Related papers: Supersymmetry and the generalized Lichnerowicz for…
Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing,…
We study Dirac operators on resolutions of Riemannian orbifolds by developing a uniform elliptic theory. The key idea is to view orbifolds as conically fibred singular (CFS) spaces and resolve them by gluing asymptotically conical…
We use the Dirac operator method to prove a scalar-mean curvature comparison theorem for spin manifolds which carry iterated conical singularities. Our approach is to study the index theory of a twisted Dirac operator on such singular…
We discuss a generalization of Clifford algebras known as generalized Clifford algebras (in particular, ternary Clifford algebras). In these objects, we have a fixed higher-degree form (in particular, a ternary form) instead of a quadratic…
We explore a new simple N=2 SQM model describing the motion over complex manifolds in external gauge fields. The nilpotent supercharge Q of the model can be interpreted as a (twisted) exterior holomorphic derivative, such that the model…
In this paper we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure $\mathbb{J}$ on the canonical symplectic manifold $(\mathbb {R}^{2n},\omega_0)$. This gives rise to two symplectic Dirac…
We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a $(2+1)$-dimensional Riemann (curved) spacetime.…
The diffeomorphism symmetry of general relativity leads in the canonical formulation to constraints, which encode the dynamics of the theory. These constraints satisfy a complicated algebra, known as Dirac's hypersurface deformation…
Recent developments in the construction of generalized Dirac duals have revealed, within the structure of the Clifford algebra $\mathbb{C}\otimes\mathcal{C}\ell_{1,3},$ the existence of distinct algebraic formulations of spinors duals with…
In this paper we derive the most general first-order symmetry operator commuting with the Dirac operator in all dimensions and signatures. Such an operator splits into Clifford even and Clifford odd parts which are given in terms of odd…
We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied in \cite{Ji2}, \cite{CH}, \cite{CMO1},…
We provide a comprehensive lattice formulation of various types of the Dirac operator indices, employing $K$-theory to classify the Wilson Dirac operator via its spectral flow. In contrast to the index of the overlap Dirac operator defined…
This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics--in particular…
The aim of the present paper is to introduce the notion of first order (supersymmetric) Dirac operators on discrete and metric (``quantum'') graphs. In order to cover all self-adjoint boundary conditions for the associated metric graph…
We prove a wave trace singularity formula for a family of generalised Laplacians defined by a Riemannian fibre bundle; for example, the superconnection curvature operator associated to the Bismut superconnection. It is explained how this…
We study a relation between certain extensions of the Clifford bundle and Finsler type structures that naturally generalize the standard Clifford relation between (pseudo)-Riemannian metric structures and Dirac matrices. We show for flat…
Let $M$ be a globally hyperbolic manifold with complete spacelike Cauchy hypersurface $\Sigma$. We prove well-posedness of the Cauchy problem for the Dirac operator on globally hyperbolic manifolds with complete Cauchy hypersurfaces. This…
In this paper we give a proof of the Lefschetz fixed point formula of Freed$^{\rm [1]}$ for an orientation-reversing involution on an odd dimensional spin manifold by using the direct geometric method introduced in [2] and then we…
By requiring unambiguous symmetric quantization leading to the Dirac equation in a curved space, we obtain a special representation of the spin connections in terms of the Dirac gamma matrices and their space-time derivatives. We also…
Quaternionic and octonionic realizations of Clifford algebras and spinors are classified and explicitly constructed in terms of recursive formulas. The most general free dynamics in arbitrary signature space-times for both quaternionic and…