共形映射流形上的微积分与CFT
Calculus on manifolds of conformal maps and CFT
摘要：
在黎曼球面单连通域上的共形场论（CFT）中，自映射下的自然共形对称性以某种方式推广到一般共形映射下的局部对称性，这是CFT强大技巧的基础。单连通域的共形映射自然具有无限维广群的结构，它推广了有限维自映射群。在单连通域上的共形映射空间上引入一种拓扑结构，使之成为拓扑广群。此外，我们（几乎）扩展到一个局部流形结构的基础上的无限维Frechet拓扑向量空间的全纯函数在一个给定的区域A。由此，我们发展了共形A-可微的概念。我们的主要结论是，二次微分特征余切元素的局部流形上享有类似的全纯应力能量张量的CFT的属性，这些属性的基础CFT的局部对称性。应用一般形式主义的CFT关联函数，我们表明，应力-能量张量正是这样一个二次微分。这是构造共形圈系综中应力-能量张量的基础。它还澄清了Cardy的边界条件CFT单连通域，和表达的应力-能量张量的度量变化之间的关系。
英文摘要：
In conformal field theory (CFT) on simply connected domains of the Riemann sphere, the natural conformal symmetries under self-maps are extended, in a certain way, to local symmetries under general conformal maps, and this is at the basis of the powerful techniques of CFT. Conformal maps of simply connected domains naturally have the structure of an infinite-dimensional groupoid, which generalizes the finite-dimensional group of self-maps. We put a topological structure on the space of conformal maps on simply connected domains, which makes it into a topological groupoid. Further, we (almost) extend this to a local manifold structure based on the infinite-dimensional Frechet topological vector space of holomorphic functions on a given domain A. From this, we develop the notion of conformal A-differentiability at the identity. Our main conclusion is that quadratic differentials characterizing cotangent elements on the local manifold enjoy properties similar to those of the holomorphic stress-energy tensor of CFT; these properties underpin the local symmetries of CFT. Applying the general formalism to CFT correlation functions, we show that the stress-energy tensor is exactly such a quadratic differential. This is at the basis of constructing the stress-energy tensor in conformal loop ensembles. It also clarifies the relation between Cardy's boundary conditions for CFT on simply connected domains, and the expression of the stress-energy tensor in terms of metric variations.
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