Ashoke Sen and S-Duality Winner of Fundamental Physics Prize

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GENERAL ¨ ARTICLE Ashoke Sen and S-Duality Winner of Fundamental Physics Prize

Dileep Jatkar

The Fundamental Physics Prize Foundation de- clared names of nine scientists in a varietyof ar- easof theoreticalphysicsas recipients of the Fun- damentalPhysicsPrize.Thisarticle is a quick introduction to Ashoke Sen's scienti¯c achievements. In addition, this article containssome discussion of his S-duality proposal and the Sen Dileep Jatkar is a conjecture, which fetched him the Fundamental Professor at Harish- Physics Prize. Chandra Research Institute, Allahabad. He works in theoretical physics and his area of interest is string theory.

The poor man says,\O King, both of us are Loknath; while I am theone whose masters are thepeople, you are master of thepeople". 1. Citation of the FundamentalPhysics Prize The year 2012 marks the beginning of the Fundamen- talPhysicsPrizefounded by Russianentrepreneur Yuri Milner. In the inauguralyear the Foundation announced a list of nine recipients for their outstanding contribu- tion to various disciplines in theoreticalphysics.Asper their website, http://fundamentalphysicsprize.org, TheFundamental Physics Prize recognizes transforma- tive advances in the ¯eldof fundamental physics ... in- cluding advances in closelyrelated ¯elds with deep con- nections tophysics.

Ashoke Sen, a professor attheHarish-Chandra Research Keywords Institute, Allahabad, India is one of the nine recipients Fundamental Physics Prize of this Prize. The citation for which he is awarded this 2012, S-duality, Sen conjecture.

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Ashoke Sen prize (asstated onthewebsite) is received the [Ashoke Sen is awarded the Fundamental Physics Prize] Fundamental for uncovering strikingevidenceof strong-weakduality Physics Prize for in certain supersymmetric string theories andgauge the- providinga ories,opening thepath to therealization that all string definitiveevidence theories are di®erent limits of the same underlying in favour of S- theory. duality. Other recipients of the FundamentalPhysicsPrizeare: Nima Arkani-Hamed, JuanMaldacena,NathanSeiberg, Edward Witten all from Institute for Advanced Study, Princeton, USA, AlanGuth from MIT,Boston,USA, Andre Linde from Stanford University, Stanford, USA, Alexei Kitaev fromCalTech,Pasadena,USAand Maxim Kontsevich from IHES, France. It would perhaps be appropriate to say thattheFundamentalPhysicsPrize hasbecomeprestigious solely due to the stature of these inauguralrecipients. Certainly each of these recipients is a stalwart in his Ashoke Sen ¯eld of interest but it will not be anexaggeration to say thatAshokeSen being one of the recipients hasgiven further credence to the Prize. It is not becauseheisthe only Indianinthelistof recipients butbecause of his phenomenal achievements in string theory. He hasbeen spearheading severalnewdevelopments in string theory for over two decades. As mentioned above, Sen received this award for his work on strong{weakduality(also known asS-duality), which he did in the early 1990s, with a de¯nitive evidence for such a duality being provided by him in Febru- ary 1994. In additiontothis,someof his pioneering works include, formulation of string theory in background ¯elds, classi¯cation of conformal¯eld theories, string ¯eld theory (for open strings aswellas closed strings), tachyon condensation, and anenormousbodyof work on developing understandingof a class of black holes in string theory. Clearly it would be impossible to give even

