# nLab symbol of a differential operator

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

# Contents

## 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

$\sigma(D) \;:\; T^* X \times_X E \to E$

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

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

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).

## 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)