Related papers: Index Theorems and Loop Space Geometry

The Atiyah-Patodi-Singer (APS) index theorem relates the index of a Dirac operator to an integral of the Pontryagin density in the bulk (which is equal to global chiral anomaly) and an $\eta$ invariant on the boundary (which defines the…

The Atiyah-Singer index theorem is generalized to a two-dimensional SO(3) Yang-Mills-Higgs (YMH) system. The generalized theorem is proven by using the heat kernel method and a nonlinear realization of SU(2) gauge symmetry. This theorem is…

The quantization of the superclassical system used in the proof of the index theorem results in a factor of $\hbar^{2}R/8 $ in the Hamiltonian. The path integral expression of the kernel is analyzed up to and including 2-loop order. The…

Field theoretic and geometric ideas are used to construct a chiral supersymmetric field theory whose ground state is a specified irreducible representation of a centrally extended loop group. The character index of the associated…

We study here a complete quantization of a Callan-Giddings-Harvey-Strominger (CGHS) vacuum model following loop quantum gravity techniques. Concretely, we adopt a formulation of the model in terms of a set of new variables that resemble the…

The split involution quantization scheme, proposed previously for pure second--class constraints only, is extended to cover the case of the presence of irreducible first--class constraints. The explicit Sp(2)--symmetry property of the…

We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.

These are notes from a talk at the 2010 Talbot Workshop on Twisted K-theory and Loop Groups. This particular talk is an overview of index theory from the point of view of topological K-theory. Assuming little background in analysis, but…

The semiclassical solution of quantum Dirac constraints in generic constrained system is obtained by directly calculating in the one-loop approximation the gauge field path integral with relativistic gauge fixing procedure. The gauge…

Constrained Hamiltonian systems fall into the realm of presymplectic geometry. We show, however, that also Poisson geometry is of use in this context. For the case that the constraints form a closed algebra, there are two natural Poisson…

Symplectic unitary representations for the Poincar\'{e} group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space…

Using a K-theory point of view, Bott related the Atiyah-Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This article explains how to prove the local index theorem for compact…

Motivated by group-theoretical questions that arise in the context of asymptotic symmetries in gravity, we study model spaces and their quantization from the viewpoint of constrained Hamiltonian systems. More precisely, we propose that a…

In this article we study generalised Dirac-Schr\"odinger operators in arbitrary signatures (with or without gradings), providing a general KK-theoretic framework for the study of index pairings and spectral flow. We provide a general…

We associate a half-integer number, called {\em the quantum index}, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum…

Recently various new concepts for the construction of Dirac operators in lattice Quantum Chromodynamics (QCD) have been introduced. These operators satisfy the so-called Ginsparg-Wilson condition (GWC), thus obeying the Atiyah-Singer index…

Using the Arthur-Selberg trace formula we express the index of a Dirac operator on an arithmetic quotient over a totally real field with at least two real embeddings as the integral over the index form plus a sum of orbital integrals. For…

Variables adapted to the quantum dynamics of spherically symmetric models are introduced, which further simplify the spherically symmetric volume operator and allow an explicit computation of all matrix elements of the Euclidean and…

Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in…

The goal of this paper is to propose and discuss a practical way to implement the Dirac algorithm for constrained field models defined on spatial regions with boundaries. Our method is inspired in the geometric viewpoint developed by Gotay,…