David Olive: his life and work
DDavid Olive his life and work
Edward Corrigan
Department of Mathematics, University of York, YO10 5DD, UK
Peter Goddard
Institute for Advanced Study, Princeton, NJ 08540, USASt John’s College, Cambridge, CB2 1TP, UK
Abstract
David Olive, who died in Barton, Cambridgeshire, on 7 November 2012, aged 75, wasa theoretical physicist who made seminal contributions to the development of stringtheory and to our understanding of the structure of quantum field theory. In early workon S -matrix theory, he helped to provide the conceptual framework within which stringtheory was initially formulated. His work, with Gliozzi and Scherk, on supersymmetryin string theory made possible the whole idea of superstrings, now understood as thenatural framework for string theory. Olive’s pioneering insights about the dualitybetween electric and magnetic objects in gauge theories were way ahead of their time;it took two decades before his bold and courageous duality conjectures began to beunderstood. Although somewhat quiet and reserved, he took delight in the companyof others, generously sharing his emerging understanding of new ideas with studentsand colleagues. He was widely influential, not only through the depth and vision of hisoriginal work, but also because the clarity, simplicity and elegance of his expositionsof new and difficult ideas and theories provided routes into emerging areas of research,both for students and for the theoretical physics community more generally.[A version of section I Biography is to be publishedin the
Biographical Memoirs of Fellows of the Royal Society .] a r X i v : . [ phy s i c s . h i s t - ph ] S e p Biography
Childhood
David Olive was born on 16 April, 1937, somewhat prematurely, in a nursing home in Staines,near the family home in Scotts Avenue, Sunbury-on-Thames, Surrey. He was the only child ofLilian Emma (n´ee Chambers, 1907-1992) and Ernest Edward Olive (1904-1944). Ernest workedas a clerk in the Bank of Belgium in the City of London. David believed the Olive family to be ofHuguenot origin, although he was unable to establish this definitely. However the male line of thefamily could be traced back to the eighteenth century in the London area through his grandfatherThomas Henry Olive (1866-1948), an interior decorator, his great grandfather, William HenryOlive, a publican in Richmond, and William’s father, James John Olive (1797-1869), a brazierand gas fitter.His maternal grandfather, Frederick McCall Chambers, was a music hall artist, who apparentlydisappeared mysteriously from the family scene. David had two cousins on his mother’s side andeight on his father’s but only one of them received a university education and there seems to beno scientific, or particularly intellectual, background in his family.By 1940, David’s family had moved to Walton-on-Thames, where he began his schooling at a localkindergarten. In 1941, to contribute to the war effort, Ernest resigned his post with the Bankof Belgium and volunteered for service in the Royal Air Force, which left the family in need ofincome. They moved to a bungalow in Govett Avenue, Shepperton, and Lilian took in a lodgerand acquired a part-time job. David briefly attended Shepperton Grammar School before hismother decided in early 1943 that they should move to Edinburgh, partly to escape the bombingraids and partly to be near her mother. There, they lived in rented accommodation in Joppa, asuburb of Edinburgh, eventually taking rooms in the house of Mrs Susan Gage, the widow of theformer steward of the Muirfield Golf Club. David struck up a friendship with Leslie, the youngerof Mrs Gage’s two sons, even though he was five years older than David. He joined Leslie at theRoyal High Preparatory School and their friendship lasted a lifetime.Late in 1943, Mrs Gage invited the Olives to move with her temporarily into the Muirfield ClubHouse, where she had been invited to stand in for the steward. Ernest joined them there on leaveover Christmas. This happy occasion was the last time that David saw his father, because, on25 February 1944, Ernest died when the Lancaster bomber, on which he was serving as a flightengineer, was shot down over Germany. Lilian, understandably distraught, could not bring herselfto give the news that Ernest was missing in action to the six-year-old David and eventually sheleft a letter announcing his father’s death for David to read. Other disasters followed: the familyfurniture stored in London was destroyed in a fire, a loss not covered by insurance; a deposit thathad been paid on a house being built was forfeited; and, in a particularly cruel stroke, no widow’spension was forthcoming from the Bank of Belgium because Ernest had resigned his position thereto volunteer for the RAF, rather than waiting to be conscripted.1oon after the loss of her husband, Lilian decided to return to England, to Birchington in Kent,where David attended Woodford House School, but the move did not last. To get away from theV-1 and V-2 rocket bombs, and also because they found the education at Woodford House inferiorto that provided by the Royal High School, David and his mother went back to Scotland in thespring of 1945. Lilian bought a bungalow in Newhailes Avenue, Musselburgh, near Edinburgh,next to friends, Jimmie and Mabel Weatherhead, parents of one of David’s classmates. Jimmie, alocal bank manager, was in a position to help, at least with financial advice.The Royal High School had the further advantage that no fees would be charged provided thatDavid performed sufficiently well academically. This was a condition he had no difficulty inmeeting, ending up top of his class every year from 1946 onwards. He left the junior part ofthe Royal High School in 1949 and, subsequently, in 1955, the senior part, as
Dux (the leadingacademic student). He remembered two of the women who taught him as really inspiring: HilarySpurgeon, who gave David his first science courses, and Letitia Whiteside, who provided extrascience classes after hours. But the others were, in his view, lacklustre: the man who taughtphysics just read from a text book, while the head of science was ‘awful’ (although, it seems headded some colour by standing on a stool at the end of the last lesson of the day and proclaiming,after Horace,
Odi profanum vulgus ).For all this, it was not at school that David found the direction of his future scientific career.
David Olive aged about 11.Photograph kindly provided by the Olive family.
In David’s view, more important than anything he learnt atschool was the time he spent with his Meccano set, buildingits little metal ‘girders’, together with nuts, bolts, wheelsand gears into relatively complicated mechanical devices inthe form of cranes, lorries, cars, etc., sometimes powered byelectric motors. The more complicated, the longer was thetime that might be taken for the construction, up to monthsin some cases. David felt that this encouraged him later, inhis research career, to conceptualize long-term programmes,aimed at achieving complicated objectives.As a teenager, David developed other interests that wouldstay with him throughout his life. Beginning in 1952, Lilianencouraged David to take an interest in golf. Perhaps theseed had also been sown by that last family Christmas withhis father at Muirfield. He joined the local club as a juniormember, had lessons with the club professional, and becamesecretary of its junior section. Inspite of his love of the game,in his own estimation, David was never very distinguishedas a golfer, but a number of his contemporaries at the club became quite well known.An even more consuming interest, music, which was to become a life-long passion, had its begin-nings in these years. This was encouraged by his friend, Leslie Gage, who had a large radiogram.2tarting with Beethoven, David soon progressed to Mozart, Berlioz, Delius and many others,both the widely familiar and others rarely heard. He became a great admirer of the conductor SirThomas Beecham and, at one Edinburgh Festival, he and Leslie Gage managed to slip into theUsher Hall to sit in on some of the great man’s rehearsals.David left the Royal High School laden with academic honours, including bursaries and schol-arships for Edinburgh University. In his farewell speech as Dux of the school, foreshadowinghis future intellectual path, he chose to talk about theoretical physics, a subject which alreadyfascinated him.
Undergraduate Years
After matriculating at Edinburgh University in September 1955, David continued to live at home.He began by taking the courses in Mathematics, Mathematical Physics and Natural Philosophy(as physics was then known there). The Mathematical Physics lectures took place in the TaitInstitute, newly opened in 1955, which had been established for Nicholas Kemmer. Appointed asTait Professor in 1953, initially within the Natural Philosophy (physics) department, Kemmer hadfound that Norman Feather, the holder of the more ancient chair of Natural Philosophy, thoughtthat there was no such thing as theoretical physics, because, in his view, physics was in essenceexperimental. So, Kemmer had secured an independent institute for theoretical physics. Againstthis background, when David had to choose two of the three courses he was studying to specializein for his degree, he decided to drop physics, rather than mathematics or mathematical physics,even though he continued to attend physics lectures throughout his second year.David found Kemmer’s lectures on hydrodynamics, given without reference to notes, wonderfully,indeed enviably, clear. He also attended lectures by John Polkinghorne, whose first lecturing post(1956-8) was in the Tait Institute, and who was later to become David’s colleague and collaboratorin Cambridge. He continued to excel academically, as he had at school, only held back at timesby his nearly illegible handwriting, which caused at least one examiner to mark him down.A large part of David’s social life centred on the Edinburgh University Physical Society, whichorganized hiking expeditions in addition to lectures on physics. Through the society and otheractivities, he met many of his future colleagues and friends, such as Keith Moffatt, David Fairlie,Jim Mirrlees, Ian Drummond, Ian Halliday, Tom Kibble, and Alan MacFarlane, all exact or nearcontemporaries. They were just part of a remarkable succession of students in theoretical physicsand mathematics at Edinburgh at that time, and Kemmer’s charismatic influence inspired manyof them to follow research careers in fundamental physics.A lecturer who sometimes joined the physical society hikes was William Edge, whose specialtywas projective geometry. Edge, who was perhaps not completely at home in the twentieth cen-tury, encouraged many of the more talented students, including David, to go to Cambridge aftergraduation from Edinburgh, to follow a path, well-trodden at least since the time of James Clerk3axwell, of reading the Mathematical Tripos for a second undergraduate degree. In December1957, before he finished his Edinburgh degree, which he completed in three years rather than theusual four, obtaining first class honours, David competed successfully for an open scholarship atSt John’s College, Cambridge, on Edge’s advice.After Edinburgh, David spent the summer working at Metropolitan Vickers, an electrical engi-neering company in Manchester, and it was from there he put his trunk on the train to go up toCambridge in October, 1958. As a Scholar of St John’s, he was given rooms in College, at C2Chapel Court, in a part of the College with the relatively modern convenience of a toilet on eachstaircase, although it was still necessary to cross the court in order to get a bath. In his first year,he studied for Part II of the Mathematical Tripos, then taken by students coming to Cambridgedirectly from school in either their second or, more commonly, third year, and he again obtainedfirst class honours. Having experienced the intensity of the Cambridge course, he came to feelthat he had benefited from the somewhat slower pace of the Scottish system, and he said thatthis led him to think of himself as Scottish, in spite of having been teased at school for being a‘sassenach’.At some point during his first term, he attended a meeting of the Cambridge University HereticsSociety, at which, perhaps appropriately enough, the speakers did not show up. David and someother members of the society decided to go out to a pub instead. In this way, David met JennyTutton, then in her second year reading mathematics at Girton College, whom he was to marryfour and a half years later.In his second year at St John’s, in order to complete the requirements for the BA degree, whichCambridge allows graduates of other universities to take in two years rather than three, Davidstudied for Part III of the Mathematical Tripos, taking courses by John Ziman on Solid StatePhysics, Christopher Zeeman on Algebraic Topology, Fred Hoyle on General Relativity and Cos-mology, among others. The courses David found most inspiring included one on Quantum FieldTheory by John Polkinghorne, who had returned to Cambridge from Edinburgh in 1958, and, es-pecially, two courses on Quantum Mechanics by Paul Dirac, whose expositions and contributionsto physics were to have a very profound influence on David’s approach to research in theoreticalphysics. The Part III examinations at the end of the academic year took place at the height ofthe hay fever season and, perhaps because of this, although David passed, he failed to be awardeda distinction, the mark necessary for an assured place to remain in Cambridge to undertake re-search for a PhD. Nevertheless, his special talents had been recognized, the Science ResearchCouncil awarded him a research studentship and, not withstanding his somewhat disappointingexamination performance, he was allowed to stay on.David spent his last summer before starting research in Austria. He had obtained a grant fromthe Austrian Institute to attend a German course at the University of Vienna, which lasted formost of July, but David stayed on until late September. His interest in music had not diminishedat all during his undergraduate years and here he had the leisure to attend many performancesat the Vienna State Opera, including the whole
Ring cycle and
Tristan und Isolde , conducted by4erbert von Karajan,
Der Rosenkavalier , Capriccio and (at the Redoutensaal)
Cosi Fan Tutte ,conducted by Karl Bohm, as well as
Aida , conducted by Lovro Von Matacic, as he recorded inthe very detailed notes that he kept on the concerts and opera performances he attended over theyears.
Beginning Research at Cambridge
David began research in October 1960, working within the Department of Applied Mathematicsand Theoretical Physics (DAMTP) in Cambridge, which had been established just a year earlier.
David Olive as a student in Cambridge.Photograph kindly provided by the Olive family.
At first, David had lodgings in Park Parade, buthis landlady objected to his habit of playing clas-sical music loudly and he moved to Alpha Roadat the end of the term. One term later, he joinedDHJ (Ben) Garling and Johnson (Joe) Cann, fel-low graduate students at St John’s College, in aflat in Newnham Road. Both Ben and Joe foundDavid to be very undomesticated; his motherhad looked after him so well that he had acquiredno practical skills in cooking or housekeeping.The particle theory research students inDAMTP were housed in the Austin Wing of theold Cavendish Laboratory off Free School Lanein the centre of Cambridge. His contemporariesthen included David Bailin and Ian Drummondand, in the year ahead of him, Peter Landshoff.John C Taylor, who had just returned to Cam-bridge from Imperial College, was appointed asDavid’s research supervisor. Weekly seminarswere attended by Paul Dirac, then Lucasian Pro-fessor of Mathematics, as well as the other fac-ulty: Richard Eden, John Polkinghorne and John C Taylor.Taylor initially suggested to David that he try to find a renormalizable theory of the weak inter-actions based on Yang-Mills gauge theory. This was an extremely ambitious objective, eventuallysuccessfully attained through the work of Glashow, Salam and Weinberg, and of ’t Hooft andVeltman (for which they received Nobel Prizes in 1979 and 1999, respectively). Convinced by anearlier paper by Salam and Komar (1960) that it was impossible to construct a renormalizabletheory of massive gauge particles, David’s interest shifted to what was then the local speciality ofstudying the analytic properties of perturbative quantum field theory. His first paper (1), written5n collaboration with JC Taylor, early in his second year as a research student, was a contribu-tion to the understanding of the complicated singularity structures that can occur, in particularacnodes and cusps.From this work, his interest developed towards the then current attempts to formulate an axiomatictheory of the scattering matrix, known as the S -matrix, and this was the first area in which Davidwould make important contributions. The leading proponent internationally of this approach wasGeoffrey Chew of the University of California at Berkeley, who fortuitously spent the academicyear 1962-63 in Cambridge as a Visiting Fellow of Churchill College.On 15 April 1963, David married Jenny Tutton at The Catholic Church of Our Lady, Belper,Derbyshire, with Joe Cann acting as best man. After a honeymoon in Paris, David and Jennyreturned to an upstairs flat in Devonshire Road, near the railway station in Cambridge, convenientfor Jenny’s daily journey to work as a mathematics teacher at the Hertfordshire and Essex CountyHigh School for Girls in Bishop’s Stortford. [Jenny’s two younger sisters, Rodie and Clare, werealso to marry Cambridge theoretical physicists: Rodie married Tony Sudbery, who, after taking hisPhD in Cambridge, spent his career in the University of York, and Clare married Ian Drummond,who was to be David’s faculty colleague in DAMTP.]Just before his PhD examination, conducted by Gordon Screaton and (following the practice thenusual in Cambridge) his supervisor, JC Taylor, David was elected to a Research (i.e. Postdoctoral)Fellowship at Churchill College, along with Ian Drummond, possibly (as David thought) thanksto the backing of Chew. His doctoral dissertation was entitled Unitarity and S -matrix Theory .On September 7th, David and Jenny set sail on the Berlin from Southampton to New York totravel to Pittsburgh, to spend the academic year at Carnegie Tech (now Carnegie Mellon Univer-sity) at the invitation of Dick Cutkosky. It was to be David’s only year away from Cambridge asa postdoctoral fellow. Whilst he was there, David restructured the unpublished paper, Towardsan axiomatization of S -matrix theory , which he had written just before leaving Cambridge, andpublished it as An exploration of S -matrix theory (5), an approach to a self-consistent determi-nation of the singularity structure of the S -matrix. As well as the research life in the physicsdepartment and social contacts with its members, the Olives enjoyed the concerts available inPittsburgh and David took the opportunity to acquire new amplifiers, speakers and turntable forhis audio system. However, when Jenny became pregnant with their first child, they decided toreturn to Cambridge, where David’s fellowship at Churchill College would provide accommodationand a secure salary for a few years.Arriving back in Cambridge in August 1964, David and Jenny moved into a newly built flat atChurchill College. David took part in the social and intellectual life of the College, presidedover by its founding Master, Sir John Cockroft, until his death in 1967. Cockcroft attracted manydistinguished physicists to the College as visitors, including Peter Kapitza, Mark Oliphant, GeorgeGamow and Murray Gell-Mann. 6n December 5th, 1964, Jenny gave birth by Caesarean section to a daughter, Katie, who weighedless than 3 pounds and spent some weeks in an incubator. With the family growing, and Davidappointed to an Assistant Lectureship in the University of Cambridge from October 1965, heand Jenny purchased a house with a sizable garden in the village of Barton, about 5 miles fromCambridge, which they continued to own for the rest of their lives and to which they returned inretirement. Earlier, in the first months of 1965, David gave a lecture course, his first, a graduatecourse on S -matrix theory, based on the paper he had written in Pittsburgh (5). These lectures inturn became the basis for David’s contribution to the book, The Analytic S -Matrix (12), writtentogether with his Cambridge colleagues, Richard Eden, Peter Landshoff and John Polkinghorne.Completed against a tight timetable in July 1965, and published by Cambridge University Pressthe following year, the book’s four chapters were each assigned principally to one of the authors,with David contributing the final chapter, S -matrix theory. Exceptionally, for a book on particlephysics, The Analytic S -matrix , known affectionately by the initials of its authors, ELOP, hasremained in active use as the standard reference on the subject. The first three chapters comprisea general introduction, a discussion of the singularity structure of the Feynman graphs describingthe perturbative treatment of quantum field theory, and an account of methods for analyzing thehigh-energy behavior of Feynman graphs, these two topics being ones to which their respectiveauthors had made important contributions, as David had to S -matrix theory. Characteristically,rather than rely on the material of his co-authors, David’s treatment in the fourth chapter beginspractically de novo , and so can be read independently of the rest of the book.Throughout his career, nearly all of David’s research was done in collaboration, often workingwith a particular colleague periodically for over a decade or more, but he always sought to buildup his own understanding of a topic from a set of basic principles or assumptions, which he wouldanalyze and simplify, repeatedly going through the arguments to find the simplest, most elegantand rigorous discussion of the topic.In 1966, David was promoted to a Lectureship in the University, an effectively tenured position,and, on 25 October, Jenny gave birth to their second child, Rosalind. He continued to work on S -matrix theory and extended his interest to encompass the Regge theory of the high energybehavior of scattering amplitudes.In August 1968, he travelled again to Vienna to attend the International Conference on HighEnergy Physics, the fourteenth in a series of conferences, then held every two years, which broughttogether leading experimentalists and theoreticians from around the world. As usual, he took fulladvantage of what was available musically while he was there, including a performance of TheMagic Flute , with Dietrich Fischer-Dieskau, and a Franco Zeffirelli production of
La Boh`eme , withRenate Holm and Cesare Siepi. But, as he noted (131), it was an unexpected talk given in themarbled ballroom of the Hofburg Palace that was, despite the poor acoustics of the conferencelocation, to change the direction of his research and, soon after, of his life.The talk, by Gabriele Veneziano, then not quite 26 years old, introduced his soon-to-be famousformula for a two particle scattering amplitude. Veneziano’s objective was to illustrate how ana-7ytic properties could be combined with Regge asymptotic behavior in an explicit mathematicalfunction (Veneziano 1968), but it quickly stimulated attempts at generalization and eventuallyproved to be the seed that grew into string theory, one of the most influential developments infundamental physics in the twentieth century, and one in which David was to play a leading role.Back in Cambridge, David gave a general talk in DAMTP on Veneziano’s work, and rememberedbeing taken aback by Dennis Sciama’s prescient suggestion that this might be the start of anew theory. He began collaborating with David Campbell and Wojtek Zakrzewski, then researchstudents, on finding extensions of Veneziano’s formula to the scattering of particles with spin(18–20). While this work did not find a permanent place in the development of the subject, itserved to focus David’s interest.
