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a rXiv:h ep-th/02615v117J un22June 17,2002YITP-SB-02-33SUPERWAVES W.Siegel 1C.N.Yang Institute for Theoretical Physics State University of New York,Stony Brook,NY 11794-3840ABSTRACT We give some supersymmetric wave solutions,both chiral (selfdual)and nonchiral,to interacting supersymmetric theories in four dimensions.21.IntroductionExact wave solutions to interacting theories[1]have recently drawn attention for application to the AdS/CFT correspondence[2].In Yang-Mills theory they take the form(neglecting matter sources)dx m A m=dx+A+(x+,x i),(∂i)2A+=0and in gravitydx m dx n g mn=dx m dx nηmn+dx+dx+g++(x+,x i),(∂i)2g++=0In the absence of sources,they describe free particles in interacting theories;in the presence of sources,they describe free particles produced by the interaction of parti-cles of lower spin.In this paper we consider supersymmetric generalizations of such solutions.They describe particles of various spins,related by supersymmetry;in some solutions there are no interactions,in others,some act as sources for the highest spin.plex solutionsIn quantumfield theory one begins with a classical solution to thefield equations as a vacuum,and perturbs about it:The perturbations include both tree and loop graphs.Generally real Euclidean solutions are considered,as appropriate to the Euclidean path integral.Two types of real Euclidean solutions to interactingfield theories have been con-sidered:(1)Those that have time-reversal invariance with respect to the coordinate to be Wick rotated remain real after analytic continuation to Minkowski space.(The symmetry t→−t becomes complex conjugation invariance it→−it,i.e.,reality.) This includes static solutions,such as monopoles or spherical black holes.(But see, e.g.,the use of Euclidean space to eliminate black hole singularities[3].)Nonstatic solutions include certain cosmological ones,with appropriate choice of time coordi-nate.(Euclidean space has also been proposed to eliminate singularities in cosmology [4].)(2)More complicated time-dependent solutions,such as instantons,can become complex upon continuation,so their validity has sometimes been questioned.(Instan-tons are complex in Minkowski space because there selfduality is a complex condition on thefield strength.)Since wave solutions by definition propagate at the speed of light(as distinguished from solitons,etc.),we need to consider the general validity of3such complex solutions.We begin by looking at such solutions that are themselves perturbative.Since we are considering classical solutions,such perturbative solutions corre-spond to tree graphs.It is generally stated that tree graphs of quantumfield theory are equivalent to classicalfield theory[5].However,there are two important differ-ences:(1)Classicalfield theory requires retarded(or advanced)propagators to pre-serve reality,while quantumfield theory requires the complex St¨u ckelberg-Feynman propagator.(In time-3-momentum space,compare the retarded2θ(t)sin(ωt)or ad-vanced−2θ(−t)sin(ωt)to the St¨u ckelberg-Feynman ie−iω|t|.)This requirement is equivalent to analytic continuation from real solutions in Euclidean space.(Consider ie−iω|t|vs.e−ω|t|under complex conjugation,including p→− p.)(2)While classicalfield theory uses the same realfield for each external line, quantumfield theory uses different one-particle wave functions for each line,and they are complex in Minkowski space,from choosing either positive(initial)or neg-ative(final)energies for each line,by the same considerations as for the propagator. (Similar remarks apply for theories with multiplefields,or with respect to reality properties for complexfields,which can always be re-expressed in terms of real ones.) But any such solutions that are,e.g.,traveling in a given direction are also complex in Euclidean space,since spatial momentum is not Wick rotated.(However,such so-lutions can be real in two space and two time dimensions.)Thus,external line factors can result in complex solutions in both Minkowski and Euclidean space in quantum field theory.In particular,for solutions propagating at the speed of light it is not possible to eliminate the problem by choice of time axis.(In classicalfield theory, adding the complex conjugate solution to make a real solution can still give a wave propagating in a given direction,since changing the signs of both the momentum and energy leaves the velocity invariant.)Thus we are led to consider only complex solutions that are consistent with the requirements of perturbation theory.In fact,wave solutions are perturbative:Their contributions to thefield are linear in the coupling,and satisfy the non-self-interacting Laplace equation(perhaps with sources)in two fewer dimensions;they take the ex-act same form as in the Abelian theory.(Yang-Mills wave solutions are the same as Abelian ones;gravitational wave solutions are the same as theflat metric plus an Abelian spin-2field.)However,this requirement generalizes to nonperturbative solu-tions,by analytic continuation from regions of spacetime where they can be expressed perturbatively.(For example,the Schwarzschild solution can be derived perturba-tively,at least outside the event horizon[6].)43.Selfdual solutionsIn particular,in four dimensions these wave solutions to the two-dimensional Laplace equation decompose into the sum of parts analytic and antianalytic in the complex coordinates for those two dimensions.Treated as a classical theory,the analytic and antianalytic terms must be complex conjugates to preserve reality;but in quantumfield theory the two terms correspond to positive and negative helicity, and can be treated as independent.In practice,we set one term to vanish to describe a particle of definite helicity.Truncation to states of given helicity corresponds directly to truncation of the theory to the selfdual theory,as has been demonstrated both in Feynman diagrams [7]and directly in the action[8].(This differs from early actions for selfdual theories [9],which not only did not correspond to truncations of nonselfdual theories,but violated Lorentz invariance and even dimensional analysis.Also,when coupling self-dual gravity to selfdual Yang-Mills,they violated Lorentz invariance and selfduality of thefield equations themselves.)Here we will concentrate our attention to max-imally symmetric theories,since in the lightcone they can be described by a single (super)field.In the lightcone,the action for(super)Yang-Mills theory can be written as the sum of(1)the kinetic term,(2)the selfdual(3-point)vertex,(3)the antiselfdual vertex,and(4)the4-point vertex[10].The(polynomial form of)the lightcone gauge for the selfdual theory[11]is described in terms of the samefields,but keeping just thefirst two terms in the action[12]:S=tr d4x d4θ[13φ(∂[−φ)(∂t]φ)]where∂t is the derivative with respect to the complex coordinate.Since any so-lution to the selfdual theory is also a solution to the full theory,it is sufficient to consider this truncated action.Since the action contains no derivatives with respect to the anticommuting coordinates,thefield equations take the same form for all spins (“spectralflow”):Only the labels on the componentfields change,since the helicities at the vertex must add to1.Consequently,the wave solution is identical to that of nonsupersymmetric selfudal Yang-Mills theory(i.e.,one helicity of the nonselfdual theory),except that thefield has arbitrary dependence on the anticommuting coor-dinates.Since thefield strength is defined by taking the same derivatives(∂−and∂t) that appear in the action,not both of them can vanish.We will choose∂−to vanish; the other choice is related by parity.Furthermore,since15wave solution is thenφ(x,θ)=φ(x+,x t,θι)and the only nonvanishing components of the gaugefield andfield strength areA+=∂tφ,F t+=∂2tφExcept for the arbitraryθdependence,this is the same as the wave solution stated in the introduction,specialized to one of the two chiralities in D=4(analytic,and not antianalytic).These superfields yield the corresponding componentfields atθ=0; other componentfields(for spinors and scalars)reside inφat higher orders inθ(ψι=∂ι∂tφ,ϕικ=∂ι∂κφ,etc.).Similar remarks apply to(super)gravity,except that the action[13]is polynomial only in the lightcone gauge[14]:S= d4x d8θ[16iφ(∂[−∂[−φ)(∂t]∂t]φ)+g16nonsupersymmetric waves (as bosonic solutions in supersymmetric theories;see,e.g.,[2]).Here we consider the simplest case in D=4,namely N=1super Yang-Mills.We first consider the theory in components:In two-component spinor notation,the non-vanishing component of the vector is A +.