Automorphic spectra and the conformal bootstrap (Q6547222)

The Laplace spectrum on a hyperbolic manifold has been studied using a varied collection of tools coming from fields including number theory, geometry, and analysis. The present paper investigates a connection between the Laplace spectrum and the conformal bootstrap from mathematical physics. The focus is on upper bounds for the lowest nontrivial eigenvalue \(\lambda_1\) for closed, connected, orientable hyperbolic two-dimensional orbifolds (henceforth hyperbolic 2-orbifolds), including hyperbolic surfaces. For example, a rigorous upper bound of \(\lambda_1< 44.8883537\) holds on any hyperbolic 2-orbifold. When the topology of the orbifold is taken into account, the bounds are considerably smaller. The authors compare their bounds to values of \(\lambda_1\) that have been numerically calculated for certain surfaces. For instance, the orbifold of genus zero with three orbifold singularities of orders \(2\), \(3\), and \(7\) has \(\lambda_1 \approx 44.88835\). They also compare their bounds, as a function of the genus of the 2-orbifold, to those found by others, including [\textit{P. C. Yang} and \textit{S.-T. Yau}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 7, 55--61 (1980; Zbl 0446.58017); \textit{A. El Soufi} and \textit{S. Ilias}, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble 1983--1984, No.VII, 15 p. (1984; Zbl 0757.53028); \textit{A. Ros}, J. Math. Soc. Japan 74, No. 3, 813--828 (2022; Zbl 1496.35268); \textit{M. Karpukhin} and \textit{D. Vinokurov}, Math. Z. 301, No. 3, 2733--2746 (2022; Zbl 1491.58012); \textit{H. Huber}, Proc. Symp. Pure Math. 36, 181--184 (1980; Zbl 0448.58023)]; in almost all cases, their bounds are an improvement on those previously known. Finally, the authors use the conformal bootstrap and the Yang-Yau bound to make substantial progress toward proving the following conjecture on the structure of the set of values of \(\lambda_1\) across all hyperbolic two-orbifolds: it is the union of an interval that is roughly (0, 15.7902) with three discrete points.\N\NThe conformal bootstrap appears in the study of conformal field theories in mathematical physics, and key ingredients are a set of correlation functions and an operator product expansion. To obtain the bounds, the authors interpret these ingredients in the relevant context using the representation theory of \(PSL_2(\mathbb{R})\), quadruple integrals of holomorphic forms, and semidefinite programming. A general overview of the method is given in section 2, with more technical details provided in section 3; section 4 investigates the bounds on \(\lambda_1\) that result from the conformal bootstrap. In concluding remarks, the authors note that their approach can be used to give estimates on higher eigenvalues when the bound on \(\lambda_1\) is nearly attained by an actual hyperbolic 2-orbifold. They also mention possible extensions to higher-dimensional hyperbolic manifolds and orbifolds, as well as to the Dirac operator on hyperbolic orbifolds.\N\NThere is overlap between this article and [\textit{J. Bonifacio}, J. High Energy Phys. 2022, No. 3, Paper No. 93, 19 p. (2022; Zbl 1522.81460)]; both papers use the conformal bootstrap to derive bounds on \(\lambda_1\) for hyperbolic surfaces.