Supersymmetric phases of 4d $$ \mathcal{N} $$ = 4 SYM at large N

Abstract

We find a family of complex saddle-points at large N of the matrix model for the superconformal index of SU(N ) $$ \mathcal{N} $$ N = 4 super Yang-Mills theory on S3× S1 with one chemical potential τ . The saddle-point configurations are labelled by points (m, n) on the lattice Λτ = 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/Λτ . 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 AdS5 and the supersymmetric AdS5 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.

Gravitational Cardy limit and AdS black hole entropy

“…This expression has recently been extracted directly from the boundary CFT in [3][4][5] with further clarifying field theory work presented in [31][32][33][34][35][36][37][38][39][40]. We show below that this boundary CFT result can also be obtained from a particular near-horizon Cardy formula.…”

“…The Cardy-like limit for the N = 4 SYM index was defined in [4]. This limit has been discussed in the context of N = 4 SYM also in [31,32,39]. In the more general context of N = 1 superconformal theories, it has been discussed in [33][34][35].…”

The large-N limit of the 4d $$ \mathcal{N} $$ = 1 superconformal index

“…These indices were first calculated in [6,7] in the form of integrals over unitary matrices, and the recent progress involves a detailed study of these integrals. Independent studies in the last couple of years have reached the conclusion that the 1 4 -BPS index in N = 1 theories (or its direct lifts like the 1 16 -BPS in N = 4 SYM) indeed captures the entropy of the dual black hole at large N [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The basic idea of all the approaches is the same and can be paraphrased as follows: one calculates the index of the BPS states, and shows that it agrees with the "entropy function" of the BPS black hole.…”

“…The third approach, which we use here, is a direct saddle-point analysis of the matrix integral that was developed for N = 4 SYM in [21]. The integral over unitary matrices in [6,7] reduces, in a completely standard manner, to an integral over the corresponding eigenvalues which live on a circle.…”

“…The integral over unitary matrices in [6,7] reduces, in a completely standard manner, to an integral over the corresponding eigenvalues which live on a circle. The essence of the approach of [21] is to extend the range of eigenvalues of the unitary matrix from a circle to a torus, one of whose cycles is the original circle. This prompts us to refer to this approach as that of elliptic extension.…”