On elliptic genera of 6d string theories

TL;DR

This paper develops a method to determine the elliptic genera of 6d string theories using their modular properties and initial BPS data, enabling a comprehensive understanding of their structure and anomalies.

Contribution

It introduces an ansatz for elliptic genera based on modularity and anomaly data, allowing complete determination from limited initial BPS information.

Findings

Elliptic genera are weak Jacobi forms with indices from 2d anomalies.

The ansatz successfully determines elliptic genera across various 6d SCFTs.

Additional contributions from Coulomb branches are identified and incorporated.

Abstract

We study the elliptic genera of 6d strings based on their modular properties. They are weak Jacobi forms of weight 0, whose indices are determined from the 2d chiral anomalies. We propose the ansatz for the elliptic genera which reflects the analytic structure of instanton partition functions. Given a finite amount of initial BPS data, we completely determine the elliptic genera of 6d strings in various 6d SCFTs. We also apply our ansatz to study $\mathcal{N}=(2,0)$ and $(1,1)$ little strings as well as $\mathcal{N}=(1,0)$ heterotic little strings, for which T-duality of little string theories supplies a sufficient number of initial BPS data. The anomaly polynomials of 6d little strings are worked out, which is needed for the elliptic genera bootstrap. In some little string theories, the elliptic genera must have the extra contributions from the Coulomb branch, which correspond to the additional zero modes for the full strings. The modified ansatze for such elliptic genera are also discussed.