Moving to CERN
Early in 1969, David gave a seminar at CERN, Geneva, at the invitation of Andr´e Martin, and thismotivated David to apply to spend his upcoming term of sabbatical leave there. In September,David and his family began a three-month stay, renting a CERN apartment in Rue du Livron,Meyrin, and taking the opportunity to get to know the Swiss countryside, through visits to Chillon,Gruy`eres, Lauterbrunnen and, particularly, Annecy, nearby in France.He soon met Daniele Amati, then a CERN staff member, who introduced him to Michel Le Bellacand suggested that the three of them collaborate. Amati was an Italian physicist, with greatcharisma, who had grown up in Argentina: he had a poster of Che Guevara on his office door, anddrove a Bentley, which he never locked and readily loaned to David and others in temporary needof a car. Amati had a great talent for stimulating lively research discussions and for encouragingyounger physicists, a talent strangely lacking in some of the other staff members in the CERNTheory Division at the time. He was to have a major influence on David.David wrote two papers (23, 24) with Amati and Le Bellac on the operator formalism for dualmodels, as the development of Veneziano’s breakthrough had become known. Amati arranged forDavid’s stay at CERN to be extended by a further three months at the beginning of 1970, forwhich David took unpaid leave from Cambridge. This time Jenny and their daughters stayed inCambridge. David took a room in Chemin du Vieux Bureau, Meyrin, and devoted his leisure timeto skiing and, as always, music. A recital of Beethoven sonatas by Wilhelm Kempff captivatedhim and he remained a devotee of the great German pianist for the rest of his life.The collaboration with Amati and Le Bellac was joined by Victor Alessandrini, a younger Ar-gentinian physicist, and the four agreed to give a series of six lectures on dual models, whichthen became the basis for an influential review article,
The operator approach to dual multiparticletheory (26). When he returned to Cambridge, David gave a version of these lectures in DAMTPduring May, which were attended by Ed Corrigan, Peter Goddard, Michael Green and, probably,Jeffrey Goldstone, all of whom went on to make substantial contributions to the subject.8uring David’s second term at CERN, Amati raised with him the possibility of his spending alonger time there as a staff member, for three years in the first instance. There was no possibilityof obtaining leave from Cambridge for such an extended period under the University’s policies atthe time. John Polkinghorne, the leader of the theoretical physics group within DAMTP, madeit clear to David that it might be possible for him to return to a faculty position there, but thatwas by no means guaranteed. As David later put it, he was making an enormous gamble, becausehe would be giving up a tenured post in Cambridge for a fixed-term one at CERN; but, he wasprepared to make this sacrifice, in order to be able to spend all his time on the theory of dualmodels, because he thought this might well be the theory of the future.At the end of June 1971, David left Cambridge to take up what was eventually to be a six-yearstaff position at CERN. For the first month, David stayed in the hostel on the top floor of thebuilding housing the CERN Theory Division. He quickly met other new arrivals with interestsin physics similar to his own, including Lars Brink from Gothenburg, who was just beginninga postdoctoral fellowship and who was to become one of David’s collaborators and a life-longfriend. He was joining the group of mainly young theoretical physicists working on dual modelsthat had gathered around Daniele Amati, possibly the largest group in the world working on thesubject. It was big enough to sustain a weekly seminar, meeting on Thursdays at 2 pm, with thoseattending regularly at times during David’s first year at CERN including Alessandrini, Amati,Brink, Corrigan, Di Vecchia, Frampton, Goddard, Rebbi, Scherk and Thorn.The atmosphere was informal and collaborative, with a very free exchange of ideas. There wasthe feeling of participation in a shared enterprise to construct a new and radically different theory,as well as a camaraderie engendered by the active disapproval of many of the senior physicists atCERN and elsewhere. It seemed possible that a fully consistent theory of the strong interactionsmight be fashioned out of dual models, with the very requirement of consistency narrowing downthe range of dual models that should be considered as physically relevant. There was the sense thatthe theory was thus defining itself, rather being crafted by those working on it, and it was excitingto see its form, different from quantum field theory, emerge before one’s eyes, from within itself.David’s background in S -matrix theory, which provided the conceptual context within which thetheory could be defined, together with his ability to find precise, elegant and simple arguments,made it an ideal research area for him.When David’s family arrived, they moved into a spacious, newly built apartment on the ninthfloor at Le Lignon, with expansive views towards the Sal`eve and the Alps beyond. David was tofind living in Geneva somewhat of a culture shock for someone brought up in Edinburgh, thougha shock that was not completely unwelcome. At a mundane level, David found it a relief not tohave his Barton garden to tend and, without the teaching and administrative duties he had hadin Cambridge, he was free to concentrate on research, while also having time for his usual leisurepursuits. He joined the CERN golf club and he purchased a season ticket for the opera and balletseason at the Grand Th´eˆatre in Geneva. 9t weekends the family frequently visited the mountains, hiking in summer and skiing in winter,often in the nearby Jura mountains. While Jenny and Katie preferred cross country skiing,David took Rosalind downhill skiing, where he struck a characteristically nice balance betweenencouraging adventure and providing reassurance and caution when appropriate. David remaineda keen skier past his years at CERN, taking advantage of the opportunities provided by conferencesin Les Houches and other mountain resorts whenever he could.Ed Corrigan, then David’s research student, came to CERN for two months in late 1971, and theyworked together on building a consistent dual model that included fermions. At the beginning ofthe year, Pierre Ramond had introduced fermions into the theory in a way that looked extremelypromising (Ramond 1971), and so it was to turn out, but many steps would be necessary toensure that a full theory could be constructed that did not harbour inconsistencies. Most ofDavid’s efforts in the first half of his six-year tenure at CERN were directed towards this end,working mainly with Lars Brink.In 1972, through work at CERN and elsewhere, definitive progress was made on determining thephysical states occurring in dual models, leading to an understanding of how the picture of dualmodels as describing the scattering of one-dimensional objects, referred to at the time as ‘rubberbands’, ‘threads’ or ‘strings’, suggested two or three years earlier by Nambu (1969), Nielsen (1969)and Susskind (1970), could be made precise in terms of the quantum theory of a what was nowtermed a ‘relativistic string’ (Goddard et al. David Olive, at a workshop at the Aspen Center for Physics in Colorado, inAugust, 1974, seated next to Jo¨el Scherk (with head bowed) and with Eug`eneCremmer in the foreground.Photograph kindly provided by Lars Brink. models at the 17th InternationalConference on High Energy Physics,held at Imperial College, London, atthe beginning of July 1974. In hisreview (38), David sought not onlyto emphasize the conceptual formu-lation of dual models as a quantumtheory of strings but also that dual(or string) theory contained withinit electrodynamics, Yang-Mills gaugetheory and Einstein gravity. Indeed,as he stressed at the beginning of histalk, consistency demanded that thetheory contain massless spin 2, spin1 and spin particles, correspondingto the graviton, photon, and the neu-trino, as seen in nature. Dual theory,conceived as a theory of strong interactions, ironically had produced from within itself the emerg-ing theories of all the other fundamental interactions, offering the prospect of a unified theory. An Apparent Change of Direction
The London conference of 1974 was a turning point in David’s research, not only because of theimpact of his review talk, but even more as a result of a talk given by Gerard ’t Hooft on magneticmonopoles in gauge theories (’t Hooft 1974). David later said he did not really understand what’t Hooft had done until he heard Murray Gell-Mann discuss it during a meeting at the AspenCenter for Physics, Colorado, which took place just after the London conference. This meeting,organized by John Schwarz, brought together many of those then working on string theory for11hat seemed in retrospect to be just about the last hurrah of the early years of string theory,before it was eclipsed by the rise of QCD and the standard model. Although, outwardly, thefocus of his interests moved on to other subjects, like a number of the early dual model devotees,David’s fascination with string theory never left him, and the major contributions that he madeduring the remainder of his career, even when they appeared unrelated to string theory, ended upplaying a central role in its development down to the present day.After David returned to Geneva in the autumn of 1974, he began trying to understand the ’tHooft-Polyakov monopole, as it came to be known following a paper from Alexander Polyakov(1974), which appeared at roughly the same time and covered similar ground to that of ’t Hooft.Indeed, David spent most of his remaining three years at CERN elucidating the structure ofmonopoles in gauge theories, working initially with Ed Corrigan, who had come back to CERNas a postdoctoral fellow for two years, and with David Fairlie and Jean Nuyts, who joined thecollaboration in 1975 (40, 42).Nuyts had a familiarity with the theory of Lie algebras, and their roots and weights, and, inthe summer of 1976, he and David began to realize that this part of mathematics provided theappropriate mathematical language for describing magnetic monopoles in general non-abeliangauge theories. Peter Goddard also came to CERN, as a visitor for the summer of 1976, andtheir discussions led to the paper,
Gauge Theories and Magnetic Charge (46), which has had along-term and continuing influence. The magnetic monopoles Goddard, Nuyts and Olive classifiedbecame known as ‘GNO’ monopoles.David later recalled how it was while driving with his family to spend some days in Wengen inthe Bernese Oberland that the key idea came to him. Just as the electric charges were associatedwith points on the weight lattice of symmetry group, the magnetic charges were associated withanother lattice, the lattice dual to the weight lattice. David realized that this dual lattice wasitself the weight lattice of another Lie group, which GNO called the ‘dual group’. In the springof 1977, Goddard met Michael Atiyah at a conference on mathematical education in Nottingham.Atiyah was becoming interested in theoretical physics, though he was not yet familiar with thethen recent developments on monopoles in gauge theories. He immediately realized that the GNOdual group was the same as the dual group introduced by Robert Langlands (1970) within thecontext of what was known as the Langlands program, one of the major developments in puremathematics in the second half of the twentieth century. However, it was nearly thirty yearsbefore the relationship between electric-magnetic duality in gauge theories and Langlands dualityin the theory of automorphic forms began to be understood in depth.The magnetic monopoles identified by ’t Hooft and Polyakov, and the generalizations studied byDavid and his collaborators, are extended objects in a gauge theory, and so have a different statusa priori from the electrically charged particles, which are quanta of the fundamental fields in thetheory. David formed the bold and prescient vision that there should be a dual formulation of thetheory in which magnetic and electric charges and fields interchanged their roles.12hen Claus Montonen, who had been David’s graduate student for a year before he left Cam-bridge, visited CERN in the spring and summer of 1977, David began working with him to findevidence for his duality conjecture. Together they considered the simplest theory containing ’tHooft-Polyakov monopoles and showed that the spectrum of magnetic charges of the monopoleswas just like that of the electric charges of the fundamental quantum particles of the theory
David Olive in Kyoto in April 1976.Photograph kindly provided by the Olive family. and, moreover, the mass of themonopoles was given in terms of themass and electric charge of thesefundamental particles by exactly thesame expression as that relating thefundamental particle mass to themonopole mass and magnetic charge.All this provides evidence for a du-ality symmetry between electricallycharged fundamental particles andmagnetic monopoles, which are ex-tended objects in the theory. Theelectrically charged particles are themassive gauge bosons produced bythe Higgs mechanism in the sponta-neously broken theory, so Montonenand Olive conjectured that there wasa dual formulation of the theory inwhich the magnetic monopoles became fundamental particles, the massive gauge bosons associ-ated with a spontaneously broken dual gauge symmetry.Montonen and Olive refined their ideas over the months preceding David’s departure from CERNin September 1977, but they felt that several problems remained. Characteristically, David wasreluctant to write a paper before he felt that the arguments had reached their most elegant andsuccinct form. However, they were both convinced that they were on to something, and theydrafted a paper,
Magnetic monopoles as gauge particles? (47), which became one of David’s mostfamous and influential research contributions, and the proposal that it put forward, of the existenceof a dual magnetic formulation of a gauge theory, became known as Montonen-Olive duality orthe Montonen-Olive conjecture. The formulation of this conjecture exemplified very well David’sextraordinary ability to find precise, elegant and deep relations, encapsulating the essence of aphysical situation, and build on them a bold imaginative vision of what further structures mightawait discovery. 13 ast Years at CERN
Although the study of monopoles dominated David’s last three years at CERN, his collaborationwith J¨oel Scherk drew him again to dual theory for a significant interlude in 1976. At the ´EcoleNormale Sup´erieure, Scherk had begun collaborating with a visitor, Fernando Gliozzi, and theyhad noticed that, after making a certain projection in the fermion dual model, which both removedsomething like half the states, and solved one of the model’s problems by eliminating the unwantedtachyon (faster than light particle), the number of fermion states at any given mass equalled thenumber of boson states, which strongly suggested that the theory was spacetime supersymmetric.This projection as a means of removing the tachyon had been discussed informally since 1974 butit had not been realized that it resulted in an equal number of fermion and bosons states. Ona visit to CERN to give a talk, Scherk told David about their results. David, who had earlierstudied the properties of the Dirac equation in various dimensions of space-time, pointed out tothem that the consistency of their projection required this dimension to differ from 2 by a multipleof 8, which fortunately included the space-time dimension 10, which was needed for consistencyof the theory for other reasons.Building on their earlier work together, Scherk invited David to join the collaboration and togetherGliozzi, Scherk and Olive wrote two papers on these ideas (44, 45). In essence, these papers definedwhat became, following the work of Michael Green and John Schwarz, superstring theory, andthe projection has become known as the Gliozzi-Scherk-Olive, or GSO, projection, now one of thecornerstones of string theory.By 1976 David had come to accept that there was no real prospect of a permanent appointment atCERN and also that no effort was being made in Cambridge to find a tenured post there to which hecould return, positions which may seem very difficult to understand given the significance his workwas to acquire. David was offered a permanent post at the Niels Bohr Institute in Copenhagen,and he made a number of visits there, but he eventually concluded, for family reasons, that hewould prefer to go back to the UK. Then, in 1977, he was offered a lectureship at Imperial College,London, and at the end of September, David and his family left Geneva to return to England.When David later reflected on his time in the Theory Division at CERN, he viewed it as a par-ticularly happy and fruitful period, despite the shadow thrown towards the end by the temporaryeclipse of string theory, and the implications that had for his job prospects. It was an exciting timeto be there, with inspiring colleagues and a very pleasant atmosphere that was highly conduciveto research. He came to regard his last academic year there, 1976-77, with GNO monopoles, theGSO projection, and Montonen-Olive duality, as the high point of his research career.
