Necessary and sufficient conditions for non-perturbative equivalences of large N orbifold gauge theories

TL;DR

<3-5 sentence high-level summary>This paper formalizes when non-perturbative large-$N_c$ equivalence holds between a parent $U(N_c)$ gauge theory with adjoint matter and its orbifold daughter theories. By constructing large-$N_c$ dynamics as classical mechanics on a coherence-group–generated phase space, it shows that equivalence holds in the neutral sectors provided the orbifold projection symmetry (parent) and the theory-space translation symmetry (daughter) are not spontaneously broken, and it proves the sufficiency of these symmetry conditions. The key results include an isomorphism between neutral coherence subalgebras, identical neutral-observable dynamics, and identical neutral large-$N$ Hamiltonians, extending previous perturbative and strong-coupling proofs to arbitrary phases. The findings clarify how ground-state properties, spectra, and scattering amplitudes align across parent and daughter theories in the large-$N_c$ limit and delineate when equivalence can fail due to symmetry breaking.

Abstract

Large N coherent state methods are used to study the relation between U(N) gauge theories containing adjoint representation matter fields and their orbifold projections. The classical dynamical systems which reproduce the large N limits of the quantum dynamics in parent and daughter orbifold theories are compared. We demonstrate that the large N dynamics of the parent theory, restricted to the subspace invariant under the orbifold projection symmetry, and the large N dynamics of the daughter theory, restricted to the untwisted sector invariant under "theory space'' permutations, coincide. This implies equality, in the large N limit, between appropriately identified connected correlation functions in parent and daughter theories, provided the orbifold projection symmetry is not spontaneously broken in the parent theory and the theory space permutation symmetry is not spontaneously broken in the daughter. The necessity of these symmetry realization conditions for the validity of the large N equivalence is unsurprising, but demonstrating the sufficiency of these conditions is new. This work extends an earlier proof of non-perturbative large N equivalence which was only valid in the phase of the (lattice regularized) theories continuously connected to large mass and strong coupling.