Cartan Geometry & Generalised Structures

• Cartan-geometric generalised geometry is a unified algebraic and differential framework for encoding geometric structures via Cartan connections on model spaces.
• It extends classical geometries by incorporating duality groups and tensor hierarchies to address symmetry in physics and moduli problems.
• Applications include exceptional field theory, sigma model dynamics, and integrated geometric structures with implications in moduli stabilization and current algebra.

The Cartan-geometric framework for generalised geometries provides a unified algebraic and geometric language for understanding geometric structures via Cartan connections and their analogues, extending beyond classic Riemannian, symplectic, or complex geometries. Modern developments reformulate both the foundational geometric PDEs and new symmetry principles as Cartan-style connections and curvature, leading to deep connections with duality-invariant field theories, moduli problems, and the geometry underlying physical field theories such as exceptional field theory, double field theory, and mechanics on homogeneous spaces.

1. Structural Foundations of Cartan-Geometric Generalised Geometry

At its core, Cartan geometry models a manifold’s infinitesimal structure on a homogeneous space $G/H$, encoding curvature and torsion via a $\mathfrak{g}$-valued Cartan connection $\omega$ on a principal $H$-bundle. The modern generalisation extends this by allowing the model to be a more intricate homogeneous or even quasi-homogeneous space: for example, a bundle or an extended algebraic structure with higher symmetries or duality groups, such as $O(d,d)$, $E_{n(n)}$, or their generalisations in exceptional field theory.

Generalised geometries in this context are constructed either as reductions of the structure group on generalised tangent bundles—such as $TM \oplus T^*M$ for $O(d,d)$-generalised geometry—or as explicit extensions involving the structure and automorphism groups of the models (e.g., "skeletons" in the sense of $(\mathfrak{t},L,p)$, or groupoids with multiplicative 1-forms) (Gregorovič, 2016, Cattafi, 2019, Wang, 2015, Hassler et al., 30 Aug 2024).

A quintessential framework is provided by the notion of a Cartan bundle, where a geometric structure is packaged as a principal bundle with a vector-valued 1-form satisfying certain properties that ensure surjectivity or injectivity at each point, and possible involutivity or bracket-compatibility, thus encompassing both classical Cartan geometries and more general $G$-structures (Cattafi, 2019).

2. Generalised Cartan Calculus and Tensor Hierarchies

A substantial extension of traditional Cartan calculus arises in the context of exceptional and double field theory, where one requires a calculus compatible with the hierarchical gauge structure (tensor hierarchy) of supergravity. In these settings, one introduces a generalised exterior derivative $\widehat{\partial}$, a generalised Lie derivative $\mathcal{L}_A$, and a “magic formula”: $$\mathcal{L}_A X = A \cdot \widehat{\partial} X + \widehat{\partial}(A \cdot X)$$ This calculus is closed only under a strong "section condition" linking the extra coordinates' derivatives, enforced via invariant $Z$-tensors that encode the failure of naive closure (Wang, 2015).

The tensor hierarchy is constructed recursively through these generalised operators: 1-form gauge fields are paired with 2-form field strengths, but these field strengths fail to close until 2-form potentials are introduced, and so on. The generalized Poincaré lemma is subtle—exactness can depend on the choice of solution to the section condition (e.g., M-theory versus type IIB solution), leading to nontrivial local cohomology properties in exceptional field theory.

In more algebraic settings, such as with differential graded Lie algebras, this recursive structure is realised as a brane current algebra or “phase space Poisson structure of $p$-branes”, systematically constructing higher torsion and curvature tensors via an infinite generation ladder (Hassler et al., 4 Sep 2025).

3. Generalised Connections, Curvature, and Torsion

A key insight is that Cartan's classical concept of curvature,

$\Theta = -d\theta + \tfrac{1}{2}[\theta, \theta]$

generalises to the setting of $T \oplus T^*M$ or higher generalised tangent objects: here, the generalised Cartan connection may be realised as a map $\theta:(T \oplus T^*)P\to \mathfrak{d}$, with $\mathfrak{d}$ a relevant (quasi-)Lie bialgebra carrying $O(d,d)$ or larger symmetry (Hassler et al., 30 Aug 2024).

