# Supersymmetry and the Atiyah-Singer index theorem

@article{AlvarezGaum1983SupersymmetryAT, title={Supersymmetry and the Atiyah-Singer index theorem}, author={Lu{\'i}s Alvarez-Gaum{\'e}}, journal={Communications in Mathematical Physics}, year={1983}, volume={90}, pages={161-173} }

Using a recently introduced index for supersymmetric theories, we present a simple derivation of the Atiyah-Singer index theorem for classical complexes and itsG-index generalization using elementary properties of quantum mechanical supersymmetric systems.

## 384 Citations

Supersymmetry and the Atiyah-Singer Index Theorem I: Peierls Brackets, Green's Functions, and a Supersymmetric Proof of the Index Theorem

- Physics
- 1993

The Peierls bracket quantization scheme is applied to the supersymmetric system corresponding to the twisted spin index theorem. A detailed study of the quantum system is presented, and the Feynman…

Derivation of Index Theorem by Supersymmetry

- Physics
- 2007

The present paper gives calculations in detail to prove several special cases of Atiyah-Singer theorem through supersymmetric σ-models. Some technical tricks are employed to calculate the…

Derivation of Index theorem by Supersymmetry

- Physics
- 2007

The present paper gives calculations in detail to prove several special cases of Atiyah-Singer theorem through supersymmetric $\sigma$-models. Some technical tricks are employed to calculate the…

Supersymmetry and the Atiyah–Singer index theorem. I. Peierls brackets, Green’s functions, and a proof of the index theorem via Gaussian superdeterminants

- Physics
- 1994

The Peierls bracket quantization scheme is applied to the supersymmetric system corresponding to the twisted spin index theorem. A detailed study of the quantum system is presented, and the Feynman…

Quantization of supermanifolds and an analytic proof of the Atiyah-Singer index theorem

- Mathematics
- 1993

An analytic proof of the Atiyah-Singer index, theorem is given with the help of the tools of supermathematics. The index formula for the Dirac operator on a spinor manifold is obtained here by direct…

Derivation of index theorems by localization of path integrals

- Mathematics
- 1994

We review the derivation of the Atiyah-Singer and Callias index theorems using the recently developed localization method to calculate exactly the relevant supersymmetric path integrals.

Gauge fixing and equivariant cohomology

- Physics
- 2005

The supersymmetric model developed by Witten (1982 J. Differ. Geom. 17 661–92) to study the equivariant cohomology of a manifold with an isometric circle action is derived from the BRST quantization…

The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs

- Mathematics
- 1986

SummaryThe purpose of this paper is to give two heat equation proofs of the Index Theorem of Atiyah-Singer for a family of Dirac operators.

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