Dimer models with boundary

二聚体模型是一个图，意味着一组由边连接的节点，绘制在表面上。图的节点被涂上黑色和白色，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，但二聚体模型记录了惊人数量的数学和物理数据。虽然二聚体模型首先出现在统计力学中，例如在研究具有两种不同大小分子的液体的化学行为时，但它们已被证明在广泛的领域中有用，后来在弦理论和代数几何中再次出现。从代数学家的角度来看，二聚体模型中最重要的信息是二聚体代数，即曲面上的路径集合，这些路径可以根据几何规则相乘。迄今为止，这一领域的大多数研究都涉及封闭曲面上的二聚体模型，例如环面（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;用技术语言来说，它们的二聚体代数是3-Calabi-Yau。这也是弦理论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，宇宙的四维时空被这样一个流形产生的额外的六维所增强。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，已经导致了数学和理论物理领域的大量有趣研究。（如鲍尔，国王和马什）和物理学家（例如Franco和合作者）已经独立地考虑了具有边界的表面上的二聚体模型，例如圆盘，在数学方面的集群代数和表示论的背景下，以及各种物理问题，如散射振幅的计算。这种二聚体模型有许多自然的例子，例如那些从研究最大非交叉集合及其与格拉斯曼簇代数的关系中产生的例子。边界的引入导致了许多新的现象，因为与二聚体模型相关的许多数据，例如其二聚体代数，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，这一性质并没有完全消失，二聚体代数仍然与3-Calabi-Yau代数共享许多性质。我最近的工作给出了“内部3-Calabi-Yau代数”的精确定义，它捕捉了一个代数在其内部是Calabi-Yau的想法，但在边界处具有不同的行为。这一新概念开辟了可能性，扩大了许多富有成效的领域的研究二聚体模型在封闭的表面到一个更广泛的背景下，和奖学金打算利用这个机会，通过发展一个理论一致的二聚体模型的边界，二聚体代数的内部3-Calabi-Yau，并调查这种对称性的后果。此外，它将解决只有在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，并探索与新兴的充满活力的簇代数数学理论的联系，这在边界情况下更加明显。

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.