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Celebratio Mathematica

Michael F. Atiyah

Papers on gauge theories

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.

Works

[1]M. F. Atiyah and R. S. Ward: “In­stan­tons and al­geb­ra­ic geo­metry,” Comm. Math. Phys. 55 : 2 (1977), pp. 117–​124. MR 0494098 Zbl 0362.​14004 article

[2]M. F. Atiyah, N. J. Hitchin, and I. M. Sing­er: “De­form­a­tions of in­stan­tons,” Proc. Nat. Acad. Sci. U.S.A. 74 : 7 (1977), pp. 2662–​2663. MR 0458424 Zbl 0356.​58011 article

[3]M. F. Atiyah: “Geo­metry of Yang–Mills fields,” pp. 216–​221 in Math­em­at­ic­al prob­lems in the­or­et­ic­al phys­ics (Rome, 6–15 June 1977). Edi­ted by G. Dell’Ant­o­nio. Lec­ture Notes in Phys­ics 80. Spring­er (Ber­lin), 1978. See also Geo­metry of Yang–Mills fields (1979). MR 518436 incollection

[4]M. F. Atiyah, N. J. Hitchin, and I. M. Sing­er: “Self-du­al­ity in four-di­men­sion­al Rieman­ni­an geo­metry,” Proc. Roy. Soc. Lon­don Ser. A 362 : 1711 (1978), pp. 425–​461. MR 506229 Zbl 0389.​53011 article

[5]M. F. Atiyah, N. J. Hitchin, V. G. Drin­fel’d, and Yu. I. Man­in: “Con­struc­tion of in­stan­tons,” Phys. Lett. A 65 : 3 (1978), pp. 185–​187. MR 598562 article

[6]M. F. Atiyah and J. D. S. Jones: “To­po­lo­gic­al as­pects of Yang–Mills the­ory,” Comm. Math. Phys. 61 : 2 (1978), pp. 97–​118. MR 503187 Zbl 0387.​55009 article

[7]M. F. Atiyah: Geo­metry of Yang–Mills fields. Lezioni Fer­mi­ane. Ac­ca­demia Nazionale dei Lincei & Scuola Nor­male Su­peri­ore Pisa (Pisa), 1979. Re­prin­ted in Atiyah’s Col­lec­ted works, vol. 5 (1988). See also art­icle in Math­em­at­ic­al prob­lems in the­or­et­ic­al phys­ics (1978). MR 554924 Zbl 0435.​58001 book

[8]M. F. Atiyah: “Real and com­plex geo­metry in four di­men­sions,” pp. 1–​10 in The Chern Sym­posi­um 1979 (Berke­ley, CA, June 1979). Edi­ted by W. Y. Hsiang. Spring­er (New York), 1980. MR 609554 Zbl 0454.​53043 incollection

[9]M. F. Atiyah and R. Bott: “Yang–Mills and bundles over al­geb­ra­ic curves,” pp. 11–​20 in Geo­metry and ana­lys­is: Pa­pers ded­ic­ated to the memory of V. K. Pat­odi. In­di­an Academy of Sci­ences (Ban­galore), 1980. Re­pub­lished in Proc. In­di­an Acad. Sci. Math. Sci. 90:1 (1981). MR 592249 Zbl 0482.​14007 incollection

[10]M. F. Atiyah: “Geo­met­ric­al as­pects of gauge the­or­ies,” pp. 881–​885 in Pro­ceed­ings of the In­ter­na­tion­al Con­gress of Math­em­aticians (Hel­sinki, 15–23 Au­gust 1978), vol. 2. Edi­ted by O. Le­hto. Aca­demia Sci­en­tiar­um Fen­nica (Hel­sinki), 1980. MR 562703 Zbl 0428.​58018 incollection

[11]M. F. Atiyah: “Re­marks on Morse the­ory,” pp. 1–​5 in Re­cent de­vel­op­ments in gauge the­or­ies. Edi­ted by G. ’t Hooft, C. Itzyk­son, A. Jaffe, H. Lehmann, P. K. Mit­ter, I. M. Sing­er, and R. Stora. NATO Ad­vanced Study In­sti­tutes Series B. Phys­ics. 59. Plen­um Press (New York and Lon­don), 1980. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 5. incollection

