Intrinsic Torsion Research Articles

A Class of Eternal Solutions to the G$$_{\mathbf 2}$$-Laplacian Flow

We explicitly describe the solution of the G$_2$-Laplacian flow starting from an extremally Ricci-pinched closed G$_2$-structure on a compact 7-manifold and we investigate its properties. In particular, we show that the solution exists for all real times and that it remains extremally Ricci-pinched. This result holds more generally on any 7-manifold whenever the intrinsic torsion of the extremally Ricci-pinched G$_2$-structure has constant norm. We also discuss various examples.

S^1-quotient of Spin(7)-structures

If a Spin(7)-manifold N^8 admits a free S^1 action preserving the fundamental 4-form, then the quotient space M^7 is naturally endowed with a G_2-structure. We derive equations relating the intrinsic torsion of the Spin(7)-structure to that of the G_2-structure together with the additional data of a Higgs field and the curvature of the S^1-bundle; this can be interpreted as a Gibbons–Hawking-type ansatz for Spin(7)-structures. In particular, we show that if N is a Spin(7)-manifold, then M cannot have holonomy contained in G_2 unless the N is in fact a Calabi–Yau fourfold and M is the product of a Calabi–Yau threefold and an interval. By inverting this construction, we give examples of SU(4) holonomy metrics starting from torsion-free SU(3)-structures. We also derive a new formula for the Ricci curvature of Spin(7)-structures in terms of the torsion forms. We then describe this S^1-quotient construction in detail on the Bryant–Salamon Spin(7) metric on the spinor bundle of S^4 and on flat mathbb {R}^8.

The exterior derivative of the Lee form of almost Hermitian manifolds

The exterior derivative dθ of the Lee form θ of almost Hermitian manifolds is studied. If ω is the Kähler two-form, it is proved that the Rω-component of dθ is always zero. Expressions for the other components, in [λ01,1] and in [[λ2,0]], of dθ are also obtained. They are given in terms of the intrinsic torsion. Likewise, it is described some interrelations between the Lee form and U(n)-components of the Riemannian curvature tensor.

Systematics of consistent truncations from generalised geometry

We present a generalised geometry framework for systematically constructing consistent truncations of ten- and eleven-dimensional supergravity preserving varying fractions of supersymmetry. Truncations arise when there is a reduced structure group GS of the exceptional generalised geometry, such that the intrinsic torsion is a GS -singlet. The matter content of the truncated theory follows from group-theoretical arguments, while the gauging is determined by the sub-algebra of generalised diffeomorphisms generated by the GS -singlet vectors. After discussing the general ideas across different spacetime dimensions and amounts of supersymmetry, we provide detailed formulae for truncations to gauged half-maximal supergravity in five dimensions. In particular, we establish an expression for the generalised metric on the exceptional tangent bundle, which determines the scalar truncation ansatz. As applications, we show that this formalism gives a simple derivation of a new consistent truncation of type IIB supergravity on β-deformed Lunin-Maldacena geometries, yielding half-maximal supergravity coupled to two vector multiplets, and of the truncation of eleven-dimensional supergravity on Maldacena-Núñez geometries, given by S4 twisted over a Riemann surface, which leads to half-maximal supergravity coupled to three vector multiplets.

An integral formula for Riemannian G-structures with applications to almost Hermitian and almost contact structures

For a Riemannian G-structure, we compute the divergence of the vector field induced by the intrinsic torsion. Applying the Stokes theorem, we obtain the integral formula on a closed oriented Riemannian manifold, which we interpret in certain cases. We focus on almost Hermitian and almost contact metric structures.

Supersymmeric Gauge Theory on Curved 7‐Branes

AbstractWe construct novel 7d supersymmetric gauge theories which include a Chern‐Simons‐like term on curved spaces. In order to so, we examine the supersymmetry constraints for E7‐branes in type IIA* theory, rather than making use of an off‐shell supergravity. We find two classes of solutions to the constraints, expressed in terms of a G2‐structure with non‐vanishing intrinsic torsion. The supersymmetric gauge theories are then obtained by coupling flat gauge theory to such supersymmetric backgrounds. Various examples are given, including round and squashed S7 as well as with M4 hyperkähler.

On the classification of almost contact metric manifolds

On connected manifolds of dimension higher than three, the non-existence of 132 Chinea and González-Dávila types of almost contact metric structures is proved. This is a consequence of some interrelations among components of the intrinsic torsion of an almost contact metric structure. Such interrelations allow to describe the exterior derivatives of some relevant forms in the context of almost contact metric geometry.

