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May 26, 2005
Talking to myself
Posted by Robert H.
Since nobody answered my question about how to properly generalize the calibration condition for BPS-branes if the gauge field has curvature I have to do it myself.
Until this very minute, I have been preparing for today’s joint math/physics block seminar in Hamburg where we’re going to find out what Generalized Complex Geomtery is really about (Lubos has chatted about it in his reference frame). Not to arrive completely clueless I have been reading Gualtieri’s thesis which I can strongly recommend to everybody. It is an excelent read even for physicists! And there, in chapter 7 my question is answered:
You probably know that this generalized business works by considering the tangent and co-tangent bundles together. Then a generalized complex structure is maps to itself and squares to and fulfills some integrability condition. It’s easy to see that this condition contains complex, symplectic and Poisson geometry and interpolates between these. Furthermore it is co/invariant under transformations by closed 2-forms and can be twisted by closed threeforms , e.g. for not closed .
Now consider a submanifold of this space on which there is a 2-form with (0 without twist). The trick now is to look at the subbundle of on the submanifold such that the vector component is tangent to the submanifold and the form component is given by . The condition for this to be a generalized complex submanifold is now to require that this bundle is stable under . And, as promised, this generalizes complex, Lagrangian and self-duality for as BPS conditions. And there is also a spinorial description.
I must say, this story is one of those that is so beautiful that it can really foster your belief that there must be some truth to string theory!
May 18, 2005
String(n), Part II
Posted by Urs Schreiber
In the last entry I have listed some facts related to the group . Here is the literature that my discussion was mainly based on as well as a review of what has to do with 2-groups and 2-bundles.
String(n), Part I
Posted by Urs Schreiber
I was asked to say something about the meaning of the group and about manifolds with string structure.
So here I’ll try to give a somewhat more comprehensive discussion than last time that we talked about this.
May 17, 2005
Nonabelian Weak Deligne Hypercohomology
Posted by Urs Schreiber
What I described last time is really best thought of in the context of what I propose to call nonabelian weak Deligne hypercohomolgy.
Unless I am hallucinating the following is the correct formalism to generalize the well-known Deligne hypercohomology formulation of strict abelian -gerbes to weak and nonabelian -gerbes.
May 16, 2005
PSM and Algebroids, Part V
Posted by Urs Schreiber
[Update: The following has become section 13 of hep-th/0509163.]
Last time I discussed how the functorial definition of a -bundle with -connection can locally be differentiated to yield morphisms between -algebroids. Now I think I have figured out the differential version of the transition law describing the transformation of these algebroid morphisms from one patch to the other. The result is a formalism that allows you to derive the infinitesimal cocylce relations of a nonabelian -gerbe with curving and connection, etc. using just a couple of elementary steps.
May 13, 2005
PSM and Algebroids, Part IV
Posted by Urs Schreiber
Last time I discussed how Lie -algebroids (and hence Lie -algebras) and dg-algebras on graded vector spaces of maximal grade are two aspects of the same thing. This goes a long way towards merging the study of -bundles with -connections with the study of algebroid morphisms as they arise in the Poisson -model, Dirac -models and other field theories.
Here are more details.
May 4, 2005
PSM and Algebroids, Part III
Posted by Urs Schreiber
I have just returned from visiting Thomas Strobl at Jena University, where we talked about algebroids, gerbes, categorified gauge theory, and generalized geometry and how it all fits together. I have learned a lot in these discussions and have gotten a little closer to seeing the big picture, also thanks to the valuable pointers to the literature by Melchior Grützmann and Branislav Jurčo. Here I’ll list some useful and interesting facts – except for those that are top-secret…
(Please note that all my attributions in the following reflect only my level of awareness of the literature. I’d be grateful for corrections and further pointers to the literature.)
[Note: Users of non-Mac machines might have to download a new font in order to properly view all mathematical symbols in the following. More general information can be found here.]