Signature operator

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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[edit]

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[edit]

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[edit]

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[edit]

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

See also[edit]

Notes[edit]

References[edit]

  • Atiyah, M.F.; Bott, R. (1967), "A Lefschetz fixed-point formula for elliptic complexes I", Annals of Mathematics 86: 374–407 
  • Atiyah, M.F.; Bott, R.; Patodi, V.K. (1973), "On the heat equation and the index theorem", Inventiones Math. 19: 279–330 
  • Gilkey, P.B. (1973), "Curvature and the eigenvalues of the Laplacian for elliptic complexes", Advances in Mathematics 10: 344–382 
  • 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", Journal of Operator Theory 14: 113–127