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 Ricci tensor of almost parahermitian manifolds

We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi-Civita connection. The formula uses the intrinsic torsion of an underlying SL(n,R)-structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As an application, we construct Einstein and Ricci-flat examples on Lie groups. We disprove the parak\"ahler version of the Goldberg conjecture, and obtain the first compact examples of a non-flat, Ricci-flat nearly parak\"ahler structure. We study the paracomplex analogue of the first Chern class in complex geometry, which obstructs the existence of Ricci-flat parak\"ahler metrics.

$${\mathrm {SU}}(4)$$ SU ( 4 ) -holonomy via the left-invariant hypo and Hitchin flow

The Hitchin flow constructs eight-dimensional Riemannian manifolds (M, g) with holonomy in $${\mathrm {Spin}}(7)$$ starting with a cocalibrated $$\mathrm {G}_2$$ -structure on a seven-dimensional manifold. As $${\mathrm {Sp}}(2)\subseteq {\mathrm {SU}}(4)\subseteq {\mathrm {Spin}}(7)$$ , one may also obtain Calabi–Yau fourfolds or hyperkahler manifolds via the Hitchin flow. In this paper, we show that the Hitchin flow on almost Abelian Lie algebras and on Lie algebras with one-dimensional commutator always yields Riemannian metrics g with $$Hol(g)\subseteq {\mathrm {SU}}(4)$$ but $$Hol(g)\ne {\mathrm {Sp}}(2)$$ . We investigate when we actually get $$Hol(g)={\mathrm {SU}}(4)$$ and obtain many new explicit examples of Calabi–Yau fourfolds. The results rely on the connection between cocalibrated $$\mathrm {G}_2$$ -structures and hypo $${\mathrm {SU}}(3)$$ -structures and between the Hitchin and the hypo flow and on a systematic study of hypo $${\mathrm {SU}}(3)$$ -structures and the hypo flow on Lie algebras. This study gives us many other interesting results: We obtain full classifications of hypo $${\mathrm {SU}}(3)$$ -structures with particular intrinsic torsion on Lie algebras. Moreover, we can exclude reducible or $${\mathrm {Sp}}(2)$$ -holonomy for the Riemannian manifolds obtained by the hypo flow on Lie algebras with initial values lying in certain intrinsic torsion classes and show that for initial values in other torsion classes we always get Riemannian metrics with holonomy equal to $${\mathrm {SU}}(4)$$ .

7-dimensional mathcal{N} = 2 consistent truncations using SL(5) exceptional field theory

We show how to construct seven-dimensional half-maximally supersymmetric consistent truncations of 11-/10-dimensional SUGRA using SL(5) exceptional field theory. Such truncations are defined on generalised SU(2)-structure manifolds and give rise to seven-dimensional half-maximal gauged supergravities coupled to n vector multiplets and thus with scalar coset space {mathbb{R}}^{+}times mathrm{O}left(3, nright)/mathrm{O}(3)times mathrm{O}(n) . The consistency conditions for the truncation can be written in terms of the generalised Lie derivative and take a simple geometric form. We show that after imposing certain “doublet” and “closure” conditions, the embedding tensor of the gauged supergravity is given by the intrinsic torsion of generalised SU(2)-connections, which for consistency must be constant, and automatically satisfies the linear constraint of seven-dimensional half-maximal gauged supergravities, as well as the quadratic constraint when the section condition is satisfied.

Pure spinors, intrinsic torsion and curvature in odd dimensions

We study the geometric properties of a (2m+1)-dimensional complex manifold M admitting a holomorphic reduction of the frame bundle to the structure group P⊂Spin(2m+1,C), the stabiliser of the line spanned by a pure spinor at a point. Geometrically, M is endowed with a holomorphic metric g, a holomorphic volume form, a spin structure compatible with g, and a holomorphic pure spinor field ξ up to scale. The defining property of ξ is that it determines an almost null structure, i.e. an m-plane distribution Nξ along which g is totally degenerate.We develop a spinor calculus, by means of which we encode the geometric properties of Nξ and of its rank-(m+1) orthogonal complement Nξ⊥ corresponding to the algebraic properties of the intrinsic torsion of the P-structure. This is the failure of the Levi-Civita connection ∇ of g to be compatible with the P-structure. In a similar way, we examine the algebraic properties of the curvature of ∇.Applications to spinorial differential equations are given. Notably, we relate the integrability properties of Nξ and Nξ⊥ to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we conjecture a Goldberg–Sachs-type theorem on the existence of a certain class of almost null structures when (M,g) has prescribed curvature.We discuss applications of this work to the study of real pseudo-Riemannian manifolds.

Electron-theoretical study on the influences of torsional deformation on electrical and optical properties of O atom absorbed graphene

The effects of torsional deformation on the structural stability, the electronic structures and the optical properties, including adsorption energy, band gap, absorption coefficient and reflectivity of O atom adsorbed graphene are studied by using the first-principles calculations. Our results indicate that the C atom closest to O atom is pulled up, causing the graphene plane to be distorted after the O atom has been adsorbed. The adsorption energy calculations show that due to the adsorption of O atom, the structural stability of graphene system decreases, but the degree of torsion has a weak effect on the structural stability. The analysis of band structure shows that the adsorption of O atom causes the graphene to convert into a semiconductor from a metal. Torsional deformation makes it change from a semiconductor to a metal, and to a semiconductor. The O atom adsorption system with a torsion angle of 12° has an indirect band gap but the band gaps of other systems are all direct bandgaps. Compared with the intrinsic graphene torsion system, the adsorbed O atom system has an electronic structure that is less sensitive to torsional deformation. When the torsion angle changes from 10° to 16°, the bandgap is always stable at around 0.11 eV. And the adsorption system always corresponds to a narrow bandgap semiconductor in this torsion angle range. For optical properties, comparing with the O atoms adsorbed on graphene with the 0° torsion angle, the peaks of the absorption coefficient and the reflectivity of the system are reduced, and have a transform of red shift into blue shift in a torsion angle ranging from 2° to 20°.

