  Variables parametrized by closed and open curves are defined to reformulate
compact U(1) Quantum Electrodynamics in the circle with a massless fermion
field. It is found that the gauge invariant nature of these variables
accommodates into a regularization scheme for the Hamiltonian and current
operators that is specially well suited for the study of the compact case. The
zero mode energy spectrum, the value of the axial anomaly and the anomalous
commutators this model presents are hence determined in a manifestly gauge
invariant manner. Contrary to the non compact case, the zero mode spectrum is
not equally spaced and consequently the theory does not lead to the spectrum of
a free scalar boson. All the states are invariant under large gauge
transformations. In particular, that is the case for the vacuum, and
consequently the $\theta$-dependence does not appear.
