W-Algebras and Universal Objects in Vertex Algebra Theory

在过去的半个世纪里，许多受物理学启发的结构和概念对数学产生了深远的影响。例如，流形的弦不变量，如量子上同调，导致了计数几何的惊人进步。在代数方面，一种基本的新结构称为顶点算子代数（VOA），在20世纪80年代从量子场论中出现，并由Borcherds在他关于月光猜想的工作中公理化。VOA是交换环的自然推广，在过去的三十年里，它们在各种学科中得到了应用，包括有限群论、表示论、组合学、数论和代数几何。在这个项目中，PI将研究VOA的结构和表示理论，以及VOA和代数几何之间的一些联系。这些项目将推进该主题，并为PI的现任和前任研究生提供教育和合作机会。首先，PI在A，D和E类型的主W-代数的陪集实现方面的工作解决了一个30年前的猜想，该猜想一直是物理学中的一个关键假设。它有许多引人注目的推论，如主W-代数的所有离散级数表示的酉性，以及仿射VOA在容许水平上模的模张量范畴的存在性。PI将构造其他李类型的主W-代数的陪集实现，以及非主W-代数和W-超代数。第二，PI最近构造了一个通用的双参数VOA，它插在所有A型主W-代数之间，从某种意义上说，它们是这种结构的单参数子代数。PI将构造更一般的泛对象，它们是定义在某个簇X上的函数环上的VOA。通常，有趣的VOA是通过沿着X的某些子变种专门化通用对象来获得的，这些VOA的意外同构对应于这些子变种上的交点。第三，PI将研究两个仿射计划，已附加到一个VOA的荒川：相关联的计划和奇异支持。在相应的格式的弧空间中，奇异支撑总是有一个封闭的嵌入。PI将调查这个嵌入是同构的好标准。这个奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many structures and concepts that were inspired by physics have had a profound influence on mathematics during the last half century. For example, stringy invariants of manifolds such as quantum cohomology have led to spectacular advances in enumerative geometry. On the algebraic side, a fundamental new structure called a vertex operator algebra (VOA) emerged from quantum field theory in the 1980s, and was axiomatized by Borcherds in his work on the Moonshine Conjecture. VOAs are natural generalizations of commutative rings, and in the last thirty years, they have found applications in a diverse range of subjects including finite group theory, representation theory, combinatorics, number theory, and algebraic geometry. In this project, the PI will investigate the structure and representation theory of VOAs, as well as some connections between VOAs and algebraic geometry. These projects will advance the subject and provide educational and collaborative opportunities for the PI's current and former graduate students.First, the PI's work on the coset realization of principal W-algebras of A, D, and E types resolved a 30-year-old conjecture which had been a key starting assumption in physics. It has many striking corollaries such as the unitarity of all discrete series representations of principal W-algebras, and the existence of modular tensor categories of modules for affine VOAs at admissible levels. The PI will construct coset realizations of principal W-algebras of other Lie types, as well as non-principal W-algebras and W-superalgebras. Second, the PI recently constructed a universal two-parameter VOA which interpolates between all the type A principal W-algebras, in the sense that they arise as one-parameter quotients of this structure. The PI will construct more general universal objects, which are VOAs defined over the ring of functions on some variety X. Typically, interesting VOAs are obtained by specializing the universal object along certain subvarieties of X, and unexpected isomorphisms of these VOAs correspond to intersection points on these subvarieties. Third, the PI will study two affine schemes that have been attached to a VOA by Arakawa: the associated scheme and the singular support. There is always a closed embedding of the singular support in the arc space of the associated scheme. The PI will investigate good criteria for this embedding to be an isomorphism. This has many applications in VOA theory and is connected to questions about the geometry of arc spaces that are of independent interest.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.