General instanton counting and 5d SCFT

TL;DR

The authors develop a general, contour-based framework for counting instantons in 5d N=1 gauge theories by interpreting Nekrasov’s partition functions as Witten indices of ADHM quantum mechanics with (0,4) SUSY, and derive Jeffrey–Kirwan contour prescriptions for various representations. They apply the formalism to 5d SCFTs realized at UV fixed points, notably Sp(N) theories with N_f fundamentals and an antisymmetric hypermultiplet, demonstrating E_{N_f+1} symmetry enhancements and clarifying the role of small instantons in UV-incomplete theories. A central theme is the factorization Z_QFT = Z_QM / Z_extra, separating the QFT content from extra UV/stringy states, with Z_extra extracted from string theory constructions such as D0–D8–O8 and D4–D8–O8 systems; the paper provides both JK-residue and unit-circle contour analyses and checks against direct D0–D8–O8 computations. The results illuminate how UV/IR decoupling and continuum states shape the 5d/6d SCFT observables and establish concrete connections between instanton counting, enhanced symmetries, and brane-engineered UV fixed points.

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 \leq 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.

Figures (6)

• Figure 1: Deformations of the contours $\Gamma_\pm$ to compute $f_\pm(0)$
• Figure 2: Choice of charge vectors for $U(N)$ index at $k=2$ with $\eta=(1,1)$
• Figure 3: Shaded boxes form the border of a Young diagram. Dotted boxes are at the corners.
• Figure 4: (a) 5-brane web for the pure $SU(2)$ theory; (b) $SU(2)$ at $\kappa=1$; (c) $SU(2)$ at $\kappa=2$; (d) $SU(2)$ with $N_f=4$ at $\kappa=0$. Horizontal lines are D5-branes on which 5d QFTs live. Red horizontal lines denote D1-branes which can escape to infinity by developing a continuum.
• Figure 5: 5-brane web for the pure $SU(3)$ theory at $\kappa=3$