nLab Uhlenbeck's compactness theorem

Idea

Uhlenbeck’s compactness theorem is a theorem about sequences of (weak Yang-Mills) connections with uniformly bounded curvature having weakly or uniformly convergent subsequences up to gauge.

This is an important theorem used in the compactification of the anti self-dual Yang-Mills moduli space (ASDYM moduli space), which is central to the construction of Donaldson invariants on 4-manifolds or monopole Floer homology on 3-manifolds.

Uhlenbeck's singularity theorem was also first published in the same journal, Communications in Mathematical Physics. In 2019, Karen Uhlenbeck, after whom the theorem is named, became the first woman to be awarded the Abel Prize, in part for her contributions to partial differential equations and gauge theory.

Uhlenbeck’s weak compactness theorem

Let $X$ be a $n$-dimensional compact Riemannian manifold and $P\twoheadrightarrow X$ be a principal $G$-bundle with a compact Lie group $G$. Let $1\lt p\lt\infty$ with $p\gt n/2$ and let $(A_m)_{m\in\mathbb{N}}\in\mathcal{A}^{1,p}(P)\coloneqq W^{1,p}(X,Ad(P))\subset L^p(X,Ad(P))$ be a sequence of Sobolev principal connections with uniform bound for $\|F_{A_m}\|_p$, the norm of their curvatures. Then there exists a sequence $u_m\in\mathcal{G}^{2,p}(P)$ of gauge transformations, so that $u_m^*A_m$ converges weakly?. In other words, any $L^p$-bounded? subset of $\mathcal{A}^{1,p}(P)/\mathcal{G}^{2,p}(P)$ is weakly compact?.

(Uhlenbeck 82, Theorem 1.5 (3.6) Wehrheim 03, Theorem A)

Uhlenbeck’s strong compactness theorem

Let $X$ be a $n$-dimensional compact Riemannian manifold and $P\twoheadrightarrow X$ be a principal $G$-bundle with a compact Lie group $G$. Let $1\lt p\lt\infty$ with $p\gt n/2$ and $p\gt 4/3$ if $n=2$. Let $(A_m)_{m\in\mathbb{N}}\in\mathcal{A}^{1,p}(P) \coloneqq W^{1,p}(X,Ad(P))\subset L^p(X,Ad(P))$ be a sequence of weak Yang-Mills connections, hence so that:

$$\int_X\langle F_{A_m},d_{A_m}\beta\rangle d vol_g =0$$

for all $m\in\mathbb{N}$ and $\beta\in\mathcal{A}^{1,p}(P)$, with uniform bound for $\|F_{A_m}\|_p$. Then there exists a subsequence, also denoted $(A_m)_{m\in\mathbb{N}}$, and a sequence $u_m\in\mathcal{G}^{2,p}(P)$ of gauge transformations, so that $u_m^*A_m$ converges uniformly to a smooth connection $A\in\mathcal{A}(P)$. (Uhlenbeck’s strong compactness theorem is not stated explicitly in Uhlenbeck’s 1982 paper, but follows from the results within.)

(Wehrheim 03, Theorem E)

Related concepts

On ordinary Yang-Mills theory (YM):

• D=2 YM, D=3 YM, D=4 YM, D=5 YM, D=6 YM, D=7 YM, D=8 YM

• Maxwell theory/electromagnetism (U(1) YM), Donaldson theory (SU(2) YM), quantum chromodynamics (SU(3) YM)

• Yang-Mills equation, linearized Yang-Mills equation, Yang-Mills instanton, Yang-Mills field, stable Yang-Mills connection, Yang-Mills moduli space, Yang-Mills flow, F-Yang-Mills equation, Bi-Yang-Mills equation

• Uhlenbeck's singularity theorem, Uhlenbeck's compactness theorem

On variants of Yang-Mills theory and on super Yang-Mills theory (SYM):

• Yang-Mills-Higgs equations, stable Yang-Mills-Higgs pair, Yang-Mills-Higgs flow

• Einstein-Yang-Mills theory, Einstein-Yang-Mills-Dirac theory, Einstein-Yang-Mills-Dirac-Higgs theory

• 3D Yang-Mills-Chern-Simons theory

• 3d superconformal gauge field theory: D=3 N=1 SYM, D=3 N=2 SYM, D=3 N=4 SYM

• 4d superconformal gauge field theory: D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM

• D=5 SYM, D=6 SYM, D=7 SYM, D=10 SYM

• topological Yang-Mills theory, topologically twisted D=4 super Yang-Mills theory

References

• Karen Uhlenbeck, Connections with $L^p$ bounds on curvature, Communications in Mathematical Physics 83 (1982) 31–42 [doi:10.1007/BF01947069]

• Alex Waldron, Uhlenbeck compactness for Yang-Mills flow in higher dimensions (2018) [arXiv:1812.10863]

• Katrin Wehrheim, Uhlenbeck compactness, EMS Series of Lectures in Mathematics. Vol. 1 (2003) [doi:10.4171/004 ISBN 978-3-03719-004-3]

See also:

• Wikipedia, Uhlenbeck’s compactness theorem