Comments on "Ether of Orbifolds"

TL;DR

This paper critiques and clarifies misconceptions in a recent work on orbifold lattice Hamiltonians, emphasizing correct interpretations of gauge invariance and related quantities.

Contribution

It corrects previous claims about gauge invariance and the interpretation of a key quantity in orbifold lattice formulations, clarifying foundational misunderstandings.

Findings

$oldsymbol{ ext{The quantity }oldsymbol{ extepsilon_g} ext{ measures lattice spacing shift, not gauge violation.}}$

$oldsymbol{ ext{Misinterpretation of }oldsymbol{ extepsilon_g} ext{ as departure from SU(N) is incorrect.}}$

$oldsymbol{ ext{Clarifies the correct gauge invariance properties in orbifold lattice Hamiltonians.}}$

Abstract

We comment on a recent manuscript "Ether of Orbifolds" by Henry Lamm. In the first version, it was mistakenly claimed that the orbifold lattice Hamiltonian is not gauge invariant, and a quantity $\epsilon_g$, which has nothing to do with a non-existent "gauge violation", was introduced. The scaling of this $\epsilon_g$ was used to claim a huge simulation cost. In fact, $\epsilon_g$ characterizes the shift of the effective lattice spacing -- because, in the orbifold lattice formulation, the lattice is generated dynamically from the vacuum expectation value of the complex matrices. In the second version, the claim about the gauge symmetry was partially corrected, based on our comments. However, $\epsilon_g$ is still mistakenly interpreted as a measure of "departure from SU($N$)", inconsistently with the foundational results by Kaplan, Katz, and \"{U}nsal, and also by Arkani-Hamed, Cohen, and Georgi. This interpretation plays a central role in sustaining the argument introduced in the first version.