Algebraic constructions related to marked Riemann surfaces

与标记黎曼曲面相关的代数构造

• 批准号: RGPIN-2014-05999
• 负责人: Brüstle, Thomas
• 金额: $ 2.04万
• 依托单位: Bishop's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588743
• 关键词: Algebraic; constructions; related; marked; Riemann

A marked Riemann surface (S,M) is formed by a Riemann surface S and a finite set M of “marked” points. Fomin, Shapiro and Thurston constructed a cluster algebra A(S,M) for each marked surface, and the work of Amiot, Cerulli Irelli, Keller, Labardini-Fragoso and Plamondon allows to define the corresponding cluster category C(S,M). The main focus of this research proposal is the interplay between these three objects: the geometry of the Riemann surface with its Mapping Class Group, the cluster algebra with its cluster automorphisms, and the triangulated category C(S,M). An important combinatorial invariant of the marked surface is its flip graph which is formed by all triangulations of (S,M) and where edges are given by the flip of an arc. The same graph occurs as the cluster exchange graph of A(S,M) or C(S,M), with respective notions of mutations which correspond to the flip of an arc. The cluster exchange graph can be endowed with an orientation, and maximal paths in this oriented graph are referred to as maximal green sequences. These maximal green sequences are studied in various contexts, as they give rise to quantum dilogarithm identities and non-commutative Donaldson–Thomas invariants. The case of a pentagon (S,M) yields the classical quantum dilogarithm identity in two skew-commuting variables, and the construction has been generalized by Reineke and Keller to many other cases. Moreover, Riemann surfaces are also studied in mathematical physics, in the context of string theory. In particular, the complete spectrum of a BPS (Bogomol’nyi–Prasad–Sommerfield) particle in string theory can be computed using maximal green sequences. The string theory approach is based on quadratic differentials on S, and recent work of Bridgeland and Smith relates this to studying stability conditions on the triangulated category C(S,M). The main objectives of this research proposal are: • To provide a definition of the cluster category C(S,M) which is independent of any triangulation, using Cohen-Macaulay modules over orders. • To study minimal paths in the exchange graph. • To characterize the existence of maximal green sequences in terms of stability conditions on C(S,M).

标记黎曼曲面 (S,M) 由黎曼曲面 S 和“标记”点的有限集 M 组成，Fomin、Shapiro 和 Thurston 为每个标记曲面构建了簇代数 A(S,M)，并且工作。 Amiot、Cerulli Irelli、Keller、Labardini-Fragoso 和 Plamondon 允许定义相应的簇类别 C(S,M)。 本研究提案的主要焦点是这三个对象之间的相互作用：黎曼曲面的几何形状及其映射类群、簇代数及其簇自同构以及三角范畴 C(S,M)。 标记表面的一个重要的组合不变量是其翻转图，该翻转图由 (S,M) 的所有三角剖分形成，并且其中边由弧的翻转给出。该图与 A(S, M) 的簇交换图相同。 M) 或 C(S,M)，分别对应于弧的翻转的突变概念。簇交换图可以被赋予一个方向，并且该有向图中的最大路径被称为最大绿色序列。 这些最大绿色序列在不同的背景下进行了研究，因为它们给出了量子二对数恒等式和非交换唐纳森-托马斯不变量。五边形 (S,M) 的情况产生了两个斜交换变量中的经典量子二对数恒等式，并且Reineke 和 Keller 已将该结构推广到许多其他情况，而且在数学物理学中，特别是在完整​​的弦理论中，也研究了黎曼曲面。弦理论中 BPS（Bogomol'nyi-Prasad-Sommerfield）粒子的谱可以使用最大绿色序列计算。弦理论方法基于 S 的二次微分，Bridgeland 和 Smith 最近的工作将其与稳定性条件的研究联系起来。在三角类别 C(S,M) 上。 本研究计划的主要目标是： • 使用Cohen-Macaulay 模来提供独立于任何三角测量的簇类别C(S,M) 的定义。 • 研究交换图中的最小路径。 • 根据C(S,M) 上的稳定性条件来表征最大绿色序列的存在性。