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Morse theory for the Yang-Mills energy function near flat connections

Paul M. N. Feehan

TL;DR

This work refines the Uhlenbeck energy-gap distance estimate for connections by incorporating Morse–Bott structure and, when necessary, a power $\lambda$ of the curvature norm $\|F_A\|_{L^p}$ to accommodate non-regular flat points. It develops a robust framework combining YM gradient flow on Coulomb-gauge slices, abstract evolution in Banach spaces, and Łojasiewicz distance inequalities to obtain global existence, convergence, and explicit rates near local minima, as well as a deformation-retraction result onto the moduli space of flat connections. The results connect non-regular points in the moduli space to quantitative distance bounds to $M(P)$, demonstrate energy gaps that force flatness for small energy, and provide a Morse–Bott–friendly landscape for YM energy in higher dimensions. The work thus advances a Morse-theoretic, analytic understanding of the Yang–Mills energy near flat connections with potential applications to higher-dimensional gauge theories and the study of moduli spaces of flat connections.}

Abstract

A result (Corollary 4.3) in an article by Uhlenbeck (1985) asserts that the $W^{1,p}$-distance between the gauge-equivalence class of a connection $A$ and the moduli subspace of flat connections $M(P)$ on a principal $G$-bundle $P$ over a closed Riemannian manifold $X$ of dimension $d\geq 2$ is bounded by a constant times the $L^p$ norm of the curvature, $\|F_A\|_{L^p(X)}$, when $G$ is a compact Lie group, $F_A$ is $L^p$-small, and $p>d/2$. While we prove that this estimate holds when the Yang-Mills energy function on the space of Sobolev connections is Morse-Bott along the moduli subspace $M(P)$ of flat connections, it does not hold when the Yang-Mills energy function fails to be Morse-Bott, such as at the product connection in the moduli space of flat $\mathrm{SU}(2)$ connections over a real two-dimensional torus. However, we prove that a useful modification of Uhlenbeck's estimate always holds provided one replaces $\|F_A\|_{L^p(X)}$ by a suitable power $\|F_A\|_{L^p(X)}^λ$, where the positive exponent $λ$ reflects the structure of non-regular points in $M(P)$. The proof of our refinement involves gradient flow and Morse theory for the Yang-Mills energy function on the quotient space of Sobolev connections and a Lojasiewicz distance inequality for the Yang-Mills energy function. A special case of our estimate, when $X$ has dimension four and the connection $A$ is anti-self-dual, was proved by Fukaya (1998) by entirely different methods. Lastly, we prove that if $A$ is a smooth Yang-Mills connection with small enough energy, then $A$ is necessarily flat.

Morse theory for the Yang-Mills energy function near flat connections

TL;DR

This work refines the Uhlenbeck energy-gap distance estimate for connections by incorporating Morse–Bott structure and, when necessary, a power of the curvature norm to accommodate non-regular flat points. It develops a robust framework combining YM gradient flow on Coulomb-gauge slices, abstract evolution in Banach spaces, and Łojasiewicz distance inequalities to obtain global existence, convergence, and explicit rates near local minima, as well as a deformation-retraction result onto the moduli space of flat connections. The results connect non-regular points in the moduli space to quantitative distance bounds to , demonstrate energy gaps that force flatness for small energy, and provide a Morse–Bott–friendly landscape for YM energy in higher dimensions. The work thus advances a Morse-theoretic, analytic understanding of the Yang–Mills energy near flat connections with potential applications to higher-dimensional gauge theories and the study of moduli spaces of flat connections.}

Abstract

A result (Corollary 4.3) in an article by Uhlenbeck (1985) asserts that the -distance between the gauge-equivalence class of a connection and the moduli subspace of flat connections on a principal -bundle over a closed Riemannian manifold of dimension is bounded by a constant times the norm of the curvature, , when is a compact Lie group, is -small, and . While we prove that this estimate holds when the Yang-Mills energy function on the space of Sobolev connections is Morse-Bott along the moduli subspace of flat connections, it does not hold when the Yang-Mills energy function fails to be Morse-Bott, such as at the product connection in the moduli space of flat connections over a real two-dimensional torus. However, we prove that a useful modification of Uhlenbeck's estimate always holds provided one replaces by a suitable power , where the positive exponent reflects the structure of non-regular points in . The proof of our refinement involves gradient flow and Morse theory for the Yang-Mills energy function on the quotient space of Sobolev connections and a Lojasiewicz distance inequality for the Yang-Mills energy function. A special case of our estimate, when has dimension four and the connection is anti-self-dual, was proved by Fukaya (1998) by entirely different methods. Lastly, we prove that if is a smooth Yang-Mills connection with small enough energy, then is necessarily flat.

