Exact Results for SYM on $Y^{p,q}$ and $S^2\times S^2$ with Conical Singularities

TL;DR

This paper derives exact partition functions for supersymmetric theories on five-dimensional toric Sasakian manifolds with conical singularities, extending to orbifold spaces and product of spindles.

Contribution

It introduces a method to compute exact partition functions on $Y^{p,q}$ manifolds with conical singularities, including flux and instanton effects, generalizing previous approaches.

Findings

Partition functions computed for $Y^{p,q}$ manifolds with flux and instantons.

Extended results to theories on spaces with orbifold singularities.

Provided explicit partition functions for theories on $S^2 imes S^2$ with conical defects.

Abstract

Starting from a theory on $S^3\times S^3$ and dimensionally reducing, we compute the full partition function, including flux and instanton contributions, for an $\mathcal{N}=1$ theory of vector multiplets and hypermultiplets on five-dimensional toric Sasakian manifolds $Y^{p,q}$. Dimensionally reducing, we obtain the partition function for Pestun-like theories on a class of manifolds whose topology is $S^2\times S^2$. Generalizing the procedure starting from branched covers of $S^3\times S^3$, we reduce to a theory on $Y^{p,q}$ with codimension two twist defects. Exploiting a proposed equivalence with partition functions on spaces with orbifold singularities, our results provide the partition function of an $\mathcal{N}=2$ theory on the product of two spindles.