFI-Ghent University-Warsaw String Theory” (GQFI-UGent-WST) is a joint initiative between the strings/holography groups at the Albert Einstein Institute (AEI) in Potsdam, Ghent University and the University of Warsaw. The calendar for upcoming GQFI-UGent-WST seminars can be found here: https://gqfi.aei.mpg.de/node/3​​​​​​​​​. This talk is primarily based on arXiv:2111.12716 [hep-th] (https://arxiv.org/abs/2111.12716). This talk was recorded on February 1st, 2022 at the Albert Einstein Institute in Potsdam-Golm.