Interview with Edward Witten - ias.edu · Interview with Edward Witten Hirosi Ooguri This is a slightly edited version of an interview with Edward Witten that appeared in the December

Interview withEdward Witten

Hirosi Ooguri

This is a slightly edited version of an interview with Edward Witten that appeared in theDecember 2014 issue of Kavli IPMU News, the news publication of the Kavli Institute forthe Physics and Mathematics of the Universe (IPMU). An abridged Japanese translationhas also appeared in the April and May 2015 issues of Sugaku Seminar, the popularmathematics magazine. The interview is published here with permission of the KavliIPMU.

The interview took place in November 2014 on the occasion of Witten’s receipt of the2014 Kyoto Prize in Basic Sciences. Witten is Charles Simonyi Professor in the School ofNatural Sciences at the Institute for Advanced Study in Princeton. Yukinobu Toda andMasahito Yamazaki, two junior faculty members of the Kavli IPMU, also participated inthe interview.

Ooguri: First I would like to congratulate Edward onhis Kyoto Prize. Every four years, the Kyoto Prize inBasic Sciences is given in the field of mathematicalsciences, and this is the first prize in this categoryawarded to a physicist.

Witten: Well, I can tell you I’m deeply honoredto have this prize.

Ooguri: It is wonderful that your work in thearea at the interface of mathematics and physicshas been recognized as some of the most importantprogress in mathematics as well as in physics. Forthose of us working in this area, this is also verygratifying.

Witten: Actually, in my acceptance speech acouple of days ago, I remarked that I regard it alsoas a recognition of the field, not just of me.

Generalizing Chern-SimonsOoguri: This conversation will appear as an articlein the Japanese magazine, Sugaku Seminar, as

Hirosi Ooguri is Fred Kavli Professor of Theoretical Physicsand Mathematics, the director of the Walter Burke Institutefor Theoretical Physics, and the deputy chair of the Divisionof Physics, Mathematics and Astronomy at the CaliforniaInstitute of Technology. He is also a principal investigator ofthe Kavli Institute for the Physics and Mathematics of theUniverse at the University of Tokyo. His email address [email protected].

All photos provided by and used with permission of KavliIPMU.

DOI: http://dx.doi.org/10.1090/noti1248

Edward Witten.

well as in the Kavli IPMUNews. You have alreadygiven two interviews forSugaku Seminar. In 1990,at the International Con-gress of Mathematiciansin Kyoto, you received theFields Medal. On that oc-casion, Tohru Eguchi hadan interview with you. Youalso had a discussion withVaughan Jones, anotherFields Medalist at the con-gress, and I rememberyou expressed interest ingeneralizing your work inthe Chern-Simons theorywith a spectral parameter,which is very natural fromthe point of view of integrable models.

Witten: Yes, I very much wanted to find anexplanation along these lines of the “integrability”that makes it possible to get exact solutions oftwo-dimensional lattice models such as the Isingmodel. I was completely unsuccessful, but just inthe last couple of years, something in the spirit ofwhat I wanted to do was done by Kevin Costello.

Ooguri: We were just talking about Costello’swork before you arrived here. Do you think that itachieved what you wanted to do at that time?

Witten: Yes. Integrable models have many facets,and there is no one way to understand everything.

May 2015 Notices of the AMS 491

But I would say that specifically the kind ofexplanation I was looking for is what Costellofound. What Costello did involves a very simplebut beautiful twist on the three-dimensional Chern-Simons theory, in which he simply replaced one ofthe three real dimensions of space by a complexvariable z.

Ooguri: It means going to four dimensions.Witten: This is a four-dimensional world with

two real coordinates and a complex coordinate z.Costello defined a 4-form, which was the wedgeproduct of the Chern-Simons 3-form with the1-form dz. He studied this as the action of a four-dimensional theory. There is a crucial technicaldetail: for this theory to make any sense, thedifferential operator that is obtained by linearizingthe equations of motion must be elliptic modulo thegauge group. I think that this is a little surprising,but it is true. And given that, he then has ageneralization of the Chern-Simons theory thatdoes not have the full three-dimensional symmetry,but it does have a complex variable, namely z.

If you think carefully, you will see that in-tegrability, the Yang-Baxter equation, involvestwo-dimensional symmetry but not really three-dimensional symmetry. The reason I was unableto incorporate the spectral parameter was that Iwas working in the context of three-dimensionaltopological field theory. In three-dimensional topo-logical field theory, in addition to the moves whereknots cross—that is, in addition to the Yang-Baxterrelation—you have further relations involving cre-ation and annihilation. There are Reidemeistermoves that are valid in topological field theory butare not relevant for integrable systems. I couldn’tfind the spectral parameter because I was trying touse topological field theory. Costello made a verysimple twist, replacing a real variable by a complexvariable, and then everything worked beautifully.I definitely regard that as the explanation I wastrying to find, unsuccessfully, around 1990.

Ooguri: I see. So, after twenty-three years, finallyyour question was answered.

At the time of your second visit to Kyoto in 1994,you were just finishing your work on the Seiberg-Witten theory and also the Vafa-Witten theory. Iremember a discussion session we had at the Re-search Institute of Mathematical Sciences, KyotoUniversity, with you and Hiraku Nakajima, whereNakajima explained his work on the action of anaffine Lie algebra on the cohomology of the modulispace of instantons. In the second interview you hadwith Tohru Eguchi for Sugaku Seminar, you men-tioned progress in mirror symmetry and S-dualityat the time and expressed a hope of a more unifiedview on duality encompassing gauge theory andstring theory. Some of this hope has been achievedin the last twenty years, I think.

Witten: Definitely some of it has been achieved.One thing that was achieved in the couple of yearsafter that second interview was simply that thereemerged a picture of nonperturbative dualities instring theory, generalizing what happens in fieldtheory. However, there are other aspects that arestill mysterious and not clearly understood.

On the bright side, the fact that four-dimensionalgauge dualities and a lot of dualities in lowerdimensions come from the existence of a six-dimensional conformal field theory is a majorinsight in understanding dualities better. Wehaven’t gotten to the bottom of things becausewe don’t really understand the six-dimensionaltheory, but just knowing that the matter should beunderstood in terms of the properties of the six-dimensional theory is an advance in understandingduality that certainly wasn’t there at the time ofthis last interview.

I Was a Skeptic about DualityOoguri: I should introduce our discussants today.Yukinobu Toda is a mathematician and an associateprofessor at the Kavli IPMU, and he received hisPhD in 2006. Masahito Yamazaki is a physicist anda new assistant professor at the Kavli IPMU, andhe received his PhD in 2010. They represent theyoung generation of mathematicians and physicistsworking at the interface of string theory and gaugetheory.

At your Kyoto Prize Commemorative Lecture,you reviewed your career in theoretical physics.You entered graduate school in 1973, when theasymptotic freedom was just theoretically beingdiscovered. You came to Japan the second time in1994 and gave the interview we were just talkingabout, and it was roughly twenty years after youstarted in graduate school. Another twenty yearshave passed since that time, so I thought we shouldstart by trying to catch up with the second twentyyears of your career and see what your thoughtshave been on some of the most important progress.

We have already started to talk about some ofthe developments since the interview in 1994, butmaybe you can expand on that and tell us what youthink have been the highlights in this area in thelast twenty years.

Witten: Certainly one of the highlights has beenthe understanding of nonperturbative dualitiesin string theory, as a result of which we havea much wider picture of what string theoryis. In 1994 we knew about mirror symmetryand other two-dimensional dualities that arise instring perturbation theory, and we were reallyjust beginning to think that there are similardualities in space-time: four-dimensional gaugetheory dualities that are analogous to the two-dimensional dualities. But in 1994 it was really just

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a guess that something analogous might happenin string theory.

By this time there were clues in the literature,and a number of new ones had been discovered inthe early 1990s. The clue that influenced me themost was the work of John Schwarz and Ashoke Sen,who showed that the low-energy effective action ofthe heterotic string on a six-torus had propertiesconsistent with the existence of a nonperturbativeSL(2, Z) duality. They didn’t have what I regardedas really decisive evidence for that conjecture, buttheir ideas were very suggestive.

