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Superconformal Algebras and Mock Theta Functions 2. Rademacher Expansion for K3 Surface

Tohru Eguchi, Kazuhiro Hikami

TL;DR

The paper uses mock theta functions and harmonic Maass forms to study the K3 elliptic genus, recasting it as a decomposition into $ abla$N=4 level-1 characters and a massive sector with coefficients $A_n$. By completing the Lerch sums to harmonic Maass forms and constructing a weight-$\tfrac{1}{2}$ Poincaré--Maass series, the authors derive an exact, convergent Rademacher-type expansion for the non-BPS Fourier coefficients, including the ALE/decompactified limit with coefficients $A_n^ ext{circ}$. They establish the equality $\widehat{\Sigma} = P_{3/4}$ and provide explicit Fourier expansions involving Bessel and Kloosterman-type sums, along with numerical checks validating convergence and asymptotics. The work also connects these modular phenomena to topological invariants via the WRT theory and discusses implications for entropy in string-theoretic contexts, with potential extensions to higher-level algebras and higher-dimensional hyperKähler manifolds.

Abstract

The elliptic genera of the K3 surfaces, both compact and non-compact cases, are studied by using the theory of mock theta functions. We decompose the elliptic genus in terms of the N=4 superconformal characters at level-1, and present an exact formula for the coefficients of the massive (non-BPS) representations using the Poincare-Maass series.

Superconformal Algebras and Mock Theta Functions 2. Rademacher Expansion for K3 Surface

TL;DR

The paper uses mock theta functions and harmonic Maass forms to study the K3 elliptic genus, recasting it as a decomposition into N=4 level-1 characters and a massive sector with coefficients . By completing the Lerch sums to harmonic Maass forms and constructing a weight- Poincaré--Maass series, the authors derive an exact, convergent Rademacher-type expansion for the non-BPS Fourier coefficients, including the ALE/decompactified limit with coefficients . They establish the equality and provide explicit Fourier expansions involving Bessel and Kloosterman-type sums, along with numerical checks validating convergence and asymptotics. The work also connects these modular phenomena to topological invariants via the WRT theory and discusses implications for entropy in string-theoretic contexts, with potential extensions to higher-level algebras and higher-dimensional hyperKähler manifolds.

Abstract

The elliptic genera of the K3 surfaces, both compact and non-compact cases, are studied by using the theory of mock theta functions. We decompose the elliptic genus in terms of the N=4 superconformal characters at level-1, and present an exact formula for the coefficients of the massive (non-BPS) representations using the Poincare-Maass series.

Paper Structure

This paper contains 13 sections, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $\mathcal{N}=4$ Superconformal Algebras and Mock Theta Functions
  3. Elliptic Genus of K3 Surface and Harmonic Maass Form
  4. Decomposition of Elliptic Genus into $\mathcal{N}$=4 Characters
  5. Elliptic Genus of ALE space
  6. Poincaré--Maass Series and Character Decomposition
  7. Multiplier System for Harmonic Maass Forms
  8. The Whittaker Function and the Poincaré--Maass Series
  9. Non-Compact Case
  10. Numerical Checks
  11. Chern--Simons Theory
  12. Concluding Remarks
  13. Jacobi Theta Functions

Figures (1)

  • Figure 1: Exact values of $A_n$\ref{['l1_coefficient_massive']} and values of the asymptotic formula \ref{['asymptotics_A']} are given by blue dots and blue line, respectively. We have also plotted exact absolute values $|A_n^\circ|$\ref{['l1_coefficient_decompose']} together with their asymptotic values \ref{['asymptotics_AC']} in red.