Topologically twisted indices in five dimensions and holography

We provide a formula for the partition function of five-dimensional N= 1 gauge theories on ℳ4 × S1, topologically twisted along ℳ4 in the presence of general background magnetic fluxes, where ℳ4 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: N = 2 SYM and the USp(2N) theory with Nf flavors and an antisymmetric matter field. For ℙ1×ℙ1×S1, which can be easily generalized to Σg2×Σg1×S1, we conjecture the form of the relevant saddle point at large N. The resulting partition function for N = 2 SYM scales as N3 and is in perfect agreement with the holographic results for domain walls in AdS7 × S4. The large N partition function for the USp(2N) theory scales as N5/2 and gives a prediction for the entropy of a class of magnetically charged black holes in massive type IIA supergravity.