Entanglement Negativity in Flat Holography

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 four-point twist correlator in a (1+1)-dimensional GCFT may be obtained by performing the Inonu-Wigner 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.