324 RESONANCE ¨April 2013 GENERAL ¨ ARTICLE a °avour of some of these works in this article. We will During PhD and therefore restrict ourselves to a discussion of S-duality, post-doctoral the work cited for the Prize. However, I cannot resist fellowship he did giving a not-so-brief account of Ashoke Sen's academic pioneering work on timeline! QCD, grand 2. Tracing Academic Worldline unified theories, magnetic Ashoke Sen studied up to his matriculation in Shailendra monopoles and SircarVidyalaya in Calcutta (Kolkata). He obtained his string theory. Bachelor of Science (BSc) degree from Presidency Col- lege, Calcutta. He then joined the IndianInstitute of Technology, Kanpur, where he got his Masters degree in Physics. After that, he went to the State University of New York, Stonybrook, USA (now Stonybrook Uni- versity) to pursue his PhD. Here he worked under the guidance of George Sterman, who hadthenjustjoined the Institute for TheoreticalPhysics,Stonybrook, as a faculty and was working on QCD, the theory of strong interactions. Sen worked on QCD during his PhD, which wasentirelybasedonworkdoneall by himself.Ibelieve his ¯rst work, an80-page manuscript on `Asymptotic be- havior of the Sudakov form-factor in QCD',byitself was good enough to ful¯l the requirement of a PhD degree. It was a thorough study of a certain classof logarith- miccorrectionsinQCD,whoseimportance wasbetter appreciated by the QCD community almost a decade later. Sen then took up a post-doctoralposition atFermilab, Batavia, USA. During this period he worked on a variety of subjects and wrote about 23 papers within a spanof three years. This, in my opinion, was more of anexploratory period during which he worked on QCD, magnetic monopoles and their implications onproton stability, grand uni¯ed theories, supergravity, and string theory, which he worked on towards the end of his stay in Fermilab. His pioneering work on `The heterotic string in arbitrary background ¯eld 'was just the declaration of

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Ever since his his arrival in the ¯eld of string theory. After completing return to India, his post-doctoralpositionatFermilab hejoinedSLAC, Ashoke Sen has Menlo Park, USA where he spent a littlemorethantwo been an years. This was another period of proli¯cworkbutwas undisputed leader quite unlike thatinFermilab. By now hehadmade up in the field of string hismindtoworkinstring theory and all his 20 publica- tions from SLAC are onvarious aspects of string theory. theory. While severalof them are important contributions, con- ditions on space-time supersymmetry on the heterotic string compacti¯cations and analysis of various issues in string perturbation theoryhave had a lasting impact on the ¯eld. After spending two years at SLAC, Sen joined the faculty of TIFR atMumbai. His work in TIFR canbe broadly classi¯ed into three parts: Study and classi¯cation of rationalconformal¯eld theories;² Alarge body of workonstring ¯eld theory; ² Duality symmetries in string theory. ² It is the programme of studying duality symmetries, which fetched him the FundamentalPhysics Prize. There are di®erent kinds of duality symmetries in string theory. It mayseemodd to use bothduality and symmetry at the same time; however, string theory hascertain interesting properties due to which a single phenomenon has two descriptions. These dual descriptions are not dis- tinguishable for a low energy observer. Among a variety of dualities, strong{weakduality is one thatismoredif- ¯cult to understand. This is because it relates a weakly coupled theory to a strongly coupled theory. While one has better controlovercomputations in the weakly coupled theory, computations in the strong coupling limit are notoriously di±cult. We will see below the trick employed by Ashoke Sen, of studying a specialclass of states, called the BPSstates, to gain insight into this symmetry.

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After spending about seven and a half years in TIFR, Ashoke Sen moved to Harish-Chandra Research Insti- tute (HRI) (atthattimeknownasMehta Research Insti- tute of Mathematics and MathematicalPhysics),Alla- habad. His work on duality symmetries in string theory hadjustledtowhatisnowcalled `the second super- stringrevolution'. However, change of locationhadno adverse impact on research output. For example, some of his early works in HRI are now known as`The Sen limit of F-theory', `The Sen four manifolds' and `Sen{ Seiberg de¯nition of Matrix theory'. Althoughhewas the one who convinced the string theory community to pay attention to the spectrum of BPS states, he started focussing his attention on non-BPS states! In a series of papers on tachyon condensation and non-BPS states, he convinced the community thatthereismore interesting physics hidden there, just when most of thecommunity hadbarely begun to appreciate the importanceof BPS states. The work on tachyon condensation and its understanding in terms of a new formulation of string ¯eld theory was followed by another big programme of understanding entropy of a specialclass of black holes in string theory. A programme which is still being pursued! 3. S-duality We now focus our attentionontheS-duality.Before we get into the technicaldetails let us take a semi-technical tour of the implications of S-duality. Action of the S- duality transformation takes us from a strongly coupled description of one theory to a weakly coupled description of another theory. A coupling parameter in a theory is a measure of how constituents of the theory interact with each other. If this parameter is small, then constituents are weakly interacting but if this parameter is large, then they are strongly interacting. There are techniques like perturbation theory which give reliable results if the coupling parameter is small, orequiva- lently if the theory is weakly coupled, but this technique

