We report on recent advances in the understanding of non-perturbative
phenomena in the quantum theory of fields and strings.Comment: 6 pages, latex, uses stwol.sty; Plenary talk given at International
  Conference on High Energy Physics (ICHEP), Warsaw, July 199

Ferrara, Sergio

English

arXiv

We report on recent advances in the understanding of non-perturbative phenomena in the quantum theory of fields and strings.We report on recent advances in the understanding of non-perturbative phenomena in the quantum theory of fields and strings

CERN Document Server

arXiv:hep-th/9610085v1  11 Oct 1996RECENT DEVELOPMENTS IN NON-PERTURBATIVE QUANTUM FIELD THEORYSergio FERRARATheoretical Physics Division, CERN, 1211 Geneva 23, SwitzerlandWe report on recent advances in the understanding of non-perturbative phenomena in the quantum theory offields and strings.1 IntroductionIn the last couple of years we have been witnessinga series of impressive results in the framework oftwo different but related types of quantum theo-ries: supersymmetric Yang–Mills theories in fourdimensions1 and superstrings2. They have in com-mon space-time supersymmetry. They also sharethe property that a formal perturbative expansioncan be defined for these theories, and “renormal-ization” can be used in order to extract finite an-swers from any perturbative calculation.These theories enjoy non-renormalization the-orems3 depending on the degree of supersymmetryof the vacuum around which we define the quan-tum perturbative series.Let us recall the spin content of the lightstates: the gauge theory includes massless quantawith spin 0, 1/2, 1; superstrings include masslessquanta with spin 0, 1/2, 1, 3/2 and 2; in a suitablelimit (α′ → 0, gauge-coupling fixed) superstringsreproduce ordinary gauge theories.Exact non-perturbative results of SQCD havebeen obtained. Seiberg and Witten 4 suggested anexact expression for the low-energy effective actionfor the Coulomb phase of an N = 2 SU(2) SYMtheory, which may be regarded as an extension ofthe Georgi–Glashow SU(2) gauge theory.Electric–magnetic duality plays a crucial roleto solve the theory, to compute strong couplingphases of this theory (where massless monopolesand dyons appear), and to prove colour confine-ment through a magnetic Higgs mechanism witha monopole condensation (analogue to Meissnereffects in superconductors).The solution of the problem is possible thanksto non-renormalization theorems, making thecomplete perturbative computation affordable.The non-perturbative part, conjecturally dueto processes from multi-instanton transitions, isobtained from a mathematical hypothesis thatidentifies electric and magnetic massive stateswith windings (n,m) of a genus 1 (torus) ellip-tic Riemann surface (genus n for an SU(n) gauge-theory as shown by Klemm, Lerche, Theisen andYankielowicz and by Argyres and Faraggi 5). Twotopologically different cycles correspond to elec-tric and magnetic charges. Quantum massive BPSstates ψ(n,m) correspond to distinct topologicalconfigurations (n,m) of the elliptic surface, in par-ticular ψ(1, 0) = W -boson, ψ(0, 1) = monopole,ψ(−1,+1) = dyon.2 Electric–Magnetic Duality and Super-symmetryLet us recall the duality of Maxwell equations∇ · (E + iB) = ρe + iρm = ρ∇ ∧ (E + iB)− i ∂∂t (E + iB) = Je + iJm = JL = Re(E + iB) · (E + iB) = E2 −B2 .Here E,B are the electric and magnetic fields;ρe(ρm), Je(Jm) denote electric (magnetic) chargedensity and current, respectively. A topologicalterm E · B may eventually be added to L. Thephysical observables such as the energy density(E+iB) ·(E−iB) = E2+B2, and the momentumdensity (E + iB) ∧ (E − iB) are invariant under(continuous) U(1) duality rotations:(E+ iB)→ eiϕ(E+ iB) , ρ→ eiϕρ , J → eiϕJ .In particular the Z2 symmetry, which is the rem-nant of U(1), acting on discrete charged states,exchanges electric with magnetic fields E →B andB → −E, and electric and magnetic chargesq → g, g → −q, accordingly.The simultaneous occurrence of electric andmagnetic sources implies a charge quantization,which reads:qg = 2pik(Dirac 1931) 6(for monopoles)1andq1g2 − q2g1 = 2pik(Schwinger, Zwanziger 1968) 7(for dyons).In the Coulomb phase the Georgi–Glashow SU(2)gauge theory has a monopole with mass (’t Hooft,Polyakov 1974) 8:Mmonopole ≥ 1/λ〈φ〉(Bogomolny bound, 1975) 9while the classical vector boson mass is MW =λ〈φ〉. In the Prasad–Sommerfeld (1976) limit 10(supersymmetry)Mmonopole = 1/λ〈φ〉 satisfies theduality conjecture (Montonen, Olive 1977) 11:M2(q, g) =M2(q2 + g2) = 〈φ〉2(q2 + g2) .This generalizes when a topological term θE · Bis included by defining a complex parameter τ =θ + i/λ2 and then writing:M2(φ, τ, n,m) =|φ2|lmτ|n+ τm|2invariant under SL(2, Z):Z2 : τ →−1τ, n→ m , m→ −nθ − shift : τ → τ + 1 , n→ n−m , m→ m .This means that the dual theory obtained by a Z2symmetry E → B,B → −E has τd = −1/τ, nD =m,mD = −n. The N = 4 supersymmetric Yang–Mills theory realizes the Montonen–Olive dualityconjecture 11. The theory has an exact SL(2,Z)symmetry 12,4, which is possible in virtue of avanishing β function, in the full quantum theory.Electric states are fundamental, while magneticstates are solitons in the theory T , but their roleis reversed in the dual theory TD.Seiberg and Witten 4 extended the duality toN = 2, SYM quantum field theories undergoingrenormalization (β 6= 0), which gives correctionsto a ‘holomorphic prepotential’, F (φ); this is theappropriate tool to build up N = 2 effective ac-tions. The BPS states (which lie in hypermul-tiplets) have mass M(φ, n,m, λ) ∝ |φn + Fφm|where13 F (φ) = (i/2pi)φ2 ln(φ2/λ2)+ . . . (the dotsdenote the non-perturbative contributions).They also extended the duality conjecture.This came by identifying the pair (φ, Fφ) with theperiods of an hyper-elliptic surface, which allowsus to give a closed expansion for F (φ). As a resultof this at strong coupling φ2/λ2 = ±1, one gets amassless monopole (0,1) and a dyon (−1,+1).The dual (U(1) magnetic) theory is weaklycoupled in the strong coupling of the electrictheory and describes a magneto-dynamic of acharged monopole. In the weakly coupled mag-netic Higgs phase, monopole condensation de-scribes confinement of the original (strongly cou-pled) electric theory. It is worth mentioning that,for BPS states, their mass appears in the centralextension of the supersymmetry algebra (Haag–Lopuszanski–Sohnius) 14 and this allows one, us-ing supersymmetry 15, to compute their mass interms of the low-energy data. The duality hasbeen further extended to N = 1 super-Yang–Millstheories 16, in particular to SQCD with colourgroup SU(Nc) and Nf flavours. This theory hasan anomaly-free global symmetry:SUL(Nf )× SUR(Nf )× U(1)B × U(1)R .Seiberg suggested that there is a non-AbelianCoulomb phase for 3/2Nc < Nf < 3Nc. Atthe non-trivial infra-red fixed point, the theoryof quarks and gluons has a dual description interms of an interacting conformal invariant the-ory with magnetic gauge group SU(Nf −Nc) andNf flavours. Quarks and gluons are solitons in thedual picture.3 Supergravity, Strings and M-TheoryDuality symmetries in the context of supergrav-ity theories 17, further extended to superstrings 18,allow us to prove exact equivalences of differentstring theories 