+.Therefore,we choose for the nonvanishing spinor components ψ+and its complex conjugate ¯ψ.+.Again dropping dependence on x −.−,the field equations reduce to∂−.+ψ+=∂+.−¯ψ.+=0,∂+.−∂−.+A +.+=i ¯ψ.+ψ+We can then write the solution in terms of¯φ.+(x +.+,x +.−),φ+.+(x +.+,x +.−)(where x +.−is x t )asψ+=∂+.−¯φ.+,¯ψ.+=∂−.+φ+;A +.+=φ+.++¯φ+.++iφ+¯φ.+The superfield formulation is interesting in its own right.It is not as simple as in the selfdual case,but may suggest generalizations.As in the component analysis,the only nonvanishing components of the spinor field strength are W +and7 5.Future directionsAn interesting application of perturbation about selfdual wave solutions would be a description of the full theory by a perturbation in helicity.This could be particularly useful for gravity,where only the selfdual action is polynomial.Generalizations to higher dimensions might be relevant for further generalizations of the AdS/CFT correspondence.For example,for super Yang-Mills in D=3,4,6,10, where there are no scalars,we again keep only the A+component of the vector,and just theγ+ψpart of the spinor,tofindγi∂iψ=0,(∂i)2A+=¯ψγ+ψso that a free solution to the Dirac equation in the transverse dimensions acts as source to the Abelian A+,as we have already seen for D=4(and is somewhat trivial in D=3,where such wave solutions reduce to the usual plane waves).AcknowledgmentsI thank Martin Roˇc ek for pointing out references[3-4].This work was supported in part by the National Science Foundation Grant No.PHY-0098527.REFERENCES1H.W.Brinkmann,Math.Annalen94(1925)119;I.Robinson,unpublished lectures(1956);J.H´e ly,Compt.Rend.Acad.Sci.249(1959)1867;A.Peres,hep-th/0205040,Phys.Rev.Lett.3(1959)571;J.Ehlers and W.Kundt,Exact solutions of gravitationalfield equations,in Gravitation: An introduction to current research,ed.L.Witten(Wiley,1962)p.492M.Blau,J.Figueroa-O’Farrill,C.Hull,and G.Papadopoulos,hep-th/0110242,JHEP 0201(2002)047,hep-th/0201081,Class.Quant.Grav.19(2002)L87;R.R.Metsaev,hep-th/0112044,Nucl.Phys.B625(2002)70;D.Berenstein,J.M.Maldacena,and H.Nastase,hep-th/0202021,JHEP0204(2002)0133J.B.Hartle and S.W.Hawking,Phys.Rev.D13(1976)21884J.B.Hartle and S.W.Hawking,Phys.Rev.D28(1983)29605Y.Nambu,Phys.Lett.26B(1968)626;D.G.Boulware and L.S.Brown,Phys.Rev.172(1968)16286M.J.Duff,Phys.Rev.D7(1973)23177W.A.Bardeen,Prog.Theor.Phys.Suppl.123(1996)1;D.Cangemi,hep-th/9605208,Nucl.Phys.B484(1997)5218G.Chalmers and W.Siegel,hep-th/9606061,Phys.Rev.D54(1996)76289H.Ooguri and C.Vafa,Mod.Phys.Lett.A5(1990)1389,Nucl.Phys.B361(1991) 469,367(1991)83;N.Marcus,hep-th/9207024,Nucl.Phys.B387(1992)263810S.Mandelstam,Nucl.Phys.B213(1983)149;L.Brink,O.Lindgren,and B.E.W.Nilsson,Nucl.Phys.B212(1983)40111 A.N.Leznov,Theor.Math.Phys.73(1988)1233,A.N.Leznov and M.A.Mukhtarov,J.Math.Phys.28(1987)2574;A.Parkes,hep-th/9203074,Phys.Lett.286B(1992)26512W.Siegel,hep-th/9205075,Phys.Rev.D46(1992)R323513W.Siegel,hep-th/9207043,Phys.Rev.D47(1993)250414J.F.Pleba´n ski,J.Math.Phys.16(1975)2395。
a rXiv:h ep-th/9611152v12N ov1996CERN-TH/96-268hep-th/9611152ON NON-PERTURBATIVE RESULTS IN SUPERSYMMETRIC GAUGE THEORIES –A LECTURE 1Amit Giveon 2Theory Division,CERN,CH-1211,Geneva 23,Switzerland ABSTRACT Some notions in non-perturbative dynamics of supersymmetric gauge theories are being reviewed.This is done by touring through a few examples.CERN-TH/96-268September 19961IntroductionIn this lecture,we present some notions in supersymmetric Yang-Mills(YM) theories.We do it by touring through a few examples where we face a variety of non-perturbative physics effects–infra-red(IR)dynamics of gauge theories.We shall start with a general review;some of the points we consider follow the beautiful lecture notes in[1].Phases of Gauge TheoriesThere are three known phases of gauge theories:•Coulomb Phase:there are massless vector bosons(massless photonsγ;no confinement of both electric and magnetic charges).The behavior of the potential V(R)between electric test charges,separated by a large distance R,is V(R)∼1/R;the electric charge at large distance behaves like a constant:e2(R)∼constant.The potential of magnetic test charges separated by a large distance behaves like V(R)∼1/R,and the magnetic charge behaves like m2(R)∼constant,e(R)m(R)∼1 (the Dirac condition).•Higgs Phase:there are massive vector bosons(W bosons and Z bosons), electric charges are condensed(screened)and magnetic charges are confined(the Meissner effect).The potential between magnetic test charges separated by a large distance is V(R)∼ρR(the magnetic flux is confined into a thin tube,leading to this linear potential witha string tensionρ).The potential between electric test charges isthe Yukawa potential;at large distances R it behaves like a constant: V(R)∼constant.•Confining Phase:magnetic charges are condensed(screened)and elec-tric charges are confined.The potential between electric test charges separated by a large distance is V(R)∼σR(the electricflux is confined1into a thin tube,leading to the linear potential with a string tensionσ).The potential between magnetic test charges behaves like a constant at large distance R.Remarks1.In addition to the familiar Abelian Coulomb phase,there are theorieswhich have a non-Abelian Coulomb phase[2],namely,a theory with massless interacting quarks and gluons exhibiting the Coulomb poten-tial.This phase occurs when there is a non-trivial IRfixed point of the renormalization group.Such theories are part of other possible cases of non-trivial,interacting4d superconformalfield theories(SCFTs)[3,4]. 2.When there are matterfields in the fundamental representation of thegauge group,virtual pairs can be created from the vacuum and screen the sources.In this situation,there is no invariant distinction between the Higgs and the confining phases[5].In particular,there is no phase with a potential behaving as V(R)∼R at large distance,because the flux tube can break.For large VEVs of thefields,a Higgs description is most natural,while for small VEVs it is more natural to interpret the theory as“confining.”It is possible to smoothly interpolate from one interpretation to the other.3.Electric-Magnetic Duality:Maxwell theory is invariant underE→B,B→−E,(1.1) if we introduce magnetic charge m=2π/e and also interchangee→m,m→−e.