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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611966
• 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).

明显的Riemann表面（S，M）由Riemann表面S和一个“标记”点的有限集M形成。 Fomin，Shapiro和Thurston为每个标记的表面构建了一个集群代数A（S，M），Amiot，Cerulli Irelli，Keller，Labardini-Fragoso和Plamondon的工作允许定义相应的群集类别C（S，M）。 这项研究建议的主要重点是这三个对象之间的相互作用：黎曼表面与其映射类组的几何形状，群集代数及其群集自动形态，而三角形C类C（S，M）。 标记表面的重要组合不变是其翻转图，该图由（S，M）的所有三角形形成，并由弧的翻转给出边缘。与a（s，m）或c（s，m）的群集交换图发生相同的图，其突变的相对音符与弧的翻转相对应。群集交换图可以赋予方向，并且该方向图中的最大路径称为最大绿色序列。 这些最大的绿色序列在各种情况下都是研究的，因为它们会产生量子差异身份和非交通性的唐纳森thomas不变性。五角大楼（S，M）的情况在两个偏斜的变量中产生了经典的量子Diologarithm身份，Reineke和Keller已将构造推广到许多其他情况下。此外，在字符串理论的背景下，Riemann表面也是数学物理学的研究。特别是，可以使用最大绿色序列计算出BPS（Bogomol’nyi – Prasad -Sommerfield）粒子的完整频谱。字符串理论方法基于S的二次差异，而Bridgeland和Smith的最新工作将其与研究三角剖分C（S，M）的稳定性条件有关。 该研究建议的主要目标是： •使用Cohen-Macaulay模块在订单上提供了与任何三角剖分无关的群集类别C（S，M）的定义。 •在交换图中研究最小路径。 •以C（S，M）的稳定性条件来表征最大绿色序列的存在。