Recent advances in the classical Double Copy (DC) procedure have revealed a profound connection between gauge theories and T-duality invariant frameworks, with Double Field Theory (the classical DC of Yang-Mills theory) emerging as the first explicit example. Extending this procedure to higher-derivative gauge theories predicts the existence of a Higher-Derivative Double Theory (HDDT), which incorporates Weyl gravity along with b-field and dilaton contributions, all in a T-duality invariant manner. In this work, we show that combining both mappings leads to DFT+, a T-duality invariant model related to the bosonic string, incorporating first-order $\alpha'$ corrections upon parameterization. Our results expand the potential applications of the DC program towards constructing perturbative $\alpha'$-corrected Lagrangians, while also opening up possibilities for reversing the map by considering the single and zeroth copies.

This article shows that the approach to generalised curvature and torsion pioneered by Polacek and Siegel [arXiv:1308.6350] is a generalisation of Cartan Geometry - rendering latter natural from the point of view of O($d$,$d$)-generalised geometry. We present this approach in the generalised metric formalism and show that almost all parts of the additional higher generalised tensors appearing in this approach correspond to covariant derivatives of the generalised Riemann tensor. As an application, we use this framework to phrase sigma model dynamics in an explicitly covariant way - both under generalised diffeomorphisms and local gauge transformations.

We follow the classical Double Copy (DC) procedure that links Yang-Mills and Double Field Theory (DFT), and we apply it on a four-derivative gauge theory which is known to be related to Weyl gravity at the level of the amplitudes. We obtain a perturbative T-duality invariant theory on a double geometry, or Conformal Double Field Theory (CDFT), incorporating Weyl gravity plus $b$-field and dilaton contributions at quadratic order, without the need to impose a gauge fixing condition. We also extend the formulation to cubic order for the case of vanishing generalized dilaton, which still incorporates Weyl gravity when $\square h_{\mu\nu} = h = 0$. CDFT, together with ordinary DFT, are examples of T-duality invariant theories constructed through classical DC maps, revealing a promising and deep connection between gauge theories and T-duality invariant models.

We discuss the bicrossproduct structure of the quantum group $\rho$-Poincaré and of the dual quantum universal enveloping algebra, expanding the construction to general Lie algebra-type deformations of Poincaré coming from classical $r$-matrices. We review the relation between different bases of the quantum universal enveloping algebra of $\rho$-Poincaré and noncommutative $\star$-products defined on the $\rho$-Minkowski spacetime, analysing some of their relevant features. Furthermore, we comment on the role of physical bases and introduce the $\rho$-Poincaré quantum Lie algebra.

We combine techniques from kinetic theory and string dualities to couple statistical matter to a non-relativistic (NR) supergravity background, enabling the theory to be formulated in a T-duality invariant form. Similarly to the relativistic case, we explicitly demonstrate that, in the NR limit, the many-strings system necessitates a viscous fluid description for the statistical matter. This is achieved using the generalized energy-momentum tensor of a perfect fluid, which remains covariant under T-duality.

The known stringy non-relativistic (NR) limit of the universal NS-NS sector of supergravity has a finite Lagrangian due to non-trivial cancellations of divergent parts coming from the metric and the $B$-field. We demonstrate that in Double Field Theory (DFT) and generalised geometry these cancellations already happen at the level of the generalised metric, which is convergent in the limit $c \rightarrow \infty$. On the level of bosonic DFT this implies that the NR limit can be taken before imposing the strong constraint and breaking the duality symmetry. At the (finite) leading order this reproduces the Non-Riemannian formulation of DFT. Assuming a convergent generalised metric for heterotic DFT, we derive a novel $c$-expansion for heterotic supergravity which is, by construction, O$(D,D)$-invariant.

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.

We study generic matter coupled to a $D$-dimensional supergravity using a formulation of Double Field Theory (DFT), where all the fields are encoded in O$(D,D)$ 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 non-dissipative deformation. Using the generalized fluid model, we revisit the issue of non-covariance of perfect fluids under T-dualities and we show how to resolve it in our DFT model with matter.

