Non-invertible bosonic chiral symmetry on the lattice

TL;DR

This paper constructs a lattice operator for a 3+1D non-invertible ${f Z}_N$ chiral symmetry using $U(1)$ rotor Hilbert spaces, revealing its non-extendability and anomaly structure through duality.

Contribution

It introduces a lattice realization of a non-invertible chiral symmetry in 3+1D and analyzes its properties via duality and anomaly considerations.

Findings

The chiral symmetry generator cannot be extended to the full Hilbert space while preserving locality and unitarity.

A dual description relates the symmetry to an electric one-form symmetry in a charge $1/N$ gauge theory.

The symmetry exhibits a mixed anomaly with the electric one-form symmetry in the dual formulation.

Abstract

In this work we realize the 3 + 1 dimensional non-invertible ${\mathbb{Z}}_N$ chiral symmetry generator as an operator in a many body lattice Hilbert space. A crucial ingredient in our construction is the use of infinite dimensional $U(1)$ rotor site Hilbert spaces. Specifically, our Hilbert space is that of a $U(1)$ lattice gauge theory coupled to a charge $1$ scalar in the Villain formulation, which allows for direct access to monopoles and for a simple definition of a magnetic ${\mathbb{Z}}_N$ one-form symmetry $Z^{(1)}_m$ , at the lattice Hamiltonian level. We construct the generator of the ${\mathbb{Z}}_N$ chiral symmetry as as a unitary operator in the subspace of $Z^{(1)}_m$-invariant states, and show that it cannot be extended to the entire Hilbert space while preserving locality and unitarity. Using a lattice-level duality based on gauging $Z^{(1)}_m$, we find a dual description of this subspace, as the subspace of a charge $1/N$ gauge theory invariant under an electric one-form symmetry $Z^{(1)}_e$. We show that in this dual formulation, the chiral symmetry generator does extend unitarily to the entire Hilbert space, but has a mixed anomaly with the $Z^{(1)}_e$ symmetry.