The Index Theorem of Lattice Wilson–Dirac Operators via Higher Index Theory

Superconnections and Higher Index Theory

Let M be a smooth closed spin manifold. The higher index theorem computes the pairing between the group cohomology of π1(M) and the Chern character of the “higher” index of a Dirac-type operator on M. Using superconnections, we give a heat equation proof of this theorem on the level of differential forms on a noncommutative base space. As a consequence, we obtain a new proof of the Novikov conj...

On the Higher Randić Index

Let G be a simple graph with vertex set V(G) {v1,v2 ,...vn} . For every vertex i v , ( ) i  v represents the degree of vertex i v . The h-th order of Randić index, h R is defined as the sum of terms 1 2 1 1 ( ), ( )... ( ) i i ih  v  v  v  over all paths of length h contained (as sub graphs) in G . In this paper , some bounds for higher Randić index and a method for computing the higher R...

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...

Criterium for the index theorem on the lattice

We study how far the Index Theorem can be extrapolated from the continuum to finite lattices with finite topological charge densities. To examine how the Wilson action approximates the Index theorem, we specialize in the lattice version of the Schwinger model. We propose a new criterion for solutions of the Ginsparg-Wilson Relation constructed with the Wilson action. We conclude that the Neuber...

Lectures on Dirac Operators and Index Theory