Cartan Geometry Research Articles

The classification of general affine connections in Newton–Cartan geometry: Towards metric-affine Newton–Cartan gravity

Abstract We give a full classification of general affine connections on Galilei manifolds in terms of independently specifiable tensor fields. This generalises the well-known case of (torsional) Galilei connections, i.e. connections compatible with the metric structure of the Galilei manifold. Similarly to the well-known pseudo-Riemannian case, the additional freedom for connections that are not metric-compatible lies in the covariant derivatives of the two tensors defining the metric structure (the clock form and the space metric), which however are not fully independent of each other.

P -brane Galilean and Carrollian geometries and gravities

We study D-dimensional p-brane Galilean geometries via the intrinsic torsion of the adapted connections of their degenerate metric structure. These non-Lorentzian geometries are examples of G-structures whose characteristic tensors consist of two degenerate ‘metrics’ of ranks (p+1) and (D−p−1) . We carry out the analysis in two different ways. In one way, inspired by Cartan geometry, we analyse in detail the space of intrinsic torsions (technically, the cokernel of a Spencer differential) as a representation of G, exhibiting for generic (p, D) five classes of such geometries, which we then proceed to interpret geometrically. We show how to re-interpret this classification in terms of ( D−p−2 )-brane Carrollian geometries. The same result is recovered by methods inspired by similar results in the physics literature: namely by studying how far an adapted connection can be determined by the characteristic tensors and by studying which components of the torsion tensor do not depend on the connection. As an application, we derive a gravity theory with underlying p-brane Galilean geometry as a non-relativistic limit of Einstein–Hilbert gravity and discuss how it gives a gravitational realisation of some of the intrinsic torsion constraints found in this paper. Our results also have implications for gravity theories with an underlying ( D−p−2 )-brane Carrollian geometry.

A new gravitational theory and dark matter problem

We propose a new gravitational theory with torsion based on Riemann–Cartan geometry, in which all physical quantities are dynamical. In addition to the spacetime metric, the gravitational degrees of freedom in this theory also include the torsion and two scalar fields. The energy-momentum tensor of the matter fields in this theory is also proposed. A spherically symmetric static vacuum solution of the theory is obtained. It turns out that this solution can fit the observational data of the rotation curve outside the stellar disk in the Milky Way. Therefore, the galactic dark matter may just be the gravitational effect of the theory with torsion.

Groups of basic automorphisms of chaotic Cartan foliations with Eresmann connection

The purpose of the work is to study the groups of basic automorphisms of chaotic Cartan foliations with Ehresmann connection. Cartan foliations form a category where automorphisms preserve not only the foliation, but also its transverse Cartan geometry. The group of basic automorphisms of a foliation is the quotient group of the group of all automorphisms of this foliation by the normal subgroup of leaf automorphisms with respect to which each leaf is invariant. Cartan foliations include such wide classes of foliations as pseudo-Riemannian, Lorentzian, and foliations with transversal affine connection. No restrictions are imposed on the dimension of either the foliation or the foliated manifold. Compactness of the foliated manifold is not assumed. Methods. The proof of the structure theorem for chaotic Cartan foliations is based on the application of the foliated bundle construction, commonly used in the theory of foliations with transverse geometries. Results. The main result of this paper is the theorem stating that the group of basic automorphisms of any chaotic Cartan foliation with Ehresmann connection admits the structure of a Lie group and finding estimates for the dimension of this group. In particular, it is proved that if the set of closed leaves is countable, then the group of basic automorphisms of such a foliation is countable. Conclusion. In this paper, we prove a criterion according to which the chaoticity of a Cartan foliation of type (G, H) is equivalent to the chaoticity of a locally free action of the group H on the associated parallelizable manifold. Thus, the problem of the existence of chaos in Cartan foliations with Ehresmann connection reduces to the same problem for locally free actions of a Lie group on parallelizable manifolds.

Combined Screw and Wedge Dislocations

Elastic media with defects are considered manifold with nontrivial Riemann–Cartan geometry in the geometric theory of defects. We obtain the solution of three-dimensional Euclidean general relativity equations with an arbitrary number of linear parallel sources. It describes elastic media with parallel combined wedge and screw dislocations.

