Mirror Symmetry for Cluster Varieties

本提案的中心研究对象（称为集群品种）在丰富性和简洁性之间取得了很好的平衡。它们是在复数上定义的一类空间，具有自然的体积概念。此外，它们是由相当简单的构建块组成的——它们是由粘在一起的环面构成的，以确保集群多样性中的体积概念与构成它的环面中的体积概念一致。此外，集群品种成对出现。每个环面都有一个对偶环面。这同样适用于簇的变种，其中双环实际上是由双环面构成的。我们可以通过回答似乎与双集群完全不同的问题来了解集群多样性。这种对偶是一种更为普遍的现象，即镜像对称的一个例子，在过去的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。簇的多样性和簇的对偶性在镜像对称的广泛设置中开辟了一些领域，在那里我们可以用显式计算来弄得很脏乱，并证明那些仍然无法全面推广的定理。所以，至少从几何的角度来看，我们可以通过研究簇的变化学到很多东西。但是簇变理论的另一个令人惊奇的方面是它们在数学中的广泛出现。事实上，集群品种的镜像对称联系是最近的发展。簇代数最初是由Fomin和Zelevinsky为了研究量子群的正则基而发明的，它与代数群的表示理论、颤栗的表示理论、双曲几何和泊松几何有着密切的联系。我们倾向于研究的集群品种从许多不同的角度来看都很有趣，每一个角度都提供了对其他角度的洞察。我们的建议处理镜像对称的集群品种，自然出现在代数群的表示理论设置。在这种情况下，集群多样性以一种精确的方式嵌入到一个更大的空间中——我们实际上试图研究的空间。我们提出构建和研究镜像簇的对偶嵌入。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联系起来。我们希望知识在这种二元性中向两个方向流动，希望包括双方关系在内的一幅完整的图画比任何一方单独存在更美丽。

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.