nLab
generalized Scherk-Schwarz reduction

Contents

Context

String theory

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×RU \times R, where RR is the underlying vector space of the fundamental representation of some duality group GG (e.g. U-duality or T-duality). Let us call xx the local coordinates of the base manifold UU and (y,y˜)(y,\tilde{y}) the ones of the fiber RR, called internal manifold.

A generalized Scherk-Schwarz reduction is performed by introducing twisting GG-valued matrices E M A(y,y˜)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 1M n(x,y,y˜)=E M 1 A 1(y,y˜)E M n A n(y,y˜)T A 1A n(x) 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 1M nT_{M_1 \dots M_n} on the extended space. The GG-valued matrices E M A(y,y˜)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 2nT^{2n}, where the T-duality group O(n,n)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

MN=(gBg 1B Bg 1 g 1B g 1) \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)O(n,n)-valued matrices

E M A=(e μ a 0 e a ρB ρμ e a μ) 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

MN(y,y˜)=E M 1 A 1(y,y˜)E M n A n(y,y˜) AB \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 AB=diag(δ ab,δ ab)\mathcal{H}_{A B} = \mathrm{diag}(\delta_{a b},\,\delta^{a b}). This simple example captures the analogy with ordinary frame fields.

References

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

Last revised on November 12, 2020 at 23:57:24. See the history of this page for a list of all contributions to it.