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arXiv · 2112.02771

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

Victor Kalvin
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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 $β_j\in(-1,0)$ (or, equivalently, of angle $2π(β_j+1)$). We show that among the metrics with a fixed value of the sum $β_1+β_2+β_3$ and a fixed surface area, those with $β_1=β_2=β_3$ 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.

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Victor Kalvin. 2022-02-03. Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere. https://doi.org/10.1007/s00526-022-02399-x

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