Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities

Kalvin, Victor

doi:10.48550/arxiv.1910.00104

Cited by 3 publications

(30 citation statements)

References 33 publications

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“…when β j → 0 − while β k = β ℓ with k, ℓ = j) and of a standard round sphere (i.e. when β j → 0 − for j = 1, 2, 3) our formula reproduces the corresponding explicit formulae for the determinant known from [22,28,39].…”

Section: Introduction and Main Results 1introductionsupporting

confidence: 59%

“…We believe that this explicit formula for the determinant will allow us to evaluate the determinant of Laplacian on the smooth and singular surfaces obtained by cutting and gluing genus zero constant curvature surfaces with three conical singularities. For some results in this direction see [22,23,25,26,27]. The results can also be of interest in various areas of theoretical physics.…”

Section: Introduction and Main Results 1introductionmentioning

confidence: 92%

“…As a crucial step in our study of the determinant, we express it in terms of the Liouville action and some other explicit functions of β j and S, where S is the area of the surface. On this step we rely on the Polyakov-Alvarez type formula and the BFK (Burghelea-Friedlander-Kappeler [8]) formula proven in [22].…”

Section: Introduction and Main Results 1introductionmentioning

confidence: 99%

“…It is also interesting to note that the celebrated partially heuristic Aurell-Salomonson formula for determinants of Laplacians on polyhedra with spherical topology [5, eqn. (50)] returns an equivalent result; the formula received a rigorous mathematical proof in [22,Sec. 3.2].…”

Section: Introduction and Main Results 1introductionmentioning

confidence: 99%

“…Without going into details, let us mention that there is a certain similarity between the functional H β β β [φ] and the Kähler potentials of the elliptic and cusp metrics in [42], a similar functional also appears in [9,22]. By L 2 β β β we denote the space of functions on C with finite norms…”

Section: Resultsmentioning

confidence: 99%

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

Kalvin¹

2021

Preprint

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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 ∈ (−1, 0) (or, equivalently, of angle 2π(β j +1)). We show that among the metrics of (fixed) area S and (fixed) Gaussian curvature K, the one with β 1 = β 2 = β 3 = SK 6π − 2 3 corresponds to a stationary point of the determinant. 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. The Liouville action satisfies a system of differential equations that can be easily integrated.

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Section: Introduction and Main Results 1introductionsupporting

confidence: 59%

Section: Introduction and Main Results 1introductionmentioning

confidence: 92%

Section: Introduction and Main Results 1introductionmentioning

confidence: 99%

Section: Introduction and Main Results 1introductionmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

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

Kalvin¹

2021

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Metrics of Constant Positive Curvature with Conical Singularities, Hurwitz Spaces, and Determinants of Laplacians

Kalvin

Kokotov

2017

International Mathematics Research Notices

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Abstract. Let f : X → CP 1 be a meromorphic function of degree N with simple poles and simple critical points on a compact Riemann surface X of genus g and let m be the standard round metric of curvature 1 on the Riemann sphere CP 1 . Then the pullback f * m of m under f is a metric of curvature 1 with conical singularities of conical angles 4π at the critical points of f . We study the ζ-regularized determinant of the Laplace operator on X corresponding to the metric f * m as a functional on the moduli space of the pairs (X, f ) (i.e. on the Hurwitz space H g,N (1, . . . , 1)) and derive an explicit formula for the functional.

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Spectral determinant on Euclidean isosceles triangle envelopes of fixed area as a function of angles: absolute minimum and small-angle asymptotics

Kalvin

2020

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We study extremal properties of the determinant of Friederichs selfadoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. Small-angle asymptotics show that the determinant grows without any bound as an angle of triangle envelope goes to zero. We prove that the equilateral triangle envelope (the most symmetrical geometry) always gives rise to a critical point of the determinant and find the critical value. Moreover, if the area of envelopes is not too large, then the determinant achieves its absolute minimum only on the equilateral triangle envelope and there are no other critical points, whereas for sufficiently large area the equilateral triangle envelope corresponds to a local maximum of the determinant.

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Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities

Cited by 3 publications

References 33 publications

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

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

Metrics of Constant Positive Curvature with Conical Singularities, Hurwitz Spaces, and Determinants of Laplacians

Spectral determinant on Euclidean isosceles triangle envelopes of fixed area as a function of angles: absolute minimum and small-angle asymptotics

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