Exploring exceptional Drinfeld geometries
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Springer
Received: July 2, 2020 Accepted: August 21, 2020 Published: September 23, 2020
Chris D.A. Blair,a Daniel C. Thompsona,b and Sofia Zhidkovaa a
Theoretische Natuurkunde, Vrije Universiteit Brussel, and the International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium b Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom
E-mail: [email protected], [email protected], [email protected] 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 Tduality. 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 “threealgebra 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. Keywords: M-Theory, String Duality ArXiv ePrint: 2006.12452
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)151
JHEP09(2020)151
Exploring exceptional Drinfeld geometries
Contents 1
2 The 2.1 2.2 2.3
SL(5) Exceptional Drinfeld Algebra The algebra The generalised geometry realisation The geometry
3 3 5 7
3 Three-algebra geometries 3.1 Non-Abelian T-duality revisited and CSO(3, 0, 2) 3.2 Euclidean 3-algebra and CSO(4, 0, 1) 3.3 A Leibniz geometry: τab 6= 0
8 9 13 15
4 Embedding Drinfeld doubles 4.1 Decomposing the Embedding Drinfeld Algebra 4.2 Example: Bianchi II and V
16 16 19
5 Discussion
20
A SL(5) exceptional geometry A.1 Generalised Lie derivative and generalised frames A.2 Generalised frames and their algebra A.3 Dictionary to 11- and 10-dimensional geometries
22 22 24 25
B Embedding Drinfeld doubles in SL(5) B.1 Half-maximal truncation B.2 Drinfeld doubles B.3 Spinors and gamma matrices
26 26 27 29
1
Introduction
The textbook T-duality symmetry of string theory that applies in backgrounds with Abelian isometries is a cornerstone of the duality web that ultimately leads to M-theory [1, 2]. Less standard is the application of T-duality to backgrounds whose isometry group is nonAbelian [3]. While its status as a precise duality in either α0 and gs expansions is not fully resolved, at the very least non-Abelian T-duality (NATD) is a useful tool as a solution generating symmetry of Type II supergravity (for a review see [4]). More exotic still are applications of T-duality to backgrounds which have no isometries at all. Poisson-Lie (PL) T-duality, introduced by Klimˇc´ık and Severa [5, 6], provides situations where such a non-isometric duality can be realised. This is made possible when the target spaces
–1–
JHEP09(2020)151
1 Introduct
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