Review of localization for 5d supersymmetric gauge theories

TL;DR

This work addresses exact localization for 5d supersymmetric gauge theories on toric Sasaki-Einstein manifolds by building a cohomological complex from supersymmetry and exploring toric deformations with all equivariant parameters. It identifies the localization locus (notably vector “contact instantons” and vanishing hypermultiplets under SE, with $F_H^+=0$, $\iota_R F=0$, $D\sigma=0$) and computes the perturbative partition function as a matrix model controlled by a generalized triple sine $S_3^C$, with a conjectured full answer including Nekrasov factors for each closed Reeb orbit. The analysis hinges on the Kohn–Rossi complex and transversely elliptic index theory, linking geometry to quantum fluctuations via determinant calculations and Weitzenböck-type vanishing theorems, and it recovers the flat-space 1-loop structure in appropriate limits. The results illuminate the deep connection between localization techniques and toric Sasaki-Einstein geometry, offering a concrete path toward nonperturbative understanding on curved backgrounds and guiding future checks against Nekrasov-type constructions on $\mathbb{C}^2\times S^1$-like patches.

Abstract

We give a pedagogical review of the localization of supersymmetric gauge theory on 5d toric Sasaki-Einstein manifolds. We construct the cohomological complex resulting from supersymmetry and consider its natural toric deformations with all equivariant parameters turned on. We also give detailed discussion on how the Sasaki-Einstein geometry permeates every aspect of the calculation, from Killing spinor, vanishing theorems to the index theorems.

Figures (3)

• Figure 1: One starts from super Yang-Mills on flat space without the ${\mathcal{A}}^3$ term in the prepotential. The one loop contribution generates an ${\mathcal{A}}^3$ term, then the whole system can still be put on $S^5$ and evaluated in Coulomb branch and will produce $\sigma^3$ term. On the other hand, one can first place the theory on $S^5$, perform localisation, take $r\to\infty$ limit and obtain the $\sigma^3/r^2$ term.
• Figure 2: The 1-loop diagrams. In the third diagram $\bullet$ represents the insertion $F_{pq}J^{pq}$. And the momentum in the loop is to run clockwise.
• Figure 3: The moment map cone $C_{\mu}(M)$