We study the most general conditions under which the computations of the index of a perturbed Dirac operator Ds = D + sZ localize to the singular set of the zeroth order operator Z in the semi-classical limit s → ∞. We use Witten’s method to compute the index of D by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator Z. The paper contains...

We study the most general conditions under which the computations of the index of a perturbed Dirac operator Ds = D + sZ localize to the singular set of the zeroth order operator Z in the semi-classical limit s → ∞. We use Witten’s method to compute the index of D by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator Z. The paper contains...

We prove some sharp Hardy type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.

We review a procedure of factorizing the Minkowski space Dirac operator over a~suitable superspace, discuss its Euclidean version and apply worked out formalism in case od an almost-commutative operator. The presented framework is attempt to reconcile non-commutative geometry supersymmetry.

The set of constraints under which the eigenvalues of the Dirac operator can play the role of the dynamical variables for Euclidean supergravity is derived. These constraints arise when the gauge invariance of the eigenvalues of the Dirac operator is imposed. They impose conditions which restrict the eigenspinors of the Dirac operator. PACS 04.60.-m, 04.65.+e During the last years, the Dirac op...

An expression is found for the L 2-index of a Dirac operator coupled to a connection on a Un vector bundle over S 1 × R 3. Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fred-holm. Callias' index theorem is used to calculate the index when the connection is independent of the coordinate on S 1. An excision theorem due to Gromov, Lawson, and Anghel re...

1. The SpinC-Dirac operator and the SpinC-quantization In this section we reformulate the “SpinC-quantization commutes with cobordism” principal in a more analytic language. In Subsection 1.1, we briefly recall the notion of SpinC-Dirac operator. We refer the reader to [2, 6] for details. We also express the SpinC-quantization of an orbifold M = X/G in terms of the index of the lift of the Spin...

A significant amount of the computational time in large Monte Carlo simulations of lattice quantum chromodynamics (QCD) is spent inverting the discrete Dirac operator. Unfortunately, traditional covariant finite difference discretizations of the Dirac operator present serious challenges for standard iterative methods. For interesting physical parameters, the discretized operator is large and il...

The chiral Jacobian, which is defined with Neuberger’s overlap Dirac operator of the lattice fermion, is explicitly evaluated in the continuum limit without expanding it in the gauge coupling constant. Our calculational scheme is simple and straightforward. We determine a coefficient of the chiral anomaly for general values of the mass parameter and the Wilson parameter of the overlap Dirac ope...