Generalised cosets

Saskia Demulder,Falk Hassler,Daniel C Thompson,Giacomo Piccinini

doi:10.1007/jhep09(2020)044

Abstract

Recent work has shown that two-dimensional non-linear σ-models on group manifolds with Poisson-Lie symmetry can be understood within generalised geometry as exemplars of generalised parallelisable spaces. Here we extend this idea to target spaces constructed as double cosets M = tilde{G} \\\U0001d53b/H. Mirroring conventional coset geometries, we show that on M one can construct a generalised frame field and a H -valued generalised spin connection that together furnish an algebra under the generalised Lie derivative. This results naturally in a generalised covariant derivative with a (covariantly) constant generalised intrinsic torsion, lending itself to the construction of consistent truncations of 10-dimensional supergravity compactified on M . An important feature is that M can admit distinguished points, around which the generalised tangent bundle should be augmented by localised vector multiplets. We illustrate these ideas with explicit examples of two-dimensional parafermionic theories and NS5-branes on a circle.

Highlights

Recent work has shown that two-dimensional non-linear σ-models on group manifolds with Poisson-Lie symmetry can be understood within generalised geometry as exemplars of generalised parallelisable spaces
We show that on M one can construct a generalised frame field and a H-valued generalised spin connection that together furnish an algebra under the generalised Lie derivative
The torsion FABC is in one-to-one correspondence with the embedding tensor which fixes the gauge group of the lower dimensional maximal SUGRA

Summary

Cosets and generalized parallelisable spaces

As a group manifold, is equipped with vector fields kA and vA constructed as duals to the left- and right-invariant Maurer-Cartan forms namely g g gg ιkA −1d = TA , ιvAd −1 = TA ,. D We are still working with quantities on the full group , but the generalised frame D fields we want to construct are defined on the coset G\ For this coset, we have the D D D D projection π : → G\ and the local sections σ : G\ → which are chosen such that the pullback σ∗A vanishes. Φa is dual to the ka which generate right translations and π : G → g∧g defines a Poisson bi-vector that obeys the analogue of a cocycle condition making G a Poisson-Lie group

Dressing cosets and generalised coset spaces

Localised sources and extended generalised tangent space

Examples

Parafermions and their deformations

NS5-branes on a circle

Conclusions and future directions