Despite remarkable success in describing supergravity reductions and backgrounds, generalized geometry and the closely related exceptional field theory are still lacking a fundamental object of differential geometry, the Riemann tensor. We explain that to construct such a tensor, an as of yet overlooked hierarchy of connections is required. They complement the spin connection with higher representations known from the tensor hierarchy of gauged supergravities. In addition to solving an important conceptual problem, this idea allows to define and explicitly construct generalized homogeneous spaces. They are the underlying structures of generalized U-duality, admit consistent truncations and provide a huge class of new backgrounds for flux compactifications with non-trivial generalized structure groups.

We study the renormalisation of a large class of integrable $\sigma$-models obtained in the framework of affine Gaudin models. They are characterised by a simple Lie algebra $\mathfrak{g}$ and a rational twist function $\varphi(z)$ with simple zeros, a double pole at infinity but otherwise no further restrictions on the pole structure. The crucial tool used in our analysis is the interpretation of these integrable theories as $\mathcal{E}$-models, which are $\sigma$-models studied in the context of Poisson-Lie T-duality and which are known to be at least one- and two-loop renormalisable. The moduli space of $\mathcal{E}$-models still contains many non-integrable theories. We identify the submanifold formed by affine Gaudin models and relate its tangent space to curious matrices and semi-magic squares. In particular, these results provide a criteria for the stability of these integrable models under the RG-flow. At one loop, we show that this criteria is satisfied and derive a very simple expression for the RG-flow of the twist function, proving a conjecture made earlier in the literature.

Using a recently developed formulation of double field theory in superspace, the graviton, $B$-field, gravitini, dilatini, and Ramond-Ramond bispinor are encoded in a single generalized supervielbein. Duality transformations are encoded as orthosymplectic transformations, extending the bosonic O($D$,$D$) duality group, and these act on all constituents of the supervielbein in an easily computable way. We first review conventional non-abelian T-duality in the Green-Schwarz superstring and describe the dual geometries in the language of double superspace. Since dualities are related to super-Killing vectors, this includes as special cases both abelian and non-abelian fermionic T-duality. We then extend this approach to include Poisson-Lie T-duality and its generalizations, including the generalized coset construction recently discussed in arXiv:1912.11036. As an application, we construct the supergeometries associated with the integrable $\lambda$ and $\eta$ deformations of the AdS$_5 \times$S$^5$ superstring. The deformation parameters $\lambda$ and $\eta$ are identified with the possible one-parameter embeddings of the supergravity frame within the doubled supergeometry. In this framework, the Ramond-Ramond bispinors are directly computable purely from the algebraic data of the supergroup.

We discuss the quantum Poincaré symmetries of the $\rho$-Minkowski spacetime, a space characterised by an angular form of noncommutativity. We show that it is possible to give them both a bicrossproduct and a Drinfel’d twist structure. We also obtain a new noncommutative $\star$-product, which is cyclic with respect to the standard integral measure.

Recent progress in generalised geometry and extended field theories suggests a deep connection between consistent truncations and dualities, which is not immediately obvious. A prime example is generalised Scherk-Schwarz reductions in double field theory, which have been shown to be in one-to-one correspondence with Poisson-Lie T-duality. Here we demonstrate that this relation is only the tip of the iceberg. Currently, the most general known classes of T-dualities (excluding mirror symmetry) are based on dressing cosets. But as we discuss, they can be further extended to the even larger class of generalised cosets. We prove that the latter give rise to consistent truncations for which the ansatz can be constructed systematically. Hence, we pave the way for many new examples of T-dualities and consistent truncations. The arising structures result in covariant tensors with more than two derivatives and we argue how they might be key to understand generalised T-dualities and consistent truncations beyond the leading two derivative level.

Recent progress in generalised geometry and extended field theories suggests a deep connection between consistent truncations and dualities, which is not immediately obvious. A prime example is generalised Scherk-Schwarz reductions in double field theory, which have been shown to be in one-to-one correspondence with Poisson-Lie T-duality. Here we demonstrate that this relation is only the tip of the iceberg. Currently, the most general known classes of T-dualities (excluding mirror symmetry) are based on dressing cosets. But as we discuss, they can be further extended to the even larger class of generalised cosets. We prove that the latter give rise to consistent truncations for which the ansatz can be constructed systematically. Hence, we pave the way for many new examples of T-dualities and consistent truncations. The arising structures result in covariant tensors with more than two derivatives and we argue how they might be key to understand generalised T-dualities and consistent truncations beyond the leading two derivative level.