Gauged double field theory

The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analog of the Drinfel’d double of Lie algebroids. Based on a geometric realization of doubled space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure of the Drinfel’d double behind the DFT gauge symmetry. Similar to the Courant algebroid in the generalized geometry, Lagrangian sub-bundles (L,L̃) in a para-Hermitian manifold play Dirac-like structures in the Vaisman algebroid. We find that an algebraic origin of the strong constraint in DFT is traced back to the compatibility condition needed for (L,L̃) to be a Lie bialgebroid. The analysis provides a foundation toward the “coquecigrue problem” for the gauge symmetry in DFT.

Previous constructions of supersymmetry for double field theory have relied on the so-called strong constraint. In this paper, the strong constraint is relaxed and the theory is shown to possess supersymmetry once the generalised Scherk–Schwarz reduction is imposed. The equivalence between the generalised Scherk–Schwarz reduced theory and the gauged double field theory is then examined in detail for the supersymmetric theory. As a biproduct we write the generalised Killing spinor equations for the supersymmetric double field theory.

A new version of double field theory (DFT) is derived for the exactly solvable background of an in general left-right asymmetric WZW model in the large level limit. This generalizes the original DFT that was derived via expanding closed string field theory on a torus up to cubic order. The action and gauge transformations are derived for fluctuations around the generalized group manifold background up to cubic order, revealing the appearance of a generalized Lie derivative and a corresponding C-bracket upon invoking a new version of the strong constraint. In all these quantities a background dependent covariant derivative appears reducing to the partial derivative for a toroidal background. This approach sheds some new light on the conceptual status of DFT, its background (in-)dependence and the up-lift of non-geometric Scherk-Schwarz reductions.

We study the O(D, D + n) generalized metric and the gauge symmetries in the gauged double field theory (DFT) in view of current algebras and sigma models. We show that the O(D, D + n) generalized metric in the gauged DFT is consistent with the heterotic sigma models at the leading order in the α′-corrections. We then study the non-Abelian gauge symmetries and current algebras of heterotic string theories. We show that the algebras exhibit the correct diffeomorphism, the B-field gauge transformations of the background fields together with the non-Abelian gauge transformations possibly with the appropriate local Lorentz transformations.

The Double Field Theory (DFT) is formulated on a certain class of a metric algebroid, where the structure functions satisfy the pre-Bianchi identity. The derived bracket formulation of the metric algebroid by the Dirac generating operator is applied. The action is formulated by a projected generalized Lichnerowicz formula which is a generalization of the analogous formulation in gravity and supergravity. The flux of the dilaton in this setting is introduced via an ambiguity in the Dirac generating operator, while the dilaton is introduced via an ambiguity of the divergence on the metric algebroid. This talk is based on [1].

Double field theory, twistors, and integrability in 4-manifolds

Double Field Theory (DFT) has emerged as a comprehensive framework for gravity, presenting a testable and robust alternative to General Relativity (GR), rooted in the $$\textbf{O}(D,D)$$ O ( D , D ) symmetry principle of string theory. These lecture notes aim to provide an accessible introduction to DFT, structured in a manner similar to traditional GR courses. Key topics include doubled-yet-gauged coordinates, Riemannian versus non-Riemannian parametrisations of fundamental fields, covariant derivatives, curvatures, and the $$\textbf{O}(D,D)$$ O ( D , D ) -symmetric augmentation of the Einstein field equation, identified as the unified field equation for the closed string massless sector. By offering a novel perspective, DFT addresses unresolved questions in GR and enables the exploration of diverse physical phenomena, paving the way for significant future research.

The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid that are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose several realizations of pre-rackoid structures: One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the integration of the Courant algebroid when the strong constraint of DFT is imposed. Finally, for a physical application, we show an implementation of the (pre-)rackoid in a three-dimensional topological sigma model.

