Vafa: supergroups, non-unitary cousins of CFT, and black hole puzzles

Cumrun Vafa has a very interesting new paper,

Non-Unitary Holography

You may start with a simple question. What happens if you replace gauge groups derived from \(U(N)\) by their supergroup counterparts, \(U(N+k|k)\)?

Well, the supergroup has more degrees of freedom – many of those have a negative norm (anticommuting but spin-one components of the "gauge bosons") and may produce negative probabilities. But Cumrun says that it doesn't affect anything you encounter at any order of the perturbative expansion in \(1/N\) i.e. in the string loop expansion. The effect of the extra \(k\) bosonic dimensions added to the original \(N\) is cancelled against the new \(k\) fermionic ones.




If true, this statement itself is surprising, at least to me (even though I know that such an addition of fermionic+bosonic dimension pairs is inconsequential in some topological theories).

So Cumrun's thesis is that the perturbative parts of theories such as the most intensely studied example of the AdS/CFT correspondence, one involving the \(\NNN=4\) gauge theory in four dimensions, heavily underdetermine the theory. At the non-perturbative level, there is a significant ambiguity about what the theory does, at least if you allow the Hilbert space to become indefinite at some point.




This may be bad news or good news. Cumrun also says that in these "evil cousins" with supergroups as gauge groups, there may exist worlds in which Stephen Hawking was (originally) right about the information loss. The evolution may be non-unitary.

These issues should be settled soon but I am confused about some statements made by Cumrun. First, there are different aspects of "unitarity". The word often means that we demand a positively definite Hilbert space – which is not the case here. But it also means that a bilinear (more accurately: sesquilinear) form is being preserved. These are independent conditions. Spaces with pseudounitary symmetries violate the former condition but obey the latter.

Another question is whether these non-unitary theories are sick or "in the swampland". I would tend to say "Yes" – it is OK for you to dismiss them entirely. But there may be ways to get meaningful theories out of them – theories with non-negative probabilities. If all the predictions are the same as in the "totally healthy" theories to all orders of perturbation theory, it's already pretty dramatic.

What I also have in mind is the possibility that some theories with ghosts may be made either exactly consistent or consistent up to a very satisfactory accuracy which could justify their usage in physics.

And of course, Cumrun spends significant time with the black hole puzzles. The matrix models are, to a large extent, physically equivalent to black hole physics (at the quantum level) so one should be able to learn a lot of things about the quantum dynamics of black holes from these considerations, too.