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), 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 mathcal{N}=left(0,4right) worldsheet theories of such BPS instanton strings by a simple geometric engineering argument. These are given by a twisted S2 compactification of the 4d mathcal{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 mathcal{N}=2 models.
Highlights
Often, such 2d worldsheet theories can be determined using brane engineerings in IIA or IIB superstrings [44,45,46,47]; these perturbative brane engineerings are less helpful in the case of 6d (1,0) systems with exceptional gauge groups, a fact which is related to the absence of an ADHM construction for exceptional instanton moduli spaces [48,49,50,51]
It is well-known that systems with exceptional gauge symmetries are ubiquitous in the landscape of 6d (1, 0) models realized within F-theory [53], which rely upon the gauge symmetries of non-perturbative seven-brane stacks [54,55,56,57]
For backgrounds without global symmetry fluxes (other than the U(1) R-symmetry monopole) and with a choice of U(1) R-symmetry such that all the elementary fields have non-negative R-charges, this sum turns out to consist of a single term [68], which one can identify with a RR elliptic genus for a 2d (0, 2) theory, of the kind defined in [127, 128]
Summary
Many new results have been obtained in the context of 6d (1, 0) theories [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]; many of their properties remain rather mysterious. This paper is organized as follows: in section 2 we briefly review some salient features of the F-theory backgrounds that engineer the 6d SCFTs we study in this paper; section 3 contains a review of the main properties of the 4d N = 2 theories of type HG(k) and the geometric engineering argument identifying the twisted compactification leading to the 2d (0, 4) worldsheet theories; in section 4 we discuss general properties of the 2d SCFTs which follow from the engineering: the central charges, the anomaly polynomial, and the elliptic genera; in section 5 we review the topological string argument sketched above; in section 6 we derive our Ansatz from the modularity properties of the elliptic genera; in section 7 we remark on an intriguing relation among the elliptic genera derived in section 6 and the Schur index of the corresponding N = 2 theories
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