Supersymmetric Rényi entropy and defect operators

Abstract

We describe the defect operator interpretation of the supersymmetric Rényi entropies of superconformal field theories in three, four and five dimensions. The operators involved are supersymmetric codimension-two defects in an auxiliary Z n gauge theory coupled to n copies of the SCFT. We compute the exact expectation values of such operators using localization, and compare the results to the supersymmetric Rényi entropy. The agreement between the two implies a relationship between the partition function on a squashed sphere and the one on a round sphere in the presence of defects. 4 Discussion 34 A Conventions 35 B Special functions 35 -1 -C The instanton partition function 37

Entanglement entropy: Holography and renormalization group

“…• The description by codimension-two defects living on the entangling surface is manifest (Nishioka and Yaakov, 2017).…”

Instantons and entanglement entropy

“…(See, for example, [5,6].) In those studies, their contributions either happen to cancel out, or are sufficiently localized near the conical singularity that no modification of the area law is observed.…”

Spinning conformal defects

“…Then correlation functions can depend on Z only through the gauge invariant tensor. The gauge invariance is equivalently represented as the transversality condition, 9) and the dimension and spin are written as the homogeneity conditions,…”

“…The homogeneity conditions with respect to the spins of a spinning defect and bulk local operators give rise to three constraints on the non-negative exponents n k (k = 1, · · · , 12), n 9 + n 10 + n 11 + n 12 = s , (4.54) n 1 + n 2 + n 5 + n 6 + 2n 7 + n 9 = l 1 , (4.55) n 3 + n 4 + n 5 + n 6 + 2n 8 + n 10 = l 2 , (4.56) 9 In the version 1, we included the invariantŴ • C1 • C2 •Ŵ as a linearly independent basis, but it can be decomposed by the others due to the identity (A.3). We thank Sunny Guha and Balakrishnan Nagaraj for informing us of this point.…”

“…Another situation where defects come into play in an unexpected way is entanglement entropy in QFT that is commonly defined for a spacial region at a given time slice. It has an alternative description as a codimension-two defect bounding the region [4][5][6][7], which can be made more manifest in a supersymmetric analogue of entanglement Rényi entropies [8,9]. While a complete classification of defects is far from our reach so far, there are a particularly nice class of defects called conformal defects that preserve a part of the conformal symmetry in conformal field theories (CFTs).…”