$$ \mathcal{N} $$ = 2 consistent truncations from wrapped M5-branes

We discuss consistent truncations of eleven-dimensional supergravity on a six-dimensional manifold M, preserving minimal $$ \mathcal{N} $$ N = 2 supersymmetry in five dimensions. These are based on GS ⊆ USp(6) structures for the generalised E6(6) tangent bundle on M, such that the intrinsic torsion is a constant GS singlet. We spell out the algorithm defining the full bosonic truncation ansatz and then apply this formalism to consistent truncations that contain warped AdS5×wM solutions arising from M5-branes wrapped on a Riemann surface. The generalised U(1) structure associated with the $$ \mathcal{N} $$ N = 2 solution of Maldacena-Nuñez leads to five-dimensional supergravity with four vector multiplets, one hypermultiplet and SO(3) × U(1) × ℝ gauge group. The generalised structure associated with “BBBW” solutions yields two vector multiplets, one hypermultiplet and an abelian gauging. We argue that these are the most general consistent truncations on such backgrounds.

On certain classes of $$\mathbf{Sp}(4,\mathbb{R})$$ symmetric $$G_2$$ structures

We find two different families of $$\mathbf{Sp}(4,\mathbb{R})$$ Sp ( 4 , R ) symmetric $$G_2$$ G 2 structures in seven dimensions. These are $$G_2$$ G 2 structures with $$G_2$$ G 2 being the split real form of the simple exceptional complex Lie group $$G_2$$ G 2 . The first family has $$\tau _2\equiv 0$$ τ 2 ≡ 0 , while the second family has $$\tau _1\equiv \tau _2\equiv 0$$ τ 1 ≡ τ 2 ≡ 0 , where $$\tau _1$$ τ 1 , $$\tau _2$$ τ 2 are the celebrated $$G_2$$ G 2 -invariant parts of the intrinsic torsion of the $$G_2$$ G 2 structure. The families are different in the sense that the first one lives on a homogeneous space $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_l$$ Sp ( 4 , R ) / SL ( 2 , R ) l , and the second one lives on a homogeneous space $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_s$$ Sp ( 4 , R ) / SL ( 2 , R ) s . Here $$\mathbf{SL}(2,\mathbb{R})_l$$ SL ( 2 , R ) l is an $$\mathbf{SL}(2,\mathbb{R})$$ SL ( 2 , R ) corresponding to the $$\mathfrak{sl}(2,\mathbb{R})$$ sl ( 2 , R ) related to the long roots in the root diagram of $$\mathfrak{sp}(4,\mathbb{R})$$ sp ( 4 , R ) , and $$\mathbf{SL}(2,\mathbb{R})_s$$ SL ( 2 , R ) s is an $$\mathbf{SL}(2,\mathbb{R})$$ SL ( 2 , R ) corresponding to the $$\mathfrak{sl}(2,\mathbb{R})$$ sl ( 2 , R ) related to the short roots in the root diagram of $$\mathfrak{sp}(4,\mathbb{R})$$ sp ( 4 , R ) .

Generalised cosets

Recent work has shown that two-dimensional non-linear σ-models on group manifolds with Poisson-Lie symmetry can be understood within generalised geometry as exemplars of generalised parallelisable spaces. Here we extend this idea to target spaces constructed as double cosets M = $$ \tilde{G} $$ G ˜ \𝔻/H. Mirroring conventional coset geometries, we show that on M one can construct a generalised frame field and a H -valued generalised spin connection that together furnish an algebra under the generalised Lie derivative. This results naturally in a generalised covariant derivative with a (covariantly) constant generalised intrinsic torsion, lending itself to the construction of consistent truncations of 10-dimensional supergravity compactified on M . An important feature is that M can admit distinguished points, around which the generalised tangent bundle should be augmented by localised vector multiplets. We illustrate these ideas with explicit examples of two-dimensional parafermionic theories and NS5-branes on a circle.

The Role of Spin(9) in Octonionic Geometry

Starting from the 2001 Thomas Friedrich’s work on Spin ( 9 ) , we review some interactions between Spin ( 9 ) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin ( 9 ) canonical 8-form and its analogies with quaternionic geometry as well as the role of Spin ( 9 ) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin ( 9 ) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley–Rosenfeld planes and to three series of Grassmannians.

The Role of Spin(9) in Octonionic Geometry

Starting from Thomas Friedrich’s work “Weak Spin(9) structures on 16-dimensional Riemannian manifolds”, we review several interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry, the role of Spin(9) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley-Rosenfeld planes and to three series of Grassmannians.

An analytic cosmology solution of Poincaré gauge gravity

A cosmology of Poincaré gauge theory is developed. An analytic solution is obtained. The calculation results agree with observation data and can be compared with the [Formula: see text]CDM model. The cosmological constant puzzle is the coincidence and fine tuning problem are solved naturally at the same time. The cosmological constant turns out to be the intrinsic torsion and curvature of the vacuum universe, and is derived from the theory naturally rather than added artificially. The dark energy originates from geometry, includes the cosmological constant but differs from it. The analytic expression of the state equations of the dark energy and the density parameters of the matter and the geometric dark energy are derived. The full equations of linear cosmological perturbations and the solutions are obtained.