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.{{Harvnb|Atiyah|Bott|1967}} 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 $M$ .{{Harvnb|Atiyah|Bott|1967}}

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,{{Harvnb|Hirzebruch|1995}} and the $p_i\$ the Pontrjagin forms on $M$ .{{Harvnb|Gilkey|1973}}, {{Harvnb|Atiyah|Bott|Patodi|1973}}

Homotopy invariance of the higher indices

Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.{{Harvnb|Kaminker|Miller|1985}}

Notes

References

{{Citation | last1 = Atiyah | first1 = M.F. | last2 = Bott | first2 = R. | title = A Lefschetz fixed-point formula for elliptic complexes I | journal = Annals of Mathematics | volume = 86 | issue = 2 | year = 1967 | pages = 374–407 | doi=10.2307/1970694| jstor = 1970694 }}
{{Citation | last1 = Atiyah | first1 = M.F. | last2 = Bott |first2= R. | last3 = Patodi |first3 = V.K.| title = On the heat equation and the index theorem |journal = Inventiones Mathematicae | volume = 19 | issue = 4 | year = 1973 | pages = 279–330 | doi=10.1007/bf01425417| bibcode = 1973InMat..19..279A | s2cid = 115700319 }}
{{Citation |last1 = Gilkey | first1 = P.B. | title = Curvature and the eigenvalues of the Laplacian for elliptic complexes | journal = Advances in Mathematics | year = 1973 | volume = 10 | issue = 3 | pages = 344–382 | doi=10.1016/0001-8708(73)90119-9| doi-access = free }}
{{Citation | last = Hirzebruch | first = Friedrich | title = Topological Methods in Algebraic Geometry, 4th edition|year = 1995 | publisher = Berlin and Heidelberg: Springer-Verlag. Pp. 234|isbn= 978-3-540-58663-0}}
{{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}}

Category:Elliptic partial differential equations