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

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

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define important - what would you say is the importance of the Dirac operator in pure mathematics? –  lurscher Nov 14 '11 at 12:32
Let's be fair: it was invented within physics, for physicists, by a physicist. It may have ancillary relevance to maths, but not huge. @lurscher is also right; what's "important" here? –  Noldorin Nov 14 '11 at 22:07

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.