Chiral Lattice Gauge Theories Via Mirror-Fermion Decoupling: A Mission (im)Possible?

TL;DR

The paper surveys the longstanding problem of formulating chiral gauge theories on the lattice and analyzes the mirror-fermion decoupling strategy as a viable path when implemented with Ginsparg-Wilson fermions. It explains the fundamental obstacles—measure ambiguities, sign problems, and fermion doubling—and how GW fermions address doubling while introducing new measure- and anomaly-related challenges. Through the 1-0 2d model and the splitting theorem, the authors demonstrate that strong-mirror dynamics can yield a heavy mirror sector while preserving the light chiral spectrum in a way compatible with ’t Hooft anomaly matching, and they connect these results to domain-wall formalisms. The findings provide partial, yet substantive, evidence that anomaly matching can be satisfied in the mirror sector and suggest concrete directions for extending these ideas to anomaly-free 4d theories, with 2d simulations offering a tractable proving ground. Overall, the work highlights a promising framework for lattice chiral gauge theories that respects anomalies and unitarity, while outlining key open questions and future research directions.

Abstract

This is a review of the status and outstanding issues in attempts to construct chiral lattice gauge theories by decoupling the mirror fermions from a vectorlike theory. In the first half, we explain why studying nonperturbative chiral gauge dynamics may be of interest, enumerate the problems that a lattice formulation of chiral gauge theories must overcome, and briefly review our current knowledge. We then discuss the motivation and idea of mirror-fermion decoupling and illustrate the desired features of the decoupling dynamics by a simple solvable toy model. The role of exact chiral symmetries and matching of 't Hooft anomalies on the lattice is also explained. The second, more technical, half of the article is devoted to a discussion of the known and unknown features of mirror-decoupling dynamics formulated with Ginsparg-Wilson fermions. We end by pointing out possible directions for future studies.

Figures (6)

• Figure 1: Spectrum of the single-site hamiltonian (\ref{['toysinglesite']}) of the $\lambda \rightarrow \infty$ limit our toy model (\ref{['4fermitoy']}, \ref{['kinetictoy']}). The ground state is unique, has a gap $\lambda$ in lattice units, and preserves $SU(4)$. Taking hopping terms into account causes massive excitations to propagate between adjacent sites on the lattice.
• Figure 2: In order to lift the mirror-fermion zero modes in an instanton background, the mirror interactions have to break the anomalous mirror global symmetries as well.
• Figure 3: Scalar-field susceptibility (\ref{['chiscalar']}) of the mirror sector of the "1-0" model for $\kappa=0.1$, at $y\rightarrow \infty$. The dashed line indicates the susceptibilities for the pure $XY$ model with the same $\kappa$ (undistinguishable, within errors for $N=4,8,16$). Large errors at $h =$ 0.7 and 0.8 are due to the sign problem at $h<1$.
• Figure 4: The real parts of $\Pi_{00}$ and $\Pi_{10}$ of the mirror for $\kappa = 0.1$, $h=2$, as a function of momentum approaching the origin in different directions. The value of the discontinuity indicates that when the system is in the strong-coupling symmetric phase, a single massless charged chiral fermion exists.
• Figure 5: The real parts of $\Pi_{00}$ and $\Pi_{10}$ of the mirror for $\kappa = 5$, $h=2$ indicating the system is in a "broken" (algebraically ordered) phase where a single massless "goldstone" scalar appears, as explained inPoppitz:2009gt.