Nonperturbative Renormalization as Riemann-Hilbert Decomposition of Schwinger-Dyson D-Module

TL;DR

The paper addresses the nonperturbative content of renormalization in quantum field theory by reformulating the regularized Schwinger–Dyson hierarchy as a meromorphic connection (a $\mathscr{D}$-module) on spacetime × regulator disc. It uses the irregular Riemann–Hilbert correspondence to decompose this connection into a formal submodule carrying all UV poles and an analytic submodule finite at the regulator origin, identifying counterterms with Stokes data and the renormalized theory with the analytic submodule; the Callan–Symanzik flow emerges as an isomonodromic deformation. Perturbative Connes–Kreimer renormalization is recovered as the loop-group Birkhoff factorization within this geometric framework, with the divergent graph data encoded in the formal part and renormalized amplitudes in the analytic part. The Malgrange–Sibuya condition provides a nonperturbative criterion for renormalizability, and the authors discuss finite-rank and infinite-rank (trans-series) strategies to access nonperturbative QFT dynamics, outlining substantial open challenges for a fully nonperturbative theory.

Abstract

We present a non-perturbative formulation of renormalization by viewing the regularized Schwinger-Dyson hierarchy as a meromorphic connection, that is, as a D-module on the product of spacetime with the regulator disc. The irregular Riemann-Hilbert correspondence splits this connection into a purely formal submodule that contains every ultraviolet pole and a holomorphic submodule that is finite at the regulator origin. In this setting, counterterms coincide with the formal Stokes data, the renormalized theory is identified with the analytic submodule, and the Callan-Symanzik flow appears as an isomonodromic deformation in the physical scale. Viewed through this lens, the Connes-Kreimer construction with its graph Hopf algebra, Bogoliubov recursion, and Birkhoff factorisation is simply the perturbative shadow cast by the global geometric decomposition of the full Schwinger-Dyson system.