Sebastian Fischetti (left) is a graduate student of Professor Don Marolf (middle) at UCSB. Aron Wall (right) is a member of the School of Natural Sciences at the Institute for Advanced Study.

One of the most useful features of gauge/gravity duality is that it converts difficult problems in certain types of gauge theories into (relatively) simple geometric problems in gravity in one higher dimension. For example, the Hubeny-Rangamani-Takayagani (HRT) conjecture says that the entropy associated with a region of a holographic gauge theory is just given by the area of a certain extremal surface (i.e. a surface that extremizes the area functional) that lives in the dual gravitational theory.

A conformal diagram of an AdS-dS-wormhole. The lack of intersections between the dashed red and dotted blue lines anywhere in the spacetime implies that the HRT conjecture cannot compute the entanglement between the two boundaries.

In our CQG paper, we show that the original statement of the HRT conjecture becomes ill-defined in certain spacetimes having two anti-de Sitter (AdS) boundaries connected by an inflating (but non-traversable) wormhole. This occurs because the inflating region within the wormhole can prohibit the existence of the supposed HRT surfaces that should be used to compute the dual gauge theory entanglement between the two AdS boundaries. In related contexts, an HRT surface exists but predicts an entanglement so large that it conflicts with the known density of states in the dual gauge theory. The latter effect happens because, in a certain sense, the existence of the “correct” (and smaller) HRT surface has been obstructed by the inflating region.

But there is also good news: we suggest several modifications of the conjecture that, at least in our so-called AdS-dS-wormholes, render it well-defined. Some are straightforward: we introduce two limiting procedures that regulate the divergences associated with the inflating region. The modified prescriptions each give physically well-behaved results, though it is unclear whether they agree with one another and whether they are regulator-dependent.

Another possibility is that the resolution involves using \textit{complex} surfaces in the complexified dual spacetime. This idea appears natural in the context of the Lewkowycz-Maldacena argument which gives a partial proof of HRT. Their approach makes use of a saddle point approximation, which can often give rise to complex saddles. As an exploration in this direction, we show that many complex external surfaces exist in pure de Sitter space.

It remains to be determined which of these modifications (if any) is correct. Until then, exploring AdS-dS-wormholes is a promising way to test the limits of gauge/gravity duality.

Read the full article in Classical and Quantum Gravity:
A paucity of bulk entangling surfaces: AdS wormholes with de Sitter interiors
Sebastian Fischetti, Donald Marolf and Aron C Wall
Sebastian Fischetti et al 2015 Class. Quantum Grav. 32 065011

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