Modular invariants from subfactors
Modular invariants from subfactors
复制标题
子因子的模不变量
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
David E. Evans
中科院分区：
文献类型：
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作者：
Jens Böckenhauer;David E. Evans
在这些讲座中，我们将解释共形场论中的模不变量与算子代数中的辫子子因子之间的密切关系。具有编织的子因子确定矩阵Z，其作为比较两种编织扇区感应（“α-感应”）的耦合矩阵而获得。它具有非负整数项，是归一化的，并与由编织产生的S矩阵和T矩阵交换。因此，它是一个物理模不变量在通常意义上的有理共形场论。共形场论模型的代数处理，例如SU（n）_k$模型，产生子因子，子因子实现它们已知的模不变量。模不变量的几个性质到目前为止已经被经验地注意到，并被认为是神秘的，例如它们与图的密切关系，例如SU（2）_k$的A-D-E分类。在子因子上下文中，这些属性可以在非常一般的环境中严格推导出来。此外，由于Moore-Seiberg，Dijkgraaf-Verlinde发现了最大扩展手征代数的融合规则同构，通过中间子因子，甚至没有提到$S$和$T$的模性，找到了一个清晰而非常一般的证明和解释。最后，我们给出了一个概述的现状，有关的辫子子因子的分类和二维共形场论之间的关系。我们特别演示了如何实现扭曲（II型）后裔模不变量的共形包含子因子，并说明了新的例子的方法。
In these lectures we explain the intimate relationship between modular invariants in conformal field theory and braided subfactors in operator algebras. A subfactor with a braiding determines a matrix $Z$ which is obtained as a coupling matrix comparing two kinds of braided sector induction ("alpha-induction"). It has non-negative integer entries, is normalized and commutes with the S- and T-matrices arising from the braiding. Thus it is a physical modular invariant in the usual sense of rational conformal field theory. The algebraic treatment of conformal field theory models, e.g. $SU(n)_k$ models, produces subfactors which realize their known modular invariants. Several properties of modular invariants have so far been noticed empirically and considered mysterious such as their intimate relationship to graphs, as for example the A-D-E classification for $SU(2)_k$. In the subfactor context these properties can be rigorously derived in a very general setting. Moreover the fusion rule isomorphism for maximally extended chiral algebras due to Moore-Seiberg, Dijkgraaf-Verlinde finds a clear and very general proof and interpretation through intermediate subfactors, not even referring to modularity of $S$ and $T$. Finally we give an overview on the current state of affairs concerning the relations between the classifications of braided subfactors and two-dimensional conformal field theories. We demonstrate in particular how to realize twisted (type II) descendant modular invariants of conformal inclusions from subfactors and illustrate the method by new examples.