E + E $\rightarrow$ H

TL;DR

This work derives explicit expressions for the elliptic genus of two E-strings, $Z^{\mathrm{E-str}}_2$, via a chain of M-theory dualities and shows how to recover the two-string heterotic genus $Z^{\mathrm{Het}}_2$ using Horava–Witten boundaries. Central to the method is the domain-wall decomposition involving M5 and M9 boundaries, enabling exact results up to two strings and revealing a bound-state structure for E-strings not present for heterotic strings. The authors connect M-theory configurations to refined topological strings on the half-K3 surface, deriving holomorphic (and modular) anomaly relations that fix multi-string sectors through Weyl$[E_8]$-invariant Jacobi forms. They provide explicit formulas for the two-string E- and heterotic elliptic genera, demonstrate consistency with known topological-string data and the orbifold formula, and outline a path to general $n$ through domain-wall techniques and anomaly constraints. The results illuminate the intricate interplay between M2/M5/M9 configurations, open string/topological-string interpretations, and modular constraints in 6d SCFTs with tensionless strings, with potential for extending to arbitrary $n$-string sectors.

Abstract

E-strings arise from M2 branes suspended between an M5 brane and an M9 plane. In this paper we obtain explicit expressions for the elliptic genus of two E-strings using a series of string dualities. Moreover we show how this can be used to recover the elliptic genus of two $E_8\times E_8$ heterotic strings using the Horava-Witten realization of heterotic strings in M-theory. This involves highly non-trivial identities among Jacobi forms, and is remarkable in light of the fact that E-strings are 'sticky' and form bound states whereas heterotic strings do not form bound states.

Figures (6)

• Figure 1: An M2 branes suspended between M9 planes corresponding to the heterotic string. The worldvolume of the M2 branes and M9 planes share a common $T^2$ which is suppressed in the picture. The directions orthogonal to the torus are represented as the separation $X^6$ and the quaternionic subspaces $X^{2345}$ and $X^{78910}$.
• Figure 2: An M2 brane suspended between an M9 and an M5 brane corresponding to the E-string. The worldvolume of the branes share a common $T^2$ which is suppressed in the picture. The directions orthogonal to the torus are represented as the separation $X^6$ and the quaternionic subspaces $X^{2345}$ and $X^{78910}$.
• Figure 3: An M2 brane suspended between two M5 branes corresponding to the M-string. The worldvolume of the branes share a common $T^2$ which is suppressed in the picture. The M5 branes are extended along the $X^{012345}$ directions.
• Figure 4: A M2 domain wall with a M5 defect.
• Figure 5: A M2 domain wall with a M5 defect where the $X^6$ direction is taken to be circular.