Mirror Symmetry for Cluster Varieties

这项建议的中心研究对象(称为集群品种)在丰富性和简单性之间取得了很好的平衡。它们是定义在复数上的一类空间，这些复数带有一个自然的体积概念。此外，它们是由相当简单的构建块组成的--它们是由圆环粘在一起的，这种方式确保了簇中的体积概念与组成它的圆环中的体积概念一致。此外，集群品种成对出现。每个圆环都有一个对偶圆环。同样的道理也适用于簇生变种，其中二元体实际上是由双环体构建而成的。我们可以通过回答看起来与双星团变种完全不同的问题来了解星系团变种。这种二元性是一种更普遍的现象的例子，这种现象被称为镜像对称，在过去30年里，镜像对称一直是数学和物理学中一个活跃的研究课题。簇变种和簇对偶性在镜像对称的广泛设置中开辟了一些领域，在那里我们可以通过显式计算来弄脏我们的手，并证明仍然无法完全概括的定理。因此，至少从几何学的角度来看，我们可以通过研究集群品种学到很多东西。但集群变种理论的另一个令人惊讶的方面是，它们在数学中的出现范围如此之广。事实上，与集群品种的镜像对称连接是最近的发展。簇代数最初是由Fomin和Zlevinsky发明的，用于研究量子群的标准基，它们与代数群的表示理论、箭图的表示理论以及双曲和泊松几何有着密切的联系。我们倾向于研究的集群变种从许多不同的角度来看很有趣，每个角度都提供了对其他的洞察力。我们的建议讨论了在代数群表示理论中自然出现的簇簇的镜像对称性。在这种背景下，星系团的多样性以一种精确的方式嵌入了更大的空间--我们实际上试图研究的空间。我们建议构造和研究镜像簇簇的双重嵌入。我们的主要问题是原始空间的表示理论如何与这个对偶空间的几何有关。我们希望知识在这种二元性中双向流动，希望包括双方关系在内的完整图景比任何一方单独存在更美好。

The central objects of study in this proposal (called cluster varieties) strike a nice balance of richness and simplicity. They are a class of spaces defined over the complex numbers that come with a natural notion of volume. Moreover, they are made up of rather simple building blocks-- they are built out of tori glued together in a way that ensures the notion of volume in the cluster variety agrees with the notion of volume in the tori that make it up. Furthermore, cluster varieties come in pairs. Every torus has a dual torus. The same holds for cluster varieties, where duals are in fact built out of dual tori. We can learn a lot about a cluster variety by answering questions that seem entirely different for the dual cluster variety. This duality is an instance of a far more general phenomenon known as mirror symmetry that has been a vigorous research topic in mathematics and physics for the past 30 years. Cluster varieties and cluster duality carve out some territory within the broad setting of mirror symmetry where we can get our hands dirty with explicit computations and prove theorems that remain out of reach in full generality. So, at least from a geometric point of view, we can learn a lot by studying cluster varieties. But another amazing aspect of the theory of cluster varieties is how widely they appear in mathematics. In fact, the mirror symmetry connection to cluster varieties is a recent development. Cluster algebras were originally invented by Fomin and Zelevinsky to study canonical bases for quantum groups, and they have close connections to representation theory of algebraic groups, representation theory of quivers, and hyperbolic and Poisson geometry. The cluster varieties we tend to study are interesting from many different perspectives, with each of these perspectives providing insight into the others. Our proposal deals with mirror symmetry for cluster varieties that appear naturally in the setting of representation theory of algebraic groups. In this context, the cluster variety is embedded in a larger space-- the space we are actually trying to study-- in a precise way. We propose to construct and study a dual embedding of the mirror cluster variety. Our principal question is how representation theory of the original space is related to geometry of this dual space. We hope knowledge will flow in both directions in this duality, and that a complete picture including relations between the two sides will be more beautiful than either side standing alone.