  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.
