The geometry behind double geometry

Generalised diffeomorphisms in double field theory rely on an O(d,d) structure defined on tangent space. We show that any (pseudo-)Riemannian metric on the doubled space defines such a structure, in the sense that the generalised diffeomorphisms defined using such a metric form an algebra, provided a covariant section condition is fulfilled. Consistent solutions of the section condition gives further restrictions. The case previously considered corresponds to a flat metric. The construction makes it possible to apply double geometry to a larger class of manifolds. Examples of curved defining metrics are given. We also comment on the role of the defining geometry for the symmetries of double field theory, and on the continuation of the present construction to the U-duality setting.

The usual definition of the double diffeomorphisms contains a parameter ξ M on the doubled space. Fields transform under double diffeomorphisms so that, in addition to the translation generated by ξ = ξ M ∂ M , they are rotated by an so(d, d) transformation generated by a − a t , where a M N = ∂ M ξ N . Here, the transpose is defined using a constant metric η MN , Acting on a covector (which is equivalent to a vector, using η), the standard form of a double diffeomorphism thus becomes It is then straightforward to verify that, when all fields, including the transformation parameters, obey the section condition (the "⊗" notation meaning that the two derivative may act on the same field or any pair), the commutator of two double diffeomorphisms is again a double diffeomorphism: In ref. [39], this was elucidated by the observation that where is a non-translating parameter, i.e., one for which χ M ∂ M = 0 using the section condition. Therefore, L χ ξ,η = ∆ ξ,η is a specific local so(d, d) transformation, which also turns out to be nilpotent.
The commutator can thus also be written This observation was used in ref. [39] to explain the abelian gerbe structure encoded in double diffeomorphisms.
We thus observe that the double diffeomorphisms rely on the existence of a flat metric η MN . This implies no restriction locally, but limits the choice of double manifolds to those globally allowing such a metric structure. These of course include tori, which are of special interest since they lead to the ordinary (discrete) T-duality. It seems to be of no immediate interest to introduce curvature locally, since the defining metric is non-dynamical, but it may be important to be able to include topologies that demand a non-flat metric. This is the main subject of this paper.
A local O(d, d) structure is induced by any metric H MN . We therefore ask the questions: To what extent is it possible to use a (non-flat) metric H on the double space? What are the restrictions on such a metric implied by the existence of an algebra of double diffeomorphisms? As we will see, any (pseudo-)Riemannian metric (of signature (d, d)) is algebraically allowed, and further restriction follow only from the existence of solutions to the section condition.
We would like to stress that the introduction of the metric H has nothing to do with equipping the double manifold with a generalised metric, containing the metric and B-field on a subspace obtained by solving the section condition. The metric H is thought of as an ordinary metric on the double space, whose purpose is to define a local O(d, d) structure. It will define an ordinary torsion-free affine connection Γ and a Riemann tensor R.
The Ansatz we will use for the double diffeomorphisms is the natural one that reduces to eq. (1) when H = η. In order to obtain covariance, we should then use covariant derivatives D, containing Γ, throughout, and use the covariantly constant metric H to raise and lower indices (i.e., to define the transpose of a matrix). Consider therefore a transformation defined as where As usual, the connection terms in the Lie derivative cancel, and the choice of defining metric is only reflected in the last term. It is clear that H itself is conserved by such a transformation (extended to tensors), since it is covariantly constant and In order to check the algebra of these generalised diffeomorphisms, we also need a section condition, which will be the natural generalisation of eq. (2), namely Note that there is no need of a section condition involving the metric, since it is covariantly constant.
Let us now commute two transformations of the type (7). Noting that (where of course also b is defined with the covariant derivative, b M N = D M η N ), it becomes clear that any obstruction containing curvature will reside in terms containing a t or b t . A careful calculation, now keeping track of the order of covariant derivatives, yields In the second step, the section condition has been used in the form a t b = 0 etc. The question now is whether this remainder can be written as a non-translating transformation L χ as in the case of flat metric. Consider a parameter χ M ξ,η = 1 Comparing eqs. (11) and (12), we see that they are equal modulo the (vanishing) torsion Bianchi identity R [P QM] N = 0.
This shows the somewhat surprising result that there is no curvature obstruction to the algebra of double diffeomorphisms. The metric H defining the local O(d, d) structure can be taken as any (pseudo-) Riemannian metric. The commutator is still formally given by eq.
with a set of d light-like Killing vectors ∂m. It is not obvious to us whether this provides an exhaustive list of allowed defining metrics. In any case, this class is general enough to give room for any topology of a physical compactification space (with coordinates x m ).
There should be analogous structures in exceptional extended geometry. Unlike the case of doubled space, both the generalised diffeomorphisms and the section condition involve a structure which is not a metric, but a tensor defining an E n(n) × R + structure, the so called Y tensor [27]. The transformations look formally the same as in eq. (1), but with a t replaced by a Y , with components (a Y ) M N = Y MP QN a Q P , and the "flat" section condition reads Y MN P Q ∂ P ⊗ ∂ Q = 0. The Y tensor does not factorise into a product of a metric and its inverse, so the structure sought for is not a metric structure. Nevertheless, it should be possible to pursue a similar investigation in these cases.
We would finally like to comment on the symmetries of double geometry (the remarks apply to extended geometry in general). It is known that the generalised diffeomorphisms are not general enough to accommodate overlaps that would give truly non-geometric solutions [39,41]. This is because of the section condition. Once a solution to the section condition is chosen, it is preserved by the generalised diffeomorphisms, which effectively prevents some duality transformations, namely those that would act on the extended space in way that changes the solution to the section condition (the section condition itself is of course preserved). Therefore, the whole (discrete) T-duality group of some compactification can not, with the present understanding, be constructed as generalised isometries. (This situation is by no means improved by the construction of the present paper, rather the opposite, since the possibilities of changing the section condition tend to be fewer.) A proper understanding of the section condition, and of possible ways to relax it (see e.g. refs. [42,43]) is thus one of the key problems in extended geometry. A final solution to this problem will probably have to await a formulation where the section condition is not applied "by hand", but arises dynamically, as does the string theory level matching condition. In the meantime, it is reasonable to expect that in such a formulation, although the defining metric (or exceptional structure) is not dynamical, its symmetries, which in the case of double field theory are the isometries of H, can be included as gauge symmetries. If this is the case, this will suffice to fill out the T-duality group. We note that, while ordinary diffeomorphisms in general do not have good commutators with the generalised diffeomorphisms L ξ , since they change the defining metric, isometries do. If u M is a Killing vector of H it is straightforward to check that [L u , L ξ ] = L [u,ξ] (the analogous statement for a finite isometry is obviously true as well). Here, u M does not need to obey the solution to the section condition, and it is only Killing vectors not obeying it that generate transformations not contained in the L 's.