One of the key ingredients of A. Connes’ noncommutative geometry is a generalized Dirac operator which induces a metric(Connes’ distance) on the state space. We generalize such a Dirac operator devised by A. Dimakis et al , whose Connes’ distance recovers the linear distance on a 1D lattice, into 2D lattice. This Dirac operator being “naturally” defined has the “local eigenvalue property” and i...

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 describe a construction of fibrewise inner products on the cotangent bundle of the smooth free loop space of a Riemannian manifold. Using this inner product, we construct an operator over the loop space of a string manifold which is directly analogous to the Dirac operator of a spin

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.

Using the Arthur-Selberg trace formula we express the index of a Dirac operator on an arithmetic quotient manifold as the integral over the index form plus a sum of orbital integrals. For the Euler operator these orbital integrals are shown to vanish for products of certain rank one spaces. In this case the index theorem looks exactly as in the compact case.

We study the moduli space of instantons on a simply connected positive definite four manifold by analyzing the classifying map of the index bundle of a family of Dirac operators parametrized by the moduli space. As applications we compute the cohomology ring for the charge 2 moduli space in the rank stable limit.

A transformation is devised to convert any chirally non-invariant lattice Dirac fermion operator into Ginsparg-Wilson Dirac fermion operator. For the standard Wilson-Dirac lattice fermion operator, the transformed new operator is local, free of O(a) lattice artifacts, has correct axial anomaly in the vacuum sector, and is not plagued by the notorious problems ( e.g., additive mass renormalizati...

The Schrödinger functional formalism is given as a field theory in a finite volume with a Dirichlet boundary condition in temporal direction. When one tries to construct this formalism with the Ginsparg-Wilson fermion including the overlap Dirac operator and the domain-wall fermion one easily runs into difficulties. The reason is that if the Dirichlet boundary condition is simply imposed on the...

The Schrödinger functional formalism is given as a field theory in a finite volume with a Dirichlet boundary condition in temporal direction. When one tries to construct this formalism with the Ginsparg-Wilson fermion including the overlap Dirac operator and the domain-wall fermion one easily runs into difficulties. The reason is that if the Dirichlet boundary condition is simply imposed on the...

We define the notion of a co-Riemannian structure and show how it can be used to define the Dirac operator on an appropriate infinite dimensional manifold. In particular, this approach works for the smooth loop space of a so-called stringmanifold.