It still was not clear to me how one could finddecisive evidence for nonperturbative dualities inspace-time. At least to me, the first such evidenceappeared in a short but brilliant paper by AshokeSen on a two monopole bound state in N = 4 superYang-Mills theory. To me, that was fundamentallynew evidence for the Montonen-Olive dualityconjecture. It convinced me that the duality had tobe right and, equally important, it convinced methat it was possible to understand it better.

Ooguri: I thought that Sen’s paper gave strongevidence for the S-duality, but it was your paperwith Vafa that convinced us.

Witten: Thank you. Sen’s paper showed thatyou could actually go well beyond the suggestivebut somewhat limited arguments about electric-magnetic duality that had been known and learnsomething fundamentally new. Until Sen’s paper, Ihad felt that what we understood about electric-magnetic duality, even the work of Sen andSchwarz, which had definitely influenced me, wasin the framework of what Montonen and Olivehad understood twenty years before. But Sendid a simple and elegant calculation, finding abound state of two monopoles whose existencewas predicted by the duality. That inspired me tobelieve that one could do more.

With this inspiration, and trying to find moreevidence for the duality conjectures, Cumrun Vafaand I started to study the Euler characteristics ofinstanton moduli spaces. It was not too hard to seethat electric-magnetic duality of supersymmetricYang-Mills theory implied that the generatingfunction of those Euler characteristics should be amodular function. Luckily for us, mathematiciansin a number of cases—and this includes thework of Nakajima that you mentioned before—had computed these Euler characteristics or hadobtained closely related results from which theEuler characteristics could be understood. Wefound that the expected modularity held in allcases. (In one case—the four-manifold CP2—weran into a “mock modular form,” a concept thatwas new to us at the time but has made manysubsequent appearances in gauge theory and stringtheory.)

Also during this period, Nathan Seiberg hadbeen using holomorphy as a tool to analyze thedynamics of supersymmetric gauge theories. Hewanted to understand what happened in N = 2theories. We started talking about it, and the Senpaper inspired us to think that duality would playa role. That was one of the clues that actually led toour work on what became Seiberg-Witten theory.

Ooguri: It may be hard to believe for young peo-ple like Masahito Yamazaki or Yukinobu Toda, butbefore 1994 S-duality, at least for me, was some-thing very hard to believe. It was like a beautifuldream. It would be nice to have, but you cannotrealistically hope that something like that could pos-sibly happen. As I said, the first evidence was Sen’spaper, and in some sense Edward’s work with Vafanailed it. After that, everybody believed it.

Masahito Yamazaki: That’s surprising, becauseI thought that the paper of Claus Montonen andDavid Olive is quite old. Were people skeptical aboutthe idea?

Witten: This might make you laugh, but I’ll tellyou my early history with the Montonen and Olivepaper. First of all, I hadn’t heard of it until I wasvisiting Oxford at the end of 1977. Michael Atiyahshowed this paper to me and said I should go toLondon to discuss it with Olive. So, I looked atthe paper and got in touch with David Olive andarranged to visit him. But by the time I got toLondon, I was pretty skeptical. Have you looked attheir original paper?

Masahito Yamazaki: Yes, I have.Witten: In their original paper, they considered

a bosonic theory of a gauge field and a realscalar (valued in the adjoint representation). Theyassumed that the potential energy for the scalarfield is identically zero, and they found remarkableformulas for particle masses that are valid preciselyin this case. Their proposal of electric-magneticduality was based on the fact that their massformula was symmetrical between electric andmagnetic charge.

However, I knew quantum field theory wellenough to know that saying that the potentialenergy for the scalar field is zero is not a meaningfulstatement quantum mechanically. If it were, wewould not have a gauge hierarchy problem inparticle physics. So, by the time I got to Londonto see Olive, I definitely was a skeptic. But sinceI was there to see him, I didn’t want to justsay that his idea was nonsense. We tried tomake sense out it. So we discussed it in thecontext of supersymmetry, simply because withsupersymmetry the mass renormalization (andeven the full effective potential) of a scalar canbe zero. This was the only context in which itseemed to me that the brilliant idea of Montonenand Olive could make sense. By the end of the


Ooguri and Witten.

day, we found that their formulas are valid in thecontext of N = 2 supersymmetry. So we wrote apaper on that, and it was quite a satisfying paperto write, but I drew the wrong conclusion fromthat paper. The conclusion I drew was that wehad explained their formulas without needing toassume nonperturbative duality.

Ooguri: Right. That was the message I got byreading your paper with Olive, too. The seeminglymiracle phenomenon was explained simply bysupersymmetry.

Witten: So at the time, and for many years after,I felt that there was not really a lot of evidence fornonperturbative duality in four dimensions.

Thus, to return to Masahito’s question, I wasa skeptic about electric-magnetic duality duringthese years, but actually I was a skeptic on twolevels. One, I was skeptical that it was true, and two,I was skeptical that you could really say anythingabout it even if it were true.

To give a more complete picture, in the early1990s there were various novel clues, some fromthe work of people like Mike Duff, and also CurtCallan, Jeff Harvey, and Andy Strominger, studyingsolitons in string theory, and then also therewas the work of Sen and Schwarz that I alreadymentioned. I remember that at the Strings 1993meeting in Berkeley, John Schwarz was moreexcited than I had ever seen him since January1984. In January 1984, telling me about his latestwork with Michael Green, he said, “We are gettingclose,” but I didn’t understand what he thoughthe was getting close to. It turned out, however,that that was a few months before they canceledthe anomalies. When John was so excited at theStrings meeting at Berkeley, I decided that I hadbetter take him seriously.

If you had looked at it with the same skepticismI had had since the Montonen and Olive paper, youwould have said that Sen and Schwarz were just

discussing low energy physics and did not havesolid evidence about strong coupling behavior. ButJohn’s enthusiasm put enough of a dent in myskepticism that I started looking more closely atthe papers of Duff and other authors on solitonsin string theory. At some point, I think in the fallof 1993, Duff sent me an assortment of his papers,and I took them to heart. I don’t remember rightnow all of the papers on solitons in string theorythat I looked at during this period, but certainlyone important one was by Callan, Harvey, andStrominger.

There is another part of the background tothis period that I should explain. Mike Duff,Paul Townsend, and other physicists working onsupermembranes had spent a couple of years in themid-1980s saying that there should be a theory offundamental membranes analogous to the theoryof fundamental strings. That wasn’t convincingfor a large number of reasons. For one thing, athree-manifold doesn’t have an Euler characteristic,so there isn’t a topological expansion as there isin string theory. Moreover, in three dimensionsthere is no conformal invariance to help us makesense of membrane theory; membrane theory isnonrenormalizable just like general relativity.

There are all kinds of technical objections, but atsome point around 1990 or 1991, instead of tryingto think of membranes as fundamental objects,people working in this area started thinking ofmembranes and other p-branes as nonperturbativeobjects that might exist in string theory. In generalterms, this idea did make sense. In more detail, thesituation was more complicated. If you actuallylooked at the papers, some of them made a lot ofsense because they had a classical soliton solutionwith good properties. (Even then, the solutionsusually had unusual properties that in some caseswere clues to later discoveries.) Other papers madea little bit less sense, because the classical solutioninvolved a singularity that appeared in a regionin which the classical approximation wasn’t good.But the idea of membranes as nonperturbativesoliton-like objects in string theory made a lotof sense even if the details in some papers weredubious. I was still a bit of a skeptic about whatone can do with this idea, but for reasons I’ve beenexplaining, I was paying a lot more attention. Andthat is actually why, when the Sen paper on thetwo-monopole bound state came out, I was readyto completely change my outlook.

Sen’s paper showed that one can do somethingnew about strong coupling, and it was clear that ifone had been inspired the way Sen was, one couldhave done what he did ten or fifteen years before. So,it showed that we had been missing opportunities.That definitely changed the direction in my work. Itled to the paper with Vafa that you’ve been kindly


mentioning, and it helped put Seiberg and me onthe right track for doing what we did in 1994 andthen…

Ooguri: This is a great story that shows thatchance favors the prepared mind, as Pasteur said.After that, you even went to string dualities.

