| Cartan connection | |
|---|---|
| Name | Cartan connection |
| Caption | Schematic depiction of a Cartan geometry modeled on a homogeneous space |
| Field | Differential geometry |
| Introduced by | Élie Cartan |
| Introduced | Early 20th century |
Cartan connection
A Cartan connection is a geometric structure that generalizes the notion of an affine or principal connection by modelling a manifold locally on a homogeneous space. In the context of Quantum Physics, Cartan connections provide a natural language for coupling internal gauge symmetries with spacetime geometry, underpinning approaches to geometric quantization of constrained systems and offering geometric frameworks for theories such as Chern–Simons theory and formulations of gravity used in attempts at quantum gravity.
Cartan connections arise from the program of Élie Cartan to study manifolds by comparison with a model homogeneous space G/H where G is a Lie group and H a closed subgroup. In physics this viewpoint aligns with the use of symmetry groups such as the Poincaré group, de Sitter group, and internal gauge groups like SU(2), SU(3), or U(1). The ability of Cartan geometry to encode both local gauge fields and curvature in a single object makes it relevant to the geometric formulation of quantum field theories studied at institutions such as CERN and research programs in loop quantum gravity and string theory.
Formally, a Cartan connection on a principal H-bundle P → M is a Lie algebra–valued one-form ω ∈ Ω^1(P, 𝔤) satisfying equivariance, reproducing fundamental vector fields, and providing a pointwise linear isomorphism between the tangent space of P and 𝔤. Here 𝔤 = Lie(G) and 𝔥 = Lie(H). The curvature of ω is the 2-form Ω = dω + 1/2[ω,ω], analogous to the curvature of a principal connection but interpreted relative to the model homogeneous space. Key mathematical references and developments include work by Charles Ehresmann, Cartan geometry, and modern expositions in texts by R.W. Sharpe and papers by Kobayashi and Nomizu.
Typical physical models use Cartan connections with G chosen as the Poincaré group (or its covers) and H as the Lorentz group, yielding formulations equivalent to Einstein–Cartan theory where torsion couples to spinors. In three dimensions, Cartan connections underlie the equivalence between 3D gravity and Chern–Simons theory with gauge group chosen as ISO(2,1), SO(3,1), or SL(2,R). In gauge theory contexts, Cartan geometry clarifies the relationship between gauge symmetry breaking patterns and emergent geometry, relevant to models explored in Condensed matter physics and topological phases studied at centers like the Perimeter Institute.
Cartan connections contribute to geometric quantization by providing canonical one-forms and symplectic structures on associated bundles and moduli spaces. The Cartan form can be used to construct prequantum line bundles and moment maps for group actions, connecting to works by Kostant and Souriau on prequantization. In the context of phase space formulations, Cartan geometric language aids in describing reduced phase spaces of gauge systems, moduli of flat connections (important in Chern–Simons theory and topological quantum field theory), and aspects of quantization on curved backgrounds encountered in semiclassical approximations and path integral methods advocated in research at Perimeter Institute and Institute for Advanced Study.
Cartan connections play a central role in first-order formulations of gravity such as Palatini formulation and Einstein–Cartan theory, where the connection and frame field (vielbein) appear as independent fields. These formulations are foundational to quantization attempts: canonical quantization approaches like loop quantum gravity use connection variables (e.g., Ashtekar variables) whose geometric interpretation relates to Cartan-type structures; path integral approaches to spin foam models also exploit Cartan-geometric data. In high-energy theory, generalized Cartan connections appear in gauge/gravity correspondences and in attempts to encode supersymmetry via super-Cartan geometries tied to supergravity and supersymmetry.
Computational techniques for Cartan connections involve representation theory of Lie algebras (e.g., highest-weight methods for SU(n)), cohomological methods (e.g., BRST/BV formalisms), and discretizations used in numerical relativity and lattice gauge theory. Discrete analogues such as Regge calculus and spin network/spin foam representations approximate Cartan data by group elements on edges and faces; these are implemented in software frameworks developed in mathematical physics groups at CERN, Max Planck Institute for Gravitational Physics (Albert Einstein Institute), and university research groups. Symbolic computation for Cartan geometry often leverages packages for differential forms and Lie algebra calculations available in systems like SageMath and Mathematica.
Active research questions concern rigorous quantization of Cartan-geometric gravity, the role of torsion in quantum regimes, and classification of Cartan connections with applications to anomaly cancellation and topological phases. Connections to twistor theory, holographic dualities (e.g., AdS/CFT correspondence), and incorporation of matter fields in Cartan frameworks remain under study. Bridging continuum Cartan geometry with discrete and computational models to produce testable predictions in quantum gravity and condensed-matter analogues is an ongoing interdisciplinary endeavor involving researchers at Perimeter Institute, CERN, Institute for Advanced Study, and major universities.
Category:Differential geometry Category:Mathematical physics