Minimal Areas from Entangled Matrices

In this article, the authors study a notion of entanglement entropy within the internal degrees of freedom in a large N gauged matrix model. Various proposals for such a notion of entanglement have been discussed previously in the literature, and this paper builds on some of this previous work to propose a Ryu-Takayanagi like minimization formula for the entanglement entropy. The essential subtlety in defining entanglement entropy in such a model is that the gauge-invariant Hilbert space of the matrix model does not admit any canonical factorization with respect to which the entropy can be computed, and thus the entropy is naturally associated to a choice of a factorization map, i.e., a map of the gauge-invariant Hilbert space into an extended Hilbert space which does admit factorization. The main result of this paper is a proposal for the factorization map that involves an integration over gauge transformations that "move the subregion" around while preserving its volume -- the resulting entropy in the saddle point approximation then gives an RT-like minimization formula for the entropy. In my opinion, the paper is extremely clear and very well-written. The results are very nice, and of broad potential interest to the AdS/CFT and quantum gravity community. I strongly recommend publication of the article.

One or two minor things that I personally found confusing: in AdS/CFT, there is a canonical UV definition of the entropy in terms of spatial subregions in the boundary CFT, which can then be computed in the semi-classical limit by the RT formula. In the matrix model, however, it seems like there is no canonical choice -- one must compute the entropy w.r.t to a choice of factorization map. Is there a sense in which the factorization map defined by the authors a canonical one? Or is it the case that this is a choice which gives a pleasing answer involving the minimization formula? Another general confusion is that in these matrix models, U(N) gauge transformations on the matrix side end up becoming area-preserving diffeomorphisms in the effective geometric description, and this was important in the interpretation of the entropy formula. How general is this fact? For instance, do we expect the U(N) gauge invariance of a general holographic field theory (say, N=4 SYM) to be related to diffeomorphism invariance in the dual gravity description? I feel some discussion on these issues could be useful, but as I said above, the paper is already very well-written and very readable, so this is more of a suggestion, not a requirement.

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Summary:
This paper is a continuation of previous efforts to define a Hilbert space factorization in matrix models that leads to an emergent area law entanglement entropy. The matrix model consists of NxN Hermitian matrices subject to a gauge symmetry given by conjugation by U(N) matrices. It is known that factorization in gauge theories requires a lifting of the Gauss law constraint, leading to an extended Hilbert space on which certain gauge symmetries are promoted to physical ``edge mode" symmetries. Physical states satisfying the Gauss law constraint are singlets under these edge mode symmetries, and their presence leads to a Gauss law entanglement entropy that counts the corresponding edge modes. These transforms under different representations of the symmetry, and the entropy involves dimensions of these representations.

For a generic matrix model, a preliminary (partially gauge fixed) notion of a subsystem can be given by a fixed MxM block, in which case the edge mode symmetry is U(M)xU(N-M). It was explained in previous work that for semi-classical states Fuzzy sphere state of the mini-BMN model, the U(N) symmetry has a low energy description in terms of area preserving diffeomorphisms of a two-sphere, and the MxM block can be viewed as a subregion of a two sphere. In the low energy description, the edge mode symmetry U(M)xU(N-M) are volume preserving diffeos that fix the subregion. The saddle point approximation to the entropy produces an area law on the sphere ( in this case area= perimeter) , which counts the dimensions of the dominant irrep.

In this work, the authors incorporate the edge mode symmetries U(N)/U(M)xU(N-m) that do not fix the MxM block: these ``wiggle" modes move the subregion on the sphere around in an volume preserving away. A gauge fixed subsystem then corresponds to any region of fixed volume. The authors show that reduced density matrix
involves direct sum over such subregions of fixed volume, and a saddle point approximation pics out the region with minmimal area. This is then compared to the area minimization of the Ryu-Takayanagi formula in holographic theories.

Recommendation:
The paper provides a compelling proposal for the factorization map and extended Hilbert space construction for matrix models, and illustrates in detail how this works in the mini-BMN model. I am happy to recommend its publication, provided the questions below are answered.

1.) Section 2.3 on the interpretation discussion of quantum reference frames is interesting but I would like to clarify how it goes beyond the usual paradigm of edge modes as ``Stuckelberg" degrees of freedom

For example consider the discussion around eq 2.16, that is supposed to make the matrix elements of X a gauge invariant quantity, despite that fact it manifestly transforms under conjugation by a U(N) matrix U. In the "Stuckelberg" edge mode paradigm, we would say that to make X gauge invariant, we need to introduce edge modes g in U(N), which is a choice of gauge promoted to a physical degree of freedom. Then we dress X by these edge modes by introducing gXg^{-1} as the new gauge invariant degree of freedom, where the gauge transformations map X to UXU^{-1} and g to gU^{-1}. Then gXg^{-1} is gauge invariant, but only after introducing g as a physical edge modes. Do the authors agree that what I said above is equivalent to the construction below 2.16?

If so, is the quantum reference frame the Stuckelberg formalism ?

2.) I am puzzeld by the definition of F in 2.38. According to the first two definitions, H and H' are both in G. What does it mean to take there direct product and then delete G?

Or did you mean by G\cap U(M) something different than the intersection of two sets? Like taking elements in G and restricting to their action on the first M indices?

Also Flag manifolds are quotients, did you mean HxH'/G instead of HxH'\G?
3.) Equation 2.39 is difficult to parse. Can an explicit definition of |\psi_{V}> be given?

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