Instantons on the 5-sphere and M5-branes

TL;DR

This work derives the nonperturbative partition function of 5d maximal SYM on $S^5$ as a route to the 6d $(2,0)$ index, using localization around self-dual instantons on the $\mathbb{CP}^2$ base of the Hopf fibration. The authors implement equivariant localization with two Omega-deformation parameters $\epsilon_1,\epsilon_2$ and three fixed points, producing a factorized structure $Z=\frac{1}{N!}\int d\lambda\, e^{-S_{cl}} Z_{\mathrm{pert}}^{(1)}Z_{\mathrm{inst}}^{(1)} Z_{\mathrm{pert}}^{(2)}Z_{\mathrm{inst}}^{(2)} Z_{\mathrm{pert}}^{(3)}Z_{\mathrm{inst}}^{(3)}$, with detailed Nekrasov-like instanton sums. The paper demonstrates exact Abelian and large-$N$ results: the Abelian 6d index matches the free tensor multiplet index, and ADE theories at large $N$ reproduce AdS$_7$ gravity indices, while Wilson loops on $S^5$ correspond to M2-brane Wilson surfaces in the gravity dual. Ambiguities tied to background curvature couplings and small instantons are discussed, with proposed routes to fix them via off-shell supergravity and 6d-7d matching. These results strengthen the 5d localization approach as a powerful tool for probing 6d $(2,0)$ physics and M5-brane holography.

Abstract

We calculate the partition functions of supersymmetric gauge theories on S^5, which acquire non-perturbative contributions from instanton loops wrapping its Hopf fiber. The instantons on the CP^2 base equivariantly localize to 3 fixed points of SU(3) Cartans. Using our results, we study the index of the 6d (2,0) theory. We first study the partition functions of maximal SYM with AD(E) gauge groups, which agree with the gravity dual indices on AdS_7 in the large N limits. We also show that the most general partition function of the Abelian theory agrees with the index for a free 6d tensor supermultiplet. We explain possible ambiguities in our 5d approach, role of maximal SUSY, and the Wilson loops on S^5 dual to M2-brane Wilson surfaces in AdS_7.