Charged Eigenstate Thermalization, Euclidean Wormholes and Global Symmetries in Quantum Gravity

We thank the referees for the detailed reading of our papers and for the useful comments, to which we reply below. We have replied to the referee concerns in the same order as they were raised.

{\bf Comments for referee 1}
1 - We have fixed the references.

2- We have added a more detailed explanation of this formula as requested by the referee.

3- We have added a definition of $W(x,y)$ below equation (3.4), and have added several remarks in the paragraphs that follow. The simplest way to see that this Wilson line is non-zero is to remember that we are working in Euclidean signature. Consider first a theory in Lorentzian signature on a compact manifold times time. One can have correlation functions of the form $\langle O(x,t=0)W(x,y)O(y,t=0) \rangle$. These are in fact the only type of operators that make sense since on a compact manifold the total net charge must be 0. In Euclidean signature however, there is no preferred notion of spatial slice (in particular the gravity computation we are doing does not have a clear Lorentzian counterpart that we should be matching). Therefore, as long as there is a Euclidean topological sphere that can encircle the two operators and the Wilson line one should imagine that such a correlation need no longer vanish. We have added an explanation clarifying this fact.

4- We have shown that assuming both that a global symmetry exists in the bulk and that the wormhole geometry computes the variance leads to a contradiction. Therefore, something has to give. But we agree with the referee that this is not really a paradox, but indeed more like a constraint that must be satisfied by consistency. We have changed the wording of the paragraph accordingly.

5- We agree with the referee, we have given a more explicit presentation of the two possibilities and have restated equation (1.2) there. This is the last paragraph before the discussion.

6- The fact that charge conservation is violated in correlation functions to us suggests an explicit breaking of the symmetry. Such effects occur in the standard model, where for example only the difference of the baryonic and leptonic $U(1)$ is a true global symmetry. We are not aware of an explicit model in quantum mechanics where something of this nature happens, but one could consider a complex SYK model, where we add by hand one coupling affecting only 4 fermions that breaks the symmetry. We don't expect this one coupling to be very important in the IR, but it certainly explicitly breaks the symmetry for high energy states.

{\bf Comments for referee 2}
1- We thank the referee for bringing up this issue. Indeed, our discussion of the Wilson line was incorrect. The standard product of CFT one-point functions (which is charged under $U(1)\times U(1)$) does not involve the Wilson line. We have adapted the text around equation (3.4) taking this into account. We have pushed the discussion of the Wilson line that extends through the wormhole to the discussion section.

2- We thank the referee for bringing up these references and agree that they should be mentioned. We have added a sentence in the introduction to this effect.

3- We believe it is natural to label states by their relevant quantum numbers, which in this case involves both energy and charge. However, we agree with the referee that the sums were not clear enough so we have modified them accordingly to make the selection rule in charge clearer.

4- The referee brings up an interesting point, but this is not very different from the usual ETH without charge. There, the smooth functions are functions of the energy differences which are discrete numbers as well. The point is simply that the sum over states can be replaced by an integral to a good approximation. There is however an important difference between charge and energy quantum numbers, namely that the charge quantum numbers are typically much more sparsely spaced (typically $\mathcal{O}(1)$) than the energy levels. For states with macroscopic charge, one should still be able to replace the sum over charges by an integral, which is what we have in mind here. We added a comment to clarify this.

5- This question is about the notion of "simple" operators. Respectfully, we disagree with the fact that ETH should be closed under products of operators, because at some point one can product enough operators together so as to make a macroscopic operator which is no longer simple. So the notion of simple operator means $q\ll S$ (or potentially some other power of $S$).

6- We have fixed the reference issue.

Apart from the changes incorporated to address the referee comments, we have also included an appendix to discuss some of the comments following equation (2.6). This in particular demonstrates more clearly the prediction for the 2-point function of the charged operators due to the ETH ansatze that we have discussed in Section 2.