D-branes and doubled geometry

Abstract

We define the open string version of the nonlinear sigma model on doubled geometry introduced by Hull and Reid-Edwards, and derive its boundary conditions. These conditions include the restriction of D-branes to maximally isotropic submanifolds as well as a compatibility condition with the Lie algebra structure on the doubled space. We demonstrate a systematic method to derive and classify D-branes from the boundary conditions, in terms of embeddings both in the doubled geometry and in the physical target space. We apply it to the doubled three-torus with constant H-flux and find D0-, D1-, and D2-branes, which we verify transform consistently under T-dualities mapping the system to f -, Q-and R-flux backgrounds.We also require the Neumann projector to be integrable, so that it locally defines the brane as a smooth submanifold of the target space,The projectors are moreover required to be orthogonal with respect to the doubled metric M IJ , 0 = Ξ I K M IJ Ξ

D-brane Wess-Zumino terms and U-duality

“…Together they transform as a vector of the T-duality group SO(10 − D, 10 − D). This doubling of compactified dimensions is typical for doubled geometry [32][33][34][35] but now applied to the worldvolume of the D-branes in a curved background. In the same way that in dimensions lower than ten the Wess-Zumino term of the D = 10…”

Open exceptional strings and D-branes

“…In the IIB case, they are conjugate to winding modes of the F1 string and Dp branes with p odd: in fact the F1 and D1 winding coordinates appear together as the SL(2) doubletỸ i α . There will also be coordinates conjugate to winding modes of the NS5 brane, Kaluza-Klein monopole, and (for high enough d) other "exotic" branes, denoted by the ellipsis in (7). The non-zero components of the charge, assuming the standard 10-dimensional solutions of the section condition, are always:…”

“…Hull's paper [4], and further studied in [6,7]. We now follow this approach and apply it to the exceptional sigma model (1).…”

“…As in [6,7], we can also require the Dirichlet and Neumann projectors to be orthogonal with respect to the generalised metric, and that the Neumann subbundle is integrable,…”

“…T-duality relates Dp-branes to D(p ± 1) branes, interchanging Neumann and Dirichlet boundary conditions on the string worldsheet. If one uses the doubled approach to the string worldsheet [1][2][3][4][5], an elegant picture emerges whereby all Dp-branes can be viewed as a single D-dimensional brane in the 2D-dimensional doubled target space: this can then intersect with the D-dimensional physical subspace in different number of directions in order to reproduce all standard p-branes [4,6,7]. In generalised geometry, [8,9], which underlies reformulations of supergravity such as [10] and the related formalism of double field theory (DFT) where the spacetime coordinates are doubled [11][12][13], this translates into the statement that D-branes are maximally isotropic subspaces of the doubled tangent bundle [9,14].…”