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Dimer models with boundary

带边界的二聚体模型

基本信息

批准号：
EP/T001771/1
负责人：
Matthew Pressland
金额：
$ 38.93万
依托单位：
University of Leeds
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT001771%2F1
关键词：
Dimer models boundary

项目摘要

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.

二聚体模型是一种图形，意味着一组通过边连接的节点，绘制在曲面上。图中的节点被涂上黑白颜色，边只能连接不同颜色的节点。尽管定义看起来很简单，但二聚体模型记录了惊人数量的数学和物理数据。虽然二聚体模型最早出现在统计力学中，例如在研究具有两个不同大小的分子的液体的热力学行为时，但它们被证明在广泛的领域中很有用，后来在弦理论和代数几何中重新出现。从代数学家的观点来看，二聚体模型中编码的最重要的信息是二聚体代数，它是曲面上的路径的集合，可以根据几何激励规则相乘在一起。到目前为止，这一领域的大多数研究都是关于闭合曲面上的二聚体模型，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；在专业语言中，它们的二聚体代数是3-Calabi-Yau。这就是弦理论家感兴趣的二维弦模型的一部分，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从这样的流形中额外增加了六个维度。最近，数学家(例如Baur、King和Marsh)和物理学家(例如Franco和合作者)被独立地引导在具有边界的表面(例如圆盘)上，在数学方面的团簇代数和表示理论的背景下，以及在诸如散射幅度的计算等各种物理问题中，研究相容二聚体模型。有许多这样的二聚体模型的自然例子，例如那些起源于极大非交叉集合及其与Grassman簇代数关系的研究。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，如它的二聚体代数，在边界附近的表现与在曲面内部的表现非常不同。特别是，这种不同的边界行为意味着二聚体代数不会是严格意义上的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代数有许多共同的性质。我最近的工作给出了‘内部3-Calabi-Yau代数’的精确定义，它抓住了代数在其内部是Calabi-Yau，但在边界具有不同行为的概念。这一新概念开启了将闭曲面上二聚体模型研究的许多卓有成效的领域扩展到更广泛的背景下的可能性，该研究组打算通过发展具有边界的相容二聚体模型理论并研究这种对称性的结果来利用这一机会，其中二聚体代数的内部是3-Calabi-Yau。此外，它将解决只对有边界的二聚体模型才会出现的问题，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，并探索与新兴的和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。