# Renormalization of determinant lines in Quantum Field Theory

## TL;DR

This paper develops a comprehensive theory of renormalized determinants for elliptic operators on compact manifolds, connecting quantum field theory concepts with spectral and geometric methods, and providing new analytic trivializations of determinant line bundles.

## Contribution

It constructs and classifies local renormalized determinants on elliptic operators, linking quantum field theory renormalization with spectral and geometric frameworks, and relates to Quillen's conjectural line bundle trivializations.

## Key findings

- Constructed all local renormalized determinants vanishing on non-invertible operators.
- Connected renormalized determinants with spectral zeta functions and Feynman amplitudes.
- Proved these determinants define trivializations of holomorphic line bundles over the space of perturbations.

## Abstract

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trivializations of some holomorphic line bundle over $A$ relating our results to a conjectural picture from some unpublished notes by Quillen [52] from April 1989.