Commentary by M. Atiyah
From 1977 onwards my interests moved in the direction of gauge theories and the interaction between geometry and physics. I had for many years had a mild interest in theoretical physics, stimulated on many occasions by lengthy discussions with George Mackey. However, the stimulus in 1977 came from the two other sources. On the one hand Singer told me about the Yang–Mills equations, which through the influence of Yang were just beginning to percolate into mathematical circles. During his stay in Oxford in early 1977, Singer, Hitchin and I took a serious look at the self-duality equations. We found that a simple application of the index theorem gave the formula for the number of instanton parameters, and our result appeared in the brief note [2]. At about the same time A. S. Schwarz in the Soviet Union had independently made the same discovery. From this period the index theorem began to become increasingly familiar to theoretical physicists, with far-reaching consequences.
The second stimulus from theoretical physics came from the presence in Oxford of Roger Penrose and his group. Roger and I had been research students together in Cambridge, at the time when he was an algebraic geometer. We lost touch after that but renewed contact on his arrival in Oxford as Coulson’s successor. In fact, I encouraged him to come to Oxford, at a time when there were other possibilities. I also remember discussing Penrose’s work with Freeman Dyson in Princeton. After describing the work on black holes which he could understand, Dyson went on to talk about twistor theory which he found mysterious. He ended by saying to me “perhaps you will understand it”. At that time I knew nothing about twistors, but on Penrose’s arrival in Oxford we had lengthy discussions which were mutually educational. The geometry of twistors was of course easy for me to understand, since it was the old Klein correspondence for lines in \( P_3 \) on which I had been brought up. The physical motivation and interpretation, I had to learn.
In those days Penrose made great use of complex multiple integrals and residues, but he was searching for something more natural and I realized that sheaf cohomology groups provided the answer. He was quickly converted and this made subsequent dialogue that much easier, since we now had a common framework.
Richard Ward was at this time a student of Penrose and he gave a seminar explaining how twistors could be used to reinterpret the self-dual Yang–Mills equations. This seminar made a big impact on me. I realized that this was something significant, and spent several days thinking hard about it. Penrose and Ward were working of course with Minkowski space (or its complexification), but I was already interested (through my contacts with Singer) in the Euclidean case. Fortunately some time earlier Penrose had discussed the Euclidean version of twistor space with me, and I had discovered then its quaternionic interpretation. With this understood it was not long before I saw how to put Ward’s ideas to global use for the problem of instantons. At that state only the original 1-instanton had been explicitly constructed, and I saw how to generalize this to give \( k \)-instantons by using disjoint lines in twistor space. Almost immediately I was disappointed to receive preprints from physicists who had (by different methods) discovered the same solutions! This was to be my first experience of the different tempo of the world of theoretical physics, in which new ideas spread like wildfire and stimulate the production of preprints on a lavish scale.
Stimulated by this competitive atmosphere, Ward and I pushed on, showing how algebraic geometry gave in principle methods for the construction of all multi-instantons. Our paper [1], brought this whole subject to the attention of a wider-mathematical audience, and thereby helped to increase the general interaction between physicists and geometers.
Although [1] reduced the instanton problem to one in algebraic geometry, namely the construction of suitable vector bundles on \( P_3 \), it did not really yield an explicit solution. However, vector bundles had been much studied by algebraic geometers, and various methods of construction had been developed. In particular Geoffrey Horrocks, another of my contemporaries at Cambridge, had given an interesting construction which I was trying to understand. With the help of Nigel Hitchin I finally saw how Horrocks’ method gave a very satisfactory and explicit solution to the general instruction problem. I remember our final discussion one morning, when we had just seen how to fit together the last pieces of the puzzle. We broke off for lunch feeling very pleased with ourselves. On our return we found a letter from Manin (with whom I had earlier corresponded on this subject) outlining essentially the same solution to the problem and saying “no doubt you have already realized this!” We replied at once and proposed that we should submit a joint note [5] from the four of us (Drinfeld was collaborating with Manin).
These brief publications were in due course expounded into more substantial form. In [4] Hitchin, Singer, and I gave a full account of the Penrose twistor theory in the context of Riemannian geometry and gave detailed proofs of the results announced in [2]. Our aim was in part to bring Penrose’s work to the attention of differential geometers, presenting in it a purely mathematical form without reference to the physical background. In my Fermi Lectures [7] given in Pisa in 1978, I attempted to bridge the gap between algebraic geometry and the physicists’ view of gauge theory. As well as elaborating on my paper with Ward, I also gave a fuller account of the construction of instantons in the four-author note [5]. Around this time I was in fact giving lectures in many parts of the world on the geometry of gauge theories. I think it is fair to say that the papers [1] and [5] in particular had caused quite a flurry of interest on the mathematics/physics interface. It is reported that Polyakov had described [5] as the first time abstract modern mathematics had been of any use! The two papers [7], [8] are examples of expository lectures given during this period (1978–80).
