Supersymmetry and the Superconductor-Insulator Transition

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).。

arXiv:hep-th/9911versity of British Columbia 6224 Agricultural Rd, Vancouver BC, Canada, V6T 1Z1
Abstract AdS/CFT-correspondence establishes a relationship between supersymmetric gravity (SUGRA) on Anti-de-Sitter (AdS) space and supersymmetric Yang-Mills (SYM) theory which is conformaly invariant (CFT). The AdS space is the solution of the Einstein-Hilbert equations with a constant negative curvature. Why is this relationship important? What kind of relationship is this? How does one find it? The purpose of this text is to answer these questions. We try to present the main ideas and arguments underlying this relationship, starting with a brief sketch of old string theory statements and proceeding with the definition of D-branes and a description of their main features. We finish with the observation of the correspondence in question and the arguments that favor it.

1
Alexander von Humboldt fellow at Leibniz Universit¨ at Hannover.
1
Introduction
The N =4 super Yang-Mills (SYM) field theory, being the maximally extended rigid supersymmetric model, possesses many remarkable properties. The symmetry of this model is so large that the only freedom in the classical action is the choice of the gauge group, and the quantum dynamics is free of divergences. It worth pointing out that this theory has profound relations with superstring theory, particularly due to the AdS/CFT correspondence (see, e.g., [1]). The problems of N =4 SYM theory in the quantum domain are mainly related to the effective action and correlation functions of composite operators. The superfield approaches seem to be more efficient for these purposes, since they allow one to use the supersymmetries in explicit form. However, a description of N =4 SYM theory in terms of unconstrained N =4 superfields is still missing. For various applications, formulations in terms of N =1 superfields (see, e.g., [2]), in terms of N =2 superfields [3, 4], or in terms of N =3 superfields [5] are used. All attempts to find an unconstrained N =4 harmonic superfield formulation for the N =4 SYM theory have been futile so far [7, 8, 9, 10, 11, 12], for a number of different types of harmonic variables originating from various cosets of the SU (4) group. However, one may still hope that there exists some other harmonic superspace, not based on some SU (4) coset, which is better suited for a superfield realization of N =4 supergauge theory.2 In other words, we need new superfield representations of the known irreducible multiplets of the N =4 superalgebra realized in an appropriate harmonic superspace. The main purpose of this paper is to construct N =4 superparticle models in harmonic superspace, to quantize them and to derive superfield representations of the N =4 superalgebra as a result of their quantization. It is well known that the quantization of superparticles is closely related to superfield formulations of the corresponding field theories. Indeed, the superparticle models are rich of symmetries such as reparameterization invariance, supersymmetry and, in particular cases, the kappa-symmetry (see, e.g., [13] for a review). All these symmetries are accompanied by constraints in the Hamiltonian formulation. Upon quantization, these constraints turn into equations of motion as well as superfield differential constraints, which together define superfield representations of irreducible multiplets of supersymmetry. For instance, the quantization of the N =1 superparticle was achieved in [14, 15, 16, 17], the N =2 gauge multiplet and hypermultiplets were obtained in [17, 18, 19, 20, 21] by quantizing the N =2 superparticle and, finally, the N =3 superparticle was recently studied by two of us [22], where massive and massless N =3 vector multiplets as well as the N =3 gravitino multiplet were derived. A particular case of massless superparticles with arbitrary N >2 extended supersymmetry in SU (N ) harmonic superspace was analyzed in [18], where the corresponding superfield strengths were derived. We point out the significance of harmonic superparticles with N =2 and N =3 extended supersymmetries [18, 19, 22], since they yield equations of motion for the corresponding field theories in harmonic superspaces which possess unconstrained superAs is well known, an unconstrained superfield formulation of N =4 SYM theory is impossible in standard N =4 superspace.

SU-ITP 97Biblioteka 51 hep-th/9712072
Review of Matrix Theory
D. Bigatti† and L. Susskind
Stanford University
Abstract
In this article we present a self contained review of the principles of Matrix Theory including the basics of light cone quantization, the formulation of 11 dimensional M-Theory in terms of supersymmetric quantum mechanics, the origin of membranes and the rules of compactification on 1,2 and 3 tori. We emphasize the unusual origins of space time and gravitation which are very different than in conventional approaches to quantum gravity. Finally we discuss application of Matrix Theory to the quantum mechanics of Schwarzschild black holes. This work is based on lectures given by the second author at the Cargese ASI 1997 and at the Institute for Advanced Study in Princeton.

