Ed Witten on M theory, supersymmetry and appreciating calculus

Ed Witten, who is currently visiting Caltech, is a professor at the Institute for Advanced Study in Princeton, New Jersey (the place where Einstein used to work.) Witten was not one of the creators of string theory, but since he joined "Team String" in the mid-eighties, he's been a leading visionary, pausing here and there to make revolutionary observations and discoveries in mathematics using observations gleaned from high energy theoretical physics. He recently spent two years at Caltech, a development that had East Coast string theorists fearing that the United States was tilting to the West. We interviewed him in John Schwarz's office at Caltech, which also happened to Richard Feynman's office when he was alive.

What is K theory and what does it mean for string theory?
K theory is a mathematical theory that studies topology using matrices, using operators that don't commute with each another. What topology is, first of all, is the branch of mathematics where you don't care about the shape, so for example, a lumpy ball is equivalent to a round ball. But if there are holes, you do care about that, so a donut is different from either of these two. So, mathematicians learned, around 1960, that there was a very powerful tool in topology based on matrices, and that tool was K theory. And since quantum mechanics is about non-commuting operators, or matrices, there has always been a kind of naive analogy between K theory and quantum mechanics. An analogy that seemed naive to most physicists, but was often drawn by mathematicians such as Michael Atiyah.
However, we learned in the last few years that some questions about string theory, but slightly specialized questions usually, are usefully addressed using K theory. What K theory really addresses is a little bit subtle to explain. If you want to understand the charges carried by the D-branes, that's a question that leads to K theory. Or I might say at an even more basic level, D-branes are these strange objects whose positions are measured by matrices, and studying those matrices leads to K theory.
So K theory is the sort of topological underpinning of D-brane theory. But as physicists we're interested very much in whether the ball is round or lumpy, as are different things in physics. We wouldn't want to play baseball with a lumpy ball. So, the topology is just one side of the story..

What is noncommutative geometry and why is it important in string theory?
Well, one thing which we know about for sure in string theory is that the ordinary classical ideas about geometry are approximations, and don't really work precisely. But what you should really replace them with is not clear. However, there's a naive ideas about strings which really only works for open strings. Open strings are strings with endpoints, like in the original Type I superstring, where a particle was represented by a piece of string with charges at the ends. I've labeled the charges as q and q-bar for quark and antiquark, but that's modern terminology that might not have been present in the early says of string theory.
Once you've got open strings, they can join together, I'm going to call my open strings A or B, and they join end to end. But there are two ways of joining them. I could join them with A on the left and B on the right, or I could join them with B on the left and A on the right, and I get two different outputs. And it's very much like taking two matrices A and B and multiplying them together. So there's some noncommutativity in the interactions.
And when you take account of the fact that string theory is all about geometry, somehow this is geometry where noncommutative objects are built in. In fact I've mentioned now a couple portions of it. There's the noncommutativity of joining strings, and there's the matrices that don't commute, which are related to K theory and also to the D-brane positions and so on.
Anyway, coming back here, you can try to systematically describe open string physics at least in terms of noncommutative ideas introduced in geometry,and you can get a general answer of some kind, but it's rather abstract and very hard to use. However, in the last couple of years, it was discovered that there's a certain limit with a very strong background magnetic field in which things simplify, and you can actually say something simple and useful based on the noncommutative geometry. That's a case where the rather abstract and hard to use noncommutative geometrical concepts actually come down to Earth and become useful.

Why is it so hard to break supersymmetry in string theory?
Well, if I knew the answer, if I knew how Nature has done supersymmetry breaking, then I could tell you why humans had such trouble figuring it out. But I can say one thing about it. When supersymmetry is not broken, it's easy to get a zero cosmological constant in string theory. And although a zero cosmological constant might not be the truth, it's incredibly close to the truth. If you break supersymmetry, if you do it the wrong way, you're going to get a cosmological constant that's much too big, and then you may well get associated problems, such as instabilities, runaways and so on. So it's easy to find ways that string theory could break supersymmetry, but they all have bad consequences. So I assume we're missing something, which is the answer to your question.

How can the cosmological constant be so close to zero but not zero?
I really don't know. It's very perplexing that astronomical observations seem to show that there is a cosmological constant. It's definitely the most troublesome, for my interests, definitely the most troublesome, observation in physics in my lifetime. In my career that is.

What has been the most surprising or interesting thing that you have learned in physics?
I'm going to interpret the question to be what's the most interesting thing I've learned in my career, whether I discovered it or not. It's something I've learned, perhaps through the work of other people or from textbooks. So in that sense, the most surprising thing I've learned, even though I had nothing to do with discovering it, is that strings can describe quantum gravity.

What has been the most surprising or interesting thing that you have learned in science outside of physics?
Well it's not that amazing that to me, a lot of science is physics. So, for example, I can't give you an answer in terms of chemistry, because physics underlies chemistry. I could give you an answer in biology. Biologists have learned lots of wonderful things. But it's hard to properly maintain one's sense of wonder about them, for some things that were known so long that we all remember so little that we take them for granted. But there's the theory of evolution, which is an amazing insight. And there's the understanding of the genetic code, that's a marvelous insight.
Of course, if we move on to math, which you might think isn't physics, but which is much closer to what I know, then there are lot's of fun and exciting things there. I hardly know what to tell you because, again, there are lots of things that are really wonderful but which we take for granted because it's all known. Like there's calculus. Calculus is pretty amazing.
But... it's not the first thing that comes to mind in answering such a question, because such a question tends to make you think of more recent discoveries. But... if I just have to ask , of everything I've ever learned in math, what's the most amazing and surprising -- it might by that calculus should win the prize, even though it's not so new any more.