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 [math]\displaystyle{ M }[/math] be a compact Riemannian manifold of even dimension [math]\displaystyle{ 2l }[/math]. Let
- [math]\displaystyle{ d : \Omega^p(M)\rightarrow \Omega^{p+1}(M) }[/math]
be the exterior derivative on [math]\displaystyle{ i }[/math]-th order differential forms on [math]\displaystyle{ M }[/math]. The Riemannian metric on [math]\displaystyle{ M }[/math] allows us to define the Hodge star operator [math]\displaystyle{ \star }[/math] and with it the inner product
- [math]\displaystyle{ \langle\omega,\eta\rangle=\int_M\omega\wedge\star\eta }[/math]
on forms. Denote by
- [math]\displaystyle{ d^*: \Omega^{p+1}(M)\rightarrow \Omega^p(M) }[/math]
the adjoint operator of the exterior differential [math]\displaystyle{ d }[/math]. This operator can be expressed purely in terms of the Hodge star operator as follows:
- [math]\displaystyle{ d^*= (-1)^{2l(p+1) + 2l + 1} \star d \star= - \star d \star }[/math]
Now consider [math]\displaystyle{ d + d^* }[/math] acting on the space of all forms [math]\displaystyle{ \Omega(M)=\bigoplus_{p=0}^{2l}\Omega^{p}(M) }[/math]. One way to consider this as a graded operator is the following: Let [math]\displaystyle{ \tau }[/math] be an involution on the space of all forms defined by:
- [math]\displaystyle{ \tau(\omega)=i^{p(p-1)+l}\star \omega\quad,\quad\omega \in \Omega^p(M) }[/math]
It is verified that [math]\displaystyle{ d + d^* }[/math] anti-commutes with [math]\displaystyle{ \tau }[/math] and, consequently, switches the [math]\displaystyle{ (\pm 1) }[/math]-eigenspaces [math]\displaystyle{ \Omega_{\pm}(M) }[/math] of [math]\displaystyle{ \tau }[/math]
Consequently,
- [math]\displaystyle{ d + d^* = \begin{pmatrix} 0 & D \\ D^* & 0 \end{pmatrix} }[/math]
Definition: The operator [math]\displaystyle{ d + d^* }[/math] with the above grading respectively the above operator [math]\displaystyle{ D: \Omega_+(M) \rightarrow \Omega_-(M) }[/math] is called the signature operator of [math]\displaystyle{ M }[/math].[2]
Definition in the odd-dimensional case
In the odd-dimensional case one defines the signature operator to be [math]\displaystyle{ i(d+d^*)\tau }[/math] acting on the even-dimensional forms of [math]\displaystyle{ M }[/math].
Hirzebruch Signature Theorem
If [math]\displaystyle{ l = 2k }[/math], so that the dimension of [math]\displaystyle{ M }[/math] is a multiple of four, then Hodge theory implies that:
- [math]\displaystyle{ \mathrm{index}(D) = \mathrm{sign}(M) }[/math]
where the right hand side is the topological signature (i.e. the signature of a quadratic form on [math]\displaystyle{ H^{2k}(M)\ }[/math] defined by the cup product).
The Heat Equation approach to the Atiyah-Singer index theorem can then be used to show that:
- [math]\displaystyle{ \mathrm{sign}(M) = \int_M L(p_1,\ldots,p_l) }[/math]
where [math]\displaystyle{ L }[/math] is the Hirzebruch L-Polynomial,[3] and the [math]\displaystyle{ p_i\ }[/math] the Pontrjagin forms on [math]\displaystyle{ M }[/math].[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
Notes
References
- Atiyah, M.F.; Bott, R. (1967), "A Lefschetz fixed-point formula for elliptic complexes I", Annals of Mathematics 86 (2): 374–407, doi:10.2307/1970694
- Atiyah, M.F.; Bott, R.; Patodi, V.K. (1973), "On the heat equation and the index theorem", Inventiones Mathematicae 19 (4): 279–330, doi:10.1007/bf01425417, Bibcode: 1973InMat..19..279A
- Gilkey, P.B. (1973), "Curvature and the eigenvalues of the Laplacian for elliptic complexes", Advances in Mathematics 10 (3): 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 978-3-540-58663-0
- Kaminker, Jerome; Miller, John G. (1985), "Homotopy Invariance of the Analytic Index of Signature Operators over C*-Algebras", Journal of Operator Theory 14: 113–127, http://jot.theta.ro/jot/archive/1985-014-001/1985-014-001-006.pdf
 |  |
