2026/07/20 by Osama Khlaif, Boris Pioline, Alessandro Tanzini
#hep-th #math.AG #math.DG
We study the partition function of five-dimensional N=1 U(N) supersymmetric Yang-Mills (SYM) theory with an adjoint hypermultiplet of mass m\rm adj on a toric Kähler surface S times a circle of radius \boldsymbolβ. Extending earlier work in N=2^* SYM theory on S, and in pure N=1 SYM on S× \mathbbS1\boldsymbolβ, we find that the path integral localizes to an integral along the Cartan torus of the product of Nekrasov 5D partition functions for each affine patch. Restricting to the gauge group U(2) for simplicity, the integrand has an infinite set of poles of degree at most χ(S)-2. With a natural prescription for integrating around such poles, we find that the contributing poles are in one-to-one correspondence with the torus-fixed points in the moduli space of semi-stable torsion-free sheaves on S. Moreover, for non-even first Chern class, their contributions are independent of the equivariant parameters ε1,ε2 and add up to the χy2-genus of that moduli space, where y2=e^-\boldsymbolβ m\rm adj, and hence coincide with the refined Vafa-Witten invariants. For even Chern class, the partition function depends on the equivariant parameters ε1,ε2 as well as y, and its relation to rational, refined Vafa-Witten invariants remains unclear.