nLab generalized Scherk-Schwarz reduction

Idea

A generalized Scherk-Schwarz reduction is a dimensional reduction combined with a duality twist, such as for exceptional field theory and double field theory.

Consider an exceptional field theory on a certain extended space, i.e. a smooth manifold locally isomorphic to $U \times R$, where $R$ is the underlying vector space of the fundamental representation of some duality group $G$ (e.g. U-duality or T-duality). Let us call $x$ the local coordinates of the base manifold $U$ and $(y,\tilde{y})$ the ones of the fiber $R$, called internal manifold.

A generalized Scherk-Schwarz reduction is performed by introducing twisting $G$-valued matrices $E^{A}_{\;\;M}(y,\tilde{y})$ and by encoding all the dependence of the fields on the coordinates of the internal manifold with the following ansatz:

$$T_{M_1 \dots M_n}(x,y,\tilde{y}) = E^{A_1}_{\;\;M_1}(y,\tilde{y}) \cdots E^{A_n}_{\;\;M_n}(y,\tilde{y}) T_{A_1\dots A_n}(x)$$

on any generalized tensor field $T_{M_1 \dots M_n}$ on the extended space. The $G$-valued matrices $E^{A}_{\;\;M}(y,\tilde{y})$ can be seen as generalized frame fields for the internal manifold.

Examples

Double field theory on the torus

Consider a double field theory on the torus $T^{2n}$, where the T-duality group $O(n,n)$ acts in the fundamental representation. We are in the case where the external space is just a point. Let the doubled metric be

$$\mathcal{H}_{M N} = \begin{pmatrix} g -B g^{-1}B & B g^{-1} \\ -g^{-1}B & g^{-1}\end{pmatrix}$$

Now, if we define $O(n,n)$-valued matrices

$$E^A_{\;\;M} = \begin{pmatrix}e^a_{\mu} & 0 \\ e^\rho_a B_{\rho\mu} & e^\mu_a \end{pmatrix}$$

we can immediately write the generalized Scherk-Schwarz reduction over the point by

$$\mathcal{H}_{M N}(y,\tilde{y}) = E^{A_1}_{\;\;M_1}(y,\tilde{y}) \cdots E^{A_n}_{\;\;M_n}(y,\tilde{y}) \mathcal{H}_{A B}$$

where $\mathcal{H}_{A B} = \mathrm{diag}(\delta_{a b},\,\delta^{a b})$. This simple example captures the analogy with ordinary frame fields.

Related concepts

• doubled geometry, double field theory

• exceptional field theory

• gauged supergravity

References

General

The original Scherk-Schwarz mechanism:

• Joël Scherk, John Schwarz, How to get masses from extra dimensions, in: Supergravities in Diverse Dimensions, pp. 1282-1309 (1989) (doi:10.1142/9789814542340_0083)

The duality-twisted generalized version:

• Atish Dabholkar, Chris Hull, Duality Twists, Orbifolds, and Fluxes, JHEP 0309:054, 2003 (arXiv:hep-th/0210209)

• Chris Hull, A Geometry for Non-Geometric String Backgrounds, JHEP 0510:065, 2005 (arXiv:hep-th/0406102)

In M-theory

A lift of D8-branes to M-theory M-branes by generalized Scherk-Schwarz reduction, relating to D7-branes in F-theory, is proposed in

• Chris Hull, Massive String Theories From M-Theory and F-Theory, JHEP 9811:027, 1998 (arXiv:hep-th/9811021)