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.
Highlights
More radically Poisson-Lie (PL) T-duality [8, 9] dispenses with the requirement of isometry of a target space but does assume some underlying algebraic structure given by a Drinfeld double
Critical to us will be that the most natural understanding of Poisson-Lie T-duality and its associated target spacetimes is provided by the tools and techniques of Double Field Theory (DFT) [22] and generalised geometry [23,24,25]
Though some partial descriptions have been recently suggested in the literature [33], far an algebraically robust description has been lacking. It is this that we address in the current article in the context of exceptional field theory (ExFT)/exceptional generalised geometry [34,35,36,37,38,39,40], the M-theoretic analogues to Double Field Theory/generalised geometry
Summary
We provide a brief recap of Poisson-Lie T-duality. We will flip the conventional exposition by starting with algebraic considerations to eventually arrive at an associated NLSM describing the NS sector of a closed string; this will serve as a road map for what follows. We could swap the role of g and gin the entire discussion above constructing Πab, πa, va, ̃la as well as generalised frame fields EA This results in a dual σ-model, S, defined on G = exp gthat is canonically equivalent to the first [9, 45, 46]. Should the target space of the original theory define a solution of the appropriate (super)gravity (or part thereof) under normal circumstances so too will the dual, and this procedure defines a solution generating technique Generate diffeomorphisms, two-form and 5-form gauge transformations on the internal space M via the generalised Lie derivative As it is most amenable for direct calculation, we will consider the case of d = 4.
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