SciPost Phys. 19, 156 (2025) ·
published 17 December 2025
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We discuss a one-parameter non-Abelian GLSM with gauge group $(U(1)× U(1)× U(1))\rtimes\mathbb{Z}_3$ 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 $\mathbb{P}^2×\mathbb{P}^2×\mathbb{P}^2$. 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.
SciPost Phys. 17, 165 (2024) ·
published 12 December 2024
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A hybrid model is a fibration of a Landau-Ginzburg orbifold over a geometric base. We study B-type D-branes in hybrid models. Imposing B-type supersymmetry on the boundary action, we show that D-branes are specified by matrix factorisations in the fibre direction, together with some geometric data associated to the base. We also deduce conditions for the compatibility of these branes with the bulk orbifold actions and R-symmetry. We construct examples of hybrid B-branes which are generalisations of well-studied branes in geometric and Landau-Ginzburg models. Hybrid models can arise at limiting points of the stringy Kähler moduli space of Calabi-Yaus, and can be realised as phases of the corresponding gauged linear sigma models (GLSMs). Using GLSM techniques, we establish connections between geometric branes and hybrid branes. As explicit examples, we consider one- and two-parameter Calabi-Yau hybrids with a $\mathbb{P}^1$-base.
