Nongeometric flux vacua

Brian Wecht has explained us - a J453 standing room only crowd at Harvard - their (Jessie Shelton, Wati Taylor, and Brian Wecht) work on

• Non-geometric flux compactifications

Their approach is meant to generalize the flux compactifications into a broader class that is closed under the action of T-duality. Brian first invited everyone to MIT whose physics department has a new temporary building, water cooler, and Dan Freedman. His physics talk started with a six-torus in type II that can be written as a product of three two-tori. You orbifold this six-torus by a "Z3 x Z2 x Z2" group where the "Z3" permutes the three two-tori cyclically and Z2's change the signs of the coordinates on two of the three two-tori. Brian Wecht called it "Z3 x Z2" only but he asked us to use our notation for his careful, but incorrect description of the orbifold group. :-)

So my notation for his "Z3 x Z2" is "Z3 x Z2 x Z2" even though the group he generated is really a non-Abelian, semi-direct product but at least it has 12 elements. ;-) Also, he did not care about the orbifold singularities that probably may be blown up, but he wants to ignore all the new degrees of freedom that live there.

Start with a three-torus with "N" units of H3-flux. Once you choose a B-field gauge (gauge-dependence is another source of controversy) and write the "B_{xy}" field as "N.z", you may T-dualize this to get a three-torus whose two-torus gets tilted ("tau" goes to "tau+N") if you rotate around the third circle: it's a twisted torus. The amount of twisting is considered by the authors to be a new kind of flux - it's a terminology that may be confusing especially because many of these "fluxes" are some topological, but purely geometrical invariants of their compactification manifold that they can't really define in the general case. Then they T-dualize - or mirror symmetrize - these configurations in many other ways and obtain new kind of non-geometric fluxes; the last ones are supposed to be the most surprising ones - the extremely non-geometric ones. The previous ones measure which T-duality appears as a monodromy around one of the circles, but the last ones are probably even more unusual.




Brian eventually listed a long table of different kinds of objects - integer-valued "generalized fluxes" - and their pairing in type IIA and type IIB theory. He also wrote various NS-NS and R-R tadpole cancellation conditions and claimed that all these objects (counterparts of the Bianchi identities for their class of backgrounds) and equations were now closed under T-duality. I personally remain unconvinced that these objects are closed; the map how they relate them is too Abelian and one-to-one while T-duality tends to be much more non-Abelian.

Another uncertainty is whether there actually exists a stringy background - solving the generalized Einstein's equations with the appropriate sources - for each choice of the integers that satisfy the tadpole cancellation conditions. The answer - a counterpart of Yau's theorem - is not known but in my opinion, it is pretty important. For example, I am sceptical even about the existence of the three-torus background with a nonzero H3-field (this doubt would disappear for the three-sphere where the CFT description involves the WZW models), and the more general backgrounds with arbitrary NS-NS and R-R fluxes and "twists" (that they also call "fluxes") and with monodromies around circles taking values in the T-duality groups are almost definitely more uncertain than a simple three-torus with the H3-flux.

Also, it is not known how to run the same procedure for a more general initial topology and how to complete the rules so that they're also invariant under the S-duality.

He was also showing how to find some superpotentials that were manifestly invariant under T-dualities. It created some new debates with the infidels - especially the question why there are no modular functions appearing in these formulae that would make the SL(2,Z) symmetries manifest. After the period for questions, Brian Wecht asked everyone to thank him again, and we responded with another applause. :-)

One of their general ideas that they want to promote is that non-geometric vacua are "generic". Are the "typical" backgrounds - whatever it exactly means - in the "landscape" interpretable geometrically, or are they non-geometric? Do we live in a Universe where the "hidden dimensions" have a more or less well-defined geometric shape? Of course, they were relatively far from getting realistic vacua, and the conjectured appearance of some non-Abelian gauge symmetries in their pictures also remained slightly controversial, for example because they could not describe the origin of the new light states. We had a similar feeling with Cumrun that the appearance of non-Abelian gauge groups in string theory remains a special feature of several known mechanisms, and the statement that they have a new one is a pretty strong claim.

Note that many apparently non-geometric vacua may be presented as compactifications on manifolds whose size is stringy. In some sense, it is the center of the configuration space and you may think that the things in the middle are more unique than those in the asymptotic regions. But this simple expectation may be wrong, of course, and one needs to do a proper counting.

A comment unrelated to Brian's talk:

One thing that still attracts me are the potential non-geometric vacua of M-theory. M-theory is the most geometric regime of string/M-theory that we have: the more dimensions you have, the more geometric interpretation the phenomena admit, of course. But are there some M-theoretical counterparts of the minimal models in CFT? I decided that there could be a matrix cousin of the Virasoro algebra that I call the Matrixoro algebra. The basic idea is to replace the scaling maps from CFT by the maps that scale the rank "N" in a BFSS-like Matrix theory. This scaling may be one generator in a much larger group.

A basis of its generators could perhaps be in one-to-one correspondence with the Young diagrams or the irreducible representations of U(N). Matrixoro minimal models would be made of Matrixoro primary fields that are annihilated by certain polynomials of the Matrixoro generators. If you're optimistic, there could be matrix-valued replacements for OPEs, state-operator correspondence, a central extension of the Matrixoro algebra, Matrixoro Gepner models, and other things, too. Anyone can rule it out? ;-)