Non-Rational Conformal Field Theory

二维共形场论（CFT）由于其无穷维对称代数，提供了一类非常容易理解的量子场论。有理CFT在数学、弦理论和统计物理中被证明是非常有用的。一个重要的例子是怪物月光有关有限简单的群体，模块化的形式，无限维李代数，共形场理论和玻色弦彼此。到目前为止，现代人认为，良好的CFTs的基本类别远远大于理性理论。事实上，非理性的CFT也出现在刚才提到的所有领域，它连接了最近和现代的发展。一个例子是Mathieu月光连接有限简单群，超弦理论，（模拟）模形式和K3曲面的几何。另一个例子是将对数CFT与Schramm Loewner Evolution相关联的渗滤。李群及其陪集的高阶WZW理论提供了有理共形场论的自然家族。对数的CFTs是非理性的，它们远不如理性的同类那样被探索。我期望对数理论的自然族是分数容许水平WZW理论及其陪集。理解分数阶WZW理论一直是一个长期存在的开放问题（自1988年以来）。与大卫Ridout一起，最近发现了分数阶SL（2）WZW理论的一致和令人满意的描述。这最终打开了大门，研究这些CFTs非常详细，肯定会发现新的和有趣的结构。我的目标是探索分数水平WZW理论和他们的陪集，朝着这个目标的目标是：1）描述的对称超代数的一般类陪集的超群WZW理论。在这些陪集和看起来非常不同的CFTs的例子之间建立同构。即，CFTs被表征为自由理论内的屏蔽电荷的联合核以及量子哈密顿约化。2)有一些有趣的类模对象作为对数CFTs的特征出现。在第二个目标中，我将考虑对数CFTs陪集的例子。任务是理解如何从父理论得到陪集特征（分支函数）以及逆问题，即使用分支函数重构父理论的特征。3)最终目标是理解OSP（1）的分数超群WZW理论|2）和SL（2| 1）。重要的步骤将是找到所有的不可约模与有界的共形维数以及使用目标二的思想来重建WZW理论从它的陪集。这个目标包含了许多可行的问题与有趣的结果，我想培养高素质的个人（HQP，即研究生和博士后）在这方面。特别是研究生可以在对数CFT以及模块化和表示理论方面获得宝贵的经验。前两个目标在建议书一节中详细概述，而第三个目标在高素质人才一节中描述。我也有一个雄心勃勃的长期目标。证明Verlinde公式的一类对数的CFTs将是壮观的。在有理情形下，黄的一般证明是在Falting在WZW情形下的几何证明之后十多年才给出的。换句话说，对分数阶WZW理论的Verlinde公式的证明的重要一步是理解它们的几何解释。在阿尔伯塔大学，有特里·甘农和伊曼纽尔·迪亚科内斯库的同事，他们有着理想的背景，一起问这个问题。

Due to its infinite dimensional symmetry algebra, two-dimensional conformal field theory (CFT) provides a class of very accessible quantum field theories. Rational CFTs have proven to be extremely useful in mathematics, string theory and statistical physics. An important example being the monster moonshine relating finite simple groups, modular forms, infinite dimensional Lie algebras, conformal field theory and the bosonic string to each other. By now the modern belief is that the fundamental class of nice CFTs is much larger than only rational theories. Indeed, non-rational CFT also appears in all areas just mentioned and it connects very recent and modern developments. One example is the Mathieu moonshine connecting finite simple groups, super string theory, (mock) modular forms and geometry of K3 surfaces. Another example is percolation relating logarithmic CFT to Schramm Loewner Evolution. Integer level WZW theories of Lie groups and their cosets provide the natural family of rational conformal field theory. Logarithmic CFTs are non-rational and they are far less explored than their rational cousins. The natural families of logarithmic theories I expect to be fractional admissible level WZW theories and their cosets. It has been a long-standing open problem (since 1988) to understand fractional level WZW theories. Together with David Ridout a consistent and satisfactory description has recently been found for the case of the fractional level SL(2) WZW theory. This finally opens the door to study these CFTs in great detail, and surely new and interesting structures will be found. My goal is to explore fractional level WZW theories and their cosets, and the objectives towards this goal are: 1) Describe the symmetry super algebra of a general class of cosets of super group WZW theories. Establish isomorphisms between examples of these cosets and very different looking CFTs. I.e. CFTs that are characterized as joint kernel of screening charges inside a free theory and also quantum Hamiltonian reductions. 2) There are interesting modular-like objects appearing as characters of logarithmic CFTs. In the second objective I will consider examples of cosets of logarithmic CFTs. The task is to understand how to get coset characters (branching functions) from the parent theory as well as the inverse problem, that is using the branching functions to reconstruct characters of the parent theory. 3) The aim of the final objective is to understand the fractional super group WZW theories of OSP(1|2) and SL(2|1). Important steps will be finding all irreducible modules with bounded conformal dimension as well as using the ideas of objective two to reconstruct the WZW theory from its cosets. This objective contains many feasible problems with interesting outcome and I want to train highly qualified personal (HQP, i.e. graduate students and postdocs) in this area. Especially graduate students can gain valuable experience in logarithmic CFT as well as modularity and representation theory in working on this objective. The first two objectives are outlined in detail in the proposal section, while the third one is described in the section on highly qualified personal. I also have ambitious long-term goals. Proving the Verlinde formula for a class of logarithmic CFTs would be spectacular. In the rational case, the general proof by Huang was given more than ten years after Falting's geometric proof in the WZW case. In other words, an important step towards a proof of the Verlinde formula for fractional level WZW theories is to understand their geometric interpretation. Here at the University of Alberta, there are with Terry Gannon and Emanuel Diaconescu colleagues with the ideal background to ask this question together.