Noncommutative resolutions and CICY quotients from a non-abelian GLSM

We discuss a one-parameter non-abelian GLSM with gauge group $(U(1)\times U(1)\times U(1))⋊{\mathbb{Z}}_3$ and its associated Calabi-Yau phases. The large volume phase is a free ${\mathbb{Z}}_3$-quotient of a codimension $3$ complete intersection of degree-$(1,1,1)$ hypersurfaces in ${\mathbb{P}}^2\times {\mathbb{P}}^2\times {\mathbb{P}}^2$. The associated Calabi-Yau differential operator has a second point of maximal unipotent monodromy, leading to the expectation that the other GLSM phase is geometric as well. However, the associated GLSM phase appears to be a hybrid model with continuous unbroken gauge symmetry and cubic superpotential, together with a Coulomb branch. Using techniques from topological string theory and mirror symmetry we collect evidence that the phase should correspond to a non-commutative resolution, in the sense of Katz-Klemm-Schimannek-Sharpe, of a codimension two complete intersection in weighted projective space with $63$ nodal points, for which a resolution has ${\mathbb{Z}}_3$-torsion. We compute the associated Gopakumar-Vafa invariants up to genus $11$, incorporating their torsion refinement. We identify two integral symplectic bases constructed from topological data of the mirror geometries in either phase.