A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index

Abstract

The superconformal index of the $\mathcal{N}=4$ SU(N) supersymmetric Yang-Mills theory counts the 1/16-BPS states in this theory, and has been used via the AdS/CFT correspondence to count black hole microstates of 1/16-BPS black holes. On one hand, this index may be related to the Euclidean partition function of the theory on $S^3\times S^1$ with complex chemical potentials, which maps by the AdS/CFT correspondence to a sum over Euclidean gravity solutions. On the other hand, the index may be expressed as a sum over solutions to Bethe Ansatz Equations (BAEs). We show that the known solutions to the BAEs that have a good large $N$ limit, for the case of equal chemical potentials for the two angular momenta, have a one-to-one mapping to (complex) Euclidean black hole solutions on the gravity side. This mapping captures both the leading contribution from the classical gravity action (of order $N^2$), as well as non-perturbative corrections in $1/N$, which on the gravity side are related to wrapped D3-branes. Some of the BA solutions map to orbifolds of the standard Euclidean black hole solutions (that obey exactly the same boundary conditions as the other solutions). A priori there are many more gravitational solutions than Bethe Ansatz solutions, but we show that by considering the non-perturbative effects, the extra solutions are ruled out, leading to a precise match between the solutions on both sides.

46 pages + appendices; v2: added references

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Cited by in corpus (29)

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• Exact Stringy Microstates from Gauge Theories
• 4d N=2 supergravity observables from Nekrasov-like partition functions
• BPS Chaos
• Supersymmetric phases of AdS${}_4$/CFT${}_3$
• Exact QFT duals of AdS black holes
• Higher-derivative corrections to flavoured BPS black hole thermodynamics and holography
• A quantum mechanics for magnetic horizons
• Dual Dressed Black Holes as the end point of the Charged Superradiant instability in $\mathcal{N}=4$ Yang Mills
• Quantum Phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM
• Magnetic charge and black hole supersymmetric quantum statistical relation
• On the uniqueness of supersymmetric AdS${}_5$ black holes with toric symmetry
• A note on the admissibility of complex BTZ metrics
• The large-$N$ limit of 4d superconformal indices for general BPS charges
• Supersymmetric Grey Galaxies, Dual Dressed Black Holes and the Superconformal Index
• Root of unity asymptotics for Schur indices of 4d Lagrangian theories
• S-duality in the Cardy-like limit of the superconformal index
• Thermodynamics of BPS and Near-BPS AdS6 Black Holes
• Towards the construction of multi-centered black holes in AdS
• Conformal Mapping of Non-Lorentzian Geometries in SU(1,2) Conformal Field Theory
• Hawking-Page transition on a spin chain
• Giant Gravitons and Volume Minimisation
• On the Search for Multicenter AdS Black Holes from M-theory
• Gauging the complex SYK model
• Off-shell phase diagram of BPS black holes in AdS${}_5$
• Bubbling saddles of the gravitational index
• Wrapped $M5$-branes and complex saddle points