Chiral Charged Fermions, One-Dimensional Quantum Field Theory and Vertex Algebras

We give an explicit L2-representation of chiral charged fermions using the Hardy–Lebesgue octant decomposition. In the ‘pure’ case such a representation has already been used by M. Sato in holonomic field theory. We study both ‘pure’ and ‘mixed’ cases. In the compact case, we rigorously define unsmeared chiral charged fermion operators inside the unit circle. Using chiral fermions, we orient our findings towards a functional analytic study of vertex algebras as one-dimensional quantum field theory.