Imperial College
While his family returned to their house in Barton, near Cambridge, which they had let whileat CERN, David at first lodged in a house near Kew Station in south west London, convenient14or travel to Imperial. He chose Kew because his father had lived there at one stage and nearbyRichmond had family associations. Soon, however, he found a house to buy in Kew, into whichhe moved in mid March 1978. He spent the weekdays in London, returning to Cambridge atweekends, a pattern he was to follow for much of the next fifteen years, except when on leave.This gave him ample opportunity to follow his musical interests in the evenings, often visitingCovent Garden for the opera and ballet, and the Royal Festival Hall for concerts.Soon after his arrival at Imperial College, David heard that his paper with Claus Montonen (47)had been accepted for publication. Although this paper was to become famous, David had beenunhappy that they had not been able to find tighter arguments, and had worried that it mightnot be accepted by the journal. He wrote to his collaborator, “I am glad Physics Letters acceptedour paper without any embarrassment, but further papers must be more solid.” Later it seemedto Montonen that this raised the bar so high that, in spite of many future discussions, no furtherpapers were ever written.One day in April 1978, David answered a knock on his office door to find Edward Witten, thena Junior Fellow at Harvard, who was visiting Oxford at the time at Michael Atiyah’s invitation.Atiyah had told Witten that he thought that there might be something deep in David’s work onmonopoles, and advised him to seek David out at Imperial. Witten, who had not come across thesepapers before, was inclined to be skeptical of very speculative conjectures like that of Montonen-Olive. The key observation supporting the duality conjecture was that the same formula gaveboth the masses of the elementary electric charged particles and those of the magnetic monopoles,which are extended objects, and Witten could not see how this could survive the renormalizationprocedure necessary to define the quantum field theory.It occurred to them in discussion that there was some hope that supersymmetry might provide arescue. By the end of the day of Witten’s visit, they had understood that the supersymmetry alge-bra in a supersymmetric quantum field containing monopole solutions is modified by the presenceof central charges. They were then able to show that, in the context of certain supersymmetricgauge theories, the supersymmetry algebra actually implies the Montonen-Olive mass formula, sothat the prime result motivating the duality conjectures necessarily holds in suitable theories withsupersymmetry.The joint paper (50) that they wrote was to become another of David’s most influential contri-butions. Witten much later remarked that although he was very pleased with the result, withhindsight, he could see that he drew the wrong conclusion from it. In effect using Occam’s razor,he took the fact that supersymmetry implies the Montonen-Olive mass formula to remove theneed for any deeper explanation, such as duality. It was not until the work of Ashoke Sen (1994),showing the existence of certain two monopole states in N = 4 supersymmetric gauge theories,predicted by Montonen-Olive duality, that Witten became convinced of the depth and importanceof the conjecture and, in large part through his influence, it made a seminal contribution to thereconceptualizing of string theory in the mid 1990s, in what has become known as “the secondsuperstring revolution”. 15uring his time at Imperial, David would usually go back at weekends to the house he and Jennyhad kept in Barton, near Cambridge, and often, on Saturday mornings, he would meet with PeterGoddard, by then a lecturer in DAMTP, to discuss physics in Goddard’s office. These discussionsprovided the basis for a continuing collaboration that lasted from the late 1970s to the early1990s. Soon after David moved to Imperial, they completed a review (49) on
Magnetic monopolesin gauge field theories , which provided a systematic account of the ideas David had been centrallyinvolved in developing in the previous four years. For some years thereafter, David was much indemand as a review speaker on this subject at conferences and summer schools.In his first years at Imperial, David’s research mainly focused on studying magnetic monopolesolutions to spontaneously broken gauge theories in greater detail, in particular the conditions
David Olive in Bechynˇe, Czechoslovakia, in June 1981.Photograph kindly provided by the Olive family. on their charges for their stabil-ity, finding further circumstantial evi-dence for the Montonen-Olive dualityconjectures (60, 61). In general, thenonlinear equations describing mag-netic monopoles cannot be solved ex-actly in closed form, but it had beenfound that if attention is restricted tospherically symmetric solutions in anappropriate limit, the equations be-come integrable. Leznov and Saveliev(1979) had observed that where thegauge symmetry group is SU( N ), theequations are just those associatedwith the (finite) lattice of particlesin a line interacting through suitablenonlinear springs introduced fifteenyears earlier by Morikazu Toda (1967). For a general (semisimple) gauge symmetry group, G ,the equations can be characterized in terms of the Dynkin diagram of G , which encodes thegroup’s structure (65).The way the study of spherically symmetric monopoles brought together solutions to gauge the-ories, the theory of Lie algebras and integrable systems struck David as deep and important. Itwas the initial motivation for his continuing interest in the Toda equations and their algebraicproperties, a subject he returned to repeatedly over the next dozen years. At Imperial, Davidcould easily attract and take on graduate students, in a way that had not been possible at CERN,and his work on Toda theories and other research was done in collaboration with a successionof current and former students, including Regina Arcuri, Luiz Ferreira, Andreas Fring, FrankGomes, Marco Kneipp, Peter Johnson and Jonathan Underwood, and most notably Neil Turok.Turok, who was later to hold professorships in both Princeton and Cambridge, before becomingthe Director of the Perimeter Institute in Waterloo, Ontario, started as David’s research student16n 1980. Over the following thirteen years he wrote a series of influential papers with David onToda field theories ( e.g.
70, 75, 102).
Leave in Charlottesville
When David went on leave to the University of Virginia in Charlottesville in 1982-83, he took NeilTurok with him, carrying on their work on Toda theories. In Charlottesville, David gave a courseof graduate lectures on the theory of Lie algebras, the mathematics that had underlain his seminalwork on magnetic monopoles, but otherwise he used the freedom of his visiting appointment todevelop the ideas that he had been evolving over some years in collaboration with Peter Goddard,who joined him there for some months at the beginning of 1983. Through discussions with GraemeSegal in Oxford and others, they had become aware that evidence of deep connections had begunto emerge between string theory and the theory of Kac-Moody algebras, a mathematical theorywhich had been initiated in 1967 by Victor Kac (1967) and Robert Moody (1967), coincidentallyjust about when the seeds of string theory were being sown by Veneziano’s famous paper.Around 1980, mathematicians realized that, in constructing representations of Kac-Moody alge-bras, they had rediscovered the vertex operators that physicists had been using to describe theinteractions of strings. The connection to the formalism of dual models was spelled out by IgorFrenkel and Victor Kac (Frenkel & Kac 1980) and by Segal (1981). The algebraic properties ofthese vertex operators had already featured prominently in work of both Goddard and Olive, andhad been a central tool in understanding the structure of dual models and their detailed interpre-tation as string theory. Together in Charlottesville, and stimulated by what they had learned inthe context of magnetic monopoles about the weight and root lattices of Lie algebras, Goddardand Olive studied how the vertex operator construction could be used to associate a Lie algebrato each integral lattice (i.e., a lattice such that the scalar product of any two lattice points is aninteger). The nature of the Lie algebra so defined depended on whether the lattice is Euclideanor has some other signature. Although it seemed clear that these results should have some role toplay in string theory, by producing symmetries of the spectrum, for example for strings movingin a space in which some of the dimensions have been compactified to form a torus formed byassuming periodicity under displacements corresponding to the lattice. However, at first sight,there seemed to be obstacles to using this to incorporate symmetry into string theory in a realisticway.Goddard and Olive paid particular attention to even self-dual lattices, in part because of theirconnection to modular invariance, which had already proved important in string theory. Theynoted that, in the Euclidean case, these lattices only existed in dimensions that were multiplesof 8. In dimension 8, there was only one, the root lattice of the group E , while in dimension16, there were two, the root lattice of E × E and a sublattice of the weights of the Lie algebraof SO(32). This observation proved prescient because just over a year later, in the autumn of1984, the 16-dimensional even self-dual Euclidean lattices, and the associated vertex operator17onstruction of Kac-Moody algebras, were to play a key role in the developments that led to thedramatic revival of interest in string theory.When Igor Frenkel visited Charlottesville in March 1983, Goddard and Olive were able to dis-cuss their results with him and they discovered a strong overlap with results he had recentlyobtained. Frenkel arranged for them to be invited to speak at the conference on Vertex Operatorsin Mathematics and Physics that was held the following November in Berkeley at the recentlyestablished Mathematical Sciences Research Institute. Afterwards they circulated the paper, Al-gebras, Lattices and Strings (72), which they had prepared for the proceedings and which wasthen influential in introducing many physicists to Kac-Moody algebras and their construction interms of the vertex operators of string theory.
Coset Construction and Conformal Field Theory
At the Berkeley conference, they also learned from the talk of Daniel Friedan about his work withQiu and Shenker, and about then unpublished work of Belavin, Polyakov and Zamolodchikov, onconformal field theory and the representations of the Virasoro algebra. The Virasoro algebra is theLie algebra of the conformal symmetry group and plays a central role both in string theory and inconformal field theory; indeed, in a sense, conformal field theory describes the structure of stringtheory. Friedan, Qiu and Shenker (FQS) had established necessary conditions for a representationof the Virasoro algebra to be unitary, showing that there was a continuum and an infinite discreteseries (Friedan et al.
Non-abelian bosonization in two dimensions , which demonstrated an equiva-lence between certain boson theories, associated with Lie groups, now called Wess-Zumino-Witten(WZW) theories, and certain free fermion theories, by exploiting the equivalence of representa-tions of isomorphic Kac-Moody algebras contained in the two theories. Looking for general condi-tions under which Witten’s equivalence might hold, they sought to demonstrate that the energy-momentum tensors of the boson and fermion theories were the same. In such two-dimensionaltheories, the moments of energy-momentum tensor generate the conformal symmetry of the the-ory, and they provide a representation of the Virasoro algebra. The aim of Goddard and Olive wasto determine when the representations of the Virasoro algebra in the two theories were equivalentbecause this would be a necessary condition for the complete equivalence of the theories.Goddard and Olive spent some weeks working together at the Aspen Center for Physics in thesummer of 1984, and found many instances where the two energy-momentum tensors differed.However, they realized that the difference between them was often very interesting. The boson18heory could always be imbedded within the fermion theory and the question of equivalence waswhether it was, in effect, all of that theory: subtracting the boson energy-momentum tensor fromthe fermion one gave zero if the theories were equivalent but otherwise the difference was itself anenergy momentum tensor, and, in a sense described, what needed to be added to the boson theoryto yield the full free fermion theory. The difference provided a representation of the Virasoroalgebra, one which commuted with the boson representation of the Virasoro algebra, and suchthat their sum equaled the fermion representation. Further, explicit calculation showed that, inmany instances, these ‘difference’ Virasoro representations provided missing representations fromthe discrete series of FQS, thus proving their existence and giving an explicit construction in thesecases.Goddard and Olive (74) did not manage to construct the whole of the infinite discrete series whilein Aspen, but on their return to England they explained what they had done to Goddard’s researchstudent, Adrian Kent, and together they generalized the construction by replacing the fermiontheory and considering instead a WZW theory associated with a Lie group, G , and the theorycontained within it associated with a subgroup, H , of G . Any such pair, H ⊂ G , defines a Vira-soro representation given by the difference of the Virasoro representations associated with the G David Olive with Arthur and Ludmilla Wightman at a summer school in Erice,Sicily, in August, 1985. Photograph kindly provided by Arthur Jaffe. and H theories. Goddard, Kent andOlive (GKO) associated this repre-sentation with the coset G/H and ithas become known as the coset orGKO construction (76, 81). Theywere able to show that all the repre-sentations of the FQS discrete seriescould be obtained in this way, thuscompleting the classification of uni-tary representations of the Virasoroalgebra. The WZW theory associ-ated with a Lie group G is a confor-mal field theory and, for H ⊂ G , thecoset construction associates a con-formal field theory associated with G/H with the corresponding energy-momentum tensor. This coset construction has remained one of the main ways of constructingand characterizing conformal field theories.The following year, 1985, in collaboration with Werner Nahm, Goddard and Olive succeeded insolving the original problem from which they had been fruitfully diverted by the coset construction,by showing that the condition for the fermion theory in Witten’s non-Abelian bosonization to beequivalent to a WZW theory was that the fermions should transform according to a representationof G that could be used to extend it to form a larger group, G (cid:48) , in such a way that the pair definewhat is called a symmetric space (78). 19 evival of String Theory Also at Aspen in the summer of 1984 were Michael Green and John Schwarz, who were studyinganomaly cancellations in supersymmetric gauge theories coupled to gravity in the hope of estab-lishing that, under suitable conditions, string theory might satisfy this consistency requirement.They first found that the cancellation occurred for the gauge group with the Lie algebra of SO(32),provided that its weights lay on the lattice which is even and self-dual, and then they realizedthat it also holds for E × E (Green & Schwarz 1984). Aware that these two groups had beensingled out by Goddard and Olive a year before in their paper, Algebras, Lattices and Strings ,they sought to construct a string theory with E × E symmetry, unknown at the time, basedon ideas from that paper on the incorporation of symmetry into string theory using Kac-Moodyalgebras. But further ideas were necessary, about treating left and right moving waves on closedstrings very differently, which came with the construction by Gross, Harvey, Martinec and Rohm(1985) of what they called the heterotic string.It was these developments, initiated by the work of Green and Schwarz, and promulgated throughthe unique influence of Edward Witten, that gave rise to the renaissance of interest in string theorybeginning in 1984. In them, Kac-Moody algebras, and two-dimensional conformal field theorymore generally, played a key role. As David continued working with Peter Goddard, WernerNahm, Adam Schwimmer and others on conformal field theory and related infinite-dimensionalalgebras, the mushrooming interest in string theory meant that he was much in demand as alecturer at conferences and summer schools. This prompted him to write a long pedagogicalreview, Kac-Moody and Virasoro Algebras in Relation to Quantum Physics (82), in collaborationwith Peter Goddard, which was widely read and has remained a standard reference. Alongsidehis work on conformal field theory, David continued to make contributions to Toda field theorywith Turok and many of his other students. With a postdoctoral fellow, Michael Freeman, he alsoreturned to the basic calculation of the one loop contribution in bosonic string theory, using theBRS formalism to simplify his earlier work with Lars Brink (87, 88).From the mid 1980s, David began to receive long overdue formal recognition of his achievements.He had been made a Reader at Imperial in 1980 at the age of 43, and four years later he waspromoted to a Professorship of Theoretical Physics. In 1987, he was elected a Fellow of the RoyalSociety in recognition of his contributions to S -matrix theory, dual models, the classification ofmagnetic monopoles in gauge field theories and on the concept of electric-magnetic duality.David spent the academic year 1987-88 as a Member of the Institute for Advanced Study inPrinceton. When he returned to Imperial, he became head of the theoretical physics group,but the administrative duties of this post did not really suit his temperament. He discharged hisresponsibilities perhaps too meticulously, even down to counting out rations of photocopying paperfor students when there was a funding crisis. The very qualities of mind, the almost obsessiveneed to resolve possibly tell-tale discrepancies in understanding, that led to David’s remarkablyoriginal and prescient research, nearly drove David and those around him to distraction when hetried to reconcile, down to the last penny, the different financial systems Imperial deployed at20he group, department and college levels. When a new administrator with some understanding ofaccounts arrived to support the group, it was not only David who was enormously relieved. Swansea and Later Years
