$E_{d(d)} \times \mathbb{R}^+$ Generalised Geometry, Connections and M theory

Doubled α ′-geometry

“…Global aspects of double field theory are discussed in [22] where a formula for large gauge transformations was proposed and examined in detail. There are numerous other developments in double field theory and the closely related M-theory (see [39][40][41]); for a recent review with further references see [42].…”

E 8 duality and dual gravity

“…The E 8 duality group is particularly interesting from the point of view that the potential of the dual gravity field is expected to appear in the generalised metric for the first time. 1 As has been observed in [36,37], the presence of dual gravity poses difficulties for the formulation of an E 8 generalised geometry.…”

“…This is in contrast to the E 6 case considered in reference [20] where the 6-form field could have appeared in the potential but didn't because there was an antisymmetrisation over 7 indices. In the potential given in equation ( 18 Our difficulty in establishing the gauge-invariance of the field strength is related to the difficulty in formulating a generalised geometry for E 8 [36,37]. In both cases, knowledge of the transformation of C a 1 ...a 8 ,b under gauge transformations is a requisite.…”

“…The generalised geometries associated with the M-theory dualities admit similar, but richer structures given the existence of higher rank p-forms sourced by various branes [14]. As with double field theory, the generalised geometries in this context also contain notions of a generalised diffeomorphism algebra that unifies spacetime diffeomorphims and gauge symmetries and closes on a section condition [35][36][37], as well as other structures [38][39][40][41].…”

U-folds as K3 fibrations

“…(see [16][17][18][19][20][21][22][23] for previous works on Tfolds). A related construction of string vacua where fluxes are codified into geometry has been recently developed in [24][25][26][27][28]. This approach has been pioneered in [29,30], see also [31,32] for an embedding of these ideas into a broader perspective.…”