The geometry, branes and applications of exceptional field theory

Abstract

This is a review of exceptional field theory: a generalisation of Kaluza–Klein theory that unifies the metric and [Formula: see text]-form gauge field degrees of freedom of supergravity into a generalised or extended geometry, whose additional coordinates may be viewed as conjugate to brane winding modes. This unifies the maximal supergravities, treating their previously hidden exceptional Lie symmetries as a fundamental geometric symmetry. Duality orbits of solutions simplify into single objects, that in many cases have simple geometric interpretations, for instance as wave or monopole-type solutions. It also provides a route to explore exotic or nongeometric aspects of M-theory, such as exotic branes, [Formula: see text]-folds, and more novel sorts of non-Riemannian spaces.

Kaluza-Klein fermion mass matrices from exceptional field theory and $$ \mathcal{N} $$ = 1 spectra

“…Here, Y Λ denotes the infinite tower of scalar spherical harmonics on the round S 7 or S 6 spheres. These lie in the representations (2.5) of SO (8) or SO (7). Of…”

“…where the constant, real matrices (T N ) Λ Σ are the generators of SO (8) or SO (7) in the infinite-dimensional, reducible representations (2.5), normalised as [13,14] […”

“…By construction, the duality-covariant reformulations [1][2][3][4][5][6][7] (see [8] for a review) of D = 11 [9] and type II [10][11][12] supergravities are particularly helpful to investigate the imprint that these higher-dimensional theories instill on their lower-dimensional counterparts. In particular, Exceptional Field Theory (ExFT) [3][4][5] has been recently shown to provide a powerful framework to compute the spectrum of Kaluza-Klein (KK) perturbations above a certain class of anti-de Sitter (AdS) backgrounds of string and M-theory [13,14].…”

“…We have then used our mass matrices to compute the KK gravitino and spin-1/2 spectra above concrete AdS 4 solutions in this class: those that preserve at least SU (3) internal symmetry. Prior to the general scans of [36,37], these particular solutions were classified in [38] and [35] for the SO (8) and the ISO(7) gaugings, respectively. Their corresponding D = 11 [39][40][41][42][43][44][45] and type IIA uplifts [28,29,46] are all known.…”

E6(6) exceptional Drinfel’d algebras

“…In general one can allow for EDAs as subalgebras of E n(n) × R + , see[18]. However, we will not deal with the extra R + factor here.2 See[28] for a contemporary review of ExFT 3. There is another inequivalent way to maximally solve the condition eq.…”