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Dirichlet Scalar Determinants On Two-Dimensional Constant Curvature Disks

Soumyadeep Chaudhuri, Frank Ferrari

TL;DR

This work computes the determinants $\det_{\text{D}}(\Delta_{\eta}+M^{2})$ on finite-area 2D disks of constant curvature ($\eta=0,\pm1$) using $\zeta$-function regularization, for Dirichlet boundary conditions and arbitrary boundary length $\ell$ and mass $M$. It reduces the problem to radial Sturm–Liouville determinants, analyzes large-$M$ and large-$\ell$ expansions via heat-kernel methods, and provides exact infinite-product representations that are well-suited for numerical evaluation. The authors derive explicit closed forms in special mass values $M^{2}=-\eta q(q+1)$, expressing results through Legendre and Meixner polynomials, hypergeometric functions, and Barnes $\mathsf{G}$-functions; in the hemisphere ($r_{0}=1$) they recover known results from the literature. A detailed treatment of zero-mass determinants via the conformal anomaly, along with a careful discussion of the ratio of determinants in 2D, reveals subtle finite-size effects and the nontrivial distinction between large-$\ell$ and large-$M$ limits on curved disks. The results have potential applications to finite-size quantum field theory and finite-cutoff holography, with connections to two-dimensional Liouville and Jackiw–Teitelboim gravity.

Abstract

We compute the scalar determinants $\det(Δ+M^{2})$ on the two-dimensional round disks of constant curvature $R=0$, $\mp 2$, for any finite boundary length $\ell$ and mass $M$, with Dirichlet boundary conditions, using the $ζ$-function prescription. When $M^{2}=\pm q(q+1)$, $q\in\mathbb N$, a simple expression involving only elementary functions and the Euler $Γ$ function is found. Applications to two-dimensional Liouville and Jackiw-Teitelboim quantum gravity are presented in a separate paper.

Dirichlet Scalar Determinants On Two-Dimensional Constant Curvature Disks

TL;DR

This work computes the determinants on finite-area 2D disks of constant curvature () using -function regularization, for Dirichlet boundary conditions and arbitrary boundary length and mass . It reduces the problem to radial Sturm–Liouville determinants, analyzes large- and large- expansions via heat-kernel methods, and provides exact infinite-product representations that are well-suited for numerical evaluation. The authors derive explicit closed forms in special mass values , expressing results through Legendre and Meixner polynomials, hypergeometric functions, and Barnes -functions; in the hemisphere () they recover known results from the literature. A detailed treatment of zero-mass determinants via the conformal anomaly, along with a careful discussion of the ratio of determinants in 2D, reveals subtle finite-size effects and the nontrivial distinction between large- and large- limits on curved disks. The results have potential applications to finite-size quantum field theory and finite-cutoff holography, with connections to two-dimensional Liouville and Jackiw–Teitelboim gravity.

Abstract

We compute the scalar determinants on the two-dimensional round disks of constant curvature , , for any finite boundary length and mass , with Dirichlet boundary conditions, using the -function prescription. When , , a simple expression involving only elementary functions and the Euler function is found. Applications to two-dimensional Liouville and Jackiw-Teitelboim quantum gravity are presented in a separate paper.
Paper Structure (25 sections, 6 theorems, 182 equations)

This paper contains 25 sections, 6 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. The case of infinite area
  3. Euclidean space
  4. Hyperbolic space
  5. Generalities on the cases of finite area
  6. Spectra and Sturm-Liouville operators
  7. The large $M$ and large $\ell$ expansions
  8. Flat case
  9. Curved cases
  10. Zero-mass determinants
  11. Sturm-Liouville determinants
  12. Ratio of determinants
  13. Naive discussion
  14. Correct discussion
  15. Determinants on the flat disk
  16. ...and 10 more sections

Key Result

Theorem 1.1

The functions $D^{0}(M^{2})$ and $D^{\eta}(M^{2},r_{0})$ are given by where $\zeta_{\text{R}}$ is the Riemann $\zeta$-function, $\gamma$ is Euler's constant, $I_{n}$ is the modified Bessel function of the first kind, the functions $f_{n}^{\eta}$ are hypergeometric functions defined in Eq. eigenminus and eigenplus, and $A^{\eta}$ is the area, defined in ellminrel.

Theorems & Definitions (11)

  • Theorem 1.1
  • Proposition 5.1
  • proof
  • Proposition 9.1
  • proof
  • Lemma 9.1
  • proof
  • Lemma 9.2
  • proof
  • Proposition 9.2
  • ...and 1 more