19–21, to obtain a dynamical un-derstanding of the Seiberg–Witten conjecture inthe point-particles limit 22–23 and finally to pos-sibly merge these theories in the context of M -theory, a supposedly existing quantum theory ofmembranes and five-branes24,25, whose low-energyeffective action is 11D supergravity 26.There are five known types of superstring the-ories in 10 dimensions 2:Type Gauge groupType I SO(32)Heterotic SO(32), E8 × E8Type IIA U(1)Type IIB None2The first three have N = 1 supersymmetry,while the last two have N = 2, non-chiral typeIIA and chiral IIB. There is also a conjecturedM -theory in 11 dimensions 21 (no gauge group).Upon reduction on a circle, this is equivalent totype IIA, at the non-perturbative level. A furtherspeculative theory may exist in twelve dimensions,which gives, upon reduction on a two-torus, thetype IIB theory 27.The previous theories, and their compactifi-cation to lower dimensions, reduce at low energiesto supergravity theories in diverse dimensions 28with underlying supersymmetry algebras as clas-sified by Nahm 29. In the highest and lowest di-mensions of interest we have for instance:D = 11, N = 1, 128boson + 128fermions(b = 44 + 84, f = 128)D = 10, N = 1 (chiral)N = 1 (matter)(G = E8 × E8, SO(32))N = 2 : type A (non-chiral),type B (chiral)D = 4, N = 1 (chiral) : obtained as M -theoryon M7 = CY3 × S1/Z2N = 8 (non-chiral) (b = 56 + 70 + 2,f = 112 + 16) : obtained asM -theory on T7(U(1)28 gaugegroup)or on S7(SO(8) gauge group) .Let us summarize some of the main basic re-sults of the years 94–96, in the context of stringtheory and its non-perturbative regime.1) The Seiberg–Witten solution of rigid N = 2theory generalizes to heterotic-type II dual-ity relating K3 × T2 vacua of heterotic toCalabi–Yau vacua of type II strings 30,31.The second quantized mirror symmetry 30gives exact non-perturbative results in N =2 superstrings, D = 4. In particular, dualityrelates world-sheet instanton effects on thetype II side to space-time instantons on theheterotic side 22,32. Dual pair heterotic-typeII theory constructions were proposed.2) The implication of string–string duality insix dimensions for S-T duality at D = 4was first shown by Duff 19, and U -dualityas a non-perturbative symmetry of differentstring theories was formulated by Hull andTownsend 20.3) Witten21 proved the equivalence of differentstring theories in higher dimension and theduality of type IIA at strong coupling with11D supergravity at large radius (M -theoryon M10 × S1). Type IIB is self-dual at D =10 (SL(2, Z) duality) 34.4) The E8 × E8 heterotic string at strong cou-pling is dual to theM -theory onM10×S1/Z2(Horava–Witten) 33.5) The SO(32) Type I and SO(32) heterotic atD = 10 are interchanged by weak–strongcoupling duality (Polchinski–Witten) 35.6) Open strings naturally arise, by the mecha-nism of tadpole cancellations, as sectors oftype IIB closed strings on orientifolds 36,37.Their end-points end on D-branes 38, carry-ing R–R charges. Phase transitions in sixdimensions are possible 39, and evidence fora non-perturbative origin of gauge symme-tries 40 was substantiated 41.7) T -duality atD = 9 relates type IIA and typeIIB theories, as well as SO(32) and E8 ×E8 heterotic strings in their broken phaseSO(16)×SO(16) 35.8) M -theory and strings may undergo a fur-ther unification in twelve dimensions (F -theory) 27.New predictions of non-perturbative stringtheories can be derived from these non-perturbative relations between the five seeminglydifferent superstring theories. As a circumstantialexample42, strongly coupled heterotic string meetsthe agreement