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

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.