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.^[1] 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 ^{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 ,\eta \rangle =\int _{M}\omega \wedge \star \eta

on forms. Denote by

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

Now consider $d+d^{*}$ acting on the space of all forms $\Omega (M)=\bigoplus _{p=0}^{2l}\Omega ^{p}(M)$ . 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 \quad ,\quad \omega \in \Omega ^{p}(M)

It is verified that $d+d^{*}$ anti-commutes with $\tau$ and, consequently, switches the $(\pm 1)$ -eigenspaces $\Omega _{\pm }(M)$ 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 $D:\Omega _{+}(M)\rightarrow \Omega _{-}(M)$ is called the signature operator of $M$ .^[2]

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 {index} (D)=\mathrm {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:

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

where $L$ is the Hirzebruch L-Polynomial,^[3] and the $p_{i}\$ the Pontrjagin forms on $M$ .^[4]

Homotopy invariance of the higher indices

Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.^[5]

Notes

References

Atiyah, M.F.; Bott, R. (1967), "A Lefschetz fixed-point formula for elliptic complexes I", Annals of Mathematics, 86: 374–407, doi:10.2307/1970694
Atiyah, M.F.; Bott, R.; Patodi, V.K. (1973), "On the heat equation and the index theorem", Inventiones Math., 19: 279–330, doi:10.1007/bf01425417
Gilkey, P.B. (1973), "Curvature and the eigenvalues of the Laplacian for elliptic complexes", Advances in Mathematics, 10: 344–382, doi:10.1016/0001-8708(73)90119-9
Hirzebruch, Friedrich (1995), Topological Methods in Algebraic Geometry, 4th edition, Berlin and Heidelberg: Springer-Verlag. Pp. 234, ISBN 3-540-58663-6
Kaminker, Jerome; Miller, John G. (1985), "Homotopy Invariance of the Analytic Index of Signature Operators over C*-Algebras" (PDF), Journal of Operator Theory, 14: 113–127

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