Why is the Dirac operator so important - in both physics and mathematics?

Question

Why is the Dirac operator considered so important - in both physics and (pure) mathematics?

Answer 1

It should be obvious why the Dirac operator is important in physics because of fermions. In mathematics, one could mention the following incomplete list.

1. Atiyah-Singer Index Theorem for the (twisted) spin complex, see e.g., Nakahara, Geometry, Topology and Physics, 1990; or Berline, Getzler and Vergne, Heat Kernels and Dirac Operators, 2004.

2. Connes's noncommutative differential geometry.

3. Schrödinger–Lichnerowicz formula.

4. Kostant's cubic Dirac operator.

For further information, see also nLab.

Answer 2

Because it describes spinors (i.e. behavior of fermions). String Theory speaks about both bosons and fermions... when passing to a QFT, fermionic wavefunctions become spinor fields, and their dynamics require the Dirac operator. And this is where all the mathematical importance stems from: QFT and String Theory (topological field theories) is a pure math theory. In particular, I disagree completely with Noldorin's seemingly naive comment... Clifford algebras and spinors on manifolds is a huge field.

Sorry if this too brief of an explanation (for immediate purposes).