Talking Points: Edward Witten on Geometric Langlands
The Langlands program is incredibly vast and far-reaching. The deepest aspect of it, as far as we know, involves the number theory setting where Langlands started close to forty years ago. However, the Langlands program has all kinds of manifestations. The part that I have tried to understand personally is the “geometric’’ form of the Langlands program, where some of the ideas are converted from number theory into statements in geometry. For a long time, mathematicians working on the geometric Langlands program have made great use of ideas from mathematical physics—notably an area called conformal field theory that is important both in condensed matter physics and in string theory. But the physics ideas were always rearranged in ways that—to a physicist—looked strange. This bothered me a lot for years, really for decades. I felt that if physics-based ideas are relevant to the geometric Langlands program, then it should be possible to reformulate the geometric Langlands program in terms that would be more recognizable to a physicist. Eventually, after a lot of false starts, I did have some success with this. Despite all the hard work, I personally only understand a tiny bit of the Langlands program. I think, however, that this probably puts me in the majority among researchers who work on it. It is such a vast subject that few can really have an overview. And where it will ultimately lead, it is way too soon to say.