arXiv research

Study pin manifolds using Clifford linear Dirac operator and KO-theory.

Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.

Formula derived for torsion of modified Dirac operator.

The paper calculates spectral torsion for rescaled Dirac operators on manifolds.

We present a generalization of the Clifford action for other representations spaces of $Spin(n)$, which is called the Clifford homomorphism. Their properties extend to the ones for the higher spin Dirac operators on spin manifolds. In particular, we have general Bochner identities for them, and an eigenvalue estimate o…

This manuscript attempts to present a way in which the classical construction of the Dirac operator can be carried over to the setting of diffeology. A more specific aim is to describe a procedure for gluing together two usual Dirac operators and to explain in what sense the result is again a Dirac operator. Since vers…

This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on represen…

Study relates Finsler structures to Clifford bundles for flat metrics.

A classical result in differential geometry due to Lichnerowicz [8] is concerned with the decomposition of the square of Dirac operators defined by Clifford connections on a Clifford module ${\cal E}$\ over a Riemannian manifold $M$. Recently, this formula has been generalized to arbitrary Dirac operators [2]. In this …

Researchers solve the Calderón problem for fractional Dirac operators.

We construct the spectral curve and the Baker--Akhiezer function for the Dirac operator which corresponds to the Clifford torus via the Weierstrass representation. By constructing this Baker--Akhiezer function we demonstrate a general procedure for constructing Dirac operators and their Baker--Akhiezer functions corres…

In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the square of these operators, which play an important role i…

This is the second part in a series of two papers. The $k$-Dirac complex is a complex of differential operators which are natural to a particular $|2|$-graded parabolic geometry. In this paper we will consider the $k$-Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove …

The paper constructs a complex for the Dirac operator in 4 dimensions.

The paper studies twistor sections of Dirac bundles and proves vanishing theorems.

Abstract: Study Kähler identities on almost complex manifolds.

We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal metrics and linear and nonlinear connections define different types of Finsler, …

We apply the Cartan-Kahler theorem for the k-Dirac operator studied in Clifford analysis and to the parabolic version of this operator. We show that for k = 2 the tableaux of the first prolongations of these two operators are involutive. This gives us a new characterization of the set of initial conditions for the 2-D…

We calculate the equivariant index formula for an infinite dimensional Clifford module canonically associated to any Riemannian manifold. It encompasses the fractional index formula of the projective Dirac operator by Mathai--Melrose--Singer.

We prove the rigidity and vanishing of several indices of "geometrically natural" twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.

A general theory of quantum spinor structures on quantum spaces is presented, within the conceptual framework of the formalism of quantum principal bundles. Quantum analogs of all basic objects of the classical theory are constructed and analyzed. This includes Laplace and Dirac operators, quantum versions of Clifford …

In this largely expository paper we give a self-contained treatment of the Dirac operator. Emphasizing the algebraic point of view we first sketch the necessary prerequisites from Clifford algebras and their representations and then define (and characterize) spin structures and the corresponding Spin-Dirac operator pur…

We provide explicit spinor representations for Clifford algebras.

The geometry of nonholonomic bundle gerbes, provided with nonlinear connection structure, and nonholonomic gerbe modules is elaborated as the theory of Clifford modules on nonholonomic manifolds which positively fail to be spin. We explore an approach to such nonholonomic Dirac operators and derive the related Atiyah-S…

A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted to represent internal structure of matter (e.g. classical or quantum spin). The …

Formulas for spectra of higher spin operators on sphere subbundles.

The principal group of a Klein geometry has canonical left action on the homogeneous space of the geometry and this action induces action on the spaces of sections of vector bundles over the homogeneous space. This paper is about construction of differential operators invariant with respect to the induced action of the…

We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such estimates are presented for modified Dirac operators with a $k$-degree form $0\leq k\leq…

The use of bundle gerbes and bundle gerbe modules is considered as a replacement for the usual theory of Clifford modules on manifolds that fail to be spin. It is shown that both sides of the Atiyah-Singer index formula for coupled Dirac operators can be given natural interpretations using this language and that the re…

Local index formula for Lorentzian Dirac operators on spacetimes.

This article investigates local properties of the further generalized Weierstrass relations for a spin manifold $S$ immersed in a higher dimensional spin manifold $M$ from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over $S$, which we call submanifold Dir…

Derives an index formula for families of end-periodic Dirac operators.

We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact Kähler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of Kählerian Killing ${\rm spin}^c$ spinors in a certain subbundle of…

Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces im…

In this thesis, we show the existence of a sequence of differential operators starting with with the Dirac operator in k Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\sum_j e_j\cdot \partial_{ij}: C^\infty((\R^n)^k,§)\to C^\infty((\R^n)^k,§)$ ($§$ is the spinor module). This operator is the Cauchy-Riemann operato…

Constructs Dirac generating operators for split Courant algebroids.

We give explicit formulas for all odd order differential intertwinors on the subbundle of the bundle of spinor-$k$-forms that are annihilated by the Clifford multiplication over the odd dimensional standard sphere. The Dirac and Rarita-Schwinger operators appear in the case of $k=0$ and $k=1$, respectively.

Maps are shown to be Riemannian products with Ricci-flat fibers.

This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of $|2|$-graded parabolic geometries of some particular type. We call them $k$-Dirac complexes. More explicitly, we will show that each $k$-Dirac complex arises as…

Study quantum diffusion on spectral triples and spinor bundles.

We study the index theory of a class of perturbed Dirac operators on non-compact manifolds of the form $\mathsf{D}+\mathrm{i}\mathsf{c}(X)$, where $\mathsf{c}(X)$ is a Clifford multiplication operator by an orbital vector field with respect to the action of a compact Lie group. Our main result is that the index class o…

Class lecture notes at a beginning graduate level on the mathematical background needed to understand classical gauge theory. Covers group actions, fiber bundles, principal bundles, connections, gauge transformations, parallel transport, curvature, covariant derivatives, pseudo-riemannian manifolds, lagrangians, cliffo…

In this paper we study Clifford and harmonic analysis on some conformal flat spin manifolds. In particular we treat manifolds that can be parametrized by $U / Γ$ where $U$ is a simply connected subdomain of either $S^{n}$ or $R^{n}$ and $Γ$ is a Kleinian group acting discontinuously on $U$. Examples of such manifolds t…

The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan' theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin^C-structures, Dirac operators, exterior algebra bundles and Connes' differential algebras in the commutative case…

In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strateg…

The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\gencomp (E)$ as the set of Clifford bundles $E'$ which have finite distance to $E$. If $D$, $D'$ are the associated generalized Dirac ope…

We obtain a vanishing theorem for the kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the kernel of a $\spi…

In this paper we examine a new class of five dimensional (5D) exact solutions in extra dimension gravity possessing Lie algebroid symmetry. The constructions provide a motivation for the theory of Clifford nonholonomic algebroids elaborated in Ref. hep-th/0501217. Such Einstein-Dirac spacetimes are parametrized by gene…