Noncommutative resolutions and CICY quotients from a non-Abelian GLSM

We discuss a one-parameter non-Abelian GLSM with gauge group (U(1)× U(1)× U(1))\rtimes\mathbb{Z}_3 ( U ( 1 ) × U ( 1 ) × U ( 1 ) ) ⋊ ℤ 3 and its associated Calabi-Yau phases. The large volume phase is a free \mathbb{Z}_3 ℤ 3 -quotient of a codimension 3 3 complete intersection of degree- (1,1,1) ( 1 , 1 , 1 ) hypersurfaces in \mathbb{P}^2×\mathbb{P}^2×\mathbb{P}^2 ℙ 2 × ℙ 2 × 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 63 nodal points, for which a resolution has \mathbb{Z}_3 ℤ 3 -torsion. We compute the associated Gopakumar-Vafa invariants up to genus 11 11 , incorporating their torsion refinement. We identify two integral symplectic bases constructed from topological data of the mirror geometries in either phase.