[Paper Review] Supersymmetric Renyi Entropy

Tatsuma Nishioka, Yaakov, Itamar|arXiv (Cornell University)|Jun 12, 2013

Black Holes and Theoretical Physics3 citations

TL;DR

This paper introduces a supersymmetric generalization of Rényi entropy, called super Rényi entropy, for 3d $χ\geq 2$ superconformal field theories on a branched covering of $S^3$. By coupling to an $R$-symmetry gauge field to preserve supersymmetry, the authors compute the partition function via localization, showing the super Rényi entropy is duality-invariant and reduces to entanglement entropy in the $q\to 1$ limit.

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.

Motivation & Objective

• To define a supersymmetric generalization of Rényi entropy in 3d $χ\geq 2$ superconformal field theories.
• To address the breakdown of supersymmetry in the replica trick due to conical singularities in Rényi entropy calculations.
• To construct a supersymmetric observable—super Rényi entropy—that remains well-defined and computable via localization.
• To demonstrate that the super Rényi entropy reduces to entanglement entropy in the $q\to 1$ limit.
• To establish duality invariance of the super Rényi entropy as a key consistency check for the construction.

Proposed method

• Consider the theory on a branched covering of $S^3$ with conical singularities, breaking supersymmetry in the bulk.
• Introduce a compensating $R$-symmetry gauge field to restore supersymmetry on the singular space.
• Apply localization techniques to reduce the path integral to a finite-dimensional matrix model on $S^3$.
• Define the super Rényi entropy as the logarithmic derivative of the partition function with respect to the replica parameter $q$.
• Use the $R$-symmetry gauge field to ensure the partition function is computed in a supersymmetric fashion.
• Leverage known results on $S^3$ partition functions in $χ\geq 2$ theories to compute the super Rényi entropy explicitly.

Experimental results

Research questions

• RQ1 Can a supersymmetric version of Rényi entropy be defined in 3d $χ\geq 2$ superconformal field theories where conical singularities break supersymmetry?
• RQ2 How can the $R$-symmetry gauge field be used to restore supersymmetry in the replica construction?
• RQ3 Is the resulting super Rényi entropy invariant under dualities of the underlying SCFT?
• RQ4 Does the super Rényi entropy reduce to the standard entanglement entropy in the $q\to 1$ limit?
• RQ5 Can the super Rényi entropy be computed exactly using localization techniques in supersymmetric theories?

Key findings

• The super Rényi entropy is defined as a supersymmetric observable on a branched $S^3$ with $R$-symmetry gauge field, preserving supersymmetry despite conical singularities.
• The super Rényi entropy is duality-invariant, providing a non-trivial consistency check for the construction.
• In the limit $q\to 1$, the super Rényi entropy reduces to the standard entanglement entropy, recovering the von Neumann entropy.
• The partition function on the branched covering space is computed via localization, reducing the problem to a finite matrix model.
• The method applies to any 3d $χ\geq 2$ superconformal field theory with a conserved $R$-symmetry current.
• The construction provides a new, exact observable for studying entanglement in strongly coupled supersymmetric field theories.