Abstract
Employing a conformal map to hyperbolic space cross a circle, we compute the universal contribution to the vacuum entanglement entropy (EE) across a sphere in even-dimensional conformal field theory. Previous attempts to derive the EE in this way were hindered by a lack of knowledge of the appropriate boundary terms in the trace anomaly. In this paper we show that the universal part of the EE can be treated as a purely boundary effect. As a byproduct of our computation, we derive an explicit form for the A-type anomaly contribution to the Wess-Zumino term for the trace anomaly, now including boundary terms. In d=4 and 6, these boundary terms generalize earlier bulk actions derived in the literature.
Highlights
Entanglement entropy has played an increasingly important role in theoretical physics
At least regarding logarithmic terms, we have specified a separation between bulk and boundary terms by insisting that the only place in which τ appears without a derivative in the boundary action is multiplying Qd. This split has the advantage of giving the boundary contribution a topological interpretation when the reference metric is flat. Given this choice, it becomes manifest for the two maps we considered that both W[δμν, e−2σδμν]|Boundary and W [δμν] will yield the Euler characteristic of the flat space multiplied by a logarithm of the UV cut-off
We resolved the puzzle described in ref. [14]: the universal logarithmic term in the entanglement entropy (1.2) across a sphere in flat space can be recovered by a Weyl transformation to hyperbolic space, provided one keeps careful track of boundary terms
Summary
Entanglement entropy has played an increasingly important role in theoretical physics. (In six dimensions, our boundary action is only valid in a conformally flat space time, while in four dimensions, the answer provided is completely general.) In section 5, we resolve the puzzle of how to compute the entanglement entropy of the ball through a map to hyperbolic space in general dimension. The resolution of this puzzle constitutes the main result of the paper. Appendix D provides a corresponding holographic calculation of entanglement entropy through a map to hyperbolic space
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