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Topology of weak $G$-bundles via Coulomb gauges in critical dimensions

Swarnendu Sil

TL;DR

This work defines a robust topology for Sobolev G-bundles in the critical dimension by attaching a topological class to a pair (P, A) through Coulomb gauges, ensuring C^0-continuity and Hölder regularity of the resulting bundles. It proves strong density: any W^{k,p} bundle with a W^{k,p} connection at kp=n can be approximated by smooth bundles with smooth connections while preserving an appropriate equivalence, and in the Abelian case, even in supercritical settings. The paper also shows that the topology stabilizes along sequences with uniformly bounded $n/2$-Yang-Mills energy under equiintegrability of the curvature, providing a criterion to detect topological flatness through Yang-Mills energy. Overall, the approach encodes topological data into the connection, enabling analysis in critical dimensions that would otherwise lose topological control, and links this topology to smoothing results and curvature concentration phenomena. The results have implications for higher-dimensional gauge theory, energy gap phenomena, and the precise interaction between topology and analysis in Sobolev settings.

Abstract

The transition maps for a Sobolev $G$-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle $P$ is equipped with a Sobolev connection $A$, then one can associate a topological isomorphism class to the pair $\left( P, A\right),$ which is invariant under Sobolev gauge changes and coincides with the usual notions for regular bundles and connections. This is based on a regularity result which says any bundle in the critical dimension in which a Sobolev connection is in Coulomb gauges are actually $C^{0,α}$ for any $α< 1.$ We also show any such pair can be strongly approximated by smooth connections on smooth bundles. Finally, we prove that for sequences $(P^ν,A^ν)$ with uniformly bounded $n/2$-Yang-Mills energy, the topology stabilizes if the $n/2$ norm of the curvatures are equiintegrable. This implies a criterion to detect topological flatness in Sobolev bundles in critical dimensions via $n/2$-Yang-Mills energy.

Topology of weak $G$-bundles via Coulomb gauges in critical dimensions

TL;DR

This work defines a robust topology for Sobolev G-bundles in the critical dimension by attaching a topological class to a pair (P, A) through Coulomb gauges, ensuring C^0-continuity and Hölder regularity of the resulting bundles. It proves strong density: any W^{k,p} bundle with a W^{k,p} connection at kp=n can be approximated by smooth bundles with smooth connections while preserving an appropriate equivalence, and in the Abelian case, even in supercritical settings. The paper also shows that the topology stabilizes along sequences with uniformly bounded -Yang-Mills energy under equiintegrability of the curvature, providing a criterion to detect topological flatness through Yang-Mills energy. Overall, the approach encodes topological data into the connection, enabling analysis in critical dimensions that would otherwise lose topological control, and links this topology to smoothing results and curvature concentration phenomena. The results have implications for higher-dimensional gauge theory, energy gap phenomena, and the precise interaction between topology and analysis in Sobolev settings.

Abstract

The transition maps for a Sobolev -bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle is equipped with a Sobolev connection , then one can associate a topological isomorphism class to the pair which is invariant under Sobolev gauge changes and coincides with the usual notions for regular bundles and connections. This is based on a regularity result which says any bundle in the critical dimension in which a Sobolev connection is in Coulomb gauges are actually for any We also show any such pair can be strongly approximated by smooth connections on smooth bundles. Finally, we prove that for sequences with uniformly bounded -Yang-Mills energy, the topology stabilizes if the norm of the curvatures are equiintegrable. This implies a criterion to detect topological flatness in Sobolev bundles in critical dimensions via -Yang-Mills energy.

Paper Structure

This paper contains 19 sections, 21 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Smooth principal $G$ bundles with connections
  4. Sobolev bundles and connections
  5. Sobolev principal $G$ bundles
  6. Sobolev spaces of connections
  7. Strong Density in the critical dimension
  8. Smooth Approximation in subcritical regime
  9. Coulomb gauges and elliptic estimates
  10. Regularity of Coulomb bundles
  11. Smooth approximation theorems: critical dimension
  12. Circle bundles in arbitrary dimension
  13. Topology of bundles in the critical dimension
  14. Coulomb bundles and gauge transformations
  15. Definition of the topology
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\left\lbrace U_{\alpha}\right\rbrace_{\alpha \in I}$ be a good cover of $M^{n}$ and let $\left\lbrace g_{\alpha\beta} \right\rbrace_{\alpha, \beta \in I}$ be a collection of maps such that $g_{\alpha\beta} \in W^{k,p}\left( U_{\alpha}\cap U_{\beta}; G\right)$, with $kp=n,$ for every $\alpha, \b for every $\alpha, \beta, \gamma \in I$ with $U_{\alpha}\cap U_{\beta}\cap U_{\gamma} \neq \emptyse

Theorems & Definitions (56)

  • Theorem 1: cocycle smoothing
  • Theorem 2: Stability of topology if curvatures does not concentrate
  • Theorem 3: $n/2$-Yang-Mills energy gap
  • Definition 5: $W^{k,p}$ principal $G$-bundles
  • Definition 6: $W^{k,p}$ equivalence
  • Definition 7: The $\mathcal{U}^{k,p}$ spaces of connections
  • Theorem 8: Smooth approximation in subcritical regime
  • Remark 9
  • Lemma 11
  • proof
  • ...and 46 more