Non-Abelian discrete gauge symmetries in 4d string models

TL;DR

The paper addresses the origin of non-Abelian discrete gauge symmetries in 4d string compactifications, showing they arise from gauging non-Abelian axion isometries and from field identifications, leading to residual discrete groups $P$ with nontrivial relations such as Heisenberg-type commutators ($ ilde{T}_1 ilde{T}_2 = ilde{T}_3 ilde{T}_2 ilde{T}_1$). It develops a macroscopic Lagrangian framework and provides concrete realizations: (i) torsion-homology compactifications where the Hanany–Witten effect yields non-Abelianity; (ii) compactifications with discrete isometries (twisted tori); (iii) magnetized brane settings on ${f T}^2$ and ${f T}^6$ that realize non-Abelian discrete flavour symmetries constraining Yukawa couplings. The results show that these symmetries survive non-perturbative effects, imposing exact selection rules such as Yukawa textures and the suppression of certain proton-decay operators in MSSM-like models. The work thus provides a versatile framework for embedding exact non-Abelian discrete gauge symmetries in string models with potential phenomenological implications for flavor and stability.

Abstract

We study the realization of non-Abelian discrete gauge symmetries in 4d field theory and string theory compactifications. The underlying structure generalizes the Abelian case, and follows from the interplay between gaugings of non-Abelian isometries of the scalar manifold and field identifications making axion-like fields periodic. We present several classes of string constructions realizing non-Abelian discrete gauge symmetries. In particular, compactifications with torsion homology classes, where non-Abelianity arises microscopically from the Hanany-Witten effect, or compactifications with non-Abelian discrete isometry groups, like twisted tori. We finally focus on the more interesting case of magnetized branes in toroidal compactifications and quotients thereof (and their heterotic and intersecting duals), in which the non-Abelian discrete gauge symmetries imply powerful selection rules for Yukawa couplings of charged matter fields. In particular, in MSSM-like models they correspond to discrete flavour symmetries constraining the quark and lepton mass matrices, as we show in specific examples.