Related papers: A Note on Coupled Dirac Operators
This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions…
This paper has two main objectives. The first one is to show that the Connes formulation of Dirac theory can be applied in the framework of quantum principal bundles for any n dimensional spectral triple, any quantum group, any quantum…
This paper is based on my talk at ICM on recent progress in a number of classical problems of linear algebra and representation theory, based on new approach, originated from geometry of stable bundles and geometric invariant theory.
We construct one Yang-Mills measure on a compact surface for each isomorphism class of principal bundles over this surface. For this, we define a new discrete gauge theory which is essentially a covering of the usual one. We prove that the…
We show that the exterior algebra bundle over a curved spacetime can be used as framework in which both the Dirac and the Einstein equations can be obtained. These equations and their coupling follow from the variational principle applied…
The Dirac equation is considered in the background of potentials of several types, namely scalar and vector-potentials as well as "Dirac-oscillator" potential or some of its generalisations. We investigate the radial Dirac equation within a…
We study the Dirac equation coupled to scalar and vector Klein-Gordon fields in the limit of strong coupling and large masses of the fields. We prove convergence of the solutions to those of a cubic non-linear Dirac equation, given that the…
This is an expository paper which gives a proof of the Atiyah-Singer index theorem for Dirac operators, presenting the theorem as a computation of the K-homology of a point. This paper and its follow up ("K-homology and index theory II:…
We study the logarithmic vector bundles associated to arrangements of smooth irreducible curves with small degree on the blow-up of the projective plane at one point. We then investigate whether they are Torelli arrangements, that is, they…
In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes…
We study the Ginzburg-Landau equations on Riemann surfaces of arbitrary genus. In particular: - we construct explicitly the (local moduli space of gauge-equivalent) solutions in a neighbourhood of the constant curvature ones; - classify…
The Courant bracket defined originally on the sections of the vector bundle $TM \oplus T^*M \to M$ is extended to the direct sum of the 1-jet vector bundle and its dual. The extended bracket allows to interpret many structures encountered…
Given an holomorphic Higgs bundle on a compact Riemann surface of genus greater than one, we construct a DQ-module supported by the spectral curve associated to this bundle. Then, we relate quantum curves arising in various situations…
A residue formula which evaluates any correlation function of topological $SU_n$ Yang-Mills theory with arbitrary magnetic flux insertion in two dimensions are obtained. Deformations of the system by two form operators are investigated in…
In this paper we determine solutions for the L\'evy-Leblond operator or a parabolic Dirac operator in terms of hypergeometric functions and spherical harmonics. We subsequently generalise our approach to a wider class of Dirac operators…
A lemma from elliptic theory is used to improve a recent result by Li concerning the removability of an isolated point singularity from solutions of the coupled Yang-Mills-Dirac equations.
It is shown that the calculation of Dirac operator for the spherical coordinate system with spherical Dirac matrices and using the spin connection formalism is in the contradiction with the definition of standard Dirac operator in the…
We define a diagram associated to any algebraic connection on a vector bundle on a Zariski open subset of the Riemann sphere, extending the definition of Boalch-Yamakawa to the general case featuring several irregular singularities,…
We study the index bundle of the Dirac-Ramond operator associated with a family $\pi: Z \to X$ of compact spin manifolds. We view this operator as the formal twisted Dirac operator $\dd \otimes \bigotimes_{n=1}^{\infty}S_{q^n}TM_{\C}$ so…
We give a survey on the Weierstrass representations of surfaces in three- and four-dimensional spaces, their applications to the theory of the Willmore functional and on related problems of spectral theory of the two-dimensional Dirac…