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Generalized Complex Geometry and Theoretical Physics

+ 9 like - 0 dislike
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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.

Thanks!


This post has been migrated from (A51.SE)

asked Oct 18, 2011 in Theoretical Physics by Tarun Chitra (165 points) [ revision history ]
retagged Mar 25, 2014 by dimension10
Ah, thanks for the clarification!

This post has been migrated from (A51.SE)
@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"

This post has been migrated from (A51.SE)
My comment refers to this paper: http://inspirehep.net/record/652492 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.

This post has been migrated from (A51.SE)
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 http://inspirehep.net/record/901102 for more details

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)

2 Answers

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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.

This post has been migrated from (A51.SE)
answered Oct 19, 2011 by José Figueroa-O'Farrill (2,125 points) [ no revision ]
+ 7 like - 0 dislike

As you note, the algebraic structure on $T X \otimes T^* X$ 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.

This post has been migrated from (A51.SE)
answered Oct 18, 2011 by Urs Schreiber (5,085 points) [ no revision ]

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