The String Coffee Table

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 $J$ is maps $T\oplus T^*$ to itself and squares to $-1$ 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 $B$ and can be twisted by closed threeforms $H$, e.g. $dB$ for not closed $B$.

Now consider a submanifold of this space on which there is a 2-form $F$ with $dF=H$ (0 without twist). The trick now is to look at the subbundle of $T\oplus T^*$ on the submanifold such that the vector component $X$ is tangent to the submanifold and the form component is given by $i_X F$. The condition for this to be a generalized complex submanifold is now to require that this bundle is stable under $J$. And, as promised, this generalizes complex, Lagrangian and self-duality for $F$ 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!