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Signature operator

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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.

Contents

Definition in the even-dimensional case

Let M be a compact Riemannian manifold of even dimension 2 l . Let

d : Ω p ( M ) → Ω p + 1 ( M )

be the exterior derivative on i -th order differential forms on M . The Riemannian metric on M allows us to define the Hodge star operator ⋆ and with it the inner product

⟨ ω , η ⟩ = ∫ M ω ∧ ⋆ η

on forms. Denote by

d ∗ : Ω p + 1 ( M ) → Ω p ( M )

the adjoint operator of the exterior differential d . This operator can be expressed purely in terms of the Hodge star operator as follows:

d ∗ = ( − 1 ) 2 l ( p + 1 ) + 2 l + 1 ⋆ d ⋆ = − ⋆ d ⋆

Now consider d + d ∗ acting on the space of all forms Ω ( M ) = ⨁ p = 0 2 l Ω p ( M ) . One way to consider this as a graded operator is the following: Let τ be an involution on the space of all forms defined by:

τ ( ω ) = i p ( p − 1 ) + l ⋆ ω , ω ∈ Ω p ( M )

It is verified that d + d ∗ anti-commutes with τ and, consequently, switches the ( ± 1 ) -eigenspaces Ω ± ( M ) of τ

Consequently,

d + d ∗ = ( 0 D D ∗ 0 )

Definition: The operator d + d ∗ with the above grading respectively the above operator D : Ω + ( M ) → Ω − ( M ) is called the signature operator of M .

Definition in the odd-dimensional case

In the odd-dimensional case one defines the signature operator to be i ( d + d ∗ ) τ acting on the even-dimensional forms of M .

Hirzebruch Signature Theorem

If l = 2 k , so that the dimension of M is a multiple of four, then Hodge theory implies that:

i n d e x ( D ) = s i g n ( M )

where the right hand side is the topological signature (i.e. the signature of a quadratic form on H 2 k ( M )   defined by the cup product).

The Heat Equation approach to the Atiyah-Singer index theorem can then be used to show that:

s i g n ( M ) = ∫ M L ( p 1 , … , p l )

where L is the Hirzebruch L-Polynomial, and the p i   the Pontrjagin forms on M .

Homotopy invariance of the higher indices

Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.

References

Signature operator Wikipedia
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