Noncommutative resolutions and CICY quotients from a non-abelian GLSM

Abstract

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.

58 pages. Ancillary notebook contains GV invariants. Referee comments included, typos fixed

References in corpus (60)

• Phases of $N=2$ Theories In Two Dimensions
• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
• Holomorphic Anomalies in Topological Field Theories
• Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces
• Curve counting via stable pairs in the derived category
• Black Hole Condensation and the Unification of String Vacua
• Mirror Symmetry, Mirror Map and Applications to Complete Intersection Calabi-Yau Spaces
• On Orbifolds with Discrete Torsion
• Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N=(2,2) Theories
• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• Counting higher genus curves in a Calabi-Yau manifold
• Non-commutative Donaldson-Thomas theory and the conifold
• Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections
• Non-birational twisted derived equivalences in abelian GLSMs
• On Free Quotients of Complete Intersection Calabi-Yau Manifolds
• Homological projective duality for Grassmannians of lines
• Tables of Calabi--Yau equations
• New Calabi-Yau Manifolds with Small Hodge Numbers
• Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary
• Exact Results in Supersymmetric Field Theories on Manifolds with Boundaries
• Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties
• M-Theory and Topological Strings--I
• M-Theory and Topological Strings--II
• 13/2 ways of counting curves
• Worldsheet Instantons and Torsion Curves, Part B: Mirror Symmetry
• Polynomial Structure of the (Open) Topological String Partition Function
• Topological String Theory on Compact Calabi-Yau: Modularity and Boundary Conditions
• Linear Sigma Models With Strongly Coupled Phases -- One Parameter Models
• Decompactifications and Massless D-Branes in Hybrid Models
• Stable Singularities in String Theory
• Worldsheet Instantons and Torsion Curves, Part A: Direct Computation
• Hodge Numbers for All CICY Quotients
• Completing the Web of $Z_3$ - Quotients of Complete Intersection Calabi-Yau Manifolds
• Calabi-Yau Threefolds With Small Hodge Numbers
• Reaching for Squarks and Gauginos at a 100 TeV p-p Collider
• Categories of Massless D-Branes and del Pezzo Surfaces
• Hodge Numbers for CICYs with Symmetries of Order Divisible by 4
• Dual Pairs of Gauged Linear Sigma Models and Derived Equivalences of Calabi-Yau threefolds
• Hybrid conformal field theories
• Higher genus Gromov-Witten invariants of the Grassmannian, and the Pfaffian Calabi-Yau threefolds
• Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges
• Black Hole Condensation and Duality in String Theory
• GLSM realizations of maps and intersections of Grassmannians and Pfaffians
• Topological Strings on Grassmannian Calabi-Yau manifolds
• Quantum geometry, stability and modularity
• GLSMs, joins, and nonperturbatively-realized geometries
• Topological Strings on Non-Commutative Resolutions
• A pair of Calabi-Yau manifolds from a two parameter non-Abelian gauged linear sigma model
• The 3 x 3 x 3 hyperdeterminant as a polynomial in the fundamental invariants for SL(3,C) x SL(3,C) x SL(3,C)
• Geometry of symmetric determinantal loci
• Explicit formulae for rank zero DT invariants and the OSV conjecture
• B-brane transport in anomalous (2,2) models and localization
• Classical Weight-Four L-value Ratios as Sums of Calabi--Yau Invariants
• Counting Calabi-Yau Threefolds
• Gromov-Witten theory and Donaldson-Thomas theory, I
• The fundamental invariants of 3 x 3 x 3 arrays
• A GLSM view on Homological Projective Duality
• Calabi-Yau Period Geometry and Restricted Moduli in Type II Compactifications
• Gromov-Witten theory and Donaldson-Thomas theory, II
• Topological String on elliptic CY 3-folds and the ring of Jacobi forms