Chiral charged fermions, one-dimensional quantum field theory and vertex algebras

We give an explicit L-2-representation of chiral charged fermions using the Hardy-Lebesgue octant decomposition. In the 'pure' case such a representat ion has already been used by M. Sato in holonomic field theory. We study bo th 'pure' and 'mixed' cases. In the compact case, we rigorously define unsm eared chiral charged fermion operators inside the unit circle. Using chiral fermions, we orient our findings towards a functional analytic study of ve rtex algebras as one-dimensional quantum field theory.