String Duality RevolutionWitten: By the end of 1994, we had the experienceof nonperturbative dualities in field theory bothin two dimensions and in four dimensions. In thetwo-dimensional case, for example, if one studiesa sigma-model with Calabi-Yau target space (suchas is important in studying the compactification ofstring theory), one finds that the quantum theorycan be extended far beyond the classical geometryof the Calabi-Yau manifold. One finds a web ofphase transitions between different geometricaland nongeometrical descriptions of the sigma-model, which represent different semi-classicallimits of the theory. The Montonen-Olive dualityconjecture, as refined by later authors, said thatsomething similar happens in N = 4 super Yang-Mills theory in four dimensions, and Seiberg and Iin 1994 had found something somewhat similarfor N = 2.

Certainly there was a dream that somethingsimilar might happen in string theory. Not onlywas there a dream, but there were a lot of papersin which people had pointed out pieces of sucha story. I have already mentioned some of thesepapers. Another important paper was written byChris Hull and Paul Townsend in the spring of1995. They wanted to say that Type IIA superstringtheory is the same as M-theory on a circle. Theonly thing they really didn’t do was to try tomake it more quantitative. There’s a potentialcontradiction, which is that in Type IIA superstringtheory you don’t see eleven dimensions. But it turnsout, as I realized a little later, that this questionhas a very simple answer. The eleven-dimensionallimit is a region of strong coupling from the pointof view of Type IIA superstring theory, and theeleventh dimension isn’t visible for weak coupling.

It soon became clear that the same thing wastrue in other cases. For example, one might hopethat Type I superstring theory and the SO(32)heterotic string will be the same. There is anobvious immediate contradiction: the theorieshave the same massless spectrum and low energyinteractions, but beyond the low energy limit theylook completely different. The answer is simplythat if you match up the low energy field theories,you will find that weak coupling in one is strongcoupling in the other.

Once one starts thinking along those lines,it turns out that everything works. What werethe implications? This way of thinking certainly

led to a more unified picture of what stringtheory is. But very soon, there were furtherdevelopments showing that the traditional ways ofasking questions were probably inadequate. In the1980s I was really convinced that in some sensestring theory should be based on a Lagrangian thatwould generalize the Einstein-Hilbert Lagrangianfor gravity; it would have a symmetry group whichwould generalize the diffeomorphism group. Sothere would be a new classical theory of geometry—with nonperturbative two-dimensional dualitiesbuilt in as classical symmetries. One would thengenerate string theory by quantizing this classicaltheory.

But by the early 1990s, there was a troublesomedetail that I personally did not pay much attentionto. In the moduli space of Calabi-Yau manifolds,there is a variety of singularities. Some questionsinvolving such singularities had been important inmy own work.

Ooguri: You are referring to the work involvinglinear sigma-models.

Witten: That is correct, and also my work (withHarvey, Vafa, and Lance Dixon) on orbifolds. I hadbeen interested in cases in which the classicalgeometry has a singularity but the quantum sigma-model does not; these cases illustrate the differencebetween ordinary geometry and its generalizationin the classical limit of string theory. What I had nottaken seriously is that in general, as one deformsthe moduli of a Calabi-Yau manifold, one canfind singularities of the classical geometry thatdo also lead to singularities of the correspondingsigma-model.

Such a singularity appears in string theory evenin the classical limit, so if you try to interpret stringtheory as a classical theory that then gets quantized,it looks like the classical theory has a singularity,which is strange. I personally didn’t focus onthat question, but Strominger explained that sucha singularity actually reflects a nonperturbativequantum effect. The singularity arises when acharged black hole becomes massless, and itshows that quantizing a classical theory can’t dojustice to string theory: there are nonperturbativequantum effects even in what one might havewanted to call the classical limit.

Ooguri: You say there’s no analogous result infield theory, this is a genuinely string-theoretic phe-nomenon.

Witten: I think so.Ooguri: So, did you think this was evidence that

there is no Lagrangian description in string theory?Witten: It is evidence that you can’t fully do

justice to string theory in terms of quantizing aclassical theory. I don’t want to say that there isn’ta classical theory, because I believe that from somepoint of view there is.


Ooguri: Yes, as an approximate description, butyou’re saying that you cannot start from classicaltheory and apply a quantization procedure…

Witten: We can’t fully understand string the-ory by quantizing an underlying classical theory.In some sense it is an intrinsically quantummechanical theory.

I don’t want to say you can’t derive string theoryby quantizing a classical theory, but you can’t fullydo justice to it that way, I think.

But let us remember that even in field theory,Montonen-Olive duality means that the same theoryhas different classical limits, showing that no oneclassical limit is really distinguished.

Ooguri: But in that case, you have a Lagrangiandescription.

Witten: Yes, in the Montonen-Olive case, onehas a classical Lagrangian, in fact many of them.String theory is a little bit worse because even inwhat you want to call the classical limit, there arephenomena that you really can’t make much senseof from the classical point of view.

Ultimately, Strominger’s work illuminated some-thing I’d missed. In the talk I gave at the StringsConference in 1995 about nonperturbative duali-ties in string theory, and also in the correspondingpaper [“String theory dynamics in various dimen-sions”, arXiv:hep-th/9503124, Nuclear Physics B,volume 443, issue 1–2, June 5, 1995, pages 85–126],there was one detail that didn’t completely makesense. Type IIA superstring theory on a K3 mani-fold was supposed to be dual to the heterotic stringon a four-torus, and in that context I could see thatenhanced gauge symmetry resulted from the K3surface developing an ADE singularity. But an ADEsingularity in classical geometry is just an orbifoldsingularity, and perturbation theory remains validin string theory at an orbifold. The orbifold doesnot generate a nonperturbative gauge symmetry.For a few months I was puzzled. Actually, I wasmaking a simple mistake which was corrected byPaul Aspinwall in a paper that he wrote in the sum-mer of 1995. Aspinwall explained the following:In M-theory at an ADE singularity, you have onlythe hyper-Kähler moduli, but in string theory at anADE singularity, there also are B-field moduli. Theconformal field theory becomes singular when theB-field moduli are zero; the orbifold describes anonsingular situation in which the B-field moduliare not zero.

When the B-field moduli vanish, there is abreakdown of the classical description that’sjust analogous to what Strominger had shownin his paper on the Calabi-Yau singularity. Itleads to enhanced gauge symmetry that, from thestandpoint of Type IIA superstring theory, has anonperturbative origin.

Strominger had considered a charged black holethat arises from a wrapped three-brane, while herethe relevant particle is a wrapped two-brane. Butthe idea is similar.

Ooguri: So this was the beginning of the inter-action between the gauge-theoretic idea and thestring-theoretic idea where non-Abelian, nonpertur-bative dynamics of gauge theory can emerge fromlimits of string theory.

Witten: Right. Another important but extremelysimple paper that helped show the implications ofstring theory for nonperturbative duality in gaugetheory was written by Michael Green in 1996. Bythis time, Joe Polchinski and his collaborators hadbasically shown that in modern language a systemof n parallel branes has U(n) gauge symmetry. Ihad written a paper at the end of 1995 showingwhy that was useful, but Green wrote a very simplepaper with the following observation. Type IIBsuperstring theory has a nonperturbative dualitysymmetry—a fact which we were convinced of bythis time—and on the other hand N = 4 superYang-Mills theory in four dimensions with gaugegroupU(n) can arise from a system ofn parallel D3-branes in Type IIB superstring theory. Combiningthese two facts and taking the low energy limit,Green was able to deduce the Montonen-Oliveduality of N = 4 super Yang-Mills theory withgauge group U(n). It is simply inherited fromthe Type IIB superstring theory specialized to theD3-branes. That was an important early exampleof deducing a gauge theory duality from a stringtheory duality.

Even before all of this had happened, Mike Duffand Ramzi Khuri in 1993 had written a paper onwhat they called string/string duality. They hadsaid there should be a self-dual string theory insix dimensions that, looked at in two differentways, would give electric-magnetic duality of gaugetheory in four dimensions. It was actually a brilliantidea. The only trouble was they didn’t have anexample in which it worked.

