On exceptional instanton strings

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), F₄, E₆, E₇, and E₈. These exceptional models have BPS strings which are also instantons for the corresponding gauge groups. For G simply-laced, we determine the 2d N= (0 4 ) worldsheet theories of such BPS instanton strings by a simple geometric engineering argument. These are given by a twisted S² compactification of the 4d N= 2 theories of type H₂, D₄, E₆, E₇ and E₈ (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 T²× S² 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 F₄ instanton. These results unveil a rather surprising relation with the Schur index of the corresponding 4d N= 2 models.