Noncommutative resolutions and CICY quotients from a non-abelian GLSM

TL;DR

This work analyzes a one-parameter non-abelian GLSM with gauge group $(U(1)^3)\rtimes\mathbb{Z}_3$, revealing two Calabi–Yau-like phases that share a moduli space yet are not birationally related. The large-volume phase $Y$ is the $\mathbb{Z}_3$-quotient of a codimension-3 CICY in $(\mathbb{P}^2)^3$, while the small-volume phase is a strongly coupled hybrid suggesting a noncommutative resolution of a singular codimension-two intersection, completed via a torsion-refined Gopakumar–Vafa framework. Mirror symmetry is used to identify a smooth deformation $X_{\text{def}}$ as $\mathbb{W}\mathbb{P}^5_{111223}[4,6]$ with 63 nodal points and $\mathbb{Z}_3$ torsion, enabling integral symplectic bases and genus-by-genus GV invariants up to genus $11$. The results illuminate how noncommutative resolutions and torsion refine BPS counting in non-birational Calabi–Yau pairs, and they outline avenues for understanding D-brane categories and phase structure in nonregular GLSMs. The analysis provides concrete data connecting GLSMs, mirror symmetry, and torsion-refined invariants in a novel non-abelian, non-geometric regime.

Abstract

We discuss a one-parameter non-abelian GLSM with gauge group $(U(1)\times U(1)\times U(1))\rtimes\mathbb{Z}_3$ 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 $\mathbb{P}^2\times\mathbb{P}^2\times\mathbb{P}^2$. 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.

Figures (3)

• Figure 1: Relations between this paper's geometries.
• Figure 2: The dashed circle indicates the region of convergence of the Frobenius basis of solutions $\varpi^{(LV)}$ centred at $\varphi=0$. Each X denotes a singularity in the $\varphi$-plane, and the monodromy matrices in \ref{['eq:LVmonodromy']} are computed for the red contours circling each of the singularities. To compute $n^{(LV)}_{\infty}$ we continue along the green curve through the upper-right $\varphi$-quadrant to $\varphi=i\infty$, circle this singularity counter-clockwise (not pictured) and then return along the same green curve.
• Figure 3: Here the dashed circle contains the region of convergence for the Frobenius solutions $\varpi^{(H)}$ expanded about $\phi=0$. Each X is a singularity in the $\phi$-plane, which we encircle with the red contours to compute the monodromy matrices \ref{['eq:PHMonodromies']}. To compute $n^{(H)}_{\infty}$ we continue along the green curve in the lower-right $\phi$-quadrant to $\phi=-i\infty$, which we then encircle counter-clockwise (not pictured) before returning along the green curve.