DFT is constructed from the idea to incorporate the properties of T-duality into a field theory Conserved momentum and winding quantum numbers have associated coordinates in T d Double all coordinates Every object in a duality invariant theory must belong to some representation of the duality group. In particular, x i have to be supplemented with X M fundamental rep. O(D,D) Raise and lower indices with the O(D,D) metric Introduce doubled fields and write with manifest global O(D,D) symmetry DOUBLE FIELD THEORY
Field content Focus on bosonic universal gravity sector G ij, B ij, Fields are encoded in a 2D × 2D GENERALIZED METRIC, O(D,D) INVARIANT GENERALIZED DILATON
The generalized metric spacetime action Hull and Zwiebach (2009) O. Hohm, C. Hull and B. Zwiebach (2010) DFT also has a gauge invariance generated by a pair of parameters Gauge invariance and closure of the gauge algebra lead to a set of differential constraints that restrict the theory. In particular, the constraints can be solved enforcing a stronger condition named strong constraint O(D,D) symmetry is manifest
Strong constraint All fields, gauge parameters and products of them satisfy It implies there is some dual frame where fields are not doubled Strongly constrainted DFT displays the O(D,D) symmetry but it is not physically doubled Gauge invariance and closure of gauge transformations weaker condition Certain backgrounds allow relaxations of the strong constraint, producing a truly doubled theory: – Massive type IIA O. Hohm, S. Kwak (2011) – Suggested by Scheck-Schwarz compactifications of DFT G. Aldazabal, W. Baron, D. Marqués, C.N. (2011) D. Geissbhuler (2011) – Sufficient but not necessary for gauge invariance and closure of gauge algebra M. Graña, D. Marqués (2012) – Explicit double solutions found in D. Geissbhuler, D. Marqués, C.N., V. Penas (2013)
Applications of DFT DFT has been a powerful tool to explore string theoretical features beyond supergravity and Riemanian geometry Some recent developments include: – Geometric interpretation of non-geometric gaugings in flux compactifications of string theory G. Aldazabal, W. Baron, D. Marqués, C.N. (2011) D. Geissbhuler (2011) – Identification of new geometric structures D. Andriot, R. Blumenhagen, O. Hohm, M. Larfors, D. Lust, P. Patalong (2011, 2012) – Description of exotic brane orbits F. Hassler, D., Lust (2013) J. de Boer, M. Shigemori (2010, 2012), T. Kikuchi, T. Okada, Y. Sakatani (2012) – Non-commutative/non-associative structures in closed string theory R. Blumenhagen, E. Plauschinn, D. Andriot, C. Condeescu, C. Floriakis, M. Larfors, D. Lust, P. Patalong (2010-2012) – New perspectives on ‘ corrections, O. Hohm, W. Siegel, B. Zwiebach (2012,2013) – New possibilities for upliftings, moduli fixing and dS vacua, Roest et al. (2012)
10D string sugra SS reduction on twisted T 6 4D gauged sugra T-duality 4D gauged sugra geometric fluxes in d-dim all dual (geometric & ab c & H abc non- geometric) gaugings Moduli fixing & dS vacua T-duality in D-dim Double Field Theory SS reduction on twisted T 6,6 ??? Application I: Missing gaugings in geometric compactifications (see Aldazabal’s talk)
Application II: New geometric structures Non geometry, Generalized Geometry Diffeomorphisms of GR and gauge transformations of 2-form are combined in generalized diffeomorphisms and Lie derivatives ; New term needed so that Gauge transformations The action of the generalized metric formulation is gauge invariant because R( H, d ) is a generalized scalar under the strong constraint
DFT vs Generalized Geometry The double geometry underlying DFT differs from ordinary geometry. DFT is a small departure from Generalized Geometry (Hitchin, 2003; Gualtieri, 2004) Given a manifold M, GG puts together vectors V i and one-forms i as V + TM T*M. Structures on this larger space The Courant bracket generalizes the Lie bracket V and are not treated symmetrically DFT puts TM and T*M on similar footing by doubling the underlying manifold. Gauge parameters and then C-bracket For non-doubled M the C-bracket reduces to the Courant bracket
Geometry, connections and curvature The action was tendentiously written as It can be shown that the action and EOM of DFT can be obtained from traces and projections of a generalized Riemann tensor R MNPQ The construction goes beyond Riemannian geometry because it is based on generalized rather than standard Lie derivatives The notions of connections, torsion and curvature have to be generalized E.g. the vanishing torsion and compatibility conditions do not completely determine the connections and curvatures, but only fix some of their projections I. Jeon, K. Lee, J. Park (2011), O. Hohm, B. Zwiebach (2012) Strong constraint was assumed in these constructions. Can it be relaxed?