In 1991, when the University College of Wales Swansea was seeking to fill a chair in physics,Aubrey Truman, a mathematical physicist and the Dean of Science, argued in favour of appointinga theoretical physicist, particularly in view of the fertile cross connections with mathematics thathad developed in the previous two decades in particular. Truman had got to know David Olivewhen David had served on the scientific advisory committee for the International Congress ofMathematical Physics that had taken place in Swansea in 1988. They had both been researchstudents of JC Taylor, Olive in Cambridge and Truman later in Oxford, and, although they hadnot met, Truman had long admired Olive’s contributions to S -matrix theory as well as his laterwork. He approached David about the vacant chair and was delighted when he expressed aninterest in moving from Imperial to Swansea.It turned out that David’s colleague at Imperial, Ian Halliday, was also attracted by the idea ofmoving to Swansea. At the time the future of the Swansea Physics Department had looked quiteuncertain, and the Vice Chancellor, Brian Clarkson, became interested in the idea of recruitingOlive and Halliday as the nucleus of a theoretical particle physics group that might substantiallyraise the international standing of the Physics Department. The plan was formed of appointingDavid as a Research Professor in the Department of Mathematics and Halliday in Physics, withthe group they were to lead linking the two departments. However, this initial arrangement didnot last, because there were tensions with some other members of the Department of Mathematics.David was not a political animal and he just got very agitated when he thought others were notbehaving straightforwardly. As a result, the idea of linking the two departments was abandonedand the whole group moved into Physics.David demonstrated his characteristic carefulness in the negotiations with the University overthe group’s move. Here his meticulousness proved an asset because he ensured that every lastpromised provision was written out in exquisite detail in letters signed by the Vice Chancellor,which proved extremely useful to the group in ensuring that commitments were kept. David wasvery engaged in the whole development of the new group and his presence was a decisive factor inthe recruitment of a number of outstanding young lecturers, Nick Dorey, Tim Hollowood, WarrenPerkins, and Graham Shore, soon followed by others. The new group brought highly valuablerecognition to the physics department at a time when pure science was being cut in universitieslike Swansea; the ranking of the Physics Department improved dramatically. The group Hallidayand Olive created, now with a strength of about twelve faculty members, has continued to thriveover the last twenty-five years as one of the leading theoretical particle physics groups in the UK.21avid had spent the second half of 1992 participating in one of the first programmes at the NewtonInstitute in Cambridge. When the Olives moved to Swansea at the beginning of 1993, they retainedtheir home in Barton, near Cambridge. Although Jenny may have made the move somewhat David and Jenny Olive in Finland in May 1999. Photograph kindly provided byClaus Montonen. reluctantly, she continued to sup-port David in his work, even helpingwith the production of diagrams forhis public lectures. She also helpednew physics students in the univer-sity who were having difficulties withbasic mathematics. She shared thetechniques she had developed for thiswith a wider audience in her book,
Maths: A Student’s Survival Guide.A Self-Help Workbook for Scienceand Engineering Students , publishedby Cambridge University Press.David had had four Brazilian re-search students, Regina Arcuri, LuisFerreira, Frank Gomes and MarcoKneipp. In these years he made anumber of visits to Brazil, with Jennysometimes accompanying him, to participate in workshops, research schools, etc. , contributing tofurthering the development of theoretical physics in the country. He developed a strong interest inBrazilian culture, particularly the food and the music. He loved to have lunch in the restaurantsclose to the Paulista Avenue in S˜ao Paulo, where one pays by the weight of the food taken from agenerous buffet. Characteristically, David developed models for how to get best value for money.But it was Brazilian music that most engaged him, from Villa-Lobos to much less known com-posers, such as Lorenzo Fernandez. He became extremely interested in a recording of Fernandez’smusic by the Amazonia Quartet, but it seemed that it was sold out in S˜ao Paulo and had beendiscontinued. However, he tracked it down to a small shop in a distant part of the city and soacquired a rare CD, of which he was very proud.Further public recognition of David’s achievements came with the award in 1997 of the DiracPrize and Medal of the International Centre for Theoretical Physics in Trieste, shared with Pe-ter Goddard, “in recognition of their far-sighted and highly influential contributions to theoreticalphysics”. The announcement further cited their contribution of “many crucial insights that shapedour emerging understanding of string theory and have also had a far-reaching impact on our un-derstanding of 4-dimensional field theory”, and went on to explain that “Olive’s work on spacetimesupersymmetry of the spinning string theory (with F. Gliozzi and J. Scherk) made possible thewhole idea of superstrings, which we now understand as the most natural framework for super-symmetry and string theory. Goddard and Olive introduced key ideas about the use of current22lgebra in string theory which were very important in the subsequent discovery of attractive waysto incorporate space-time gauge symmetry in string theory, thus making it possible for stringtheory to incorporate the standard model of particle physics. These discoveries, made in the years1973-83, were among the most crucial steps in making possible the ‘superstring revolution’ of1984-5. The ‘second superstring revolution’ of the last few years has been equally dependent onpioneering insights about magnetic monopoles made in 1977 by Goddard, Olive, and J. Nuyts,and further extended by Olive and C. Montonen. Their ideas concerning a dual interpretationof magnetic charge, and then about electric-magnetic duality in non-abelian gauge theory, wereway ahead of their time and have proved to have a far-reaching importance, which we are onlynow beginning to understand, in governing the dynamics of four-dimensional field theory and ofsuperstring theory.”The ‘second superstring revolution’, to which the Dirac Medal citation referred, was initiated bythe work of Nathan Seiberg and Edward Witten (1984a,b). Central to it was the generalization ofMontonen-Olive duality to string theory, in the form of S-duality, and its extension to a modulargroup of dualities by the work of Ashoke Sen (1994). After twenty years, it had at last been realizedwidely how visionary David’s work on monopoles in the mid 1970s had been. These developmentsrekindled David’s own interest in electromagnetic duality. With Pierre Van Baal and Peter West,David organized a programme at the Newton Institute on Non-Perturbative Aspects of QuantumField Theory, centred on ideas of duality and supersymmetry, in the first six months of 1997(116). His deep and beautifully clear expositions of electromagnetic duality and its generalizationsbecame in demand at conferences and graduate schools. With Marco Alvarez, who had joinedhim as a postdoctoral fellow in Swansea, in some of his last work, David characteristically soughta deeper understanding of the quantization of magnetic flux, by considering gauge theories onsmooth four-dimensional manifolds of arbitrary topology. They found that the quantization offluxes in these theories provided a physical interpretation of some mathematical concepts, such asthe Stiefel-Whitney classes (119, 122, 130).David received national recognition at the beginning of 2002 with his appointment as a Commanderof the Order of the British Empire, for services to theoretical physics, in the UK New YearHonours, and, in 2007, he was made a Foreign Member of the Royal Society of Arts and Sciencesin Gothenburg, Sweden. In many ways, David enjoyed his time in Swansea more than his yearsat Imperial, which were made somewhat fraught by commuting on crowded trains and havingto lecture to very large audiences of students. With his colleague Colin Evans, David began tocompile a history of the Swansea Physics Department and he became an expert on the history ofWelsh science, particularly in relation to the history of cosmic rays, operations research, radar,the atomic bomb, and the internet. He was a Founding Fellow of the Learned Society of Wales in2010, and a strong supporter of it.In the last six years of his life David suffered from increasingly poor health as a result of heartdisease. He and Jenny first became aware of a problem when David became exceedingly tiredwhen they were walking in the Lauterbrunnen Valley. Later, his golfing partner, a physician,advised him to consult his doctor because of the difficulties he was experiencing while they were23laying. His deteriorating heart condition led eventually to kidney failure, but the weakness ofhis heart meant that dialysis was not possible. Through his last years, David remained mentallyengaged, and, in spite of his difficulty in walking any distance, he continued to travel for specialoccasions and conferences up to a few months before his death.In late October 2012, David’s health suddenly deteriorated further and he and Jenny realizedthat he was gravely ill. One kidney had failed and efforts to keep the other kidney workingwere proving ineffective. His doctors concluded that nothing more could be done. David, thenin hospital, phoned Jenny and told her in a rather matter of fact way that he was finally beingallowed to come home for palliative care, because no further treatment would be effective. Aswas then expected, he only had a week to live, during which he showed grace and courage as hereceived many greatly appreciated telephone calls and emails from physicists and mathematiciansaround the world, who had heard of his condition.David died on 7 November in the house in King’s Grove, Barton, that he and Jenny had ownedsince 1965. At his funeral service twelve days later, the village’s 14th century parish church, StPeter’s, was full with David’s friends, colleagues and relatives. David did not have a religiousbelief, but the kindly vicar told the congregation that we could be sure that David was now in aplace where there were no more physics problems to worry about, not aware that that would havebeen no paradise at all for David. Following his wishes, he was buried in the village churchyard.
Interests, Personality and Influence
David had a wide range of interests, but his life-long love of music was exceptional. From his youth,he built up an outstanding collection of long playing records, which were gradually replaced fromthe mid 1980s onwards by an even more extensive collection of over four thousand CDs, housedin nine cabinets, which he had constructed personally with great care and skill. He carried overfrom his LP collection his practice of meticulously making a note on the sleeve of each time heplayed a recording, in part to assess wear.He had an encyclopedic knowledge of music, and of his collection of recordings in particular, whichoften astonished his friends and colleagues. Adam Schwimmer, who grew up in the little-knowntown of Grosswardein-Oradea in Transylvania, recalled a characteristic incident illustrating boththe depth of David’s musical knowledge and his particular sense of humour. One evening Davidproduced from one of his CD cabinets a recording of symphonies by Michael Haydn, Joseph’s muchless famous younger brother, played by the Oradea Philarmonic in what Schwimmer described asan awful performance. David knew that Michael Haydn had been the court composer of theBishop of Grosswardein, Schwimmer’s home town, and he had in his collection the only availablerecording of his works.David’s notes on and knowledge of the concerts and opera performances he had attended overthe years paralleled, even rivaled, the notes he made setting out his evolving understanding of24is investigations in theoretical physics. Filed in numerous loose leaf folders, and annotated withunderlinings in various colours and styles, which David did not even attempt to explain to others,these notes were written and rewritten as he sought an every deeper and clearer understandingof the topic at hand, whether this was an established branch of mathematics or a new physicaltheory. In either case, he would strive to gain his own understanding, one that met his ownexacting standards of concision and clarity, often producing novel insights into existing theories.David’s frequent comment was “I think we can understand this better”, but, even when scepticalof an idea, he would not simply dismiss it but rather look for ways in which it might make sense.He was always seeking mathematically deep, elegant equations that encapsulated physical ideas.It was quite evident that his approach emulated naturally that of his great intellectual hero, PaulDirac, whose lectures he had attended as a student at Cambridge.The many files of handwritten notes that he amassed seemed an end in themselves for him, a
David Olive accepting the Dirac Medal from Miguel Virasoro, Director ofICTP, in Trieste on 26 March 1998. The briefcase that David had usedfrom his schooldays, for nearly fifty years, can be seen at the right. record and reference for what he hadunderstood. They would form thebasis for his research publications, hislecture notes and his reviews, andtheir honing underpinned David’s ex-ceptionally clear expository style, buthe would be very reluctant to publisha result before he was convinced thatthe argument was sufficiently clear,sometimes to the frustration of hiscolleagues. He was justifiably proudof the seminal contributions that hehad made to theoretical physics, buthe never had a desire to rush intoprint to seek priority on an idea thathad not yet been elucidated to hisown personal satisfaction. Althoughhe was determined to understand everything himself de novo , he readily shared the results of hisendeavours with others.The openness and generosity with which David would share his new ideas and insights with hiscollaborators, colleagues and students was in amusing contrast to the minute care with which hewould dissect the bill for a meal shared with a colleague, for example in one of the Indian restau-rants he loved in London, evoking the cultural stereotype of a Scotsman. The leather briefcase,acquired as a schoolboy at the Royal High School, Edinburgh, was retained until nearly the endof his career. Progressively disintegrating, but kept in service by David’s own highly imaginativerepairs, performed with characteristic economy, e.g. by use of bits from an old Fairy Liquid bottleto mend the handle, it provided an outward symbol of David’s ingenuity and endearing frugalityand it was only abandoned with the greatest reluctance in his later years.25or some years, his entry in
Who’s Who seemed to contain a perhaps Freudian reference to hisconcern to conserve financial resources: his interests were listed there as ‘music and gold’, the lattera misprint occasioned by his at times almost illegible handwriting; eventually it was corrected to‘golf’, the other passion, together with music, that he carried throughout his life from his teenageyears onwards. In Swansea, he was able to find the opportunities to play the game to an extentthat he had not enjoyed since his youth in Scotland. He and Ian Halliday joined Pennard GolfClub, and for many years they played together every Saturday morning. Halliday found David’sgolf style to be the antithesis of his careful, though inspirational, approach to physics: he wouldoccasionally hit enormous drives but with very little control, and on one memorable occasionhis drive hit the ladies’ tee marker thirty yards ahead and, following a trajectory reminiscentof Rutherford scattering, the ball whistled back past Halliday’s and Olive’s ears to come to restbehind them, a drive of minus 150 yards.He was in many ways a quiet and reserved person but he enjoyed the company of his friendsand colleagues and he loved the process of collaboration, as the papers containing his majorcontributions to physics make clear. These were ahead of their time and helped to shape theunderstanding of the structures of string theory and quantum field theory gained in the last fortyyears. His relentless and uncompromising search for rigour and clarity permanently influenced allthose he taught or worked with. 26
I Scientific Contributions S -matrix Theory The Analytic S-Matrix (12) begins with the rather arch sentence, “One of the most importantdiscoveries in elementary particle physics has been that of the complex plane.” The analyticityof two-particle scattering amplitudes in energy, treated as a complex variable, had been usedfor some decades to derive dispersion relations, expressing the amplitude as an integral of itsimaginary part, under suitable assumptions. However, just before David began research at thebeginning of the 1960s, Stanley Mandelstam (1959) showed how the scattering amplitudes forany number of particles could be considered as analytic functions of the invariants formed fromthe momenta of the particles. He demonstrated that the amplitudes had singularities, poles andbranch cuts, whose presence could be seen as following directly from the interrelation of analyticityand unitarity, and that the perturbation series in a quantum field theory could be reconstructedfrom these requirements.Building on Mandelstam’s ideas, Geoffrey Chew formulated the bootstrap hypothesis, that is theproposition that the requirements of analyticity (reflecting causality) and unitarity (seeminglyessential for the probabilistic interpretation of quantum theory), together with some asymptoticassumptions at high energy, determine the scattering amplitudes, i.e. the S -matrix, uniquely. ForChew’s most zealous followers, the bootstrap hypothesis became almost an article of faith, andChew was a charismatic evangelist for it himself. For others, including Mandelstam as well asOlive, it remained a hypothesis that merited exploration.David’s first contributions in this area (2–4,6) were concerned relations between analytic contin-uations, discontinuities, and unitarity. At first, he worked within the context of quantum fieldtheory, progressing in (6) to discuss the possibility of derivations within an S -matrix theory, whosestructure he had begun to explore in An exploration of S -matrix theory (5). In The Analytic S Matrix (1966), published at the same time as (12), but without the hyphen, Chew gave his fullestaccount of his philosophy. He cited David’s work (5), as the most ambitious attempt to providean axiomatic treatment of the programme, summarizing his treatment and employing the ‘bubble’notation that David had developed and which became standard in the subject.David built on earlier work of Stapp (1962) and, particularly, that of Gunson (1965), which wasavailable as a preprint in 1963. The main ingredient that had to be included as a starting point,in addition to analyticity and unitarity, was connectedness structure, taken to be a consequenceof the assumed short-range nature of the fundamental forces. For the simplest case of two-to-twoscattering, a + b → a + b , with momenta p a , p b in the initial state and p (cid:48) a , p (cid:48) b in the final state, thestatement of connectedness in ‘bubble’ notation is27 ab ab ab ab ab ab [1]where the lefthand side denotes the S -matrix element (cid:104) p (cid:48) a , p (cid:48) b | S | p a , p b (cid:105) and, on the righthand side,the first term denotes (cid:104) p (cid:48) a , p (cid:48) b | p a , p b (cid:105) , essentially the product of delta functions between initial andfinal three-momenta for each of the two particles a and b , and, up to normalization factors, thesecond term equals iδ ( p a + p b − p (cid:48) a − p (cid:48) b ) A . Here A is the ‘connected’ part of the scatteringamplitude, a function of the invariants s = ( p a + p b ) and t = ( p a − p (cid:48) a ) , which is assumed to beas analytic as possible, consistent with the requirements of unitarity,Unitarity for the S -matrix, S , takes the form SS † = 1, which in ‘bubble’ notation reads S S † [2]where the lines joining indicating integration over momenta and the particle labels have beenomitted. So, substituting for S from [1], [3]where the second term on the lefthand side is the hermitian conjugate of the first. In (2), withinthe context of quantum field theory, David had argued that this hermitian conjugate, the ‘minus’amplitude, is the analytic continuation of the first term in [3], the ‘plus’ amplitude, through theunphysical region below s = ( m a + m b ) , the two-particle normal threshold, more generally thanhad previously been established. [Here m a , m b denote the masses of a, b , and so on.] In (5), heproduced a more general argument for the validity of this property, known as hermitian analyticity,within the context of S -matrix theory. [See also (12), p. 223.] For s real just below this thresholdand for suitable real t , the ‘plus’ and ‘minus’ amplitudes are equal and real, and so [3] holds withthe expression on the righthand side replaced by zero.Above the two-particle threshold, [3] gives an expression for the discontinuity across a cut, withthe ‘plus’ and ‘minus’ amplitudes being the boundary values of a single analytic function fromdifferent sides of the cut. If, at higher energy, it becomes possible to produce a particle, c , i.e. a + b → a + b + c , an extra term appears in the unitarity equation, which takes the form [4]in ‘bubble’ notation. The threshold at which this process becomes possible corresponds to anotherbranch point at s = ( m a + m b + m c ) , and [4] gives the total discontinuity across the two cutsalong the real axis. And so, as and when new processes become possible at higher energies, theappearance of corresponding additional terms in the unitarity equations implies a sequence ofbranch point singularities, termed normal thresholds.28he unitarity equations can also imply the presence of poles, as well as cuts, in the physicalregion, i.e. for real, physical values of the momenta. At suitable values of momenta, the unitarityequation for three-to-three scattering, a + b + c → a + b + c , the unitarity equation contains a term bc ca ab [5]which corresponds to a δ ( q − m b ) discontinuity in the amplitude, where q = p a + p b − p (cid:48) a , where p a , p b denote incoming momenta, and p (cid:48) a an outgoing momentum, for the appropriate particles.David argued that this implies that the amplitude, viewed as an analytic function, has a pole at q = m b . abc abc ∼ bc ca ab at q = m b , [6]where the middle line on the right hand side denotes a factor of ( q − m b ) − , rather than a δ -function.The unitarity equation for the a + b + c → a + b + c scattering amplitude also contains the term(a) shown in [7]. If the pole in [6] is inserted in this term, the structure corresponding to the term(b) in [7] is obtained [(11);(12), pp. 206, 266–278].