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.

This post imported from StackExchange MathOverflow at 2014-12-20 12:50 (UTC), posted by SE-user José Figueroa-O'Farrill