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mitchell porter

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There was a discussion of what "AdS/LQG" could be, https://www.physicsforums.com/showthread.php?t=485800&page=7" the appearance of Macdowell-Mansouri gravity in Dmitri Polyakov's construction of AdS4 Vasiliev gravity from string vertex operators, I had the following thoughts.

A string theory can be expressed as a theory with an infinite number of fields of arbitrarily high spin; these are the modes of the string. Vasiliev's "higher spin" gravity theory also has this property, the difference being that the high-spin modes are massive in string theory and massless in Vasiliev gravity. Vasiliev gravity also shows up in a type of AdS duality.

First of all, I am wondering if both theories can be expressed in terms of spin foams, in which there are "two-complexes" (the transition processes which change the spin-net) of arbitrary complexity, which can be grouped into levels corresponding to the various high-spin fields (or perhaps, more plausibly, corresponding to combinations of the high-spin fields). What I'm thinking is that perhaps every UV-completion of quantum gravity is indeed a string theory, but that it will also have a spin-foam formulation. This hypothesis would be more plausible if we could show that the only way for a spin foam to produce a classical limit is to encode a string theory solution in this way.

Next, Vasiliev gravity so far is only defined for AdS space. So if it can be expressed as a spin foam, what corresponds to the boundary theory? Any answer to this question might also tell us about the "AdS/LQG" representation of stringy AdS/CFT.

In AdS5/CFT4, the boundary theory is a Yang-Mills gauge theory, but in AdS4/CFT3, the boundary theory is a Chern-Simons theory. These aren't disconnected facts; I read somewhere that these Chern-Simons theories are like a dimensional reduction of a Yang-Mills theory in which the kinetic term disappears.

Now let us think about dS4/CFT3 for a moment. Although no concrete examples of dS/CFT are known, one of the standard ideas for how it works is that a gravitational theory in a dS space is dual to a Euclidean CFT in a space of one less dimension, which corresponds to the past infinity of the dS space. Sometimes there are thought to be two dual CFTs for dS, the other one at future infinity, with the cosmological evolution in between somehow corresponding to an RG flow between the CFT in the past and the CFT in the future.

In any case, what I want to suggest is that the Hilbert space of states which is constructed in loop quantum gravity (loop states, spin-network states being two alternative bases) is the Hilbert space of the Euclidean CFT3 on the boundary of a dS4, and that dS4 time evolution should arise somehow from an RG flow on that Euclidean CFT; and that this is the appropriate way to think about dynamics in LQG.

In other words: in AdS/LQG, the boundary theory is a (2+1)-dimensional Chern-Simons theory, and in dS/LQG, the boundary theory is a "timeless" 3-dimensional gauge theory which is just the Euclidean version of the Chern-Simons theory. In both cases, it should be possible to express the bulk theory as a spin foam encoding RG flow in the boundary theory, though in the AdS case it will be "radial" flow rather than timelike.

A string theory can be expressed as a theory with an infinite number of fields of arbitrarily high spin; these are the modes of the string. Vasiliev's "higher spin" gravity theory also has this property, the difference being that the high-spin modes are massive in string theory and massless in Vasiliev gravity. Vasiliev gravity also shows up in a type of AdS duality.

First of all, I am wondering if both theories can be expressed in terms of spin foams, in which there are "two-complexes" (the transition processes which change the spin-net) of arbitrary complexity, which can be grouped into levels corresponding to the various high-spin fields (or perhaps, more plausibly, corresponding to combinations of the high-spin fields). What I'm thinking is that perhaps every UV-completion of quantum gravity is indeed a string theory, but that it will also have a spin-foam formulation. This hypothesis would be more plausible if we could show that the only way for a spin foam to produce a classical limit is to encode a string theory solution in this way.

Next, Vasiliev gravity so far is only defined for AdS space. So if it can be expressed as a spin foam, what corresponds to the boundary theory? Any answer to this question might also tell us about the "AdS/LQG" representation of stringy AdS/CFT.

In AdS5/CFT4, the boundary theory is a Yang-Mills gauge theory, but in AdS4/CFT3, the boundary theory is a Chern-Simons theory. These aren't disconnected facts; I read somewhere that these Chern-Simons theories are like a dimensional reduction of a Yang-Mills theory in which the kinetic term disappears.

Now let us think about dS4/CFT3 for a moment. Although no concrete examples of dS/CFT are known, one of the standard ideas for how it works is that a gravitational theory in a dS space is dual to a Euclidean CFT in a space of one less dimension, which corresponds to the past infinity of the dS space. Sometimes there are thought to be two dual CFTs for dS, the other one at future infinity, with the cosmological evolution in between somehow corresponding to an RG flow between the CFT in the past and the CFT in the future.

In any case, what I want to suggest is that the Hilbert space of states which is constructed in loop quantum gravity (loop states, spin-network states being two alternative bases) is the Hilbert space of the Euclidean CFT3 on the boundary of a dS4, and that dS4 time evolution should arise somehow from an RG flow on that Euclidean CFT; and that this is the appropriate way to think about dynamics in LQG.

In other words: in AdS/LQG, the boundary theory is a (2+1)-dimensional Chern-Simons theory, and in dS/LQG, the boundary theory is a "timeless" 3-dimensional gauge theory which is just the Euclidean version of the Chern-Simons theory. In both cases, it should be possible to express the bulk theory as a spin foam encoding RG flow in the boundary theory, though in the AdS case it will be "radial" flow rather than timelike.

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