T-duality with H-flux: Non-commutativity, T-folds and structure

Abstract

Various approaches to T-duality with NSNS three-form flux are reconciled. Non-commutative torus fibrations are shown to be the open-string version of T-folds. The non-geometric T-dual of a three-torus with uniform flux is embedded into a generalized complex six-torus, and the non-geometry is probed by D0-branes regarded as generalized complex submanifolds. The noncommutativity scale, which is present in these compactifications, is given by a holomorphic Poisson bivector that also encodes the variation of the dimension of the world-volume of D-branes under monodromy. This bivector is shown to exist in SU(3) × SU(3) structure compactifications, which have been proposed as mirrors to NSNS-flux backgrounds. The two SU(3)-invariant spinors are generically not parallel, thereby giving rise to a non-trivial Poisson bivector. Furthermore we show that for non-geometric T-duals, the Poisson bivector may not be decomposable into the tensor product of vectors. IntroductionCompactifications with H-flux are known to give rise to topology changes and even to nongeometric situations when T-duality is performed along directions which have non-trivial support of the NSNS H-flux [1,2,3,4,5,6,7,8]. Non-geometry occurs for example in the very simple situation of a three-torus endowed with an H-flux proportional to its volume form. Consider namely the three-torus as a trivial T 2 -fibration over a circle. Upon T-duality along the fibre, the metric picks up a factor that makes it shrink under monodromy around the base circle. The monodromy around the base is a non-trivial element of the O(2, 2; Z) group acting on the two-torus. This prevents a three-dimensional global Riemannian description from existing. Further T-dualizing along the base leads to more pathological situations, where points do not exist even in a local coordinate patch, and the fibres are conjectured to become non-associative [9,10]. We will restrict ourselves to the case of two T-dualities, and assume that local coordinate patches do exist. Progress in the description of non-associative T-duals was achieved in the recent paper [11], which also contains observations on the open-string metric and non-commutativity for two T-dualities that have some overlap with ours.Essentially three conjectures have been put forward for the description of the T-dual of a torus with H-flux: 1 (I) Field of non-commutative tori: Mathai and Rosenberg proposed that T-dualizing along a two-torus with non-zero H-flux yields a fibration by (or more precisely: field of) noncommutative tori. In particular, this fibration is encoded in a closed one-form, which is obtained by integrating the NSNS flux along the fibre directions [7,12,13].(II) T-folds: these are spaces where T-dualities can act as transition functions between local patches [8]. The T-dualized directions are doubled, and T-duality transformations may patch the doubled fibres together. A sigma model with a T-fold as its target space was proposed, and its boundary conditions were studied in [14,15,16,17,18].(III) G × G struc...

A ten-dimensional action for non-geometric fluxes

“…Hereg C andφ C are both well-defined, so the non-geometricity only shows up in the non-trivial β C , whose form was first derived in [12]. The non-zero components ofQ C are (Q C ) z xy = −(Q C ) z yx = 1.…”

“…10 It can be argued, following in particular the T-duality chain (2.1) and the toroidal example, that such non-geometric situations would lead to (at least) an Rflux 11 while the locally geometric situation should give a Q-flux. 12 As we will discuss further, we will not consider R-fluxes in this paper, so we restrict ourselves to the T-fold case. The T-fold construction is related to the mathematical set-up of Generalized Complex Geometry (GCG) [21,22], which provides another way to consider a space of doubled dimension.…”

“…They are only defined up to [12,60,5,23,24] that the appearance of such a β block can be a sign of non-geometry. Before giving details on this point, let us mention that such a bivector first appeared in a particular…”

“…The NSNS degrees of freedom would be the only ones involved, but they would be mixed into new variables which would reveal non-geometric fluxes in the action. To find the good variables, we were inspired by the Generalized Complex Geometry (GCG) [21,22] approach, where a particular object has been related [12,5,23,24] to non-geometry: the antisymmetric bivector β mn . We come back to these ideas in detail in section 2.…”

Nonassociative geometry and twist deformations in non-geometric string theory

“…In this instance the metric and B-field are well-defined locally but not globally; in particular, the T 2 -fibre is glued back to itself by an SL(2, Z) transformation as one winds around the base S 1 . In [51,28,16] it is argued that this non-geometry at the topological level can be regarded globally as a fibration over S 1 by noncommutative two-tori T 2 θ with the commutation relations [x i , x j ] = i Q i j k x k among local coordinates. Here θ i j (x) = Q i j k x k determine a field of local noncommutativity parameters parametrized by the coordinates of the base S 1 ; this bivector is naturally dual to the B-field which is a potential for the original H-flux in the T-duality chain.…”