nLab spin connection

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Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id ⊣ id ∨ ∨ fermionic ⇉ ⊣ ⇝ bosonic ⊥ ⊥ bosonic ⇝ ⊣ Rh rheonomic ∨ ∨ reduced ℜ ⊣ ℑ infinitesimal ⊥ ⊥ infinitesimal ℑ ⊣ & étale ∨ ∨ cohesive ʃ ⊣ ♭ discrete ⊥ ⊥ discrete ♭ ⊣ ♯ continuous ∨ ∨ ∅ ⊣ * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

The term spin connection is traditionally used in physics – for instance in first-order formulation of gravity – to denote a connection on the tangent bundle of a manifold with spin structure given as a special orthogonal Lie algebra-valued connection on the underlying special orthogonal group-principal bundle. The term is mainly used to distinguish from the the connection expressed in Christoffel symbols. Since for this connection alone the spin structure is not important, the term spin connection is often used for 𝔰𝔬\mathfrak{so}-connecition on any orientable manifold.

See at field (physics) the section Ordinary gravity.

References

See also at:

Last revised on April 10, 2024 at 07:51:14. See the history of this page for a list of all contributions to it.

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