Duality covariant curvature and torsion tensors in double field theory/generalized geometry are central in analyzing consistent truncations, generalized dualities, and related integrable $\textit{σ}$-models. They are constructed systematically with the help of a larger, auxiliary space in a procedure inspired by Cartan geometry originally proposed by Poláček and Siegel for bosonic strings. It pivots around a maximally isotropic group that captures the generalized structure group of the physical space. We show how dropping the isotropy condition on this group allows us to describe heterotic/type I strings. As an immediate application, we construct a new family of heterotic backgrounds that interpolates between the two-dimensional cigar and trumpet backgrounds.

Poland

Falk Hassler

David Osten

Japan

Yuho Sakatani

Duality covariant curvatures for the heterotic string

We define generalized dualities for heterotic and type I strings based on consistent truncations to half-maximal gauged supergravities in more than three dimensions. The latter are constructed from a generalized Scherk-Schwarz ansatz in heterotic double field theory that satisfies the strong constraint. Necessary and sufficient conditions on the resulting embedding tensor are discussed, showing that only certain gaugings, called geometric, can arise from this procedure. For all of them, we explicitly construct the internal geometry and gauge potentials. In general, this construction is not unique and permits different uplifts which are used to define generalized T-duality. Two examples are worked out underlying the utility of our approach to explore new dualities and uplifts of half-maximal gauged supergravities.

Luca Scala

Generalized dualities for heterotic and type I strings

Despite remarkable success in describing supergravity reductions and backgrounds, generalized geometry and exceptional field theory are still lacking a fundamental object of differential geometry, the Riemann tensor. We show that to construct it, a hierarchy of connections is required. They complement the spin connection with higher representations known from the tensor hierarchy. This approach allows to define generalized homogeneous spaces which underlie generalized U-duality, admit consistent truncations and provide a huge class of new flux backgrounds with nontrivial structure groups.

Hierarchy of curvatures in exceptional geometry

We study generic matter coupled to a <math><mi>D</mi></math>-dimensional supergravity using a formulation of double field theory (DFT), where all the fields are encoded in <math><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>,</mo><mi>D</mi><mo>)</mo></mrow></math> multiplets. We study both the case when the matter comes from a variational principle, as well as the case where the matter comes from a statistical or thermodynamic approach. For the latter, we construct the distribution function for the perfect fluid and its entropy current, which is a conserved quantity. We then include general viscous and elastic terms in the generalized energy-momentum tensor which, in the general case, lead to entropy production. We consistently deform the conservation law of the generalized entropy current and identify a particular nondissipative deformation. Using the generalized fluid model, we revisit the issue of noncovariance of perfect fluids under T-dualities and we show how to resolve it in our DFT model with matter.

Croatia

Eric Lescano

Argentina

Nahuel Mirón-Granese

<math><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>,</mo><mi>D</mi><mo>)</mo></mrow></math>-covariant formulation of perfect and imperfect fluids in the double geometry

We study Poisson-Lie T-duality of the Wess-Zumino-Novikov-Witten (WZNW) models which are obtained from a class of Drinfel’d doubles and its generalization. In this case, the resultant WZNW models are known to be classically self-dual under Poisson-Lie T-duality. We describe an explicit construction of the associated currents, and discuss the conformal invariance under this duality. In a concrete example of the SU(2) WZNW model, we find that the self-duality is represented as a chiral automorphism of the   <math> <mover> <mi>su</mi> <mo>̂</mo> </mover> </math>  $$ \hat{\mathfrak{su}} $$ (2) affine Lie algebra, though the transformation of the currents is non-local and non-linear. This classical automorphism can be promoted to the quantum one through the parafermionic formulation of   <math> <mover> <mi>su</mi> <mo>̂</mo> </mover> </math>  $$ \hat{\mathfrak{su}} $$ (2), which in turn induces an isomorphism of the WZNW model. We thus find a full quantum equivalence of the dual pair under Poisson-Lie T-duality. The isomorphism is represented by a sign-change of a chiral boson or the order-disorder duality of the parafermionic conformal field theory as in Abelian T-duality on tori or in the mirror symmetry of the Gepner model.

