Elliptic Genus of E-strings

TL;DR

This work provides a detailed gauge-theory realization of E-strings as 2d $(0,4)$ $O(n)$ quivers arising from a Type IIA brane setup with NS5, O8, and D8-branes, enabling an explicit computation of the elliptic genus for $n$ E-strings. The authors perform exhaustive calculations for $n=1$–$4$ using JK-residue techniques, and show consistency with partial topological-string results and 5d instanton calculations, including an all-genus link to the refined topological string on ${1 ext{/}2}K3$. An alternative 5d $Sp(1)$ instanton approach with a broken $E_8$ via Wilson lines reproduces the same spectra, confirming the IR $E_8$ symmetry enhancement and offering a powerful cross-check. The paper also outlines the pole structure and computational scaling for higher $n$, illustrating that the method extends to arbitrary numbers of E-strings, and discusses broader implications for 6d $(1,0)$ theories and domain-wall constructions. Overall, the results yield a robust, all-genus description of E-string elliptic genera and establish a bridge between 2d gauge theories, M-/F-theory dualities, and topological-string amplitudes.

Abstract

We study a family of 2d N=(0,4) gauge theories which describes at low energy the dynamics of E-strings, the M2-branes suspended between a pair of M5 and M9 branes. The gauge theory is engineered using a duality with type IIA theory, leading to the D2-branes suspended between an NS5-brane and 8 D8-branes on an O8-plane. We compute the elliptic genus of this family of theories, and find agreement with the known results for single and two E-strings. The partition function can in principle be computed for arbitrary number of E-strings, and we compute them explicitly for low numbers. We test our predictions against the partially known results from topological strings, as well as from the instanton calculus of 5d Sp(1) gauge theory. Given the relation to topological strings, our computation provides the all genus partition function of the refined topological strings on the canonical bundle over 1/2 K3.

Figures (2)

• Figure 1: The type IIA brane configuration for the E-strings.
• Figure 2: The quiver diagram of the 2d $\mathcal{N}=(0,4)$ gauge theory for E-strings: solid/dotted lines denote hyper/Fermi multiplets, respectively.