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

${\displaystyle d:\Omega ^{p}(M)\rightarrow \Omega ^{p+1}(M)}$

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

${\displaystyle \langle \omega ,\eta \rangle =\int _{M}\omega \wedge \star \eta }$

on forms. Denote by

${\displaystyle d^{*}:\Omega ^{p+1}(M)\rightarrow \Omega ^{p}(M)}$

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

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

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

${\displaystyle \tau (\omega )=i^{p(p-1)+l}\star \omega \quad ,\quad \omega \in \Omega ^{p}(M)}$

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

Consequently,

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

Definition: The operator ${\displaystyle d+d^{*}}$ with the above grading respectively the above operator ${\displaystyle D:\Omega _{+}(M)\rightarrow \Omega _{-}(M)}$ 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 ${\displaystyle i(d+d^{*})\tau }$ acting on the even-dimensional forms of $M$.

Hirzebruch Signature Theorem

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

${\displaystyle \mathrm {index} (D)=\mathrm {sign} (M)}$

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

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

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

where $L$ is the Hirzebruch L-Polynomial,^[3] and the ${\displaystyle p_{i}\ }$ 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]

See also

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

Notes

1. Atiyah & Bott 1967
2. Atiyah & Bott 1967
3. Hirzebruch 1995
4. Gilkey 1973, Atiyah, Bott & Patodi 1973
5. Kaminker & Miller 1985

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