The corresponding generalised curvature tensor then encodes both the familiar pieces (tension, Riemann, Ricci) and hierarchies of higher-order tensors. For example, in the generalised metric formalism, the torsion $\mathcal{T}_{KL,M}$ (the antisymmetric part of the generalised connection) and the symmetric projection $\mathcal{R}_{KL,MN}$ (generalised Riemann) may appear as distinct pieces of the full Cartan curvature. Crucially, many of the additional higher generalised curvature tensors, such as the “$\rho$-torsion” $\mathcal{A}$ and the five-index tensor $\Xi_{P,KL,MN}$, can often be written (after imposing necessary compatibility conditions or equations of motion) as covariant derivatives of the generalised Riemann tensor, though some components probe the undetermined, shift-symmetric part of the connection (Hassler et al., 30 Aug 2024).

4. Model Spaces and Example Geometries

Several concrete models illustrate the Cartan-geometric generalisation principle:

• Generalised Geometries from Differential Forms: On the cotangent bundle $T^*\mathcal{B}$ of a 2D base, pairs of forms $(\Omega, \alpha)$ (with $\alpha \wedge \Omega = 0$ and Pfaffian conditions) encode Monge–Ampère (M-A) equations. These induce “generalised almost structures” (endomorphisms on $T\mathcal{M}\oplus T^*\mathcal{M}$ squaring to $\pm$Id depending on the sign of $\mathrm{Pf}(\alpha)$), and the possible structure space arising from $(\Omega,\alpha)$ and associated anticommuting structures is naturally parameterised by quadric surfaces in $\mathbb{R}^3$ (Suchánek, 2023). Integrability conditions (e.g., closedness of $\alpha/\sqrt{|\mathrm{Pf}(\alpha)|}$) play the role of vanishing torsion.
• Exceptional Generalised Geometry: In $D=11$ supergravity and related constructions, structure is built upon the coset $E_{7(7)}/SU(8)$ using a 56-bein $\mathcal{V}$, generalised affine connections with composite and flux terms, and a generalised curvature tensor. These encode both geometric and $\mathcal{G}$-gauge (e.g., dual graviton) fluxes. The embedding tensor of gauged supergravity arises as a projection of the higher-dimensional generalised connection (Godazgar et al., 2014).
• Lightlike Cartan Geometry: Geometries modelled on the future lightlike cone of Minkowski spacetime exhibit degenerate metrics $h$ with radical vector field $Z$, with additional geometric data encoded in tractor bundles and their decompositions. These "lightlike Cartan geometries" expand the tradition of parabolic/tractor calculus outside the parabolic or reductive context, with canonical correspondences to data such as families of metric linear connections on screen bundles and bundle morphisms encoding the extra structure (Palomo, 2020, Morón et al., 27 Aug 2025).
• Mechanics on Homogeneous Spaces and Cartan Media: By viewing a field’s configuration as a map into a homogeneous space $\mathbb{X}$ on which a Lie group $G$ acts transitively ($q:W\to\mathbb{X}$, with $W$ the base), kinematics and dynamics are encoded via the Maurer–Cartan form and variational principles. Generalised strains and velocities (i.e., Lie algebra-valued fields) encode the geometry, and field equations can be recovered via Lagrange–d'Alembert or reduced Euler–Poincaré principles. This framework yields, for example, the continuum mechanics of Cosserat media, rods, and surfaces as specific cases, and leads to structure-preserving numerical integrators exploiting the group geometry (Kikuchi et al., 2023).

