Simeon proves an inequality that may be interpreted as a special case of the weak gravity conjecture, namely that the lightest state in a quantum theory of gravity can't be heavier than a certain value proportional to the Planck mass.Simeon Hellerman:A universal inequality for CFT and quantum gravity (PDF).

To study 3D gravity in AdS space, he looks at the dual 2D CFTs instead. He assumes no supersymmetry, factorization, strings, or semiclassical gravity. Nevertheless, unitarity and modular invariance (yes, I am a bit uncertain whether he should use the normal "worldsheet" modular invariance for boundary CFTs) is enough for him to prove that there must exist a non-trivial operator whose dimension is smaller than a certain bound. For factorized CFTs, his inequality simplifies.

But I think it's kind of amusing to mention his "real" inequality that he could derive for a general CFT. The dimension of the lightest non-identity operator has to be smaller than

Delta < (cWhenever an AdS/CFT dictionary is possible, it means that the lightest state must be lighter than_{L}+c_{R})/12+0.4736949789...

M < 1/4Gwhere L is the AdS3 radius and Newton's constant has the dimension of length in 3D. If you care what the crazy numerical constant is, it actually equals_{N}+ 0.47369/L

0.47...=[12-Pi+(13 Pi-12)*exp(-2 Pi)] /The 1/4G part of the mass has been found to be special by Ashtekar and Varadarajan in 1994 although they couldn't offer a real "inequality" of Simeon's type. The 0.47 part is new and Simeon seems to be pretty serious about it, indicating that it can be saturated by some very special (or degenerate) CFTs. However, in the conclusions, he raises a suspicion that the bound can be improved which means that the old one couldn't be saturated. ;-) If you look at the formula for the numerical constant and imagine that it should be a dimension, the CFT has to be really strange (at least for a newbie). The strange numerical constant becomes even weirder in the bulk.

/ [6 Pi(1-exp(-2 Pi)]

In the flat space limit, when the 0.47 terms are neglected, Simeon's upper bound is twice the classical mass of the lightest BTZ black hole (the latter is 1/8G).

I deliberately added the word "classical" to make this statement clear from the very beginning - because otherwise you could ask why he couldn't improve his bound to 1/8G. The formula 1/8G for the BTZ mass is classical and is going to be modified by quantum corrections. Simeon has shown that even if you include all such corrections, the lightest object can't be heavier than twice the classical lightest BTZ mass (but the lightest object can be a "different one").

Simeon says that such a proof would be impossible in the bulk (with the methods we know today) except that he also says what happens for the 1/4G mass: the space closes off entirely into a sphere. Couldn't there exist a proof based on this observation? By the way, I am a little bit confused by the question what Simeon does with the resonances i.e. unstable states in the bulk (like evaporating black holes, when matter fields are added) that should correspond to complex-dimension operators in the CFT.

Simeon says that such a proof would be impossible in the bulk (with the methods we know today) except that he also says what happens for the 1/4G mass: the space closes off entirely into a sphere. Couldn't there exist a proof based on this observation? By the way, I am a little bit confused by the question what Simeon does with the resonances i.e. unstable states in the bulk (like evaporating black holes, when matter fields are added) that should correspond to complex-dimension operators in the CFT.

I think that it is likely that similar inequalities exist in other spacetime dimensionalities. For example, I am extremely curious what is the numerical constant determining the mass of the lightest M-theoretical black hole microstate. At any rate, as Simeon nicely says, we began to uncover the "jungle" full of flora and fauna that share the genetic code of quantum gravity and it is already the right time to learn what these animals and plants actually share - i.e. what criteria have to hold universally in all the vacua.

**Fiol's no-go quasi-theorem**

Concerning the "swampland" arguments about theories of quantum gravity, I also liked Tomeu Fiol's paper in September 2008 that argued that (and why) the anomaly-free 10D SUSY theories with gauge groups U(1)^496 and U(1)^248 x E8 might not exist.

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