Poisson-Lie U-duality in exceptional field theory

Abstract

Poisson-Lie duality provides an algebraic extension of conventional Abelian and non-Abelian target space dualities of string theory and has seen recent applications in constructing quantum group deformations of holography. Here we demonstrate a natural upgrading of Poisson-Lie to the context of M-theory using the tools of exceptional field theory. In particular, we propose how the underlying idea of a Drinfeld double can be generalised to an algebra we call an exceptional Drinfeld algebra. These admit a notion of "maximally isotropic subalgebras" and we show how to define a generalised Scherk-Schwarz truncation on the associated group manifold to such a subalgebra. This allows us to define a notion of Poisson-Lie U-duality. Moreover, the closure conditions of the exceptional Drinfeld algebra define natural analogues of the cocycle and co-Jacobi conditions arising in Drinfeld double. We show that upon making a further coboundary restriction to the cocycle that an M-theoretic extension of Yang-Baxter deformations arise. We remark on the application of this construction as a solution-generating technique within supergravity.

Exploring exceptional Drinfeld geometries

“…Until recently, there has been no hint of whether U-duality admits non-Abelian or generalised versions. A proposal for the algebraic structure that would underlie such dualities has been introduced in [20,21] and called the Exceptional Drinfeld Algebra (EDA).…”

“…For the case of n ≤ 4, that shall be our concern here, the data of an EDA can be interpreted as consisting of a Liealgebra g together with a three-algebrag that are restricted to obey a cocycle compatibility condition. A key point of [20,21] was that the EDA can be realised by a generalised Leibniz parallelisation for the exceptional tangent bundle T G ⊕ ∧ 2 T G thus echoing the set up of Poisson-Lie T-duality and allowing this framework to be used to generate solutions using the ideas of generalised Scherk-Schwarz reductions. Some features of the geometry, and the membrane interpretation, were then given in [22], while a classification of all possible EDAs for the case of n = 3 was made in [23].…”

“…We can achieve such a realisation of our Exceptional Drinfeld Algebra. First, we decompose our 10-dimensional generalised frame as 21) and specify that, in terms of pairs of vectors and two-forms, these are given by…”

Tri-vector deformations in d = 11 supergravity

“…Supergravity formulations that are natural to look at in this context are Double [26] and Exceptional [27] Field Theories (DFT and ExFT, respectively). Specifically designed to render supergravities in various dimensions covariant under T-and U-duality groups at the expense of extending the spacetime dimension, they are useful in describing Yang-Baxter deformations [15,[28][29][30][31] and Poisson-Lie T-duality [32][33][34][35][36][37]. The proof of [23] that (1.1), (1.2) is a supergravity symmetry relied upon DFT techniques, in particular the β-supergravity formalism [38][39][40].…”

T-folds as Poisson–Lie plurals

“…Natural language for their description has become Double Field Theory [26][27][28][29]. It turns out that Double Field Theory can describe not only Abelian T-duality but also Poisson-Lie T-duality [30][31][32][33] and may help investigate quantum aspects of Poisson-Lie T-duality [34] or its extension to U-duality [35,36].…”