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Renormalization and resolution of singularities

Christoph Bergbauer, Romeo Brunetti, Dirk Kreimer

TL;DR

The paper develops a geometric framework for perturbative renormalization in position space by resolving diagonals of Feynman graphs via De Concini–Procesi wonderful models. It shows that a canonical extension of Feynman distributions to the smooth model reproduces Epstein–Glaser renormalization, with pole structure captured by residues on the exceptional divisors and counterterms encoded by local subtractions along irreducible divisors. The approach unifies the forest formula, the locality of counterterms, and Connes–Kreimer Hopf-algebra ideas within a single geometric setting, and yields a combinatorial description of Laurent coefficients via nested sets and contracted graphs. These results illuminate how renormalization can be performed locally on a resolution of singularities and suggest concrete links to motivic structures and Hopf-algebra formalisms, with potential applications to curved spacetimes and broader representations of Feynman amplitudes. The framework also clarifies the role of minimal versus maximal models and sets the stage for extending to non-logarithmic divergences and other regularization schemes.

Abstract

Since the seminal work of Epstein and Glaser it is well established that perturbative renormalization of ultraviolet divergences in position space amounts to extension of distributions onto diagonals. For a general Feynman graph the relevant diagonals form a nontrivial arrangement of linear subspaces. One may therefore ask if renormalization becomes simpler if one resolves this arrangement to a normal crossing divisor. In this paper we study the extension problem of distributions onto the wonderful models of de Concini and Procesi, which generalize the Fulton-MacPherson compactification of configuration spaces. We show that a canonical extension onto the smooth model coincides with the usual Epstein-Glaser renormalization. To this end we use an analytic regularization for position space. The 't Hooft identities relating the pole coefficients may be recovered from the stratification, and Zimmermann's forest formula is encoded in the geometry of the compactification. Consequently one subtraction along each irreducible component of the divisor suffices to get a finite result using local counterterms. As a corollary, we identify the Hopf algebra of at most logarithmic Feynman graphs in position space, and discuss the case of higher degree of divergence.

Renormalization and resolution of singularities

TL;DR

The paper develops a geometric framework for perturbative renormalization in position space by resolving diagonals of Feynman graphs via De Concini–Procesi wonderful models. It shows that a canonical extension of Feynman distributions to the smooth model reproduces Epstein–Glaser renormalization, with pole structure captured by residues on the exceptional divisors and counterterms encoded by local subtractions along irreducible divisors. The approach unifies the forest formula, the locality of counterterms, and Connes–Kreimer Hopf-algebra ideas within a single geometric setting, and yields a combinatorial description of Laurent coefficients via nested sets and contracted graphs. These results illuminate how renormalization can be performed locally on a resolution of singularities and suggest concrete links to motivic structures and Hopf-algebra formalisms, with potential applications to curved spacetimes and broader representations of Feynman amplitudes. The framework also clarifies the role of minimal versus maximal models and sets the stage for extending to non-logarithmic divergences and other regularization schemes.

Abstract

Since the seminal work of Epstein and Glaser it is well established that perturbative renormalization of ultraviolet divergences in position space amounts to extension of distributions onto diagonals. For a general Feynman graph the relevant diagonals form a nontrivial arrangement of linear subspaces. One may therefore ask if renormalization becomes simpler if one resolves this arrangement to a normal crossing divisor. In this paper we study the extension problem of distributions onto the wonderful models of de Concini and Procesi, which generalize the Fulton-MacPherson compactification of configuration spaces. We show that a canonical extension onto the smooth model coincides with the usual Epstein-Glaser renormalization. To this end we use an analytic regularization for position space. The 't Hooft identities relating the pole coefficients may be recovered from the stratification, and Zimmermann's forest formula is encoded in the geometry of the compactification. Consequently one subtraction along each irreducible component of the divisor suffices to get a finite result using local counterterms. As a corollary, we identify the Hopf algebra of at most logarithmic Feynman graphs in position space, and discuss the case of higher degree of divergence.

Paper Structure

This paper contains 26 sections, 35 theorems, 135 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Subspace arrangements associated to Feynman graphs
  3. Configurations and subspace arrangements of singularities
  4. Subspace arrangements of divergences
  5. Subspaces and polydiagonals
  6. Regularization, blowing up, and residues of primitive graphs
  7. Distributions and densities on manifolds
  8. Distributions and birational transformations
  9. Analytic regularization
  10. Primitive graphs, their residues and renormalization
  11. Models for the complements of subspace arrangements
  12. Smooth models and normal crossing divisors
  13. The Wonderful Models
  14. Irreducibility and building sets
  15. Nested sets
  16. ...and 11 more sections

Key Result

Proposition 2.1

Let $\Gamma$ be at most logarithmic. If $\gamma\subseteq\Gamma$ is divergent then it is saturated.

Figures (4)

  • Figure 1: A picture of $\mathbb{R}^{V_0}_{sing}(K_4).$
  • Figure 2: (Spherical) blowup of the origin in $\mathbb{R}^{V_0}_{sing}(K_4),$ where projective spaces are replaced by spheres. The maximal wonderful model would proceed by blowing up all strict transforms of lines incident to the exceptional divisor, and finally the strict transforms of the planes.
  • Figure 3: Minimal (spherical) model of $\mathbb{R}^{V_0}_{sing}(K_4),$ corresponding to the Fulton-MacPherson compactification of the configuration space of 4 points in $\mathbb{R}.$ After the central blowup, only those strict transforms of lines are blown up which are not a normal crossing intersection in the first place.
  • Figure 4: The edges of $s$ are broken lines, the edges of $t\setminus s$ full lines. $p_{t,s}(\{v_0,v_1,v_2,v_3\})=v_0,$$p_{t,s}(v_4)=v_4,$$p_{t,s}(\{v_5,v_6,v_7\})=v_5,$$p_{t,s}(v_8)=v_8,$$p_{t,s}(v_9)=v_9.$

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.5
  • ...and 41 more