I realized in mid-1995 that if one took theheterotic/Type II duality on K3 and T 4 andcompactified on another two-torus, one would getan example rather similar to what Duff and Khurihad suggested. They had had in mind self-dualityof a string theory, but the example that I consideredwas a duality between two different string theories.Still, the idea was similar. By the end of 1995, Duffand I and some other authors had an examplethat followed even more precisely what had beenproposed two years before. That involved theE8 ⇥ E8 heterotic string on a K3 surface with equalinstanton numbers in the two E8’s. In all of thesecases, one could deduce Montonen-Olive dualityfrom a string theory duality.


As we’re talking, I remember more and morepapers from the years 1995–96 that were verydramatic at that time but that were also, honestly,pretty easy to do in most cases. I was reminded ofthis yesterday by the lecture by Hiraku Nakajimaat the Kyoto Prize Workshop. Hiraku started bykindly remembering three lectures I gave at theNewton Institute in 1996. The lectures concernedthree papers that I had written (respectively withcoauthors Lev Rozansky, Ami Hanany, and NathanSeiberg). The papers fit together nicely. They werefun to write, and the lectures were also fun to give.But what stands out in my memory is that at thattime insights like that were more or less out onthe surface. It was quite a fun time to be workingin this field. I am hoping that, during my activecareer, there will be another period like that.

Difference between Knowing What Is Trueand Why It Is TrueYukinobu Toda: I am an algebraic geometer. Iwas originally working on some classical aspectsof algebraic geometry, but I became interested insome relationships between algebraic geometry andstring theory inspired by your work. You have beendiscussing S-duality and modular forms. This wassurprising from a mathematical point of view—Icannot see why modularity appears. Do you havean insight about that from a mathematical point ofview?

Witten: Vafa and I, of course, had a reason,which was the Montonen-Olive duality conjecture.What we did was to show that modularity of acertain generating function of Euler characteristicsis a kind of corollary of Montonen-Olive duality.This is somewhat analogous to saying that somestatement in number theory follows from theRiemann hypothesis. If somebody shows thatsomething follows from the Riemann hypothesis,one may or may not view this as an explanation ofthe statement in question, but at least it puts thestatement in a bigger framework. Montonen-Oliveduality provided an analogous bigger frameworkin my work with Vafa, and soon afterwards therewas a still bigger framework, which was that theMontonen-Olive duality follows from the existenceof a certain six-dimensional theory. It also followsin various other ways from string theory dualities,and I have mentioned a few of these constructions.But most physicists would probably say the mostcomplete framework that we have for Montonen-Olive duality is its relation to the six-dimensiontheory.

Ooguri: Yukinobu was asking for some mathe-matical explanation. At that time, some hint of themathematical explanation was Nakajima’s workabout the symmetry of the moduli spaces of instan-tons. From the mathematical point of view, what the

YukinobuToda and Masahito Yamazaki.

Vafa-Witten theory was computing were generatingfunctions of the Euler characteristics of instantonmoduli spaces.

Witten: Nakajima’s discovery of the affine Liealgebras was a kind of proof and actually a mirac-ulous discovery. But it still leaves one wonderingwhere the affine Lie algebra symmetry came from.

Yukinobu Toda: Right. After computations ofthe Euler characteristics, we know that it’s a modu-lar form, but we don’t know why it is modular, evenfor the simplest example.

Witten: I completely agree. What you’re sayingis actually something I tried to say in my Com-memorative Lecture for the Kyoto Prize. There isa difference between knowing what is true andknowing why it is true. In this case, you have amathematical proof, but you’re still asking why,and physicists ultimately don’t know. All that wecan do is to offer bigger conjectures, of which thisis a manifestation. But we don’t really understandthe bigger conjectures.

Ooguri: In physicists’ perspective, this dualityhas been geometrized as symmetry in six dimen-sions.

Witten: But the six-dimensional theory is prettymysterious.

Yukinobu Toda: Is it not difficult to under-stand the relationship between S-duality andsix-dimensional theory?

Ooguri: The relation is very clear, but then youhave to make sense of the six-dimensional theoryitself.

Witten: We actually know quite a lot about thebehavior of the six-dimensional theory, though wedo not understand much about how it should beconstructed or understood microscopically.

One of the deepest discoveries about the be-havior of the six-dimensional theory was made byJuan Maldacena in 1997. He showed that it couldbe solved for large N in terms of supergravity


(N refers to the rank of an SU(N) gauge group).Unfortunately, the regime in which the theory issolved by supergravity isn’t the same regime inwhich we usually have to study it to understandthe questions you’re asking.

Ooguri: I understand that the large N limit isnot S-duality invariant.

Witten: Yes, that is correct. Maldacena’s solutionof the theory for large N works and makescomplete sense. It does not directly help us inunderstanding Montonen-Olive duality, because itinvolves studying the theory in a different region ofparameters that is not invariant under duality. Or,to put it differently, if one tries to apply Maldacena’ssolution to understand Montonen-Olive duality,one has to work in a region of parameters in whichthe description that Maldacena gave is not useful.

But the existence and success of Maldacena’ssolution definitely increases the confidence ofphysicists that the six-dimensional theory existsand that all the canonical statements about it aretrue, even though we don’t understand everything.It’s a little bit like mathematicians discovering thatsome new consequences of the Riemann hypothesisare true. This gives one more confidence in theRiemann hypothesis, but it doesn’t mean oneunderstands the Riemann hypothesis.

Ooguri: Yukinobu, what is your view on currentactivities in physics? For example, yesterday Naka-jima was saying that it took him eighteen years tounderstand what Edward was doing in his lecturesin Cambridge, and Kenji Fukaya was saying thatsometimes he doesn’t understand even the state-ment because you don’t understand the right-handside and left-hand side of equations physicists write,for example. You have been at the Kavli IPMU forseveral years, interacting with physicists, so do youhave any perspective to offer…?

Yukinobu Toda: Of course, I don’t know any-thing about string theory, but sometimes I look atpapers and some calculations and try to translatephysics words into mathematics, say D-branes tosheaves or BPS states to stable objects. Then I havelots of things to learn from the physics side andlots of problems to solve, although I don’t under-stand their physics origin. I also found that theyare related to the classical problem in algebraicgeometry.

Ooguri: You also attend string theory seminars.What do you gain by attending them and interactingwith physicists?

Yukinobu Toda: I think there are many kindsof people in string theory. Some people’s works areclose to me, like Donaldson-Thomas invariants andderived category of coherent sheaves. In their sem-inars, I can learn something, but that is almost aseminar of mathematics.

Ooguri: A mathematician told me that physi-cists are like generating functions of conjectures.Some physicists are more useful for mathematiciansthan others. For example, Hiraku Nakajima wastelling me that he particularly likes Edward’s lec-tures, because even though he doesn’t understandthe motivations and where ideas come from, someof the statements Edward makes have sharp math-ematical meanings to them, just like the equationthat Tachikawa was quoting yesterday at the work-shop, and these are something that mathematicianscan work on.

Masahito Yamazaki: But then sometimes peo-ple want to know the logic behind it. I can make astatement that makes sense mathematically, andmathematicians can try to prove it. But they defi-nitely want to know what’s happening.

Witten: In any given case, I can’t guarantee thatthere isn’t a simpler answer. But the view of mostphysicists about many of the problems that wehave been discussing is going to be that the bestsetting for these questions is in the quantum fieldtheories that are important in physics.

Quantum EntanglementOoguri: We are still discussing what was happeningin the 1990s. Now, we should move on to the newmillennium. What do you think have been highlightsin the past fourteen years?

Witten: Part of the answer is that the gauge-gravity duality that was introduced by Maldacenais very deep. Even today people are still discoveringinteresting new facets of it. An important examplewas the work of Shinsei Ryu and Tadashi Takayanagion entanglement entropy in gauge/gravity duality.They discovered a really interesting generalizationof the Bekenstein-Hawking entropy of a blackhole. Although I have not personally worked onthis subject, the developments have been prettyinteresting and may contain deeper clues aboutquantum gravity. If I could see the right way todo this, then I would probably work in this areamyself, but at least so far I don’t. But it’s one ofthe things I’d recommend watching most closely.

