nLab Uhlenbeck's singularity theorem

Idea

Uhlenbeck’s singularity theorem is a theorem about the removal of a singularity in four-dimensional Yang-Mills fields with finite energy using a suitable gauge fixing. It states as a consequence that Yang-Mills fields with finite energy on euclidean space $\mathbb{R}^4$ arise from Yang-Mills fields on its one-point compactification, the 4-sphere $S^4$.

Uhlenbeck's compactness 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.

Statement

For the closed disk $B^n\setminus\{0\} \coloneqq\{x\in\mathbb{R}^n|\|x\|\leq 1\}$ and a vector bundle $\eta\twoheadrightarrow B^4\setminus\{0\}$ with structure group $G$, a Yang-Mills connection $A\in\Omega^1(B^4\setminus\{0\},Ad(\eta))$ with finite energy:

$$\int_{B^4}\|F_A\|^2 d vol_g \lt\infty$$

the vector bundle $\eta\twoheadrightarrow B^4\setminus\{0\}$ extends to a smooth vector bundle $\overline\eta\twoheadrightarrow B^4$ and the Yang-Mills connection $A\in\Omega^1(B^4\setminus\{0\},Ad(\eta))$ extends to a smooth Yang-Mills connection $\overline{A}\in\Omega^1(B^4,Ad(\overline\eta))$.

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, Removable Singularities in Yang-Mills Fields, Communications in Mathematical Physics 83 (1982) 11–29 [doi:10.1007/BF01947068 bibcode:1982CMaPh..83…11U]

• Terence Tao, Gang Tian, A singularity removal theorem for Yang-Mills fields in higher dimensions (2002) [arXiv:math/0209352]

See also:

• Wikipedia, Uhlenbeck’s singularity theorem