Linearity of Cartan and Wasserstein means

The convex cone of positive definite Hermitian matrices has two important Riemannian geometries, where the parametrized weighted geometric (alternatively, Cartan) and Wasserstein means appear as the corresponding geodesics. The major problem with which this paper is concerned is linearity of Cartan and Wasserstein geodesics; when the Cartan (resp. Wasserstein) geodesic between two positive definite matrices A and B does lie in the space spanned by them and what the path in the plane to which it corresponds is. We settle the problem completely for the Cartan geometry and partially for the Wasserstein geometry. We show that their linearity problems for linearly independent A and B are equivalent to the solvability of the following equations for positive reals x and y with xy<14 respectively(xA+yB)−1=yA−1+xB−1,(AB)1/2+(BA)1/22=xA+yB.

On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class

The fundamental invariants for vector ODEs of order $\ge 3$ considered up to point transformations consist of generalized Wilczynski invariants and C-class invariants. An ODE of C-class is characterized by the vanishing of the former. For any fixed C-class invariant ${\mathcal U}$, we give a local (point) classification for all submaximally symmetric ODEs of C-class with ${\mathcal U} \not \equiv 0$ and all remaining C-class invariants vanishing identically. Our results yield generalizations of a well-known classical result for scalar ODEs due to Sophus Lie. Fundamental invariants correspond to the harmonic curvature of the associated Cartan geometry. A key new ingredient underlying our classification results is an advance concerning the harmonic theory associated with the structure of vector ODEs of C-class. Namely, for each irreducible C-class module, we provide an explicit identification of a lowest weight vector as a harmonic 2-cochain.

Revisiting the electronic properties of disclinated graphene sheets

The interplay between topological defects, such as dislocations or disclinations, and the electronic degrees of freedom in graphene have been extensively studied. In the literature, for the study of this kind of problems, it is in general used either a gauge theory or a curved spatial Riemannian geometry approach, where, in the geometric case, the information about the defects is contained in the metric and the spin-connection. However, these topological defects can also be associated with a Riemann–Cartan geometry where curvature and torsion plays an important role. In this article, we study the interplay between a wedge disclination in a planar graphene sheet and the properties of its electronic degrees of freedom. Our approach relies on its relation with elasticity theory through the so called elastic-gauge, where their typical coefficients, as for example the Poisson’s ratio, appear directly in the metric, and consequently also in the electronic spectrum.

Super Cartan Geometry and the Super Ashtekar Connection

This work is devoted to the geometric approach to supergravity. More precisely, we interpret mathcal {N}=1, D=4 supergravity as a super Cartan geometry which provides a link between supergravity and Yang–Mills theory. To this end, we first review important aspects of the theory of supermanifolds and we establish a link between various different approaches. We then introduce super Cartan geometries using the concept of so-called enriched categories. This, among other things, will enable us to implement anticommutative fermionic fields. We will then also show that non-extended D=4 supergravity naturally arises in this framework. Finally, using this gauge-theoretic interpretation as well as the chiral structure of the underlying supersymmetry algebra, we will derive graded analoga of Ashtekar’s self-dual variables and interpret them in terms of generalized super Cartan connections. This gives canonical chiral supergravity the structure of a Yang–Mills theory with gauge supergroup similar to the self-dual variables in ordinary first-order Einstein gravity. We then construct the parallel transport map corresponding to the super connection in mathematical rigorous way using again enriched categories. This provides the possibility of quantizing gravity and matter degrees of freedom in loop quantum gravity in a unified way.Kindly check and confirm inserted city and country name are correctly identified.City and country name are correct.

Quantum state reduction, and Newtonian twistor theory

We discuss the equivalence principle in quantum mechanics in the context of Newton–Cartan geometry, and non-relativistic twistor theory.

