Wess-Zumino-Witten term on the lattice

TL;DR

This paper constructs the Wess-Zumino-Witten term on the lattice using Ginsparg-Wilson Dirac operators, reproducing continuum topological properties and exploring implications for gauge anomalies and global SU(2) anomaly.

Contribution

It introduces a lattice formulation of the WZW term that preserves continuum topological features and clarifies the role of gauge anomalies and multivaluedness in lattice gauge theory.

Findings

Lattice WZW term reproduces continuum topological properties.

Gauge anomaly affects the basis of Weyl fermions and expectation values.

Multivaluedness of the Witten term relates to homotopy groups and anomalies.

Abstract

We construct the Wess-Zumino-Witten (WZW) term in lattice gauge theory by using a Dirac operator which obeys the Ginsparg-Wilson relation. Topological properties of the WZW term known in the continuum are reproduced on the lattice as a consequence of a non-trivial topological structure of the space of admissible lattice gauge fields. In the course of this analysis, we observe that the gauge anomaly generally implies that there is no basis of a Weyl fermion which leads to a single-valued expectation value in the fermion sector. The lattice Witten term, which carries information of a gauge path along which the gauge anomaly is integrated, is separated from the WZW term and the multivaluedness of the Witten term is shown to be related to the homotopy group $\pi_{2n+1}(G)$. We also discuss the global $\SU(2)$ anomaly on the basis of the WZW term.