The conformal anomaly and the Virasoro algebra are fundamental aspects of 2D conformal field theory and conformally covariant models in planar random geometry. In this article, we explicitly derive the Virasoro algebra from an axiomatization of the conformal anomaly in terms of real determinant lines, one-dimensional vector spaces associated to Riemann surfaces with analytically parametrized boundary components. Here, analytical orientation-preserving diffeomorphisms and deformations of the circle naturally act on the boundary components. We introduce a sewing operation on the real determinant lines over the semigroup of annuli, which then induces central extensions of the diffeomorphism group, as well as of the complex deformations.
Our main theorem shows that on the one hand, the cocycle associated to the central extension of diffeomorphisms is trivial, while on the other hand, the Lie algebra cocycle associated to the central extension of complex deformations is nontrivial, yielding the imaginary part of the Gel'fand-Fuks cocycle. We thus answer a question, partly negatively and partly affirmatively, discussed by Andre Henriques and Dylan Thurston in 2011. The proof uses concrete computations, which we aim to be accessible to a wide audience.
We also show an explicit relation to loop Loewner energy, anticipating the real determinant lines to be pertinent to locally conformally covariant (Malliavin-Kontsevich-Suhov) measures on curves and loops, as well as to Kähler geometry and geometric quantization of moduli spaces of Riemann surfaces. Inherently, the conformal anomaly and real determinant line bundles are expected to be universal, following a classification of modular functors.
Hidden global conformal symmetry without Virasoro extension in theory of elasticity
弹性理论（Theory of Elasticity）Riva-Cardy模型）被认为是尺度不变而非共形场论的一个例子。我们认为，在$d=2$尺寸，该理论隐藏了全球共形对称的$SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$没有其Virasoro扩展。更精确地说，我们可以把位移矢量的所有关联函数嵌入到一个关于两个标量势变量的具有四阶导数作用的整体共形场论中，这必然违反反射正性。势变量的能量-动量张量不能改进为无迹的，因此即使存在全局特殊共形流，它也不表现出Virasoro对称性。
WKB-asymptotics for multipoint Virasoro conformal blocks and applications
Landau Institute for Theoretical Physics（朗道理论物理研究所）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
Irreducible Virasoro representations in Liouville conformal field theory
Aix–Marseille Université（艾克斯-马赛大学）；Institut de Mathématiques de Marseille（马赛数学研究所）；Beijing International Center for Mathematical Research (BICMR)（北京国际数学研究中心）
共形引导适用于渗流和稀释自回避聚合物，两个理论Virasoro中心荷$c=0$在二维空间。在这两种情况下，我们提出了一个频谱的运营商的动机Virasoro对称性，这是缺乏一个应力能量张量作为一个近似的手段，强制执行$c=0$。逾渗在$2\leq D \leq 6$维中处理，自避免行走在$2 \leq D \leq 4$中处理。
Virasoro Conformal Blocks and modular functor from Liouville CFT
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay（巴黎萨克雷大学，法国国家科学研究中心，奥赛数学实验室）；University of Helsinki, Department of Mathematics and Statistics（赫尔辛基大学数学与统计系）；Aix-Marseille University, Institut de Mathématiques de Marseille (I2M), and Institut Universitaire de France (IUF)（艾克斯-马赛大学，马赛数学研究所（I2M），法国大学研究院（IUF））