of αGUT as measured (at LEP) fromlow-energy data.In weakly coupled heterotic string, compacti-fied on a Calabi–Yau threefold of size V ≈M−6GUTwith GN =e2φ(α′)64piV ,αGUT = e2φ(α′)3/16piV → GN = αGUTα′/4 .If e2φ ≤ 1, GN ≥ α4/3GUT /M2GUT , which is too largecompared to experiment.3In type I string (weak coupling) ,αGUT =eφI (α′)316piV, GN =e2φI (α′)64piV→ GN = eφIαGUTα′/4 .Here, GN can be small.In the M -theory set-up (κ is the 11D gravita-tional coupling and ρ the compactification radius)GN =k216pi2V ρ, αGUT =(4pik2)2/32V≪ 1 .So no disagreement with the experimental inputexists in principle.Finally, supersymmetry breaking can be de-scribed in a natural way both through a stronglycoupled hidden gauge sector leading to gauginocondensation 43 and through the no-scale struc-ture44 ofM -theory. The decompactification prob-lem may be avoided 45–47.4 ConclusionsIn the talk he gave at the Stony-Brook conferencein 1979, Murray Gell-Mann 48 outlined the prob-lems encountered with N = 8 supergravity 49:1) “Predict a gauge group SO(8) that does notcontain SU(3)× SU(2) × U(1);2) ‘Sign error’ in the prediction of the cosmo-logical constant λ ≈M4Planck!3) We can use lower-N supersymmetric the-ories, but then we have a lack of unifica-tion ...”It may be surprising to realize that the un-physical theory discussed by Gell-Mann 17 yearsago is just a different vacuum of the very same the-ory whose dynamics hopefully encodes the stan-dard model of our low-energy world!References1. For reviews on Supersymmetric Gauge The-ories, see for instance:J. Bagger and J. Wess, Supersymmetry andSupergravity (Princeton University Press,1991);S. Ferrara, Supersymmetry (North Holland–World Scientific, 1987).2. For a review on Superstrings, see: M. Green,J. Schwarz and E. Witten, Superstring The-ory (Cambridge University Press, 1987).3. J. Iliopoulos and B. Zumino, Nucl. Phys.B76 (1974) 310;S. Ferrara, J. Iliopoulos and B. Zumino,Nucl. Phys. B77 (1974) 413;M.T. Grisaru, W. Siegel and M. Rocˇek,Nucl. Phys. B159 (1979) 429;L. Girardello and M.T. Grisaru, Nucl. Phys.B194 (1982) 65.4. N. Seiberg and E. Witten, Nucl. Phys.B426 (1994) 19 and B431 (1994) 484.5. A. Klemm, W. Lerche, S. Theisen and S.Yankielowicz, Phys. Lett. B344 (1995) 169;P. Argyres and A. Faraggi, Phys. Rev. Lett.74 (1995) 3931.6. P.A.M. Dirac, Proc. Roy. Soc. A33 (1931)60.7. J. Schwinger, Science 165 (1969) 757;D. Zwanziger, Phys. Rev. 176 (1968) 1489.8. G. ’t Hooft, Nucl. Phys. B79 (1974) 276;A.M. Polyakov, JETP. Lett. 20 (1974) 194.9. E.B. Bogomolny, Sov. J. Nucl. Phys. 24(1976) 449.10. M.K. Prasad and C.M. Sommerfeld, Phys.Rev. Lett. 35 (1975) 760.11. C. Montonen and D. Olive, Phys. Lett. 72B(1977) 117.12. L. Girardello, A. Giveon, M. Porrati and A.Zaffaroni, Phys. Lett. B334 (1994) 331.13. N. Seiberg, Phys. Lett. B206 (1988) 75.14. R. Haag, J.T. Lopuszanski and M. Sohnius,Nucl. Phys. B88 (1975) 257.15. E. Witten and D. Olive, Phys. Lett. 78B(1978) 97.16. N. Seiberg, Phys. Rev. D49 (1994) 6857.17. E. Cremmer, S. Ferrara and J. Scherk, Phys.Lett. 74B (1978) 61;S. Ferrara, J. Scherk and B. Zumino, Nucl.Phys. B121 (1977) 393.18. A. Sen, Int. J. Mod. Phys. A9 (1994) 3707;J.H. Schwarz, Lett. Math. Phys. 34 (1995)309.19. M. Duff, Nucl. Phys. B442 (1995) 47.20. C. Hull and P. Townsend, Nucl. Phys. B438(1995) 109.21. E. Witten, Nucl. Phys. B443 (1995) 85.22. A. Ceresole, R. D’Auria and S. Ferrara,Phys. Lett. 339B (1994) 71;A. Ceresole, R. D’auria, S. Ferrara and A.4van Proeyen, Nucl. Phys. B444 (1995) 92.23. S. Kachru, A. Klemm, W. Lerche, P. Mayrand C. Vafa, Nucl. Phys. B459 (1996) 537.24. M.J. Duff and J.X. Lu, Nucl. Phys. B426(1994) 