Supersymmetric phases of 4d $$ \mathcal{N} $$ = 4 SYM at large N
Alejandro Cabo-Bizet, Sameer Murthy
Abstract
A bstract We find a family of complex saddle-points at large N of the matrix model for the superconformal index of SU( N ) $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 = ℤ τ + 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.