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Stable AdS flux vacua must be supersymmetric: a conjecture

Famous physicists Hiroši Ooguri and Cumrun Vafa proposed a new branch of the Swampland program in their new paper

Non-supersymmetric AdS and the Swampland
In 2005, Cumrun Vafa coined the term swampland to describe would-be theories or their low-energy effective field theory limits that look consistent according to the rules of effective quantum field theory but that are banned according to the more stringent rules of string theory or quantum gravity (which are ultimately equivalent concepts) i.e. that have no realization within string/M-theory.

The swampland (TRF) is the "larger" but messier realm surrounding the stringy landscape. The swampland shouldn't be confused with the related but inequivalent technical notion of the part of the Internet and media that is critical towards string theory. It's not called a "swampland" but rather a "cesspool" and the technical term for the individuals in the cesspool is "scumbags". The largest and stinkiest two scumbags are known as "Šmoits" but I don't want to overwhelm this blog post with the review of the standard terminology.

The extra constraints imposed by string/M-theory may be interpreted as "general predictions of string/M-theory". They're usually qualitative. Our weak gravity conjecture is the most intensely studied example of such extra constraints. It says that there have to exist particles light enough so that the repulsive electric force between them trumps the gravitational attraction. In this sense, gravity is the weakest force and it has to be.

This statement may be justified by various arguments using many viewpoints and it generally seems nontrivial yet at least much more correct than you would expect if it were a random guess. However, we couldn't settle – and the researchers still haven't agreed – on many of the details. Do the light charged particle have to exist for every type of an electric force? Every direction in the charge space? Every site in the lattice of charges allowed by the Dirac quantization rules and so on? Should the terms in the inequality be modified in some way?

And isn't there a deeper insight or structure from which the principle may be derived – much like Heisenberg's uncertainty relation (inequality) may be derived from the nonzero commutators?

In the new paper, Hiroši and Cumrun assume a somewhat stronger version of the weak gravity conjecture. It surely has to apply to every kind of an electric-like force, including forces between branes. If I understand well, they basically say that the weak gravity conjecture should imply the existence of branes of low enough tension, not just point-like charged particles, for every kind of a \(p\)-form field i.e. a generalization of electromagnetism with many indices.

Now, the existence of these charged branes has consequences for the flux vacua.

The flux vacua carry some nonzero values of \(\int_C F_p\), the integral of a field strength \(p\)-form over some \(p\)-cycle in the compactification manifold. These fluxes could be used as just some sourceless, electromagnetic-style fields. However, the weak gravity conjecture says that the charged sources actually have to exist and their tension has to be low enough. When it's true, it's being shown that such branes may nucleate inside the anti de Sitter space, get to the boundary in a finite time, and reduce the flux by a unit.

The whole vacuum spontaneously changes in this way. So the original vacuum we started with was unstable. The authors conclude that every anti de Sitter vacuum supported by fluxes has to be either unstable or supersymmetric. Supersymmetric vacua get an "exception" because the total attractive gravitational force exactly cancels against the repulsive generalized electromagnetic force – because of the well-known BPS relations. The BPS condition is nothing else than the example in which the defining inequality of the weak gravity conjecture is saturated – gravity is as strong as the non-gravitational force. Consequently, the spontaneous nucleation of the brane cannot occur with a finite probability and/or the brane isn't driven to the AdS boundary at a finite time.

It's also my understanding that all stable AdS flux vacua that have been found are supersymmetric. Well, my preferred example of a non-supersymmetric AdS vacuum with a CFT dual would be Witten's pure AdS3 gravity whose CFT dual carries the monster group symmetry (it only exists for the minimum radius so the curvature is "Planckian" and you might say that it's not a "full-fledged" example of a gravitational theory with a nearly flat space). Either the low spacetime dimension or the absence of the appropriate "flux" is probably what allows an exception for this case. I am not sure about the precise logic here.

You could say that even if it is true, the derived restriction doesn't have "practical" consequences because vacua may be unstable but very long-lived. They argue that while the lifetime could be long for a CFT, the near-horizon dual geometry in the bulk sees the instability as a much stronger one. So the restriction is harsh and real.

If these statements are right – and ideally derivable in a more rigorous way – then you might interpret the result as a prediction of string theory. We have observed the world to be rather stable and non-supersymmetric – so it cannot be AdS. We may use the stability to predict that the cosmological constant is non-negative. It's about one predicted bit of information but it's a prediction nevertheless. Needless to say, the implications for our understanding of the inner structure of string/M-theory and its configuration space could be more far-reaching.

The conditions look conceptually different. Whether the gravitational force is stonger than another one seems to be just a "boring technicality". On the other hand, this technicality may imply, like in this case, that a whole class of candidate vacua – non-supersymmetric stable AdS flux vacua – is actually non-existent. So even previously overlooked technicalities such as the weak gravity conjecture may have the potential to solve the vacuum selection problem and other seemingly insurmountable hurdles.

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