On instantons as Kaluza-Klein modes of M5-branes

TL;DR

This work establishes a precise BPS index for instantons in circle-compactified 6d (2,0) theory, recasting instantons as Kaluza–Klein modes of M5-branes and relating the index to Nekrasov’s 5d N=2* partition function. Through D0–D4 quantum mechanics and localization, it demonstrates a unique threshold bound state in the U(1) sector and reproduces the self-dual string degeneracies for SU(2) and SU(N) via momentum-carrying worldsheet degrees; the results further imply an interpretation of the symmetric-phase index as a superconformal index of the instanton sigma-model on the moduli space. The analysis reveals universal worldsheet structures and ties to anomaly coefficients, highlighting a rich 2d/4d/6d interplay and supporting the view that 5d SYM captures key aspects of the 6d (2,0) theory on a circle. It also discusses potential large-N dynamics and future avenues toward exact expressions, SL(2, Z) properties, and connections to holography via AdS7×S^4 gravity.

Abstract

Instantons and W-bosons in 5d maximally supersymmetric Yang-Mills theory arise from a circle compactification of the 6d (2,0) theory as Kaluza-Klein modes and winding self-dual strings, respectively. We study an index which counts BPS instantons with electric charges in Coulomb and symmetric phases. We first prove the existence of unique threshold bound state of (noncommutative) U(1) instantons for any instanton number, and also show that charged instantons in the Coulomb phase correctly give the degeneracy of SU(2) self-dual strings. By studying SU(N) self-dual strings in the Coulomb phase, we find novel momentum-carrying degrees on the worldsheet. The total number of these degrees equals the anomaly coefficient of SU(N) (2,0) theory. We finally show that our index can be used to study the symmetric phase of this theory, and provide an interpretation as the superconformal index of the sigma model on instanton moduli space.

Figures (2)

• Figure 1: Instantons and elementary particles uplift to momenta and M2 self-dual strings.
• Figure 2: An $N$-colored Young diagram. Boxes map to the basis of $k$ dimensional vector space.