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In this survey article, we discuss the quantum \(\mathcal {A}\) and \(\mathcal {X}\) cluster varieties and indicate how to relate the two settings. We further state the counter-example to quantum positivity of quantum theta functions. At the end, we illustrate that the quantization of cluster \(\mathcal {X}\) varieties can be extended to the deformation families of the \(\mathcal {X}\) varieties and their fibers have Poisson structures.

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Quantum Cluster Varieties, Scattering Diagrams, and the Deformed Poisson Families

  • 会议论文

  • 发表日期：

    2026年

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  • Man Wai Mandy Cheung

摘要

In this survey article, we discuss the quantum \(\mathcal {A}\) and \(\mathcal {X}\) cluster varieties and indicate how to relate the two settings. We further state the counter-example to quantum positivity of quantum theta functions. At the end, we illustrate that the quantization of cluster \(\mathcal {X}\) varieties can be extended to the deformation families of the \(\mathcal {X}\) varieties and their fibers have Poisson structures.