## Abstract

We find a family of complex saddle-points at large N of the matrix model for the superconformal index of SU(N) N = 4 super Yang-Mills theory on S^{3}× S^{1} with one chemical potential τ. The saddle-point configurations are labelled by points (m, n) on the lattice Λ_{τ} = ℤτ + ℤ with gcd(m, n) = 1. The eigenvalues at a given saddle are uniformly distributed along a string winding (m, n) times along the (A, B) cycles of the torus ℂ/Λ_{τ}. The action of the matrix model extended to the torus is closely related to the Bloch-Wigner elliptic dilogarithm, and the related Bloch formula allows us to calculate the action at the saddle-points in terms of real-analytic Eisenstein series. The actions of (0, 1) and (1, 0) agree with that of pure AdS_{5} and the supersymmetric AdS_{5} black hole, respectively. The black hole saddle dominates the canonical ensemble when τ is close to the origin, and there are new saddles that dominate when τ approaches rational points. The extension of the action in terms of modular forms leads to a simple treatment of the Cardy-like limit τ → 0.

Original language | English |
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Article number | 184 |

Pages (from-to) | 1-55 |

Number of pages | 56 |

Journal | Journal of High Energy Physics |

Volume | 2020 |

Issue number | 9 |

Early online date | 29 Sept 2020 |

DOIs | |

Publication status | Published - 29 Sept 2020 |

## Keywords

- AdS-CFT Correspondence
- Black Holes in String Theory
- Supersymmetric Gauge Theory