(1.2) Similarly,Mandelstam and‘t Hooft suggested that under electric-magnetic duality the Higgs phase is interchanged with a confining phase.Con-finement can then be understood as the dual Meissner effect associated with a condensate of monopoles.2Dualizing a theory in the Coulomb phase,one remains in the same phase.For an Abelian Coulomb phase with massless photons,this electric-magnetic duality follows from a standard duality transforma-tion,and is extended to SL(2,Z)S-duality,acting on the complex gauge coupling byτ→aτ+b2π+i4πBy effective,we mean in Wilson sense:[modes p>µ]e−S=e−S ef f(µ,light modes),(2.3) so,in principle,L eff depends on a scaleµ.But due to supersymmetry,the dependence on the scaleµdisappear(except for the gauge couplingτwhich has a logµdependence).When there are no interacting massless particles,the Wilsonian effec-tive action=the1PI effective action;this is often the case in the Higgs or confining phases.2.1The Effective SuperpotentialWe will focus on a particular contribution to L eff–the effective superpo-tential term:L int∼ d2θW eff(X r,g I,Λ)+c.c,(2.4) where X r=light chiral superfields,g I=various coupling constants,and Λ=dynamically generated scale(associated with the gauge dynamics): log(Λ/µ)∼−8π2/g2(µ).Integrating overθ,the superpotential gives a scalar potential and Yukawa-type interaction of scalars with fermions.The quantum,effective superpotential W eff(X r,g I,Λ)is constrained by holomorphy,global symmetries and various limits[9,1]:1.Holomorphy:supersymmetry requires that W eff is holomorphic in thechiral superfields X r(i.e.,independent of the X†r).Moreover,we will think of all the coupling constants g I in the tree-level superpotential W tree and the scaleΛas background chiral superfield sources.This implies that W eff is holomorphic in g I,Λ(i.e.,independent of g∗I,Λ∗).2.Symmetries and Selection Rules:by assigning transformation laws bothto thefields and to the coupling constants(which are regarded as back-ground chiral superfields),the theory has a large global symmetry.This implies that W eff should be invariant under such global symmetries.43.Various Limits:W eff can be analyzed approximately at weak coupling,and some other limits(like large masses).Sometimes,holomorphy,symmetries and various limits are strong enough to determine W eff!The results can be highly non-trivial,revealing interesting non-perturbative dynamics.2.2The Gauge“Kinetic Term”in a Coulomb Phase When there is a Coulomb phase,there is a term in L eff of the formL gauge∼ d2θIm τeff(X r,g I,Λ)W2α ,(2.5) where Wα=gauge supermultiplet(supersymmetricfield strength);schemat-ically,Wα∼λα+θβσµνβαFµν+....Integrating overθ,W2αgives the term F2+iF˜F and its supersymmetric extension.Therefore,τeff=θeffg2eff(2.6)is the effective,complex gauge coupling.τeff(X r,g I,Λ)is also holomorphic in X r,g I,Λand,sometimes,it can be exactly determined by using holomorphy, symmetries and various limits.2.3The“Kinetic Term”The kinetic term is determined by the K¨a hler potential K:L kin∼ d2θd2¯θK(X r,X†r).(2.7) If there is an N=2supersymmetry,τeff and K are related;for an N=2 supersymmetric YM theory with a gauge group G and in a Coulomb phase, L eff is given in terms of a single holomorphic function F(A i):L eff∼Im d4θ∂F2 d2θ∂2FA manifestly gauge invariant N =2supersymmetric action which reduces to the above at low energies is[10]Imd 4θ∂F2 d 2θ∂2F3Some of these results also appear in the proceedings [13]of the 29th International Symposium on the Theory of Elementary Particles in Buckow,Germany,August 29-September 2,1995,and of the workshop on STU-Dualities and Non-Perturbative Phe-nomena in Superstrings and Supergravity ,CERN,Geneva,November 27-December 1,1995.6N A supermultiplets in the adjoint representation,Φab α,α=1,...,N A ,and N 3/2supermultiplets in the spin 3/2representation,Ψ.Here a,b are fundamental representation indices,and Φab =Φba (we present Ψin a schematic form as we shall not use it much).The numbers N f ,N A and N 3/2are limited by the condition:b 1=6−N f−2N A −5N 3/2≥0,(3.1)where −b 1is the one-loop coefficient of the gauge coupling beta-function.The main result of this section is the following:the effective superpoten-tial of an (asymptotically free or conformal)N =1supersymmetric SU (2)gauge theory,with 2N f doublets and N A triplets (and N 3/2quartets)isW N f ,N A (M,X,Z,N 3/2)=−δN 3/2,0(4−b 1) Λ−b 1Pf 2N f X det N A (Γαβ)2 1/(4−b 1)+Tr N A ˜mM +1√4Integrating in the “glueball”field S =−W 2α,whose source is log Λb 1,gives the non-perturbative superpotential:W (S,M,X,Z )=S log Λb 1S 4−b 1Here,the a,b indices are raised and lowered with anǫab tensor.The gauge-invariant superfields X ij may be considered as a mixture of SU(2)“mesons”and“baryons,”while the gauge-invariant superfields Zαij may be considered as a mixture of SU(2)“meson-like”and“baryon-like”operators.Equation(3.2)is a universal representation of the superpotential for all infra-red non-trivial theories;all the physics we shall discuss(and beyond) is in(3.2).In particular,all the symmetries and quantum numbers of thevarious parameters are already embodied in W Nf,N A .The non-perturbativesuperpotential is derived in refs.[11,12]by an“integrating in”procedure, following refs.[14,15].The details can be found in ref.[12]and will not be presented here5.Instead,in the next sections,we list the main results concerning each of the theories,N f,N A,N3/2,case by case.Moreover,a few generalizations to other gauge groups will be discussed.4b1=6:N f=N A=N3/2=0This is a pure N=1supersymmetric SU(2)gauge theory.The non-perturbative effective superpotential is6W0,0=±2Λ3.(4.1) The superpotential in eq.(4.1)is non-zero due to gaugino(gluino)conden-sation7.Let us consider gaugino condensation for general simple groups[1].Pure N=1Supersymmetric Yang-Mills TheoriesPure N=1supersymmetric gauge theories are theories with pure superglue with no matter.We consider a theory based on a simple group G.The theorycontains vector bosons Aµand gauginosλαin the adjoint representation of G.There is a classical U(1)R symmetry,gaugino number,which is broken tosubgroup by instantons,a discrete Z2C2(λλ)C2 =const.Λ3C2,(4.2) where C2=the Casimir in the adjoint representation normalized such that, for example,C2=N c for G=SU(N c).This theory confines,gets a mass gap,and there are C2vacua associ-symmetry to Z2by gaugino ated with the spontaneous breaking of the Z2C2condensation:λλ =const.e2πin/C2Λ3,n=1,...,C2.(4.3) Each of these C2vacua contributes(−)F=1and thus the Witten index is Tr(−)F=C2.This physics is encoded in the generalization of eq.(4.1)to any G,givingW eff=e2πin/C2C2Λ3,n=1,...,C2.(4.4) For G=SU(2)we have C2=2.Indeed,the“±”in(4.1),which comes from the square-root appearing on the braces in(3.2)when b1=6,corresponds, physically,to the two quantum vacua of a pure N=1supersymmetric SU(2) gauge theory.The superpotentials(4.1),(4.3)can be derived byfirst adding fundamen-tal matter to pure N=1supersymmetric YM theory(as we will do in the next section),and then integrating it out.5b1=5:N f=1,N A=N3/2=0There is one case with b1=5,namely,SU(2)with oneflavor.The superpo-tential isΛ5W1,0=vacuum degeneracy of the classical low-energy effective theory is lifted quan-tum mechanically;from eq.(5.1)we see that,in the massless case,there is no vacuum at all.SU(N c)with N f<N cEquation(5.1)is a particular case of SU(N c)with N f<N c(N f quarks Q i and N f anti-quarks¯Q¯i,i,¯i=1,...,N f)[1].In these theories,by using holomorphy and global symmetries,U(1)Q×U(1)¯Q×U(1)RQ:100¯Q:010(5.2)Λ3N c−N f:N f N f2N c−2N fW:002onefinds thatW eff=(N c−N f) Λ3N c−N f N c−N f,(5.3) whereX i¯i≡Q i¯Q¯i,i,¯i=1,...,N f.