On a compact manifold
,
we consider the affine space
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 Fréchet space
vanishing
exactly over noninvertible elements, having minimal growth at infinity along complex
rays in
and which are obtained by local renormalization, 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 products and
renormalized Feynman amplitudes in perturbation theory in position space à 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
.
This relates our results to a conjectural picture from some unpublished notes by Quillen
from April 1989.
Keywords
determinant lines, renormalization, quantum field theory
