Spectral description of non-commutative local systems on surfaces and non-commutative cluster varieties
Spectral description of non-commutative local systems on surfaces and non-commutative cluster varieties
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表面上非交换局域系统和非交换簇簇的谱描述
DOI：
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发表时间：
2021
期刊：
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通讯作者：
M. Kontsevich
中科院分区：
文献类型：
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作者：
A. Goncharov;M. Kontsevich
设R为非交换域。我们证明了m维R-向量空间中的标志的一般三元组是由具有(m-1)(m-2)/2个洞的蜂窝图上的平坦R-线丛来刻画的。推广这一点，我们证明了装饰曲面上S的框架秩m平坦R向量丛的非对易堆叠X(m，Poisson)与赋于S上某些二部图的谱表面上的平坦线丛的模空间是生同的。它们具有规范的非对易泊松结构。空间X(m，S)在映射类群的作用下具有非交换簇Poisson簇的等变结构。对于环面上的二部图，我们得到了非对易二聚簇的可积系。我们定义了与二部带状图相关的非对易簇A-簇。它们带有规范的非对易2-形式。S上的扭曲装饰局部系的非对易堆栈A(m，S)在映射类群的作用下具有簇A簇结构，是等变的。空间A(m，S)上的非对易星团A坐标是Gelfand-Retakh拟行列式的比值。在m=2的情况下，这恢复了与曲面相关的Berenstein-Retakh非交换簇代数。我们引入了允许dg-层的堆栈，并用它们给出了主要结果的另一种证明。我们证明了任何Stokes数据堆栈都是某种类型的允许dg-Sheet的堆栈。利用这一点，我们证明了所有成帧的Stokes数据堆栈都具有簇Poisson结构，在野生映射类群下是等变的。因此，它们可以等变地量子化。装饰性斯托克斯数据的相似堆栈具有等变的簇A-VARIZE结构。
Let R be a non-commutative field. We prove that generic triples of flags in an m-dimensional R-vector space are described by flat R-line bundles on the honeycomb graph with (m-1)(m-2)/2 holes. Generalising this, we prove that non-commutative stacks X(m,S) of framed rank m flat R-vector bundles of on decorated surfaces S are birationally identified with the moduli spaces of flat line bundles on a spectral surface assigned to certain bipartite graphs on S. We introduce non-commutative cluster Poisson varieties related to bipartite ribbon graphs. They carry canonical non-commutative Poisson structure. The space X(m,S) has a structure of a non-commutative cluster Poisson variety, equivariant under the action of the mapping class group. For bipartite graphs on a torus, we get the non-commutative dimer cluster integrable system. We define non-commutative cluster A-varieties related to bipartite ribbon graphs. They carry canonical non-commutative 2-form. The non-commutative stack A(m,S) of twisted decorated local systems on S carries a cluster A-variety structure, equivariant under the action of the mapping class group. The non-commutative cluster A-coordinates on the space A(m,S) are ratios of Gelfand-Retakh quasideterminants. In the case m=2 this recovers the Berenstein-Retakh non-commutative cluster algebras related to surfaces. We introduce stacks of admissible dg-sheaves, and use them to give an alternative proof of main results. We show that any stack of Stokes data is a stack of admissible dg-sheaves of certain type. Using this we prove that all stacks of framed Stokes data carry a cluster Poisson structure, equivariant under the wild mapping class group. Therefore they can be equivariantly quantized. Similar stacks of decorated Stokes data carry an equivariant cluster A-variety structure.