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Bulk entanglement entropy for photons and gravitons in AdS$_3$
by Alexandre Belin, Nabil Iqbal, Jorrit Kruthoff
- Published as SciPost Phys. 8, 075 (2020)
|As Contributors:||Alexandre Belin · Jorrit Kruthoff|
|Submitted by:||Kruthoff, Jorrit|
|Submitted to:||SciPost Physics|
|Subject area:||High-Energy Physics - Theory|
We study quantum corrections to holographic entanglement entropy in AdS$_3$/CFT$_2$; these are given by the bulk entanglement entropy across the Ryu-Takayanagi surface for all fields in the effective gravitational theory. We consider bulk $U(1)$ gauge fields and gravitons, whose dynamics in AdS$_3$ are governed by Chern-Simons terms and are therefore topological. In this case the relevant Hilbert space is that of the edge excitations. A novelty of the holographic construction is that such modes live not only on the bulk entanglement cut but also on the AdS boundary. We describe the interplay of these excitations and provide an explicit map to the appropriate extended Hilbert space. We compute the bulk entanglement entropy for the CFT vacuum state and find that the effect of the bulk entanglement entropy is to renormalize the relation between the effective holographic central charge and Newton's constant. We also consider excited states obtained by acting with the $U(1)$ current on the vacuum, and compute the difference in bulk entanglement entropy between these states and the vacuum. We compute this UV-finite difference both in the bulk and in the CFT finding a perfect agreement.
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Published as SciPost Phys. 8, 075 (2020)
Author comments upon resubmission
We thank the referee for their careful reading and valuable and much appreciated comments, questions and suggestions for improvement. We reply to them in turn below.
1) We thank the referee for this very interesting and important question. We agree that from an effective field theory description the Maxwell term will generically exist. However let us denote by L the (proper) bulk size of interest; though k may be very large, we are actually always interested in a regime where L ≫ k/e2. The reason is that L is a distance to the AdS boundary; thus its size is controlled by the CFT UV cutoff, which is chosen (independent of k) such that L ≫ k/e2 (though k is large too). In particular, the UV-cutoff is not constrained by the central charge of the CFT. Thus for the purposes of this problem we believe it suffices to focus purely on the Chern-Simons term.
We agree with the referee that understanding whether the transition of the two regimes happens for a spherical region in AdS is an interesting question in its own right, and would be worth investigating in the future. We have added a comment about this on p9.
2.1) Taking k large is not necessary at all for the purposes of our calculation. We simply mean that the application to holography would imply a large value of k. We have changed the sentence before the footnote accordingly.
2.2) We thank the referee for this comment and we have clarified this sentence and added a footnote in the revised manuscript. We would like to emphasize that the state operator correspondence does work in the bulk: Any state of the bulk Chern-Simons theory is given by the insertion of the appropriate operator at the south pole of the boundary sphere. Concerning the metaphor of oranges, we actually do think it is helpful (e.g. see the next point).
2.3) The bulk tube cuts out two circles at the boundary sphere and so the resulting surface is a torus. It is like cutting a cylindrical hole through an orange, resulting in a surface that is topologically a solid torus.
3.1,3.2) We thank the referee for spotting these typos. We have changed them in the revised manuscript.
3.3) We thank the referee for the feedback on the title, but we feel it is clear from the abstract what is meant with the title and so we refrain from any changes to the current title.
3.4) This is a very interesting comment, but we have not considered such ’islands’ in this manuscript. This also connects to the question raised before about the Maxwell term. This would be the appropriate way to address that question, namely by considering islands.
3.5) That is a great question and certainly worth investigating, but we have not done so in this manuscript.
3.6) The first equality is just a restatement of the FLM relation (given also in 1.1), which is then worked out the second equality. We therefore believe it is not needed to replace A with L there.
3.7) That is a (yet another) great question. It is important to emphasize that the CFT answer must be blind to ε_bulk: It cannot know about the effective bulk cut- off introduced, since by definition it only has access to bulk UV-finite quantities. Formulating a way to understand what a bulk cutoff is in the CFT (for example an upper bound on scaling dimensions of operators) is an important question in AdS/CFT that remains unanswered. From the bulk side, it would seem natural to pick L_planck ≪ ε_bulk ≪ L_AdS but it was not really relevant in our computations.
3.8) We had in mind specific examples where the UV regulator breaks a shift symmetry whereas the edge mode does not; however the idea is more general than this example, and so we have simply removed the words “symmetry-breaking.”
3.9) We thank the referee for this comment. The figure has been changed.
We hope that this answers the questions and comments raised by the referee.
Alexandre Belin, Nabil Iqbal and Jorrit Kruthoff