Besides having applications to Yang–Mills instantons the Penrose twistor theory also applied to their gravitational counterparts, self-dual Einstein manifolds. This was discovered by Penrose and represents perhaps the deepest illustration of his ideas. After Hawking and Gibbons had constructed their gravitational instantons, Hitchin showed how to rederive their results using twistor methods. This was a very elegant approach and surprisingly was closely related to Brieskorn’s work on rational double points. In [12] I show how twistor methods could be pushed one stage further to derive the Green’s function for such manifolds. I found the translation of delta functions into homological algebra particularly appealing, and similar ideas have come to the fore more recently in the fundamental work of Donaldson.
My paper [6] with John Jones had an unusual origin. Although we had by this time complete information on instantons, i.e., solutions of the self-dual Yang–Mills equations, it was unknown (and still is in 1986) whether (for \( SU(2) \)), there were any solutions of the second-order Yang–Mills equations which were not self-dual (or anti-self-dual). I then heard that this question had been settled by a physicist, but the argument depended on an assertion which I did not believe. Paper [6] developed out of my attempt to clarify this question. It was also related to the famous Gribov ambiguity which Singer and others had analyzed topologically. The main result in [6] was a proof that the homology of the moduli spaces of \( k \)-instantons, as \( k\to\infty \), “contained” all the homology of the relevant function space.
I was at this time very intrigued by variational problems where the Morse theory failed, but where nevertheless the minima of the functional carried much of the homology. Graeme Segal had, in answer to a question of mine, established a remarkable theorem of this type for rational functions, as a minima of the energy functional for maps \( \mathbb{S}^3\to\mathbb{S}^2 \), and I was able to use this in my paper with Jones. I had many discussions with Segal about the role of “particles” in topology and physics and it was interesting to see the way instantons (or pseudo-particles as they were once called) entered into the topological picture. My general ideas, and speculations, on these Morse-theory questions were described in a conference report [11]. In subsequent years I also explained my ideas to Cliff Taubes who eventually produced the appropriate analytical refinement of Morse theory to explain the phenomena which had puzzled me. This refined Morse theory applies in some “limiting exponent” cases and is very subtle. It is (in 1986) just beginning to be understood and developed.
Another example of Morse theory, but this time of a more conventional kind, was the subjet of my long paper [19] with Bott, summarized in [13]. This arose in the following way. Bott was visiting Oxford for a while, having just returned from the Tata Institute in Bombay where he had been studying moduli spaces of vector bundles over Riemann surfaces with Ramanan. Meanwhile, I had been engrossed with the Yang–Mills equations in dimension 4. I realized that these questions were essentially trivial in dimension 2 but, one day, walking across the University Parks with Bott it occurred to me that one might nevertheless be able to use the Yang–Mills equations to study the moduli spaces. The essential point was the theorem of Narasimhan and Sesgadru stating that stable bundles arose from unitary representations of the fundamental group. Bott and I soon became convinced that this method would work but there were many technical problems. A key idea, due to Bott, was that we should use equivariant cohomology in the Morse theory. This turned out to be very productive and later, in the hands of my student, Frances Kirwan, it was put to extensive use.
A formal point which Bott and I noticed in our work was that the curvature could be viewed as a moment map for a symplectic group action (all in infinite dimensions). This was very significant in view of the role of moment maps in Mumford’s geometric invariant theory (this role had just been observed by Mumford and Sternberg). After our successful treatment of the moduli-space problem by these methods, Bott and I wondered whether they might also apply in some finite-dimensional situations. This was the problem which Frances Kirwan disposed of with such finality in her thesis.
In the course of writing [19] I had encountered some convexity questions, and in attempting to understand their significance I was led to the formulation described in [14]. Here I put some old results of Schur, Horn, Kostant, and others into a more general symplectic geometry context. When I visited Harvard I lectured on this material in Bott’s seminar and was mystified by the looks of amusement on several faces in the audience. It transpired that Guillemin and Sternberg had almost simultaneously found the same result!
After [14] appeared I received a letter from Arnold explaining how my result could be applied to simplify a result of Koushnirenko, expressing the number of zeros of a set of polynomials in terms of the volume of the associated “Newton polyhedron”. He raised the question of deriving Bernstein’s generalization, a “polarized form” of this result involving the Minkowski mixed volumes. I found these results quite beautiful and fascinating. They were also linked to Teissier’s proof of the Alexandroff–Fenchel inequalities based on the Hodge signature theorem. After I had understood how to prove Bernstein’s result it seemed to me that this material deserved some publicity, and that it was suitable for the lecture I was asked to give at the centenary meeting of the Edinburgh Mathematical Society [20].
In my paper with Bott [19] we had already noted the appearance of the moment map in an infinite-dimentional setting. In my joint paper with Pressley [18] we extended the convexity results of [14] to the loop space of a compact group. By this time these loop groups (and their associated Lie algebras) had become intensively studied by mathematicians and physicists. I had been familiar with them for some time in view of their appearance in Bott’s proof of the periodicity theorem, but my more detailed understanding of their geometry was the result of extensive discussions with Graeme Segal. He and his former student Pressley were just in the process of writing up their book on the subject (Clarendon Press, 1987).