Basic fields are generalized vielbeins E A M and dilaton E A M can be parametrized in terms of vielbein of D-dimensional metric D-dimensional Minkowski metric Arrange the fields in dynamical fluxes: Field dependent and non-constant fluxes, that give rise to gaugings or constant fluxes upon compactification (e.g. F abc =H abc ) Flux formulation of DFT W. Siegel (1993) D. Geissbhuler, D. Marqués, C.N., V. Penas (2013)
Vanishes under strong constraint Generalized metric DFT action modulo one strong constraint violating term The action The action takes the form of the electric sector of the scalar potential of N=4 D=4 gauged supergravity This action generalizes the generalized metric formulation, including all terms that vanish under the strong constraint
Generalized diffeomorphisms The closure constraints (generalized Lie derivatives generate closed transformations) take the form: and they asure that F ABC, F A transform as scalars and S is gauge invariant Imposing these conditions only requires a relaxed version of strong constraint the theory admits truly double fields Constraints can be interpreted as Bianchi identities for generalized Riemann tensor
Geometric formulation of DFT Define covariant derivative on tensors Determine the connections imposing set of conditions: – Compatibility with generalized frame: – Compatibility with O(D,D) invariant metric – Compatibility with generalized metric – Covariance under generalized diffeomorphisms: – Covariance under double Lorentz transformations: Lorentz scalar – Vanishing generalized torsion: Standard torsion non covariant – Compatibility with generalized dilaton Only determine some projections of the connections
Generalized curvature The standard Riemann tensor in planar indices is not a scalar under generalized diffeomorphisms It can be modified adding new terms, leading to Projections with give and similarly EOM Bianchi identities
Scherk-Schwarz solutions All the constraints can be solved restricting the fields and gauge parameters as where and quadratic constraints of N=4 gauged sugra For these configurations all the consistency constraints are satisfied. The dynamical fluxes become: This ansatz contains the usual decompactified strong contrained case (U=1, =0, x i, i=1,…, D ). It is a particular limit in which all the compact dimensions are decompactified.
Conclusions Presented formulation of DFT in terms of dynamical and field dependent fluxes. The gauge consistency constraints take the form of quadratic constraints for the fluxes, that admit solutions that violate the SC allows to go beyond supergravity Computed connections and curvatures on the double space under assumption that covariance is achieved upon generalized quadratic constraints, rather than SC, which can be interpreted as BI. Interestingly, this procedure gives rise to all the SC-violating terms in the action, which are gauge invariant and appear systematically This completes the original formulation of DFT, incorporating the missing terms that allow to make contact with half-maximal gauged sugra, containing all duality orbits of non-geometric fluxes (F ABC F ABC ).
Open questions Some elements of the O(D,D) geometry have been understood, but it is important to better understand the geometry underlying DFT Can this construction be extended beyond tori? Calabi-Yau? ’ corrections. Inner product and C-bracket are corrected deformation of Courant bracket and other structures in GG Beyond T-duality? U-duality? Relation between DFT and string theory. Is this a consistent truncation of string theory? No massive states, but fully consistent Worldsheet theory?