Related papers: Supersymmetric Dirac Operator on the Noncommutativ…
We review the construction of the Dirac operator and its properties in Riemannian geometry and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also…
Given a symplectic manifold $(M,\omega)$ admitting a metaplectic structure, and choosing a positive $\omega$-compatible almost complex structure $J$ and a linear connection $\nabla$ preserving $\omega$ and $J$, Katharina and Lutz Habermann…
We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators $D$ starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of $M_2(\Bbb C)$, and also applies…
We present a derivation of the general form of the scalar potential in Yang-Mills theory of a non-commutative space which is a product of a four-dimensional manifold times a discrete set of points. We show that a non-trivial potential…
We obtain the spectrum of the Dirac operator on the three-dimensional Heisenberg nilmanifold $\mathcal{M}_3$, and its complete dependence on the metric moduli. As an application, we construct the four-dimensional low-energy effective action…
Spectral triples over noncommutative principal $\T^n$-bundles are studied, extending recent results about the noncommutative geometry of principal U(1)-bundles. We relate the noncommutative geometry of the total space of the bundle with the…
A new approach to the study of spectral asymmetry for systems of partial differential equations (PDEs) on closed manifolds was proposed in a recent series of papers by the first author and collaborator. They showed that information on…
We consider super Yang-Mills theory on supermanifolds $\mathcal{M}^{(D|m)}$ using integral forms. The latter are used to define a geometric theory of integration and are essential for a consistent action principle. The construction relies…
A finite non-commutative geometry consists of a fuzzy space together with a Dirac operator satisfying the axioms of a real spectral triple. This paper addreses the question of how to extract information about these geometries from the…
This paper is a follow-up on the \emph{noncommutative differential geometry on infinitesimal spaces} [15]. In the present work, we extend the algebraic convergence from [15] to the geometric setting. On the one hand, we reformulate the…
We study four dimensional supersymmetric gauge theory on the noncommutative superspace, recently proposed by Seiberg. We construct the gauge-invariant action of N=1 super Yang-Mills theory with chiral and antichiral superfields, which has…
In this paper we introduce and study generally non-self-adjoint realizations of the Dirac operator on an arbitrary finite metric graph. Employing the robust boundary triple framework, we derive, in particular, a variant of the Birman…
This article surveys the noncommutative-geometric (NCG) approach to fundamental physics, in which geometry is encoded spectrally by a generalized Dirac operator and where dynamics arise from the spectral action. I review historically how…
This paper establishes a link between Noncommutative Geometry and canonical quantum gravity. A semi-finite spectral triple over a space of connections is presented. The triple involves an algebra of holonomy loops and a Dirac type operator…
We introduce the notion of a semi-Riemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of semi-Riemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in…
This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in $n$-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical…
We review the spectral flow techniques for computing the index of the overlap Dirac operator including results relevant for SUSY Yang-Mills theories. We describe properties of the overlap Dirac operator, and methods to implement it…
We construct the $q$-deformed Clifford algebra of $\mathfrak{sl}_2$ and study its properties. This allows us to define the $q$-deformed noncommutative Weil algebra $\mathcal{W}_q(\mathfrak{sl}_2)$ for $U_q(\mathfrak{sl}_2)$ and the…
We construct a Connes spectral triple or `Dirac operator' on the non-reduced fuzzy sphere $C_\lambda[S^2]$ as realised using quantum Riemannian geometry with a central quantum metric $g$ of Euclidean signature and its associated quantum…
Using principles of quantum symmetries we derive the algebraic part of the real spectral triple data for the standard Podle\'s quantum sphere: equivariant representation, chiral grading $\gamma$, reality structure $J$ and the Dirac operator…