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S-duality relates fails miserably if the theory is not weakly coupled. If S- strongly coupled duality transformationrelated a strongly coupled theory description of one to some other theory which is weakly coupled, then re- theory to weakly sults of the strongly coupled theory could be obtained by coupled description employing perturbation theory to the dualweakly cou- of another theory. pled description. To show thatS-duality gives us this This duality, along dual description, one needs to show that the physics of with other duality the original (strongly coupled) description and the dual symmetries in string (weakly coupled) descriptionisidentical. Since there theory, has shown are no controlled techniques of doing computations in that different string the strongly coupled theory,establishingsuchanequiv- theories are simply alence is no simpler than actually solving the strongly different weak coupled theory. The trick here is not to give the proof coupling descriptions of S-duality, for it would require solving both strongly coupled theory and weakly coupled theory exactly and of a single underlying showing thatthe results match, but to provide unequiv- theory, dubbed as the ocalevidenceof it. M-theory. Sen provided thiscrucial evidence by highlighting the fact thatthespectrum of a class of states, known asthe BPS states, in certain supersymmetrictheoriesdoesnot change aswechange the coupling parameter such that the theory changes fromweakly coupled to strongly coupled. This result provided the evidence that the theory considered by Sengets transformed into itself under S- duality transformation. His result led to a °urry of ac- tivity in understanding implications of S-duality in various theories and in many examples it wasshownthat S-duality maps the strong coupling limit of one theory to the weakcoupling limit of a di®erent theory. The most important and interesting implication of S-duality was found in string theory. Before the S-duality, there were ¯ve di®erent superstring theories and aneleven- dimensionalsupergravity, but S-duality combined with other duality symmetries of string theory showed that these ¯ve string theories and the eleven-dimensional supergravity are di®erent `weakcoupling limits' of a single master theory, theM-theory. After gathering a

328 RESONANCE ¨April 2013 GENERAL ¨ ARTICLE reasonable amount of evidence of S-duality relation be- Dirac proposed tween two theories, one can assume such a relation to magnetic monopoles be true and explore its implications and determine the in 1931; however, physics of strongly coupled system. Clearly the number there has been no of possibilities are unlimited. experimental We will now get into more technicaldetails of S-duality. signatures of them To understand S-duality, we need to preparesomeback- so far. ground. Let us start with classical electrodynamics, i.e., with Maxwell's equations. ~ E~ = ½; (1) r¢ ~ B~ =0; (2) r¢ @B~ ~ E~ = ; (3) r£ ¡ @t @E~ ~ B~ = J~ + ; (4) r£ @t where E~ is the electric ¯eld, B~ is the magnetic ¯eld, ½ is the electric charge density and J~ is the electric current density. If we compare (1) and (2), it is easy to see that on the right-hand side of (2) we do not have magnetic charge density. It implies thatthereis no magnetic analog of electric charges and consequently no magnetic current density in (3). This is also equivalently stated as: There are no magnetic monopoles in Nature. It is, however, interesting to note thatin the absence of electric charges, Maxwell's equations look quite symmetric. ~ E~ =0; (5) r¢ ~ B~ =0; (6) r¢ @B~ ~ E~ = ; (7) r£ ¡ @t @E~ ~ B~ = : (8) r£ @t

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They are invariant under theso-called electric{magnetic duality transformation, E~ B~ and B~ E~ : (9) ! !¡ Thus in free space, i.e. in the absence of both electric and magnetic sources, electric{magnetic duality transformation is a symmetry of Maxwell's equations. How- ever, this symmetry is explicitly broken in the presence of electric sources. It canbeformally restored if we pos- tulate the existence of magnetic monopoles, which are magnetic analogs of electric charges. Precisely this was proposed by Diracin1931, and hence goes under the name `Dirac monopole'. Let us look attheconsequences of Dirac's proposal. To understand Dirac's proposal, let us¯rstconsider a simpler situation of an electric ¯eld generated by a point electric charge e situated at the origin of three- dimensionalspace. This can be deduced by solving (1) with ½ = e±(~r), where ±(~r)istheDirac ±-function which ensures thatthecharge is located at ~r = 0, which is the origin of three-dimensionalspace. The solution is given by e~r E~ = : (10) 4¼r3 Let us now assume that, like the electric charge e,we have a magnetic charge g.Themagnetic ¯eld generated by this magnetic charge is g ~r B~ = : (11) 4¼r3 To determine the e®ectof this magnetic monopole charge, letusconsiderthemotionof a particle of mass m and electric charge e in the magnetic ¯eld generated by the monopole. The forceexperienced by this particle is given by the Lorentz force, d2~r e g d~r m = e~v B~ = ~r; (12) dt2 £ 4¼r3 dt £

