satisfying the integrability condition . The pair of data is then called a Higgs bundle.
Higgs bundle can be considered as a limiting case of a flatconnection in the limit in which its exterior differential tends to zero, be obtained by rescaling. So the equation where is a matrix of connection can be rescaled by putting a small parameter in front of .
For a Higgs bundle to admit a harmonic metric (…) it needs to be stable (…).
N.J. Hitchin, Stable bundles and integrable systems, Duke Math. J. , 54 (1987) pp. 91–114 MR89a:32021doieuclid; The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126, MR887284doi; Flat connections and geometric quantization, Comm. Math. Phys. 131, n 2 (1990), 347-380, euclid