Methods of supergeometry in representation theory of supergroups

超群表示论中的超几何方法

• 批准号: 0901554
• 负责人: Vera Serganova
• 金额: $ 15.32万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901554&HistoricalAwards=false
• 关键词: Methods; supergeometry; representation; theory; supergroups

The project addresses structure problems in representation theory of supergroups and suggests geometric methods to approach them. Two fundamental questions of representation theory of any algebraic structure are the description of irreducible representations and the description of indecomposable representations. The second question is trivial in semi-simple case; however, since finite-dimensional representations of a simple algebraic supergroup are not completely reducible, both questions turns out to be very difficult: naive constructions give representations which are not irreducible, and irreducibles allow nontrivial extensions. (The situation is similar to one for modular representations.) In the last 25 years works of many mathematicians brought a lot of progress. Now, the characters of irreducible representations are known for general linear and orthosymplectic supergroups. The proposal suggests to adopt the geometric ideas of localization and support variety for supergroups to address the question of extensions. In particular, the proposal constructs a graph which conjecturally encodes a lot of information about extensions and characters. This graph appears naturally in calculation of cohomology groups of invertible sheaves on flag supervarieties and in attempt to generalize a Borel--Weil--Bott theory for supervarieties. A complementary geometric approach is via a functor associating to each module over a Lie superalgebra a quasicoherent sheaf on the cone of selfcommuting odd elements. The support of this sheaf has two counterparts in other branches of representation theory: the associated varieties of Harish-Chandra modules and the rank varieties in modular case. The conjecture is that the complexity of a simple module grows with the dimension of the support of the corresponding sheaf, in particular, one can prove the Kac-Wakimoto conjecture on superdimension this way. The proposal also suggests first steps for "odd" geometric quantization of the self-commuting cone. Recently, supersymmetric spaces became very popular in relation to sigma models; several physical papers develop particular examples of harmonic analysis on such spaces. The proposal contains a conjecture about the structure of modules of regular functions on supersymmetric spaces. The final part of the proposal concerns infinite-dimensional Lie superalgebras; character formulae for affine superalgebras were conjectured by Kac and Wakimoto. The aim here is to single out the cases when the conjectures hold, and to prove them in these cases.In modern particle physics, the principle of supersymmetry plays a very important role already for decades. A relatively new development is that, during the last decade, physicists realized that using supergroup symmetries makes feasible solutions of certain problems in the theory of condensed matter as well, e.g., in (super)conductivity. The methods of supersymmetry factor the questions interesting for physicists through the theory of representations of supergroups and superalgebras. The proposal puts forward several geometric methods, which, when developed, would answer a lot of currently pending concrete questions needed by physicists.

该项目解决了超群表示论中的结构问题，并提出了解决这些问题的几何方法。 任何代数结构的表示论的两个基本问题是不可约表示的描述和不可分解表示的描述。 第二个问题在半简单情况下是微不足道的；然而，由于简单代数超群的有限维表示并不是完全可约的，因此这两个问题都变得非常困难：朴素构造给出了不可约的表示，而不可约则允许非平凡的扩展。 （这种情况与模表示类似。）在过去 25 年里，许多数学家的工作取得了很大进展。现在，不可约表示的特征对于一般线性和正交辛超群来说是已知的。该提案建议采用定位的几何思想并支持超群的多样性来解决扩展问题。特别是，该提案构建了一个图，该图推测编码了有关扩展名和字符的大量信息。该图自然地出现在计算旗形超簇上可逆滑轮的上同调群以及试图推广超簇的 Borel-Weil-Bott 理论时。 互补的几何方法是通过函子将李超代数上的每个模块关联到自交换奇数元素锥体上的准相干束。 该束的支持在表示论的其他分支中有两个对应物：Harish-Chandra 模的关联变体和模情况下的等级变体。 猜想是简单模的复杂度随着相应层的支撑维数的增加而增长，特别是可以通过这种方式证明超维上的 Kac-Wakimoto 猜想。该提案还提出了自交换锥体“奇数”几何量化的第一步。 最近，超对称空间在 sigma 模型中变得非常流行；几篇物理论文开发了此类空间上调和分析的特定示例。 该提案包含关于超对称空间上正则函数模结构的猜想。该提案的最后部分涉及无限维李超代数； Kac 和 Wakimoto 猜想了仿射超代数的特征公式。 这里的目的是挑出猜想成立的情况，并在这些情况下证明它们。 在现代粒子物理学中，超对称原理几十年来一直扮演着非常重要的角色。 一个相对较新的发展是，在过去十年中，物理学家意识到使用超群对称性也可以为凝聚态理论中的某些问题（例如（超）导性）提供可行的解决方案。超对称方法通过超群和超代数的表示理论来解决物理学家感兴趣的问题。该提案提出了几种几何方法，这些方法一旦开发出来，将回答物理学家目前需要解决的许多悬而未决的具体问题。