The Index of Elliptic Operators: I

Abstract

This compression is particularly true of the new section on ' Gauge theoretical physics' added for the English edition. Abstract ideas in geometry are now intensively studied as models for the physics of elementary particles. Index theory is central here, particularly the analytical 'heat equation' proof, which is briefly mentioned in the book. This last section contains quantum mechanics and field theory, Riemannian geometry and connections, all packed in together. The material is not so well polished as the earlier sections and much of it is available in standard references, but I found the introductory paragraphs 'Physical motivation and overview' very stimulating.

Introductory lectures to large- QCD phenomenology and lattice results

“…This formulation realizes the (modified) chiral symmetry exactly at every value of the lattice spacing, satisfies the Atiya-Singer theorem [107], and leads to exactly one massless physical flavor in the continuum limit, with no need for parameter fine-tuning. A different formulation of lattice fermions satisfying the Ginsparg-Wilson relation is based on the domain-wall construction [108], whereby a chiral fermion is obtained by introducing an unphysical fifth dimension (on which the gauge fields do not depend), along which the bare fermion mass changes sign.…”

A Class of Toeplitz Operators in Several Complex Variables

“…σ 0 (E Ψ )(z, ξ) is invertible for all (z, ξ) ∈ T * S with ξ ̸ = 0. Moreover, the index of this operator is actually independent of the particular choice of s and it can be evaluated by the familiar Atiyah-Singer formula [10].…”

An index formula for Toeplitz operators

“…σ 0 (E Ψ )(x, ξ) is invertible for all (x, ξ) ∈ T * S with ξ = 0. Moreover, the index of this operator is actually independent of the particular choice of s and it can be evaluated by the familiar Atiyah-Singer formula [AS68].…”

“…In [Fed70] Fedosov derived from the formula of [AS68] an expression for the index of an elliptic operator on a compact closed manifold in the form of integral of a differential form. In the case where the Todd class of the manifold is equal to 1, this form is written explicitly through the symbol of the operator.…”

“…If the projection Π is pseudodifferential, any generalised Toeplitz operator T Ψ is specified within the framework of pseudodifferential operators between sections of the vector bundle ⊕F j on S. To evaluate the index of an elliptic operator T Ψ we can therefore apply the Atiyah-Singer index formula [AS68]. This formula expresses the index by means of pairing the Chern character of the virtual bundle associated with the symbol of T Ψ , and the Todd class of the tangent bundle of S. In contrast to this cohomological formula [Fed70] gives an expression for the index in the form of the integral of a differential form over a cosphere bundle S * S of S. If the Todd class is equal to 1, then this form is explicitly written through the symbol of T Ψ .…”