Universal Features of Four-Dimensional Superconformal Field Theory on Conic Space

TL;DR

This paper demonstrates that the universal part of supersymmetric R'enyi entropy in 4d N=1 SCFTs on conic spaces is proportional to a linear combination of central charges, confirmed through free field and holographic methods.

Contribution

It establishes a universal relation between supersymmetric R'enyi entropy and central charges for 4d N=1 SCFTs, validated by multiple independent derivations.

Findings

Universal R'enyi entropy proportional to 3c-2a in the q→0 limit.

Derived asymptotic formula via free field computations with background R-symmetry.

Confirmed formula for interacting N=1 SCFTs using primary Hopf surface analysis.

Abstract

Following the set up in arXiv:1408.3393, we study 4d N=1 superconformal field theories in conic spaces. We show that the universal part of supersymmetric R\'enyi entropy S_q across a spherical entangling surface in the limit q goes to 0 is proportional to a linear combination of central charges, 3c-2a. This is equivalent to a similar statement about the free energy of SCFTs on conic space or hyperbolic space S^1_q*H^3 in the corresponding limit. We first derive the asymptotic formula by the free field computation in the presence of a U(1) R-symmetry background and then provide an independent derivation by studying N=1 theories on a primary Hopf surface S^1_\beta*S^3_b with a particular scaling \beta~1/\sqrt{q} and b=\sqrt{q}, which thus confirms the validity of the formula for general interacting N=1 SCFTs. Finally we revisit the supersymmetric R\'enyi entropy of general N=2 SCFTs and find a simple formula for it in terms of central charges a and c.