Yuji Satoh

On quantum Poisson-Lie T-duality of WZNW models

Abstract Generalized parallelizable spaces permit the uplift of many maximal gauged supergravities to ten or eleven dimensions. Although some of the former are explicitly known, the literature is still lacking their systematic construction. We resolve this issue by going beyond the known existence proofs and present the explicit construction of all generalized parallelizable spaces for maximal gauged supergravities in four or more dimensions. In the process, all embedding tensors that can be realized without breaking the section condition of exceptional field theory are identified. This addresses the old question: “Which maximal gauged supergravities can be embedded in the low-energy limit of superstring/M-theory” and provides the basis to explore solution-generating techniques based on generalized dualities.

All maximal gauged supergravities with uplift

Abstract The Poisson–Lie (PL) T-duality is a generalized T-duality based on the Lie algebra of the Drinfel’d double. In particular, when we consider the PL T-duality of a coset space, the dual space is found to be a generalized coset space, which is called the dressing coset. We investigate an extension of the dressing cosets to the U-duality setup. We propose the gauged actions for various branes in M-theory and type IIB theory, where the generalized metric is constructed by using the exceptional Drinfel’d algebra (EDA) and the gauge algebra is a certain isotropic subalgebra of the EDA. By eliminating the gauge fields, the gauged action reduces to the standard brane action on a certain reduced background, which we call the exceptional dressing coset. We also propose an alternative definition of the exceptional dressing cosets based on Sfetsos’s approach and reproduce a known example of non-Abelian T-duality in the U-duality-covariant formulation.

Shozo Uehara

Gauged sigma models and exceptional dressing cosets

Abstract The Poisson–Lie T-plurality is an equivalence of string theories on various cosets , where  is a Drinfel’d double and ,  are maximal isotropic subgroups. This can be extended to the equivalence for dressing cosets, i.e., , where F is an isotropic subgroup of . We explore this extended Poisson–Lie T-plurality, clarifying the relation between several previous approaches. We propose a gauged sigma model for a general gauge group F and obtain the formula for the metric and the B-field on the dressing coset. Using this formula and an ansatz for the dilaton, we show that the Poisson–Lie (PL) T-plurality for dressing cosets (with spectator fields) is a symmetry of double field theory. The formula for the Ramond–Ramond field strength is also proposed such that the equations of motion for the Neveu–Schwarz–Neveu–Schwarz fields are transformed covariantly. In addition, we provide specific examples of the PL T-plurality for dressing cosets.

Poisson–Lie -plurality for Dressing Cosets

Abstract The extended Drinfel’d algebra (ExDA) is the underlying symmetry of non-Abelian duality in the low-energy effective theory of string theory. Non-Abelian U-dualities in maximal supergravities have been studied well, but there has been no study on non-Abelian dualities in half-maximal supergravities. We construct the ExDA for half-maximal supergravities in d ≥ 4. We also find an extension of the homogeneous classical Yang–Baxter equation in these theories.

Half-maximal extended Drinfel’d algebras

Non-Abelian <math><mi>U</mi></math> duality originates from the construction of exceptional Drinfel’d algebra (EDA), which extends the constriction of the classical Drinfel’d double. This symmetry is a natural extension of Poisson–Lie <math><mi>T</mi></math> duality and is believed to be a symmetry of type II string/M-theory or their low-energy effective theories. In this paper, we consider non-Abelian <math><mi>U</mi></math> dualities of 11- or 10-dimensional backgrounds starting with, <math><msub><mi>E</mi><mrow><mi>n</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msub></math> EDA with <math><mi>n</mi><mo>≤</mo><mn>6</mn></math> with vanishing trombone gauging. The latter guarantees that all dual backgrounds satisfy the standard supergravity equations of motion. In particular, when the duality includes a timelike <math><mi>T</mi></math> duality, we obtain solutions of <math><msup><mrow><mi>M</mi></mrow><mo>*</mo></msup></math>-theory or type <math><msup><mrow><mi>II</mi></mrow><mo>*</mo></msup></math> background equations, as expected. Also starting with coboundary EDA’s we provide examples of generalized Yang–Baxter deformations of M-theory and type IIB backgrounds. The obtained results provide explicit examples when non-Abelian <math><mi>U</mi></math> duality works well as a solution generating transformation.