abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
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:
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.
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
Now, if we define $O(n,n)$-valued matrices
we can immediately write the generalized Scherk-Schwarz reduction over the point by
where $\mathcal{H}_{A B} = \mathrm{diag}(\delta_{a b},\,\delta^{a b})$. This simple example captures the analogy with ordinary frame fields.
The original Scherk-Schwarz mechanism:
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)
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.