Paper Structure

This paper contains 55 sections, 73 theorems, 494 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Existence of a flat connection on a principal bundle supporting a connection with $L^p$-small curvature
  4. Łojasiewicz distance inequality for functions on Banach spaces
  5. Global existence and convergence of Yang--Mills gradient flow on a Coulomb-gauge slice around a local minimum
  6. Almost strong deformation retraction of a neighborhood in the quotient space of Sobolev connections onto the moduli subspace of flat connections
  7. Łojasiewicz distance inequality for the Yang--Mills energy function
  8. $L^{d/2}$ and $L^2$ energy gaps for Yang--Mills connections
  9. Nonlinear estimate for distance to moduli subspace of flat connections
  10. Optimal Łojasiewicz distance inequality for a Yang--Mills energy function implies its Morse--Bott property
  11. Outline
  12. Acknowledgments
  13. Preliminaries
  14. Uhlenbeck's Weak Compactness Theorem Revisited
  15. Transformation to Coulomb gauge
  16. ...and 40 more sections

Key Result

Theorem 1

Let $G$ be a compact Lie group, $P$ be a smooth principal $G$-bundle over a closed, smooth Riemannian manifold $(X,g)$ of dimension $d \geq 2$, and $p \in (d/2,\infty)$, and $A_1$ be a $C^\infty$ reference connection on $P$. Then there is a constant $\varepsilon=\varepsilon(A_1,g,G,p,[P]) \in (0,1]$ then there are a $W^{1,p}$ flat connection $\Gamma$ on $P$ and a $W^{2,p}$ gauge transformation $v

Figures (2)

  • Figure A.1: The left-hand panel describes a fundamental domain for the action of the semidirect product $\mathbb{Z}^2\rtimes \mathbb{Z}/2\mathbb{Z}$ on $\mathbb{R}^2$ while on the right-hand panel shows the "pillowcase" obtained by performing the identifications described on the left. The pillowcase is homeomorphic to the two-dimensional sphere. (This is Hedden_Herald_Kirk_2014 due to Hedden, Herald, and Kirk and used by permission of those authors.)
  • Figure A.2: Example of an immersion and an embedding of $\mathop{\mathrm{Aut}}\nolimits^{2,p}(P)/\operatorname{Stab}(A_0)$ in ${\mathscr{A}}^{1,p}(P)$ as the orbit ${\mathscr{O}}(A_0) = \{u(A_0): u \in \mathop{\mathrm{Aut}}\nolimits^{2,p}(P)\}$.

Theorems & Definitions (146)

  • Theorem 1: Existence of a flat connection on a principal bundle supporting a connection with $L^p$-small curvature
  • Corollary 2: Refined estimate near a flat connection with zero-dimensional Zariski tangent space
  • Theorem 1.1: Łojasiewicz distance inequality for analytic functions on Euclidean space
  • Theorem 3: Łojasiewicz distance inequalities for functions on Banach spaces
  • Remark 1.2: On the hypothesis that $\mathop{\mathrm{Crit}}\nolimits{\mathscr{E}}^2=\mathop{\mathrm{Zero}}\nolimits{\mathscr{E}}^2$ in Item \ref{['item:Distance_zero_noncritical_set']} of Theorem \ref{['mainthm:Lojasiewicz_distance_inequality_hilbert_space']}
  • Theorem 4: Global existence and convergence of Yang--Mills gradient flow on a Coulomb-gauge slice around a local minimum
  • Remark 1.3: Comparison with results on Yang--Mills gradient flow due to Råde, Kozono, Maeda, and Naito, and Schlatter and Struwe
  • Theorem 1.4: Finite-time blow-up for Yang-Mills gradient flow over a sphere of dimension greater than or equal to five
  • Corollary 5: Global existence and convergence of Yang--Mills gradient flow on a slice for initial connections with small energy over low-dimensional manifolds
  • Remark 1.5: Extension of Corollary \ref{['maincor:Yang-Mills_gradient_flow_global_existence_and_convergence_started_small_energy']} to higher dimensions and critical Sobolev exponent
  • ...and 136 more