General instanton counting and 5d SCFT

Abstract

Instanton partition functions of 5d N = 1 gauge theories are Witten indices for the ADHM gauged quantum mechanics with (0, 4) SUSY. We derive the integral contour prescriptions for these indices using the Jeffrey-Kirwan method, for gauge theories with hypermultiplets in various representations. The results can be used to study various 4d/5d/6d QFTs. In this paper, we study 5d SCFTs which are at the UV fixed points of 5d SYM theories. In particular, we focus on the Sp(N ) theories with N f ≤ 7 fundamental and 1 antisymmetric hypermultiplets, living on the D4-D8-O8 systems. Their superconformal indices calculated from instantons all show E N f +1 symmetry enhancements. We also discuss some aspects of the 6d SCFTs living on the M5-M9 system. It is crucial to understand the UV incompleteness of the 5d SYM, coming from small instantons in our problem. We explain in our examples how to fix them. As an aside, we derive the index for general gauged quantum mechanics with (0, 2) SUSY.

Little strings and T-duality

“…The gauge theory elliptic genus can be computed by performing a suitable contour integral of [18][19][20], with the integral measure given by the contributions from the fields listed in table 1. The integral representation for Z k is given by [21] …”

“…The contour integral can be done by following precisely the same steps as explained in [21], which did the Witten index calculus for the 1d reduction of our (4, 4) theory, to count instantons in the 5d N = 1 * theory. One can repeat the analysis in the elliptic version.…”

“…Firstly, the contour integral in (2.6) is a sum over the so-called Jeffrey-Kirwan residues (JK-Res) [20]. The poles with nonzero JK-Res for (2.6) has been studied in [21]. It was first shown in [21] that all the poles from Z hyper never give nonzero JK-Res: whenever we pick a pole from the denominator term θ 1 (±m − ǫ + + u IJ ) according to the rules of [20,21], a term of the form θ 1 (±m − ǫ − + u IJ ) in the numerator always vanishes.…”

“…First of all, classifying the poles with nonzero JK-Res following [21], one again finds that no poles are to be kept from…”

“…However, after turning on ξ m , it often happens that there appear extra bound states of the continuum with the remaining 2d strings of our interest. For instance, see [21] and references therein for many occasions in which extra bound states occur at nonzero FI parameters. Another potential reason for discrepancies between Z IIA string andẐ IIA string is that our gauge theories have one D6-brane inserted.…”