330 RESONANCE ¨April 2013 GENERAL ¨ ARTICLE where ~v is the velocity of the electrically charged particle. We will now determine the change in the orbital an- gularmomentumof the charged particle duetothemag- netic ¯eld generated by the magnetic monopole. The orbital angularmomentumisgivenby d~r L~ = ~r p~ = m~r : (13) £ £ dt The change in the orbital angularmomentum is com- puted by studying the time variation of L~ ,

dL~ d2~r e g d~r d e g ~r = m~r = ~r ~r = : (14) dt £ dt2 4¼r3 £ dt £ dt 4¼ r µ ¶ This implies that the conserved angularmomentum of the charge{monopole system is e g ~r J~ = L~ : (15) ¡ 4¼r In classicalmechanics, we cansetthisresult aside as an interesting curiosity of the charge{monopole system, but in quantum mechanics, it leads to profound conse- quences. In quantum mechanics, angularmomentum is quantized. The change in the angularmomentum ef- fected by the monopole also hastosatisfythis quantization rule, e g 1 = n ; (16) 4¼ 2 ~ where ~ is the Planck's constant.Thisisknownas the Diracquantization condition. An immediate con- sequence of this result is quantization of the electric charge due to the presence of the magnetic monopole. Equation (16) can then be interpreted as follows. If the magnetic monopole of charge g exists in Nature, then all electric charges are quantized in units of 2¼~=g.In other words, only electric charges which are integer mul- tiples of 2¼~=g are quantum-mechanically well de¯ned in the background of a magnetic monopole of charge g.

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The Diracquantization condition can be used to understand how electromagnetic ¯elds would interact with a magnetic monopole. If we consider the motion of a free particle in three dimensions, we write the Hamiltonian as (p~)2 H = ; (17) free 2m where ~p is the momentum of the particle of mass m. If this particle has a charge e and it couples to the background electromagnetic ¯eld, then the Hamiltonian takes the form,

(~p eA~)2 Hem = ¡ + eA0 ; (18) 2m

where A~ is the electromagnetic vector potential and A0 is the Coulomb potential, which, fromnowon,willbe set to zero just for ease of presentation. Notice that coupling of the charged particle to the electromagnetic ¯eld is proportional to e, which we will refer to asthe electromagnetic coupling constant. If wecarry the out same procedure for a particle carrying magnetic charge insteadof electriccharge, then we will write the Hamil- tonian as(18)withe replaced by g andthevectorpo- tential A~ replacedbythedualvector¯eldA~.Wecan thus write the Hamiltonian for a particle with magnetic charge,

~ 2 2 (~p gA) 1 2¼ n ~ ~ Hdual = ¡ = ~p A ; (19) 2m 2m ¡ e µ ¶ wherewehave used (16) to write the last expression. If we now compare (18) and (19), we see that when electric coupling e is small, magnetic coupling g is large and vice versa.Wethus see that when electrically charged particles are weakly coupled, magnetically charged particles are strongly coupled. This implies thatif we want to study strong coupling phenomenon for electrically charged particles,weakcoupling description in terms of

332 RESONANCE ¨April 2013 GENERAL ¨ ARTICLE magnetic charges would be most suitable. This is known as the electric{magnetic duality, alternatively also known as the strong{weakduality. For this duality to work one needs magnetically charged objects to give a dual description of the theory of electrodynamics. Unfortunately no such magnetically charged objects exist in electrodynamics. However, moregeneral theories from which electrodynamics can bederived do admit magnetic monopole solutions. Although they are solutions of more generaltheories,theYang{Mills{Higgs theories, they behave exactly like the Dirac monopole in electrodynamics. Thus, if we want to test if strong{weak duality exists, the Yang{Mills{Higgs type theories provide an appropriate arena to set up testable questions and seek answers to them. The problem, in fact,ismore di±cult thanthisbecauseevenintheYang{Mills{Higgs system no reliable strong coupling computationcanbe done to test strong{weakduality. Luckily, there exist theories in which duality canbetested.These theories possess an additional symmetry known assupersymme- try. A detailed description of supersymmetry is beyond the scope of this article. Su±ce it to saythatitisa symmetry which relatesbosonstofermions andviceversa. Although supersymmetry is not observed in Nature, it is a very powerful computationaltooland plays a piv- otalroleingathering evidence for strong{weak coupling duality. 4. The Sen Conjecture Sen realized thatthepowerof supersymmetrycanbe put to good use for testing duality. He showedthat if strong{weak coupling duality is a symmetry of a supersymmetric theory then it imposes constraints onmulti- monopole solutions. He made a speci¯c conjecture about multi-monopole solutions in the maximally supersymmetric Yang{Mills theory. To understand this conjecture, let us consider anexample of motion of two parti-

