The intriguing structure of non‐geometric frames in string theory

Abstract

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...

T-duality and α′-corrections

“…Understanding the role of supersymmetry would also be of interest, since one should expect obstructions when attempting to supersymmetrize this theory for a choice of parameters leaving only even Z 2 -parity corrections. Generalized Scherk-Schwarz reductions like those considered in [47,48] would also be interesting to examine in order to find higher-derivative corrections in gauged supergravities and to clarify the relation between α ′ -corrections and non-geometry (see for example [49][50][51][52][53][54][55] and references therein). Due to the field redefinitions involved in this construction, we expect the duality covariant scalars of the reduced theory to be related to the diffeomorphism and Lorentz covariant scalars through O(α ′ ) redefinitions that are quadratic in gaugings.…”

Effective actions of nongeometric five-branes

“…As for the rewriting of the NSNS action, this was essentially done in Ref. [32] (further approaches include [33,34]). The effective action is written in terms of the metric g in the β-parametrization of the generalized metric H, the corresponding modified dilatonφ and the field strength (5.21) of the two-vector,…”

Gauged supergravities and non-geometric Q/R-fluxes from asymmetric orbifold CFT’s

“…From the point of view of the (X I ,X I ) basis, the orbifold action mixes coordinates and dual coordinates as: 21) and can be seen to correspond to the following O(2, 2) transformation:…”