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

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.

We construct a supersymmetric extension of double field theory that realizes the ten-dimensional Majorana-Weyl local supersymmetry. In terms of a stringy differential geometry we proposed earlier, our action consists of five simple terms -two bosonic plus three fermionic -and manifests not only diffeomorphism and one-form gauge symmetry of B-field, but also O(10, 10) T-duality as well as a direct product of two local Lorentz symmetries, SO(1, 9) × SO(9, 1). A gauge fixing that identifies the double local Lorentz groups reduces our action to the minimal supergravity in ten dimensions.PACS numbers: 04.60. Cf, 04.65.+e Without resorting to vector notation, Maxwell's original equations consisted of twenty formulas. It was the rotational or Lorentz symmetry that reorganized them into four or two compact equations. Recent developments in string theory indicate that supergravity theories -at least those which have stringy origin -may undergo a similar reformulation, and be greatly simplified with the renewed understanding of their stringy structure or T-duality.T-duality is a genuine stringy effect such that string theory effective actions or ten-dimensional supergravities should feature O(10, 10) structure [1][2][3][4]. The O(10, 10) T-duality can be manifestly realized if we formally double the spacetime dimension, from ten to twenty with coordinates x µ → y A = (x µ , x ν ) [5][6][7][8], and reformulate the ten-dimensional effective action in terms of twenty dimensional language i.e. tensors equipped with O(10, 10) metric,This kind of reformulation was coined Double Field Theory (DFT) [9][10][11][12], and has attracted much attention in recent years [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. In DFT, as a field theory counterpart to the level matching condition of closed string theories, the O(10, 10) d'Alembertian operator must be trivial, acting on arbitrary fields as well as their products,Hence locally, up to O(10, 10) rotation, all the fields are independent of the dual coordinates, ∂ ∂xµ ≡ 0, and the theory is not truly doubled [11].In a sense, the O(10, 10) structure in DFT is a "metasymmetry" rather than a Noether symmetry, since only after dimensional reductions can it generate a Noether symmetry. Another feature of DFT is that, the diffeomorphism and the one-form gauge symmetry of B-field are naturally unified into what we may call "double-gauge symmetry," as they are generated by the generalized Lie derivative [8,12,[32][33][34],where ω T is the weight of T A1···An . Since this differs from the ordinary Lie derivative, the underlying differential geometry of DFT is not Riemannian [7, 8, 14-16, 18, 32-37] (see [38,39] for extensions to M-theory). Namely, while doubling the spacetime dimension is sufficient to manifest the O(10, 10) structure, the double-gauge symmetry (3) calls for novel mathematical treatment.In this paper, we construct a supersymmetric extension of double field theory that manifests simultaneously O(10, 10) T-duality and various gauge symmetries lis...

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