Abstract
Following the set up in arXiv:1408.3393, we study 4d \( \mathcal{N}=1 \) superconformal field theories on conic spaces. We show that the universal part of supersymmetric Rényi entropy S q across a spherical entangling surface in the limit q → 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 \( {\mathbb{S}}_q^1\times {\mathrm{\mathbb{H}}}^3 \) 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 \( \mathcal{N}=1 \) theories on \( {\mathbb{S}}_{\beta}^1\times {\mathbb{S}}_b^3 \) with a particular scaling \( \beta \sim \frac{1}{\sqrt{q}},b=\sqrt{q} \), which thus confirms the validity of the formula for general interacting \( \mathcal{N}=1 \) SCFTs. Finally we revisit the supersymmetric Rényi entropy of generel \( \mathcal{N}=2 \) SCFTs and find a simple formula for it in terms of central charges a and c.
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Zhou, Y. Universal features of four-dimensional superconformal field theory on conic space. J. High Energ. Phys. 2015, 52 (2015). https://doi.org/10.1007/JHEP08(2015)052
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DOI: https://doi.org/10.1007/JHEP08(2015)052