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1
1
The bottom–up mapping is presented in section 2. Section 3 containes some general results obtained with this method and its application to the low energy region with light chargino and stop. More detailed presentation of the bottom–up mapping and more results can be found in reference [3].
In the small or moderate tan β regime, the dependence of the RG running of these parameters on slepton (and the two first generation sfermion) masses comes only through the small hypercharge D-term contributions which we neglect here. Their contribution can be found in [3].
Abstract We present a bottom-up approach to the question of supersymmetry breaking in the MSSM. Starting with the experimentally measurable low energy supersymmetry breaking parameters which can take any values consistent with present experimental constraints, we evolve them up to an arbitrary high energy scale. Approximate analytical expressions for such an evolution, valid for low and moderate values of tan β , are presented. We then discuss qualitative properties of the high energy parameter space and in particular, identify the conditions on the low energy spectrum which are necessary for the parameters at high energy scale to satisfy simple regular pattern such as universality or partial universality. Talk given at the 28th International Conference on High Energy Physics, Warsaw, Poland, July 1996 † On leave of absence from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland

1
Jefferson Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
2
Physics Department, Stanford University, Stanford, California 94305, USA CERN, Theory Division, CH-1211 Geneva 23, Switzerland
1
Introduction
Our concept of Naturalness – the principle that Nature abhors fine-tunings – is based on theories with a few vacua. It has led to the proposal of low-scale supersymmetry, in order to avoid tuning short-distance parameters to 30-decimals. But this principle faces a serious challenge from the cosmological constant problem, where we see no new physics at the scale ∼ 10−3 eV required by naturalness. Recent developments in string theory [1] suggest the existence of an enormous “landscape” of long-lived metastable vacua. These can have a significant impact on the criterion of naturalness, and may provide the key to resolving the cosmological constant problem along the lines of ref. [2]. They may also have a bearing on the solution to the hierarchy problem. For example, if the number of vacua that break supersymmetry at high-scale is more than 1030 times larger than those with low-scale supersymmetry, then the breaking of supersymmetry at high scales is favored [3]. The 30-decimal tuning is compensated by the enormous “entropy factor” favoring high-scale breaking of supersymmetry. In this case, the simplest possibility would be that the Standard Model (SM) is preferred to the usual low-energy supersymmetric SM [4]. This would then account for why we have not seen any evidence for supersymmetry – either in the spectrum or in rare process, such as FCNCs, CP violation, proton decay, etc. – at the expense of giving up two major successes of the supersymmetric SM: gauge coupling unification [5] and natural dark-matter candidate [6]. A more interesting possibility that preserves these successes is that approximate chiral symmetries protect the fermions of the supersymmetric SM down to the TeV scale [7, 8]. So, the sparticle spectrum of these theories is “split” in two: (i) the scalars (squarks and sleptons) that get a mass at the high-scale of supersymmetry breaking m ˜ , which can be as large as the GUT scale, and (ii) the fermions (gauginos and higgsinos) which remain near the electroweak scale and can account for both gauge-coupling unification and DM. The only light scalar in this theory is a finely-tuned Higgs. So, rather than the dull prediction that the LHC will discover just the Higgs, these theories – coined Split Supersymmetry – predict gauginos and higgsinos at a TeV, maintain the successes of the supersymmetric SM, and account for the absence of problematic flavor and CP-violation, of fast proton decay, and of an excessively light Higgs, caused by the presence of light squarks and sleptons in the supersymmetric SM. In this paper we address some novel theoretical, cosmological and phenomenological aspects of Split Supersymmetry1 . An important theoretical issue is how the spectrum of Split Supersymmetry can naturally emerge from a high-energy theory. In sects. 2 and 3 we

S´ eminaire Poincar´ e XI (2007) 1 – 50
S´ eminaire Poincar´ e
SPIN OR, ACTUALLY: SPIN AND QUANTUM STATISTICS∗
arXiv:0801.2724v3 [math-ph] 29 Feb 2008
J¨ urg Fr¨ ohlich Theoretical Physics ´ † ETH Z¨ urich and IHES
Abstract. The history of the discovery of electron spin and the Pauli principle and the mathematics of spin and quantum statistics are reviewed. Pauli’s theory of the spinning electron and some of its many applications in mathematics and physics are considered in more detail. The role of the fact that the tree-level gyromagnetic factor of the electron has the value ge = 2 in an analysis of stability (and instability) of matter in arbitrary external magnetic fields is highlighted. Radiative corrections and precision measurements of ge are reviewed. The general connection between spin and statistics, the CPT theorem and the theory of braid statistics, relevant in the theory of the quantum Hall effect, are described.

