On Exceptional Instanton Strings

TL;DR

This paper classifies and analyzes the worldsheet theories of BPS instanton strings in certain 6d superconformal theories, revealing new relations with 4d theories, topological strings, and Schur indices.

Contribution

It determines the 2d worldsheet theories of exceptional 6d instanton strings using geometric engineering and relates them to 4d SCFTs, topological strings, and modular bootstrap methods.

Findings

Derived anomaly polynomials for the instanton strings.

Established relations between 6d instanton strings and 4d SCFT partition functions.

Computed elliptic genera for single instanton strings, including the F4 case.

Abstract

According to a recent classification of 6d (1,0) theories within F-theory there are only six "pure" 6d gauge theories which have a UV superconformal fixed point. The corresponding gauge groups are $SU(3),SO(8),F_4,E_6,E_7$, and $E_8$. These exceptional models have BPS strings which are also instantons for the corresponding gauge groups. For $G$ simply-laced, we determine the 2d $\mathcal{N}=(0,4)$ worldsheet theories of such BPS instanton strings by a simple geometric engineering argument. These are given by a twisted $S^2$ compactification of the 4d $\mathcal{N}=2$ theories of type $H_2, D_4, E_6, E_7$ and $E_8$ (and their higher rank generalizations), where the 6d instanton number is mapped to the rank of the corresponding 4d SCFT. This determines their anomaly polynomials and, via topological strings, establishes an interesting relation among the corresponding $T^2 \times S^2$ partition functions and the Hilbert series for moduli spaces of $G$ instantons. Such relations allow to bootstrap the corresponding elliptic genera by modularity. As an example of such procedure, the elliptic genera for a single instanton string are determined. The same method also fixes the elliptic genus for case of one $ F_4 $ instanton. These results unveil a rather surprising relation with the Schur index of the corresponding 4d $\mathcal{N}=2$ models.