Conformal invariance in 2-dimensional discrete field theory

TL;DR

This paper demonstrates that certain two-dimensional discrete field theories exhibit exact solutions to their continuous counterparts due to a discrete form of conformal invariance, extending to more general models including bosonic and fermionic fields.

Contribution

It introduces a class of lattice field theories with exact solutions to continuous equations, highlighting the role of discrete conformal invariance and generalizing to models akin to the Wess-Zumino-Witten model.

Findings

Discrete massless wave equation solutions match continuous solutions.

Discrete conformal invariance explains exact solvability.

Generalized models include coupled bosonic and fermionic fields.

Abstract

A discretized massless wave equation in two dimensions, on an appropriately chosen square lattice, exactly reproduces the solutions of the corresponding continuous equations. We show that the reason for this exact solution property is the discrete analog of conformal invariance present in the model, and find more general field theories on a two-dimensional lattice that exactly solve their continuous limit equations. These theories describe in general non-linearly coupled bosonic and fermionic fields and are similar to the Wess-Zumino-Witten model.