Related papers: Index Theorem in Finite Noncommutative Geometry
This is an introduction to an algebraic construction of a gravity theory on noncommutative spaces which is based on a deformed algebra of (infinitesimal) diffeomorphisms. We start with some fundamental ideas and concepts of noncommutative…
A detailed description of the infinite-dimensional Lie algebra of star-gauge transformations in noncommutative Yang-Mills theory is presented. Various descriptions of this algebra are given in terms of inner automorphisms of the underlying…
Noncommutative geometric gauge theory is reconstructed based on the superconnection concept. The bosonic action of the Connes-Lott model including the symmetry breaking Higgs sector is obtained by using a new generalized derivative, which…
We investigate the invertible and non-invertible symmetries of topological finite-group gauge theories in general spacetime dimensions, where the gauge group can be abelian or non-abelian. We focus in particular on the 0-form symmetry. The…
This contribution describes the mathematical theory of topological indices in solid state systems composed of non-interacting Fermions. In particular, this covers the spectral localizer and the bulk-boundary correspondence.
A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie…
We prove an implicit function theorem and an inverse function theorem for free noncommutative functions over operator spaces and on the set of nilpotent matrices. We apply these results to study dependence of the solution of the initial…
An introduction is given to some selected aspects of noncommutative geometry. Simple examples in this context are provided by finite sets and lattices. As an application, it is explained how the nonlinear Toda lattice and a discrete time…
Noncommutative versions of theories with a gauge freedom define (when they exist) consistent deformations of their commutative counterparts. General aspects of Seiberg-Witten maps are discussed from this point of view. In particular, the…
We study K-theory classes of Hamiltonian loop group spaces represented by admissible Fredholm complexes. We prove various equivariant index formulae in this context. In a sequel to this article we show that, when specialized to a family of…
We clarify the role of gauge invariance for the computation of quantum non-Gaussian correlators in inflation. A gauge invariant generating functional for n-point functions is given and the special status of the spatially flat gauge is…
We consider the use of interpolating gauges (with a gauge function (F[A;alpha ]) in gauge theories to connect the results in a set of different gauges in the path-integral formulation. We point out that the results for physical observables…
The index theorem, discovered by Atiyah and Singer in 1963, is one of most important results in the twentieth century mathematics. It found numerous applications in analysis, geometry and physics. Since it was discovered numerous attempts…
A recently proposed connection between closed string field and an open Wilson line defined on an arbitrary contour is further explored here. We suggest that reparametrization invariance of a Wilson line is the principle which determines the…
Properties of pure gauge theories in thermal equilibrium as calculated via standard functional integral treatments are mathematically identical to ground state properties of a theory with spatially-periodic boundary conditions imposed on…
Endomorphisms algebras can replace the concept of principal fiber bundle. Gauge theories are reformulated within this algebraic framework and further generalized to unify ordinary connections and Higgs fields. A 'noncommutative Maxwell'…
A novel approach to the analysis of a noncommutative Chern--Simons gauge theory with matter coupled in the adjoint representation has been discussed. The analysis is based on a recently proposed closed form Seiberg--Witten map which is…
Gauge symmetries emerge from a redundant description of the effective action for light degrees of freedom after the decoupling of heavy modes. This redundant description avoids the use of explicit constraints in configuration space. For…
Noncommutative Chern-Simons gauge theory coupled to nonrelativistic scalars or spinors is shown to admit the ``exotic'' two-parameter-centrally extended Galilean symmetry, realized in a unique way consistent with the Seiberg-Witten map.…
We offer streamlined proofs of fundamental theorems regarding the index theory for partial self-maps of an infinite set that are bijective between cofinite subsets.