AdS/CFT as the swampland/bootstrap duality

Last June, I discussed machine learning approaches to the search for realistic vacua.

Computers may do a lot of work and lots of assumptions that some tasks may be "impossibly hard" may be shown incorrect with some help of computers that think and look for patterns. Today, a new paper was published on that issue, Deep learning in the heterotic orbifold landscape. Mütter, Parr, and Vaudrevange use "autoencoder neural networks" as their brain supplements.



The basic idea of the bootstrap program in physics.

But I want to mention another preprint,

Putting the Boot into the Swampland

The authors, Conlon (Oxford) and Quevedo (Trieste), have arguably belonged to the Stanford camp in the Stanford-vs-Swampland polemics. But they decided to study Cumrun Vafa's conjectures seriously and extended it in an interesting way.



Cumrun's "swampland" reasoning feels like a search for new, simple enough, universal principles of Nature that are obeyed in every theory of quantum gravity – or in every realization of string theory. These two "in" are a priori unequivalent and they represent slightly different papers or parts of papers as we know them today. But Cumrun Vafa and others, including me, believe that ultimately, "consistent theory of quantum gravity" and "string/M-theory" describe the same entity – they're two ways to look at the same beast. Why? Because, most likely, string theory really is the only game in town.



Some of the inequalities and claims that discriminate the consistent quantum gravity vacua against the "swampland" sound almost like the uncertainty principle, like some rather simple inequalities or existence claims. In one of them, Cumrun claims that a tower of states must exist whenever the quantum gravity moduli space has some extreme regions.

Conlon and Quevedo assume that this quantum gravitational theory lives in the anti de Sitter space and study the limit \(R_{AdS}\to\infty\). The hypothesized tower on the bulk side gets translated to a tower of operators in the CFT, by the AdS/CFT correspondence. They argue that some higher-point interactions are fully determined on the AdS side and that the constraints they obey may be translated, via AdS/CFT, to known, older "bootstrap" constraints that have been known in CFT for a much longer time. Well, this is the more "conjectural" part of their paper – but it's the more interesting one and they have some evidence.

If that reasoning is correct, string theory is in some sense getting where it was 50 years ago. String theory partly arose from the "bootstrap program", the idea that mere general consistency conditions are enough to fully identify the S-matrix and similar things. That big assumption was basically ruled out – especially when "constructive quarks and gluons" were accepted as the correct description of the strong nuclear force. String theory has basically violated the "bootstrap wishful thinking" as well because it became analogously "constructive" as QCD and many other quantum field theories.

However, there has always been a difference. String theory generates low-energy effective field theories from different solutions of the same underlying theory. The string vacua may be mostly connected with each other on the moduli space or through some physical processes (topology changing transitions etc.). That's different from quantum field theories which are diverse and truly disconnected from each other. So string theory has always preserved the uniqueness and the potential to be fully derived from some general consistency condition(s). We don't really know what these conditions precisely are yet.

The bootstrap program has been developed decades ago and became somewhat successful for conformal field theories – especially but not only the two-dimensional conformal field theories similar to those that live on the stringy world sheets. Cumrun's swampland conditions seem far more tied to gravity and the dynamical spacetime. But by the AdS/CFT, some of the swampland conditions may be mapped to the older bootstrap constraints. Conlon and Quevedo call the map "bootland", not that it matters. ;-)

The ultimate consistency-based definition of quantum gravity or "all of string/M-theory" could be some clever generalization of the conditions we need in CFTs – and the derived bootstrap conditions they obey. We need some generalization in the CFT approach, I guess. Because CFTs are local, we may always distinguish "several particles" from "one particle". That's related to our ability to "count the number of strings" in perturbative string theory i.e. to distinguish single-string and multi-string states, and to count loops in the loop diagrams (by the topology of the world sheet).

It seems clear to me that this reduction to the one-string "simplified theory" must be abandoned in the gravitational generalization of the CFT calculus. The full universal definition of string theory must work with one-object and multi-object states on the same footing from the very beginning. Even though it looks much more complicated, there could be some analogies of the state-operator correspondence, operator product expansions, and other things in the "master definition of string/M-theory". In the perturbative stringy limits, one should be able to derive the world sheet CFT axioms as a special example.