Thursday, April 28, 2005

Generalized geometry

The book on the left contains almost everything you need to know about algebraic geometry and Calabi-Yau manifolds in the context of string theory and closely related fields...

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?
And he answered the question by the phrase "generalized geometry" and the associated equations and concepts that I will mention below. Consequently, 20 physicists at Harvard had to spend 2 hours tonight trying to answer the following question:
  • What the heck is generalized geometry?
What's the answer? Well, surprisingly, it seems that it is a crazy mathematical construction that is supposed to incorporate the B-field. :-)

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*
where T* is the cotangent bundle. There is a natural contraction between the vectors and covectors that is preserved by a SO(d,d,R) group. It's a mathematically analogous contraction to the contraction of the momentum and winding, although the latter two quantities are discrete, and similarly, the SO(d,d,R) group is analogous to the discrete T-duality group SO(d,d,Z) that occurs for string theory on tori.

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?
I wonder whether he will replace "T+T*" by "T+(T* wedge T*)". ;-) Update 2008: See Exceptional generalized geometry.


  1. dear lubos,
    as you mentioned, Witten and Pestun found an "anomaly" at one loop level, ie. the resulting partition function depended on the Kahler transformation.
    also, it seems that Naqvi,de Boer and Shomer were able to obtain a twisted sigma model but not the corresponding topological string theory.
    these problems seem to have similar implications as for the existence of a topological M theory.
    what is the status of the understanding of this problem?

  2. I can see that the B-field (or rather theta, the associated Poisson tensor) appear naturally as parts of tensors of the tangent (+) cotangent bundle (see for example the recent papers by Anton Kapustin). However, the impression that Mariana Grana left with me after her talks at Cambridge and at Strings in Paris (see also her paper) was that one needs these generalized geometries to describe what replaces the reduced holonomy manifolds in the case where you have fluxes so they somehow go together with G-structures. Can you comment on this?

  3. Dear Ladies and Gentlemen, thank you for your remarks. I would still prefer if someone else gave more reliable comments than what I could give you...

    All the best

  4. I don't have anything useful to say about the question of whether there is or isn't a string theory underlying the Hitchin action -- other than to remark that one doesn't necessarily expect one.

    Regarding the one-loop anomaly found by Pestun and Witten, it's worth noting that (if I understood them correctly) this issue afflicts all versions of the B model, and is logically independent of the conjecture that the (square of the) B model can be reformulated in terms of Hitchin's holomorphic volume functional. It's certainly something that should be better understood, but there is by now a lot of evidence that the topological string makes sense, so I would be surprised if the conclusion were that the B model is a fiction. What seems slightly more possible is that the B model will turn out to be a fiction on manifolds with trivial canonical bundle which do not admit Kahler metrics. (Nobody ever claimed that the B model _did_ make sense on such manifolds as far as I know, but prior to this result, I think one might have speculated that it did.) As Pestun and Witten also comment in their paper, in some sense this is consonant with the philosophy of topological M-theory, since that theory involves both the Kahler metric and the complex structure, and the time-independent classical solutions are honest CY manifolds. I don't have anything more precise to say at the moment, unfortunately.

  5. maybe I can add something concerning mirror symmetry and generalized complex geometry. some people have thought about this (see e.g., hep-th/0406046, hep-th/0408169) and it seems that mirror symmetry/T-duality can be quite naturally formulated in the generalized setup. About the generalized stuff and bihermitian geometry, there is a theorem by Gualtieri (in his thesis) who proved that a bihermititan structure and a generalized Kaehler structure (what you get if you have two commuting generalized complex structures) are equivalent.

  6. For those with generalized perspectives, that excludes you higher math types. Is there room here?

    I know this has nothing to do with your post Lubos, but I spent time trying to find the source of this mathematical adventure.

    There is a mathematical figure in the article, and when I listen I wonder sometimes how such a complex field(B field?) could have ever been explained.

    You say generalized geometry, yet I see this road leading to a complex issue and had to follow some method? So generalized geometry is something of a strange thing, knowing well that the Riemanian was part and parcel of this progressive view.

    Klein's Ordering of Geometries

    Leading from earlier historical references Gauss was instrumental, as with other predeccessors in this long line of developement?

    But yes, you are all way beyond this. How do you relate to dimensional analysis, as topological forms and relate to such dynamical features?

    Still, Gauss understood this, and the road leading to non-euclidean geometries. So it is stil a viable feature that people will be able to understand what you are doing when we look at the wonderful electromagnetic field or the dynamical nature of the gravitational field?

    Back to regular programming. Thanks

  7. Hi Lumo,
    just a simple question is the b-field here the magnetic field or not?

  8. Hi CGM,

    good question. I wanted to post an explanation because it was clear that someone would ask.

    B-field in electromagnetism is the magnetic field but in string theory we use it with a different meaning.

    B-field is a tensor B_{mn} which is completely antisymmetric - also called a 2-form. It generalizes the electromagnetic potential A_{m}: while the particles charged electromagnetically are pointlike (1 index), the objects charged under the field whose potential is the B-field are one-dimensional (strings, i.e. two indices).

    They're nothing else than the fundamental strings of string theory.


  9. Hi! I am new to this generalized world. I was having a hard time trying to understand some concepts in Gualtieri's tesis (I am afraid I am stuck at the very begining of it!). I have also some general questions about it... I will appeciate a lot any kind of help!
    1) Gualtieri's thesis, page 8: I realy don't know how the volume element of the Clifford algebra is defined here, and how it acts on the exterior algebra of V* to decompose it in S+ and S-. Some hint? (I suppose he is considering the real representations of the real algebra CL(V+V*).. right?)
    2) Gualtieri's thesis, page 9: how is he defining d(\rho)_x(v)? Initially x belonged to Spin(V+V*), and then he says that x belongs to so(V+V*)? How can that be possible?
    What I do know is this. If \rho: Spin(V+V*) --> SO(V+V*), then d\rho: T{Sping(V+V*)} ---> T(SO(V+V*)), where T is for tangent spaces.. right?
    Is it then a fact that so(V+V*) = T(Spin(V+V*)) ?
    3) This is general question: Why are string theory people so interested in generalized geometry?
    4) Is the definition of T-duality given by Gualtiei the same as in string theory?
    Any reference papers will be appreciated.
    Thanks a lot!