Three-family supersymmetric Pati-Salam models from intersecting D6-branes on rigid cycles

Intersecting D6-brane models without discrete torsion typically suffer from unstabilized open-string moduli arising from D-brane positions and Wilson lines. These moduli generate additional massless adjoint fields, obstructing the realization of negative <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>β</a:mi></a:math> functions necessary for asymptotic freedom unless they are decoupled around the string scale. A viable solution involves utilizing rigid cycles, which eliminate these unwanted adjoint fields. In this work, we for the first time present a class of consistent three-family supersymmetric Pati-Salam models from rigid intersecting D6-branes on the factorizable <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi mathvariant="double-struck">T</c:mi><c:mn>6</c:mn></c:msup><c:mo>/</c:mo><c:mo stretchy="false">(</c:mo><c:msub><c:mi mathvariant="double-struck">Z</c:mi><c:mn>2</c:mn></c:msub><c:mo>×</c:mo><c:msubsup><c:mi mathvariant="double-struck">Z</c:mi><c:mn>2</c:mn><c:mo>′</c:mo></c:msubsup><c:mo stretchy="false">)</c:mo></c:math> orientifold with discrete torsion. These models satisfy all the known consistency conditions, including <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi mathvariant="script">N</j:mi><j:mo>=</j:mo><j:mn>1</j:mn></j:math> supersymmetry, K-theory constraints, tadpole cancellation, and recent swampland bounds on the maximal gauge group rank. We provide detailed particle spectra, analyze their phenomenological implications, and discuss the decoupling of exotic states through strong dynamics in the hidden sector.