Signature operator explained

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.

Definition in the even-dimensional case

Let

M be a compact Riemannian manifold of even dimension 2 l . Let d : \Omega ^{p(M) →} \Omega ^{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 \star and with it the inner product\langle\omega, η \rangle = \int _{M\omega\wedge\starη}

on forms. Denote by

d ^{*:} \Omega ^{p+1} (M) → \Omega ^{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 ) ^{2l(p+1)} \star d \star = - \star d \star

Now consider

d + d ^{*} acting on the space of all forms \Omega(M) = o plus ^{2 l} _{p =0} \Omega ^{p} (M) .One way to consider this as a graded operator is the following: Let \tau be an involution on the space of all forms defined by:\tau(\omega) = i ^{p(p-1)+l} \star \omega , \omega \in \Omega ^{p(M)}

It is verified that

d + d ^{*} anti-commutes with \tau and, consequently, switches the (\pm 1 ) -eigenspaces \Omega _{\pm} (M) of \tau

Consequently,

d + d ^{*} = \begin{pmatrix} 0 & D \ D ^{*} & 0 \end{pmatrix}

Definition: The operator

d + d ^{*} with the above grading respectively the above operator D: \Omega _{+(M)} → \Omega _{-(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 ^{*)\tau} 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:index (D) = sign (M)

where the right hand side is the topological signature (i.e. the signature of a quadratic form on

H ^{2k} (M) defined by the cup product).

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

sign (M) = \int _{M} L(p _{1,\ldots,p} _{l)}

where

L is the Hirzebruch L-Polynomial, and the p _{i} the Pontrjagin forms on M .^[1]

Homotopy invariance of the higher indices

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

See also

• Hirzebruch signature theorem
• Pontryagin class
• Friedrich Hirzebruch
• Michael Atiyah
• Isadore Singer

Notes and References

1. ,