  On a compact manifold $M$, we consider the affine space $A$ of non
self-adjoint perturbations of some invertible elliptic operator acting on
sections of some Hermitian bundle, by some differential operator of lower
order. We construct and classify all complex analytic functions on the
Fr\'echet space $A$ vanishing exactly over non invertible elements, having
minimal order and which are obtained by local renormalizations, a concept
coming from quantum field theory, called renormalized determinants. The
additive group of local polynomial functionals of finite degrees acts freely
and transitively on the space of renormalized determinants. We provide
different representations of the renormalized determinants in terms of spectral
zeta determinants, Gaussian Free Fields, infinite product and renormalized
Feynman amplitudes in perturbation theory in position space \`a la
Epstein-Glaser. Specializing to the case of Dirac operators coupled to vector
potentials and reformulating our results in terms of determinant line bundles,
we prove our renormalized determinants define some complex analytic
trivializations of some holomorphic line bundle over $A$ relating our results
to a conjectural picture from some unpublished notes by Quillen [52] from April
1989.
