Algebraic constructions related to marked Riemann surfaces

标记的黎曼曲面（S，M）由黎曼曲面S和一个有限集M的“标记”点组成。Fomin， Shapiro和Thurston为每个标记曲面构造了一个聚类代数a (S，M), Amiot, Cerulli Irelli, Keller， Labardini-Fragoso和Plamondon的工作允许定义相应的聚类类别C（S，M）。本研究计划的主要焦点是这三个对象之间的相互作用：黎曼曲面的几何及其映射类群，簇代数及其簇自同构，以及三角化范畴C（S，M）。标记曲面的一个重要的组合不变量是它的翻转图，它由（S，M）的所有三角构成，其中的边由弧的翻转给出。同样的图出现在A（S，M）或C（S，M）的簇交换图中，它们各自具有对应于弧翻转的突变概念。聚类交换图可以被赋予一个方向，这个方向图中的最大路径称为最大绿色序列。这些极大绿序列在不同的背景下进行了研究，因为它们产生了量子二对数恒等式和非交换Donaldson-Thomas不变量。五边形（S，M）的情况得到了两个斜交换变量的经典量子二对数恒等式，Reineke和Keller将这种构造推广到许多其他情况。此外，在弦理论的背景下，黎曼曲面也在数学物理中被研究。特别是，弦理论中BPS （Bogomol 'nyi-Prasad-Sommerfield）粒子的全谱可以用极大绿序列来计算。弦理论的方法是基于S上的二次微分，而bridgeeland和Smith最近的工作将其与研究三角化范畴C（S，M）上的稳定性条件联系起来。本研究计划的主要目标是：•使用Cohen-Macaulay模块对订单提供独立于任何三角测量的群集类别C（S，M）的定义。•研究交换图中的最小路径。•用C（S，M）上的稳定性条件来表征最大绿色序列的存在性。

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).