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Concept
Signature operator
Summary
In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four. It is an instance of a Dirac-type operator.
Let be a compact Riemannian manifold of even dimension . Let
be the exterior derivative on -th order differential forms on . The Riemannian metric on allows us to define the Hodge star operator and with it the inner product
on forms. Denote by
the adjoint operator of the exterior differential . This operator can be expressed purely in terms of the Hodge star operator as follows:
Now consider acting on the space of all forms .
One way to consider this as a graded operator is the following: Let be an involution on the space of all forms defined by:
It is verified that anti-commutes with and, consequently, switches the -eigenspaces of
Consequently,
Definition: The operator with the above grading respectively the above operator is called the signature operator of .
In the odd-dimensional case one defines the signature operator to be acting
on the even-dimensional forms of .
If , so that the dimension of is a multiple of four, then Hodge theory implies that:
where the right hand side is the topological signature (i.e. the signature of a quadratic form on defined by the cup product).
The Heat Equation approach to the Atiyah-Singer index theorem can then be used to show that:
where is the Hirzebruch L-Polynomial, and the the Pontrjagin forms on .
Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.
Official source
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