Geometric aspects of the Kapustin-Witten equations. (English) Zbl 1260.53002

Gagliardo, Michael; Uhlenbeck, Karen

J. Fixed Point Theory Appl. 11, No. 2, 185-198 (2012).

Summary: This expository paper introduces the Kapustin-Witten equations to mathematicians. We discuss the connections between the complex Yang-Mills equations and the Kapustin-Witten equations. In addition, we show the relation between the Kapustin-Witten equations, the moment map condition and the gradient Chern-Simons flow. The new results in the paper correspond to estimates on the solutions to the Kapustin-Witten equations given an estimate on the complex part of the connection. This leaves open the problem of obtaining global estimates on the complex part of the connection.

MSC:

53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C80	Applications of global differential geometry to the sciences
58E15	Variational problems concerning extremal problems in several variables; Yang-Mills functionals

Keywords:

Kapustin-Witten equations; complex Yang-Mills equations

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