Factorization of 5D super Yang-Mills on $Y^{p,q}$ spaces

TL;DR

The authors demonstrate that the perturbative partition function of 5D super Yang-Mills on toric Sasaki-Einstein Y^{p,q} spaces factorises into four copies of the perturbative answer on R^4 × S^1, enabling the identification of equivariant parameters directly from toric data. This factorisation motivates a conjectured full partition function that includes instantons, expressed as four Nekrasov-type instanton sectors centered at the Reeb-orbit corners. They provide explicit factorisation results in the S^5 and T^{1,1} cases and derive general Y^{p,q} formulas, with parameter identifications obtained both from geometric and toric diagram analyses. While a complete first-principles proof for the absence of smooth instantons is lacking, the work yields an exact, infinitely many-manifold partition function framework in this class and suggests extensions to broader toric SE geometries and connections to higher-dimensional indices and topological strings.

Abstract

We continue our study on the partition function for 5D supersymmetric Yang-Mills theory on toric Sasaki-Einstein $Y^{p,q}$ manifolds. Previously, using the localisation technique we have computed the perturbative part of the partition function. In this work we show how the perturbative part factorises into four pieces, each corresponding to the perturbative answer of the same theory on $\mathbb{R}^4 \times S^1$. This allows us to identify the equivariant parameters and to conjecture the full partition functions (including the instanton contributions) for $Y^{p,q}$ spaces. The conjectured partition function receives contributions only from singular contact instantons supported along the closed Reeb orbits. At the moment we are not able to prove this fact from the first principles.

Figures (1)

• Figure 1: The polytope cone for $p=3$, $q=2$, projected onto the plane $\mu_1=1+\mu_2+\mu_3$, the cone is the area enclosed by the four lines.