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Global gauge fixing for connections with small curvature on \(\mathbb T^{2}\). (English) Zbl 1127.53020

It is a natural question whether for connections on a given vector bundle over a manifold \(M\), there always exist gauges such that the size of the connection matrix is (universally) bounded by the size of the curvature. For bundles over the ball \(B^4\), it was proved by K. Uhlenbeck in [Commun. Math. Phys. 83, 31–42 (1982; Zbl 0499.58019)] that this is possible for connections with sufficiently small curvature. The result there is that if the curvature \(F_A\) of a connection \(A\) is sufficiently small then the \(C^{r+1}\)-norm of \(A\) is bounded above by a universal constant times the \(C^r\)-norm of \(F_A\).
In the paper under review, the author studies this problem in the case of a trivial complex rank two bundle over the torus \(\mathbb T^2\). Assuming that \(F_A\) is sufficiently small, it is shown that the \(C^{r+1}\)-norm of \(A\) is bounded above by a universal constant times the square root of the \(C^r\)-norm of \(F_A\). Examples of constant connection matrices are used to show that the square root in this estimate is necessary.
The proof is based on the fact that a unitary connection endows a trivial rank two bundle with a holomorphic structure. For small curvature, this structure has to be semi-stable, and such bundles can be classified. This is used to find explicit representatives in gauge equivalence classes of connections with sufficiently small curvature. Using these explicit forms, the result is derived via an implicit function theorem and a result of Fukaya, which implies that a connection with small curvature is close to a flat connection.
Reviewer: Andreas Cap (Wien)

MSC:

53C05 Connections (general theory)
32L05 Holomorphic bundles and generalizations
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems

Citations:

Zbl 0499.58019
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References:

[1] DOI: 10.1017/CBO9780511543098 · doi:10.1017/CBO9780511543098
[2] DOI: 10.1007/978-1-4612-1688-9 · doi:10.1007/978-1-4612-1688-9
[3] DOI: 10.1007/s000390050064 · Zbl 0914.53015 · doi:10.1007/s000390050064
[4] DOI: 10.1007/BF01947069 · Zbl 0499.58019 · doi:10.1007/BF01947069
[5] DOI: 10.1007/978-1-4757-3946-6 · doi:10.1007/978-1-4757-3946-6
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