c − a from the N = 1 $$ \mathcal{N}=1 $$ superconformal index

Abstract

We present a prescription for obtaining the difference of the central charges, c−a, of a four dimensional superconformal quantum field theory from its single-trace index. The formula is derived from a one-loop holographic computation, but is expected to be valid independent of holography. We demonstrate the prescription with several holographic and non-holographic examples. As an application of our formula, we show the AdS/CFT matching of c − a for arbitrary toric quiver CFTs without adjoint matter that are dual to smooth Sasaki-Einstein 5-manifolds.

Supersymmetric Casimir energy and the anomaly polynomial

“…Making these replacements and dividing by the equivariant Euler class e(T M ) = ω 1 ω 2 ω 3 , we find 11) in agreement with our proposal (up to a conventional minus sign in the definition of the anomaly polynomial). …”

“…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 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…”

“…The result for the chiral multiplet is simply 18) where the second equality is obtained from the identity in (B.12). 11 Using the eta and theta functions defined in appendix B and their modular properties, we can rewrite this 1-loop determinant as follows: 20) with…”

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])…”