<p>We compute the scalar determinants <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\det (\Delta +M^{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">det</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on the two-dimensional round disks of constant curvature <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mp } 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∓</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, for any finite boundary length <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> and mass <i>M</i>, with Dirichlet boundary conditions, using the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation>-function prescription. When <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$M^{2}=\pm q(q+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>=</mo> <mo>±</mo> <mi>q</mi> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$q\in {\mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, a simple expression involving only elementary functions and the Euler <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> function is found. Applications to two-dimensional Liouville and Jackiw–Teitelboim quantum gravity are presented in a separate paper.</p>

Dirichlet Scalar Determinants on Two-Dimensional Constant Curvature Disks

期刊论文
发表日期：

2025年5月5日

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Soumyadeep Chaudhuri,
Frank Ferrari

摘要

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