Topics in Chiral Symmmetry on the Lattice

Abstract

There currently does not exist a good regulator for chiral gauge theories on the lattice. A number of approaches have been attempted, including domain wall fermions and overlap fermions. It has been shown that these fermions both satisfy the Ginsparg-Wilson (GW) relation, a relation constraining chiral symmetry on the lattice. In this thesis, we will discuss various generalizations of GW fermions, as well as developing a novel (continuum) theory of domain wall fermions which may evade some of the shortcomings of ordinary domain wall fermions in describing chiral gauge theories. In Chapter 2, we give a general derivation of Ginsparg-Wilson relations for both Dirac and Majorana fermions in any dimension. These relations encode continuous and discrete chiral, parity, and time-reversal anomalies and will apply to the various classes of free-fermion topological insulators and superconductors (in the framework of a relativistic quantum field theory in Euclidean spacetime). We show how to formulate the exact symmetries of the lattice action and the relevant index theorems for the anomalies. In Chapter 3, we derive the Hamiltonian for a fermion satisfying the GW equation. We work with a solution to the GW equation which is fractional linear in time derivatives. The resulting Hamiltonian is non-local and has ghosts, but is free of doublers and has the correct continuum limit. This construction works in general odd spatial dimensions, and we provide an explicit expression for the Hamiltonian in 1 spatial dimension. In Chapter 4, the theory of fermions in odd dimensional bulk with radial domain wall mass profile is discussed. Edge states localized near the boundary describe the theory of double-valued Weyl fermions on even dimensional spheres, and may be able to evade the doubling problem present on the lattice for ordinary domain wall fermions.