Tuesday, August 09, 2005

Uniqueness of quantum gravity

One of Steve Shenker's recent deep ideas - and inexhaustible sources of frustration - is the following observation:
  • Holography seems to associate a (d+1)-dimensional theory of quantum gravity with any d-dimensional quantum field theory (or quantum mechanics). This implies that theories of quantum gravity do not seem unique anymore. Start with any QFT, and you obtain a theory of quantum gravity. Consequently, Maldacena and holography may be able to bring us as much desperation as they have offered excitement.

In other words, we must ask: Is every quantum field theory a theory of quantum gravity?

Well, I would emphasize that the dual gravitational theories are typically strongly curved (and they have large gradients of fields like the dilaton) and they cannot be described as a small perturbation of a well-defined smooth geometry. This fact itself makes it inappropriate to call these backgrounds "quantum gravity" because the QFT description is always more controllable than the geometric one - and only the field theoretical description is potentially weakly coupled.

Even our intuition should be enough for us not to call the Hydrogen atom or loop quantum gravity a "theory of quantum gravity" because they don't describe a nearly flat space on which wiggles can nevertheless propagate. A spin foam, for example, is rather a piece of dirty and unpredictable material that threatens the crews of space shuttles. Fortunately, a piece of spin foam has been removed from the Discovery space shuttle and its crew safely landed in California today. (Well, I guess that this piece would fly away even if the astronaut did nothing, but it is sometimes better to be safer than sorry.)

The exceptions are the "large N" theories whose dual geometry becomes weakly curved. In these cases, one of the well-known examples of local gravitational physics emerges (eleven-dimensional supergravity, type IIA, IIB, or SO(32) or E8 x E8 type I ten-dimensional supergravity). There may be many quantum field theories whose large N limits give the same theory of gravity in the flat space; that's one of the examples of universality of string theory or examples of the famous rule

  • All roads lead to string theory.

For example, both conformal field theories dual to "AdS4 x S7" and "AdS7 x S4" locally lead to 11-dimensional physics of M-theory in the large N limit. At any rate, the rules of uniqueness have to be clarified a bit; feel free to disagree and identify errors:

  • The only consistent theory (description of local dynamics) of quantum gravity in 11 dimensions with curvature much smaller than the 11D Planck scale is M-theory. Below 11 large dimensions, the number of possible theories is larger but all of them may be described as generalized compactifications of the same string/M-theory. The information which compactifications are possible is encoded in a deep enough analysis of string/M-theory backgrounds in the highest dimensions.

Note that this new formulation accepts the observation that our theories have to be more unique if the number of large dimensions is large, and the degeneracy arises as the number of large dimensions decreases; all of these possibilities must be interpreted as compactifications of the same theory; of course, the word "compactification" is not the naive field-theoretical one as it may contain branes and other stringy phenomena.

In particular, 11 is the upper limit for the number of spacetime dimensions that may be more or less flat and the complexity of the lower-dimensional vacua is bounded from above because there is a limited number of compactifications.

The colleagues who believe that string theory predicts equally good vacua above 11 dimensions (e.g. in 18 dimensions) that can also be arranged to be essentially flat will have to modify these rules. I have no idea whether these people believe in uniqueness of quantum gravity in any sense, and if they do, what is exactly their sense. If someone believes that there is a well-defined supercritical M-theory in 2003 dimensions (I chose the dimensionality to be 3 modulo 8, to agree with M-theory), it is likely that the possible number of her compactifications down to 4 dimensions will form a pretty large landscape. Because the set of ridiculous numbers like 2003 is infinite, their landscape is more or less guaranteed to be infinite.

I choose to believe that these supercritical theories have to be inconsistent (or at least irrelevant for anything we would like to call physics) because their existence has not been established with the standards usual for the supersymmetric backgrounds, and because the assumption of their existence would render string theory meaningless in my understand of the word "meaning".

Even though the number of compactifications down to lower dimensions may be large, it is limited and all of them may furthermore share some general properties. For example, string theory will probably never generate a pure N=2 supergravity in 6 dimensions although this theory seems to be perfectly valid as an effective field theory of "something".

Those who share my belief that the only consistent quantum gravitational background with 11 flat or nearly flat dimensions is M-theory may ask many other questions, for example:

  • Can we define rules - rules independent of string/M-theory - what such a theory should look like whose unique solution is M-theory in 11 dimensions? In other words, is string/M-theory a unique solution to some general bootstrap conditions of self-consistency? What are exactly those rules, and how do we prove (at least partially) that the solution is unique?


  1. "For example, string theory will probably never generate a pure N=2 supergravity in 6 dimensions although this theory seems to be perfectly valid as an effective field theory of "something"."

    Hi Lubos,

    that's a very important and interesting point. I believe I've heard it before from Cumrun Vafa, though I remember he referred to a pure SUGRA in 4d that presumably cannot arise from string theory. Could you please remind me what the argument for this is?


  2. Hi Larry, yes! That's an observation by Cumrun. The main argument is that among the zillions of compactifications we know, neither of them is pure N=2 supergravity at low energies. Best, Lubos