Topologically twisted indices in five dimensions and holography

Abstract

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 Kähler 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.

arXiv comment80 pages. v3: minor corrections, published version

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