Apart from Ryu and Takayanagi, I also woulddefinitely recommend the papers of Horacio Casini,in some cases coauthored with Marina Huerta. Oneof these papers addressed the following question.A black hole has a Bekenstein-Hawking entropy.Roughly twenty years ago, Jacob Bekenstein con-sidered the following question. Suppose an objectfalls into a black hole. The object has an entropy.When it falls into a black hole, its entropy disap-pears into the hole. The black hole gains masswhen it absorbs the object, so its entropy goesup. The second law of thermodynamics says thetotal entropy should increase in this process, so inother words the black hole entropy increases by


at least the entropy that the in-falling object hadbefore approaching the black hole. This tells youbasically that if an object has given energy and issmall enough to fit inside a black hole of givenmass, then there’s an upper bound to its entropy.Bekenstein proposed such a bound, and peoplecalled it the Bekenstein bound, but for a long timeno one could formulate precisely what this boundwas supposed to say.

I am reminded here of what Fukaya said aboutthe relationship between physics and mathematics,where he remarked that it can be hard to formu-late precisely the terms that enter some of thestatements made by physicists. In the case of theBekenstein bound, the situation was as follows. Ina situation in which the concepts (the size, energy,and entropy of the in-falling object) have clearmeaning, the Bekenstein bound was trivially trueand not very interesting. For example, consider agas consisting of many particles bouncing aroundin a box. Here the size of the system and itsenergy and entropy all have a clear meaning. TheBekenstein bound was true but not very interestingbecause it was satisfied by a very wide margin. Youcould ask, could you find a situation where theBekenstein bound is close to being saturated? Youcan accomplish this by considering not a wholegas of particles but a single particle in a box. Moreexactly, this gets close to saturating the Bekensteinbound if we can ignore the mass of the box, butthat is an unrealistic assumption. To get close tosaturating the Bekenstein bound, we really shouldconsider a single particle that at a given time isalmost certainly contained in a certain region inspace-time even though there is no box keeping itthere. (I say “almost certainly” because relativisticquantum mechanics does not permit us to saythat a particle is definitely present in a givenregion.) Here for a single particle, we can define itsenergy, and we can identify (within general limitsof relativistic quantum mechanics) the region thatit is confined in, but it is hard to make sense of theentropy of a single particle. For a long time, therewere many papers discussing this, many dozensand probably hundreds of papers, for the mostpart with limited insight. Then, there was a simpleand quite brilliant paper by Horacio Casini, whoshowed that the right concept is entanglement en-tropy and that it can always be defined in a naturalway and does enter in a universal Bekenstein-likebound. This paper was well ahead of the prevailingthinking in the field, and it was a number of years,I think, before people widely appreciated it.

Ooguri: For example, Casini’s paper solved thespecies problem that I had been puzzled about forsome time and gave a convincing explanation thatit’s not an issue.

Witten: There were what people thought werecounterexamples to the Bekenstein bound, andso some people, and I was one, thought that ifthe bound was true, it was a statement aboutquantum field theories that can be coupled togravity, not about all quantum field theories. ButCasini showed that this was completely wrong.He gave a precise meaning to all the terms thatentered the Bekenstein bounds, and he showedthat it was a universal statement of quantum fieldtheory that follows from general principles. Thatwas extremely illuminating and like other work onentanglement entropy, one suspects it’s probablyan important clue, but it might take a youngerperson than I with fresh thinking to see what it isan important clue to.

I do want to mention one more contribution inthat direction, which is by Casini and Maldacenawith Rafael Bousso and Zachary Fisher (BCFM).Years ago, Bousso had formulated a covariantversion of the Bekenstein bound; it is well adaptedto problems in cosmology. Everything I have saidabout the Bekenstein bound has an analog forthe Bousso bound. When you understood what itmeant, it wasn’t very interesting, and when it wasinteresting, you couldn’t understand what it meant.In the recent work of BCFM, a precise formulationand proof of the Bousso bound are given, at leastfor quantum field theory in flat space-time.

Ooguri: I see that this new joint activity betweenquantum gravity and quantum information theoryhas become very exciting. Clearly entanglementmust have something to say about the emergenceof space-time in this context.

Witten: I hope so. I’m afraid it’s hard to work on,so in fact, I’ve worked with more familiar kinds ofquestions. I have spent a lot of the last decade or soworking on a succession of problems that probablywere a little bit more out of the mainstream thanmost of my previous work. Also, I simply workedon these problems much longer than I usuallyworked on any one problem in the past. I guessthe three problems that best fit what I have justsaid have been gauge theory and the geometricLanglands program, gauge theory and Khovanovhomology, and superstring perturbation theory.

Superstring perturbation theory is best under-stood in terms of super Riemann surfaces. I shouldsay that much of my knowledge about super Rie-mann surfaces comes from things I have learnedfrom Pierre Deligne, both in the 1980s and morerecently. Super Riemann surfaces are a fascinatinggeneralization of ordinary Riemann surfaces toinclude odd or anticommuting variables. There isa fascinating algebro-geometric theory that waspartly developed in the 1980s and not pursued somuch since then. It would be great if it gets revived.By the way, we are having a workshop in May 2015


Toda, Yamazaki, Ooguri, and Witten.

at the Simons Center for Geometry and Physics inStony Brook [Supermoduli, May 18–22, 2015], andalgebraic geometers might be interested in it.

Khovanov HomologyMasahito Yamazaki: I was attending your lectureyesterday, and you were explaining how you cameto the idea that Khovanov homology can be writtenas N = 4 super Yang-Mills integrated over an un-usual integration cycle. One thing that impressedme there was that your previous papers were thecrucial input, namely your work with Anton Ka-pustin in which you formulated the Kapustin-Wittenequation, and also subsequent work you did withDavide Gaiotto on boundary conditions in N = 4super Yang-Mills theory. When you worked on thesepapers, did you already have in mind applicationto Khovanov homology?

Witten: The answer is “no”: in those years, I knewabout Khovanov homology and I was frustrated tonot understand it, but I had no idea it was relatedto geometric Langlands. I was frustrated at notunderstanding Khovanov homology, because I feltthat my work on the Jones polynomial ought to bea good starting point for understanding Khovanovhomology, but I just could not see how to proceed.(From a mathematical point of view, Khovanovhomology is a refinement or “categorification” ofthe Jones polynomial of a knot.) Actually, SergeiGukov, Albert Schwarz, and Vafa had alreadygiven (in 2004) a physics-based interpretation ofKhovanov homology, drawing in part on earlierwork of Ooguri and Vafa. But I found it perplexingand a little frustrating that the relation of this togauge theory was so indirect and remote. I wantedto find a more direct route, but for several years Ifound this difficult.

Eventually, however, some developments in themathematical literature helped me understand thatKhovanov homology should be understood using

the same ingredients that are used to understandgeometric Langlands. I didn’t understand all ofthese clues, but I learned from two of them.One was the work of Dennis Gaitsgory on whatmathematicians call quantum geometric Langlands(I am not sure this is the name a physicist woulduse) showing that the q parameter of quantumgeometric Langlands is related to the q parameterof quantum groups and the Jones polynomial.The other was the work of Sabin Cautis and JoelKamnitzer constructing Khovanov homology usinga space of repeated Hecke modifications. I did notinitially know what to make of those clues, butthey were a sort of red flag hanging out there.

Hecke transformations are one of the mostimportant ingredients in geometric Langlands.What they mean in terms of physics had botheredme for a long time and eventually had been the lastmajor stumbling block in interpreting geometricLanglands in terms of physics and gauge theory.Finally, while on an airplane flying home fromSeattle, it struck me that a Hecke transformationin the context of geometric Langlands is simply analgebraic geometer’s way to describe the effects ofa “’t Hooft operator” of quantum gauge theory. Ihad never worked with ’t Hooft operators, but theywere familiar to me, as they had been introduced inthe late 1970s as a tool in understanding quantumgauge theory. The basics of how to work with’t Hooft operators and what happens to them underelectric-magnetic duality were well known, so onceI could reinterpret Hecke transformations in termsof ’t Hooft operators, many things were clearer tome.