During an Arbeitstagung in the early eighties I had discussed with Hans Duistermaat the problem of explaining why stationary-phase approximation sometimes gave exact results. Shortly afterwards Duistermaat and Heckman found an elegant result on these lines for Hamiltonians arising from circle actions on symplectic manifolds. When Bott was next in Oxford we tried together to understand one of Witten’s paper where he had introduced the operator \( d + i_X \) for a Killing field \( X \). Putting these two together we saw that new insight could be obtained in one worked with a de Rham version of equivariant cohomology. In fact this involved ideas with which we had both been familiar, and which had been widely used. The new developments however sharpened our understanding and so we wrote [22], essentially as an expository paper tieing together various points of view.
A very quick survey of the role of the moment map in various contexts is given in [22].
Papers [15] and [26] are both conference lectures on gauge theories. Paper [15] is a shortened version of my Fermi lectures [7], while [26] describes Donaldson’s now famous application of the Yang–Mills equations to the geometry of 4-manifolds.
In my talk at the Helsinki Congress [10] I had drawn attention to the problem of monopoles, i.e., solutions of the Bogomolny equations. While twistor methods and algebraic geometry had been very successful with instantons, the related problem of monopoles appeared more difficult. In the subsequent years this probelm was attacked by many people and, due to the work notably of Ward, Hitchin, and Nahm, the general nature of the solutions was well understood. In [17] and [16] I reported on this progress, mainly from the point of view of Hitchin. During the Trieste conference, where [17] was presented, I had several discussions with Nick Manton during which he explained to me his ideas on monopole dynamics. He showed me how the dynamics of slowly moving monopoles would be described by geodesics on the parameter space of static solutions. This struck me as a very attractive idea and I tried with some of my students to find or guess the metric on the 2-monopole space, but without success. Then, a couple of years later, I learnt from Hitchin of the beautiful construction of hyper-Kähler quotients and the likelihood that this would apply to the monopole spaces. The time seemed ripe therefore for another attack on the problem, and so Hitchin and I began our lengthy investigation into the geometry and dynamics of monopoles which is summarized in [27] and [32]. This has since been taken further by Gibbons and Manton who have analysed the quantization.
Around this time my former student Donaldson was taking the lead in the study of instantons and monopoles. He made a beautifully simple observation concerning the instanton construction on \( \mathbb{R}^4 \) and he then went on to deal with the case of monopoles. The two papers [25] and [33] essentially arose out of discussion with him, although a casual breakfast conversation with Howard Garland in Berkeley had started the ball rolling. The main result of [25] relating instantons on \( \mathbb{R}^4 \) to rational curves on the loop space interested me because it opened up a possible door to establishing the conjectures I had made with Jones on the topology of instanton moduli spaces. The idea was that the method Segal had used to study the topology of rational maps might be extended to the case of \( \Omega G \). It now looks as though this programme can in fact be carried out, while Taubes has developed his refined Morse theory, which makes a direct attack also possible. All in all the Morse theory questions raised in [11] have proved very fruitful.
In 1982 I was flattered to be invited to the Solvay conference in Austin, Texas. At that meeting I heard Witten explain his mod-2 anomaly. Discussions with him, and earlier discussions in Oxford with Quillen, opened my eyes to the meaning of anomalies and their relation to the index theorem for families. In the next few years this became a very hot topic amongst physicists, leading to a typically large number of papers. Papers [21] and [30] are conference lectures where I was addressing physicists and attempting to explain the relevant mathematics, while [24] is a short note with Singer summarizing our point of view. This was intended to be written up at greater leisure, but physics moves at a different pace from mathematics. The lesurely account has now been overtaken by events and is unlikely to see the light of day.
The 1984 Arbeitstagung in Bonn was a special 25th anniversary occasion, so that for the first time the proceedings were published. My talk [28] was a presentation of beautiful results of Witten and Vafa. These results had impressed me because they involved topological methods to prove inequalities for eigenvalues. Although Witten was now very well known to mathematicians, his influential papers had been those published in journals of mathematics or mathematical physics. Papers published in regular physics journals were unlikely to be read by the mathematical community and so I felt some publicity for his ideas would be a public service. My commentary [31] on Manin’s manuscript is a rare case where I put down on paper the kind of wild speculation which I usually only indulge in verbally. This is a hostage to fortune, but it may perhaps serve as a useful purpose by showing that we mathematicians are not the rigorous formalists our published papers might suggest, and that we do allow our imagination a free reign.
At the Solvay Conference in Austin, on the boat trip, Witten had explained to Singer and me his beautiful ideas on the Duistermaat–Heckman formula applied to the loop space. He showed us how this led heuristically to the index theorem for the Dirac operator! Although this was not rigorous mathematics I felt it was suitable for a lecture at the Schwartz Colloquium in Paris [29]. As it happened my words fell on fertile ground, because Bismut was in the audience and he immediately turned his attention to providing rigorous proofs of Witten’s ideas. Several other versions of Witten’s ideas have been developed and this whole area is still in a state of great activity. The interaction between physics and mathematics in this field is quite remarkable, and I am really struck by the way most of the work which Singer and I did in the 60s and 70s has become relevant to physics.
