2019/01/29 by Nguyen Viet Dang
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential operator #Field (mathematics) #Field theory (psychology) #Geometry and complex manifolds #Holomorphic function #Line bundle #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum #Quantum electrodynamics #Quantum field theory #Quantum mechanics #Renormalization #Statistical physics #Theoretical physics #math-ph #math.MP #msc:58B12 #msc:58J40 #msc:58J50 #msc:58J52 #msc:81T16 #msc:81T20
paper · pdf · doi:10.2140/apde.2022.15.1
published as Analysis & PDE 15 (2022) 1-62 · Almost totally revised paper, some mistakes corrected and many details added
openalex publication_date 2019/01/29 · openalex created_date 2019/03/22 · arxiv created 2020/05/27 · arxiv updated 2022/03/23 · openalex updated_date 2026/08/05
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 Fréchet 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 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analytic trivializations of some holomorphic line bundle over A relating our results to a conjectural picture from some unpublished notes by Quillen [52] from April 1989.