The joy of factorization at large N: five-dimensional indices and AdS black holes

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>We discuss the large <jats:italic>N</jats:italic> factorization properties of five-dimensional supersymmetric partition functions for CFT with a holographic dual. We consider partition functions on manifolds of the form <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathrm{\mathcal{M}}={\mathrm{\mathcal{M}}}_3\times {S}_{\upepsilon}^2 $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℳ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ℳ</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>ϵ</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula>, where <jats:italic>ϵ</jats:italic> is an equivariant parameter for rotation. We show that, when ℳ<jats:sub>3</jats:sub> is a squashed three-sphere, the large <jats:italic>N</jats:italic> partition functions can be obtained by gluing elementary blocks associated with simple physical quantities. The same is true for various observables of the theories on <jats:inline-formula><jats:alternatives><jats:tex-math>$$ {\mathrm{\mathcal{M}}}_3={\Sigma}_{\mathfrak{g}}\times {S}^1 $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℳ</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$ {\Sigma}_{\mathfrak{g}} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> is a Riemann surface of genus 𝔤, and, with a natural assumption on the form of the saddle point, also for the partition function, corresponding to either the topologically twisted index or a mixed one. This generalizes results in three and four dimensions and correctly reproduces the entropy of known black objects in AdS<jats:sub>6</jats:sub><jats:italic>×</jats:italic><jats:sub><jats:italic>w</jats:italic></jats:sub><jats:italic>S</jats:italic><jats:sup>4</jats:sup> and AdS<jats:sub>7</jats:sub><jats:italic>× S</jats:italic><jats:sup>4</jats:sup>. We also provide the supersymmetric background and explicitly perform localization for the mixed index on <jats:inline-formula><jats:alternatives><jats:tex-math>$$ {\Sigma}_{\mathfrak{g}}\times {S}^1\times {S}_{\upepsilon}^2 $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>ϵ</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula>, filling a gap in the literature.</jats:p>