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.

The index theorem generalizes earlier results such as the Riemann-Roch theorem.

References

General

The original articles:

• 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

Review:

• Friedrich Hirzebruch, Thomas Berger, Rainer Jung, chapter 6 of: Manifolds and Modular Forms, Aspects of Mathematics 20, Viehweg (1992), Springer (1994) [doi:10.1007/978-3-663-10726-2, pdf]

• Peter Gilkey, Section 5.2 of: Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem, 1995 (pdf)

• Rafe Mazzeo, The Atiyah-Singer Index theorem: what it is and why you should care, (slides)

• Daniel Freed, The Atiyah-Singer index theorem, (slides)

See also

• Wikipedia, Atiyah-Singer index theorem

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 lightning review of the proof is on the last pages of

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

From the point of view of physics

The index theorem has an interpretation in terms of the quantum field theory of the superparticle on the given space. See also at supersymmetric quantum mechanics.

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)

• Florian Hanisch, Matthias Ludewig, A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold, (arXiv:1709.10027)