Two different approaches to calculate the fusion rules of the c_{p,1} series of logarithmic conformal field theories are discussed. Both are based on the modular transformation properties of a basis of chiral vacuum torus amplitudes, which contains the characters of the irreducible representations. One of these is an extension, which we develop here for a non-semisimple generalisation of the Verlinde formula introduced by Fuchs et al., to include fusion products with indecomposable representations. The other uses the Verlinde formula in its usual form and gets the fusion coefficients in the limit, in which the basis of torus amplitudes degenerates to the linear dependent set of characters of irreducible and indecomposable representations. We discuss the effects, which this linear dependence has on any result for fusion rules, which are calculated from these character's modular transformation properties. We show that the two presented methods are equivalent. Furthermore we calculate explicit BPZ-like expressions for the resulting fusion rules for all p larger than 2.
The Verlinde formula computes the dimension of conformal blocks associated to simple Lie algebras and stable pointed curves. If a simply-laced simple Lie algebra admits a nontrivial diagram automorphism, then this automorphism acts on the space of conformal blocks naturally. We prove an analogue of Verlinde formula for the trace of the diagram automorphism on the space of conformal blocks. Along the way, we get an analogue of Kac-Walton formula for the trace of the diagram automorphism. We also get a twining type formula between the conformal blocks for the pair $(sl(2n+1),sp(2n))$.
The rank 1 bosonic ghost vertex algebra, also known as the $βγ$ ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor $C_2$-cofinite. We identify a module category, denoted category $\mathscr{F}$, which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category $\ma…
Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
对数极小模型不是有理数的，但是在W-扩展图中，它们类似于有理数的共形场论。我们认为，W-投射表示是基本的积木在边界和散装描述这些理论。在边界理论中，每个由基本融合产生的W-投射表示都与一个边界条件相关联。在相应的Grothendieck环上的乘法得到了一个关于A-型扭仿射图A^{（2）}_{p}及其陪集图A^{（2）}_{p，p '}=A^{（2）}_{p} x A^{（2）}_{p'}/Z_2的Verlinder类公式.这为具有W-投影边界条件的共形配分函数提供了简洁的公式。在环面上，我们提出了模不变的配分函数作为半双线性形式的W-投影和有理极小字符，并观察到它们是由相同的陪集融合图编码。
