On the conformal anomaly

The conformal anomaly is the distinguishing feature of a conformal field theory as a two-dimensional quantum field theory. Recently, it also appears in conformally covariant random geometry. This thesis studies the conformal anomaly mathematically as a real determinant line bundle over infinite-dimensional moduli spaces of Riemann surfaces with analytically parametrized boundary components, summarizing three works of the author on this topic. As an introduction, the conformal anomaly is presented in the context of mathematical physics and probability theory, including a detailed breakdown of the relevant geometry of sewing operations involving said Riemann surfaces. <br> The main results of the contained works are, respectively, the derivation of the Virasoro algebra from the conformal anomaly, generalizations of loop Loewner energy for Schramm--Loewner evolution, and a universal property for real one-dimensional modular functors. The latter is an abstraction of the real determinant line bundle inspired by Segal's definitions in the complex case, to which assumptions of locality and modular invariance are added. The main tools developed are results on complex deformations of the unit circle, which come with a local composition law integrating the Lie algebra of complex-valued vector fields on the unit circle. Since complex deformations act on the moduli spaces by deformation of boundary components, the real determinant line bundle pulls back to a central extension of the Lie algebra, which facilitates the algebraic study of the conformal anomaly.