Areas and entropies in BFSS/gravity duality

Abstract

The BFSS matrix model provides an example of gauge-theory / gravity duality where the gauge theory is a model of ordinary quantum mechanics with no spatial subsystems. If there exists a general connection between areas and entropies in this model similar to the Ryu-Takayanagi formula, the entropies must be more general than the usual subsystem entanglement entropies. In this note, we first investigate the extremal surfaces in the geometries dual to the BFSS model at zero and finite temperature. We describe a method to associate regulated areas to these surfaces and calculate the areas explicitly for a family of surfaces preserving SO(8) symmetry, both at zero and finite temperature. We then discuss possible entropic quantities in the matrix model that could be dual to these regulated areas.

Gauge invariant target space entanglement in D-brane holography

“…The paper [79] has explored general extremal surfaces in D brane geometries (as distinct from RT surfaces) and speculated on possible meanings of their areas with entanglement of degrees of freedom in the D0 brane quantum mechanics. In particular, these authors have considered subsets of operators consisting of linear combinations of traceless symmetric products of the matrices in the D0 brane theory which would correspond to functions which have support on some region of S 8 and speculated that an entropy can be associated with such a subset.…”

“…15 Unfortunately we do not see any evidence for this so far and leave it as an important question for further investigation. On a related note, one would think that area of extremal surfaces not of the RT type, as in D0 brane geometry [79] would also have some understanding in terms of target space entanglement entropy. Finally, for usual AdS/CF T duality there is evidence in favour of the conjecture that there is an intimate connection between entanglement in base space and emergence of a smooth AdS bulk with locality [85][86][87].…”

Target space entanglement in quantum mechanics of fermions and matrices

“…The Hilbert space is generally not tensor-factorized with respect to the target space. An alternative method is adopted to resolve this issue [15,16]. This is based on an algebraic approach [17] (for reviews see also [18,19]).…”

“…The function t(θ) has four discontinuities located at ±θ ± with θ ± = π(d ± r) in the range [−π, π] as and m 1 , m 2 are arbitrary integers. The Fisher-Hartwig conjecture states that the large N asymptotic behavior of det T is given by 16 det T…”

Quantum error correction and large $N$

“…Moreover, for generic matrix/tensor models the bulk is highly stringy. BFSS does have a well-defined ten-dimension bulk, but it is still not clear what is the boundary counterpart of extremal surfaces in the bulk [55].…”