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arXiv:1611.03116

Rényi Entropy for a $\bf 2d$ CFT with a gauge field: $\bf \widehat{\rm SU}(N)_1$ WZW theory on a branched torus

Howard J. Schnitzer

Published Nov 9, 2016• 5 citations
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Abstract

The Rényi entropy for the $\widehat{\rm SU}(N)_1$ WZW model as described by $N$ free fermions coupled to a $U(1)$ constraint field is computed on an $n$-sheeted branched torus. The boundary condition of the harmonic component of the gauge field on the homology cycles of the genus $g$ Riemann surface is central to the final result. This calculation is complementary to that of arXiv:$1510.05993$, which presents the bose side of the bose-fermi equivalence.

DOI
10.48550/arxiv.1611.03116

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Rényi Entropy for a 2d\bf 2d2d CFT with a gauge field: SU^(N)1\bf \widehat{\rm SU}(N)_1SU(N)1​ WZW theory on a branched torus
The Rényi entropy for the SU^(N)1\widehat{\rm SU}(N)_1SU(N)1​ WZW model as described by NNN free fermions coupled to a U(1)U(1)U(1) constraint field is computed on an nnn-sheeted branched torus. The boundary condition of the harmonic component of the gauge field on the homology cycles of the genus ggg Riemann surface is central to the final result. This calculation is complementary to that of arXiv:1510.059931510.059931510.05993, which presents the bose side of the bose-fermi equivalence.
We analytically evaluate the Rényi entropies for the two dimensional free boson CFT. The CFT is considered to be compactified on a circle and at finite temperature. The Rényi entropies S n are evaluated for a single interval using the two point function of bosonic twist fields on a torus. For the case of the compact boson, the sum over the classical saddle points results in the Riemann-Siegel theta function associated with the A n−1 lattice. We then study the Rényi entropies in the decompactification regime. We show that in the limit when the size of the interval becomes the size of the spatial circle, the entanglement entropy reduces to the thermal entropy of free bosons on a circle. We then set up a systematic high temperature expansion of the Rényi entropies and evaluate the finite size corrections for free bosons. Finally we compare these finite size corrections both for the free boson CFT and the free fermion CFT with the one-loop corrections obtained from bulk three dimensional handlebody spacetimes which have higher genus Riemann surfaces as its boundary. One-loop corrections in these geometries are entirely determined by quantum numbers of the excitations present in the bulk. This implies that the leading finite size corrections contributions from one-loop determinants of the Chern-Simons gauge field and the Dirac field in the dual geometry should reproduce that of the free boson and the free fermion CFT respectively. By evaluating these corrections both in the bulk and in the CFT explicitly we show that this expectation is indeed true.
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