Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. For the local boundary conditions, limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.

We compute the axial anomaly for the Taub-NUT metric on R. We show that the axial anomaly for the generalized Taub-NUT metrics introduced by Iwai and Katayama is finite, although the Dirac operator is not Fredholm. We show that the essential spectrum of the Dirac operator is the whole real line. Pacs: 04.62.+v

We establish an upper estimate for the small eigenvalues of the twisted Dirac operator on Kähler submanifolds in Kähler manifolds carrying Kählerian Killing spinors. We then compute the spectrum of the twisted Dirac operator of the canonical embedding CP d → CP n in order to test the sharpness of the upper bounds.

Consider a Riemannian spin manifold of dimension n ≥ 3 and denote by D the Dirac operator acting on spinor fields. A solution of the Einstein-Dirac equation is a spinor field ψ solving the equations Ric − 1 2 S · g = ± 1 4 T ψ , D(ψ) = λψ. Here S denotes the scalar curvature of the space, λ is a real constant and T ψ is the energy-momentum tensor of the spinor field ψ defined by the formula

The Seiberg-Witten equations are studied from the viewpoint of gauge potential decomposition. We find a determinant equation ∆Aμ = −λAμ for the twisting U(1) potential Aμ of the Seiberg-Witten theory, which is in itself an eigenvalue problem of the Laplacian operator, with the eigenvalue being the vacuum expectation value of the field function, λ = ‖Φ‖ /2. This establishes a direct relationship...

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.

Let M be a compact manifold with a metric g and with a fixed spin structure χ. Let λ + 1 (g) be the first non-negative eigenvalue of the Dirac operator on (M, g, χ). We set τ (M, χ) := sup inf λ + 1 (g) where the infimum runs over all metrics g of volume 1 in a conformal class [g 0 ] on M and where the supremum runs over all conformal classes [g 0 ] on M. Let (M # , χ #) be obtained from (M, χ)...

We study the adiabatic limit of the eta invariant of the Dirac operator over cofinite quotient of PSL(2,R), which is a noncompact manifold with a nonexact fibred-cusp metric near the ends.

Let M be a compact manifold with a fixed spin structure χ. The Atiyah-Singer index theorem implies that for any metric g on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and χ. We show that for generic metrics on M this bound is attained.

A remarkable feature of a lattice Dirac operator is discussed. Unlike the Dirac operator for massless fermions in the continuum, this Ginsparg-Wilson lattice Dirac operator does not possess topological zero modes for any topologically-nontrivial background gauge fields, even though it is exponentially-local, doublers-free, and reproduces correct axial anomaly for topologically-trivial gauge con...