[12]M. F. Atiyah: “Green’s func­tions for self-dual four-man­i­folds,” pp. 129–​158 in Math­em­at­ic­al ana­lys­is and ap­plic­a­tions, Part A, published as Adv. in Math. Sup­pl. Stud. 7A. Issue edi­ted by L. Nachbin. Aca­dem­ic Press (New York), 1981. MR 634238 Zbl 0484.​53036 incollection

[13]M. F. Atiyah and R. Bott: “Yang–Mills and bundles over al­geb­ra­ic curves,” pp. 11–​20 in Geo­metry and ana­lys­is: Pa­pers ded­ic­ated to the memory of V. K. Pat­odi, published as Proc. In­di­an Acad. Sci., Math. Sci. 90 : 1. In­di­an Academy of Sci­ences (Ban­galore), 1981. MR 653942 Zbl 0499.​14005 incollection

[14]M. F. Atiyah: “Con­vex­ity and com­mut­ing Hamilto­ni­ans,” Bull. Lon­don Math. Soc. 14 : 1 (1982), pp. 1–​15. MR 642416 Zbl 0482.​58013 article

[15]M. F. Atiyah: “Gauge the­ory and al­geb­ra­ic geo­metry,” pp. 1–​20 in Dif­fer­en­tial geo­metry and dif­fer­en­tial equa­tions (Beijing, 1980), vol. 1. Edi­ted by S.-S. Chern and W.-t. Wu. Sci­ence Press, Gor­don and Breach (Beijing, New York), 1982. Zbl 0544.​14012 incollection

[16]M. F. Atiyah: “Solu­tions of clas­sic­al equa­tions,” pp. 207–​219 in Gauge the­or­ies: Fun­da­ment­al in­ter­ac­tions and rig­or­ous res­ults (Poi­ana Braşov, Ro­mania, sum­mer 1981). Edi­ted by P. Dita, V. Georges­cu, and R. Purice. Pro­gress in Phys­ics 5. Birkhä user (Bo­ston), 1982. Zbl 0539.​70028 incollection

[17]M. F. Atiyah: “Geo­metry of mono­poles,” pp. 3–​20 in Mono­poles in quantum field the­ory (Trieste, Decem­ber 1981). Edi­ted by N. S. Craigie, P. God­dard, and W. Nahm. World Sci­entif­ic (Singa­pore), 1982. MR 766750 incollection

[18]M. F. Atiyah and A. N. Press­ley: “Con­vex­ity and loop groups,” pp. 33–​63 in Arith­met­ic and geo­metry: Pa­pers ded­ic­ated to I. R. Sha­far­ev­ich, vol. 2. Edi­ted by M. Artin and I. R. Ša­far­e­v­ič. Pro­gress in Math­em­at­ics 36. Birkhäuser (Bo­ston), 1983. MR 717605 Zbl 0529.​22013 incollection

[19]M. F. Atiyah and R. Bott: “The Yang–Mills equa­tions over Riemann sur­faces,” Philos. Trans. R. Soc. Lond., A 308 : 1505 (1983), pp. 523–​615. MR 702806 Zbl 0509.​14014 article

[20]M. F. Atiyah: “An­gu­lar mo­mentum, con­vex poly­hedra and al­geb­ra­ic geo­metry,” Proc. Ed­in­burgh Math. Soc. (2) 26 : 2 (1983), pp. 121–​133. MR 705256 Zbl 0521.​58026 article

[21]M. Atiyah: “An­om­alies and in­dex the­ory,” pp. 313–​322 in Su­per­sym­metry and su­per­grav­ity/non­per­turb­at­ive QCD (Ma­habalesh­war, In­dia, 5–19 Janu­ary 1984). Edi­ted by P. Roy and V. Singh. Lec­ture Notes in Math­em­at­ics 208. Spring­er (Ber­lin), 1984. MR 774599 incollection

[22]M. F. Atiyah: “The mo­ment map in sym­plect­ic geo­metry,” pp. 43–​51 in Glob­al Rieman­ni­an geo­metry (Durham Uni­versity, UK, Ju­ly 1983). Edi­ted by T. Will­more and N. J. Hitchin. Math­em­at­ics and its Ap­plic­a­tions. El­lis Hor­wood (Chichester, UK), 1984. MR 757204 Zbl 0615.​53023 incollection

[23]M. F. Atiyah and R. Bott: “The mo­ment map and equivari­ant co­homo­logy,” To­po­logy 23 : 1 (1984), pp. 1–​28. MR 721448 Zbl 0521.​58025 article

