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

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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 Rényi entropy Sq 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 S¹q*H³ 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 S¹β*S³b with a particular scaling β 1/√q and b=√q, which thus confirms the validity of the formula for general interacting N=1 SCFTs. Finally we revisit the supersymmetric Rényi entropy of general N=2 SCFTs and find a simple formula for it in terms of central charges a and c.