(5.4) Classically,SU(N c)with N f<N c is broken down to SU(N c−N f).The ef-fective superpotential in(5.3)is dynamically generated by gaugino condensa-tion in SU(N c−N f)(for N f≤N c−2)8,and by instantons(for N f=N c−1).The SU(2)with N f=1ExampleFor example,let us elaborate on the derivation and physics of eq.(5.1).An SU(2)effective theory with two doublets Q a i has one light degree of freedom: four Q a i(i=1,2is aflavor index,a=1,2is a color index;2×2=4)threeout of which are eaten by SU(2),leaving4−3=1.This single light degree of freedom can be described by the gauge singletX=Q1Q2.(5.5) When X =0,SU(2)is completely broken and,classically,W eff,class=0 (when X =0there are extra masslessfields due to an unbroken SU(2)). Therefore,the classical scalar potential is identically zero.However,the one-instanton action is expected to generate a non-perturbative superpotential.The symmetries of the theory(at the classical level and with their cor-responding charges)are:U(1)Q=number of Q1fields(quarks or squarks),1=number of Q2fields(quarks or squarks),U(1)R={number of U(1)Q2gluinos}−{number of squarks}.At the quantum level these symmetries are anomalous–∂µjµ∼F˜F–and by integrating both sides of this equation one gets a charge violation when there is an instanton background I.The instanton background behaves likeI∼e−8π2/g2(µ)= Λ9For SU(N c)with N fflavors,the instanton background I has2C2=2N c gluino zero-modesλand2N f squark zero-modes q and,therefore,its R-charge is R(I)=number(λ)−number(q)=2N c−2N f.Since I∼Λb1and b1=3N c−N f,we learn thatΛ3N c−N f has an R-charge=2N c−2N f,as it appears in eq.(5.2).11and,therefore,W eff has charges:U(1)Q(5.9)1×U(1)Q2×U(1)RW eff:002Finally,because W eff is holomorphic in X,Λ,and is invariant under symme-tries,we must haveΛ5W eff(X,Λ)=c10This is reflected in eq.(3.2)by the vanishing of the coefficient(4−b1)in front of the braces,leading to W=0,and the singular power1/(4−b1)on the braces,when b1=4, which signals the existence of a constraint.12At the classical limit,Λ→0,the quantum constraint collapses into the clas-sical constraint,Pf X=0.SU(N c)with N f=N cEquations(6.1),(6.2)are a particular case of SU(N c)with N f=N c.[1]In these theories one obtains W eff=0,and the classical constraint det X−B¯B=0is modified quantum mechanically todet X−B¯B=Λ2N c,(6.3) whereX i¯i=Q i¯Q¯i(mesons),B=ǫi1...i N c Q i1···Q i N c(baryon),¯B=ǫ¯i1...¯i N c¯Q¯i1···¯Q¯iN c(anti−baryon).(6.4)6.2N f=0,N A=1,N3/2=0The massless N A=1case is a pure SU(2),N=2supersymmetric Yang-Mills theory.This model was considered in detail in ref.[17].The non-perturbative superpotential vanishesW non−per.0,1=0,(6.5) and by the integrating in procedure we also get the quantum constraint:M=±Λ2.(6.6) This result can be understood because the starting point of the integrating in procedure is a pure N=1supersymmetric Yang-Mills theory.Therefore,it leads us to the points at the verge of confinement in the moduli space.These are the two singular points in the M moduli space of the theory;they are due to massless monopoles or dyons.Such excitations are not constructed out of the elementary degrees of freedom and,therefore,there is no trace for them in W.(This situation is different if N f=0,N A=1;in this case,monopoles are different manifestations of the elementary degrees of freedom.)137b1=3There are two cases with b1=3:either N f=3,or N A=N f=1.In both cases,for vanishing bare parameters in(3.2),the semi-classical limit,Λ→0, imposes the classical constraints,given by the equations of motion:∂W=0; however,quantum corrections remove the constraints.7.1N f=3,N A=N3/2=0The superpotential isW3,0=−Pf X2Tr mX.(7.1)In the massless case,the equations∂X W=0give the classical constraints; in particular,the superpotential is proportional to a classical constraint: Pf X=0.The negative power ofΛ,in eq.(7.1)with m=0,indicates that small values ofΛimply a semi-classical limit for which the classical constraints are imposed.SU(N c)with N f=N c+1Equation(7.1)is a particular case of SU(N c)with N f=N c+1[1].In these theories one obtainsW eff=−det X−X i¯iB i¯B¯iThis is consistent with the negative power ofΛin W eff which implies that in the semi-classical limit,Λ→0,the classical constraints are imposed. 7.2N f=1,N A=1,N3/2=0In this case,the superpotential in(3.2)readsW1,1=−Pf X2Tr mX+12TrλZ.(7.5)Here m,X are antisymmetric2×2matrices,λ,Z are symmetric2×2 matrices andΓ=M+Tr(ZX−1)2.(7.6) This superpotential was foundfirst in ref.[18].Tofind the quantum vacua, we solve the equations:∂M W=∂X W=∂Z W=0.Let us discuss some properties of this theory:•The equations∂W=0can be re-organized into the singularity condi-tions of an elliptic curve:y2=x3+ax2+bx+c(7.7)(and some other equations),where the coefficients a,b,c are functions of only thefield M,the scaleΛ,the bare quark masses,m,and Yukawa couplings,λ.Explicitly,a=−M,b=Λ316,(7.8)whereα=Λ62Γ.(7.10)15•W1,1has2+N f=3vacua,namely,the three singularities of the elliptic curve in(7.7),(7.8).These are the three solutions,M(x),of the equations:y2=∂y2/∂x=0;the solutions for X,Z are given by the other equations of motion.•The3quantum vacua are the vacua of the theory in the Higgs-confinement phase.•Phase transition points to the Coulomb branch are at X=0⇔˜m= 0.Two of these singularities correspond to a massless monopole or dyon,and are the quantum splitting of the classically enhanced SU(2) point.A third singularity is due to a massless quark;it is a classical singularity:M∼m2/λ2for large m,and thus M→∞when m→∞, leaving the two quantum singularities of the N A=1,N f=0theory.•The elliptic curve defines the effective Abelian coupling,τ(M,Λ,m,λ), in the Coulomb branch:Elliptic Curves and Effective Abelian CouplingsA torus can be described by the one complex dimensional curve in C2 y2=x3+ax2+bx+c,where(x,y)∈C2and a,b,c are complex parameters.The modular parameter of the torus isτ(a,b,c)= βdx αdxIn this form,the modular parameterτis determined(modulo SL(2,Z)) by the ratio f3/g2through the relation4(24f)3j(τ)=8b1=2There are three cases with b1=2:N f=4,or N A=1,N f=2,or N A=2.In all three cases,for vanishing bare parameters in(3.2),there are extra massless degrees of freedom not included in the procedure;those are expected due toa non-Abelian conformal theory.8.1N f=4,N A=N3/2=0The superpotential isW4,0=−2(Pf X)1Λ+12N c<N f<3N c the theory is in an interacting,non-AbelianCoulomb phase(in the IR and for m=0).In this range of N f the theory is asymptotically ly,at short distance the coupling18constant g is small,and it becomes larger at larger distance.However, it is argued that for32N c<N f<3N c,the IR theory is a non-trivial4d SCFT.The elementary quarks and gluons are not confined but appear as in-teracting massless particles.The potential between external massless electric sources behaves as V∼1/R,and thus one refers to this phase of the theory as the non-Abelian Coulomb phase.