Cautis and Kamnitzer had interpreted Khovanovhomology in terms of the B-model of a space ofrepeated Hecke transformations. Kamnitzer alsoconjectured in another paper that there would bean alternative description in terms of an A-model ofthe same space. Technically, it was hard to find theright A-model. I really wanted to understand theA-model, because that was the approach in whichone could expect to achieve manifest three- or four-dimensional symmetry. My main goal in studyingKhovanov homology was to find a description withmanifest symmetry and a clear relationship to thegauge theory description of the Jones polynomial.I eventually succeeded in doing this. One of thetrickiest elements was that the gauge fields haveto obey a subtle boundary condition that I call theNahm pole boundary condition. (The basic idea thatleads to the Nahm pole boundary condition wasintroduced by Werner Nahm more than thirty yearsago in his work on magnetic monopoles.) Luckilyfor me, I was familiar with the Nahm pole boundarycondition and its role in electric-magnetic dualitybecause of work that I had done with DavideGaiotto a few years earlier.


I suspect that the mathematics world couldappreciate my work on Khovanov homology in theshort to medium term and that the obstacle to thisis largely a lack of familiarity with the Nahm poleboundary condition. With this in mind, I have beenworking with Rafe Mazzeo trying to give a detailedmathematical theory of that boundary condition.We have written one paper formulating rigorouslythe Nahm pole boundary condition in the absenceof knots, and we are trying to generalize this toinclude the knots. The necessary inequalities areavailable, but some details are not yet in place.

Masahito Yamazaki: I see. That’s a very nicestory of the physics-mathematics interaction. Youwere partly motivated by the important papers inmathematics and interpreted them as a physicist.Then you have your own physics story and you arenow trying to bring it back to mathematics.

Witten: As I have mentioned, the version thatCautis and Kamnitzer were actually able to un-derstand was the B-model. Since it doesn’t havemanifest three-dimensional symmetry, I decidedto concentrate on the A-model, but if I ever have acouple of months to spare, I would try to explainas a physicist the Cautis and Kamnitzer B-model.I’m reasonably optimistic I could do that and Ithink it would be illuminating. The only problem isthat there are a lot of things like that—interestingloose ends that I think I could clarify if I spend afew months on them.

Langlands Correspondence and GaugeTheory DualitiesOoguri: That the Langlands correspondence hassomething to do with S-duality was there even in thelate 1970s. When was it that you actually realizedthe significance of it?

Witten: I didn’t give the complete explanation ofmy interaction with Michael Atiyah in 1977. He toldme about two things that were new to me. One wasthe Montonen-Olive paper and the other was theLanglands correspondence, which plays a centralrole in number theory but which I had never heardof. He had noticed that the dual group of Langlandsand the dual group that enters the Montonen-Oliveconjecture (and which had been introduced earlierby Peter Goddard, Jean Nuyts, and Olive) were thesame. On this basis, Atiyah suspected that theLanglands correspondence has something to dowith the Montonen-Olive conjecture.

Ooguri: So that was in the late 1970s?Witten: It was December of 1977 or January of

1978. That was when I visited Oxford for the firsttime.

Ooguri: Did you take seriously already at thattime that the Langlands correspondence had some-thing to do with this gauge theory dynamics?

Witten: Well, I didn’t forget about it, but since, asI already told you, I was skeptical about Montonen-Olive duality, I didn’t seriously try to relate it toLanglands duality and I didn’t try to learn whatLanglands duality was. I did not learn anythingmore about these matters until the late 1980s. ThenI learned just superficially about the Langlandscorrespondence. If one knows even a little bit aboutthe Langlands correspondence and a little bit aboutconformal field theory on a Riemann surface, onecan see an analogy between them. I wrote a paperthat was motivated by that, but then I realized thatmy understanding was too superficial to lead toanything deep, so I abandoned the matter for anumber of years.

Ooguri: I remember when I was a postdoc atthe Institute for Advanced Study in 1988 and 1989,Robert Langlands himself was actually quite inter-ested in conformal field theory. I am not sure exactlywhich aspect he was interested in, however.

Witten: I don’t think he was motivated by theLanglands correspondence. But I think his workwas influential. Even though in a sense he didn’tprecisely make any major breakthrough himself,he helped to find the questions that stimulated thelater development of Stochastic Loewner Evolution,which has had a major impact on mathematicsand has even enlightened physicists about newways to think about some questions in conformalfield theory. I think Langlands was an influencebehind this work, but I do not believe his interestin conformal field theory was motivated by theLanglands correspondence or by gauge theorydualities. This is my impression from interactingwith him over the years.

As I have already remarked, in the late 1980s,after spending some time trying to develop theanalogy between conformal field theory and theLanglands correspondence, I concluded reluctantlythat the analogy in the form I was developingwas way too superficial, so I stopped. But thenaround 1990, I heard about new work of AlexanderBeilinson and Vladimir Drinfeld on the geomet-ric Langlands correspondence. This had a fewconsequences. First of all, it confirmed that myunderstanding of what the duality would mean inphysics was way too superficial. What they had wasmuch more incisive and much more detailed thanmy rather primitive analogy between the Langlandscorrespondence and conformal field theory. Theirwork confirmed that physics that I knew wasrelevant. But I was troubled, because they wereusing conformal field theory in a way that didn’tmake any sense to me. They studied conformalfield theory at negative integer levels—in physicspositive integers are more natural here—and usedit in ways that looked quite strange.


Familiar Ingredients That Seemed Not to FitAs I explained yesterday in my lecture at theKyoto Prize Symposium, for a number of years, the“volume conjecture” concerning the Jones polyno-mial (formulated and developed starting around2000 by Rinat Kashaev, Hitoshi Murakami, and JunMurakami, among others, and explained to me inlarge part by Sergei Gukov) bothered me. Althoughtheir statements bore a superficial resemblance tophysically well-motivated statements—in fact, tostatements that I myself had made in my originalpaper on the Jones polynomial in 1988—there wasa crucial difference. They seemed to have com-plex critical points that made exponentially largecontributions, and this normally is not possiblein physics. I am not sure if this point botheredanyone else, but it bothered me. It turned out thatthis was a good question to think about, sinceI eventually found a nice explanation, and thiswas a turning point in enabling me to understandKhovanov homology via gauge theory.

The work of Beilinson and Drinfeld on geometricLanglands bothered me in much the same way.They were using familiar ingredients of physics,but they were using them in ways that did notseem to fit. It looked like somebody had taken abunch of chess pieces, or perhaps here in JapanI should say a bunch of shogi pieces, and placedthem on the board at random. The way that thepieces were arranged did not make any sense tome. That bothered me, but I could not do anythingabout it.

Actually, the very little bit of what Beilinson andDrinfeld were saying that I could understand mademe wonder if the work of Nigel Hitchin would berelevant to them, so I pointed out to them Hitchin’spaper in which he had constructed commutingdifferential operators on the moduli space ofbundles on a curve. Differently put, Hitchin had ina certain sense quantized the classical integrablesystem that he had constructed a few years before.Although I understood scarcely anything of whatBeilinson and Drinfeld were saying, I did put themin touch with Hitchin’s work, and actually, intheir very long, unpublished foundational paperon geometric Langlands that you can find on theWeb, Beilinson and Drinfeld acknowledged me verygenerously, far overestimating how much I hadunderstood. All that had really happened was thatbased on a guess, I told them about Hitchin’s work,and then I think that made all kinds of thingsobvious to them. Maybe they felt I knew someof those things, but I didn’t. But anyway, therewere ample reasons in those years to think thatgeometric Langlands had something to do withphysics, but as you can see I still couldn’t makeany sense out of it.

Ooguri: So, what inspired you to return to this?

Witten: A decade later there was a workshopat the Institute for Advanced Study on geometricLanglands for physicists. Were you there?

Ooguri: I was invited, but there was a conflict ofschedule, so I couldn’t go. I missed it.

Witten: There were two long series of lecturesand then there were a couple of outliers. The longseries were very well done, but they did not helpme very much. Mark Goresky gave a long seriesof lectures aiming to tell physicists what is theLanglands correspondence. The only trouble forme was that to the extent that one can explain thistopic in a couple of lectures, assuming essentiallyno knowledge of algebra beyond the definition ofa field (in the algebraic sense), I was familiar withthe Langlands correspondence already. Namely, Ididn’t really know anything about it, but I knew asmuch as one could explain in a few hours startingfrom zero. So I couldn’t really get much out ofthose lectures.

