A formulation of discrete differential geometry applied to fermionic lattice field theory and its implications for chiral symmetry

Abstract

In this thesis, we develop the Geometric Discretization formulation of Dirac-Kahler fermions. We note that the naive definition of chiral synnnetry is only approximately captured in the formulation. However, we show that we can use the two complexes associated with the definition of the Hodge star to construct chiral projection operators that exactly project a different chirality of held on to each complex. Similarly, we construct flavour projection operators that project a different flavour of field on to each complex. We also see that, in two space-time dimensions, we need four complexes to simultaneously separate the chiral and flavour components of the field.