Superconducting strings in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math>

We discuss the appearance of superconducting strings in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>E</a:mi> <a:mn>6</a:mn> </a:msub> </a:math> 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>E</c:mi> <c:mn>6</c:mn> </c:msub> </c:math> 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:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>E</e:mi> <e:mn>6</e:mn> </e:msub> </e:math> symmetry breaking leaves unbroken the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>Z</g:mi> <g:mn>2</g:mn> </g:msub> </g:math> subgroup of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>Z</i:mi> <i:mn>4</i:mn> </i:msub> </i:math> , the center of SO(10). The zero modes of the Standard Model fermions are the charge carriers in this case. Finally, the flux matching condition ensures that the Aharonov-Bohm phase change in going around the metastable strings is an integer multiple of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mn>2</k:mn> <k:mi>π</k:mi> </k:math> for all fields. The fields in the spinorial representation of SO(10) acquire a phase change of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>exp</m:mi> <m:mo stretchy="false">(</m:mo> <m:mo>±</m:mo> <m:mi>i</m:mi> <m:mi>π</m:mi> <m:mo stretchy="false">)</m:mo> </m:math> if taken around the topologically stable <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>Z</q:mi> <q:mn>2</q:mn> </q:msub> </q:math> string.