Exploring exceptional Drinfeld geometries

Abstract

We explore geometries that give rise to a novel algebraic structure, the Exceptional Drinfeld Algebra, which has recently been proposed as an approach to study generalised U-dualities, similar to the non-Abelian and Poisson-Lie generalisations of T-duality. This algebra is generically not a Lie algebra but a Leibniz algebra, and can be realised in exceptional generalised geometry or exceptional field theory through a set of frame fields giving a generalised parallelisation. We provide examples including “three-algebra geometries”, which encode the structure constants for three-algebras and in some cases give novel uplifts for CSO(p, q, r) gaugings of seven-dimensional maximal supergravity. We also discuss the M-theoretic embedding of both non-Abelian and Poisson-Lie T-duality.

E6(6) exceptional Drinfel’d algebras

“…10 It can also be the case thatẼA don't apparently follow from an EDA construction, which leads to a wider still notion of duality. This feature was seen explicitly in [21] in which an EDA uplift of certain CSO(p, q, r) gaugings was given complementing the existing uplift of [36]. 11 Including potentially actions in GL(dim R1) outside E n(n) .…”

“…The non-perturbative generalisation of Poisson-Lie T-duality to a U-duality version in M-theory, or more conservatively as a solution-generating mechanism of 11-dimensional supergravity, has long been an open problem, which was recently addressed in [17,18] and further elaborated on in [19][20][21]. Building on the interpretation of PLTD and Drinfel'd doubles within Double Field Theory (DFT), [17,18,[22][23][24] used Exceptional Field Theory (ExFT)/Exceptional Generalised Geometry to propose a natural generalisation of the Drinfel'd double for dualities along four spacetime dimensions.…”

“…To complete the process one has to use the well-established dictionary between M IJ (x, y) and conventional supergravity fields. An intuition for the sorts of internal geometries that arise can be obtained by examining the case whenM AB is constant and we refer the reader to recent works for examples of this [17,20,21].…”

Currents, charges and algebras in exceptional generalised geometry

“…The type of finite-dimensional algebras, as for example the recently discussed exceptional Drinfeld algebras [51,52,[56][57][58][59][60][61][62][63], are contained in the current algebra as the algebra of the zero modes z A = − Z A in case F is constant: 6 The SL( 5)-vielbeine and their invariance conditions, used here, are:…”

“…Success in the strive for an E-model like formulation has, by now, only been achieved for the E 3(3) -theory and partly the E 4(4) -theory [32,42,43,[50][51][52]. Generalising [51], such a formulation might -similar to Poisson-Lie T -duality as canonical transformation of classical string theory -help to understand to which extent the recently discussed Poisson-Lie U -duality, transformations of exceptional Drinfeld algebras or other interesting flux configurations [51][52][53][54][55][56][57][58][59][60][61][62][63], can be understood as some kind of classical duality transformation of world-volume theories in exceptional field theory.…”