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Russian contributions to the Dirac operator


Dirac operator
Michael Atiyah Mathematician
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Because of Hirzebruch's work coming from algebraic geometry and topology, there were all these integrality theorems which he proved.  Various combinations of characteristic numbers are divisible by various quantities because in algebraic geometry, this quantity which you want to put as integer has the interpretation as the dimension of a space of holomorphic functions. And then Hirzebruch was so skilful with all these manipulations, he was able to transfer these formulas from algebraic geometry to other contexts, where you've got things which were integers but there was nothing, nothing for them to describe. There was no... no space of which these things were the dimension. Well that was a slight irritation. You'd like to be able to find a substitute for what we did in algebraic geometry in things which weren't algebraic, that weren't complex.

But even more irritating was the one to do with the… the A-hat genus, A-roof genus, which is… you add things which are integers and sometimes divisible by powers of two, and there, again there was no… there were even an… not only did you not know how to interpret the integer, but sometimes you couldn't even prove that it was an integer. It was unsolved problems. So these problems had been around for a while with Hirzebruch's work, and he was interested in them, trying to prove them and they would turn up at the Arbeitstagung, they would play a role. So it was the kind of thing you focused on, and when we got interested in K-theory one of the initial consequences of the K-theory was to get some progress on these integrality theorems.

And then because of Grothendieck's work in algebraic geometry, the K-theory of formalism, these linked up with the algebraic geometry again, so all of this was pointing towards some… hoped to interpret things like A-roof genus as some dimension of some space somewhere. And that's about… that’s what I... I was looking for that, I was trying to find that. And well, when Singer came we sort of… that's where we sort of rediscovered for ourselves the Dirac operator. I'm not sure quite how far I'd got before he came, but obviously I had the problem trying to interpret.

I knew the formula, Hirzebruch's work, I knew what the answer was; what I had to guess was the problem. We had to find out what was this object. We knew from algebraic geometry, what it should look like in algebraic cases. We knew that it had to do with spinors because of Hirzebruch's formula. So the question was, there should be some differential equation which would play the role of the Cauchy-Riemann equations in the spinor case which ought to fit the left-hand side of the equation. And so, when Singer came, he was better than me at differential geometry, covariant derivatives and… and I suppose he knew a bit more, perhaps, of the theoretical physics. So we stumbled on rediscovering the Dirac operator and we were able to formulate the index theorem.

Simultaneous to that was the… how we started, and then at about the same time Stephen Smale came passing through Oxford and he told us he'd been talking to the Russians and Gelfond and people, and the Russian school had formulated the general question of index problem for elliptic operators, in general abstract terms, and whether there was a formula you could calculate for it, and so on. We realised that what we were doing was a special case. So somehow we had solved, we had started from the opposite extreme from the Russians. We had... we had the answer; we didn't know what the problem was. We invented the problem, and then we realised this was a special problem of a more general problem, the index problem which the Russians were concerned about. But they had no idea what the answer was. So they had a problem, you know, here there is this index which ought to have a topological formula because it's topologically invariant, but they had no machinery.  They didn't have any sort of… we had the example of coming from Hirzebruch of a formula… which had the formula all built in, and a guess from algebraic geometry what it should be. All we had to do was to plug in the little bit coming from the theoretical physics, the Dirac equation, and that made the click. And then having got the formula, then of course we had to work for quite a long time to get the proofs, but that was a different story. But that's how it started. I was working on it, coming from the Hirzebruch end. Singer came along at the right moment and was able to produce the Dirac operator, rediscover it, although I'd been to Dirac's lectures, but that didn't… and actually we didn't think that it really had much to do with Dirac's work because this was a Riemannian case and he was in with the Lorentz case, so we thought it pretty formal.

And then also, the other thing was I had to… Singer brought in, the other bit he brought in was… he first brought in more skill with differential geometry and covariant derivatives… I wasn't that good on that. Secondly he brought in the functional analysis, he knew all about Hilbert spaces and bounded operators – and that was his training – so he was able to set me right on that and Fourier transforms. So he was able to explain to me the generalities that the Russians were doing. I learnt a bit of analysis going off with him, so I learnt the analysis and I suppose he learnt the topology and the algebraic geometry from my side.

So it was an interesting development and just a bit of good fortune I suppose that he happened to be around the right time, and I happened to be talking to Steve Smale at the right time. But it arose out of all this background of things coming from Bonn. It was the answer looking for the problem. And the physics was a, well, a long way down the road yet, but… I suppose, well I don't know how much I knew, but how much even Singer knew, but we got the formulas and obviously we must have known […] the physics, we called it the Dirac operator, but we didn't really know much more than that… the name.

Eminent British mathematician Sir Michael Atiyah (1929-2019) broke new ground in geometry and topology with his proof of the Atiyah-Singer Index Theorem in the 1960s. This proof led to new branches of mathematics being developed, including those needed to understand emerging theories like supergravity and string theory.

Listeners: Nigel Hitchin

Professor Nigel Hitchin, FRS, is the Rouse Ball Professor of Mathematics and Fellow of Gonville and Caius College, Cambridge, since 1994, and was appointed to the Savilian Professorship of Geometry in October 1997. He was made a Fellow of the Royal Society in 1991 and from 1994 until 1996 was President of the London Mathematical Society.

His research interests are in differential and algebraic geometry and its relationship with the equations of mathematical physics. He is particularly known for his work on instantons, magnetic monopoles, and integrable systems. In addition to numerous articles in academic journals, he has published "Monopoles, Minimal Surfaces and Algebraic Curves" (Presses de l'Universite de Montreal, 1987) and "The Geometry and Dynamics of Magnetic Monopoles" (Princeton University Press, 1988, with Michael Atiyah).

Tags: Bonn, Friedrich Hirzebruch, Stephen Smale, Isadore Singer, Paul Dirac

Duration: 5 minutes, 40 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008