Topologically twisted indices in five dimensions and holography
Seyed Morteza Hosseini, Itamar Yaakov, Alberto Zaffaroni
TL;DR
This work constructs and analyzes the topologically twisted index of five-dimensional $ ext{N}=1$ gauge theories on $ ext{M}_4 imes S^1$, with a toric Kähler $ ext{M}_4$ and general background fluxes, by gluing $ ext{Nekrasov}$ partition functions at toric fixed points. The index reduces in the non-equivariant, large-$N$ limit to a matrix-model-like expression dominated by the classical+one-loop sector, with instantons exponentially suppressed; the authors connect the SW prepotential $ ext{F}(a)$ to the large-$N$ behavior and conjecture a universal role for its critical points, analogous to Bethe vacua in lower dimensions. For $ ext{N}=2$ SYM on $oldsymbol\Sigma_{ rak g_2} imes (oldsymbol\Sigma_{ rak g_1} imes S^1)$, the on-shell value of $ ext{F}(a)$ reproduces $F_{S^5}$ and yields a twisted index whose extremization aligns with holographic data for AdS$_7 imes S^4$ domain walls; for the $ ext{USp}(2N)$ fixed point with matter, the index scales as $N^{5/2}$ and predicts entropy for magnetically charged AdS$_6$ black holes in massive type IIA. The paper further develops the localization framework on $ ext{M}_4 imes S^1$ via fixed-point Nekrasov gluing, relates the 5D index to 2D/4D central charges through $I$-extremization, and establishes a holographic interpretation through the attractor mechanism for black strings/holes. Overall, the results extend the 3D/4D twisted indices to five dimensions, providing new quantitative bridges between 5D gauge dynamics, localization techniques, and holographic entropy/counting in AdS backgrounds.
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 $Σ_{\mathfrak{g}_2} \times Σ_{\mathfrak{g}_1} \times S^1$, 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.