From the Conformal Anomaly to the Virasoro Algebra

Abstract: 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 K\"ahler 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.