Gauge Theory Is Dead!–Long Live Gauge Theory!

NOTICES OF THE AMS 335 I n late October and early November 1994 many mathematicians received e-mail from colleagues trumpeting the death of gauge theory. More than a decade earlier, S. K. Donaldson (Oxford) had found a deep but mysterious link between Yang-Mills theory from mathematical physics on the one hand and 4-dimensional differential topology on the other. Since then, many topologists had become fascinated by gauge theory. Last autumn they found themselves on the receiving end of wry comments, when rumour had it that a new set of equations proposed by E. Witten (IAS, Princeton) had made gauge theory obsolete in topology. This was, of course, an exaggeration. Witten’s equations, originating in his joint work [SW] with N. Seiberg (Rutgers) in quantum field theory and appearing in [W], do provide shortcuts to many of the consequences of gauge theory and quickly lead to proofs of very important new results. The equations are themselves part of a gauge theory and shed new light on the Yang-Mills equations that Donaldson used. Rather than making gauge theory obsolete, Witten’s equations make gauge theory even more interesting and more powerful.