In addition, Ed Frenkel (who had been the primemover behind the occurrence of this workshop)gave a series of lectures that, as far as I wasconcerned, were basically about the shogi boardon which the pieces have been arranged at random.I really couldn’t get much out of those lectureseither, because I already knew that people workingon the geometric Langlands were taking familiarpieces from the shogi set and arranging them onthe board at random as far as I was concerned.

There were a couple of additional lectures thatweren’t part of any series. One of them was byDavid Ben-Zvi. He told us about what was supposedto be an approximation to the geometric Langlandscorrespondence. I think he was talking largely aboutthe work of another mathematician, Dima Arinkin.What was supposed to be the approximation tothe geometric Langlands correspondence was T-duality on the fibers of the Hitchin fibration. Thiswas described by Ben-Zvi in a complex structurein which the fibers of the Hitchin fibration areholomorphic, so the T-duality is a holomorphicduality. It was already known to physicists that theT-duality on the fibers of the Hitchin fibration comesfrom Montonen-Olive duality in four dimensions,and of course ever since Atiyah’s observations of1977–78, I had been aware of the possibility thatsome version of the Langlands correspondencemight be associated to Montonen-Olive duality. Butwhat about the fact that Ben-Zvi was only claimingto deduce from T-duality an approximation togeometric Langlands duality rather than the realthing? At a certain point, I started to suspect thatthe reason for this was simply that Ben-Zvi wasdescribing the situation in the wrong complexstructure. The idea was that the same T-duality ofHitchin’s moduli space, viewed differently, wouldgive a mirror symmetry between a B-model in a


certain complex structure and an A-model in acertain symplectic structure. This mirror symmetrywas supposed to be the true geometric Langlandsduality, not the approximation. Actually, the reasonI started working on geometric Langlands withAnton Kapustin was that he had studied generalizedcomplex geometry in two-dimensional dualities. Inthat world, a family of dualities can degenerate,and a mirror symmetry can degenerate to aholomorphic duality.

When one starts thinking along these lines, itsoon makes a lot of sense that the geometricLanglands duality is really a mirror symmetry,which can degenerate to a holomorphic duality,and that this is the approximation Ben-Zvi taughtus about. I became convinced that that had to beright. There were still a few hurdles to overcome.The most difficult one I already described earlier.One does not get to first base with the Langlandscorrespondence without Hecke operators, so itwas necessary to have a physical interpretation ofHecke operators in terms of ’t Hooft operators ofgauge theory. It was also necessary to know howto interpret the A-model of the cotangent bundleof a complex manifold M in terms of differentialoperators on M . This actually was fairly close tothings that Kapustin had done earlier. Once thesepoints were understood, it was pretty clear to meas a physicist what geometric Langlands duality is.

I Felt Like I Discovered the Meaning of Lifeand Couldn’t Explain ItBut it was very hard to write a paper about it.It took about a year. For that year, I felt likesomeone who had discovered the meaning of lifeand couldn’t explain it to anybody else. And in asense, I still feel that way for the following reason.Physicists with a background in string theory orgauge theory dualities can understand my paperwith Kapustin on geometric Langlands, but formost physicists this topic is too detailed to be reallyexciting. On the other hand, it is an exciting topicfor mathematicians but difficult to understandbecause too much of the quantum field theoryand string theory background is unfamiliar (anddifficult to formulate rigorously). That paper withKapustin may unfortunately remain mysterious tomathematicians for quite some time.

Masahito Yamazaki: Maybe that means that wehave to wait an extra ten or fifteen years before…

Witten: We indeed may have to. I think it’sactually very difficult to see what advance in the nearterm could make the gauge theory interpretation ofgeometric Langlands accessible for mathematicians.That’s actually one reason why I’m excited aboutKhovanov homology. My approaches to Khovanovhomology and to geometric Langlands use manyof the same ingredients, but in the case of

Ooguri and Witten.

Khovanov homology, I think it is quite feasible thatmathematicians could understand this approach inthe near future if they get excited about it. I believeit will be more accessible. If I had to bet, I think Ihave a decent chance to live to see gauge theory andKhovanov homology recognized and appreciatedby mathematicians, and I think I’d have to belucky to see that in the case of gauge theory andthe geometric Langlands correspondence—just apersonal guess.

Yokinobu Toda: Do you think your idea of theS-duality and the geometric Langlands can be some-how applied to the honest Langlands program?

Witten: I see that as being far away. For mepersonally, it’s a dream that eventually numbertheory would make contact with physics sometime,but I doubt it will be soon.

There are all kinds of areas where specificnumber theory formulas appear in physics, andthese may be clues that the dream will come trueone day. But to really get me excited, somehow thenumber theory would have to enter the physicsin a more structural way. I’m not that interestedin a specific formula that comes out of a physicscalculation in a more or less ad hoc fashion.Number theory would have to be more integratedwith the physics to get me excited, and I don’t seethat happening soon.

In my work, I concentrated on the geometricform of the Langlands correspondence because Icould see that there was hope to really understandit in the context of the physics-based tools thatwere at hand. There might be something like thatone day for the Langlands correspondence ofnumber theory, but probably a lot is missing, andwe do not know what has to happen first. I feelthat the reason I was able to make progress wasthat my focus was much more narrow than tryingto understand the Langlands correspondence ofnumber theory.


Yokinobu Toda: The relationship between S-duality and geometric Langlands was surprising tome, as number theory seems to be a research areafar from physics.

Witten: Nevertheless, there have been manydevelopments which one day may be seen asimportant clues. One of the deepest was startedroughly fifteen years ago by Savdeep Sethi andMichael Green and then continued by Green withmany collaborators. In the original work, Sethi andGreen were trying to understand certain low energyR4 interactions in Type IIB superstring theory inten dimensions (here R is the Riemann tensor).They made an amazing discovery, I would say:the answer is given by a certain nonholomorphicEisenstein series of weight 3/2. Although myknowledge of number theory is very superficial,I think that this sort of thing is much closer tothe interests of modern number theorists than theaspects of classical modular forms that usuallyappear in two-dimensional conformal field theory.

Ooguri: Those objects which are not totally mod-ular have also appeared in number theory.

Witten: That is correct. A lot of things thatnumber theorists like have appeared in physics,and some have even appeared in my own work.Plenty has been found to show that the physicstheories that we work on as string theorists areinteresting in number theory. These theories knowsomething about number theory, but personallyI don’t see an opportunity to really make contactin a structural way with number theory in theforeseeable future. I can’t even formulate what itwould mean to make such contact, so I can’t evenproperly tell you what we can’t do but I think thetime is not right to do it.

Anyway, that’s why I personally concentrated ongeometric Langlands rather than on number theory,and geometric Langlands was hard enough. It was alot of work to understand it, but I think that havingunderstood it, many things that mathematiciansdo involving geometric aspects of representationtheory are much more accessible as part of physics.For example, I did not understand what HirakuNakajima explained yesterday at the Kyoto PrizeWorkshop, but I think that an understanding mightinvolve some of the things that were clear afterworking with geometric Langlands. I can’t promise,but it is worth a try.

Just one obvious thing is that although Nakajimadid not have time to explain the whole picture,at the end of his lecture he was telling us aboutthe affine Grassmannian. Isomorphism classes of’t Hooft operators are associated to cycles in theaffine Grassmannian, so if a mathematician tellsyou about the affine Grassmannian, you probablywant to think about at least part of what you arehearing in terms of ’t Hooft operators. I can make

no promises, but I feel it would definitely be wortha try to understand what Nakajima was sayingfrom a physicist’s viewpoint.

