Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere

Abstract

We deduce an explicit closed formula for the zeta-regularized spectral determinant of the Friedrichs Laplacian on the Riemann sphere equipped with arbitrary constant curvature (flat, spherical, or hyperbolic) metric having three conical singularities of order ${\beta }_j\in (-1,0)$ (or, equivalently, of angle $2\pi ({\beta }_j+1)$). We show that among the metrics with a fixed value of the sum ${\beta }_1+{\beta }_2+{\beta }_3$ and a fixed surface area, those with ${\beta }_1={\beta }_2={\beta }_3$ correspond to a stationary point of the determinant. If, in addition, the surface area is sufficiently small, then the stationary point is a minimum. As a crucial step towards obtaining these results we find a relation between the determinant of Laplacian and the Liouville action introduced by A. Zamolodchikov and Al. Zamolodchikov in connection with the celebrated DOZZ formula for the three-point structure constants of the Liouville field theory.

References in corpus (15)

• Structure Constants and Conformal Bootstrap in Liouville Field Theory
• Two and three-point functions in Liouville theory
• Classical Conformal Blocks and Painleve VI
• Conformal field theory of Painlevé VI
• On Functional Determinants of Laplacians in Polygons and Simplices
• Proof of Polyakov conjecture for general elliptic singularities
• Liouville theory and uniformization of four-punctured sphere
• Lowest Landau level on a cone and zeta determinants
• Spectral analysis and zeta determinant on the deformed spheres
• Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities
• Local index theorem for orbifold Riemann surfaces
• On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions
• Spectral determinant on Euclidean isosceles triangle envelopes of fixed area as a function of angles: absolute minimum and small-angle asymptotics
• Determinant of Friederichs Dirichlet Laplacians on $2$-dimensional hyperbolic cones
• Metrics of constant positive curvature with conical singularities, Hurwitz spaces, and $\det \Delta$