Open Gromov-Witten theory and the crepant resolution conjecture. (English) Zbl 1273.14111
In this paper the authors combine two important aspects of Gromov-Witten (GW) theory, which they refer to as “Crepant transformation” and “Gluing”. In their own words (in “Summary of results”) these mean:
“Crepant transformation: the equivalence between GW theories of two targets related by a crepant birational transformation. In particular, when the crepant transformation is the resolution of singularities of a Gorenstein orbifold, this equivalence is referred to as the crepant resolution conjecture (CRC)”
“Gluing: the ability to recover GW invariants for a toric variety/orbifold from open invariants of open subspaces covering the target.”
The interplay between these two ideas is presented in all details in the example of a specific geometry, so that, by combining crepant transformation and gluing, Gromov-Witten invariants of \(X=\mathcal{O}_{\mathbb{P}^1}(-1) \otimes \mathcal{O}_{\mathbb{P}^1}(-1) / \mathbb{Z}_2\) are computed. In the process the authors prove four theorems for related geometries, which concern: crepant resolution conjecture for the open GW invariants, Ruan’s crepant resolution conjecture for closed GW invariants, and gluing of \(X\) and of its crepant resolution from appropriate open invariants. The paper combines several independent ideas in a nice unified context, and can be recommended not only to mathematicians (and algebraic geometers in particular), but also to mathematical and theoretical physicists (interested in the areas of topological string theory, topological vertex, etc.).
“Crepant transformation: the equivalence between GW theories of two targets related by a crepant birational transformation. In particular, when the crepant transformation is the resolution of singularities of a Gorenstein orbifold, this equivalence is referred to as the crepant resolution conjecture (CRC)”
“Gluing: the ability to recover GW invariants for a toric variety/orbifold from open invariants of open subspaces covering the target.”
The interplay between these two ideas is presented in all details in the example of a specific geometry, so that, by combining crepant transformation and gluing, Gromov-Witten invariants of \(X=\mathcal{O}_{\mathbb{P}^1}(-1) \otimes \mathcal{O}_{\mathbb{P}^1}(-1) / \mathbb{Z}_2\) are computed. In the process the authors prove four theorems for related geometries, which concern: crepant resolution conjecture for the open GW invariants, Ruan’s crepant resolution conjecture for closed GW invariants, and gluing of \(X\) and of its crepant resolution from appropriate open invariants. The paper combines several independent ideas in a nice unified context, and can be recommended not only to mathematicians (and algebraic geometers in particular), but also to mathematical and theoretical physicists (interested in the areas of topological string theory, topological vertex, etc.).
Reviewer: Piotr Sulkowski (Amsterdam)
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
Keywords:
Gromow-Witten invariants; open Gromov-Witten invariants; crepant transformation; toric geometryReferences:
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