Differential geometry with a projection: application to double field theory

Abstract

In recent development of double field theory, as for the description of the massless sector of closed strings, the spacetime dimension is formally doubled, i.e. from D to D+D, and the T-duality is realized manifestly as a global O(D, D) rotation. In this paper, we conceive a differential geometry characterized by a O(D, D) symmetric projection, as the underlying mathematical structure of double field theory. We introduce a differential operator compatible with the projection, which, contracted with the projection, can be covariantized and may replace the ordinary derivatives in the generalized Lie derivative that generates the gauge symmetry of double field theory. We construct various gauge covariant tensors which include a scalar and a tensor carrying two O(D, D) vector indices.

Doubled α ′-geometry

“…Next we do an α ′ expansion by writing 24) where Λ(H) is first order in α ′ , containing two derivatives. To leading order (7.23) gives…”

Extended geometry and gauged maximal supergravity

“…Covariant definitions of these tensors were introduced in [22,23,33,34,36,37], which interestingly contain undetermined components. Here, extending the definition of [33,34] to the exceptional case, we find a covariant (though still not uniquely defined) version of the generalized Ricci tensor. Taking its trace, the undetermined pieces go away, and we show that the generalized Ricci scalar, which coincides with that of [22,23] when the section condition is imposed, can be written purely in terms of the dynamical fluxes.…”

U-duality covariant gravity

“…massive type II theories [11][12][13][14], and their supersymmetric extensions [1,[15][16][17][18], and also leads to a compelling generalization of Riemannian geometry [1,[19][20][21][22][23][24], which in turn is closely related to (and an extension of) results in the 'generalized geometry' of Hitchin and Gualtieri [25][26][27] (see [28][29][30][31][32][33][34] for other applications and [35][36][37][38] for reviews).…”

“…With this choice, a short calculation reveals that the kinetic and the Chern-Simons term of the action (4.1) reduce to 23) reproducing the corresponding terms in the Lagrangian (5.13), with covariant derivatives and field strength from (5.6), upon the identification…”