Related papers: Dirac Operator on Complex Manifolds and Supersymme…

The second order symmetry operators that commute with the Dirac operator with external vector, scalar and pseudo-scalar potentials are computed on a general two-dimensional spin-manifold. It is shown that the operator is defined in terms of…

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…

In this paper, we consider the symmetries of the Dirac operator derived from a connection with skew-symmetric torsion. We find that the generalized conformal Killing-Yano tensors give rise to symmetry operators of the massless Dirac…

We consider Dirac-like operators with piecewise constant mass terms on spin manifolds, and we study the behaviour of their spectra when the mass parameters become large. In several asymptotic regimes, effective operators appear: the…

On complete non-compact manifolds with bounded sectional curvature, we consider a class of self-adjoint Dirac-type operators called Dirac-Schr\"odinger operators. Assuming two Dirac-Schr\"odinger operators coincide at infinity, by previous…

We give more details about an integrable system in which the Dirac operator D=d+d^* on a finite simple graph G or Riemannian manifold M is deformed using a Hamiltonian system D'=[B,h(D)] with B=d-d^* + i b. The deformed operator D(t) = d(t)…

In this lecture I will report on some recent progress in understanding the relation of Dirac operators on Clifford modules over an even-dimensional closed Riemannian manifold $M$\ and (euclidean) Einstein-Yang-Mills-Higgs models.

We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin manifolds of signature $(p,q)$, considered as an unbounded operator in the Hilbert space $L^2_\xi(S)$. The definition of $L^2_\xi(S)$ involves the choice of a…

We obtain a generalization, for a general compact almost complex manifold, of the well-known K\"{a}hler (or Hodge) identities for K\"{a}hler manifolds involving the commutators of the exterior differential and the Lefschetz operator and its…

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 introduce a family of metric-deformed Heisenberg algebras $M_1$ and $M_2$, where the commutation relations are expressed directly in terms of the components of a diagonal Lorentzian metric. We show that these algebras unify several known…

Let $M$ be an oriented even-dimensional Riemannian manifold on which a discrete group $\Gamma$ of orientation-preserving isometries acts freely, so that the quotient $X=M/\Gamma$ is compact. We prove a vanishing theorem for a half-kernel of…

We study a natural Dirac operator on a Lagrangian submanifold of a K\"ahler manifold. We first show that its square coincides with the Hodge-de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and…

In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families…

In this paper we prove that Dirac operators on non-compact complete orbifolds which are sufficiently regular at infinity, admit a unique extension. Additonally, we prove a generalized orbifold Stokes'/Divergence theorem.

We observe that the Hamiltonian H = D^2, where D is the flat 4d Dirac operator in a self-dual gauge background, is supersymmetric, admitting 4 different real supercharges. A generalization of this model to the motion on a curved conformally…

Assume that the compact Riemannian spin manifold $(M^n,g)$ admits a $G$-structure with characteristic connection $\nabla$ and parallel characteristic torsion ($\nabla T=0$), and consider the Dirac operator $D^{1/3}$ corresponding to the…

Potential algebras can be used effectively in the analysis of the quantum systems. In the article, we focus on the systems described by a separable, 2x2 matrix Hamiltonian of the first order in derivatives. We find integrals of motion of…

We assume that the manifold with boundary, X, has a Spin_C-structure with spinor bundle S. Along the boundary, this structure agrees with the structure defined by an infinite order integrable almost complex structure and the metric is…

A new class of semi-analytically solvable MHD alpha^2-dynamos is found based on a global diagonalization of the matrix part of the dynamo differential operator. Close parallels to SUSY QM are used to relate these models to the Dirac…