KOBE-TH-07-06
Supersymmetry in 5d Gravity
C. S. Lim,1, ∗ Tomoaki Nagasawa,2, † Satoshi Ohya,3, ‡ Kazuki Sakamoto,3, § and Makoto Sakamoto1, ¶
of Physics, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan National College of Technology, 265 Aoki, Minobayashi, Anan 774-0017, Japan 3 Graduate School of Science, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan (Dated: October 2, 2007)
arXiv:0710.0170v2 [hep-th] 1 Oct 2007st decade a considerable number of studies have been made on gauge/gravity theories with extra dimensions. In gauge-Higgs unification scenario, extra components of gauge fields play a role of Higgs fields [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Attractive models of grand unified theories (GUTs) on orbifolds have been constructed, avoiding common problems of four-dimensional GUTs [12, 13, 14, 15, 16]. Higgsless gauge symmetry breaking can be realized via boundary conditions of extra dimensions [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] and a large mass hierarchy can naturally be obtained in this scenario [17]. A higher dimensional scenario with a warped geometry has been proposed to solve the hierarchy problem by Randall-Sundrum [29]. In the scenario, all scales except for the scale of gravity are reduced to the weak scale by a warped factor. According to this scenario, various attempts have been made to construct realistic models [22, 23, 30, 31, 32]. Randall and Sundrum have also proposed a mechanism to localize gravity in the vicinity of a brane [33] which has attracted enormous attention [32, 34, 35, 36, 37, 38, 39, 40]. In constructing realistic models with extra dimensions, the spectrum of light Kaluza-Klein (KK) modes becomes important if those masses are accessible to future collider experiments. Thus, it will be worthwhile investigating what are characteristic features of the 4d spectrum of KK modes coming from extra dimensions. A related subject to study is to clarify the cancellation mechanism of divergences in loop corrections. Mass corrections to extra components of gauge fields are found to be finite at least at one-loop order. The finiteness is very important because finite quantities can be considered to be predictions of higher-dimensional theories though they will not be renormalizable. The cancellation of would-be divergences has not been, however, understood fully yet. 1 In a 4-dimensional point of view, the cancellation of divergences

