We study closed bosonic strings propagating both in a flat background with constant H-flux and in its T-dual configurations. We define a conformal field theory capturing linear effects in the flux and compute scattering amplitudes of tachyons, where the Rogers dilogarithm plays a prominent role. For the scattering of four tachyons, a fluxed version of the VirasoroShapiro amplitude is derived and its pole structure is analyzed. In the case of an R-flux background obtained after three T-dualities, we find indications for a nonassociative target-space structure which can be described in terms of a deformed tri-product. Remarkably, this product is compatible with crossing symmetry of conformal correlation functions. We finally argue that the R-flux background flows to an asymmetric CFT.

Non-geometric frames in string theory are related to the geometric ones by certain local O(D, D) transformations, the so-called β-transforms. For each such transformation, we show that there exists both a natural field redefinition of the metric and the Kalb-Ramond two-form as well as an associated Lie algebroid. We furthermore prove that the all-order low-energy effective action of the superstring, written in terms of the redefined fields, can be expressed through differential-geometric objects of the corresponding Lie algebroid. Thus, the latter provides a natural framework for effective superstring actions in non-geometric frames. Relations of this new formalism to double field theory and to the description of non-geometric backgrounds such as T-folds are discussed as well.R. Blumenhagen et al.: Non-geometric frames in string theory developed where the O(D, D) transformations 1 play a crucial role, namely generalized geometry [3][4][5][6] and double field theory (DFT) [7][8][9][10][11]. In the first approach, the concept of Riemannian geometry is extended from the tangent bundle T M to the generalized tangent bundle T M ⊕ T * M , whereas in the second the dimension of the space is doubled by including winding coordinates subject to certain constraints. For the latter construction, this admits a manifest global O(D, D) invariance of the action, so in particular, the action is manifestly invariant under T-duality transformations. The fundamental object in both approaches is a generalized metric which combines the usual metric and Kalb-Ramond field. The two local symmetries, diffeomorphisms and B-field gauge transformations, sit inside a subgroup of O (D, D). Their complement in O(D, D) contains so-called (local) β-transforms, which lead out of the usual geometric frame of string theory. Therefore, applying a local β-transform to the geometric frame leads to what we call a non-geometric frame.The existence of non-geometric backgrounds can be seen by analyzing the action of T-duality on the simple background of a flat three-dimensional torus with a constant H-flux [12]. Applying successive T-dualities, this H-flux is first mapped to a geometric flux [13] and by a second T-duality to the nongeometric Q-flux [14][15][16]. The latter background can be understood as a T-fold [17], where the transition functions between two charts involve T-duality transformations. A third T-duality is beyond the scope of the Buscher rules, and both non-commutative geometry [18][19][20] and conformal field theory [21-25] hint towards a non-associative structure. The effect of T-duality on brane solutions has been analyzed recently in [26].Since in DFT a global O(D, D) symmetry is manifest, the first-order effective action in at least a subset of these non-geometric frames is also described by it. What has been puzzling is that the DFT action cannot be straightforwardly interpreted as the Einstein-Hilbert action of some O(D, D) covariant differential geometry [27,28]. The problem is that the notions of torsion and curvature have to be c...

Starting from a (non-associative) quasi-Poisson structure, the derivation of a Roytenberg-type algebra is presented. From the Jacobi identities of the latter, the most general form of Bianchi identities for fluxes (H, f, Q, R) is then derived. It is also explained how this approach is related to the mathematical theory of quasi-Lie and Courant algebroids.1 A bi-vector field also appears in [15] and [16], where the expressions for the fluxes Q and R can be found as well.

Based on the structure of a Lie algebroid for non-geometric fluxes in string theory, a differential-geometry calculus is developed which combines usual diffeomorphisms with so-called β-diffeomorphisms emanating from gauge symmetries of the Kalb-Ramond field. This allows to construct a bi-invariant action of Einstein-Hilbert type comprising a metric, a (quasi-)symplectic structure β and a dilaton. As a salient feature, this symplectic gravity action and the resulting equations of motion take a form which is similar to the standard action and field equations. Furthermore, the two actions turn out to be related via a field redefinition reminiscent of the Seiberg-Witten limit. Remarkably, this redefinition admits a direct generalization to higher-order α ′ -corrections and to the additional fields and couplings appearing in the effective action of the superstring. Simple solutions to the equations of motion of the symplectic gravity action, including Calabi-Yau geometries, are discussed.

Inspired by recent studies on string theory with non-geometric fluxes, we develop a differential geometry calculus combining usual diffeomorphisms with what we call β-diffeomorphisms. This allows us to construct a manifestly bi-invariant Einstein-Hilbert type action for the graviton, the dilaton and a dynamical (quasi-)symplectic structure. The equations of motion of this symplectic gravity theory, further generalizations and the relation to the usual form of the string effective action are discussed. The Seiberg-Witten limit, known for open strings to relate commutative with non-commutative theories, makes an interesting appearance.