Multi-entropy at low Renyi index in 2d CFTs

This manuscript concerns multi-entropy, which is a recently proposed measure of multi-partite entanglement in CFTs with a definite conjectured holographic dual (Steiner trees in AdS3/CFT2). This quantity is generally difficult to calculate on the CFT side. The present manuscript manages to calculate it in several cases of interest. This is a technical feat, which will aid other researchers interested in multi-entropy to get off the ground.

The paper is clearly written, valid, useful, and on a timely subject. I recommend publication.

I have the following recommendations for the text:

(1) In equation (22), the authors wish to write $\sigma_{a^3 b^3}$ but the quantity that is actually given is $\sigma_{a b}$, i.e. the inverse of $\sigma_{a^3 b^3}$.

(2) I apologise for bringing this up but I recommend another proofreading of the text at the level of grammar and vocabulary. Some sentences are not easy to understand, for example "Along the four remaining intervals are shown the corresponding copy it will be glued to to form the replica manifold."

(3) Optional: I think it would be easier to understand definition (20) if the indices corresponded better with the letters. It is strange to associate $\alpha, \beta, \alpha', \beta'$ with $A$ and $a,b,a', b'$ with $B$. A far clearer notation would be to associate $a,a',\alpha, \alpha'$ with $A$ and $b,b', \beta, \beta'$ with $B$. I think this change does not impose a huge burden on the authors because it can be done locally (these indices do not show up in many other places in the text). On the other hand, it will make the manuscript much clearer to absorb for readers.

Multi-entropy is an interesting mutliparty generalization of entanglement entropy recently proposed by Gadde et al, with a conjectured holographic dual in terms of networks of bulk minimal surfaces. The present paper further develops the computation of multi-entropies both in CFTs and holographically. Unfortunately, with current CFT technology, multi-entropies are harder to calculate from first principles even than entanglement entropies. Nonetheless, the authors are able to compute (Renyi) multi-entropies in a number of cases, both in a general CFT and in specific ones such as free bosons and fermions, while importantly pointing out subtleties and pitfalls concerning for example the correct definition of the relevant twist operators in the fermionic case.

Investigations of the kind in this paper are essential for furthering our understanding of multi-entropy, a promising tool for understanding the structure of multiparty entanglement in quantum field theories and holography. The paper is generally clearly written. I recommend its publication in SciPost.

I have a few minor recommendations concerning the presentation.

First, a definition of multi-entropy in terms of the density matrix is not given until eqs. (19) and (20). So the formulas before that may appear somewhat arbitrary. It might be useful instead to start with the definition, then derive the twist operator formula. A small figure illustrating the tensor contractions in (20) would also be helpful for the reader’s intuition. A full review of multi-entropy isn’t necessary, but a quick reminder for a reader who hasn’t looked at the previous papers in a while would be useful.

Shortly after that, in (21), it wasn’t clear to me if the particular group representation given there was somehow unique (up to trivial equivalences like relabelling), or if some choice was being made.

A few even more minor corrections: In a few place, the cyclic group of order $n$ is written $Z^n$, where it would conventionally (in the physics literature) be written $Z_n$. Above (16), the $(q-1)$-fold power of $Z_n$ is written with a tensor product symbol, whereas it is actually just a (Cartesian) group product. On p. 19, just below (79), “Even though we are expecting” should be “Even though we are NOT expecting”. Also on p. 19, “fig. 14” should be “fig. 11” in a couple of place. In several places throughout the paper, “explicate” should be “explicit”.

1) A definition of multi-entropy in terms of the density matrix is not given until eqs. (19) and (20). So the formulas before that may appear somewhat arbitrary. It might be useful instead to start with the definition, then derive the twist operator formula. A small figure illustrating the tensor contractions in (20) would also be helpful for the reader’s intuition. A full review of multi-entropy isn’t necessary, but a quick reminder for a reader who hasn’t looked at the previous papers in a while would be useful.

2) In (21), it wasn’t clear to me if the particular group representation given there was somehow unique (up to trivial equivalences like relabelling), or if some choice was being made.

3) In a few place, the cyclic group of order $n$ is written $Z^n$, where it would conventionally (in the physics literature) be written $Z_n$.

4) Above (16), the $(q-1)$-fold power of $Z_n$ is written with a tensor product symbol, whereas it is actually just a (Cartesian) group product.

5) On p. 19, just below (79), “Even though we are expecting” should be “Even though we are NOT expecting”.

6) Also on p. 19, “fig. 14” should be “fig. 11” in a couple of place.

7) In several places throughout the paper, “explicate” should be “explicit”.