Joyce structures and their twistor spaces.(English)Zbl 07972352
This paper provides a systematic introduction to Joyce structures, a class of geometric structures that arise in connection with holomorphic generating functions for Donaldson-Thomas (DT) invariants. They serve as a nonlinear analogue of Frobenius structures and encode the complex hyper-Kähler geometry via a one-parameter family of flat and symplectic nonlinear connections on the tangent bundle of a complex manifold. The study of these structures has significant implications in areas such as integrable systems, wall-crossing phenomena, and cluster varieties.
A key contribution of this work is a detailed analysis of the twistor space associated with Joyce structures, which provides a geometric framework for studying their properties. The author introduces a novel construction of the twistor space as a leaf space of a foliation on \(P^1 \times X\), where \(X\) is the total space of the tangent bundle of a complex manifold. The paper establishes that the twistor space carries a natural holomorphic symplectic structure parametrized by \(P^1\), and the induced fibration exhibits three distinguished fibers, \(Z_0\), \(Z_1\), and \(Z_{\infty}\), which encode various aspects of the underlying geometry.
Among the main results, the paper proves that a Joyce structure naturally induces a complex hyper-Kähler structure on the total space of the tangent bundle. It also derives explicit formulae for the associated symplectic forms and their behavior under the twistor fibration. Furthermore, the author introduces the Plebański function, a generating function satisfying a system of nonlinear partial differential equations known as Plebański’s second heavenly equations, which governs the Joyce structure. The paper also defines and studies the Joyce function, which acts as a Hamiltonian generating function for the \(C^*\)-action on the twistor fiber \(Z_{\infty}\).
The paper provides interesting examples, including those arising from spaces of holomorphic quadratic differentials and their connections to supersymmetric gauge theories of class \(S[A_1]\). The construction also extends to cases related to the Painlevé I equation. Notably, the paper highlights deep connections between Joyce structures and cluster varieties, suggesting further applications in mirror symmetry and Poisson geometry.
A key contribution of this work is a detailed analysis of the twistor space associated with Joyce structures, which provides a geometric framework for studying their properties. The author introduces a novel construction of the twistor space as a leaf space of a foliation on \(P^1 \times X\), where \(X\) is the total space of the tangent bundle of a complex manifold. The paper establishes that the twistor space carries a natural holomorphic symplectic structure parametrized by \(P^1\), and the induced fibration exhibits three distinguished fibers, \(Z_0\), \(Z_1\), and \(Z_{\infty}\), which encode various aspects of the underlying geometry.
Among the main results, the paper proves that a Joyce structure naturally induces a complex hyper-Kähler structure on the total space of the tangent bundle. It also derives explicit formulae for the associated symplectic forms and their behavior under the twistor fibration. Furthermore, the author introduces the Plebański function, a generating function satisfying a system of nonlinear partial differential equations known as Plebański’s second heavenly equations, which governs the Joyce structure. The paper also defines and studies the Joyce function, which acts as a Hamiltonian generating function for the \(C^*\)-action on the twistor fiber \(Z_{\infty}\).
The paper provides interesting examples, including those arising from spaces of holomorphic quadratic differentials and their connections to supersymmetric gauge theories of class \(S[A_1]\). The construction also extends to cases related to the Painlevé I equation. Notably, the paper highlights deep connections between Joyce structures and cluster varieties, suggesting further applications in mirror symmetry and Poisson geometry.
Reviewer: Amedeo Altavilla (Ancona)
MSC:
| 53C26 | Hyper-Kähler and quaternionic Kähler geometry, “special” geometry |
Cite
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