Dimer models with boundary

二聚体模型是一个图，意思是一组由边连接的节点，画在一个表面上。图的节点是黑色和白色的，并且边只能连接不同颜色的节点。尽管这个定义看起来很简单，二聚体模型却记录了数量惊人的数学和物理数据。虽然二聚体模型最初出现在统计力学中，例如研究具有两种不同大小分子的液体的热力学行为，但它们已被证明在广泛的领域都很有用，后来又出现在弦理论和代数几何中。从代数学家的角度来看，二聚体模型中编码的最重要的信息是二聚体代数，它是表面上的路径集合，可以根据几何驱动规则相乘。到目前为止，这一领域的大多数研究都是关于封闭表面上的二聚体模型，比如环面（甜甜圈的表面）。许多这样的二聚体模型是一致的，这意味着它们具有极强的对称性；用专业语言来说，它们的二聚体代数是3-Calabi-Yau。这是弦理论学家对二聚体模型感兴趣的部分原因，因为3-Calabi-Yau代数与Calabi-Yau流形密切相关——在许多弦理论模型中，宇宙的四个时空维度被由这样一个流形产生的额外六个维度所增强。研究一致二聚体模型的性质，以及从图中检测一致性的不同方法，已经在数学和理论物理领域引发了大量有趣的研究。最近，数学家（如Baur， King和Marsh）和物理学家（如Franco和合作者）在数学方面的聚类代数和表示理论的背景下，以及在各种物理问题（如散射振幅的计算）中，被独立地引导考虑具有边界的表面（如圆盘）上的二聚体模型。有许多这样的二聚体模型的自然例子，例如那些产生于最大非交叉集合及其与格拉斯曼簇代数的关系的研究。边界的引入导致了许多新现象，因为与二聚体模型相关的许多数据，如二聚体代数，在边界附近的行为与在表面内部的行为非常不同。特别是，这种不同的边界行为意味着二聚体代数在严格意义上不会是3-Calabi-Yau。然而，二聚体代数并没有完全丧失这种性质，它仍然与3-Calabi-Yau代数具有许多相同的性质。我最近的工作给出了一个“内部3-Calabi-Yau代数”的精确定义，它抓住了代数在内部是Calabi-Yau的概念，但在边界处具有不同的行为。这一新概念为将封闭表面上的二聚体模型的许多卓有成效的研究领域扩展到更广泛的背景下开辟了可能性，该奖学金计划通过发展具有边界的一致二聚体模型理论来利用这一机会，其内部二聚体代数为3-Calabi-Yau，并研究这种对称性的后果。此外，它将解决只有在具有边界的二聚体模型中才会出现的问题，例如如何以保持一致性的方式将这种二聚体模型粘合在一起的问题，这也是物理学家感兴趣的问题，并探索与新兴和充满活力的簇代数数学理论的联系，这在边界情况下更为明显。

A dimer model is a graph, meaning a set of nodes connected by edges, drawn on a surface. The nodes of the graph are coloured black and white, and edges may only connect nodes of different colours. Despite this apparently simple definition, a dimer model records an astonishing amount of mathematical and physical data. While dimer models first appeared in statistical mechanics, for example in studying thermodynamical behaviour of liquids having molecules of two different sizes, they have turned out to be useful in a broad range of areas, reappearing later in string theory and algebraic geometry. From an algebraist's point of view, the most important piece of information encoded in a dimer model is the dimer algebra, a collection of paths in the surface that can be multiplied together according to geometrically-motivated rules.So far, most research in this area has concerned dimer models on closed surfaces, such as the torus (the surface of a doughnut). Many such dimer models are consistent, meaning that they have extremely strong symmetry properties; in technical language, their dimer algebras are 3-Calabi-Yau. This is part of what makes dimer models interesting to string theorists, since 3-Calabi-Yau algebras are closely related to Calabi-Yau manifolds---in many string theoretical models, the four spacetime dimensions of the universe are augmented by an additional six dimensions arising from such a manifold. Studying the properties of consistent dimer models, as well as different ways of detecting consistency from the graph, has led to a great deal of interesting research across mathematics and theoretical physics.More recently, mathematicians (e.g. Baur, King and Marsh) and physicists (e.g. Franco and collaborators) have been led independently to consider dimer models on surfaces with boundary, such as discs, in the context of cluster algebras and representation theory on the mathematics side, and in various physical problems such as the calculation of scattering amplitudes. There are many natural examples of such dimer models, for example those arising from the study of maximal non-crossing collections and their relationship to Grassmannian cluster algebras. The introduction of a boundary leads to many new phenomena, since much of the data associated to the dimer model, such as its dimer algebra, behaves very differently near the boundary compared to in the interior of the surface. In particular, this different boundary behaviour means that the dimer algebra will not be 3-Calabi-Yau in a strict sense. However, this property is not totally lost, and the dimer algebra still shares many properties with 3-Calabi-Yau algebras.My recent work gives a precise definition of an 'internally 3-Calabi-Yau algebra', which captures the idea of an algebra being Calabi-Yau in its interior, but with different behaviour at the boundary. This new notion opens up the possibility of extending the many fruitful areas of research on dimer models on closed surfaces into a wider context, and the fellowship intends to exploit this opportunity by developing a theory of consistent dimer models with boundary, the dimer algebras of which are internally 3-Calabi-Yau, and investigating consequences of this symmetry. Moreover, it will address questions that can only arise for dimer models with boundary, such as the problem of how to glue such dimer models together in such a way that consistency is preserved, which is also of interest to physicists, and explore links to the emerging and vibrant mathematical theory of cluster algebras, which are more pronounced in the boundary case.