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Sen combined the cles in three dimensions. Each particle is parametrized power of by its location in three-dimensionalspace. In other supersymmetry with words, we need tospecify three continuous parameters S-duality to per particle (or in totalsix parameters) for completely conjecture the specifying two-particle system. If these particles are existence of certain identical, then their positions are described by six pa- unique objects in the rameters up to exchange (of the two particles). Simi- multi-monopole larly, for a system of n identicalparticles, the parameter parameter space. space is 3n-dimensionalup to permutation of n parti- He supported his cles. Identical considerations apply to multi-monopole conjecture by systems, except thateach monopole is parametrized by explicitlyconstructing four parameters, threepositions and one phase associ- one for a ated with electromagnetic symmetry. This phase also implies thatthemonopole could carry electric charge, in 2-monopole system. addition to the magnetic charge. Thus the n-monopole system is parametrizedby4n parameters. Basedonthe assumption of S-duality symmetry, Sen [1] conjectured that For n-monopole system, there exists a unique function which can be integrated on the half dimensional subspace of 4n dimensional parameter space, for every electric charge carried by the system. He then explicitly constructed such a function for the 2- monopole system [1], providing compelling evidence for S-duality. This conjecture waspartly proved later by mathematicians [2], where they showed the existence of such functions but their uniqueness wasnotestablished. This result not only paved the waytounderstand strong coupling physics of certain supersymmetric theories but also played a pivotal role in `uni¯cation' of various string theories. In fact, the development in string theory trig- gered by this result is now referred to as`thesecond superstring revolution'. It would not be anexaggeration to saythat mostof the subsequent developments in string theory hinged on the understanding of various duality symmetries of stringtheory.

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5. The Magician Ashoke Sen is a recipient of essentially every science award in India aswellas severalinternationalhonours. Prominent among them are the Shanti Swarup Bhatna- gar Prize 1994, the Third World Academy Prize 1997, the Yukawa Medal 1989, the Infosys Prize2009, Pad- mashree 2001 and Padmabhushan 2013. He is a fellow of all three Indian science academies; INSA, INA and NAS. In addition, he is a Fellow of the Royal Society(FRS) since 1998 and a Fellow of the Third World Academy of Sciences since 2004. He has also been conferred an honorary doctorate by Calcutta University and Bengal Engineering and Science University. Apparently, the Nobel Laureate Hans Bethe hadonce said, \There are two types of genius. Ordinary geniuses do great things, but they leave you room to believe that you could do the same if only you worked hard enough. Then there are magicians, and you canhave no idea how they do it." Sen is a curious mix of these two types { while he is clearly a magician but with phenomenallog- icalclarity and mathematicalprecision,he also(while presenting his results) makes his listeners believe that they could also obtain such results if only they worked hard enough! Despite being one of the world leaders in the ¯eld of string theory, Sen is one of themost acces- sible and an extremely humble person. He is a living Address for Correspondence example of the following Sanskrit quote Dileep Jatkar Harish-Chandra Research Institute Chhatnag Road, Jhusi Allahabad 211 019, India. Branches of a tree laden with fruits benddownwards, Email: [email protected] likewise truegenius is humble. Suggested Reading

[1] Ashoke Sen, Dyon-monopole bound states, self-dual harmonic form on the multi-monopole moduli space, and SL(2,Z) invariance in string theory, Phys. Lett. B,Vol.329,p.217, 1994. [hep-th/9402032]. [2] Graeme Segal and Alex Selby,Thecohomology of the space of magnetic monopoles, Commun. Math. Phys., Vol.177, p.775, 1996.

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