How do hyperbolic tessellations like {7,3} in Anti de-Sitter space relate to our intuition of 3D space (or 4D structure if you include time)?

Question

I just ran into the AdS/CFT correspondence, as I am looking at various use-cases of hyperbolic tessellations, specifically related to the pentagrid and heptagrid as defined by Maurice Margenstern in their Cellular Automata in Hyperbolic Spaces work.

I was asking ChatGPT about it, and it led to imagining (without a deep quantum mechanics background), quantum gravity modeled using the $\{7,3\}$ hyperbolic tessellation (heptagrid). How does that work?

Just as lattice models are used in Euclidean space to study quantum chromodynamics (QCD) and other field theories, hyperbolic tessellations can be used to study quantum gravity in AdS space. The discrete heptagonal cells form a lattice that approximates the continuous space.

Just as in lattice QCD, where the continuous space-time is approximated by a finite lattice of points, the {7,3} tessellation approximates the continuous hyperbolic AdS space with a discrete lattice of heptagonal cells.

The larger the tessellation (i.e., the more heptagons included), the better the approximation of the continuous AdS space.

Main question is, how do I conceptually/mentally relate our experience of 3D space, where things are locally connected and such, to an AdS space on the hyperbolic tessellation of $\{7,3\}$? How is it used to model gravity, at a higher level, to point me in the right direction conceptually?

Images such as this make it seem like the hyperbolic Poincaré disk (forming a tube, when projected through time), is somehow a representation of 3D space (or 4D spacetime), but I don't get how the variables are translated from one space to the next. How do you conceptually make the leap from our intuition of the 3D space of the universe, to this hyperbolic tessellation representation? The hyperbolic space is only 2D, so how does it capture 3D relationships?

Answer 1

AdS/CFT does not usually involve the hyperbolic plane. The original examples due to Maldacena involve higher-dimensional hyperbolic spaces. The best known example features string theory in a five-dimensional hyperbolic space (with the five other dimensions compactified in a hypersphere), whose boundary is a four-dimensional space. The idea is that quantum correlations between locations in the four-dimensional space, can be calculated by strings which enter and exit the hyperbolic fifth dimension from those locations (interpreted as locations on the boundary of the hyperbolic space).

The connection to gravity is that in such "holographic" relationships, the boundary theory is one without gravity. Gravity emerges along with the extra dimensions.

AdS/CFT that does involve the hyperbolic plane, specifically, would be "AdS3/CFT2". Physics in a 3-dimensional universe, with two spatial dimensions and one time dimension, would be equivalent to physics in a circular "line land" with just one spatial dimension, that forms the boundary of the hyperbolic plane. There are actually a lot of papers on AdS3/CFT2, but it doesn't describe our universe. To have an AdS/CFT that even looks like ours, it would need to be AdS4/CFT3 (the best known example there is ABJM theory). But that's not our world either, since our space isn't hyperbolically curved.

In the case of a flat space with a gravitational force, such as our universe appears to be, the holographic boundary might be the "cosmological horizon". That would mean that our 3+1 dimensional physics (standard model plus gravity) has a holographic reformulation as a 2+1 dimensional theory on the "celestial sphere". This celestial holography is a nice idea but it's much less developed than AdS/CFT; as far as I know, no one is able to write the complete equations for the celestial dual theory.

In a full version of AdS/CFT, the spaces are continua. However, there are attempts to approximate or model these relationships using lattices or grids. The original example of this is "AdS/MERA", but by now there are many more, e.g. the "HaPPY" tensor network, which is based on {5,4}.

You should be aware that these lattice approaches to AdS/CFT can be controversial! The main criticism seems to be that a lot of this work is being done, without sufficient attention to a continuum completion via string theory.

As for {7,3}, I haven't found anyone using that tessellation in particular for AdS3/CFT2, and I have no idea of its prospects. It would be something if Margenstern's heptagrid cellular automaton could somehow be quantum-enhanced into an example of holography. There is "folklore" (i.e. belief among physicists) that any kind of dynamics in AdS should have a boundary formulation, but CA dynamics can be so unlike physics (e.g. lacking conservation laws), that it might not apply here.