Central Charges from theN=1Superconformal Index

Abstract

We present prescriptions for obtaining the central charges, a and c, of a four dimensional superconformal quantum field theory from the superconformal index. At infinite N , for holographic theories dual to Sasaki-Einstein 5-manifolds the prescriptions give the O(1) parts of the central charges. This allows us, among other things, to show the exact AdS/CFT matching of a and c for arbitrary toric quiver CFTs without adjoint matter that are dual to smooth Sasaki-Einstein 5-manifolds. In addition, we include evidence from non-holographic theories for the applicability of these results outside of a holographic setting and away from the large-N limit.

Supersymmetric Casimir energy and the anomaly polynomial

“…There are some tantalizing hints from the Cardy formula in four and six dimensions [69], which involve the β → 0 limit of the partition function (whereas the supersymmetric Casimir energy controls the β → ∞ limit). 12 We hope our results may help to elucidate the connection between these limits.…”

“…1 See also [9][10][11][12][13][14][15] for related work on how the superconformal index or partition function of 4d N = 1 theories encodes various anomalies. 2 See [16] for a pedagogical exposition on anomalies and the anomaly polynomial.…”

“…The anomaly polynomial of the interacting theory is [20][21][22] 12) where r g , d g and h ∨ g are the rank, dimension and dual Coxeter number of the simply-laced Lie algebra g, respectively. We should mention that, as far as we are aware, this formula for the anomaly polynomial is conjectural for the E-type theories.…”

“…The full action is then invariant under the supersymmetry variation 11) for the chiral multiplet (φ, ψ − ) and 12) JHEP09 (2015)142 for the Fermi multiplet (ψ + , G), and…”

Cardy formula for 4d SUSY theories and localization

“…Similarly to the Cardy formula in 2d CFT [24], this limit controls the asymptotic behavior at large energies of the weighted density of short states. Following the observations in [25][26][27], and assuming the existence of a weakly-coupled point in the space of couplings, 1 it was argued in [28] that for β → 0 the partition function has a universal behavior (see also [34][35][36][37][38])…”