I have been wondering about some of the different uses of Generalized Complex Geometry (GCG) in Physics. Without going into mathematical detail (see Gualtieri's thesis for reference), a Generalized Complex Geometry attempts to unify symplectic and complex geometry by considering the bundle $TM\oplus T^* M$ with its natural metric $\langle X+\xi, Y+\eta\rangle = \frac{1}{2} \left( \eta(X) + \xi(Y)\right)$ and the Courant Bracket.
The first hints of the necessity of GCGs in Physics came up in a famous paper by Gates, Hull and Roc̆ek, in which they found an 'extra' supersymmetry in the $(2,2)$ supersymmetric model. This extra symmetry turns out to be related to specifying two (integrable) complex structures $J_1, J_2$ which in turn are covariantly constant under torsionful connections. This means that the manifold need not be Kähler (which is Hermitian and Torsion-free) and led Nigel Hitchin (and his students) to propose more general geometries that could be useful in physics.
More recently, a connection between GCGs and AdS/CFT has been discovered. Recall that in AdS/CFT, we consider a spacetime that is a warped product of $AdS_4$ and a 6-manifold. It turns out that it is natural to consider a 5-manifold $Y^5$ whose cone has some special geometry. If this geometry is Calabi-Yau then such a manifold is known as a Sasaki-Einstein manifold. As such, we start out with a metric of the form,
$ g_{ij} = g_{AdS_5} + g_{Y^5} = e^{2\Delta + \phi/2}r^2 \left(g_{\mathbb{R}^{1,3}} + r^{-4} g_{C(Y^5)} \right) $
where $g_{C(Y^5)} = dr^2 + r^2 g_{Y^5}$ (the metric cone of $Y^5$). If we want to obey $\mathcal{N}=1$ supersymmetry, we must enforce on the dilatino and gravitino which eventually leads to a condition on pure spinors. In Generalized Complex Geometry, $TM\oplus T^*M$ naturally acts as a Clifford Algebra on the Clifford Module $\wedge^{\bullet} T^*M$. It turns out that in this situation, we can represent the pure spinors over a Generalized Complex Manifold as the sum of differential forms of different degree (polyforms). As such GCGs can be good candidates for $C(Y^5)$.
Related to this is the result of Graña, et. al which can be poorly paraphrased as:
All $\mathcal{N}=1$ solutions of IIB string theory are described by a
pair of pure spinors $\Omega_{\pm}$(up to $B$ transform) that satisfy
a pair of differential constaints, $d \Omega_+ = 0$, $d\Omega_- = dA \wedge \Omega_+ + \frac{i}{8}e^{3A}e^{-B}\star (F_1 – F_3 + F_5)$, where $F_k$ is the $k$-form flux and $A = 2\Delta + \phi/2$
I was wondering if there were any other significant uses of GCGs in physics that I have not mentioned. I have seen a variety of papers that do mention GCGs, but outside of these examples, I have not been particularly compelled by their usage.
Thanks!
Best Answer
As you note, the algebraic structure on $T X \oplus T^* X$ studied in generalized complex geometry is that of the standard Courant Lie 2-algebroid. Courant Lie 2-algebroids (standard or non-standard) play a role in various guises in 2-dimensional QFT, thanks to the fact that they are in a precise sense the next higher analog of symplectic manifolds (see symplectic Lie n-algebroid) and thus the direct generalization of Hamiltonian mechanics from point particles to strings. This higher symplectic geometry aspect of Courant Lie 2-algebroids -- gence of GCG -- is recently receiving more attention.
Related to this is the Courant sigma-model, which is a 3d TFT generalizing Chern-Simons theory, being the direct higher dimensional analog of the Poisson sigma-model. It has a Courant Lie 2-algebroid as its target space. Hence in particular every generalized complex geometry forms the target space of such a 3d sigma-model.