Spin Manifolds and Spin Connections
Definition 1. Let be a Riemannian manifold. The Clifford bundle over is the Clifford algebra generated with respect to the inverse Riemannian metric. A left Clifford module for a compact Riemannian manifold is the finitely generated projective -module corresponding to a complex vector bundle over together wth a -homomorphism The Clifford module is self-adjoint if
Thus a Clifford module is a bimodule where the left scalar multiplication is given by
for and Hence The chirality element is given locally by with For an orientable manifold, this induces a -grading on decomposes into -eigenspaces of which in turn yields a decomposition of
Recall that an elementary C*-algebra is one that is isomorphic to a C*-algebra of compact operators on some Hilbert space and the algebra is either finite dimensional, or infinite dimensional and separable. Then for even (or odd) dimensional, (or ) is in fact a continuous field of locally trivial elementary C*-algebras. Depending on the dimension of we’ll use for the variable Clifford bundle.
The forthcoming machinery is a bit cumbersome, some I will omit proofs.
Proposition 2. A locally trivial continuous field of elementary C*-algebras determines a Cech cohomology class called a Dixmier-Douady class, such that iff
Theorem 3 (Plymen). If M is a compact Riemannian manifold, then and are Morita equivalent iff
Recall the Picard group of a C*-algebra is defined as the group under the operation
Proposition 4. Let be a compact Riemannian manifold, and Then acts freely and transitively (on the right) on
Definition 5. If is a compact orientable manifold with and Morita equivalent (i.e. ), then a pair with an orientation of and (hence in particular, it is a Clifford module) is called a spinc-structure on , and is called a spinc-manifold.
Proposition 6. Any self-adjoint Clifford module over a compact spin^c manifold is of the form for a finitely generated projective -module
If is a continuous field of elementary C*-algebras and it turns out that they can be classified by Cech classes And just as the vanishing of was equivalent to Morita equivalence of and the vanishing of has an equivalence as well.
Theorem 7. Let be a spin^c manifold with and Then iff at least one Morita equivalence bimodule admits a bijective antilinear map satisfying
Proposition 8. Any map satisfying the above equations further satisfies
Definition 9. A spin structure is a triple with a spin^c structure on and a conjugation map satisfying properties of the above theorem. is called a spin manifold. The bundle for which will be called the spinor bundle.
Proposition 10. In fact, if the spin structures of are classified by
A Hermitian connection on an -inner product space is one that satisfies
with being the universal derivation and
Theorem 11. Let be a spin manifold with spin structure Then there exists a unique Hermitian connection satisfying
for and (Note extends to by duality).
The Hermitian connection is called the spin connection of the spin manifold The spin connection may be generalized to a Clifford connection of any self-adjoint Clifford module.
Proposition 12. If for a given spin structure and is a Hermitian connection on then is a Clifford connection on Conversely, if is a self-adjoint Clifford module over a compact spin manifold and is a Clifford connection on then there is a unique Hermitian connection on such that
Definition 13. Let be a self-adjoint Clifford module over a compact Riemannian manifold with the the induced action defined by If be a Clifford connection, the generalized Dirac operator associated to is a map defined by
After applying one embeds into In the case of a compact spin manifold and spin connection the Dirac operator is defined as
Proposition 14. If is a generalized Dirac operator on a self-adjoint Clifford module and then
So the Dirac operator on a spin manifold is ultimately defined by the Riemannian metric of the manifold. A key starting point for noncommutative geometry is the ability to recover the metric from the Dirac operator alone.
First note that the spinor module is an inner product space with the inner product
where is -valued. It’s completion is the Hilbert space called the space of L^2 spinors. Now suppose that is a shortest path between points and in That is, the length
is minimized over all with and Then for we have
Hence if then Thus we’d have That is,
The inequality is in fact an equality. And we have the following.
Proposition 15. Let with a compact spin manifold. Then
Proof. We need only to show that the condition is equivalent to the condition By proposition 14 above, we have and hence
which yields the result.
A key fact regarding the Dirac operator involves its behavior on the space of L^2 spinors. Recall an operator on an inner product space is self-adjoint if
An densely-defined operator (defined on a dense subset) is essentially self-adjoint if its closure is self-adjoint.
Theorem 16. The Dirac operator on a compact spin manifold is essentially self-adjoint on the spinor space
The -grading of then extends to a -grading of We have The chirality operator anticommutes with the Dirac operator on , and this extends to an operator on where we also have
An important result is the Lichnerowicz theorem that says on a compact spin manifold with the Dirac operator, the spinor Laplacian (induced by spin connection), and the scalar curvature, we have
We also end up obtaining that
This allows us to define the integral of an element via the Dirac operator instead of the Laplacian, obtaining
This, in turn, gives a way of defining a noncommutative integral on elements of a generic algebra once a suitable Dirac operator is chosen.
 Varilly, Joeseph et al. Elements of Noncommutative Geometry.