nLab
Atiyah-Singer index theorem

Statement

Michael Atiyah and Isadore Singer associated two differently defined numbers to an elliptic operator on a manifold: the topological index and the analytical index. The index theorem asserts that the two are equal.

References

General

The original articles are

• Michael Atiyah, Isadore Singer, The index of elliptic operators I, Ann. of Math. (2) 87 (1968) pp. 484–530; III, Ann. of Math. (2) 87 (1968) pp. 546–604; IV, Ann. of Math. (2) 93 (1971) pp. 119–138; V, Ann. of Math. (2) 93 (1971) pp. 139–149

• Michael Atiyah, Graeme Segal, The index of elliptic operators II, Ann. of Math. (2) 87 (1968) pp. 531–545

A proof of the Atiyah-Singer index theorem in terms of KK-theory/E-theory has been given by Nigel Higson, an account is in

• Nigel Higson, John Roe, Lectures on operator K-theory and the Atiyah-Singer Index Theorem (2004) (pdf)

A lighning review of the proof is on the last pages of

• Introduction to KK-theory and E-theory, Lecture notes (Lisbon 2009) (pdf slides)

See also

• Wikipedia, Atiyah-Singer index theorem

From the point of view of physics

The index theorem is supposed to have an interpretation in terms of the quantum field theory of the superparticle on the given space.

Traditional physics arguments along these lines include for instance

• P. Windey, Supersymmetric quantum mechanics and the Atiyah–Singer index theorem, Acta Physica Polonica, B15 (1984). (PDF)

• Luis Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. (EUCLID)