Algebraic perturbation theory: traversable wormholes and generalized entropy beyond subleading order

The use of operator algebras to study thermodynamic aspects of gravity has a long history. Recently, this topic has been revitalized by Witten's observation that the general structure of the von Neumann algebra of observables is modified in certain cases. For example, in AdS/CFT, when accounting for certain gravitational corrections, the so-called crossed product construction applies. This construction amounts to appropriately adjoining the generator of modular evolution. The advantage is that if the original algebra is of type III, the new algebra becomes type II, allowing one to define entropies and density matrices in a rigorous way. The crossed product construction is now understood to apply more generally in various cases.

The authors of this paper study how this construction is modified in perturbation theory when introducing deformations. They consider perturbations which can be written as the action of a unitary operator on the whole Hilbert space, but which do not act unitarily on the algebra or on its commutant. In this case the von Neumann entropy will generically change due to the perturbation.

In particular, they focus on a deformation introduced by Gao, Jafferis, and Wall (GJW) in the context of AdS/CFT, which involves terms from both the algebra and its commutant, effectively coupling the two boundaries of the eternal AdS black hole. The authors determine the general structure of the deformation of the algebras for an arbitrary perturbation of this sort, including the change in von Neumann entropy. They then apply their formalism to the GJW perturbation and discuss its physical implications.

This topic is certainly worth studying, and the authors make an important contribution. Therefore, I recommend this paper for publication. However, I have a few comments and corrections that I would like the authors to address briefly.

1) When formula (2.5) for the trace is used, at least in the context of Witten's result for the canonical ensemble, the factor of $X$ in the exponential should depend on $N$, the rank of the dual gauge theory. This dependence appears to be generic in the canonical ensemble, and formulas must be interpreted only formally as functions of $N$ (see, for example, Appendix A of the follow-up paper 2209.10454). Is this $N$-dependence present here as well? If so, does it appear in the $e^{\lambda (\lambda_0)}$ factor in equation (3.24)? How does it impact the analysis of the perturbative expansion, for instance, in Section 4.3?

I believe a comment from the authors could clarify this situation. Note that these formulas for the trace appear elsewhere in the literature, and it is commonly accepted that they can be interpreted as formal results without invalidating other arguments. That would be fine if the same applies here.

2) There is a typo in formula (3.28) on page 11: there is an extra ")" before the ket $| \Psi \rangle$.

3) In footnote 12, as well as in Section 4.2, the authors remark that their definition of the left Hamiltonian does not involve subtracting the vacuum expectation value. They justify this by referring to their previous work. I believe the readability of the paper would improve if they expanded on this remark.

4) In the first line of page 12, the authors write: "the state of the $L^2$ factor above." To what does "above" refer? Could they be more explicit?

5) In formula (3.33), the authors consider the possibility that the temperature changes after the perturbation. This is further discussed on page 22, where they consider the GJW deformation. In the deformed KMS state, the (inverse) temperature is now $\beta = \beta_0 + \delta \beta$, where $\beta_0$ is the temperature of the unperturbed state. But isn't $\delta \beta$ computable from the perturbation? In ordinary statistical mechanics, one expects that, in linear perturbation theory, if the Hamiltonian is perturbed as $H + V$, then the change in temperature at first order is controlled by $\langle V \rangle_{\beta_0}$. Could something similar hold in this case? I believe it would be helpful for the authors to add a comment on this point.

6) The authors consider deformations which act unitarily on the full Hilbert space but not unitarily on the algebra of observables on one of the boundaries. Is it possible to show directly that 4.1 is of this form? What I mean is are there particular conditions that the coupling $h$ or the operators have to satisfy, or is this true in general due to the coupling of the two boundaries?

7) There is a typo above equation (4.7): "the $U$ dependence between operators and states allows us to to take the difference..." One "to" should be removed.

8)There is a typo in the statement before equation (4.16) on page 22: should $\delta H_0 = - \delta S$ be $\delta H_0 = - \delta I$?

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