MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

On a Riemann surface irreducible unitary connections on complex vector bundles of rank $n\geq 2$ are uniquley determined by their underlying holomorphic structure $\frac{1}{2}(\nabla+i*\nabla)$, where $*$ is the adjoint of the complex structure $J$ of the Riemann surface. This holomorphic structure is stable.

Are there any special examples, where this correspondence is made more explicit?

share|cite|improve this question
Surely you mean unitary connections on complex vector bundles on a complex manifold, for otherwise there is no underlying holomorphic structure. Since gauge theory is not only studied on complex manifolds, then could I ask you to please edit your question and state clearly your assumptions? Thanks. – José Figueroa-O'Farrill Apr 19 '11 at 8:16
Many thanks for editing the question! – José Figueroa-O'Farrill Apr 19 '11 at 11:00
You have been right, I should have ask my question more carefully. Thanks for your comments. – Sebastian Apr 19 '11 at 11:26

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.