Operator algebras and mathematical physics

我们研究了von Neumann代数的局部共形网的熵。它是用“热核”半群的迹的对数展开的系数来定义的。与研究流形的拉普拉斯本征值的渐近密度分布类似，我们将这些系数视为无穷多个自由度的非对易几何不变量。在自然模数假设下，熵的前导项非对易面积与中心电荷成正比，一阶修正非对易欧拉特性与网络全局指数的对数成正比。我们还研究了它们与黑洞熵的关系。我们构造了一个局部的von Neumann代数的共形网，类似于具有Longo的框架顶点算子代数的局部共形网。作为例子，我们得到了与月光顶点算子代数相对应的局部共形网络。我们还证明了这个局部共形网的自同构群确实是Monster群。我们完全分类了中心电荷小于1的Virasoro网的不可约但可能的非局部扩张。根据Longo和Rehren的一般理论，这相当于中心电荷小于1的代数边界CFT的完全分类。我们研究了超共形场理论的算符代数方法。超Virasoro代数具有中心电荷小于3/2的离散表示序列。我们将这些情形下的算子代数的超Virasoro网实现为陪集网，并通过研究它们的扩张得到了分类结果。结合我们已建立的一般理论，我们还使用了Gannon和Walton给出的模不变量的分类技术。

We have studied entropy of local conformal nets of von Neumann algebras. It is defined in terms of the coefficients in the expansion of the logarithm of the trace of the "heat kernel" semigroup. In analogy with study on the asymptotic density distribution of the Laplacian eigenvalues of a manifold, we regard these coefficients as noncommutative geometric invariants of infinitely many degrees of freedom. Under a natural modularity assumption, the leading term of the entropy, noncommutative area, is proportional to the central charge and the first order correction, noncommutative Euler characteristic, is proportional to the logarithm of the global index of the net. We have also studied their relations to black hole entropy.We have made a construction of local conformal nets of von Neumann algebras analogous to the one of framed vertex operator algebras with Longo. As an example, we have obtained a local conformal net corresponding to the moonshine vertex operator algebra. We have also shown that the automorphism group of this local conformal net is indeed the Monster group, as expected.We completely classified irreducible, but possibly non-local extensions of the Virasoro net with central charge less than 1. By general theory of Longo and Rehren, this amounts to a complete classification of algebraic boundary CFT with central charge less than 1 satisfying the Haag duality.We have studied operator algebraic approach to super conformal field theory. It is known that the super Virasoro algebras have discrete series of representations for central charges less than 3/2. We have realized the super Virasoro nets of operator algebras for these cases as coset nets and obtained a classification result by studying their extensions. Together with general theory we have established, we also use the classification technique of modular invariants given by Gannon and Walton.