Topological strings on non-commutative resolutions. (English) Zbl 1541.81095
Summary: In this paper we propose a definition of torsion refined Gopakumar-Vafa (GV) invariants for Calabi-Yau threefolds with terminal nodal singularities that do not admit Kähler crepant resolutions. Physically, the refinement takes into account the charge of five-dimensional BPS states under a discrete gauge symmetry in M-theory. We propose a mathematical definition of the invariants in terms of the geometry of all non-Kähler crepant resolutions taken together. The invariants are encoded in the A-model topological string partition functions associated to non-commutative (nc) resolutions of the Calabi-Yau. Our main example will be a singular degeneration of the generic Calabi-Yau double cover of \({\mathbb{P}}^3\) and leads to an enumerative interpretation of the topological string partition function of a hybrid Landau-Ginzburg model. Our results generalize a recent physical proposal made in the context of torus fibered Calabi-Yau manifolds by one of the authors and clarify the associated enumerative geometry.
MSC:
| 81T10 | Model quantum field theories |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 14D06 | Fibrations, degenerations in algebraic geometry |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
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