What role do generalized geometries (in terms of Dirac structures, for instance, symplectic, Poisson, complex, and generalized complex structures in the sense of Hitchin, Cavalcanti, and Gualtieri) play in string theory?

EDIT: More generally, what role to Dirac structures (subbundles of the generalized tangent bundle $TM \bigoplus T^*M$ which are maximally isotropic to the natural pairing and closed under the Courant bracket) play?

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    $\begingroup$ As worded, this question is too vague. Please define what you mean by "generalized geometries" as there are various interpretations of this phrase. $\endgroup$ Oct 22, 2010 at 21:54
  • $\begingroup$ I hope you don't mind, but, in light of David MJC's answer, I figured out what you meant and have updated your question to be more clear. $\endgroup$ Oct 22, 2010 at 22:41
  • $\begingroup$ yes, the updated version is clearer. thanks. $\endgroup$ Oct 22, 2010 at 22:46
  • $\begingroup$ The updated version changes the scope of the question. Generalized geometries play a much broader role in string theory that generalized complex structures do. If the question concerns the specific role played by generalized complex structures, then my answer to date (already brief 23/10) is inadequate. $\endgroup$
    – David MJC
    Oct 22, 2010 at 22:50
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    $\begingroup$ Unfortunately, I cannot edit your post to clarify that you are referring to geometric structures on $T\oplus T^*$, not just orthogonal complex structures. But I am willing to elaborate on my answer in due course anyway (despite being very busy IRL). I hope other MO contributors will provide answers too. $\endgroup$
    – David MJC
    Oct 22, 2010 at 23:45

4 Answers 4


Let me add something to what David and Urs have written already, since the way those two answers are shaping up, perhaps what I'm about to say does not get mentioned.

One of the most interesting applications of generalised geometry in string theory is in the study of supersymmetric flux compactifications. Ten-dimensional superstring theories have a well-defined limit (=the effective theory of massless states) which corresponds to ten-dimensional supergravity theories. One way to view these theories is as variational problems for certain geometric PDEs which generalise the Einstein-Maxwell equations. The dynamical variables consist of a lorentzian ten-dimensional metric and some extra fields, depending on the theory in question. One set of of fields common to the ten- and eleven-dimensional supergravity theories are $p$-forms obeying possibly nonlinear versions of Maxwell equations. In the Physics literature these $p$-form fields are called fluxes and the geometric data consisting of the lorentzian manifold, the fluxes and any other fields all subject to the field equations are known as supergravity backgrounds.

These supergravity backgrounds are actually not just lorentzian manifolds, but in fact they are spin and part of the baggage of the supergravity theory is a connection (depending on the fluxes,...) on the spinor bundle, which defines a notion of parallel transport. Parallel spinor fields are known as (supergravity) Killing spinors and backgrounds admitting Killing spinors are called supersymmetric.

One way to make contact with the 4-dimensional physics of everyday experience is to demand that the ten-dimensional geometry be of the form $M \times K$, where $M$ is a four-dimensional lorentzian spacetime (usually a lorentzian spaceform: Minkowski, de Sitter or anti de Sitter spacetimes) and $K$ a compact six-dimesional riemannian manifold, known as the compactification manifold.

When all fields, except the metric, are set to zero, the connection agrees with the spin lift of the Levi-Civita connection and supersymmetric backgrounds of this type are lorentzian Ricci-flat manifolds admitting parallel spinor fields. If we demand that they be metrically a product $M \times K$ as described above, then a typical solution is $M$ being Minkowski spacetime and $K$ a six-dimensional manifold admitting parallel spinors; that is, a Calabi-Yau manifold, by which I mean simply a manifold with holonomy contained in $SU(3)$. This result, which today seems quite unassuming, was revolutionary when it was first discovered in the 1985 paper of Candelas, Horowitz, Strominger and Witten. That paper is responsible for the interest of physicists in Calabi-Yau manifolds and ensuing rapprochement between physicists and algebraic geometers, the fruits of which we're still reaping today.

But Calabi-Yau compactifications are in fact very special from the physics point of view: since most of the fields in the theory (especially the fluxes) have been turned off. Generalised geometry enters in the search for more realistic "flux compactifications". One of the fields which all ten-dimensional supergravity theories have in common is the $B$-field (also called Kalb-Ramond field). One of Hitchin's motivations for the introduction of generalised geometry was to give a natural geometric meaning to the $B$-field. For example, the automorphism group of the Courant algebroid $T \oplus T^*$ is the semidirect product of the group of diffeomorphisms and $B$-field transformations.

More generally, I think that it is still true that all known supersymmetric flux compactifications $M \times K$ (even allowing for warped metrics) of ten-dimensional supergravity theories are such that $K$ is a generalised Calabi-Yau manifold. The fluxes turn out to be related to the pure spinors in the definition of a GCY structure. There are many papers on this subject and perhaps a good starting point is this review by Mariana Graña.