(a) (b) [7]Related terms are generated by other terms in the unitarity equation and together they imply acut in ( s, t )-plane with discontinuity given by [8]just as in perturbative quantum field theory (QFT). In a series of three papers (15–17) withhis student, Michael Bloxham, and John Polkinghorne, David was able to extend this approachto show that for physical values of the momenta, unitarity requires scattering amplitudes to besingular on the arcs of curves where Landau (1959) had determined that singularities occur for29eynman diagrams in QFT, and that the discontinuities associated with these singularities aregiven by the rules obtained for perturbative QFT by Cutkosky (1960).Building on ideas of Gunson (later published in Gunson (1965); see p. 847), David also showedhow similar arguments based on analyticity and unitarity imply the existence of antiparticles, andmay allow the proof of crossing symmetry and the TCP theorem (5,12), as well as implying thestandard connection between spin and statistics (8) [ i.e. that integral spin particles are bosons andhalf-odd-integral ones are fermions]. However, these results depend on the existence of suitablepaths for analytic continuation and a knowledge of the singularity structure of the S -matrix outsidethe physical region and for complex values of the momenta.Reviewing the state of S -matrix theory nearly a decade after he ceased working on it, Davidreiterated his belief that “Properties involving analyticity well away from [the physical region]should surely not be postulated but rather deduced from more fundamental principles involvingthe physical region.” The developments of the early 1960s, in which David had played a leadingrole, had shown how one might hope to re-establish the main achievements of Axiomatic QFT (seeStreater and Wightman 1964) within the (arguably) more general framework of S -matrix theory,but, as David put it, “Although the general features of the physical region were appreciated, adetailed, precise and comprehensive mathematical treatment was lacking.”(48) Later approachesto providing such a mathematical treatment for discussing the S -matrix near the physical regionwere developed by Sato (1975) and by Iagolnitzer (1981), but no progress on establishing a rigorousanalysis of non-physical region singularity structure has been made. Fermions and the GSO Projection
At early in 1971, Pierre Ramond proposed a description of free fermions in dual models (Ramond1971) in a sort of generalization of the Dirac equation. This was quickly extended by Andr´e Neveuand John Schwarz who gave expressions for dual model amplitudes involving a single Ramondfermion interacting with a sequence of mesons (Neveu & Schwarz 1971). When he arrived atCERN some months later to take up his staff appointment, David began working with EdwardCorrigan on a programme to construct a complete consistent theory of dual fermions and mesonsworking within the operator formalism.The operator formalism of the original bosonic dual model of Veneziano, the space of states iscreated by bosonic oscillators, a µn , where µ runs over the dimensions of space-time and n runsover the integers. Neveu and Schwarz enlarged this using anti-commuting fermionic oscillators, b µr , where r runs over half odd integers, in addition to the a µn , to create a space of space-timeboson states, while the Ramond states, which are space-time fermions, are created by fermionicoscillators, d µn , where n runs over the integers, in addition to the a µn . These oscillators satisfy thecommutation and anti-commutation relations,[ a µm , a νn ] = mδ m, − n η µν , { b µr , b νs } = δ r, − s η µν , { d µm , d νn } = δ m, − n , η µν , [9]30here η µν is the space-time metric, and a µ † m = a µ − m , b µ † r = b µ − r , d µ † n = d µ − n .The Neveu-Schwarz states are created by the action of the oscillators a µm , b µr , on a vacuum state | (cid:105) that is annihilated by the operators a µm , m > , and b µr , r > . The Fock space created in this wayhas a natural scalar product which is not positive because the space-time metric η µν is not. Theconsistency of a dual model, such as the original Veneziano model or the Ramond-Neveu-Schwarzmodel, requires that the physical states that couple in amplitudes belong to a positive definitesubspace defined by an infinite set of gauge conditions, i.e. that there are no ‘ghost’ physicalstates.The first step in constructing a complete dual theory of fermions and bosons was to rewrite theamplitudes of Neveu and Schwarz in a dual form corresponding to a boson annihilating into afermion pair. In essence, this required the construction of a ‘fermion emission vertex’, an operatordescribing the process by which a fermion changes into a boson by the emission of a fermion.Such a construction had been found by Thorn (1971) and by Schwarz (1971). Corrigan and Olive(28) gave this a more elegant and manageable formulation, making the action of the vertex on thegauge conditions more transparent. A prime objective was the calculation of the amplitude forfermion-fermion scattering by combining a fermion emission vertex and its conjugate.The defining properties of this vertex operator, W χ ( z ), as formulated by Corrigan and Olive, arethat it maps boson (or meson) states into fermion states in such a way that it intertwines betweena Neveu-Schwarz field, H µ ( y ) = (cid:80) b µr y − r and a Ramond field, Γ µ ( y ) = (cid:80) d µn y − n , according to anequation of the form, W χ ( z ) H µ ( y ) √ y = λ Γ µ ( y − z ) √ y − z W χ ( z ) , [10]where λ is a suitable constant, and that it creates the fermion state, | χ (cid:105) , from the boson vacuum, | (cid:105) : W χ ( z ) | (cid:105) = e zL − | χ (cid:105) .Whereas the vertices describing the emission of bosons essentially commute with the gauge condi-tions that eliminate ghost states, the behaviour of the fermion emission vertex is more complicated.It converts an infinite linear combination of gauge conditions in the boson sector into an infinitecombination in the fermion sector. It did not immediately follow that ghost states would notcouple in the four-point fermion amplitude.Working with Lars Brink, David saw that it was necessary to introduce a projection operatoronto the space of physical states to secure this (30). As a preparation, Brink and Olive looked atthe algebraically related problem of calculating the one loop contribution in the original bosonictheory, giving a precise derivation (31) of the results earlier proposed heuristically by Lovelace(1971), that had first suggested that the space-time in bosonic string theory needed to be 26 forconsistency. Next, with Claudio Rebbi and Jo¨el Scherk, they worked out exactly how the gaugeconditions related to the fermion emission vertex, under the requirement that the lowest fermionmass m = 0 and the dimension of space-time D = 10, and found that the calculation of thefour-point fermion amplitude was surprising similar to the one loop calculation.31sing these results, Olive and Scherk (35) were able to calculate that the effect of projectingonto the physical states satisfying the gauge conditions was to introduce a factor 1 / ∆( x ) intothe integral expression for the four-point fermion amplitude. The function ∆( x ) was defined asan infinite determinant which was not readily evaluated. After John Schwarz and Cheng-ChinWu (1974) using a computer calculation to propose ∆( x ) = (1 − x ) , a result that was provedanalytically soon after by Corrigan, Goddard, Olive and Russell Smith (36). Meanwhile, StanleyMandelstam (1973) had obtained the same result very efficiently using his mastery of the light-coneformulation of string theory.The conclusion of the calculation of the four fermion amplitude not only in the s channel, where,by construction, the poles correspond to the propagation of physical states, but also in the t channel, where the poles correspond to the same spectrum. Before the calculation, it was notcertain not that the amplitude would be meromorphic: conceivably it could have had cuts in the t channel. There was however a subtle discrepancy, which proved to be an unappreciated clue toa deeper structure: the parity of the boson states was changed between the s and t channels, sothat if the lowest mass state in the s channel, a tachyon, is a pseudo-scalar, the lowest mass statein the t channel is a scalar tachyon.The structure of the four fermion amplitude gave much encouragement that a consistent Ramond-Neveu-Schwarz model could be constructed, but the parity doubling was puzzling. By 1974,it had been realized by various people that it was possible to make a projection, consistent withinteractions, that would remove something like half the states including the tachyon in the bosonicsector, and leave a massless chiral fermion as the lowest state in the fermionic sector. In 1974, thefocus of David’s interest shifted towards the study of monopoles in gauge theories, but in 1976 hewas drawn back to dual fermion theories for what proved to be a very seminal interlude.In the summer of 1975, David began thinking about the behaviour of spinors in different dimen-sions. A spinor in even dimension D of space-time in general has 2 D/ components. This canbe halved by imposing either of two conditions; that the spinor be Weyl (chiral) or that it beMajorana (real). These conditions are incompatible unless D = 2 (mod 8). David was struckby the fact that D = 10, the dimension appropriate for the Ramond-Neveu-Schwarz model, isthe smallest non-trivial dimension in which there are Majorana-Weyl spinors. Such spinors have2 / − = 8 components.In 1976, Jo¨el Scherk, who had returned to the ´Ecole Normale Sup´erieure in Paris, and FernandoGliozzi, who was visiting, had begun studying the multiplicity of the states at various mass levelsin the Ramond-Neveu-Schwarz model. They noticed, first by manual calculation, there was aremarkable coincidence between the number of states at a given mass level in the bosonic sectorand the same level in the fermionic sector, provided that they could impose both Weyl andMajorana conditions. Such a projection would also eliminate the problem with parity doublingthat had surfaced in the four fermion amplitude.32oon they discovered that there was an identity in the literature, proved by Jacobi in 1829,12 w ∞ (cid:89) m =1 (cid:32) w m − − w m (cid:33) − ∞ (cid:89) m =1 (cid:32) − w m − − w m (cid:33) = 8 ∞ (cid:89) m =1 (cid:18) w m − w m (cid:19) , [11]that guarantees equal numbers of bosons and fermions at each mass level. The coefficient of w n onthe left hand side of this relation gives the number of bosonic states at the n -th mass level (above m = 0) and the right hand side does the same for the fermionic sector, provided that the spinorsare subject to two conditions. From the beginning, two-dimensional supersymmetry, on the worldsheet of the string, had been at the heart of the structure of the Ramond-Neveu-Schwarz model,but the equality of the numbers of bosons at each mass level strongly suggested that the theorywas also supersymmetric in space-time, and this was subsequently shown to be the case.When Jo¨el Scherk visited CERN, to give a talk on developments in supergravity, he told Davidof these results and David explained to Jo¨el that the space-time dimension being 10 plays anessential role in being able to impose both Majorana and Weyl conditions, necessary for the factor8 in [11]. Because of the importance of this insight, Jo¨el invited David to join the collaboration,and the projection has become known as the Gliozzi-Scherk-Olive, or GSO, projection (44,45),and the theory it defined is what has become known as superstring theory. For the first time,there was a perturbatively unitary, tachyon free, space-time supersymmetric dual theory. Monopoles and Duality
In one of the most breathtakingly original contributions he made in the early years of QuantumMechanics, Paul Dirac (1931) showed that the consistency of quantum mechanics for a systeminvolving both electric charges, q , and magnetic charges, g , requires that each pair of such chargessatisfy qg π (cid:126) = n , n an integer . [12]This relation implies that the striking conclusion that the existence of a single magnetic chargerequires the existence of a smallest electric charge, q , with any other electric charge being anintegral multiple of q , and, similarly, that there is a smallest magnetic charge, g , of which anyother magnetic charge must be an integral multiple.Dirac was discussing point charges, singularities in the electromagnetic field, but, the mid 1970s,in nearly simultaneous papers, ’t Hooft (1974) and Polyakov (1974) showed that extended objectswith magnetic charge can occur as smooth classical solutions in certain familiar gauge field theories.They considered an SO(3) gauge theory with an adjoint representation ( i.e. triplet real) scalarfield φ , whose self-interactions are described by the Higgs potential, V ( φ ) = 14 λ ( φ − a ) . [13]33n the lowest energy state φ = φ , constant, a value on the surface M of minima of V , which isthe sphere φ = 1. All such choices of φ are gauge-equivalent and each choice corresponds to aspontaneous breaking of the gauge group down to U(1), which can be identified with the gaugegroup of electromagnetism.’t Hooft and Polyakov showed that the classical field equations of this SU(2) Higgs model havea smooth spherically symmetric solution under simultaneous rotations in ordinary space and thethree-dimensional space in which the Higgs field φ lives. This implies making a particular identifi-cation between directions in these spaces; with this, φ is everywhere radial, vanishing at the origin.Asymptotically, φ ( r ) ∼ a ˆ r , and its U(1) little group is identified with the electromagnetic gaugegroup. The corresponding component of the gauge field strength gives the electromagnetic field,which, for the ’tHooft-Polyakov solution is a purely magnetic field, B ∼ (1 /er )ˆ r asymptotically,where e is the gauge coupling constant; this is the field of a magnetic monopole of strength, g = 4 πe . [14]In the quantized SU(2) gauge theory, the electric charges q are multiples of (cid:126) e and so such chargessatisfy the Dirac quantization condition [12] with the magnetic charge g given by [14].To understand the ’t Hooft-Polyakov monopole more deeply, David began to investigate whethersimilar solutions in gauge theories might provide theories of particles, such as hadrons, workingwith Edward Corrigan, David Fairlie and Jean Nuyts (40). They sought generalizations of it bylooking for solutions to an SU(3) Higgs model that were spherically symmetric in an appropriatesense. In the case of SU(3), for a generic quartic (and so renormalizable) Higgs potential, V , thegauge group acts transitively on the vacuum manifold, M , of minima of V , and the unbrokensymmetry group, the little group of the Higgs field, φ , is SU(2) × U(1) / Z , and we can againidentify the U(1) factor with electromagnetism.The mapping between ordinary space and the space of the Higgs field, in the case of the ’t Hooft-Polyakov monopole, determines an isomorphism between SU(2), the covering group of the group ofspatial rotations SO(3), and the SU(2) gauge group. For other gauge groups, such as SU(3), givena homomorphism of SU(2) into the gauge group, spherical symmetry can be defined as invarianceunder the simultaneous applications of rotations and the corresponding gauge transformationsunder the homomorphism. If t i , i = 1 , , , are the images of a standard basis for the generators ofSU(2) in the Lie algebra of the gauge group, spherical symmetry corresponds to invariance undertransformations generated by − i r ∧ ∇ + t .David and his collaborators noted that there are two essentially distinct homomorphisms of SU(2)into SU(3), that is ones not related by SU(3) gauge transformation: (i) one in which the image isSU(2); and (ii) another in which the image is SO(3). They found spherically symmetric solutions,possessing U(1) magnetic charge, for each of these cases, but, while in case (ii) the Dirac condition[12] was satisfied, for case (i) there were solutions for which the condition had to be relaxed to34llow n to be half-integral. These solutions evaded the original Dirac condition by having a longrange magnetic type SU(2) gauge field, not part of the context of Dirac’s original argument.In his next contribution, with Edward Corrigan (42), David derived a generalization of the Diraccondition which encompassed such cases. They considered a theory with gauge group G sponta-neously broken, e.g. by an adjoint representation Higgs field, to a subgroup H = K × U(1) /Z ,where Z = K ∩ U(1) is discrete; K can be identified with a ‘colour’ gauge group and U(1) withelectromagnetism. They showed that, for any smooth solution, the magnetic charge g associatedwith the U(1) factor satisfies the generalized Dirac condition,exp( igQ ) ∈ K , [15]where Q is the electric charge operator, which generates the U(1). If k , the