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Global critical points of the Standard Model on four-dimensional spacetimes of expanding type

Volker Branding, Marko Sobak

TL;DR

The paper develops a gauge‑theoretic, geometric formulation of the classical Standard Model on globally hyperbolic four‑manifolds, expressing the coupled Yang–Mills–Higgs–Dirac system in a bundle‑valued, conformally covariant framework. Using a conformal transformation to a Gaussian foliation with bounded geometry, it proves a future‑global existence result for conformal Standard Model triplets under small initial data on expanding spacetimes, leveraging a gauge‑invariant higher‑order energy estimate and careful propagation of constraints. Central to the approach are intrinsic Sobolev spaces on spatial slices, Sobolev spaces of connections with bounded geometry, and detailed energy inequalities that control the Yang–Mills, Higgs, and Dirac sectors and their couplings, including Yukawa interactions and curvature terms. The result advances the mathematical understanding of field theories in curved spacetimes by providing a fully geometric, gauge‑invariant, global well‑posedness theory with explicit decay rates, and it integrates conformal techniques with energy methods to handle the curvature‑dependent dynamics.

Abstract

The Standard Model of elementary particle physics is one of the most successful models of contemporary theoretical physics being in full agreement with experiments. However, its mathematical structure deserves further investigations both from a geometric and an analytic point of view. The aim of this manuscript is to provide a mathematically well-defined and self-contained description of the Standard Model in terms of gauge theory and differential geometry on globally hyperbolic manifolds. Within this setup we then prove the existence of a global solution for the Euler-Lagrange equations of the Standard Model (with the conformal Higgs potential) under the assumptions that the globally hyperbolic manifold is a four-dimensional spacetime of expanding type and small initial data. This is achieved by establishing a gauge-invariant energy estimate which is of independent mathematical interest.

Global critical points of the Standard Model on four-dimensional spacetimes of expanding type

TL;DR

The paper develops a gauge‑theoretic, geometric formulation of the classical Standard Model on globally hyperbolic four‑manifolds, expressing the coupled Yang–Mills–Higgs–Dirac system in a bundle‑valued, conformally covariant framework. Using a conformal transformation to a Gaussian foliation with bounded geometry, it proves a future‑global existence result for conformal Standard Model triplets under small initial data on expanding spacetimes, leveraging a gauge‑invariant higher‑order energy estimate and careful propagation of constraints. Central to the approach are intrinsic Sobolev spaces on spatial slices, Sobolev spaces of connections with bounded geometry, and detailed energy inequalities that control the Yang–Mills, Higgs, and Dirac sectors and their couplings, including Yukawa interactions and curvature terms. The result advances the mathematical understanding of field theories in curved spacetimes by providing a fully geometric, gauge‑invariant, global well‑posedness theory with explicit decay rates, and it integrates conformal techniques with energy methods to handle the curvature‑dependent dynamics.

Abstract

The Standard Model of elementary particle physics is one of the most successful models of contemporary theoretical physics being in full agreement with experiments. However, its mathematical structure deserves further investigations both from a geometric and an analytic point of view. The aim of this manuscript is to provide a mathematically well-defined and self-contained description of the Standard Model in terms of gauge theory and differential geometry on globally hyperbolic manifolds. Within this setup we then prove the existence of a global solution for the Euler-Lagrange equations of the Standard Model (with the conformal Higgs potential) under the assumptions that the globally hyperbolic manifold is a four-dimensional spacetime of expanding type and small initial data. This is achieved by establishing a gauge-invariant energy estimate which is of independent mathematical interest.

Paper Structure

This paper contains 63 sections, 17 theorems, 310 equations, 1 table.

Table of Contents

  1. Introduction
  2. Setting
  3. Main result
  4. Past and present
  5. Organization
  6. Conventions and preliminaries
  7. Spacetime conventions
  8. Spin geometry
  9. Bundle-valued forms
  10. Principal bundles
  11. Bundle automorphisms
  12. Geometric structure of the Standard Model
  13. Bosons (Yang-Mills-Higgs sector)
  14. Yang-Mills energy density
  15. Bosonic mass and Higgs fields
  16. ...and 48 more sections

Key Result

Theorem 1.1

Let $k > 3/2$ be an integer. Let $(M, h)$ be a four-dimensional spin spacetime of expanding type with scale factor $s$, and let $P \cong \pi^\ast P^\Sigma \to M$ be a principal $G$-bundle for a compact Lie group $G$. Fix an initial connection where $\eta^0 \in H^{k+1}_{\sigma^0}(T^\ast \Sigma \otimes \mathop{\mathrm{Ad}}\nolimits P^\Sigma)$ and $\sigma^0$ is a smooth connection of bounded geometr

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • ...and 36 more