Double sigma model with strong constraints is equivalent to the ordinary sigma model by imposing a self-duality relation. The gauge symmetries are the diffeomorphism and one-form gauge transformation with the strong constraints. We consider boundary conditions in the double sigma model from three ways. The first way is to modify the Dirichlet and Neumann boundary conditions with a fully O(D,D) description from double gauge fields. We perform the one-loop β function for the constant background fields to find low-energy effective theory without using the strong constraints. The low-energy theory can also have O(D,D) invariance as the double sigma model. The second way is to construct different boundary conditions from the projectors. The third way is to combine the antisymmetric background field with field strength to redefine an O(D,D) generalized metric. We use this generalized metric to reconstruct a consistent double sigma model with the classical and quantum equivalence.

Seeing words in context: the interaction of lexical and sentence level information during reading

We construct a symplectic realisation of the twisted Poisson structure on the phase space of an electric charge in the background of an arbitrary smooth magnetic monopole density in three dimensions. We use the extended phase space variables to study the classical and quantum dynamics of charged particles in arbitrary magnetic fields by constructing a suitable Hamiltonian that reproduces the Lorentz force law for the physical degrees of freedom. In the source-free case the auxiliary variables can be eliminated via Hamiltonian reduction, while for non-zero monopole densities they are necessary for a consistent formulation and are related to the extra degrees of freedom usually required in the Hamiltonian description of dissipative systems. We obtain new perspectives on the dynamics of dyons and motion in the field of a Dirac monopole, which can be formulated without Dirac strings. We compare our associative phase space formalism with the approach based on nonassociative quantum mechanics, reproducing extended versions of the characteristic translation group three-cocycles and minimal momentum space volumes, and prove that the two approaches are formally equivalent. We also comment on the implications of our symplectic realisation in the dual framework of non-geometric string theory and double field theory.

This essay’s title is justified by discussing a class of Yang–Mills-type theories of which standard Yang–Mills theories are special cases but which is broad enough to include gravity as a double field theory. We use the framework of homotopy algebras, where conventional Yang–Mills theory is the tensor product K⊗g of a ‘kinematic’ algebra K with a color Lie algebra g. The larger class of Yang–Mills-type theories are given by the tensor product of K with more general Lie-type algebras, of which K itself is an example, up to anomalies that can be canceled for the tensor product with a second copy K¯. Gravity is then given by K⊗K¯.

Boundary terms in generalized geometry and doubled field theory

Double field theory yields a formulation of the low-energy effective action of bosonic string theory and half-maximal supergravities that is covariant under the T-duality group O(d, d) emerging on a torus Td. Upon reduction to three spacetime dimensions and dualisation of vector fields into scalars, the symmetry group is enhanced to O(d+1, d+1). We construct an enhanced double field theory with internal coordinates in the adjoint representation of O(d + 1, d + 1). Its section constraints admit two inequivalent solutions, encoding in particular the embedding of D = 6 chiral and non-chiral theories, respectively. As an application we define consistent generalized Scherk-Schwarz reductions using a novel notion of generalized parallelization. This allows us to prove the consistency of the truncations of D = 6, mathcal{N}=left(1,1right) and D = 6, mathcal{N}=left(2,0right) supergravity on {mathrm{AdS}}_3times {mathbb{S}}^3 .

We ask to what extent are the higher-derivative corrections of string theory constrained by T-duality. The seminal early work by Meissner tests T-duality by reduction to one dimension using a distinguished choice of field variables in which the bosonic string action takes a Gauss-Bonnet-type form. By analyzing all field redefinitions that may or may not be duality covariant and may or may not be gauge covariant we extend the procedure to test T-duality starting from an action expressed in arbitrary field variables. We illustrate the method by showing that it determines uniquely the first-order $\alpha'$ corrections of the bosonic string, up to terms that vanish in one dimension. We also use the method to glean information about the ${\cal O}(\alpha'^2)$ corrections in the double field theory with Green-Schwarz deformation.