In this paper, we give an optimal lower bound for the eigenvalues of the basic Dirac operator on a quaternion-Kähler foliations. The limiting case is characterized by the existence of quaternion-Kähler Killing spinors. We end this paper by giving some examples.

Let D be a homogeneous Dirac operator on the quotient M = G=H of two compact connected Lie groups. We construct a deformation ~ D of D and calculate its equivariant-invariant G (~ D) explicitly on the dense subset G 0 of G that acts freely on M. On G 0 , G (~ D) and G (D) diier only by a virtual character of G.

The talk will give an overview of our further development of the paper [7] by Robert Downes, Michael Levitin and Dmitri Vassiliev and it will also give an insight how spectrum of massless Dirac operator on a 3-manifold interplays with geometric contents of the manifold. In contrast to the Riemann flat manifold studied in [7], 3-torus, we study the massless Dirac operator on a 3-sphere equipped ...

Inspired on a decomposition of the lattice Laplacian operator into massive terms (coming from the use of the block renormalization group transformation for bosonic systems), we establish a telescopic decomposition of the Dirac operator into massive terms, with a property named \orthogonality between scales". Making a change of Grassmann variables and writing the initial elds in terms of the eig...

In the present paper we extend the methods of the Weierstrass (or spinor) representation of surfaces in R [10, 11] and SU(2) = S [12] for surfaces in the threedimensional Lie groups Nil , S̃L2, and Sol endowed with so-called Thurston’s geometries [9]. The main feature of this approach is that the geometry of a surface is related to the spectral properties of the corresponding Dirac operator. The...

Recent results on the spectral properties of the Hermitian Wilson-Dirac operator are presented.

We prove a Kastler-Kalau-Walze type theorem for the Dirac operator and the signature operator for 3, 4-dimensional manifolds with boundary. As a corollary, we give two kinds of operator theoretic explanations of the gravitational action in the case of 4-dimensional manifolds with flat boundary. Subj. Class.: Noncommutative global analysis; Noncommutative differential geometry. MSC: 58G20; 53A30...

The distribution of individual Dirac eigenvalues is derived by relating them to the density and higher eigenvalue correlation functions. The relations are general and hold for any gauge theory coupled to fermions under certain conditions which are stated. As a special case, we give examples of the lowest-lying eigenvalue distributions for QCD-like gauge theories without making use of earlier re...

In this paper we build on the framework developed in [7, 8, 9] to obtain a more complete understanding of the gluing properties for indices of boundary value problems for the SpinC-Dirac operator with sub-elliptic boundary conditions. We extend our analytic results for sub-elliptic boundary value problems for the SpinC-Dirac operator, and gluing results for the indices of these boundary problem...

It is well-known that a compact Riemannian spin manifold (M, g) can be reconstructed from its canonical spectral triple (C∞(M), L2(M,ΣM),D) where ΣM denotes the spinor bundle and D the Dirac operator. We show that g can be reconstructed up to conformal equivalence from (C∞(M), L2(M,ΣM), sign(D)).