•The Seiberg Duality:it is claimed[2]that in the IR an SU(N c)theory with N fflavors is dual to SU(N f−N c)with N fflavors but,in addition to dual quarks,one should also include interacting,massless scalars. This is the origin to the branch cut in W eff at X =0,because W eff does not include these light modes which must appear at X =0. The quantum numbers of the quarks and anti-quarks of the SU(N c) theory with N fflavors(=theory A)are11A.SU(N c),N f:The Electric TheorySU(N f)L×SU(N f)R×U(1)B×U(1)RQ:N f111−N cN f(8.2)The quantum numbers of the dual quarks q i and anti-quarks¯q¯i of the SU(N f−N c)theory with N fflavors theory(=theory B)and its mass-less scalars X i¯iareB.SU(N f−N c),N f:The Magnetic TheorySU(N f)L×SU(N f)R×U(1)B×U(1)R q:¯N f1N cN f ¯q:1N f−N c N fX:N f¯N f02 1−N ctheory A and theory B have the same anomalies:U(1)3B:0U(1)B U(1)2R:0U(1)2B U(1)R:−2N2cSU(N f)3:N c d(3)(N f)SU(N f)2U(1)R:−N2cN2f(8.5)Here d(3)(N f)=Tr T3f of the global SU(N f)symmetries,where T fare generators in the fundamental representation,and d(2)(N f)=Tr T2f2.Deformations:theory A and theory B have the same quantummoduli space of deformations.Remarks•Electric-magnetic duality exchanges strong coupling with weak cou-pling(this can be read offfrom the beta-functions),and it interchanges a theory in the Higgs phase with a theory in the confining phase.•Strong-weak coupling duality also relates an SU(N c)theory with N f≥3N c to an SU(N f−N c)theory.SU(N c)with N f≥3N c is in a non-Abelian free electric phase:in this range the theory is not asymptoti-cally ly,because of screening,the coupling constant becomes smaller at large distance.Therefore,the spectrum of the theory at large distance can be read offfrom the Lagrangian–it consists of the elemen-tary quarks and gluons.The long distance behavior of the potential between external electric test charges isV(R)∼1R,e(R→∞)→0.(8.6)For N f≥3N c,the theory is thus in a non-Abelian free electric phase; the massless electrically chargedfields renormalize the charge to zero21at long distance as e −2(R )∼log(R Λ).The potential of magnetic test charges behave at large distance R asV (R )∼log(R Λ)R ,⇒e (R )m (R )∼1.(8.7)SU (N c )with N f ≥3N c is dual to SU (˜N c )with ˜N c +2≤N f ≤3R ∼e 2(R )2˜N c ,the massless magnetic monopoles renormal-ize the electric coupling constant to infinity at large distance,with a conjectured behavior e 2(R )∼log(R Λ).The potential of magnetic test charges behaves at large distance R asV (R )∼1R ⇒e (R )m (R )∼1.(8.9)•The Seiberg duality can be generalized in many other cases,includ-ing a variety of matter supermultiplets (like superfields in the adjoint representation [20])and other gauge groups [21].8.2N f =2,N A =1,N 3/2=0In this case,the superpotential in (3.2)readsW 2,1=−2(Pf X )1ΛΓ+˜mM +1√•The equations∂W=0can be re-organized into the singularity condi-tions of an elliptic curve(7.7)(and some other equations),where the coefficients a,b,c are functions of only thefield M,the scaleΛ,the bare quark masses,m,and Yukawa couplings,λ.Explicitly[11,12],a=−M,b=−α4Pf m,c=α16detλ,µ=λ−1m.(8.12)•As in section7.2,the parameter x,in the elliptic curve(7.7),is given in terms of the compositefield:x≡1•As in section7.2,the negative power ofΛ,in eq.(8.10)with˜m= m=λ=0,indicates that small values ofΛimply a semi-classical limit for which the classical constraints are imposed.Indeed,for vanishing bare parameters,the equations∂W=0are equivalent to the classical constraints,and their solutions span the Higgs moduli space[22].•For special values of the bare masses and Yukawa couplings,some of the 4vacua degenerate.In some cases,it may lead to points where mutually non-local degrees of freedom are massless,similar to the situation in pure N=2supersymmetric gauge theories,considered in[3].For example,when the masses and Yukawa couplings approach zero,all the 4singularities collapse to the origin.Such points might be interpreted as in a non-Abelian Coulomb phase[1]or new non-trivial,interacting, N=1SCFTs.•The singularity at X=0(inΓ)and the branch cut at Pf X=0 (due to the1/2power in eq.(8.10))signal the appearance of extra massless degrees of freedom at these points;those are expected similar to references[2,20].Therefore,we make use of the superpotential only in the presence of bare parameters,whichfix the vacua away from such points.8.3N f=0,N A=2,N3/2=0In this case,the superpotential in eq.(3.2)readsdet MW0,2=±212The fractional power1/(4−b1)on the braces in(3.2),for any theory with b1≤2, may indicate a similar phenomenon,namely,the existence of confinement and oblique24theory has two quantum vacua;these become the phase transition points to the Coulomb branch when det˜m=0.The moduli space may also contain a non-Abelian Coulomb phase when the two singularities degenerate at the point M=0[18];this happens when˜m=0.At this point,the theory has extra massless degrees of freedom and,therefore,W0,2fails to describe the physics at˜m=0.Moreover,at˜m=0,the theory has other descriptions via an electric-magnetic triality[1].9b1=1There are four cases with b1=1:N f=5,or N A=1,N f=3,or N A=2, N f=1,or N3/2=1.9.1N f=5,N A=N3/2=0The superpotential isW5,0=−3(Pf X)1Λ12Tr mX.(9.1)This theory is a particular case of SU(N c)with N f>N c+1.The discussion in section8.1is relevant in this case too.9.2N f=3,N A=1,N3/2=0In this case,the superpotential in(3.2)readsW3,1=−3(Pf X)1Λ13+˜mM+1√confinement branches of the theory,corresponding to the4−b1phases due to the fractional power.It is plausible that,for SU(2),such branches are related by a discrete symmetry.25•The equations∂W=0can be re-organized into the singularity condi-tions of an elliptic curve(7.7)(and some other equations),where the coefficients a,b,c are[11,12]a=−M−α,b=2αM+α4Pf m,c=α64detλ,µ=λ−1m.(9.4) In eq.(9.3)we have shifted the quantumfield M toM→M−α/4.(9.5)•The parameter x,in the elliptic curve(7.7),is given in terms of thecompositefield:x≡12.(9.6)Therefore,as before,we have identified a physical meaning of the pa-rameter x.•W3,1has2+N f=5quantum vacua,corresponding to the5singularities of the elliptic curve(7.7),(9.3);these are the vacua of the theory in the Higgs-confinement phase.•From the phase transition points to the Coulomb branch,we conclude that the elliptic curve defines the effective Abelian coupling,τ(M,Λ,m,λ), for arbitrary bare masses and Yukawa couplings.As before,on the sub-space of bare parameters,where the theory has N=2supersymmetry, the result in eq.(9.3)coincides with the result in[7]for N f=3.•For special values of the bare masses and Yukawa couplings,some of the 5vacua degenerate.In some cases,it may lead to points where mutually non-local degrees of freedom are massless,and might be interpreted as in a non-Abelian Coulomb phase or another new superconformal theory in four dimensions(see the discussion in sections7.2and8.2).26•The singularity and branch cuts in W3,1signal the appearance of extra massless degrees of freedom at these points.•The discussion in the end of sections7.2and8.2is relevant here too.9.3N f=1,N A=2,N3/2=0In this case,the superpotential in(3.2)reads[12]W1,2=−3(Pf X)1/32Tr mX+12TrλαZα.(9.7)Here m and X are antisymmetric2×2matrices,λαand Zαare symmetric 2×2matrices,α=1,2,˜m,M are2×2symmetric matrices andΓαβis given in eq.(3.3).This theory has3quantum vacua in the Higgs-confinement branch.At the phase transition points to the Coulomb branch,namely, when det˜m=0⇔det M=0,the equations of motion can be re-organized into the singularity conditions of an elliptic curve(7.7).Explicitly,when ˜m22=˜m12=0,the coefficients a,b,c in(7.7)are[12]a=−M22,b=Λ˜m21132 2detλ2.(9.8)However,unlike the N A=1cases,the equations∂W=0cannot be re-organized into the singularity condition of an elliptic curve,in general.This result makes sense,physically,since an elliptic curve is expected to“show up”only at the phase transition points to the Coulomb branch.For special values of the bare parameters,there are points in the moduli space where (some of)the singularities degenerate;such points might be interpreted as in a non-Abelian Coulomb phase,or new superconformal theories.For more details,see ref.[12].9.4N f=N A=0,N3/2=1This chiral theory was shown to have W non−per.0,0(N3/2=1)=0;[24]perturb-ing it by a tree-level superpotential,W tree=gU,where U is given in(3.4), may lead to dynamical supersymmetry breaking[24].2710b1=0There arefive cases with b1=0:N f=6,or N A=1,N f=4,or N A= N f=2,or N A=3,or N3/2=N f=1.These theories have vanishing one-loop beta-functions in either conformal or infra-red free beta-functions and, therefore,will possess extra structure.10.1N f=6,N A=N3/2=0This theory is a particular case of SU(N c)with N f=3N c;the electric theory is free in the infra-red[1].1310.2N f=4,N A=1,N3/2=0In this case,the superpotential in(3.2)readsW4,1=−4(Pf X)1Λb12+˜mM+1√β2 2α+1β2α13A related fact is that(unlike the N A=1,N f=4case,considered next)in the(would be)superpotential,W6,0=−4Λ−b1/4(Pf X)1/4+1。
