Nonabelian 2D Gauge Theories for Determinantal Calabi-Yau Varieties

TL;DR

This work extends the GLSM toolkit to nonabelian gauge theories to realize determinantal Calabi–Yau varieties, introducing two dual constructions, the PAX and PAXY models, that impose rank constraints on a defining matrix $A(\phi)$. By analyzing axial anomalies, central charges, and geometric phases, the authors show that the two models flow to the same IR SCFT and describe the same Calabi–Yau moduli as incidence correspondences, $X_A$ and $\hat X_A$, resolving singular determinantal loci. The analysis of linear determinantal varieties reveals a detailed phase structure, including pure and mixed Coulomb branches, with a computable discriminant locus that matches expectations from mirror symmetry where available. The paper applies the framework to explicit examples (quintics and higher-codimension Calabi–Yau manifolds), obtaining topological invariants and confirming flop relations between dual phases, thereby providing new, cross-checked data for non-complete-intersection Calabi–Yau geometries and guiding future explorations of mirrors and non-square determinantal cases.

Abstract

The two-dimensional supersymmetric gauged linear sigma model (GLSM) with abelian gauge groups and matter fields has provided many insights into string theory on Calabi--Yau manifolds of a certain type: complete intersections in toric varieties. In this paper, we consider two GLSM constructions with nonabelian gauge groups and charged matter whose infrared CFTs correspond to string propagation on determinantal Calabi-Yau varieties, furnishing another broad class of Calabi-Yau geometries in addition to complete intersections. We show that these two models -- which we refer to as the PAX and the PAXY model -- are dual descriptions of the same low-energy physics. Using GLSM techniques, we determine the quantum Kähler moduli space of these varieties and find no disagreement with existing results in the literature.

Figures (1)

• Figure 1: The classical vacuum moduli space of the GLSM as a function of the FI parameters $(r_0,r_1)$: the Higgs branch is shaded in grey, and the Coulomb branch locus is the thick solid lines. The phase boundary between $Y_{A^T}$ and $Y_A$ only exists when $\lfloor\sqrt{nk-1}\rfloor \geq k+1$ and ${k_d} \geq 2$. In particular, for $(k,n)=(2,4)$ or $(4,5)$ it is absent. The phase boundary between $Y_{A^T}$ and $X_{A^T}$ is defined by $r_0 + {k_d} r_1 = 0$.