element of K definedby [15], equals 1, then this condition is equivalent to the original Dirac condition [12] and this isthe case for G = SU(2), where K is trivial. For G = SU(3), in case (ii) k = 1, but in case (i) k (cid:54) = 1, which is why the condition [12] can be violated in this case. Corrigan and Olive noted that,if k (cid:54) = 1, the generalized condition [15] implies that [12] holds for the electric charges of coloursinglet particles, establishing a link between colour and fractional electric charges.For spherically symmetric monopole solutions, David showed (43) that the image, in the Liealgebra of G , of the generator of rotations about the radial direction ˆ r · t = − gQ + κ , where κ is a generator of K . The condition [14] follows again from this relation because the eigenvaluesof ˆ r · t are half integral. Spherical symmetry implies that ˆ r · t is a generator of the little group, H , of φ ( r ) and so h ( s ) = exp(4 πis ˆ r · t ) , ≤ s ≤ , defines a closed loop in H and an element ofthe homotopy group, Π ( H ), which David show equalled the topological invariant or ‘topologicalquantum number’ (although it is a classical concept), which had earlier been associated with thesolution (Tyupkin et al. φ ∞ (ˆ r ) = lim r →∞ φ ( r ˆ r ), defines a map S → M ∼ = G/H , and so an element of thehomotopy group Π ( G/H ), which is isomorphic to Π ( H ) (assuming G to be simply connected).David showed that the element of Π ( H ) associated with the solution in this way corresponds tothe loop h .Seeking to deepen our understanding of magnetic monopoles in gauge theories, working with PeterGoddard and Jean Nuyts (GNO), David next investigated (46) the general case of a theory witha compact gauge group G , spontaneously broken to an exact gauge group H ⊂ G by a Higgs field, φ , in an arbitrary representation. They considered monopole solutions, that is solutions for whichthe spatial components of the gauge field strength G αβ , expressed as a generator of H , the littlegroup of φ ∞ (ˆ r ), the asymptotic value of φ ( r ) in the radial direction, G ij ( r ) ∼ (cid:15) ijk r k r G (ˆ r ) , as r → ∞ , ≤ i, j, k ≤ . [16]35eneralizing [12] and [15], they established thatexp( ie G ) = 1 , [17]where G = G (ˆ r ), and that the loop h ( s ) = exp( ise G ) , ≤ s ≤ , determines the topologicalinvariant associated with the solution as an element of Π ( H ). Taking a maximal set of commutinggenerators of H , T a , ≤ a ≤ m, where m is the rank of H , i.e. a basis for a Cartan subalgebraof H , G will be equivalent, under an H gauge transformation, to a linear combination, G = g a T a ,of the { T a } . G is the generalized magnetic charge of the solution, and GNO called g = ( g a ) itsmagnetic weights.The simultaneous eigenvalues q = ( q a ) of (cid:126) eT a , ≤ a ≤ m, in representations of H , are the gener-alized electric charges of quanta fields, transforming under those representations. The quantizationcondition [17] can be re-expressed in the form q · g π (cid:126) = n , n an integer , [18]more closely paralleling [12]. The possible values of q are points of the lattice (cid:126) e Λ H , where Λ H is the weight lattice of the group H , implying that g ∈ (2 π/e )Λ H ∗ , where Λ H ∗ is the lattice dualto Λ H . The weight lattice, Λ (cid:101) H , of weights of the simply connected group, (cid:101) H , with the same Liealgebra as H , consists of those λ for which 2 α · λ / α is an integer for every root α of H . Thiscondition is satisfied by the roots themselves, and thus Λ R ⊂ Λ H ⊂ Λ (cid:101) H , where Λ R is the rootlattice of H , the lattice generated by its roots, α . If H is a simple group all of whose roots havethe same length ( i.e. H is simply-laced), we can normalize so that α = 2 for each root, and thenΛ R ⊂ Λ H ∗ ⊂ Λ (cid:101) H . Thus Λ H ∗ is the weight lattice of some group, H ∨ say, called the dual of H ,with the same Lie algebra as H but a different global structure in general, e.g. SU(2) ∨ = SO(3)and conversely. The relationship between a group and its dual is reflexive: ( H ∨ ) ∨ = H .If H is not simply-laced, Λ H ∗ is still the weight lattice of some group H ∨ , the dual of H , butnow it is a group with roots α ∨ = 2 α / α , where α is a root of H , and its algebra may not beisomorphic to that of H . The simple Lie algebras that are not simply-laced are the series B n andC n , corresponding to, e.g. , the groups SO(2 n + 1) and Sp( n ), respectively, and the exceptionalalgebras G and F . For the groups with Lie algebras G and F , the dual groups have the sameLie algebra, but different global structure. However, the dual of a group with Lie algebra B n isone with Lie algebra C n , and vice versa .Thus, GNO showed that, while the generalized electric charges are, up to a factor, the weightsof the exact symmetry group, H , the generalized magnetic charges of the extended monopolesolutions are, up to a factor, weights of the dual group H ∨ . They suggested that these solutionsshould form multiplets of H ∨ and that this group should be an exact symmetry group of the theoryas well as H . Further, they speculated that the the dual relationship between the generalizedmagnetic charges, g , associated with the topological characteristics of the monopole solution andthe group H ∨ , on the one hand, and the generalized electric charges, q , associated with the quanta36f the basic fields of the theory and the group H , on the other, was analogous to that betweenthe solitons and the quanta of the basic boson field in the two-dimensional sine-Gordon theory.GNO noted that the sine-Gordon theory had been shown to be equivalent to the massive Thirringmodel at the quantum level, with the solitons, which are extended solutions in the former bosontheory, corresponding to the quanta of the basic fermion field in the Thirring model (Coleman1975, Mandelstam 1975), ideas that go back to the work of T. H. R. Skyrme (Skyrme 1961, andreferences therein). In this correspondence, the topologically conserved soliton number in thesine-Gordon theory corresponds to a conserved Noether charge associated with a U(1) symmetryof the Thirring model. But this exact equivalence of different theories is intrinsically quantummechanical and so any such equivalence between dual theories, interchanging electric and magneticcharges, is beyond the basically classical analysis of GNO (46). However, they speculated that themonopoles of the original theory might correspond to particles in a fundamental representation of H ∨ in a dual formulation; in his next investigations, David was led instead to a more elegant andprofound conjecture.David was determined to explore further what he acknowledged were highly speculative proposi-tions, that H monopoles behave as irreducible representations of H ∨ , and that the theory has a H ∨ gauge symmetry. Working with Claus Montonen (47), he looked for evidence for the conjecturesin the simplest monopole theory, that studied by ’t Hooft and Polyakov, in which the gauge group G = SO(3) is spontaneously broken to H = SO(2) ∼ = U(1) by the choice of a particular vacuumvalue for φ minimizing the Higgs potential [13]. The mass, M M , of the monopole solutions in thistheory is of the form, M M = 4 πae f ( λ/e ) , [19]where f ( λ/e ) → λ → λ = 0, in which the potential vanishes but the asymptotic condition φ → a is maintained at spatial infinity. Then, from [14], M M = ag , which is of exactly thesame form as the formula for the massive of the massive spin 1 particles, W ± , generated bythe Higgs mechanism, namely M W = ae (cid:126) = aq . Bogomolnyi (1976) showed that, for general λ , M M ≥ ag , so that the monopole mass reaches its lower bound at λ = 0, which is now known asthe Bogomolnyi-Prasad-Sommerfield (BPS) limit.Montonen and Olive conjectured that there should be dual equivalent versions of this theory,described by Lagrangians of the same form. Under this duality correspondence, electric and mag-netic properties interchange, the massive ‘gauge’ particles, W ± , exchange rˆoles with monopolesolutions, and topological and Noether quantum numbers are interchanged. In addition to thesymmetry between the mass formulae M W = aq and M M = ag , they noted that further circum-stantial evidence for the duality conjecture was provided by Manton’s calculation of the classicalmagnetic force between monopoles as g / πr , symmetric with the electric case (Manton 1977).Because [14] implies that as e becomes small, g becomes large, this Montonen-Olive duality pro-vides a (conjectured) equivalence between theories at strong and weak coupling. It was the first37xample of what became known as S-duality, which make a central concept in quantum field theoryand string theory from the mid 1990s onwards.Notwithstanding the suggestive evidence in favour of the extremely bold Montonen-Olive conjec-ture, there were significant potential obstacles to its validity: Why should the symmetry betweenthe formulae for the masses of monopoles and massive gauge particles survive renormalization?Why should the monopole have spin 1 quantum mechanically? Answers to these two questionscame quickly. With Edward Witten, David showed that in suitable N = 2 supersymmetric theo-ries, the presence of monopole solutions results in the existence of central terms that modify thesupersymmetry algebra, with the mass formulae following as a consequence of the algebra (50).Then Hugh Osborn (1979) observed that in N = 4 supersymmetric gauge theory, with sponta-neous symmetry breaking, the monopole states would have spin 1 like the massive gauge particles,making it the most suitable theory for realizing the Montonen-Olive duality conjecture. Fifteenyears later, Nathan Seiberg and Edward Witten (1994a,b) found a form of S-duality that workedin N = 2 supersymmetric theories.David continued to study the characteristics of monopole solutions to gauge theories, motivatedin large part by his duality conjecture. With Peter Goddard, he studied theories in a gauge group G is broken by an adjoint representation Higgs field to an exact symmetry gauge group, H , whosestructure is locally of the form U(1) × K (60). They found that a necessary and sufficient conditionfor K to be semi-simple is that the vacuum value of the Higgs field be on the same orbit of G as a fundamental weight, and they gave prescriptions for determining K , and for calculating thebasic units of electric and magnetic charge, from the Dynkin diagram of G . They also showedthat the global structure of H = (U(1) × K ) /Z , where Z is a cyclic group whose order can alsobe calculated from the Dynkin diagram.Next (61), they considered such theories in the BPS limit, analysing the charges of potentiallystable magnetic monopoles. They showed that they have a structure consistent with their inter-pretation as heavy gauge particles of an overall symmetry group, G ∨ , dual to G , providing furthercircumstantial evidence for the some form of Montonen-Olive duality. Toda Theories
David Olive continued his studies of solutions to gauge theories with symmetry broken by aHiggs field in the adjoint representation, working with Nikos Ganoulis and Peter Goddard (65),investigating static stable solutions to the field equations in the BPS limit that are sphericallysymmetric in a suitable sense. Andrej Leznov and Mikhail Saveliev (1980) had shown that theycould be constructed from solutions to a spatial version of the so-called Toda molecule equations, d θ i dr = exp (cid:32) R (cid:88) j =1 K ij θ j (cid:33) , [20]38here K ij = 2 α i · α j / α j is the Cartan matrix of G , α i , ≤ i ≤ R , being a basis of simple roots.The equations [20] are integrable and Ganoulis, Goddard and Olive showed how to determine theconstants of integration so as to ensure regularity at the origin, thus obtaining solutions for anysimple group G in terms of its root system. This work marked the beginning of David’s fascinationwith Toda systems, which he kept coming back to, resulting in a series of contributions over thefollowing fifteen years.Toda’s work (1967) was motivated by the numerical studies of Fermi, Pasta and Ulam (1955)of the behaviour of a system of particles joined by springs exerting forces nonlinear in theirextensions. Toda made the important observation that if the polynomial forces considered byFermi et al. were replaced by exponential ones, a one-dimensional lattice comprising an infinitenumber of such particles could be solved analytically instead of numerically. It was found thatthis integrability extended to finite systems of particles, labelled by the simple roots of finite-dimensional Lie algebras, as in [20], replacing d θ i /dr by − d θ i /dt (Kostant, 1979), and also tocorresponding systems of nonlinear relativistic wave equations, i.e. Toda field theories, replacing d θ i /dr by d θ i /dr − d θ i /dt in [20]. There are also infinite lattice systems similarly related tothe simple root systems of Kac-Moody algebras (Mikhailov, Olshanetsky and Perelomov 1981).With Neil Turok (70), David showed how the symmetries of the Dynkin diagram of a simple Liealgebra led to new integrable systems of equations, which are reductions of the Toda moleculeequations associated with the algebra, obtained by identifying variables related by the symmetry.Then, turning to the Toda lattice field theories associated with Kac-Moody algebras, termed affineToda field theory, Olive and Turok systematically constructed an infinite number of commutinglocal conserved quantities, which could be used as Hamiltonians to define evolution in associatedtimes (75, 86).David returned to the study of Toda theories in the 1990s, following work by others on exactS-matrices for two-dimensional integrable quantum field theories, starting with the sine-Gordonquantum field theory, which is actually the simplest of the Toda field theories, being based onthe group SU (2). A key feature of the sine-Gordon theory is that classically it possesses solitonsolutions, including soliton-anti-soliton bound states, known as breather solutions. These persistin the quantum field theory, becoming a finite discrete set of states whose number depends onthe coupling constant. David’s approach was to seek to use general Lie algebraic techniques,which he was familiar with from his work on magnetic monopoles and on conformal field theory,to provide unified and deeper understandings of mass spectra and couplings in affine Toda fieldtheories (100, 101), in particular the rule describing three-point couplings discovered by PatrickDorey (1991). Given David’s background, it was natural for him to apply these insights to theconjectured S -matrix of the quantum field theory.Typically the soliton solutions are complex rather than real, and so might seem uninterestingphysically. However, with Turok and David’s student, Jonathan Underwood (102, 103), he showedthat the solutions have real energy and momentum, equal to the sum of contributions of theindividual solitons. They further showed that the number of species of soliton equalled the rank39f the Lie group with which it was associated and their masses were given in terms of its algebraicstructure, and used a vertex operator formalism for constructing soliton solutions (105). WithSaveliev and Underwood (104), and also with his students Andreas Fring, Peter Johnson, andMarco Kneipp (108, 110), David used vertex operators to study solitons and their scatteringfurther, calculating time delays and showing that the process should be interpreted as transmissionslowed down by attractive forces rather than reflection. No doubt one of the attractions of thestudy of affine Toda field theories for David was that aspects of them bore some resemblanceto Yang-Mills-Higgs theories in four dimensions, with their monopole solutions analogous to thesolitons of the Toda theories, and the mass spectrum of the solitons being equal to the massspectrum of the dual theory based on the dual affine Lie algebra, but with the inverse couplingconstant, recalling Montonen-Olive duality in gauge theories. Algebras, Lattices and Strings
Having become aware of the profound connections that had been discovered between the vertexoperators developed in dual models and string theory, on the one hand, and representations ofKac-Moody algebras, on the other (Frenkel & Kac 1980, Segal 1981), about 1980 David began,with Peter Goddard, to explore the possible role of these algebras in physics, and in string theoryin particular. A further motivation was a proof by Graeme Segal (1980) of the Jacobi identityusing a fermion-boson equivalence, the isomorphism between the space of states for a single bosonfield defined on a circle and that for a pair of fermion fields. The role of integral lattices ( i.e. onessuch that the scalar product between any two points is an integer) in these constructions providedlinks with concepts David had introduced in the theory of magnetic monopoles.In Charlottesville in 1983, Goddard and Olive wrote an account synthesizing some of what theyhad learned and found out, under the title,
Algebras, Lattices and Strings (72). They used dualmodel vertex operators to provide a unified construction of finite dimensional Lie algebras, affineKac-Moody algebras, Lorentzian algebras and fermionic extensions of these algebras. Given anintegral lattice, Λ, they associated to each point, r ∈ Λ, of squared length 2, the operator e r = c r A r ,where A r is the contour integral about the origin of the dual model (or string) vertex operator, V ( r, z ), for emitting a ‘tachyon’, A r = 12 πi (cid:73) V ( r, z ) dzz , [21]and c r is a function of momentum such that c r c s = ( − r · s c s c r . They considered the Lie algebra, g Λ , generated by these e r under commutation. If Λ contains points of squared length 1, g Λ maybe extended into a superalgebra by including operators associated with those points, which quitenaturally satisfy anti-commutation relations.If the scalar product on Λ is positive definite, g Λ is a compact finite-dimensional Lie algebra, witha basis comprising the A r , with r ∈ Λ and r = 2 , together with the momenta. If the scalar40roduct is positive semi-definite, g Λ is an affine Kac-Moody algebra, which includes the contourintegrals of vertex operators for emitting ‘photons’, associated with points k ∈ Λ with k = 0 , aswell as momenta. If the scalar product is indefinite, g Λ involves the vertex operators associatedwith other dual model states.In the context of string theory, if Λ were a lattice of momenta of physical states, discrete becausethe corresponding spatial dimensions were compactified, g Λ would map the physical state spaceinto itself, i.e. the physical states would fall into representations of g Λ . Motivated in a general wayby the ideas of electromagnetic duality that he had pioneered in gauge theories, where generalizedelectric and magnetic charges correspond to points of dual lattices, David formed the intuitionthat the cases where Λ is self-dual might be particularly interesting. Further, to concentrate onthe bosonic case, Goddard and Olive considered the case where Λ is even, i.e. all the squaredlengths of all its points are even, as well as self-dual.The requirement of Λ being both even and self-dual is quite restrictive; for Λ ⊂ R m,n , it isnecessary that m − n be a multiple of 8. Thus, in the Euclidean case, the smallest possibilitiesfor the dimension of Λ are 8 , ,
