The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere

TL;DR

We address the perturbative partition function of a radius-deformed $N=1$ five-dimensional Yang-Mills theory with matter on the five-sphere and derive a concrete matrix-model representation using localization. The calculation employs a cohomological reformulation of supersymmetry and the Atiyah-Singer index theorem to evaluate one-loop determinants for vector and hypermultiplets, yielding a convergent Cartan integral with a Gaussian weight and representation-dependent determinants. In the large-$N$ limit at fixed $g_{YM}^2/r$, nonperturbative instanton sectors are exponentially suppressed, suggesting the matrix-model contribution dominates and may realize the expected $N^3$ scaling of the free energy. The work discusses connections to the six-dimensional $(2,0)$ theory, the flat-space $N=2$ theory limit via adjoint matter, and the note that subsequent results confirm $N^3$ behavior in the adjoint-vector case.

Abstract

Based on the construction by Hosomichi, Seong and Terashima we consider N=1 supersymmetric 5D Yang-Mills theory with matter on a five-sphere with radius r. This theory can be thought of as a deformation of the theory in flat space with deformation parameter r and this deformation preserves 8 supercharges. We calculate the full perturbative partition function as a function of r/g^2, where g is the Yang-Mills coupling, and the answer is given in terms of a matrix model. We perform the calculation using localization techniques. We also argue that in the large N-limit of this deformed 5D Yang-Mills theory this matrix model provides the leading contribution to the partition function and the rest is exponentially suppressed.