Supersymmetric Renyi Entropy
Tatsuma Nishioka, Itamar Yaakov
TL;DR
The paper defines a supersymmetric analogue of the Renyi entropy for 3d ${\cal N}\ge2$ SCFTs by placing the theory on a branched ${S^3}$ and compensating conical singularities with an ${R}$-symmetry background. Using supersymmetric localization on a smooth, resolved version of the branched space, the partition function reduces to a finite matrix model equivalent to the squashed sphere ${S_b^3}$ with ${b=\sqrt{q}}$, enabling exact computation of ${S_q^{\text{susy}}}$ through ${Z_{S_b^3}(b)}/{Z_{S^3}^q}$. The observable is shown to be duality-invariant, reduces to entanglement entropy at ${q\to1}$, and admits checks with free theories, IR dualities, and large-${N}$ limits; it also admits a defect-operator interpretation in terms of monodromy-like insertions along the entangling surface. The results provide a robust, computable, SUSY-protected quantity that mirrors many properties of ordinary Renyi entropy while leveraging the power of localization, with explicit examples including ${\cal N}=4$ SQED, ${\cal N}=2$ SQED, ABJM, and large-${N}$ behavior. The framework opens avenues for exploring how supersymmetry constrains entanglement-like observables and for connecting to defect constructions and holographic duals in a controlled setting.
Abstract
We consider 3d N>= 2 superconformal field theories on a branched covering of a three-sphere. The Renyi entropy of a CFT is given by the partition function on this space, but conical singularities break the supersymmetry preserved in the bulk. We turn on a compensating R-symmetry gauge field and compute the partition function using localization. We define a supersymmetric observable, called the super Renyi entropy, parametrized by a real number q. We show that the super Renyi entropy is duality invariant and reduces to entanglement entropy in the q -> 1 limit. We provide some examples.