Generalised structures for N = 1 $$ \mathcal{N}=1 $$ AdS backgrounds

We expand upon a claim made in a recent paper [arXiv:1411.5721] that generic minimally supersymmetric AdS backgrounds of warped flux compactifications of Type II and M theory can be understood as satisfying a straightforward weak integrability condition in the language of $E_{d(d)}\times\mathbb{R}^+$ generalised geometry. Namely, they are spaces admitting a generalised $G$-structure set by the Killing spinor and with constant singlet generalised intrinsic torsion.

On morphoelastic rods

Morphoelastic rods are thin bodies which can grow and can change their intrinsic curvature and torsion. We deduce a system of equations that rule accretion and remodeling in a morphoelastic rod by combining balance laws involving non-standard forces with constitutive prescriptions filtered by a dissipation principle that takes into account both standard and non-standard working. We find that, as in the theory of three-dimensional bulk growth proposed [DiCarlo, A and Quiligotti, S. Mech Res Commun 2002; 29: 449–456], it is possible to identify a universal coupling mechanism between stress and growth, conveyed by an Eshelbian driving force.

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.

Intrinsic torsion in quaternionic contact geometry

We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in seven dimensions, when this condition is equivalent to integrability in the sense of Duchemin. We prove that the choice of a reduction to Sp(n)H^* (or equivalently, a complement of the qc distribution) yields a unique K-connection satisfying natural conditions on torsion and curvature. We show that the choice of a compatible metric on the qc distribution determines a canonical reduction to Sp(n)Sp(1) and a canonical Sp(n)Sp(1)-connection whose curvature is almost entirely determined by its torsion. We show that its Ricci tensor, as well as the Ricci tensor of the Biquard connection, has an interpretation in terms of intrinsic torsion.

Supersymmetric backgrounds and generalised special holonomy

We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form . Using the language of generalised geometry, we show that, for , preserving minimal supersymmetry is equivalent to the manifold M having generalised special holonomy and list the relevant holonomy groups. We conjecture that this result extends to backgrounds preserving any number of supersymmetries. As a prime example, we consider in D = 4. The corresponding generalised special holonomy group is , giving the natural M theory extension to the notion of a G2 manifold, and, for Type II backgrounds, reformulating the pure spinor conditions as an integrable structure.

Cosmology in Poincaré gauge gravity with a pseudoscalar torsion

A cosmology of Poincare gauge theory is developed, where several properties of universe corresponding to the cosmological equations with the pseudoscalar torsion function are investigated. The cosmological constant is found to be the intrinsic torsion and curvature of the vacuum universe and is derived from the theory naturally rather than added artificially, i.e. the dark energy originates from geometry and includes the cosmological constant but differs from it. The cosmological constant puzzle, the coincidence and fine tuning problem are relieved naturally at the same time. By solving the cosmological equations, the analytic cosmological solution is obtained and can be compared with the LCDM model. In addition, the expressions of density parameters of the matter and the geometric dark energy are derived, and it is shown that the evolution of state equations for the geometric dark energy agrees with the current observational data. At last, the full equations of linear cosmological perturbations and the solutions are obtained.

Pure spinors, intrinsic torsion and curvature in even dimensions

We study the geometric properties of a 2m-dimensional complex manifold M admitting a holomorphic reduction of the frame bundle to the structure group P⊂Spin(2m,C), the stabiliser of the line spanned by a pure spinor at a point. Geometrically, M is endowed with a holomorphic metric g, a holomorphic volume form, a spin structure compatible with g, and a holomorphic pure spinor field ξ up to scale. The defining property of ξ is that it determines an almost null structure, i.e. an m-plane distribution Nξ along which g is totally degenerate.We develop a spinor calculus, by means of which we encode the geometric properties of Nξ corresponding to the algebraic properties of the intrinsic torsion of the P-structure. This is the failure of the Levi-Civita connection ∇ of g to be compatible with the P-structure. In a similar way, we examine the algebraic properties of the curvature of ∇.Applications to spinorial differential equations are given. In particular, we give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. We also conjecture a Goldberg–Sachs-type theorem on the existence of a certain class of almost null structures when (M,g) has prescribed curvature.We discuss applications of this work to the study of real pseudo-Riemannian manifolds.