5. Intrinsic Holonomy, Automorphisms, and Decomposition Theorems

The Cartan-geometric viewpoint generalises holonomy and the structures associated to product decompositions:

• Intrinsic Holonomy and Curved Cosets: For any Cartan geometry $(G,H,\omega)$, the intrinsic holonomy group is defined in terms of paths in the principal bundle and the "development" in the model group $G$. Structures called "curved cosets" generalise the notion of left cosets and subgroups in the Cartan setting, and their projections yield corresponding decompositions of the base manifold (Erickson, 2021).
• De Rham Theorem Generalisation: For manifolds whose Cartan connection curvature "splits" in an appropriate sense, the product decomposition of the model group (at the Lie algebra level) lifts to a product decomposition of the Cartan geometry, paralleling the classical de Rham theorem for Riemannian manifolds.
• Automorphism Groups: By lifting automorphisms through skeleton extensions or groupoid viewpoints, the computation of automorphism groups, including infinitesimal automorphisms realised as parallel sections with respect to bundle connections, becomes tractable. This leads to new descriptions of moduli, isomorphism classes, and classification, as in, for example, the classification of Riemannian metrics sharing the same Levi-Civita connection or the moduli of metrics with maximal symmetry algebra in teleparallel geometries (Gregorovič, 2016, McNutt et al., 5 Jan 2024).

6. Holomorphic, Branched, and Complex Cartan Geometries

The holomorphic and complex analytic generalisation of Cartan geometry brings additional richness:

• Branched Holomorphic Cartan Geometries: Structures where the Cartan connection degenerates on a divisor extend the scope of geometric structures to all compact complex projective manifolds, each admitting a branched holomorphic projective structure, and under strong rigidity results (e.g., on Calabi–Yau manifolds) force flatness or uniqueness (Biswas et al., 2019).
• Atiyah Sequences and Flatness: Interpreting the Cartan connection as an inclusion within the Atiyah exact sequence of holomorphic vector bundles provides links to the theory of connections, stability conditions, and vanishing of Chern classes, which in turn influence global properties and rigidity.

7. Applications and Implications in Mathematical Physics

The Cartan-geometric framework translates directly into modern physics:

• Duality-Invariant Theories: The generalised Cartan connections underpin exceptional field theory, double field theory, and the geometric formulation of string and M-theory. In O$(d,d)$-generalised geometry, torsion and Riemann tensors (defined via generalised Cartan curvature) become objects with manifest duality-covariance (Godazgar et al., 2014, Hassler et al., 30 Aug 2024).
• Sigma Model Dynamics: The phase-space reformulation of classical current algebras using generalised Cartan connections allows sigma model dynamical equations to be written in a manifestly generalised-diffeomorphism and local gauge-covariant form, with the master equation for the Hamiltonian flow expressed in terms of the generalised curvature tensor (Hassler et al., 30 Aug 2024).
• Moduli Stabilisation and Brane Physics: In supergravity and flux compactifications, the generalised Cartan-geometric viewpoint gives a systematic algebraic language for moduli stabilisation, duality-covariant embedding tensors, and tensor hierarchies, making contact with the underlying physical origin of gauged symmetries, as well as the current algebras arising from $p$-brane phase spaces (Hassler et al., 4 Sep 2025).

Summary Table: Generalised Structures in the Cartan-Geometric Framework

Framework/component	Key structural ingredients	Reference Example(s)
Generalised Cartan Calculus	Generalised Lie derivative, $\widehat{\partial}$, tensor hierarchy, section condition	(Wang, 2015)
Exceptional/generalised connection	Duality group valued connections/curvature, embedding tensor	(Godazgar et al., 2014, Hassler et al., 30 Aug 2024)
Generalised almost structures	Endomorphisms on $T\mathcal{M}\oplus T^*\mathcal{M}$, parameterised by quadric equations	(Suchánek, 2023)
Lie groupoid/Cartan bundle formulation	Multiplicative vector-valued 1-forms, global groupoid language	(Cattafi, 2019)
Lightlike Cartan geometry	Lightlike metric, radical field, tractor bundles, non-reductive structure	(Palomo, 2020, Morón et al., 27 Aug 2025)
Moduli and automorphism theory	Skeleton extension, groupoid automorphisms, Cartan-Karlhede algorithm	(Gregorovič, 2016, McNutt et al., 5 Jan 2024)
Mechanics on homogeneous spaces	Maurer-Cartan form, geometric action principle, Cosserat media	(Kikuchi et al., 2023)

The wide-ranging scope of the Cartan-geometric approach to generalised geometry ensures its continued relevance, unifying much of contemporary mathematics and physics under a robust, functorial framework that clarifies both geometric and symmetry-based aspects of complex structures, field theories, and moduli classification problems.