There are other uses of generalised geometry in string theory, e.g., the so-called doubled field theory formalism, as in this recent paper of Chris Hull and Barton Zwiebach.

  • $\begingroup$ There is an obvious graded version of generalized geometry. Is the definition written down formally anywhere? $\endgroup$ Jul 9, 2014 at 21:12
  • $\begingroup$ Double field theory goes beyond generalized geometry as Andreas Deser and I tried to explain in our recent posting arxiv.org/abs/1406.3601 $\endgroup$ Jul 9, 2014 at 21:17
  • $\begingroup$ @JimStasheff, I agree, but I have not seen it written down anywhere. $\endgroup$ Jul 10, 2014 at 10:34

Generalized geometry (in Hitchin's sense, following Courant and Dorfman) is adapted to the physical motion of string-like particles in the same way that traditional geometry is adapted to the physical motion of point-like particles. More general generalized geometries are useful in connection with higher dimensional objects such as membranes (and hence also M-theory). Pavol Severa's first letter to Alan Weinstein is a nice early reference point for the basic idea.

Update 1. I won't be able to add significantly to this post until Tuesday perhaps, but I want to indicate some of the relationships between my answer (as it is and to come) and Urs'. Ignoring higher dimensional objects than strings for now, generalized geometry initially concerns geometry on the generalized tangent bundle $T\oplus T^*$ (where $T=TM$ is the tangent bundle of a manifold $M$). The bundle $T\oplus T^*$ has a natural symmetric form with respect to which both $T$ and $T^*$ are maximal isotropic.

However, generalized geometry takes the point of view that $T\oplus T^*$ is an extension of $T$ by $T^*$, and is thus an example of a Courant algebroid $CA$, in that there is a short exact sequence $0\to T^*\to CA\to T\to 0$, where $CA$ has a symmetric form and other structure (the Courant bracket) making it isomorphic to $T\oplus T^*$ for suitable isotropic splittings of the exact sequence. A Dirac structure is such an isotropic splitting.

From the naive $T\oplus T^*$ viewpoint, a Dirac structure is given by an orthogonal involution of $T\oplus T^*$ whose eigenspaces do not meet $T^*$. Generalized complex geometry is a subfield of generalized geometry, in which one studies orthogonal complex structures on $T\oplus T^*$ instead of involutions. However, there are interesting structures on $T\oplus T^*$ which involve neither Dirac structures nor generalized complex structures. This should not be surprising: there is more to ordinary geometry than involutions and complex structures.

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    $\begingroup$ I am willing to elaborate on this answer considerably! $\endgroup$
    – David MJC
    Oct 22, 2010 at 22:15
  • $\begingroup$ As a rough guideline, a post here of about 10 times the length of your current post would be read; 40 times the length might not be. Links to explanations are also welcome. Gerhard "Ask Me About System Design" Paseman, 2010.10.22 $\endgroup$ Oct 23, 2010 at 5:12
  • $\begingroup$ Thanks José, and thanks for your answer above (now!), which complements the math viewpoint nicely, and will make it easier for me to elaborate on my comments when I get time! $\endgroup$
    – David MJC
    Oct 27, 2010 at 21:48

What is called "generalized complex geometry" is really the study of symplectic Lie 2-algebroids, going by the name "Courant algebroid"s.

This is the beginning of a sequence of Lie n-algebroids -- symplectic Lie n-algebroids -- that goes along with n-dimensional quantum field theory.

As with ordinary symplectic structures, they appear in two (related) roles: as curvatures of lines bundles on target space and then as curvatures on bundles on phase space, via geometric quantization. Similarly for the string: the line 2-bundle aka bundle gerbe on target space that the string is charged under has an underlying Courant Lie 2-algebroid. This witnesses notably the T-duality on this background structure. But also the geometric quantization of the string in multisymplectic geometry is governed by a gerbe and its Courant Lie 2-algebroid -- on extended phase space.

See for instance Baez-Rogers; Categorified Symplectic Geometry and the String Lie 2-Algebra Notice that the string Lie 2-algebra is a Courant Lie 2-algebroid over the point.

  • $\begingroup$ Beware of thinking generalized geometry means complex. The real case is of sufficient interest, cf. double field theory $\endgroup$ Jul 9, 2014 at 21:10

I would like to add that generalized geometry in the sense of Hitchin is also a good framework for the notion of brane (a very important notion in what physicists call M-theory) and also T-dualities one of which is mirror symmetry. Especially I have read an article basically saying that the category of generalized Calabi-Yau manifolds provide a good canditate for the Homological Mirror Symmetry conjecture but I don't remember the article anymore, sorry, maybe José Figueroa-O'Farrill does remember. See also the question: Mirror symmetries for generalized geometries ?.

EDIT: The article I was searching is: http://arxiv.org/abs/1106.1747.


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