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Entanglement Negativity in Flat Holography
by Debarshi Basu, Ashish Chandra, Himanshu Parihar, Gautam Sengupta
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Authors (as registered SciPost users):  Debarshi Basu · Ashish Chandra · Himanshu Parihar · Gautam Sengupta 
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Preprint Link:  scipost_202107_00037v2 (pdf) 
Date submitted:  20211009 18:12 
Submitted by:  Sengupta, Gautam 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We advance holographic constructions for the entanglement negativity of bipartite states in a class of $(1+1)$dimensional Galilean conformal field theories dual to asymptotically flat three dimensional bulk geometries described by Einstein Gravity and Topologically Massive Gravity. The construction involves specific algebraic sums of the lengths of bulk extremal curves homologous to certain combinations of the intervals appropriate to such bipartite states. Our analysis exactly reproduces the corresponding replica technique results in the large central charge limit. We substantiate our construction through a semi classical analysis involving the geometric monodromy technique for the case of two disjoint intervals in such holographic Galilean conformal field theories.
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The detailed response to the referee' s comments and issues raised has been described in the author' s reply section.
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Reports on this Submission
Anonymous Report 1 on 20211027 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202107_00037v2, delivered 20211027, doi: 10.21468/SciPost.Report.3750
Report
I thank the authors for their response. I would however appreciate some further clarification by the authors. To make my question more precise:
To determine the conformal block eq(93) of the article (and hence eq(105)), one needs to impose certain monodromy conditions. In the relativistic case, there is only one condition, namely that the trace of the monodromy matrix should be related in a specific way to the conformal dimension of the operator exchanged in the respective channel.
Analogous conditions in the Galilean case for the HHLL (heavyheavylightlight) four point function and for exchanged/intermediate operators of small conformal dimension, were found in ref[67]. Namely that: trM=2, accompanied by eq(97) of this article.
Can perhaps the authors comment as to why these conditions are valid here when the external operators are all light (in the replica limit) and the operator exchanged is not light?
If I understand well the authors claim that the exchanged operator is the lightest of all exchanged operators. I agree with this, but it was my impression that the derivation was based on the assumption of an exchanged operator with small conformal dimension with respect to the central charge in the large central charge limit.
Author: Gautam Sengupta on 20211123 [id 1967]
(in reply to Report 1 on 20211027)We would like to thank the referee for the comment. The clarification sought by the referee is described below and included in our revised submission and detailed in two additional appendices B and C.
In this article we consider a four point twist correlator in the GCFT$_{1+1}$ where in the replica limit, all of the external operators are light but the exchange operator in the conformal block expansion remains heavy in the large central charge limit. We agree with the referee that the perturbative analysis in the expansion parameter $\epsilon_\alpha=\frac{6}{c_M}\chi_\alpha$ requires further investigation when the exchange operator is heavy. We would like to emphasize that even if $\epsilon_\alpha$ is not infinitesimally small, we may still perform a series solution to the Fuchsian differential equation given in eq. (86) of our revised manuscript. For the case when $\chi_\alpha$ is associated with a heavy exchange operator, we need to be more careful and in principle should keep track of the full series solution.
The first subtlety faced in such a series solution is that whether the monodromy condition given in eq. (91) of our revised manuscript has the correct form. We emphasize that eq. (91) reflects the monodromy condition only in the first order in the expansion parameter $\epsilon_\alpha$. We have added the derivation of the first order monodromy condition in appendix B of our revised manuscript. In particular, the monodromy matrix in the leading order is given in eq. (174), from which eq. (91) follows.
Since the truncation of the series solution up to the first order remains questionable, we have performed the next to leading order monodromy analysis also in appendix B of our revised manuscript and the corresponding monodromy condition is given in eq. (180). Interestingly, for the four point function in question, the next to leading order monodromy analysis shows that the auxiliary parameter $c_2$ and hence the dominant conformal block $\mathcal{F}_\alpha$ has exactly the same form as obtained through the first order analysis. This is reported in equations (182, 183) and discussed briefly in footnote 10 on page 23 of our revised manuscript. The analysis in appendix B hints towards the fact that the monodromy method works at each order in the expansion parameter. We may interpret this as follows: the approximate solution to the differential equation is able to pick up the same monodromy while circling around the light operators, as would the complete solution. This provides a strong substantiation of our result for the dominant conformal block. These subtleties have been briefly discussed in footnote 10 on page 23 of our revised manuscript.
Furthermore, in appendix C of our revised submission, we have shown that the fourpoint twist correlator in a (1+1)dimensional GCFT may be obtained by performing the InonuWigner contractions of the corresponding relativistic result in the context of $AdS_3/CFT_2$ obtained in ref [23,25] of our revised manuscript. Remarkably, the result matches exactly with that obtained through the geometric monodromy method. This provides further support towards the validity of the monodromy analysis up to leading order in the exchanged dimension. Furthermore as pointed out in our earlier response an independent holographic crosscheck from the bulk entanglement wedge cross section as described in arXiv: 2106.14896 also confirms our results for the mixed state configuration in question.
We have also discussed the above issues in the summary section of our revised manuscript. In addition, we would like to point out a small typographical error in eq. (150) of our revised submission regarding a sign discrepancy that has now been corrected.