Algebraic constructions related to marked Riemann surfaces

一个标记黎曼曲面（S，M）由一个黎曼曲面S和一个“标记”点的有限集合M构成。Fomin，Shapiro和Thurston为每个标记曲面构造了一个簇代数A（S，M），Amiot，Cerulli Irelli，Keller，Labardini-Fragoso和普拉蒙东的工作允许定义相应的簇范畴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：几何的黎曼曲面及其映射类群，集群代数及其集群自同构，和三角范畴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）的所有三角剖分形成，其中边由弧的翻转给出。同样的图出现作为A（S，M）或C（S，M）的簇交换图，具有对应于弧的翻转的突变的相应概念。簇交换图可以被赋予一个方向，在这个方向图中的最大路径被称为最大绿色序列。这些最大的绿色序列在不同的背景下进行了研究，因为它们引起量子双对数身份和非交换唐纳森-托马斯不变量。五边形（S，M）的情形给出了两个斜对易变量的经典量子双对数恒等式，并且Reineke和Keller已经将该构造推广到许多其他情形。此外，黎曼曲面也在数学物理学中被研究，在弦理论的背景下。特别是，弦理论中BPS（Bogomol'nyi-Prasad-Sommerfield）粒子的完整谱可以使用最大绿色序列来计算。弦理论的方法是基于S上的二次微分，Bridgeland和Smith最近的工作将其与研究三角范畴C（S，M）的稳定性条件联系起来。本研究计划的主要目标是：·使用阶上的Cohen-Macaulay模，给出独立于任何三角范畴的簇范畴C（S，M）的定义。·研究交换图中的最小路径。·利用C（S，M）上的稳定性条件刻画了极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).