Contents

duality

# Contents

## 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.

### General

The original Scherk-Schwarz mechanism:

The duality-twisted generalized version:

### 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