[24]M. F. Atiyah and I. M. Sing­er: “Dir­ac op­er­at­ors coupled to vec­tor po­ten­tials,” Proc. Nat. Acad. Sci. U.S.A. 81 : 8 (1984), pp. 2597–​2600. MR 742394 Zbl 0547.​58033 article

[25]M. F. Atiyah: “In­stan­tons in two and four di­men­sions,” Comm. Math. Phys. 93 : 4 (1984), pp. 437–​451. MR 763752 Zbl 0564.​58040 article

[26]M. F. Atiyah: “The Yang–Mills equa­tions and the struc­ture of 4-man­i­folds,” pp. 11–​17 in Glob­al Rieman­ni­an geo­metry (Durham Uni­versity, UK, Ju­ly 1983). Edi­ted by T. Will­more and N. J. Hitchin. Series in Math­em­at­ics and its Ap­plic­a­tions. El­lis Hor­wood (Chichester, UK), 1984. MR 757200 Zbl 0614.​57007 incollection

[27]M. F. Atiyah and N. J. Hitchin: “Low en­ergy scat­ter­ing of nona­beli­an mono­poles,” Phys. Lett. A 107 : 1 (1985), pp. 21–​25. MR 778313 Zbl 1177.​53069 article

[28]M. Atiyah: “Ei­gen­val­ues of the Dir­ac op­er­at­or,” pp. 251–​260 in Arbeit­sta­gung Bonn 1984 (Max-Planck-In­sti­tut für Math­em­atik, Bonn, 15–22 June 1984). Edi­ted by F. Hirzebruch, J. Schwer­mer, and S. Suter. Lec­ture Notes in Math­em­at­ics 1111. Spring­er (Ber­lin), 1985. MR 797424 Zbl 0568.​53022 incollection

[29]M. F. Atiyah: “Cir­cu­lar sym­metry and sta­tion­ary-phase ap­prox­im­a­tion,” pp. 43–​59 in Col­loque en l’hon­neur de Laurent Schwartz (École Poly­tech­nique, Pal­aiseau, 30 May–3 June 1983). As­térisque 131. So­ciété math­ématique de France (Par­is), 1985. MR 816738 Zbl 0578.​58039 incollection

[30]M. Atiyah: “To­po­lo­gic­al as­pects of an­om­alies,” pp. 22–​32 in Sym­posi­um on an­om­alies, geo­metry, to­po­logy (Uni­versity of Chica­go, 28–30 March 1985). Edi­ted by W. A. Bardeen and A. R. White. World Sci­entif­ic (Singa­pore), 1985. MR 850843 Zbl 0651.​58035 incollection

[31]M. Atiyah: “Com­ment­ary on the art­icle of Man­in,” pp. 103–​109 in Arbeit­sta­gung Bonn 1984 (Max-Planck-In­sti­tut für Math­em­atik, Bonn, 15–22 June 1984). Edi­ted by F. Hirzebruch, J. Schwer­mer, and S. Suter. Lec­ture Notes in Math­em­at­ics 1111. Spring­er (Ber­lin), 1985. The art­icle is Yu. I. Man­in, “New di­men­sions in geo­metry,” from the same volume. MR 797417 Zbl 0595.​53071 incollection

[32]M. Atiyah, N. J. Hitchin, J. T. Stu­art, and M. Tabor: “Low-en­ergy scat­ter­ing of nona­beli­an mag­net­ic mono­poles [and dis­cus­sion],” Philos. Trans. Roy. Soc. Lon­don Ser. A 315 : 1533 (1985), pp. 459–​469. MR 836746 article

[33]M. F. Atiyah: “Mag­net­ic mono­poles in hy­per­bol­ic spaces,” pp. 1–​33 in Vec­tor bundles on al­geb­ra­ic vari­et­ies (Bom­bay, 9–16 Janu­ary 1984). Stud­ies in Math­em­at­ics 11. Tata In­sti­tute for Fun­da­ment­al Re­search (Bom­bay), 1987. MR 893593 Zbl 0722.​53063 incollection

[34]M. Atiyah: Col­lec­ted works, vol. 5: Gauge the­or­ies. Ox­ford Sci­ence Pub­lic­a­tions. The Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford, New York), 1988. MR 951896 Zbl 0691.​53003 book