Aspects of Gauge Theory

Abstract

Let us summarize briefly some of the material on integrable systems, bidifferential calculi, bicomplexes, etc. in Chapter 10 and highlight the connections to gauge theory and the SW map. We expand also the references to the SW map and cite [22, 35, 36, 55, 68, 77, 78, 109, 355, 356, 357, 358, 359, 360, 361, 362, 438, 448, 615]. In a certain sense the relation of BC to integrable systems developed in Chapter 10 seems overly contrived since one seems to essentially insert by hand the coefficients needed to produce an integrrable system. However, looking at Examples 1.6, 1.7, etc. we see that the combinations of f, f _x ,f _xx ,f _xxx ,etc. needed for KdV and other equations do actually arise by reverse engineering from simple commutativity properties of dx, dt,dt etc.