I am sure, at any rate, that there is much morethat can and should be done to understand muchmore of geometric representation theory from aphysical viewpoint. In fact, part of the original workof Beilinson and Drinfeld on geometric Langlandshas still not been understood to my satisfaction.Here I have in mind the use of conformal fieldtheory at what they call the critical level (level�h, where h is the dual Coxeter number) toconstruct the A-model dual of certain B-branes(the ones that are associated to opers, in thelanguage of Beilinson and Drinfeld). Davide Gaiottoand I obtained a few years ago a reasonableunderstanding of what electric-magnetic dualitydoes to the variety of opers, but I still do not reallyfeel I understand its relation to conformal fieldtheory. However, in the last few years physicistsworking on supersymmetric gauge theories in fourdimensions and their cousins in six dimensionshave made several discoveries involving the roleof conformal field theory at the critical level, sothe time may well be right to resolve this point.

How to Work with MathematiciansYokinobu Toda: I have a general question. Whatkinds of problems should mathematicians work on?

Witten: Well, there are lots of problems thatalgebraic geometers study that involve dualitiesstudied by physicists. In many of those cases, Iwill not be able to give much advice, as I am not anexpert on recent developments. In some cases, I amstill struggling to understand things that physicistsdid quite some time ago that are very relevant.Just to give one example, the Gopakumar-Vafa andOoguri-Vafa formulas have been very influentialfor algebraic geometers, but as a physicist, I wasnever satisfied that I understood them. So I actuallyspent a lot of time in the last year with a student(Mykola Dedushenko) trying to understand theseformulas better. In this work, I was doing some ofthe homework that I’d have to understand beforeeven trying to answer your question.

Ooguri: You will talk about it next week at theKavli IPMU [later published as a paper, “Some de-tails on the Gopakumar-Vafa and Ooguri-Vafa for-mulas,” arXiv:1411.7108].

Witten: Going back to Yukinobu’s question,although there are many areas of current intereston which I probably cannot give useful advice,there is one bit of advice that I actually would offerto algebraic geometers. I do recommend superRiemann surfaces. I’m sure there’s a deep theorythere. I can’t promise anything about how quicklyit will emerge. A deep theory will probably onlybe developed in the near term if enough people


get excited about it. Maybe the workshop we arehaving next spring at the Simons Center will helpmake that happen. We will see.

Ooguri: It’s certainly true that when people wereworking on the finiteness and vanishing of the cos-mological constant in perturbative string theorytwenty-five to thirty years ago, it was not totally sat-isfactory. The complete understanding came onlyafter your proper description in terms of geometryof super Riemann surfaces.

Witten: Thank you, Hirosi, and I’m glad youthink so. Not all physicists agree, because it ispossible to express everything in terms of picture-changing operators and so on, hiding the superRiemann surfaces. I think personally that whenone does that, one doesn’t understand properlywhat the formulas mean. But not everybody agrees.

I think one reason that the theory of superRiemann surfaces stopped developing in the 1980swas that physicists became satisfied with theirpartial understanding in which the super Riemannsurfaces were hidden. There is a tremendousbeauty to this subject that I think is simply missedif you try to understand things that way. I careabout it enough to have spent several years by nowon spelling out details of the description in termsof super Riemann surfaces.

It has seemed unclear that a lot of physicistswould really get excited about the sort of detailsthat I was trying to fill in. So one of my hopes hasbeen that mathematicians will get excited aboutdeveloping super Riemann surface theory. I can’tpromise, but I think they should.

Ooguri: Do you expect also new physics insightcoming out of the more precise understanding ofperturbative string theory?

Witten: The answer may depend on what youmean by physics insight. I think that one under-stands better what superstring perturbation theorymeans if one formulates it in terms of integrationon the moduli space of super Riemann surfaces.That is insight of a sort. However, I do not see anyevidence at the moment that incorporating superRiemann surfaces in the way that we understandperturbation theory will help us with nonperturba-tive questions, for example, or with understandingbetter the symmetry structure of string theory orwhatever may be the correct concept.

Masahito Yamazaki: Let me ask my last ques-tion. You’re working partly in the area of mathe-matical physics. You have a lot of discussion withmathematicians and also write math papers.

Witten: Well, I write math papers in very specialcases where I think something I could actually dowould be illuminating. Recent examples have beenmy work with Ron Donagi on some foundationalquestions about the moduli space of super Riemannsurfaces and the work with Rafe Mazzeo that I

mentioned before on the Nahm pole boundarycondition.

Masahito Yamazaki: I see. So, my question is,what’s the advice if a physicist wants to work witha mathematician effectively?

Witten: It’s really difficult to give advice. Usuallyproducing rigorous proofs requires very detailedmethods. That makes it hard for a physicist, and soI myself have only done that in very special caseswhere I thought something was really missing thatwas actually simple enough that I could help doit if I had the right collaborator. Some physicistswould want to go into more detail and learn thetechniques for rigorous proofs in a particular area,but most physicists I think will only be happy andsuccessful doing that in very special cases like theones I’ve picked.

Masahito Yamazaki: I see. Is it also true that inmany of your works, the conversation with somemathematicians has been an inspiration for you?

Witten: This usually happens when somethinga mathematician has done involves an aspect ofthe physics that hasn’t been understood and thatdoesn’t make sense to me. I mentioned earlierone case involving the volume conjecture. Foryears I could not understand the results in thisarea because complex critical points were makingexponentially large contributions. I kept putting itaside, not able to make progress.

Finally, in the summer of 2009, I attended aconference at the Hausdorff Institute in Bonn on thetwentieth anniversary of the Chern-Simons theory.I heard more lectures on the volume conjecture.To me, it was just embarrassing to not understandwhy exponentially large contributions were comingin. I feel vindicated in hindsight for worrying somuch about this question, because the answerturned out to be really useful.

Masahito Yamazaki: I see. In that case the feel-ing that the pieces are not in the right place led youto the question, which you eventually solved, andalso it led to new developments.

Witten: Yes. Another case was when I feltthat Beilinson and Drinfeld had the shogi piecesjumbled on the chessboard.

Message for Young StudentsOoguri: Pieces are placed in a wrong way, but ifyou look at it in different dimensions, perhaps theyare totally aligned.

I also have a final question. In the interview withTohru Eguchi twenty years ago in Sugaku Seminar,he asked you about the prospects at the interfaceof mathematics and physics, and you repliedsaying that the area had been certainly growingvery strongly and you predicted that the progresswould continue in the foreseeable future. Certainly


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your prediction has been amply verified in the lasttwenty years. Given this, my question is, again, whatis your prospect for the next twenty years? Can youalso give advice on the future of the field for youngstudents who will be reading this article?

Witten: First of all, in the last twenty years,not only has this interaction of math and physicscontinued to be very rich but it has developed insuch diversity that very frequently exciting thingsare done which I myself am able to understandembarrassingly little about, because the field isexpanding in so many directions.

I am sure that this is going to continue and Ibelieve the reason it will continue is that quantumfield theory and string theory, I believe, somehowhave rich mathematical secrets. When some ofthese secrets come to the surface, they often comeas surprises to physicists because we do not reallyunderstand string theory properly as physics—wedo not understand the core ideas behind it. At aneven more basic level, the mathematicians are stillnot able to fully come to grips with quantum fieldtheory and therefore things coming from it aresurprises. So for both of those reasons, I think thatthe physics and math ideas generated are going tobe surprising for a long time.

I think there are definitely exciting opportunitiesfor young people to come and help explain what itall means. We don’t understand this properly. Wegot a wider perspective in the 1990s when it becameclear that the different string theories are unifiedby nonperturbative dualities and that string theoryin some sense is inherently quantum mechanical.But we’re still studying many different aspects ofa subject whose core underlying principles are notclear. As long as that is true, there are opportunitiesfor even bigger discoveries by today’s young people.But if I could tell you exactly what direction youhad to go in, I would be there.

Ooguri: Thank you very much for taking timeto talk to us. It has been fun. Congratulations againon your Kyoto Prize.

Witten: Thank you so much for your kind wordson the Kyoto Prize, and also thank you for thediscussion, because the discussion has helped meremember how much we have advanced in the lasttwenty years.

Ooguri: Let’s meet again twenty years from nowto assess our progress in the next twenty years.

Witten: Let’s try. For that we will all have toexercise and keep fit.


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Interview with Edward Witten - ias.edu · Interview with Edward Witten Hirosi Ooguri This is a slightly edited version of an interview with Edward Witten that appeared in the December