nLab Bn-geometry

Idea

In the context of T-duality and in particular of differential T-duality one considers (as discussed there) fiber products of pairs of torus-fiber bundles equipped with a circle 2-bundle (with connection).

In some disguise, this has been called $B_n$-geometry (Baraglia 2012). The T-duality interpretation is made explicit by Bouwknegt 2011.

Here “$B_n$” refers to the special orthogonal group of the form $SO(n+1,n)$, which appears as the structure group of a generalized tangent bundle tensored with a line bundle (the Poincaré line bundle of the T-duality correspondence).

Properties

Interpretation in higher differential geometry

We give the interpretation of $B_n$-geometry in higher differential geometry.

For $c_{conn},c'_{conn} \colon X \to \mathbf{B}U(1)$ modulating two circle principal bundles with conection, a differential T-duality structure is a choice of trivialization of their cup product class. From this we get the pasting diagram of homotopy pullbacks of smooth $\infty$-stacks

$$\array{ P \times_X \hat P&\stackrel{\tau}{\to}& \mathbf{B}^2 U(1) &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\to& fib(\cup) &\to& (\mathbf{B}U(1)) \times (\mathbf{B}U(1)) \\ &\searrow & \downarrow && \downarrow^{\mathrlap{\cup}} \\ && * &\to& \mathbf{B}^3 U(1) } \,.$$

Here $\tau$ is the morphism that modulates the circle 2-bundle on the fiber product of the two circle bundles.

(…)

Related concepts

• differential T-duality

• generalized complex geometry

• exceptional generalized geometry

References

The term $B_n$-geometry was introduced in:

• David Baraglia: Leibniz algebroids, twistings and exceptional generalized geometry, Journal of Geometry and Physics 62 5 (2012) 903–934 [doi:10.1016/j.geomphys.2012.01.007, arXiv:1101.0856]

  (and relating in §2.3 to the supergravity C-field gauge algebra)

A review is in

• Nigel Hitchin, talks at StringMath2012 [pdf]

The relation to T-duality is made clear around slide 80 of:

• Peter Bouwknegt: Courant Algebroids and Generalizations of Geometry, talk at StringMath 2011 [pdf]

A discussion of the higher Lie theoretic aspects is in

• Ernesto Lupercio, Camilo Rengifo, Bernardo Uribe, T-duality and exceptional generalized geometry through symmetries of dg-manifolds [arXiv:1208.6048]