Global aspects of double geometry

Abstract

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.

T-duality and α′-corrections

“…However, the first order redefinition ∆e µ a can induce a non-covariant behavior. The matrices that generate the Z 2 -parity transformations adopt the following parameterization 58) and at the level of components they exchange Z 2 (ē (±) µ a ) =ē (∓) µ a . So, after the gauge fixing, they leave the bein (and thus the metricḡ µν ) invariant, but they exchange the sign of the two-form Z 2 (B µν ) = −B µν , as expected.…”

“…A pure generalized flux formulation of the theory [45] could be useful in understanding these issues. Finally, the generalized Green-Schwarz transformation might be relevant in the analysis of large gauge transformations in DFT [56][57][58][59][60][61].…”

Extended geometries

“…Among the most pressing ones is to determine the dynamics, if not in all cases, at least in some important ones, such as affine and hyperbolic cases. It is also desirable to obtain a better understanding of finite transformations, which in double field theory are reasonably well understood [22,[24][25][26], but which have been elusive in the exceptional cases, for fundamental or technical reasons. A generic treatment is desirable.…”

U-gravity: SL(N)

“…Then, like T-fold [53][54][55], by combining diffeomorphism and O(D, D) rotation as for a transition function, DFT may acquire nontrivial global aspects of non-geometry [56][57][58][59][60].…”

“…Accordingly, each equivalence class or gauge orbit represents a single physical point, and diffeomorphism symmetry means an invariance under arbitrary reparametrizations of the gauge orbits. This allows more than one finite transformation rule of diffeomorphism [56][57][58]. The idea has been pushed further to construct a completely covariant string world-sheet action on doubled-yet-gauged spacetime [61], where the coordinate gauge symmetry is realized literally as one of the local symmetries of the action.…”