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Generating Function of Single Centered Black Hole Index from the Igusa Cusp Form

Ajit Bhand, Ashoke Sen, Ranveer Kumar Singh

TL;DR

The paper addresses isolating the single-centered black hole index in heterotic string theory by subtracting two-centered contributions from the Igusa cusp form, and constructs a manifestly duality-invariant generating function $\widetilde{F}(\Omega)$. It provides a detailed analytic framework, proving convergence of the defining sums and establishing the meromorphic structure and invariance properties in the Siegel upper half-plane; crucially, the poles cancel against those of $1/\Phi_{10}$, enabling a clean analytic continuation. The work further shows that the single-centered generating function $F(\Omega)$ coincides with $\widetilde{F}(\Omega)$ upon analytic continuation and connects the resulting Fourier coefficients to mock Jacobi forms, suggesting a deep link to mock Siegel modular objects and potential extensions to broader CHL models. Overall, the results deliver a robust, duality-consistent description of single-centered black hole indices, with precise pole data and a path towards a modular-analytic interpretation of the coefficients.

Abstract

We introduce manifestly duality invariant generating function of the index of single centered black holes in the heterotic string theory compactified on a six dimensional torus. This function is obtained by subtracting, from the inverse of the Igusa cusp form, the generating function of the index of two centered black holes constructed from the Dedekind eta function. We also study the analytic properties of this function in the Siegel upper half plane.

Generating Function of Single Centered Black Hole Index from the Igusa Cusp Form

TL;DR

The paper addresses isolating the single-centered black hole index in heterotic string theory by subtracting two-centered contributions from the Igusa cusp form, and constructs a manifestly duality-invariant generating function . It provides a detailed analytic framework, proving convergence of the defining sums and establishing the meromorphic structure and invariance properties in the Siegel upper half-plane; crucially, the poles cancel against those of , enabling a clean analytic continuation. The work further shows that the single-centered generating function coincides with upon analytic continuation and connects the resulting Fourier coefficients to mock Jacobi forms, suggesting a deep link to mock Siegel modular objects and potential extensions to broader CHL models. Overall, the results deliver a robust, duality-consistent description of single-centered black hole indices, with precise pole data and a path towards a modular-analytic interpretation of the coefficients.

Abstract

We introduce manifestly duality invariant generating function of the index of single centered black holes in the heterotic string theory compactified on a six dimensional torus. This function is obtained by subtracting, from the inverse of the Igusa cusp form, the generating function of the index of two centered black holes constructed from the Dedekind eta function. We also study the analytic properties of this function in the Siegel upper half plane.

Paper Structure

This paper contains 12 sections, 18 theorems, 208 equations, 3 figures.

Table of Contents

  1. Introduction and summary
  2. Construction of the generating function $\widetilde{F}(\Omega)$
  3. Analysis of the generating function $F(\Omega)$
  4. Convergence of $\widetilde{F}(\Omega)$
  5. Convergence of ${\cal F}_1$
  6. Convergence of ${\cal F}_2$
  7. Convergence of ${\cal F}_3$
  8. Properties of $\widetilde{F}(\Omega)$
  9. Proof of Theorem \ref{['thm:S_mero_poles']}
  10. Proof of Theorem \ref{['thm:F_rel_Igusa']}
  11. Proof of Theorem \ref{['thm:sing_deg_gen']}
  12. Proof of Theorem \ref{['thm:sing_cent_att_cont']}

Key Result

Theorem 4.1

The sum over $a,b,c,d$ and $r$ in eguessfin converges absolutely and uniformly on compact subsets of ${\cal R}$.

Figures (3)

  • Figure 1: Chambers in the $x$-$y$ plane separated by pole locations of $1/\Phi_{10}$.
  • Figure 2: This figure shows, for $r=3$, the restriction on $(\sigma_2',\tau_2',z_2')$ imposed by the Heaviside functions. The Heaviside functions only allow the region bounded by the red lines. There are infinite number of such red lines shown by the dots. The $x'$-axis has been rescaled appropriately to show close points on the curve separately.
  • Figure 3: This figure shows, for $r=3$, the three chambers with one vertex at $\infty$. There are other chambers, not related to these by transformation (\ref{['e555']}), which lie close to the parabola, with vertices lying between $u_c$ and $u_c^{-1}$.

Theorems & Definitions (34)

  • Theorem 4.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • proof
  • Proposition 4.5
  • ...and 24 more