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 that treats the three objects in a unified manner, manifests not only diffeomorphism and one-form gauge symmetry but also $\mathbf{O}(D,D)$ 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 that is given by a direct product of two local Lorentz groups, $\mathbf{S}\mathbf{O}(1,D-1)\times \overline{\mathbf{S}\mathbf{O}}(1,D-1)$. We comment that the notion of cosmological constant naturally changes.

• Received 31 May 2011

DOI:https://doi.org/10.1103/PhysRevD.84.044022