I derive the set of constraints under which the eigenvalues of the Dirac operator can play the role of the dynamical variables for the Euclidean supergravity.

An elliptic Fourier integral operator of order 0, associated to a homogeneous canonical diffeomorphism, on a compact manifold is Fredholm on L2. The index may be expressed as the sum of a term, which we call the contact degree, associated to the canonical diffeomorphism and a term, computable by the Atiyah-Singer theorem, associated to the symbol. The contact degree is shown to be defined for a...

We review some recent results concerning lower eigenvalues estimates for the Dirac operator [6, 7]. We show that Friedrich’s inequality can be improved via certain well-chosen symmetric tensors and provide an application to Sasakian spin manifolds.

New exact upper and lower bounds are derived on the spectrum of the square of the hermitian Wilson Dirac operator. It is hoped that the derivations and the results will be of help in the search for ways to reduce the cost of simulations using the overlap Dirac operator. The bounds also apply to the Wilson Dirac operator in odd dimensions and are therefore relevant to domain wall fermions as well.

Let L be the differential operator Ly = i ( 1 0 0 −1 ) dy dx + ( 0 P (x) Q(x) 0 ) y, y = ( y1 y2 ) , where P (x), Q(x) are 1-periodic functions such that Q(x) = P (x). The operator L, considered on [0, 1] with periodic (y(0) = y(1)), or antiperiodic (y(0) = −y(1)) boundary conditions, is self-adjoint, and moreover, for large |n| it has, close to nπ, a pair of periodic (if n is even), or antiper...

We solve for spectrum, obtain explicitly and study group properties of eigenfunc-tions of Dirac operator on the Riemann sphere S 2. The eigenvalues λ are nonzero integers. The eigenfunctions are two-component spinors that belong to representations of SU (2)-group with half-integer angular momenta l = |λ| − 1 2. They form on the sphere a complete orthonormal functional set alternative to convent...

It is shown that on a compact spin symmetric space with a Kähler or Quaternion-Kähler structure, the first eigenvalue of the Dirac operator is linked to a “lowest” action of the holonomy, given by the fiberwise action on spinors of the canonical forms characterized by this holonomy. The result is also verified for the symmetric space F4/Spin9, proving that it is valid for all the “possible” hol...

The decomposition of polynomials in terms of spherical harmonics is widely used in various branches of analysis. In this paper we describe a set of REDUCE procedures generating this decomposition and its more general, monogenic, counterpart in Clifford analysis. We then illustrate their use by inverting the Laplacian and the Dirac operator on both Euclidean and Minkowski spaces. Let ~k denote t...