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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.


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It used to be an empirical fact (which I'm not sure still holds) that all known supersymmetric solutions with geometry a warped product of four-dimensional Minkowski spacetime and a 6-dimensional compact manifold $K$ and otherwise arbitrary fluxes, required $K$ to be a generalised Calabi-Yau manifold. –  José Figueroa-O'Farrill Oct 18 '11 at 18:25
Ah, I think its actually more general. If one doesn't have fluxes, it is generalized Calabi-Yau (e.g. for pure spinors $\Omega_{\pm}$, $d\Omega_+ = d\Omega_- = 0$). Otherwise, one has a constraint like $d\Omega_- = F_{RR}$, where $F_{RR}$ is the Ramond-Ramond flux. See for more details –  Tarun Chitra Oct 18 '11 at 23:26
My comment refers to this paper: and papers around the same time (when I last followed this approach to the subject). The backgrounds have fluxes, and the 6-geometry is still GCY. –  José Figueroa-O'Farrill Oct 19 '11 at 0:08
@Tarun: you can have $d\Omega_-=F_{RR}$ and still be GCY, its just that you're on a twisted Courant Algebroid instead of $T\oplus T^*$ as Urs also mentioned by "non-standard" –  Reimundo Heluani Oct 19 '11 at 0:46
Ah, thanks for the clarification! –  Tarun Chitra Oct 19 '11 at 2:06

2 Answers 2

up vote 11 down vote accepted

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.

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Is it \otimes or \oplus? –  NikolajK Sep 28 '12 at 9:42

One of my favourite (still) applications of generalised geometry is the derivation of the Buscher rules for T-duality, which you can read about in the relatively recent paper arXiv:1106.1747 [math.DG] by Gualtieri and Cavalcanti. I first heard about it from a colloquium by Cavalcanti here in Edinburgh a few years ago and found it to be the most transparent derivation of the Buscher rules I had ever seen.

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