nLab
symbol of a differential operator

Definition

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

$$\sigma (D):T^{*}X{\times }_{X}E\to E$$\sigma(D) \;:\; T^* X \times_X E \to E

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

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

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

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

Examples

• For $X={ℝ}^n$ a Cartesian space and $D$ the Dirac operator of the flat connection, the symbol of $D$ reproduces the symbol map between differential forms and Clifford algebra elements.

Related concepts

• elliptic differential operator

• elliptic chain complex

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)

• Wikipedia, Symbol of a differential operator