Algebraic constructions related to marked Riemann surfaces

一个有标记的黎曼曲面(S，M)是由一个黎曼曲面S和一个有限集M组成的。Fomin，Shapiro和瑟斯顿为每个标记曲面构造了一个簇代数A(S，M)，Amiot，Cerulli Irelli，Keller，Labardini-Fragoo和Plamondon的工作允许定义相应的簇范畴C(S，M)。 这一研究方案的重点是这三个对象之间的相互作用：黎曼曲面及其映射类群的几何，簇代数及其簇自同构，三角范畴C(S，M)。 被标记曲面的一个重要组合不变量是它的翻转图，它由(S，M)的所有三角剖分组成，其中边是由圆弧的翻转给出的。与A(S，M)或C(S，M)的簇交换图相同，有各自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，这个有向图中的最大路称为极大绿序列。 这些极大绿序列在不同的背景下被研究，因为它们产生了量子双对数恒等式和非交换的Donaldson-Thomas不变量。五角形(S，M)的情况产生了两个斜交变量的经典量子双对数恒等式，这种构造已被Reineke和Keller推广到许多其他情况。此外，黎曼曲面也是在弦理论的背景下，在数学物理中被研究的。特别地，弦理论中BPS(Bogomol‘nyi-Prasad-Sommerfield)粒子的全谱可以用最大格林序列来计算。弦理论的方法是基于S的二次微分，最近Bridgeland和Smith的工作将其与研究三角范畴C(S，M)上的稳定性条件联系起来。 这项研究建议的主要目标是： ·利用序上的Cohen-Macaulay模，给出了独立于任何三角剖分的簇范畴C(S，M)的定义。 ·研究交换图中的最小路。 ·利用稳定性条件刻画了C(S，M)上极大格林序列的存在性。

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).