Minimal area submanifolds in AdS × compact

Abstract

Abstract:We describe the asymptotic behavior of minimal area submanifolds in product spacetimes of an asymptotically hyperbolic space times a compact internal manifold. In particular, we find that unlike the case of a minimal area submanifold just in an asymptotically hyperbolic space, the internal part of the boundary submanifold is constrained to be itself a minimal area submanifold. For applications to holography, this tells us what are the allowed "flavor branes" that can be added to a holographic field theory. We also give a compact geometric expression for the spectrum of operator dimensions associated with the slipping modes of the submanifold in the internal space. We illustrate our results with several examples, including some that haven't appeared in the literature before.

Entanglement shadows in LLM geometries

“…In section 2, we briefly review the LLM geometries first constructed in [16], and introduce the examples we consider. In section 3 we consider extremal surfaces in the central region of our geometries, and explain the relation to the earlier work of [22][23][24]. In section 4, we argue that the boundary-anchored minimal surfaces in our spacetime close off on the central S 5 without penetrating deep into the central region.…”

“…It would be worth understanding whether there is such an interpretation for non-minimal extremal surfaces in the AdS 5 case also. Interesting earlier holographic studies of Yang-Mills theory in the Coulomb branch [22][23][24] had proposed that a minimal surface which divides the S 5 part of the boundary of an asymptotically AdS 5 × S 5 spacetime can be identified with the entanglement entropy associated to a non-spatial division of the field theory degrees of freedom. In the context of our geometries, the surfaces described by [22][23][24] can be regarded as extremal surfaces in the central AdS 5 region, which can be extended into the asymptotic AdS 5 region to describe entanglement in a conventional spatial partition of the UV theory.…”

“…Interesting earlier holographic studies of Yang-Mills theory in the Coulomb branch [22][23][24] had proposed that a minimal surface which divides the S 5 part of the boundary of an asymptotically AdS 5 × S 5 spacetime can be identified with the entanglement entropy associated to a non-spatial division of the field theory degrees of freedom. In the context of our geometries, the surfaces described by [22][23][24] can be regarded as extremal surfaces in the central AdS 5 region, which can be extended into the asymptotic AdS 5 region to describe entanglement in a conventional spatial partition of the UV theory. Our analysis shows that the particular extremal surfaces studied in [22][23][24] are not, in fact, the minimal ones that are asymptotic to the equator of the boundary S 5 .…”

“…In the context of our geometries, the surfaces described by [22][23][24] can be regarded as extremal surfaces in the central AdS 5 region, which can be extended into the asymptotic AdS 5 region to describe entanglement in a conventional spatial partition of the UV theory. Our analysis shows that the particular extremal surfaces studied in [22][23][24] are not, in fact, the minimal ones that are asymptotic to the equator of the boundary S 5 . It would be very interesting to understand what information theoretic quantity is being computed by such extremal surfaces, and also by the true minimal surfaces with these boundary conditions.…”

Areas and entropies in BFSS/gravity duality

“…Since our expectation (according to previous references [22][23][24] and our analysis of the symmetrical case) is that the surface should approach some equator of the spatial S 8 at the boundary, we will gear our parametrization to surfaces that approach the plane x 9 = 0 at the boundary. We parameterize the surface using the coordinates x 1 , .…”