Abstract
We consider the concept of a generalised manifold in the O(d,d) setting, i.e., in double geometry. The conjecture by Hohm and Zwiebach for the form of finite generalised diffeomorphisms is shown to hold. Transition functions on overlaps are defined. Triple overlaps are trivial concerning their action on coordinates, but non-trivial on fields, including the generalised metric. A generalised manifold is an ordinary manifold, but the generalised metric on the manifold carries a gerbe structure. We show how the abelian behaviour of the gerbe is embedded in the non-abelian T-duality group. We also comment on possibilities and difficulties in the U-duality setting.
Highlights
JHEP09(2014)066 processes uncovered an interesting Courant algebroid structure
Double field theory was brought into being in order to make manifest the hidden O(d, d; Z) T-duality symmetry of string theory
Given that double field theory should describe the background of a string it cannot possess any global symmetries that are not the global part of a local symmetry
Summary
Double field theory was brought into being in order to make manifest the hidden O(d, d; Z) T-duality symmetry of string theory. The double field theory action is invariant under a “local O(d, d; R) transformation” of the generalised metric This transformation is just the usual combination of d-dimensional diffeomorphisms and 2-form gauge transformations (up to section condition). The bundle is equipped with transition functions fαβ which act on the fibres and so the generalised metric In usual geometry these would be the normal diffeomorphisms acting on a tensor induced from the coordinate transformations between patches i.e.,. (The cocycle condition on quadruple intersections will be trivial .) This implies double field theory possesses a gerbe structure [48] and leads to the following question: if it is a gerbe and valued in O(d, d) is it a non-abelian gerbe [49]? We end with some comments on the extension of these ideas to the extended, exceptional geometries that occur in M-theory where the full U-duality groups are made a manifest symmetry [58, 60,61,62,63,64,65,66,67,68,69,70,71,72,73]
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