Andy Neitzke was leading the postdoc journal club, and it was exciting.
Hitchin, a famous mathematician, decided to understand the following question:
- What the heck is the B-field?
- What the heck is generalized geometry?
OK, let's start more seriously. When you talk about complex manifolds or something like that, it is useful to imagine that you have the tangent field T at every point of the manifold. And there is a group like SO(d) in the real case, or more precisely GL(d) because you're not forced to preserve any metric, acting at each point.
Hitchin makes it more complicated and tells you that you should replace
- T ... by ... T (+) T*
Also, we had a small argument whether the GL(n,Z) group of large diffeomorphisms of an n-torus is a "global" subgroup of the internal group GL(n,R) that acts at each point; I was saying that GL(n,Z) may be isomorphic to a subgroup, but this particular GL(n,Z) group is not a subgroup of the particular GL(n,R) because the latter is an internal group while the former physically acts on space as a diffeomorphism.
After this basic definition of the T+T* bundle, one applies a rather standard one-to-one dictionary between the spinors of SO(d,d) and the differential forms. And one defines some analogues of the holomorphic (n,0) form - but in this new tangent space that is doubled.
One can define generalized complex manifolds - a notion that surprisingly includes both the ordinary complex manifolds as well as the ordinary symplectic manifolds. Note that these two properties (symplectic and complex) are independent. Both of them may hold simultaneously, and if the complex structure and the symplectic structure are moreover compatible (i.e. the symplectic structure is a (1,1)-form according to the complex structure), then one obtains the Kahler manifolds - a small subgroup of which (with the vanishing first Chern class) are the Calabi-Yaus.
In terms of the generalized complex manifolds, one can rephrase the condition by having two independent mutually compatible generalized complex structures. Note that one of them remembers the complex structure and the other remembers the symplectic structure (which carries, assuming a complex structure, the same information as the Kahler form).
A similar construction allows to define generalized G2 manifolds that have an SO(7,7) group at each point that may be broken, by a differential form, to a G2 x G2. We were confused why the "generalized" character is not lost if the factorization into two G2's is imposed anyway.
There has been a lot of debates to what extent the generalized objects generalize the previous objects. A generalized Calabi-Yau manifold is too general a concept - string theory can't be compactified on it in general. However, there is a special (but still generalized) case of it - the manifold with a generalized SU(3) structure which essentially has two independent compatible generalized Calabi-Yau structures on it that satisfy some extra conditions.
One can write down generalized equations for the covariantly constant spinors i.e. for the appropriate number of preserved supercharges, and the tools of generalized geometry "package" the metric and the B-field (both of which, namely Christoffel's symbol and the torsion H appear in the covariant derivatives) into one object - well, it's nothing else than g+B, the "generalized (asymmetric) metric tensor".
However, it seems that for the "proper" Calabi-Yau spaces one does not find any new solutions. Nevertheless, there has been a debate how this stuff is related to Witten and Pestun's (WP) refinement of topological M-theory. Recall that WP argued that the precise conjecture about topological M-theory fails at one-loop level, and one can fix it if the Hitchin model is replaced by the so-called extended Hitchin model.
The extended Hitchin model turns out to have no new classical solutions for the proper Calabi-Yau spaces, but there are new massive deformations of it that modify the one-loop functional determinants and so forth. Again, we were confused by the statements that "it is suddently not known whether topological string theory describes Calabi-Yau solutions or generalized Calabi-Yau solutions; Andy argued that some of the confusion follows from a famous paper that constructed an almost correct string field theory for topological string theory where, however, some massive deformations were omitted, and most people believed this otherwise important paper including the minor flaws.
Finally, we discussed various other confusing points about topological string theory - for example the one-loop anomaly that only occurs if we put it on a wrong background, but was nevertheless unknown to everyone on the journal club before the WP paper.
Also, Shiraz Minwalla asked whether a new mathematical insight, one that does not used the word "generalized", has been found using these new tools, and Andy Neitzke answered that a student of Hitchin has solved a problem in "bihermitean geometry". This bihermitean geometry is an older concept than generalized geometry, but we could not resist the temptation to think that they're equivalent or at least closely related.
The language of generalized geometry should be good for T-duality, but we had a feeling that the integrity and topology of the base space of the bundle is preserved anyway, whatever one does with the bundles, so it does not treat the T-dual backgrounds on equal footing. Also, it seems that there has been no useful interaction between the generalized geometry and mirror symmetry - something that would otherwise be inevitable if T-duality were kind of more natural in this language.
Not too surprisingly, the next task for Hitchin is to understand
- What is the C_{MNP} field in M-theory?
