A new approach to putting chiral gauge theory on the lattice

 

Abstract: The standard model of particle physics a type of gauge theory called a chiral gauge theory.  Although we know how to study such theories in perturbation theory (using Feynman diagrams), we do not know how to define or study such theories non-perturbative.  In particular (in contrast to the the theory of the strong force, QCD), we don’t know how to formulate a realistic chiral gauge theory like the standard model on the lattice.  This is a fundamental problem that does not receive as much attention as it deserves: it is likely a clue that we are missing something important about the geometric meaning of fermions/spinors.

 

In this talk, we propose a new way to formulate certain realistic chiral gauge theories like the standard model on a lattice (or a general simplicial complex in curved spacetime), so that it has the correct continuum limit, with the correct symmetries and (co)homological properties, and no unwanted “fermion doubling.” Building on recent progress by Catterall and collaborators, our approach uses a so-called “restricted Kahler-Dirac fermions” (spinors with a dual interpretation as differential forms). To obtain the right continuum limit, we find we must appropriately couple the Kahler-Dirac fermions to the geometry of spacetime (described by the tetrad and spin connection one-forms). Conceptually, the crucial new point in our formulation is to carefully distinguish between the diffeomorphism group (which is broken by discretization) and the local Lorentz group (which is exactly preserved).  I will discuss how this formulation hints at a partial explanation why the standard model has the structure that it does.