A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model

Abstract

The standard model is a chiral gauge theory where the gauge fields couple to the right-hand and the left-hand fermions differently. The standard model is defined perturbatively and describes all elementary particles (except gravitons) very well. However, for a long time, we do not know if we can have a non-perturbative definition of standard model as a Hamiltonian quantum mechanical theory. In this paper, we propose a way to give a modified standard model (with 48 two-component Weyl fermions) a non-perturbative definition by embedding the modified standard model into a SO(10) chiral gauge theory and then putting the SO(10) chiral gauge theory on a 3D spatial lattice with a continuous time. Such a non-perturbatively defined standard model is a Hamiltonian quantum theory with a finite-dimensional Hilbert space for a finite space volume. Using the defining connection between gauge anomalies and the symmetry-protected topological orders, we show that any chiral gauge theory can be non-perturbatively defined by putting it on a lattice in the same dimension, as long as the chiral gauge theory is free of all anomalies.

Figures (1)

• Figure 1: (a) A SPT state described by a cocycle $\nu \in \cH^{d+1}(G,\R/\Z)$ in $(d+1)$-dimensional space-time. After "gauging" the on-site symmetry $G$, we get a bosonic chiral gauge theory on one boundary and the "mirror" of the bosonic chiral gauge theory on the other boundary. (b) A stacking of a few SPT states in $(d+1)$-dimensional space-time described by cocycles $\nu_i$. If $\sum_i \nu_i=0$, then after "gauging" the on-site symmetry $G$, we get a anomaly-free chiral gauge theory on one boundary. We also get the "mirror" of the anomaly-free chiral gauge theory on the other boundary, which can be gapped without breaking the "gauge symmetry".