24, and these are the only dimensions relevant to bosonic stringtheory, for which the space-time dimension is bounded by 26. Goddard and Olive noted thatthe only such lattice with dim Λ = 8 is the root lattice of E , for dim Λ = 16 there are twopossibilities: the root lattice of E × E and a sublattice of the weight lattice of the Lie algebraof SO(32) (corresponding to the weight lattice of the group Spin(32) / Z ), while for dim Λ = 24there are 24 possibilities.Applied to the quantum motion of a relativistic string, moving in a space some of whose dimensionsare compactified to form a torus by identifying points of space related by displacements formingan even self-dual lattice (the length scale being set by the characteristic length of the string), g Λ generates a gauge symmetry. For a closed string, the symmetry would be g Λ ⊕ g Λ . To avoid this,Gross, Harvey, Martinec and Rohm (1985) found it was necessary to treat left and right movingwaves on the closed string differently, so that, somewhat bizarrely, only those moving in onedirection had motion in these compactified directions. The model they constructed, the heteroticstring, enabled them to realize in string theory either of the gauge groups, E × E or Spin(32) / Z ,for which Green and Schwarz (1984) had found anomaly cancellations for supersymmetric gaugetheories coupled to gravity.Rather than pursue these applications of Kac-Moody algebras, David returned to investigationsof the equivalence of descriptions of two-dimensional quantum field theories in terms of eitherboson fields or fermion fields. Edward Witten (1984) had extended the known results on theequivalence in two dimensions between a single boson field and a pair of fermion fields to anequivalence between certain nonlinear boson theories and certain free fermion theories. Specifically,he demonstrated an equivalence between a nonlinear σ model, with a field g taking values in thegroup SO( N ), and a theory with N free fermion fields ψ i , ≤ i ≤ N. To obtain suitable equations of motion for g , Witten needed to include a Wess-Zumino termin the σ -model Lagrangian, defining what is now known as the Wess-Zumino-Witten (WZW)41heory, so that these then become ∂ + ( g − ∂ − g ) = ∂ − ( ∂ + gg − ) = 0, where ∂ ± denote the partialderivatives with respect to x ± = x ± t , implying that ∂ + gg − and g − ∂ − g depend only on x + and x − , respectively. Imposing periodic boundary conditions, identifying x with x + L , and expanding ∂ + gg − in powers of z = e πix + /L and a basis, { t a } , for the Lie algebra of SO( N ), the canonicalcommutation relations imply that, with suitable normalizations, its components, T an , satisfy theKac-Moody algebra, [ T am , T bn ] = if abc T cm + n + kmδ ab δ m, − n , [22]where k is an integer determined by the coefficient of the Wess-Zumino term, and f abc are structureconstants for SO( N ). Witten observed that a representation same algebra was provided by T an → i (cid:88) r ψ ir M aij ψ jn − r [23]where the M a are the representation matrices for the defining representation of the Lie algebraof SO( N ) and ψ ir are the modes of the free fermion fields labelled by half odd integers r (forconvenience taken to be odd under x → x + L ). The Kac-Moody algebra [22] with k = 1 followsfrom the canonical anti-commutation relations, { ψ ir , ψ js } = δ r, − s δ ij . Witten observed that [22]has only a few positive energy unitary representations for k = 1, so that the isomorphism of theKac-Moody algebras in the two theories effectively guarantees their equivalence in this case.The requirements of positive energy, i.e. that the spectrum of L be non-negative, and unitaryrequire k in [22] to be a positive integer. (In the case of the affine Kac-Moody algebra ˆ g associatedwith a general compact simple Lie algebra g , the condition is that x = 2 k/ϕ should be a positiveinteger, called the level of the representation.) For SO( N ) WZW theories with k >
1, Wittenproposed taking k copies of the N fermion theory, but, as David surmised might be the case, theequivalence is then no longer exact: there is more in the fermion theory than that WZW model.David thought that exact equivalence would entail not only the isomorphism of the Kac-Moodyalgebras possessed by the two theories, but also this should extend to an equivalence of the energy-momentum tensors in the boson and fermion theories. Working in the general context of the WZWtheory for a simple Lie group, G , with an N -dimensional real representation, M in [23], Goddardand Olive (74) sought to determine whether the energy-momentum tensors of the WZW theoryand the N fermion theory were necessarily identical, and, if not, what conditions would ensurethat they were equal.The conformal symmetry of each of these theories means that the modes of the energy momentumtensor satisfy the Virasoro algebra,[ L m , L n ] = ( m − n ) L m + n + c m ( m − δ m, − n , [24]with L n = L g n , c = c g , where c g = 2 k dim g k + Q g = x dim gx + h g , [25]42or the WZW theory, and L n = L ψn , c = c ψ = N for the fermion theory, where L g n = 12 k + Q ϕ (cid:88) m oo T a − m T an + m oo , L ψn = 12 (cid:88) r : rb j − r b jn + r : . [26][Here oo and : denote normal ordering with respect to the modes T am and b ir , respectively, and Q g is the quadratic Casimir operator for the adjoint representation of g , and h g = Q g /ϕ , the dualCoxeter of g .] For the affine algebra ˆ g to be the same for the two theories, we need k = κ , where M aij M bij = κ δ ab .If the two energy-momentum tensors are equal, it is necessary that c g = c ψ . Goddard and Olivefound that this is typically not the case, and, further, that the difference K n = L ψn − L g n itselfsatisfies a Virasoro algebra, commuting with L g m , i.e. [24] holds for L n = K n and c = c ψ − c g .The requirement that the Virasoro algebra representations provided by L ψn and L g n are unitarityand positive energy implies that K n has the same property and so that c g ≤ c ψ . If c g = c ψ , K n vanishes because the Virasoro algebra does not have any non-vanishing positive energy unitaryrepresentations with c = 0. Thus c g = c ψ provides a condition for the identity of the energy-momentum tensors and the equivalence of the theories that is both necessary and sufficient.The cases where c K = c ψ − c g > c and the lowesteigenvalue of L , h , say. Some months before Goddard and Olive had begun their study of WZWmodels, Friedan, Qiu and Shenker (FQS) had shown that, for such representations, either c ≥ c belongs to an infinite discrete sequence, c = 1 − m + 2)( m + 3) , m = 0 , , . . . , [27]for each value of which there are ( m + 1)( m + 2) possible values of h (Friedan et al. 1984).At that time, it was known whether any of these discrete series representations exist, apart from c = 0, the trivial representation, and c = , which corresponds to a single fermion field; the rest,with c = , , , . . . , were unknown. In most cases either c K = 0 or c K ≥
1, but Goddard andOlive found that taking M a to be the 7-dimensional representation of G = SO(3) gave the c K = representation of the Virasoro algebra, with all the associated 6 values of h .It turns out that the other values of c in the discrete series [27], those with m ≥
4, can not beobtained in this way and a more general approach is needed. This was found by Goddard, Kentand Olive (GKO) who changed the context from the free fermion representation of the affine al-gebra associated with so( N ) to that a positive energy unitary representation of the affine algebra,ˆ g , associated with compact simple Lie algebra, g (cid:48) , and a subalgebra g ⊂ g (cid:48) . The representation ofˆ g (cid:48) provides a representation of ˆ g , whose level x is determined by the level y of the representationof ˆ g (cid:48) . (We obtain the previous construction by taking g (cid:48) = so( n ).) We may define a Virasoro rep-resentation L g (cid:48) n associated to the representation of ˆ g (cid:48) in the same way the Virasoro representation L g n is associated to the representation of ˆ g by [26]. Now K n = L g (cid:48) n − L g n commutes with the whole43f ˆ g and so with L g n . It follows that K n provides a representation of the Virasoro algebra withcentral charge, c K = c g (cid:48) − c g = y dim g (cid:48) y + h g (cid:48) − x dim gx + h g . [28]This construction, which has become known as the coset (or GKO) construction (76), extendsstraightforwardly to cases where g (cid:48) or g are not simple (in which cases the various the level isdefined for each summand). All of the values in the series [27], and all the associated values of h ,can be obtained in this way, e.g. by taking g (cid:48) = sp( m + 1) , x = 1 , g = sp( m ) ⊕ sp(1), for both ofwhich the level is also 1 (81).Given a conformal field theory (CFT) associated with the affine Kac-Moody algebra ˆ g (cid:48) , and g ⊂ g (cid:48) ,the coset construction defines a CFT associated with g (cid:48) / g , on the subspace of states annihilatedby the positive modes of ˆ g , with Virasoro algebra K n . If c K = 0, this is trivial, and the CFTassociated with ˆ g is said to be conformally embedded in that associated with ˆ g (cid:48) . The cosetconstruction has provided one if the principal ways of constructing interesting examples of CFTsin applications.David returned to the investigation of the condition under which the energy-momentum tensors ofa WZW theory for the group G , with an N -dimensional real representation, M , and an N fermiontheory are equal, from which the coset construction arose as a by-product, and, with Peter Goddardand Werner Nahm, he found an elegant geometric interpretation, which enabled the classificationof the cases where it is satisfied (78). The condition for the vanishing of K n = L ψn − L g n is M aij M akl + M aik M alj + M ail M ajk = 0 . [29]They observed that this is the condition that g can be extended by an N -dimensional space, s , togive a Lie algebra, g (cid:48) = g ⊕ s , which is a symmetric space, i.e. [ g , g ] ⊂ g , [ g , s ] ⊂ s , [ s , s ] ⊂ g , withthe action of g on s providing the representation M . Using { s i } to denote an orthonormal basisfor s , for g (cid:48) = g ⊕ s to be a symmetric space, the communication relations must have the form,[ t a , t b ] = if abc t c , [ t a , s i ] = iM aij s j , [ s i , s j ] = iM aij t a . [30]Given that g is a Lie algebra with representation M the condition for these to be consistent isthe Jacobi identity for s i , s j , s k , which is just [29]. Symmetric spaces have been classified and soprovide a list of the instances in which L g n = L ψn . Acknowledgements
In addition to his account of his work on string theory (44), David Olive left detailed notes on his earlylife and much of his scientific career, which have been immensely valuable to us. We are very gratefulto many of David Olive’s friends and colleagues for much information and helpful correspondence and,most particularly, to David’s widow, Jenny Olive, and younger daughter, Rosalind Shufflebotham, whoalso read carefully drafts of the biography, and provided the frontispiece and other photographs. Jennydied on 1 September 2018. eferences to other authors Bogomol’nyi, E. B. 1976 The stability of classical solutions.
Sov. J. Nucl. Phys. , 389–394.Chew, G. F. 1966 The Analytic S Matrix. New York, 1966: Benjamin.Coleman, S. 1975 Quantum sine-Gordon equation as the massive Thirring model. Phys. Rev.
D11 ,2088–2097.Cutkosky, R. E. 1960 Singularities and discontinuities of Feynman amplitudes.
J. Math. Phys. ,429–433.Dirac, P. A. M. 1931 Quantised singularities in the electromagnetic field. Proc. Roy. Soc.
A33 , 60–72.Dorey, P. 1991 Root systems and purely elastic S -matrices. Nucl. Phys.
B358 , 654–676.Fermi, E., Pasta, J. & Ulam S. 1955 Studies of nonlinear problems. I. Los Alamos report LA-1940,published later in E. Segr´e (ed.) 1965
Collected Papers of Enrico Fermi , University of ChicagoPress.Frenkel, I. B. & Kac, V. G. 1980 Basic representations of affine Lie algebras and dual resonance models.
Invent. Math. , 23–66.Friedan, D., Qiu, Z. & Shenker, S. 1984 Conformal invariance, unitarity, and critical exponents in twodimensions. Phys. Rev. Lett. , 1575–1578.Goddard, P., Goldstone, J., Rebbi, C. & Thorn, C. B. 1973 Quantum dynamics of a massless relativisticstring. Nucl. Phys.
B56 , 109–135.Green, M. B. & Schwarz, J. H. 1984 Anomaly cancellations in supersymmetric D = 10 gauge theory andsuperstring theory. Phys. Lett. , 117–122.Gross, D. J., Harvey, J. A., Martinec, E. & Rohm, R. 1985 Heterotic string theory (I) The free heteroticstring.
Nucl. Phys.
B256 , 253–284.Gunson, J. 1965 Unitarity and on-mass-shell analyticity as a basis for S-matrix theories. I, II, III.
J.Math. Phys. , 827–844; 845–851; 852–858.Iagolnitzer, D. 1981 Analyticity properties of the S-matrix: historical survey and recent results in S-matrixtheory and axiomatic field theory. Acta Phys. Austriaca, Suppl., , 235–328.Kac, V. G. 1967 Simple graded algebras of finite growth. Funct. Anal. Appl. , 328–329.Kostant, B. 1979 The solution to a generalized Toda lattice and representation theory. Advances inMathematics , 195–338.Landau, L. D. 1959 On analytic properties of vertex parts in quantum field theory. Nucl. Phys. ,181–192.Langlands, R. P. 1970 Problems in the theory of automorphic forms In: Lectures in Modern Analysis andApplications III (ed. C. T. Taam)
Springer Lecture Notes in Mathematics , pp. 18–61. eznov, A. N. & Saveliev, M. V. 1980 Representation theory and integration of nonlinear sphericallysymmetric equations to gauge theories. Commun. Math. Phys. , 111–118.Lovelace, C. 1971 Pomeron form factors and dual Regge cuts. Phys. Lett.
Nucl. Phys.
B126 , 525–541.Mandelstam, S. 1959 Analytic properties of transition amplitudes in perturbation theory.
Phys. Rev.
Phys. Rev.
Phys. Rev.
D11 , 3026–3030.Moody, R. V. 1967 Lie algebras associated with generalized Cartan matrices.
Bull. Amer. Math. Soc. , 217–221.Mikhailov, A. V., Olshanetsky, M. A. & Perelomov, A. M. 1981 Two dimensional generalized Toda lattice. Commun. Math. Phys. , 473–488.Monastyrski˘i, M. I. & Perelomov, A. M. 1975 Concerning the existence of monopoles in gauge fieldtheories. ZhETF Pis. Red. , 94–96.Nambu, Y. 1969 Quark model and the factorization of the Veneziano amplitude. In: Proceedings ofthe International Conference on Symmetries and Quark Models held at Wayne State University,June 18–20, 1969 (ed. R. Chand) pp. 258–277. New York: Gordon and Breach.Neveu, A. & Schwarz, J. H. 1971 Quark model of dual pions.
Phys. Rev. D4 , 1109–1111.Nielsen, H. B. 1969 An almost Physical Interpretation of the integrand of the n -point Veneziano model(unpublished).Osborn, H. 1979 Topological charges for N = 4 supersymmetric gauge theories and monopoles of spin 1. Phys. Lett. , 321–326.Polyakov, A. M. 1974 Particle spectrum in quantum field theory.
JETP Lett. , 194–195.Prasad, M. K. & Sommerfield, C. M. 1975 An exact classical solution for the ’t Hooft monopole and theJulia-Zee dyon. Phys. Rev. Lett. , 760–762.Ramond, P. 1971 Dual theory for free fermions. Phys. Rev. D3 , 2415–2418.Salam, A. & Komar, A. 1960 Renormalization problem for vector meson theories. Nucl. Phys. ,624–630.Sato, M. 1975 Recent development in hyperfunction theory and its applications to physics (microlocalanalysis of S-matrices and related quantities) Lect. Notes Math. , 13–29. Berlin: Springer-Verlag.Schwarz, J. H. 1971 Dual quark-gluon model of hadrons.
Phys. Lett. , 315–319.Schwarz, J. H. & Wu, C. C. 1974 Functions occurring in dual fermion amplitudes.
Nucl. Phys.
B73 egal, G. 1980 Jacobi’s identity and an isomorphism between a symmetric algebra and an exterior algebra(unpublished).Segal, G. 1981 Unitary representations of some infinite dimensional groups. Commun. Math. Phys. ,301–342.Seiberg, N. & Witten, E. 1994a Electromagnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys.
B426 , 19–52; Erratum
B430 , 485–486.Seiberg, N. & Witten, E. 1994b Monopoles, duality and chiral symmetry breaking in N = 2 supersym-metric QCD.
Nucl. Phys.
B431 , 484–550.Sen, A 1994. Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole modulispace, and SL(2, Z ) invariance in string theory. Phys. Lett.
B298 , 217–221.Skyrme, T. H. R. 1961 Particle states of a quantized meson field.
Proc. Roy. Soc.
A262 , 237–245.Stapp, H. P. 1962 Derivation of the CPT theorem and the connection between spin and statistics frompostulates of the S-Matrix theory.
Phys. Rev. , 2139–2162.Streater, R. & Wightman, A. 1964 PCT, Spin and Statistics, and All That. Princeton: PrincetonUniversity Press.Susskind, L. 1970 Dual-symmetric theory of hadrons I.
Nuovo Cim. , 457–496.’t Hooft, G. 1974 Magnetic monopoles in unified gauge theories.
Nucl. Phys.
B79 , 276–281.Thorn, C. B. 1971 Embryonic dual model for pions and fermions
Phys. Rev. D4 , 1112–1116.Toda, M. 1967 Vibration of a chain with nonlinear interaction. J. Phys. Soc. Japan , 431–436.Tyupkin, Yu. S., Fateev, V. A. & Shvarts, A. S. 1975 Existence of heavy particles in gauge field theories. ZhETF Pis. Red. , 91–93.Veneziano, G. 1968 Construction of a crossing-symmetric, Regge-behaved amplitude for linearly risingtrajectories. Nuovo Cim. , 190–197.Witten, E. 1984 Non-abelian bosonization in two dimensions.
Commun. Math. Phys. , 455-472. Bibliography (1) 1962 (with J. C. Taylor) Singularities of scattering amplitudes at isolated real points.
NuovoCim. , 814–822.(2) 1962 Unitarity and evaluation of discontinuities. Nuovo Cim. , 73–102.(3) 1963 On analytic continuation of scattering amplitude through a three-particle cut. NuovoCim. , 1318–1336.(4) 1963 Unitarity and evaluation of discontinuities – II. Nuovo Cim. , 326–335.(5) 1964 Exploration of S -matrix theory. Phys. Rev. , B745–B760.(6) 1965 Unitarity and evaluation of discontinuities – III.
Nuovo Cim. , 1423–1445.(7) 1966 (with P. V. Landshoff & J. C. Polkinghorne) Hierarchical principle in perturbation theory. Nuovo Cim.
A43 , 444–453.
8) 1966 (with E. Y. C. Lu) Spin and statistics in S -matrix theory. Nuovo Cim. , 205–218.(9) 1966 (with P. V. Landshoff & J. C. Polkinghorne) Icecream-cone singularity in S -matrix theory. J. Math. Phys. , 1600–1606.(10) 1966 (with P. V. Landshoff & J. C. Polkinghorne) Normal thresholds in subenergy variables J.Math. Phys. , 1593–1599.(11) 1966 (with P. V. Landshoff) Extraction of singularities from S -matrix. J. Math. Phys. ,1464–1477.(12) 1966 (with R. J. Eden, P. V. Landshoff & J. C. Polkinghorne) The Analytic S -Matrix . Cam-bridge, UK: Cambridge University Press.(13) 1968 (with J. C. Polkinghorne) Properties of Mandelstam cuts. Phys. Rev. , 1475–1481.(14) 1969 (with R. E. Cutkosky, P. V. Landshoff & J. C. Polkinghorne) A non-analytic S -matrix. Nucl. Phys.
B12 , 281–300.(15) 1969 (with M. J. W. Bloxham, D. I. Olive & J. C. Polkinghorne) S -matrix singularity structurein the physical region. I. Properties of multiple integrals. J. Math. Phys. , 494–502.(16) 1969 (with M. J. W. Bloxham, D. I. Olive & J. C. Polkinghorne) S -matrix singularity structurein the physical region. II. Unitarity integrals. J. Math. Phys. , 545–552.(17) 1969 (with M. J. W. Bloxham, D. I. Olive & J. C. Polkinghorne) S -matrix singularity structurein the physical region. III. General discussion of simple Landau singularities. J. Math.Phys. , 553–561.(18) 1969 (with D. K. Campbell & W. J. Zakrzewski) Veneziano amplitudes for Reggeons and spin-ning particles. Nucl. Phys.
B14 , 319–329.(19) 1969 (with W. J. Zakrzewski) A Veneziano amplitude for four spinning pions.
Phys. Lett.
B30 ,650–652.(20) 1970 (with W. J. Zakrzewski) A possible Veneziano amplitude for many pions.
Nucl. Phys.
B21 , 303–320.(21) 1970 The connection between Toller and Froissart-Gribov signatured amplitudes.
Nucl. Phys.
B15 , 617–627.(22) 1970 (with V. Alessandrini, D. Amati & M. Le Bellac) Duality and gauge properties of twistedpropagators in multi-Veneziano theory.
Phys. Lett.
B32 , 285–290.(23) 1970 (with D. Amati & M. Le Bellac) Twisting-invariant factorization of multiparticle dualamplitude.
Nuovo Cim.
A66 , 815–830.(24) 1970 (with D. Amati & M. Le Bellac) The twisting operator in multi-Veneziano theory.
NuovoCim.
A66 , 831–844.(25) 1971 Operator vertices and propagators in dual theories.