a r X i v :0805.4481v 1 [h e p -t h ] 29 M a y 2008SUPERSYMMETRIC CHERN-SIMONS MODELS IN HARMONIC SU-PERSPACESB.M.ZupnikBogoliubov Laboratory of Theoretical Physics,JINR,Dubna,Moscow Region,141980,Russia;E-mail:zupnik@theor.jinr.ruAbstractWe review harmonic superspaces of the D =3,N =3and 4supersymmetries and gauge models in these superspaces.Superspaces of the D =3,N =5supersymmetry use harmonic coordinates of the SO (5)group.The superfield N =5actions describe the off-shell infinite-dimensional Chern-Simons supermultiplet.1IntroductionSupersymmetric extensions of the D =3Chern-Simons theory were discussed in [1]-[10].A superfield action of the D =3,N =1Chern-Simons theory can be interpreted as the superspace integral of the differential Chern-Simons superform dA +2iwhere i andˆk are two-component indices of the automorphism groups SU L(2)and SU R(2), respectively,αis the two-component index of the SL(2,R)group and m=0,1,2is the 3D vector index.The N=4supersymmetry transformations areδx m=−i(γm)αβ(ǫαjˆk θβjˆk−iǫβjˆkθαjˆk),(2.2)whereγm are the3Dγmatrices.The SU L(2)/U(1)harmonics u±i[11]can be used to construct the left analytic super-space[15]with the LA coordinatesζL=(x m L,θ+ˆkα).(2.3)The L-analytic prepotential V++(ζL,u)describes the left N=4vector multiplet A m,φˆkˆl ,λαiˆk,D ik.The D=3,N=4SY M action can be constructed in terms of this prepotential by anal-ogy with the D=4,N=2SY M action[14].Let us introduce the new notation for the left harmonics u±i=u(±1,0)iand the analogousnotation v(0,±1)ˆk for the right SL R(2)/U(1)harmonics.The biharmonic N=4superspaceuses the Grassmann coordinates[15]θ(±1,±1)α=u(±1,0)i v(0,±1)ˆkθiˆkα.(2.4)In this representation,we haveζL=(x m L,θ(1,±1)α),V++≡V(2,0),(2.5)D(1,±1)αV(2,0)=0,D(0,2)vV(2,0)=0.(2.6)The right analytic N=4coordinates areζR=(x m R,θ(±1,1)α),x m R=x m L−2i(γm)αβθ(−1,1)αθ(1,1)β.(2.7) The mirror R analytic prepotentialˆV(0,2)D(±1,1)ˆV(0,2)=0,D(2,0)uˆV(0,2)=0(2.8)describes the right N=4vector multipletˆA m,ˆφij,ˆλαiˆk ,ˆD ik,whereˆA m is the mirror vectorgaugefield using the independent gauge group.The right N=4SY M action is similar to the analogous left action.These multiplets can be formally connected by the map SU L(2)↔SU R(2).The N=4superfield Chern-Simons type(or BF-type)action for the gauge group U(1)×U(1)connects two mirror vector multipletsdud3x L dθ(−4,0)V(2,0)(ζL,u)D(1,1)αD(1,1)αˆV(0,−2),(2.9) where the right connection satisfies the equationD(0,2)v ˆV(0,−2)=D(0,−2)v˜V(0,2),D(2,0)uˆV(0,−2)=0.(2.10)The component form of this action was considered in[16,17].We can identify the left and right isospinor indices in the N=4spinor coordinatesθαjˆk→θαjk=θα(jk)+12D++αD++αV−−.(2.14)The action of the corresponding CS-theory can be constructed in the full or analytic N=3superspaces[8,9].3Harmonic superspaces for the group SO(5)The homogeneous space SO(5)/U(2)is parametrized by elements of the harmonic5×5 matrixU K a=(U+i a,U0a,U−ia)=(U+1a,U+2a,U0a,U−1a,U−2a),(3.1) where a=1,...5is the vector index of the group SO(5),i=1,2is the spinor index of the group SU(2),and U(1)-charges are denoted by symbols+,−,0.The basic relations for these harmonics areU+i a U+k a=U+i a U0a=0,U−ia U−ka =U−ia U0a=0,U+i a U−ka=δi k,U0a U0a=1,U+i a U−ib +U−ia U+ib+U0a U0b=δab.(3.2)We consider the SO(5)invariant harmonic derivatives with nonzero U(1)charges∂+i=U+i a ∂∂U−ia,∂+i U0a=U+i a,∂+i U−ka=−δi k U0a,∂++=U+ia∂∂U0a −U0a∂∂U+i a ,[∂−i,∂−k]=εki∂−−,∂−k∂−k=−∂−−,where some relations between these harmonic derivatives are defined.The U(1)neutral harmonic derivatives form the Lie algebra U(2)∂i k=U+i a ∂∂U−ia,[∂+i,∂−k]=−∂i k,(3.4)∂0≡∂k k=U+k a ∂∂U−ka,[∂++,∂−−]=∂0,∂i k U+l a=δl k U+i a,∂i k U−la =−δi l U−ka.(3.5)The operators∂+k,∂++,∂−k ,∂−−and∂i k satisfy the commutation relations of the Lie al-gebra SO(5).One defines an ordinary complex conjugation on these harmonicsU0a=U0a,(3.6) however,it is convenient to use a special conjugation in the harmonic space(U+i a)∼=U+i a,(U−ia)∼=U−ia,(U0a)∼=U0a.(3.7) All harmonics are real with respect to this conjugation.The full superspace of the D=3,N=5supersymmetry has the spinor CB coordi-natesθαa,(α=1,2;a=1,2,3,4,5)in addition to the coordinates x m of the three-dimensional Minkowski space.The group SL(2,R)×SO(5)acts on the spinor coordinates. The superconformal transformations of these coordinates are considered in Appendix.The SO(5)/U(2)harmonics allow us to construct projections of the spinor coordinates and the partial spinor derivativesθ+iα=U+i aθαa,θ0α=U0aθαa,θ−αi=U−iaθαa,(3.8)∂−iα=∂/∂θ+iα,∂0α=∂/∂θ0α,∂+iα=∂/∂θ−αi.The analytic coordinates(AB-representation)in the full harmonic superspace use these projections of10spinor coordinatesθ+iα,θ0α,θ−αi and the following representation of the vector coordinate:x m A ≡y m=x m+i(θ+kγmθ−k)=x m+i(θaγmθb)U+k a U−kb.(3.9)The analytic coordinates are real with respect to the special conjugation. The harmonic derivatives have the following form in AB:D+k=∂+k−i(θ+kγmθ0)∂m+θ+kα∂0α−θ0α∂+kα,D++=∂+++i(θ+kγmθ+k )∂m+θ+αk∂+kα,(3.10)D k l=∂k l+θ+kα∂−lα−θ−αl∂−kα.We use the commutation relations[D+k,D+l]=−εkl D++,D+k D+k=D++.(3.11) The AB spinor derivatives areD+iα=∂+iα,D−iα=−∂−iα−2iθ−βi∂αβ,D0α=∂0α+iθ0β∂αβ.(3.12) The coordinates of the analytic superspaceζ=(y m,θ+iα,θ0α,U K a)have the Grass-mann dimension6and dimension of the even space3+6.The functionsΦ(ζ)satisfy the Grassmann analyticity condition in this superspaceD+kαΦ=0.(3.13)In addition to this condition,the analytic superfields in the SO (5)/U (2)harmonic super-space possess also the U (2)-covariance.This subsidiary condition looks especially simple for the U (2)-scalar superfieldsD k l Λ(ζ)=0.(3.14)The integration measure in the analytic superspace dµ(−4)has dimension zerodµ(−4)=dUd 3x A (∂0α)2(∂−iα)4=dUd 3x A dθ(−4).(3.15)The SO (5)/U (1)×U (1)harmonics can be defined via the components of the real orthogonal 5×5matrix [20,21]U Ka = U (1,1)a ,U (1,−1)a ,U (0,0)a ,U (−1,1)a ,U (−1,−1)a (3.16)where a is the SO(5)vector index and the index K =1,2,...5corresponds to givencombinations of the U(1)×U(1)charges.We use the following harmonic derivatives∂(2,0)=U (1,1)b∂/∂U (−1,1)b−U (1,−1)b∂/∂U (−1,−1)b,∂(1,1)=U (1,1)b∂/∂U (0,0)b −U (0,0)b∂/∂U (−1,−1)b,∂(1,−1)=U (1,−1)b ∂/∂U (0,0)b−U (0,0)b∂/∂U (−1,1)b,∂(0,2)=U (1,1)b∂/∂U (1,−1)b−U (−1,1)b∂/∂U (−1,−1)b,∂(0,−2)=U (1,−1)b∂/∂U (1,1)b−U (−1,−1)b∂/∂U (−1,1)b.(3.17)We define the harmonic projections of the N =5Grassmann coordinatesθK α=θaαU K a =(θ(1,1)α,θ(1,−1)α,θ(0,0)α,θ(−1,1)α,θ(−1,−1)α).