Edge Modes and Dressing Fields for the Newton–Cartan Quantum Hall Effect

It is now well-known that Newton–Cartan theory is the correct geometrical setting for modelling the quantum Hall effect. In addition, in recent years edge modes for the Newton–Cartan quantum Hall effect have been derived. However, the existence of these edge modes has, as of yet, been derived using only orthodox methodologies involving the breaking of gauge-invariance; it would be preferable to derive the existence of such edge modes in a gauge-invariant manner. In this article, we employ recent work by Donnelly and Freidel in order to accomplish exactly this task. Our results agree with known physics, but afford greater conceptual insight into the existence of these edge modes: in particular, they connect them to subtle aspects of Newton–Cartan geometry and pave the way for further applications of Newton–Cartan theory in condensed matter physics.

The gauging procedure and carrollian gravity

We discuss a gauging procedure that allows us to construct lagrangians that dictate the dynamics of an underlying Cartan geometry. In a sense to be made precise in the paper, the starting datum in the gauging procedure is a Klein pair corresponding to a homogeneous space. What the gauging procedure amounts to is the construction of a Cartan geometry modelled on that Klein geometry, with the gauge field defining a Cartan connection. The lagrangian itself consists of all gauge-invariant top-forms constructed from the Cartan connection and its curvature. After demonstrating that this procedure produces four-dimensional General Relativity upon gauging Minkowski spacetime, we proceed to gauge all four-dimensional maximally symmetric carrollian spaces: Carroll, (anti-)de Sitter-Carroll and the lightcone. For the first three of these spaces, our lagrangians generalise earlier first-order lagrangians. The resulting theories of carrollian gravity all take the same form, which seems to be a manifestation of model mutation at the level of the lagrangians. The odd one out, the lightcone, is not reductive and this means that although the equations of motion take the same form as in the other cases, the geometric interpretation is different. For all carrollian theories of gravity we obtain analogues of the Gauss-Bonnet, Pontryagin and Nieh-Yan topological terms, as well as two additional terms that are intrinsically carrollian and seem to have no lorentzian counterpart. Since we gauge the theories from scratch this work also provides a no-go result for the electric carrollian theory in a first-order formulation.

Carrollian manifolds and null infinity: a view from Cartan geometry

We discuss three different (conformally) Carrollian geometries and their relation to null infinity from the unifying perspective of Cartan geometry. Null infinity per se comes with numerous redundancies in its intrinsic geometry and the two other Carrollian geometries can be recovered by making successive choices of gauge. This clarifies the extent to which one can think of null infinity as being a (strongly) Carrollian geometry and we investigate the implications for the corresponding Cartan geometries. The perspective taken, which is that characteristic data for gravity at null infinity are equivalent to a Cartan geometry for the Poincaré group, gives a precise geometrical content to the fundamental fact that ‘gravitational radiation is the obstruction to having the Poincaré group as asymptotic symmetries’.

Cartan F(R) Gravity and Equivalent Scalar–Tensor Theory

We investigate the Cartan formalism in F(R) gravity. F(R) gravity has been introduced as a theory to explain cosmologically accelerated expansions by replacing the Ricci scalar R in the Einstein–Hilbert action with a function of R. As is well-known, F(R) gravity is rewritten as a scalar–tensor theory by using the conformal transformation. Cartan F(R) gravity is described based on the Riemann–Cartan geometry formulated by the vierbein-associated local Lorenz symmetry. In the Cartan formalism, the Ricci scalar R is divided into two parts: one derived from the Levi–Civita connection and the other from the torsion. Assuming the spin connection-independent matter action, we have successfully rewritten the action of Cartan F(R) gravity into the Einstein–Hilbert action and a scalar field with canonical kinetic and potential terms without any conformal transformations. red Thus, symmetries in Cartan F(R) gravity are clearly conserved. The resulting scalar–tensor theory is useful in applications of the usual slow-roll scenario. As a simple case, we employ the Starobinsky model and evaluate fluctuations in cosmological microwave background radiation.

