Superconducting Strings in $E_6$

TL;DR

This work investigates superconducting cosmic strings within $E_6$ grand unification, focusing on flux matching between monopole fluxes and string flux tubes to ensure consistency with monopole–antimonopole tunneling on metastable strings. It identifies two concrete $E_6$ breaking paths that yield metastable superconducting strings carrying zero modes of right-handed Majorana neutrinos and a dark matter candidate, with potential signatures in gravitational waves, leptogenesis, and sterile-neutrino phenomenology depending on the breaking scale. A second result establishes a topologically stable superconducting string arising from an unbroken $Z_2$ in the $Z_4$ center of $SO(10)$, which also hosts RH neutrino zero modes and exhibits nontrivial Aharonov–Bohm phases for SM and spinorial fields. Collectively, the paper demonstrates AB-phase quantization and rich phenomenology, including PTA-detectable gravitational waves and dark matter scenarios linked to the string zero modes, thereby connecting GUT-scale topology to observable cosmology.

Abstract

We discuss the appearance of superconducting strings in $E_6$ grand unification, keeping track of the magnetic monopole flux that precedes the formation of the string flux tube. This flux matching ensures compatibility with the quantum tunneling of a monopole-antimonopole pair on a metastable string. We identify two realistic $E_6$ models with superconducting (metastable) strings that also carry zero modes of the right handed Majorana neutrinos and dark matter particles. Depending on the symmetry breaking scale associated with the strings, the latter could be a source of observable gravitational waves, intermediate scale dark matter, and the observed baryon asymmetry via leptogenesis. Topologically stable superconducting strings also appear if the $E_6$ symmetry breaking leaves unbroken the $Z_2$ subgroup of $Z_4$, the center of $SO(10)$. The zero modes of the SM fermions are the charge carriers in this case. Finally, the flux matching condition ensures that the Aharanov-Bohm phase change in going around the metastable strings is an integer multiple of $2 π$ for all fields. The fields in the spinorial representation of SO(10) acquire a phase change of $\exp(\pm iπ)$ if taken around the topologically stable $Z_2$ string.

Figures (4)

• Figure 1: $\psi$ monopole-antimonopole pair connected by the superconducting string with magnetic flux $\psi/4+\chi/4$. The right moving and left moving fermion zero modes are also shown. In the presence of an unbroken $Z_2$ symmetry, the lightest $N_i$ is a plausible dark matter candidate.
• Figure 2: $\psi'$ monopole-antimonopole pair connected by the superconducting string carrying magnetic flux $\psi'/5-Y/5$. The right moving and left moving fermion zero modes are also shown.
• Figure 3: A $\psi$ monopole with flux $\psi'/4+\chi'/4$ ($=\psi/4+\chi/4$), and a $\chi$ antimonopole with flux $\psi'/20 + Y/5$ ($=\chi/5+Y/5$), are connected by $U(1)_{\chi'}$ flux tube as described in the text. They merge together to form a $\psi'$ monopole. This arises from the breaking $E_6 \to SO(10) \times U(1)_\psi \to SU(5)\times U(1)_\chi \times U(1)_\psi \to SU(5)\times U(1)_{\psi'}$.
• Figure 4: Monopole-string system from the breaking $E_6\to SO(10)\to SU(5)\times U(1)_\chi\to SU(3)_c\times SU(2)_L\times U(1)_Y\times Z_2\to SU(3)_c\times U(1)_{\rm em}\times Z_2$. $\mathcal{B}$ denotes the broken electroweak generator orthogonal to $Q$, the electric charge. An open string is stable because of the unbroken $Z_2$ symmetry. The phase change of the matter fields in the spinorial representation of $SO(10)$ is $\exp(\pm i\pi)$ if taken around this superconducting $Z_2$ string with magnetic flux $\chi/10 - \mathcal{B}$ (see Table \ref{['tab:charges']}, second last column.)