Non-Rational Conformal Field Theory

二维共形场论（CFT）由于其无限维对称代数，提供了一类非常容易理解的量子场论。Rational cft已经被证明在数学、弦理论和统计物理中非常有用。一个重要的例子是将有限单群、模形式、无限维李代数、共形场论和玻色子弦相互联系起来的怪物月光。到目前为止，现代的信念是，良好的cft的基本类别要比理性理论大得多。事实上，非理性CFT也出现在刚才提到的所有领域，它将最近和现代的发展联系起来。其中一个例子是Mathieu moonshine，它连接了有限单群、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演化相关联的渗透。

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.