nLab
T-duality

Idea

The space of differential geometric data consisting of

• a smooth manifold $X$

• equipped with the structure of a $k$-torus bundle $X\simeq Y\times T^k$;

• and equipped with a Riemannian metric $g$

• and a $U(1)$-gerbe with connection $G$

admits a certain operation that, roughly, inverts the Riemannian circumference of the torus fibers and mixes the metric with the gerbe data. This operation is called T-duality.

It was noticed originally in the study of conformal field theories in the context of string theory: the conformal field theory sigma-models with target space $X$ turn out to be equivalent as quantum field theories for T-dual backgrounds $(X,g,G)$ and $(X\prime ,g\prime ,G\prime )$ (at least to the approximate degree to which these are realized as full CFTs in the first place).

Further generalisations let $X$ be a nontrivial torus bundle, but the T-dual is then generically a bundle of non-commutative tori?. (cite Mathai, Rosenberg and Hannabus)

topological T-duality

It turns out to be possible and useful to discuss just the topological aspects of T-duality, meaning all the aspects that depend on the $X$ as a topological space, on the topological class of the gerbe and of its 3-form curvature, but not on the Riemannian metric and not on the precise connection on the gerbe (there may be several inequivalent one for a given curvature)!

This sub-phenomenon is discussed in more detail at topological T-duality.

geometric T-duality in generalized complex geometry

Another approach to the study of T-duality takes a somewhat complementary point of view and ignores the integral cohomology class in $H^3(X,\mathbb{Z})$ of the gerbe but does consider the Riemannian metric.

In this context T-duality is described as an isomorphism of standard Courant algebroids. This picture emerged in the study of generalized complex geometry.

References

T-duality is identified as an isomorphism of standard Courant algebroids in section 4 of

• Cavalcanti, Gualtieri, Generalized complex geometry and T-duality (pdf)

A discussion of the sigma-model description of T-duality in this context is in

• Jonas Persson, T-duality and Generalized Complex Geometry (arXiv)

Further references are

• Willie Carl Merrell, Application of superspace techniques to effective actions, complex geometry and T-duality in String theory (pdf)

• Peggy Kao, T-duality and Poisson-Lie T-duality in generalized geometry (pdf)