Aspects of Nonrelativistic Strings

We review recent developments on nonrelativistic string theory. In flat spacetime, the theory is defined by a two-dimensional relativistic quantum field theory with nonrelativistic global symmetries acting on the worldsheet fields. This theory arises as a self-contained corner of relativistic string theory. It has a string spectrum with a Galilean dispersion relation, and a spacetime S-matrix with nonrelativistic symmetry. This string theory also gives a unitary and ultraviolet complete framework that connects different corners of string theory, including matrix string theory and noncommutative open strings. In recent years, there has been a resurgence of interest in the non-Lorentzian geometries and quantum field theories that arise from nonrelativistic string theory in background fields. In this review, we start with an introduction to the foundations of nonrelativistic string theory in flat spacetime. We then give an overview of recent progress, including the appropriate target-space geometry that nonrelativistic strings couple to. This is known as (torsional) string Newton–Cartan geometry, which is neither Lorentzian nor Riemannian. We also give a review of nonrelativistic open strings and effective field theories living on D-branes. Finally, we discuss applications of nonrelativistic strings to decoupling limits in the context of the AdS/CFT correspondence.

Geometric theory of Weyl structures

Given a parabolic geometry on a smooth manifold [Formula: see text], we study a natural affine bundle [Formula: see text], whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on [Formula: see text], which induces an almost bi-Lagrangian structure on [Formula: see text] and a compatible linear connection on [Formula: see text]. We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with nonzero scalar curvature, provided that the parabolic geometry is torsion-free and [Formula: see text]-graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in [Formula: see text]. For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. We also show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge–Ampere equation and thus to properly convex projective structures.

Deformation theory of holomorphic Cartan geometries, II

Abstract In this continuation of [4], we investigate the deformations of holomorphic Cartan geometries where the underlying complex manifold is allowed to move. The space of infinitesimal deformations of a flat holomorphic Cartan geometry is computed. We show that the natural forgetful map, from the infinitesimal deformations of a flat holomorphic Cartan geometry to the infinitesimal deformations of the underlying flat principal bundle on the topological manifold, is an isomorphism.

Imprints from a Riemann–Cartan space-time on the energy levels of Dirac spinors

In this work, we investigate the effects of the torsion–fermionic interaction on the energy levels of fermions within a Riemann–Cartan geometry using a model-independent approach. We consider the case of fermions minimally coupled to the background torsion as well as non-minimal extensions via additional couplings with the vector and axial fermionic currents which include parity-breaking interactions. In the limit of zero-curvature, and for the cases of constant and spherically symmetric torsion, we find a Zeeman-like effect on the energy levels of fermions and anti-fermions depending on whether they are aligned/anti-aligned with respect to the axial vector part of the torsion (or to specific combination of torsion quantities), and determine the corresponding fine-structure energy transitions. We also discuss non-minimal couplings between fermionic fields and torsion within the Einstein–Cartan theory and its extension to include the (parity-breaking) Holst term. Finally we elaborate on the detection of torsion effects related to the splitting of energy levels in astrophysics, cosmology and solid state physics using current capabilities.

Cones and Cartan geometry

We show that the extended principal bundle of a Cartan geometry of type (A(m,R),GL(m,R)), endowed with its extended connection ωˆ, is isomorphic to the principal A(m,R)-bundle of affine frames endowed with the affine connection as defined in classical Kobayashi-Nomizu volume I.Then we classify the local holonomy groups of the Cartan geometry canonically associated to a Riemannian manifold. It follows that if the holonomy group of the Cartan geometry canonically associated to a Riemannian manifold is compact then the Riemannian manifold is locally a product of cones.

Disclinations in the Geometric Theory of Defects

In the geometric theory of defects, media with a spin structure, for example, ferromagnet, is considered as a manifold with given Riemann--Cartan geometry. We consider the case with the Euclidean metric corresponding to the absence of elastic deformations but with nontrivial ${\mathbb S}{\mathbb O}(3)$-connection which produces nontrivial curvature and torsion tensors. We show that the 't Hooft--Polyakov monopole has physical interpretation in solid state physics describing media with continuous distribution of dislocations and disclinations. The Chern--Simons action is used for the description of single disclinations. Two examples of point disclinations are considered: spherically symmetric point "hedgehog" disclination and the point disclination for which the $n$-field has a fixed value at infinity and essential singularity at the origin. The example of linear disclinations with the Franc vector divisible by $2\pi$ is considered.