signature operator

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 $2l$. Let

:$d : \Omega^p(M)\rightarrow \Omega^(M)$

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

:$\langle\omega,\eta\rangle=\int_M\omega\wedge\star\eta$

on forms. Denote by

:$d^*: \Omega^(M)\rightarrow \Omega^p(M)$

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

:$d^*= (-1)^ \star d \star= - \star d \star$

Now consider $d + d^*$ acting on the space of all forms $\Omega(M)=\bigoplus_^\Omega^(M)$.
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:

:$\tau(\omega)=i^\star \omega\quad,\quad\omega \in \Omega^p(M)$

It is verified that $d + d^*$ anti-commutes with $\tau$ and, consequently, switches the $(\pm 1)$-eigenspaces $\Omega_(M)$ of $\tau$

Consequently,

:$d + d^* = \begin 0 & D \\ D^* & 0 \end$

Definition: The operator $d + d^*$ with the above grading respectively the above operator $D: \Omega_+(M) \rightarrow \Omega_-(M)$ 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 $l = 2k$, so that the dimension of $M$ is a multiple of four, then Hodge theory implies that:

:$\mathrm(D) = \mathrm(M)$

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

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

:$\mathrm(M) = \int_M L(p_1,\ldots,p_l)$

where $L$ is the Hirzebruch L-Polynomial, and the $p_i\$ the Pontrjagin forms on $M$.

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

References

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*{{Citation , last1 = Kaminker , first1 = Jerome , last2 = Miller , first2 = John G. , title = Homotopy Invariance of the Analytic Index of Signature Operators over C*-Algebras , journal = Journal of Operator Theory , year = 1985 , volume = 14 , pages = 113–127 , url = http://jot.theta.ro/jot/archive/1985-014-001/1985-014-001-006.pdf

Elliptic partial differential equations