(3.18)The SO (5)/U (1)×U (1)analytic superspace contains only spinor coordinatesζ=(x m A ,θ(1,1)α,θ(1,−1)α,θ(0,0)α),(3.19)x m A=x m +iθ(1,1)γm θ(−1,−1)+iθ(1,−1)γm θ(−1,1),δǫx m A =−iǫ(0,0)γm θ(0,0)−2iǫ(−1,1)γm θ(1,−1)−2iǫ(−1,−1)γm θ(1,1),(3.20)where ǫKα=ǫαa U Ka are the harmonic projections of the supersymmetry parameters.General superfields in the analytic coordinates depend also on additional spinor coor-dinates θ(−1,1)αand θ(−1,−1)α.The harmonized partial spinor derivatives are∂(−1,−1)α=∂/∂θ(1,1)α,∂(−1,1)α=∂/∂θ(1,−1)α,∂(0,0)α=∂/∂θ(0,0)α,(3.21)∂(1,1)α=∂/∂θ(−1,−1)α,∂(1,−1)α=∂/∂θ(−1,1)α.We use the special conjugation ∼in the harmonic superspaceU (p,q )a =U (p,−q )a ,θ(p,q )α=θ(p,−q )α, x m A =x m A,(θ(p,q )αθ(s,r )β)∼=θ(s,−r )βθ(p,−q )α, f (x A )=¯f(x A ),(3.22)where ¯fis the ordinary complex conjugation.The analytic superspace is real with respect to the special conjugation.The analytic-superspace integral measure contains partial spinor derivatives(3.21)1dµ(−4,0)=−they are traceless and anti-HermitianΛ†=−Λ.We treat these prepotentials as connections in the covariant gauge derivatives∇+i=D+i+V+i,∇++=D+++V++,δΛV+i=D+iΛ+[Λ,V+i],δΛV++=D++Λ+[Λ,V++],(4.4) D+kαδΛV+k=D+kαδΛV++=0,D i jδΛV+k=δk jδΛV+i,D i jδΛV++=δi jδΛV++,where the infinitesimal gauge transformations of the gauge superfields are defined.These covariant derivatives commute with the spinor derivatives D+kαand preserve the CR-structure in the harmonic superspace.We can construct three analytic superfield strengths offthe mass shellF++=112πdµ(−4)Tr{V+j D++V+j+2V++D+j V+j+(V++)2+V++[V+j,V+j]},(4.5)where k is the coupling constant,and a choice of the numerical multiplier guarantees the correct normalization of the vector-field action.This action is invariant with respect to the infinitesimal gauge transformations of the prepotentials(4.4).The idea of construction of the superfield action in the harmonic SO(5)/U(2)was proposed in[18],although the detailed construction of the superfield Chern-Simons theory was not discussed in this work.The equivalent superfield action was considered in the framework of the alternative superfield formalism[20].The superconformal N=5invariance of this action was proven in[19].The action S1yields superfield equations of motion which mean triviality of the su-perfield strengths of the theoryF(+3) k =D++V+k−D+kV+++[V++,V+k]=0,F++=V++−D+k V+k −V+k V+k=0.(4.6)These classical superfield equations have pure gauge solutions for the prepotentials onlyV+k=e−ΛD+k eΛ,V++=e−ΛD++eΛ,(4.7) whereΛis an arbitrary analytic superfield.The transformation of the sixth supersymmetry can be defined on the analytic N=5 superfieldsδ6V++=ǫα6D0αV++,δ6V+k=ǫα6D0αV+k,(4.8)δ6D+k V+l=ǫα6D0αD+k V+l,δ6D++V+l=ǫα6D0αD++V+l,(4.9) whereǫα6are the corresponding odd parameters.This transformation preserves the Grass-mann analyticity and U(2)-covariance{D0α,D+kβ}=0,[D k l,D0α]=0,[D+k,D0α]=D+kα,[D++,D0α]=0.(4.10)The action S1is invariant with respect to this sixth supersymmetryδ6S1= dµ(−4)ǫα6D0αL(+4)=0.(4.11)In the SO(5)/U(1)×U(1)harmonic superspace,we can introduce the D=3,N=5 analytic matrix gauge prepotentials corresponding to thefive harmonic derivativesV(p,q)(ζ,U)=[V(1,1),V(1,−1),V(2,0),V(0,±2)],(V(1,1))†=−V(1,−1),(V(2,0))†=V(2,0),V(0,−2)=[V(0,2)]†,(4.12) where the Hermitian conjugation†includes∼conjugation of matrix elements and trans-position.We shall consider the restricted gauge supergroup using the supersymmetry-preserving harmonic(H)analyticity constraints on the gauge superfield parametersH1:D(0,±2)Λ=0.(4.13) These constrains yield additional reality conditions for the component gauge parameters.We use the harmonic-analyticity constraints on the gauge prepotentialsH2:V(0,±2)=0,D(0,−2)V(1,1)=V(1,−1),D(0,2)V(1,1)=0(4.14) and the conjugated constraints combined with relations(4.12).The superfield CS action can be constructed in terms of these H-constrained gauge superfields[21]2ikS=−V(2,0)V(2,0)}.(4.15)2Note,that the similar harmonic superspace based on the USp(4)/U(1)×U(1)harmon-ics was used in[22]for the harmonic interpretation of the D=4,N=4super Yang-Mills constraints.This work was partially supported by the grants RFBR06-02-16684,DFG436RUS 113/669-3,INTAS05-10000008-7928and by the Heisenberg-Landau programme.References[1]W.Siegel,Nucl.Phys.B156(1979)135.[2]J.Schonfeld,Nucl.Phys.B185(1981)157.[3]B.M.Zupnik,D.G.Pak,Teor.Mat.Fiz.77(1988)97;Eng.transl.:Theor.Math.Phys.77(1988)1070.[4]B.M.Zupnik,D.G.Pak,Class.Quant.Grav.6(1989)723.[5]B.M.Zupnik,Teor.Mat.Fiz.89(1991)253,Eng.transl.:Theor.Math.Phys.89(1991)1191.[6]B.M.Zupnik,Phys.Lett.B254(1991)127.[7]H.Nishino,S.J.Gates,Int.J.Mod.Phys.8(1993)3371.[8]B.M.Zupnik,D.V.Khetselius,Yad.Fiz.47(1988)1147;Eng.transl.:Sov.J.Nucl.Phys.47(1988)730.[9]B.M.Zupnik,Springer Lect.Notes in Phys.524(1998)116;hep-th/9804167.[10]J.H.Schwarz,Jour.High.Ener.Phys.0411(2004)078,hep-th/0411077.[11]A.Galperin,E.Ivanov,S.Kalitzin,V.Ogievetsky,E.Sokatchev,Class.Quant.Grav.1(1984)469.[12]A.Galperin,E.Ivanov,S.Kalitzin,V.Ogievetsky,E.Sokatchev,Class.Quant.Grav.2(1985)155.[13]A.Galperin,E.Ivanov,V.Ogievetsky,E.Sokatchev,Harmonic superspace,Cam-bridge University Press,Cambridge,2001.[14]B.M.Zupnik,Phys.Lett.B183(1987)175.[15]B.M.Zupnik,Nucl.Phys.B554(1999)365,Erratum:Nucl.Phys.B644(2002)405E;hep-th/9902038.[16]R.Brooks,S.J.Gates,Nucl.Phys.B432(1994)205,hep-th/09407147.[17]A.Kapustin,M.Strassler,JHEP04(1999)021,hep-th/9902033.[18]P.S.Howe,M.I.Leeming,Clas.Quant.Grav.11(1994)2843,hep-th/9402038.[19]B.M.Zupnik,Chern-Simons theory in SO(5)/U(2)harmonic superspace,arXiv0802.0801(hep-th).[20]B.M.Zupnik,Phys.Lett.B660(2008)254,arXiv0711.4680(hep-th).[21]B.M.Zupnik,Gauge model in D=3,N=5harmonic superspace,arXiv0708.3951(hep-th).[22]I.L.Buchbinder,O.Lechtenfeld,I.B.Samsonov,N=4superparticle and super Yang-Mills theory in USp(4)harmonic superspace,arxiv:0804.3063(hep-th).。
a r X i v :n l i n /0212002v 2 [n l i n .S I ] 21 M a r 2003Nonlinear superposition formula for N =1supersymmetric KdV Equation Q.P.Liu and Y.F.Xie Department of Mathematics,China University of Mining and Technology,Beijing 100083,China.Abstract In this paper,we derive a B¨a cklund transformation for the supersymmetric Korteweg-de Vries equation.We also construct a nonlinear superposition formula,which allows us to rebuild systematically for the supersymmetric KdV equation the soliton solutions of Carstea,Ramani and Grammaticos.The celebrated Korteweg-de Vries (KdV)equation was extended into super frame-work by Kupershmidt [3]in 1984.Shortly afterwards,Manin and Radul [7]proposed another super KdV system which is a particular reduction of their general supersymmet-ric Kadomstev-Petviashvili hierarchy.In [8],Mathieu pointed out that the super version of Manin and Radul for the KdV equation is indeed invariant under a space supersymmetric transformation,while Kupershmidt’s version does not.Thus,the Manin-Radul’s super KdV is referred to the supersymmetric KdV equation.We notice that the supersymmetric KdV equation has been studied extensively in lit-erature and a number of interesting properties has been established.We mention here the infinite conservation laws [8],bi-Hamiltonian structures [10],bilinear form [9][2],Darboux transformation [6].By the constructed Darboux transformation,Ma˜n as and one of us calculated the soli-ton solutions for the supersymmetric KdV system.This sort of solutions was also obtained by Carstea in the framework of bilinear formalism [1].However,these solutions are char-acterized by the presentation of some constraint on soliton parameters.Recently,using super-bilinear operators,Carstea,Ramani and Grammaticos [2]constructed explicitly new two-and three-solitons for the supersymmetric KdV equation.These soliton solutions are interesting since they