Axial anomaly and Atiyah-Singer theorem

Notes on the Atiyah-Singer Index Theorem

This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential operators on a compact manifold, namely those operators P with the property that dim ker P, dim coker P < ∞. In general these integers are very difficult to compute without some very precise information abou...

Chen Series and Atiyah-singer Theorem

The goal of this paper is to give a new and short proof of the local Atiyah-Singer index theorem by using approximations of heat semigroups. The heat equation approach to index theorems is not new: It was suggested by Atiyah-Bott [1] and McKean-Singer [18], and first carried out by Patodi [21] and Gilkey [13]. Bismut in [8] introduces stochastic methods based on Feynman-Kac formula. For probabi...

Applications of Elliptic Operators and the Atiyah Singer Index Theorem

1. Review of Differential Geometry 2 2. Definition of an Elliptic Operator 5 3. Properties of Elliptic Operators 7 4. Example of an Elliptic Operator 9 5. Example: The Euler Characteristic 12 6. Example: The Signature Invariant 14 7. A Theorem of Atiyah, Frank and Mayer 18 8. Clifford Algebras 20 9. A Diversion: Constructing Vector Fields on Spheres using Clifford Algebras 23 10. Topological In...

An alternative stochastic proof for the Atiyah-Singer index theorem

According to Watanabe [6], by generalized Wiener functional we could give a simple stochastic proof of the index theorem for twisted Dirac operator (the Atiyah-Singer index theorem) on Clifford module. MR Subject Classification: 58J20 Chinese Library Classification: O186.12

A Gerbe formulation of the Atiyah-Singer Index Theorem