Abstract
Abstract We compute the supersymmetric Rényi entropies across a spherical entanglement surface in $$ \mathcal{N}=4 $$ N = 4 SU(N) SYM theory using localization on the four-dimensional ellipsoid. We extract the leading result at large N and λ and match its universal part to a gravity calculation involving a hyperbolically sliced supersymmetric black hole solution of $$ \mathcal{N}={4}^{+} $$ N = 4 + SU(2) × U(1) gauged supergravity in five dimensions. We repeat the analysis in the presence of a Wilson loop insertion and find again a perfect match with the dual string theory. Understanding the Wilson loop operator requires knowledge of the full ten-dimensional IIB supergravity solution which we elaborate upon.
Highlights
One first computes the traces of powers of the reduced density matrix, or equivalently, the n−th Renyi entropy
Substituting into the definition of the supersymmetric Renyi entropy given in section 1, we find, in the absence of the Wilson loop, from the result for Fn given in subsection 2.2.1, Sn
In section 2.1.2 we identified the unbroken global symmetries of the original SU(4) ∼= SO(6) R-symmetry of N = 4 SYM theory on the ellipsoid to be SU(2)H, which gets enhanced by the bonus symmetry to SU(2)H × U(1)B
Summary
The Renyi entropy of a circular region in a d dimensional conformal field theory can be computed by calculating the Euclidean path integral for the theory on a branched d-sphere [1, 21]. The N = 4 theory with gauge group G consists of a vector field, Aμ, six scalars, φI , and four two component Weyl fermions, ψa. We will be interested in the case of G = SU(N ), for which the theory has a bonus, U(1)B, symmetry at large N and ’t Hooft coupling [22, 23]. This theory has 16 real supercharges, excluding the superconformal charges, and it is possible to compute the Euclidean partition function exactly on S4 [7]. We are interested in the partition function for the round sphere, but for the singular space, (2.1)
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