are free of any constraint on soliton parameters.Furthermore,the fermionic part of these solutions is dressed through the interactions.In addition to the bilinear form approach,B¨a cklund transformation (BT)is also a powerful method to construct solutions.Therefore,it is interesting to see if the soliton solutions of Carstea-Ramani-Grammaticos can be constructed by BT approach.In this paper,we first construct a BT for the supersymmetric KdV equation.Then,we derive a nonlinear superposition formula.In this way,the soliton solutions can be producedsystematically.We explicitly show that the two-soliton solution of Carstea,Ramani and Grammaticos appears naturally in the framework of BT.To introduce the supersymmetric extension for the KdV equation,we recall some terminology and notations.The classical spacetime is(x,t)and we extend it to a super-spacetime(x,t,θ),whereθis a Grassmann odd variable.The dependent variable u(x,t) in the KdV equation is replaced by a fermionic variableΦ=Φ(x,t,θ).Now the super-symmetric KdV equation reads asΦt−3(ΦDΦ)x+Φxxx=0,(1)where D=∂∂x is the superderivative.Mathieu found the following supersymmetricversion of Gardner type mapΦ=χ+ǫχx+ǫ2χ(Dχ),(2) whereǫis an ordinary(bosonic)parameter.It is easy to show thatχsatisfies the following supersymmetric Gardner equationχt−3(χDχ)x−3ǫ2(Dχ)(χDχ)x+χxxx=0.(3) This map was used in[8]to prove that there exists an infinite number of conservation laws for the supersymmetric KdV equation(1).In the classical case,Gardner type of map was studied extensively by Kupershmidt[4].It is well known that such map may be used to construct interesting BT.We will show that it is also the case for the supersymmetric KdV equation.We notice that the supersymmetric Gardner equation(3)is invariant underǫ→−ǫ. The new solution of the supersymmetric KdV equation corresponding to with−ǫis denoted as˜Φ.Thus we have˜Φ=χ−ǫχx+ǫ2χ(Dχ).(4) From above relations(3-4),wefindΦ−˜Φ=2ǫχx,(5)Φ+˜Φ=2χ+2ǫχ(Dχ).(6) Let us introduce the potentials as followsΦ=Ψx,˜Φ=˜Ψx,thus,the equation(5)provides usχ=12(Ψ−˜Ψ)(DΨ−D˜Ψ),(8)whereλ=1/ǫis the B¨a cklund parameter.The transformation(8)is in fact the spatial part of BT.Its temporal counterpart can be easily worked out.We also remark here that the BT is reduced to the well known BT for the classical KdV equation ifη=0,as it should be.A BT can be used to generate special solutions.If we start with the trivial solution ˜Ψ=0,we obtain2a(ζ+θλ)eλx−λ3tΨ=−(Ψ1−Ψ0)(DΨ1−DΨ0),(10)2and1(Ψ0+Ψ2)x=λ2(Ψ2−Ψ0)+(Ψ3−Ψ1)(DΨ3−DΨ1),(12)2and1(Ψ2+Ψ3)x=λ1(Ψ3−Ψ2)+Subtraction (10)form (11),we have(Ψ2−Ψ1)x =λ2Ψ2−λ1Ψ1+(λ1−λ2)Ψ0+12Ψ2(D Ψ0)−12Ψ1(D Ψ1)+12Ψ0(D Ψ1),(14)similarly,from (12)and (13)we have(Ψ2−Ψ1)x = λ1−λ2+12(D Ψ2) Ψ3+12Ψ2(D Ψ2)−12(D Ψ1)−12(Ψ1−Ψ2)[(D Ψ3)+2λ1+2λ2−(D Ψ0)]=0.(16)The equation (16)is a differential equation for Ψ3.Solving it we obtainΨ3=Ψ0−(λ1+λ2)(Ψ1−Ψ2)λ1−λ2+v 1−v 2−(λ1+λ2)(η1−η2)(η1,x −η2,x )(λ1−λ2+v 1−v 2).It is clear that our nonlinear superposition formula reduces to the well-known superposition formula for the KdV as it should be.The advantage to have a superposition formula is that it is an algebraic one and can be used easily to find solutions.Using the solutions (9)as our seeds,we may construct a 2-soliton solution of (1)by means of our superposition formula.Indeed,letΨ0=0,Ψ1=−2a 1(ζ1+θλ1)e λ1x −λ31t1+a 2e λ2x −λ32tthen from our nonlinear superposition formula (17),we obtainΨ3=2 ζ1a 1e δ1−ζ2a 2e δ2+(ζ1−ζ2)a 1a 2e δ1+δ2+θ(λ1a 1e δ1−λ2a 2e δ2+(λ1−λ2)a 1a 2e δ1+δ2)whereδi=λi x−λ3i t+θζi(i=1,2).Now,takingλ1>λ2,a1=−λ1−λ2λ1+λ2we recover the2-soliton solution foundfirst by Carstea,Ramani and Grammaticos[2].As in the classical KdV case[11],we generate this2-soliton from a regular solution and a singular solution.We could continue this process to build the higher soliton solutions and the calculation will be tedious but straightforward.Acknowledgment The work is supported in part by National Natural Scientific Foun-dation of China(grant number10231050)and Ministry of Education of China. References[1]Carstea A S2000Extension of the bilinear formalism to supersymmetric KdV-typeequations Nonlinearity131645.[2]Carstea A S,Ramani A and Grammaticos B2001Constructing the soliton solutionsfor the N=1supersymmetric KdV hierarchy Nonlinearity141419.[3]Kupershmidt B A1984A super Korteweg-de Vries equation Phys.Lett.102A213.[4]Kupershmidt B A1981On the nature of Gardner transformation J.Math.Phys.22449;Kupershmidt B A1983Deformations of integrable systems Proc.R.Soc.Irish 83A45.[5]Liu Q P1995Darboux transformation for the supersymmetric KdV equations Lett.Math.Phys.35115.[6]Liu Q P and Ma˜n as M1997Darboux transformation for the Mann-Radul supersym-metric KdV equation Phys.Lett.B394337;Liu Q P and Ma˜n as M in Supersymmetry and Integrable Systems eds.H.Aratyn et al Lecture Notes in Physics502(Springer).[7]Manin Yu I and Radul A O1985A supersymmetric extension of the Kadomtzev-Petviashvili hierarchy Commun.Math.Phys.9865.[8]Mathieu P1988Supersymmetric extension of the Korteweg-de Vries equation J.Math.Phys.292499.[9]McArthur I N and Yung C M1993Hirota bilinearfform for the super KdV hierarchyMod.Phys.Lett.8A1739.[10]Oevel W and Popowicz Z1991The bi-Hamiltonian structure of fully supersymmetricKorteweg-de Vries systems Commun.Math.Phys.139441;Figueroa-O’Farril J,Mas J and Ramos E1991Integrability and biHamiltonian structure of the even order sKdV hierarchies Rev.Math.Phys.3479.[11]Wahlquist H D and Estabrook F B1974B¨a cklund transformation for solutions of theKorteweg-de Vries equation Phys.Rev.Lett.311386.。
近似解析求解Woods-Saxon势场Klein-Gordon方程陈文利;史艳维;冯晶晶【摘要】利用超几何函数方法对Pekeris近似中心项的变形Woods-Saxon势任意l态Klein-Gordon方程的束缚态和散射态进行近似解析求解,推导出径向波函数和束缚态特征值方程,得到了散射态相移公式,并通过对散射振幅在极点解析性质的研究得到了束缚态能级,最后讨论态的特例.%Within a Pekeris-type approximation to the centrifugal term,the approximate analytical bound and scattering state solutions of the arbitrary l-wave Klein-Gordon equation for the deformed Woods-Saxon potential were carried out by using hypergeometric function method.The analytical radial wave functions of the l-wave Klein-Gordon equation with the deformed Woods-Saxon potential are presented and the corresponding energy equation for bound states and phase shifts for scattering states were derived by studying analytical properties of scattering amplitude.And the special case for s-wave was also studied briefly.【期刊名称】《西华大学学报（自然科学版）》【年(卷),期】2017(036)004【总页数】6页(P87-92)【关键词】变形的Woods-Saxon势场;束缚态和散射态;Klein-Gordon方程;近似解析解【作者】陈文利;史艳维;冯晶晶【作者单位】西安培华学院通识教育中心,陕西西安 710125;西安培华学院通识教育中心,陕西西安 710125;西安培华学院通识教育中心,陕西西安 710125【正文语种】中文【中图分类】O365由Roger等[1]在1954年研究质子与重核弹性散射而提出的Woods-Saxon势是一种重要的平均场原子势，用来描述中子和重核的相互作用和表示核密度的分布，被广泛应用于核、粒子、原子、凝聚态物理和化学物理。
第 52 卷第 6 期2023 年 6 月Vol.52 No.6June 2023光子学报ACTA PHOTONICA SINICA 线性散焦PT 对称波导中饱和非线性孤子传输与控制武琦，王娟芬，杜晨锐，杨玲珍，薛萍萍，樊林林（太原理工大学 光电工程学院，太原 030600）摘要：为了研究线性散焦宇称-时间对称双通道波导中分数阶衍射饱和非线性下孤子的模式以及孤子的传输与控制，通过改进的平方算子迭代法对含有线性势的分数阶饱和非线性薛定谔方程进行数值计算得到孤子模式，傅里叶配置法判断孤子线性稳定性，并利用分步傅里叶法模拟仿真孤子的传输。
研究结果表明：在散焦饱和非线性中，该宇称-时间对称波导可支持稳定的双峰灰孤子模式。
随着饱和非线性系数和传播常数绝对值的增大，双峰灰孤子的背景强度增大，灰度值减小，功率增大。
Lévy 指数、增益/损耗系数和饱和非线性系数的增加会导致孤子的横向能流密度变化增大，但在波导通道位置处接近于0。
在聚焦饱和非线性下，线性散焦宇称-时间对称波导对亮孤子光束具有控制作用。
当光束在波导中心输入，孤子以呼吸子的形式长距离传输；在非波导中心输入，光束以初始输入位置为边界振荡传输。
随着饱和非线性系数的增大，光束的振荡频率增加，光束宽度变宽，峰值强度减小。
宇称-时间对称波导势阱深度的增加会导致光束的振荡频率增加，峰值强度增加。
该研究结果可为宇称-时间对称波导对光束的控制提供一定的理论参考。
关键词：非线性光学；宇称-时间对称光波导；灰孤子；光束控制；饱和非线性；分数阶薛定谔方程中图分类号：O437 文献标识码：A doi ：10.3788/gzxb20235206.06190010 引言宇称-时间（Parity -Time ， PT ）对称的概念起源于量子力学，它表示系统在宇称变换和时间反演变换下的对称性。
1998年，BENDER C M 等发现非厄米哈密顿量如果满足PT 对称且势函数虚部不超过对称破缺点，则其具有实的本征值谱［1-2］。
