nLab
symbol of a differential operator

Contents

Definition

For XX a smooth manifold, EXE \to X a vector bundle and D:Γ(E)Γ(E)D : \Gamma(E) \to \Gamma(E) a differential operator on sections of EE, its symbol is the bundle morphism

σ(D):T *X× XEE \sigma(D) \;:\; T^* X \times_X E \to E

given at any point xXx \in X on a cotangent vector of the form (df) xΓ(T *X) x(\mathbf{d}f)_x \in \Gamma(T^* X)_x by

σ(D) x:df x[D,f] x, \sigma(D)_x \;\colon\; \mathbf{d}f_x \mapsto [D,f]_x \,,

where in the commutator on the right we regard multiplication by ff as an endomorphism of Γ(E)\Gamma(E).

The symbol may naturally be thought of as an element in the K-theory of XX (Freed).

Examples

References

For instance chapter 2.5 of

  • Nigel Higson, John Roe, Lectures on operator K-theory and the Atiyah-Singer Index Theorem (pdf)

  • Dan Freed, Geometry of Dirac operators (pdf)

Revised on April 10, 2013 22:00:48 by Urs Schreiber (131.174.41.18)