Nuovo Cim. A3 , 399–411.(26) 1971 (with V. Alessandrini, D. Amati & M. Le Bellac) The operator approach to dual multi-particle theory. Phys. Rept. , 269–345.(27) 1971 The dual approach to the strong interaction S -matrix. In Proceedings of the Symposiumon Basic Questions in Elementary Particle. pp. 140–147. M¨unchen, Germany: Max-Planck-lnstitut f¨ur Physik und Astrophysik.(28) 1971 (with E. Corrigan) Fermion meson vertices in dual theories.
Nuovo Cim.
A11 , 749–773.(29) 1972 Clarification of the rubber string picture. In:
Proceedings of 16th International Conferenceon High-energy Physics, Batavia, IL (ed. J. D. Jackson & A. Roberts), vol. 1, pp. 472-474.Batavia, IL, USA: National Accelerator Laboratory.
30) 1973 (with L. Brink & D. I. Olive) The physical state projection operator in dual resonancemodels for the critical dimension of space-time.
Nucl. Phys.
B56 , 253–265.(31) 1973 (with L. Brink) Recalculation of the unitary single planar dual loop in the critical dimen-sion of spacetime.
Nucl. Phys.
B58 , 237–253.(32) 1973 (with L. Brink, C. Rebbi & J. Scherk) The missing gauge conditions for the dual fermionemission vertex and their consequences.
Phys. Lett.
B45 , 379–383.(33) 1973 (with L. Brink & J. Scherk) The gauge properties of the dual model pomeron-reggeonvertex – their derivation and their consequences.
Nucl. Phys.
B61 , 173–198.(34) 1973 (with J. Scherk) No ghost theorem for the pomeron sector of the dual model.
Phys. Lett.
B44 , 296–300.(35) 1974 (with J. Scherk) Towards satisfactory scattering amplitudes for dual fermions.
Nucl. Phys.
B64 , 334–348.(36) 1973 (with E. Corrigan, P. Goddard & R. A. Smith) Evaluation of the scattering amplitude forfour dual fermions.
Nucl. Phys.
B67 , 477–491.(37) 1975 Unitarity and discontinuity formulas.
Lect. Notes Math. , 133–142. Berlin: Springer-Verlag.(38) 1975 Dual Models. In:
Proceedings of 17th International Conference on High-energy Physics,London (ed. J.R. Smith), pp. 269-280. Chilton, UK: Rutherford Laboratory.(39) 1975 (with D. Bruce & E. Corrigan) Group theoretical calculation of traces and determinantsoccurring in dual theories.
Nucl. Phys.
B95 , 427–433.(40) 1976 (with E. Corrigan, D. B. Fairlie & J. Nuyts) Magnetic monopoles in SU(3) gauge theories.
Nucl. Phys.
B106 , 475–492.(41) 1976 Some topics in dual theory.
Acta Universitatis Wratislaviensis , 293–310.(42) 1976 (with E. Corrigan) Color and magnetic monopoles.
Nucl. Phys.
B110 , 237–247.(43) 1976 Angular momentum, magnetic monopoles and gauge theories.
Nucl. Phys.
B113 , 413–420.(44) 1976 (with F. Gliozzi & J. Scherk) Supergravity and the spinor dual model.
Phys. Lett.
B65 ,282–286.(45) 1977 (with F. Gliozzi & J. Scherk) Supersymmetry, supergravity theories and the dual spinormodel.
Nucl. Phys.
B122 , 253–290.(46) 1977 (with P. Goddard & J. Nuyts) Gauge theories and magnetic charge.
Nucl. Phys.
B125 ,1–28.(47) 1977 (with C. Montonen) Magnetic monopoles as gauge particles?
Phys. Lett.
B72 , 117–120.(48) 1977 Progress in analytic S -matrix theory. Publ. Res. Inst. Math. Sci. Kyoto , 347–349.(49) 1978 (with P. Goddard) Magnetic monopoles in gauge field theories. Rep. Prog. Phys. ,1357–1437.(50) 1978 (with E. Witten) Supersymmetry algebras that include topological charges. Phys. Lett.
B78 , 97–101.(51) 1979 Supersymmetric solitons.
Czech. J. Phys.
B29 , 73–80.(52) 1979 Magnetic monopoles.
Phys. Rept. , 165–172.(53) 1979 The electric and magnetic charges as extra components of four momentum. Nucl. Phys.
B153 , 1–12.(54) 1979 (with S. Sciuto & R. J. Crewther) Instantons in field theory.
Rev. Nuovo Cim. , 1–117.
55) 1979 Magnetic monopoles and nonabelian gauge theories. In:
Proceedings of the InternationalConference on Mathematical Physics, Lausanne, Switzerland,1979 (ed. K. Osterwalder),pp. 249-262. Berlin: Springer-Verlag.(56) 1980 Magnetic monopoles. In:
Proceedings of the International Conference On High EnergyPhysics, 1979 (ed. A. Zichichi), pp. 953–957. Geneva: CERN.(57) 1980 Magnetic monopoles and grand unified theories. In:
Unification of the Fundamental Par-ticle Interactions (ed. S. Ferrara et al. ), pp. 451–459. New York: Plenum Press.(58) 1981 Classical solutions in gauge theories – spherically symmetric monopoles – Lax pairs andToda lattices. In:
Current Topics in Elementary Particle Physics (ed. K. H. M¨utter & K.Schilling), pp. 199–217. New York: Plenum Press.(59) 1981 Magnetic monopoles and other topological objects in quantum field theory.
Phys. Scripta , 821–826.(60) 1981 (with P. Goddard) Charge quantization in theories with an adjoint representation Higgsmechanism. Nucl. Phys.
B191 , 511–527.(61) 1981 (with P. Goddard) The magnetic charges of stable self-dual monopoles.
Nucl. Phys.
B191 , 528–548.(62) 1982 Self-dual magnetic monopoles.
Czech. J. Phys.
B32 , 529–536.(63) 1982 The structure of self-dual monopoles. In:
Symposium on Particle Physics: Gauge Theoriesand Lepton-Hadron Interactions (ed. Z. Horvath et al. ), pp. 355–395. Budapest, Hungary:Central Research Inst. Physics.(64) 1982 Magnetic monopoles and electromagnetic duality conjectures. In:
Monopoles in QuantumField Theory (ed. N. S. Craigie et al. ), pp. 157–191. Singapore: World Scientific.(65) 1982 (with N. Ganoulis & P. Goddard) Self-dual monopoles and Toda molecules.
Nucl. Phys.
B205 , 601–636.(66) 1982 (with N. Turok) Z vortex strings in grand unified theories. Phys. Lett.
B117 , 193–196.(67) 1983 Relations between grand unified and monopole theories. In:
Unification of the Funda-mental Particle Interactions II (ed. J. Ellis & S. Ferrara), pp. 17–28. New York: PlenumPress.(68) 1983 Lectures on gauge theories and lie algebras with some applications to spontaneous sym-metry breaking and integrable dynamical systems. University of Virginia preprint 83-0529(unpublished).(69) 1983 (with P. C. West) The N = 4 supersymmetric E gauge theory and coset space dimensionalreduction. Nucl. Phys.
B217 , 248–284.(70) 1983 (with N. Turok) The symmetries of Dynkin diagrams and the reduction of Toda fieldequations.
Nucl. Phys.
B215 , 470–494.(71) 1983 (with N. Turok) Algebraic structure of Toda systems.
Nucl. Phys.
B220 , 491–507.(72) 1984 (with P. Goddard) Algebras, lattices and strings. In:
Vertex Operators in Mathematicsand Physics (ed. J. Lepowsky et al. ), pp. 51–96. New York: Springer-Verlag.(73) 1985 (with L. A. Ferreira) Non-compact symmetric spaces and the Toda molecule equations.
Commun. Math. Phys. , 365–384.(74) 1985 (with P. Goddard) Kac-Moody algebras, conformal symmetry and critical exponents. Nucl. Phys.
B257 , 226–252.(75) 1985 (with N. Turok) Local conserved densities and zero curvature conditions for Toda latticefield theories.
Nucl. Phys.
B257 , 277–301.
76) 1985 (with P. Goddard & A. Kent) Virasoro algebras and coset space models.
Phys. Lett.
B152 , 88–92.(77) 1985 (with P. Goddard & A. Schwimmer) The heterotic string and a fermionic construction ofthe E Kac-Moody algebra.
Phys. Lett.
B157 , 393–399.(78) 1985 (with P. Goddard & W. Nahm) Symmetric spaces, Sugawara’s energy momentum tensorin two-dimensions and free fermions.
Phys. Lett.
B160 , 111–116.(79) 1985 Kac-Moody algebras: an introduction for physicists. In:
Proceedings of the Winter SchoolGeometry and Physics (ed. Z. Frolk et al. ), pp. 177–198. Palermo: Circolo Matematicodi Palermo.(80) 1986 Kac-Moody and Virasoro Algebras in Local Quantum Physics. In:
Fundamental Problemsof Gauge Field Theory (ed. G. Velo &A. S. Wightman), pp. 51–92. New York: PlenumPress.(81) 1986 (with P. Goddard & A. Kent) Unitary representations of the Virasoro and SuperVirasoroalgebras.
Commun. Math. Phys. , 105–119.(82) 1986 (with P. Goddard) Kac-Moody and Virasoro algebras in relation to quantum physics.
Int.J. Mod. Phys. A1 , 303–414.(83) 1986 (with N. Turok) The Toda lattice field theory hierarchies and zero-curvature conditions inKac-Moody algebras. Nucl. Phys.
B265 , 469–484.(84) 1986 (with P. Goddard) An introduction to Kac-Moody algebras and their physical applications.In:
Proceedings of the Santa Barbara workshop on Unified String Theories (ed. M. Green& D. Gross), pp. 214–243. Singapore: World Scientific.(85) 1986 Infinite dimensional lie algebras and quantum physics. In:
Fundamental Aspects of Quan-tum Theory, Como 1985 (ed. V. Gorrini & A. Figuereido), pp. 289-293. New York:Plenum Press.(86) 1986 (with P. Goddard, W. Nahm & A. Schwimmer) Vertex operators for non-simply-lacedalgebras.
Commun. Math. Phys. , 179–212.(87) 1986 (with M. D. Freeman) BRS cohomology in string theory and the no ghost theorem.
Phys.Lett.
B175 , 151–154.(88) 1986 (with M. D. Freeman) The calculation of planar one loop diagrams in string theory usingthe BRS formalism.
Phys. Lett.
B175 , 155–158.(89) 1987 Lectures on algebras, lattices and strings. In:
Supersymmetry, Supergravity, Superstrings’86 (ed. B. de Wit et al. ), pp. 239–256. Singapore: World Scientific.(90) 1987 Infinite dimensional algebras in modern theoretical physics. In:
Proceedings of the VIIIthInternational Congress on Mathematical Physics, Marseilles, 1986 (ed. M. Mebkhout &R. S´en´eor), pp. 242-256. Singapore: World Scientific.(91) 1987 (with R. C. Arcuri & J. F. Gomes) Conformal subalgebras and symmetric spaces.
Nucl.Phys.
B103 , 327–339.(92) 1987 (with P. Goddard, W. Nahm, H. Ruegg & A. Schwimmer) Fermions and octonions.
Com-mun. Math. Phys. , 385–408.(93) 1987 (with P. Goddard & G. Waterson) Superalgebras, symplectic bosons and the Sugawaraconstruction.
Commun. Math. Phys. , 591–611.(94) 1988 The vertex operator construction for non-simply laced Kac-Moody algebras. In:
Infinite-dimensional Lie algebras and their applications, Proceedings, Montreal, 1986 (ed. S. N.Kass), pp. 181-188. Singapore: World Scientific.
95) 1988 Loop algebras, QFT and strings. In:
Strings and superstrings: XVIIIth International GIFTSeminar on Theoretical Physics Madrid 1987 (ed. J. R. Mittelbrunn et al. ), pp. 217-285.Singapore: World Scientific.(96) 1988 (ed. with P. Goddard)
Kac-Moody and Virasoro algebras: a reprint volume for physicists.
Singapore: World Scientific.(97) 1989 Introduction to string theory: its structure and its uses.
Phil. Trans. R. Soc. London
A329 , 319-328.(98) 1990 Introduction to conformal invariance and infinite dimensional algebras. In:
Physics, ge-ometry, and topology: 1989 Banff NATO ASI (ed. H.C. Lee), pp. 241-261. New York:Plenum Press.(99) 1991 (with M. -F. Chu, P. Goddard, I. Halliday & A. Schwimmer) Quantization of the Wess-Zumino-Witten model on a circle.
Phys. Lett.
B266 , 71–81.(100) 1991 (with A. Fring & H. C. Liao) The Mass spectrum and coupling in affine Toda theories.
Phys. Lett.
B266 , 82–86.(101) 1992 (with A. Fring) The Fusing rule and the scattering matrix of affine Toda theory.
Nucl.Phys.
B379 , 429–447.(102) 1993 (with H. C. Liao & N. Turok) Topological solitons in A r affine Toda theory. Phys. Lett.
B298 , 95–102.(103) 1993 (with N. Turok & J. W. R. Underwood) Solitons and the energy momentum tensor foraffine Toda theory.
Nucl. Phys.
B401 , 663–697.(104) 1993 (with M. V. Saveliev & J. W. R. Underwood) On a solitonic specialization for the generalsolutions of some two-dimensional completely integrable systems.
Phys. Lett.
B311 ,117–122.(105) 1993 (with N. Turok & J. W. R. Underwood) Affine Toda solitons and vertex operators.
Nucl.Phys.
B409 , 509–546.(106) 1993 (with M. A. C. Kneipp) Crossing and anti-solitons in affine Toda theories.
Nucl. Phys.
B408 , 565–573.(107) 1994 (with E. Rabinovici & A. Schwimmer) A Class of string backgrounds as a semiclassicallimit of WZW models.
Phys. Lett.
B321 , 361–364.(108) 1994 (with A. Fring, P. R. Johnson & M. A. C. Kneipp) Vertex operators and soliton timedelays in affine Toda field theory.
Nucl. Phys.
B430 , 597–614.(109) 1995 (with L. Ferreira & M. V. Saveliev) Orthogonal decomposition of some affine Lie algebrasin terms of their Heisenberg subalgebras.
Theor. Math. Phys. , 10–22.(110) 1996 (with M. A. C. Kneipp) Solitons and vertex operators in twisted affine Toda field theories.
Commun. Math. Phys. , 561–582.(111) 1996 Exact electromagnetic duality.
Nucl. Phys. Proc. Suppl. , 88–102; , 1–15.(112) 1997 Lectures on exact electromagnetic duality II. In: Proceedings of the IX Jorge Andr´e e SwiecaSummer School Campos de Jord˜ao (ed. J.C.A. Barata et al. ), pp. 166–195. Singapore:World Scientific.(113) 1997 Introduction to electromagnetic duality.
Nucl. Phys. Proc. Suppl. , 43–55.(114) 1998 The monopole. In: Paul Dirac: the man and his work (ed. P. Goddard), pp. 88-107.Cambridge: Cambridge University Press.(115) 1999 Exact electromagnetic duality. In:
Strings, branes and dualities, Cargese, 1997 (ed. L.Baulieu), pp. 3–31. Berlin: Springer-Verlag. Duality and supersymmetric theories . Cambridge: CambridgeUniversity Press.(117) 1999 (with P. C. West) Particle physics and fundamental theory: introduction and guide toduality and supersymmetric theories. In:
Duality and supersymmetric theories (ed. D.Olive & P. C. West), pp. 1–20. Cambridge: Cambridge University Press.(118) 1999 Introduction to duality. In:
Duality and supersymmetric theories (ed. D. Olive & P. C.West), pp. 62–94. Cambridge: Cambridge University Press.(119) 2000 (with M. Alvarez) The Dirac quantization condition for fluxes on four manifolds.
Com-mun. Math. Phys. , 13–28.(120) 2000 Aspects of electromagnetic duality. In:
Proceedings, Conference on Nonperturbativequantum effects: Paris, 2000 (ed. D. Bernard et al. ), JHEP
PoS tmr2000 021 , pp.1–7.(121) 2000 Lie algebras, integrability, and particle physics.
Theor. Math. Phys. , 659–662.(122) 2001 (with M. Alvarez) Spin and Abelian electromagnetic duality on four manifolds.
Commun.Math. Phys. , 331–356.(123) 2001 Spin and electromagnetic duality: an outline. Talk at 15th Anniversary Meeting of DiracMedallists, Trieste, 2000. hep-th/0104062 .(124) 2001 (with H. Nicolai) The principal SO(1; 2) subalgebra of a hyperbolic Kac-Moody algebra.
Lett. Math. Phys. , 141–152.(125) 2002 The quantization of charges. Italian Phys. Soc. Proc. , 173–183.(126) 2002 (with M. R. Gaberdiel & P. C. West) A class of Lorentzian Kac-Moody algebras. Nucl.Phys.
B645 , 403–437.(127) 2002 Duality and Lorentzian Kac-Moody algebras. In:
Proceedings, International Workshopon Integrable Theories, Solitons and Duality UNESP 2002 JHEP
PoS unesp2002 008 ,pp. 1–8.(128) 2003 Paul Dirac and the pervasiveness of his thinking. Talk at Dirac Centenary Conference,Waco, Texas, 2002. hep-th/0304133 .(129) 2003 Charges and fluxes for branes. In:
PoS jhw2003 019 , pp. 1–7.(130) 2006 (with M. Alvarez) Charges and fluxes in Maxwell theory on compact manifolds withboundary.
Commun. Math. Phys.
The Birth of String Theory (ed. A. Cappelli etal. ), pp. 346–360.), pp. 346–360.