Stringy differential geometry, beyond Riemann

Abstract

While the fundamental object in Riemannian geometry is a metric, closed string theories call for us to put a two-form gauge field and a scalar dilaton on an equal footing with the metric. Here we propose a novel differential geometry which treats the three objects in a unified manner, manifests not only diffeomorphism and one-form gauge symmetry but also O(D,D) T-duality, and enables us to rewrite the known low energy effective action of them as a single term. Further, we develop a corresponding vielbein formalism and gauge the internal symmetry which is given by a direct product of two local Lorentz groups, SO(1,D-1) times SO(1,D-1). We comment that the notion of cosmological constant naturally changes.Comment: 7 pages, double column; References adde

Doubled α ′-geometry

“…Using the strong constraint, the new inner product can also be written as 25) where K M N is the "field strength" of the gauge parameter:…”

“…In [20] this formalism has been related to the double field theory actions of [18,19] and to a metriclike formulation. The metric-like approach has been examined in more detail in [24,25] (in a "semi-covariant" approach that truncates connections) and in [21]. The fully "invariant" formulation in [23] provides a unifying framework for the metric-and frame-like formalisms.…”

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).…”

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.…”