Double field theory and membrane sigma-models

Abstract

We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the corresponding operations in DFT. This describes precisely how the geometric structure of DFT lies in between two Courant algebroids and is reconciled with generalized geometry. We construct the membrane sigma-model that corresponds to DFT, and demonstrate how the standard T-duality orbit of geometric and non-geometric flux backgrounds is captured by its action functional in a unified way. This also clarifies the appearence of noncommutative and nonassociative deformations of geometry in non-geometric closed string theory. Gauge invariance of the DFT membrane sigma-model is compatible with the flux formulation of DFT and its strong constraint, whose geometric origin is explained. Our approach leads to a new generalization of a Courant algebroid, that we call a DFT algebroid and relate to other known generalizations, such as pre-Courant algebroids and symplectic nearly Lie 2-algebroids. We also describe the construction of a gauge-invariant doubled membrane sigma-model that does not require imposing the strong constraint. 1 a.chatzistavrakidis@gmail.com 2 larisa@irb.hr 3 Fech.Scen.Khoo@irb.hr 4 R.J.Szabo@hw.ac.uk 29)19 In this subsection we simplify the notation for the components of the map ρ + by denoting (ρ + ) I J as ρ I J .

Double Field Theory for the A/B‐Models and Topological S‐Duality in Generalized Geometry

“…This clarifies more precisely the geometrical meaning of the model of [] in terms of a Courant algebroid structure. An alternative geometric description as a certain reduction of the standard Courant sigma‐model for the target space of double field theory is discussed in [], which we study in §5. A vanishing anchor map ρ with non‐zero R ‐flux means that the bivector field π is identically zero, and the Dorfman bracket is given solely by the three‐vector R in the simple form so that the tangent bundle decouples completely from this structure.…”

“…The C‐bracket is defined on DFT vectors of , which correspond to the degree 1 functions A in the subspace spanned by : It can be obtained from derived brackets of the QP2‐manifold together with the symmetric pairing and anchor map as for DFT vectors A , A 1 and A 2 . Since the master equation does not hold for , the algebraic structure is not that of a Courant algebroid, but called a DFT algebroid in []: A DFT algebroid on is a vector bundle of rank 2 d over …”

“…[7,8,[20][21][22] A more natural framework for describing fluxes is double field theory (DFT), [23][24][25] where the original coordinates conjugate to closed string momentum modes are extended with dual coordinates conjugate to winding modes, and the continuous version of T-duality becomes a manifest symmetry of the theory (see [26][27][28] for reviews). The algebroid structure of double field theory was studied in [12,[29][30][31][32][33][34][35]; in particular, the notion of DFT algebroid was introduced in [34] whose derived bracket is the Cbracket of double field theory, and whose corresponding membrane sigma-model naturally captures the T-duality orbit of geometric and non-geometric flux backgrounds in a single unified description.…”

“…Based on our observation that the Poisson sigma‐model on doubled spaces captures both the A‐ and B‐models with different choices of the doubled Poisson structure, we propose an open AKSZ membrane sigma‐model, inspired by the approach of [] to T‐duality between geometric and non‐geometric fluxes, which gives back the doubled Poisson sigma‐model on the boundary in a specific gauge. Then we reduce the fields in a way which can be interpreted as the same reduction performed in [], where it was called a DFT projection. The resulting AKSZ membrane sigma‐model captures a particular class of generalized complex structures given by an initial Poisson and complex structure.…”

Pre‐NQ Manifolds and Correspondence Spaces: the Nilmanifold Example

“…Originating from Buscher's observation this was motivated and extensively used in supergravity . Studying dualities from the sigma‐model point of view uses total spaces of Courant algebroids as target spaces . The mathematical study of T‐duality for principal torus bundles with H ‐flux in the geometric case and beyond showed the need for continuous fields of noncommutative and nonassociative tori .…”