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In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, a Clifford module bundle, a bundle of Clifford modules or just Clifford module is a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
whose fibers are
Clifford module In mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra ''C'' is a central simple algebra over some field extension ''L'' of the field ''K'' over which the quadratic form ''Q'' defining ''C'' is d ...
s, the representations of
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
s. The canonical example is a
spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
. In fact, on a
Spin manifold In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical ...
, every Clifford module is obtained by twisting the spinor bundle. The notion "Clifford module bundle" should not be confused with a
Clifford bundle In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian ...
, which is a bundle of Clifford algebras.


Spinor bundles

Given an oriented Riemannian manifold ''M'' one can ask whether it is possible to construct a bundle of
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
Clifford module In mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra ''C'' is a central simple algebra over some field extension ''L'' of the field ''K'' over which the quadratic form ''Q'' defining ''C'' is d ...
s over ''Cℓ''(''T''*''M''). In fact, such a bundle can be constructed if and only if ''M'' is a
spin manifold In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical ...
. Let ''M'' be an ''n''-dimensional spin manifold with
spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical ...
''F''Spin(''M'') → ''F''SO(''M'') on ''M''. Given any ''Cℓ''''n''R-module ''V'' one can construct the associated
spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
:S(M) = F_(M) \times_\sigma V\, where σ : Spin(''n'') → GL(''V'') is the representation of Spin(''n'') given by left multiplication on ''S''. Such a spinor bundle is said to be ''real'', ''complex'', ''graded'' or ''ungraded'' according to whether on not ''V'' has the corresponding property. Sections of ''S''(''M'') are called
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
s on ''M''. Given a spinor bundle ''S''(''M'') there is a natural bundle map :C\ell(T^*M) \otimes S(M) \to S(M) which is given by left multiplication on each fiber. The spinor bundle ''S''(''M'') is therefore a bundle of
Clifford module In mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra ''C'' is a central simple algebra over some field extension ''L'' of the field ''K'' over which the quadratic form ''Q'' defining ''C'' is d ...
s over ''Cℓ''(''T''*''M'').


See also

*
Orthonormal frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
*
Spin representation In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equ ...
*
Spin geometry In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in math ...


Notes


References

* * Riemannian geometry Structures on manifolds Clifford algebras Vector bundles {{differential-geometry-stub