301.25. E. Witten, preprint IASSNS-HEP-9563(hep-th/9507121), contribution to Strings’95, Los Angeles, 1995.26. E. Cremmer, B. Julia and J. Scherk, Phys.Lett. 76B (1978) 409.27. C. Vafa, Nucl. Phys. B469 (1996) 403.28. A. Salam and E. Sezgin, Supergravity indiverse dimensions (North-Holland–WorldScientific, 1989).29. W. Nahm, Nucl. Phys. B135 (178) 149.30. S. Ferrara, J. Harvey, A. Strominger and C.Vafa, Phys. Lett. B361 (1995) 59.31. S. Kachru and C. Vafa, Nucl. Phys. B450(1995) 69.32. A. Strominger, Nucl. Phys. B451 (1995)97;B. Greene, D. Morrison and A. Strominger,Nucl. Phys. 451 (1995) 109.33. P. Horava and E. Witten, Nucl. Phys. B460(1996) 506.34. For recent reviews. see:J. Schwarz, preprint CALTECH-68-2065(hep-th/9607201);A. Sen, preprint MRI-PHY-96-28 (hep-th/9609176);M. Duff, preprint CPT-TAMU-33-96 (hep-th/9608117).35. J. Polchinski and E. Witten, Nucl. Phys.B460 (1996) 525.36. A. Sagnotti, Open Strings and their Sym-metry Groups, talk at the Carge`se SummerInstitute (1987);G. Pradisi and A. Sagnotti, Phys. Lett.B216 (1989) 59;M. Bianchi, G. Pradisi and A. Sagnotti,Nucl. Phys. B376 (1992) 365.37. J. Dai, R. Leigh and J. Polchinski, Mod.Phys. Lett. A4 (1989) 2073;P. Horava, Nucl. Phys. B227 (1989) 461;J. Polchinski, Phys. Rev. D50 (1994) 6041.38. J. Polchinski, Phys. Rev. Lett. 75 (1995)4734.39. A. Sagnotti, Phys. Lett. 294B (1992) 196.40. E. Witten, Nucl. Phys. B471 (1996) 121.41. M. Duff, R. Minasian and E. Witten, Nucl.Phys. B465 (1996) 44.42. E. Witten, Nucl. Phys. B471 (1996) 121.43. P. Horava, preprint PUPT-1637 (hep-th/9608019).44. E. Cremmer, S. Ferrara, C. Kounnas andD.V. Nanopoulos, Phys. Lett. B133 (1983)61;J. Ellis, A. Lahanas, D.V. Nanopoulos andK. Tamvakis, Phys. Lett. B134 (1984) 419;J. Ellis, C. Kounnas and D.V. Nanopoulos,B241 (1984) 406.45. T. Banks and M. Dine, preprint SCIPP-96-26 (hep-th/9609046); preprint RU-96-27(hep-th/9605136).46. E. Kiritsis, C. Kounnas, P.M. Petropou-los and J. Rizos, preprint CERN-TH/96-90(hep-th/9608034).47. I. Antoniadis and M. Quiros, CPTH-S465-0996 (hep-th/9609209).48. M. Gell-Mann in Supergravity, eds.D.Z. Freedman and P. van Nieuwenhuizen(North-Holland, Amsterdam, 1979), p. 315.49. J. Cremmer and B. Julia, Nucl. Phys. B159(1979) 141;B. de Wit and H. Nicolai, Nucl. Phys. B208(1982) 323.50. L. Alvarez-Gaume´, J. Distler, C. Kounnasand M. Marin˜o, Int. J. Mod. Phys. A11(1996) 4745.QuestionsG.G. Ross, Oxford University:What progress has there been towards break-ing supersymmetry and using duality to give in-formation about normal QCD?S. Ferrara:Alvarez-Gaume´, Distler, Kounnas andMarin˜o 50 have analysed soft breaking terms pre-serving the analyticity properties of the Seiberg–Witten solution. This allows a detailed descriptionof the onset of the confinement transition and thepattern of chiral symmetry breaking. When thoseresults are extrapolated to a limit where super-symmetry decouples and then QCD is retrieved,an indication that the QCD vacuum may requirethe simultaneous occurrence of mutually non-localdegrees of freedom (monopoles and dyons) seemsto emerge.G. Veneziano, CERN: In order to get a successful5relation between the Planck scale and the GUTscale is it crucial that the original type II theoryis strongly coupled or can the coupling be just atthe self-dual value = 1?S. Ferrara:The strong coupling in heterotic theory meansthat its dual theory is weakly coupled. Thereforethe self-dual value seems not appropriate for thisregime.6

Recent developments in non-perturbative quantum field theory

Abstract

Similar works

Full text

Available Versions

CERN Document Server