On the Spectrum of Lattice Dirac Operators

Relativistic particle physics is described by relativistic quantum field theory; the theories explaining the fundamental interactions are gauge theories. QFT as it is formulated in terms of Lagrangians and functional integration has to be regularized. The only known regularization scheme that retains the gauge symmetry is replacement of the space-time continuum by a space-time lattice. For convenience and other reasons this is done in an Euclidean world. The most prominent QFT is QCD, the SU(3) gauge theory of quarks and gluons. There various non-perturbative phenomena appear intertwined: confinement and chiral symmetry breaking. The classical continuum gauge fields A that are continuous and differentiable, living on compact manifolds, may be classified by a topological quantum number (the Pontryagin index) Q(A). Changing through continuous deformations of the field from one such topologically defined sector to another is impossible. Topology is closely related to the fermion zero modes; these are eigenstates (eigenvalue E = 0) of the Dirac operator