Topologically twisted indices in five dimensions and holography

We provide a formula for the partition function of five-dimensional $\mathcal{N}=1$ gauge theories on $\mathcal{M}_4 \times S^1$, topologically twisted along $\mathcal{M}_4$ in the presence of general background magnetic fluxes, where $\mathcal{M}_4$ is a toric K\"ahler manifold. The result can be expressed as a contour integral of the product of copies of the K-theoretic Nekrasov's partition function, summed over gauge magnetic fluxes. The formula generalizes to five dimensions the topologically twisted index of three- and four-dimensional field theories. We analyze the large $N$ limit of the partition function and some related quantities for two theories: $\mathcal{N}=2$ SYM and the $\mathrm{USp}(2N)$ theory with $N_f$ flavors and an antisymmetric matter field. For $\mathbb{P}^1 \times \mathbb{P}^1 \times S^1$, which can be easily generalized to $\Sigma_{\mathfrak{g}_2} \times \Sigma_{\mathfrak{g}_1} \times S^1$, we conjecture the form of the relevant saddle point at large $N$. The resulting partition function for $\mathcal{N}=2$ SYM scales as $N^3$ and is in perfect agreement with the holographic results for domain walls in AdS$_7 \times S^4$. The large $N$ partition function for the $\mathrm{USp}(2N)$ theory scales as $N^{5/2}$ and gives a prediction for the entropy of a class of magnetically charged black holes in massive type IIA supergravity.