a r X i v :h e p -p h /0002115v 3 5 J u n 2000UWThPh-2000-06HEPHY-PUB 729/00FTUV/00-10IFIC/00-10hep-ph/0002115LC-TH-2000-031Phenomenology ofStops,Sbottoms,τ-Sneutrinos,and Stausat an e +e −Linear ColliderA.Bartl,1H.Eberl,2S.Kraml,2W.Majerotto,2W.Porod 31Institut f¨u r Theoretische Physik,Universit¨a t Wien,A–1090Vienna,Austria2Inst.f.Hochenergiephysik,¨Osterr.Akademie d.Wissenschaften,A–1050Vienna,Austria3Inst.de F´ısica Corpuscular (IFIC),CSIC,E–46071Val`e ncia,SpainAbstractWe discuss production and decays of stops,sbottoms,τ-sneutrinos,and staus in e +e −annihila-tion in the energy range√s =0.5−1.5TeV,wherethese states are expected to be pair produced.Moreover,at an e +e −Linear Collider with this energy and an integrated luminosity of about 500fb −1it will be possible to measure masses,cross sections and decay branching ratios with high precision [6].This will allow us to obtain information on the fundamental soft SUSY breaking parameters.Therefore,it is necessary to investigate how this information can be extracted from the experimental data,and how precisely these parameters can be determined.In this way it will be possible to test our theoretical ideas about the underlying SUSY breaking mechanism.Phenomenological studies on SUSY particle searches at the LHC have shown that the detection ofthe scalar top quark may be very difficult due to the overwhelming background from t ¯tproduction [7,8,9,10,11].This is in particular true for m ˜t 1<∼250GeV [7].In principle,such a light stop could be discovered at the Tevatron.The actual mass reach,however,strongly depends on the luminosity,decay modes,and the available phase–space [12,13].Thus an e +e −Linear Collider with√s =500GeV and 1TeV.We give numerical results for the production cross sectionstaking into account polarization of both the e −and e +beams.In particular,we show that by using polarized beams it will be possible to determine the fundamental SUSY parameters with higher precision than without polarization.Moreover,we discuss the decays of these particles.The production crosssections as well as the decay rates of the sfermions show a distinct dependence on the ˜f L –˜f R mixingangles.Squarks (sleptons)can decay into quarks (leptons)plus neutralinos or charginos.Squarks may also decay into gluinos.In addition,if the splitting between the different sfermion mass eigenstates is large enough,transitions between these states by emmission of weak vector bosons or Higgs bosons are possible.These decay modes can be important for the higher mass eigenstates,and lead to complicated cascade decays.In the case of the lighter stop,however,all these tree–level two–body decays may bekinematically forbidden.Then the ˜t1has more complicated higher–order decays [14,15,16].The framework of our calculation is the Minimal Supersymmetric Standard Model (MSSM)[1]whichcontains the Standard Model (SM)particles plus the sleptons ˜ℓ±,sneutrinos ˜νℓ,squarks ˜q ,gluinos ˜g ,twopairs of charginos ˜χ±i ,i =1,2,four neutralinos,˜χ0i ,i =1,...,4,and five Higgs bosons,h 0,H 0,A 0,H ±[17].In Section 2we shortly review the basic features of left–right mixing of squarks and sleptons of the 3rd generation,and present formulae and numerical results for the production cross sections with polarized e −and e +beams.In Section 3we discuss the decays of these particles and present numerical results for their branching ratios.In Section 4we give an estimate of the errors to be expected for the fundamental soft SUSY–breaking parameters of the stop mixing matrix.In Section 5we compare the situation concerning stop,sbottom,and stau searches at LHC and Tevatron with that at an e +e −Linear Collider.Section 6contains a short summary.2Production Cross SectionsLeft–right mixing of the sfermions is described by the symmetric 2×2mass matrices which in the (˜f L ,˜f R )basis (f =t,b,τ)read [2,17]M 2˜f =M 2˜f La f m f a f m f M 2˜f R.(1)The diagonal elements of the sfermion mass matrices areM 2˜f L=M 2˜F +m 2Z cos 2β(T 3f −e f sin 2ΘW )+m 2f ,(2)M 2˜f R=M 2˜F ′+e f m 2Z cos 2βsin 2θW +m 2f(3)where m f ,e f and T 3f are the mass,charge and third component of weak isospin of the fermion f ,and θW is the Weinberg angle.Moreover,M ˜F =M ˜Q for ˜f L =˜t L ,˜b L ,M ˜F =M ˜L for ˜f L =˜τL ,and M ˜F ′=M ˜U ,M ˜D ,M ˜E for ˜f R =˜t R ,˜b R ,˜τR ,respectively.M ˜Q ,M ˜U ,M ˜D ,M ˜L ,and M ˜E are soft SUSY–breaking mass parameters of the third generation sfermion system.The off–diagonal elements of the sfermion mass matrices arem t a t=m t (A t −µcot β),(4)m b a b =m b (A b −µtan β),(5)m τa τ=m τ(A τ−µtan β)(6)for stop,sbottom,and stau,respectively.A t ,A b ,A τare soft SUSY–breaking trilinear scalar couplingparameters.Evidently,in the stop sector there can be strong ˜tL -˜t R mixing due to the large top quark mass.In the case of sbottoms and staus the ˜f L −˜f R mixing effects are also non-negligible if tan β>∼10.We assume that all parameters are real.Then the mass matrices can be diagonalized by 2×2orthogonalmatrices.The mass eigenvalues for the sfermions ˜f=˜t ,˜b,˜τare m 2˜f 1,2=1(M 2˜f L−M 2˜fR)2+4m 2f a 2f ),(7)and the mass eigenstates are˜f 1=˜f L cos θ˜f +˜f R sin θ˜f ,(8)˜f 2=˜f R cos θ˜f −˜f L sin θ˜f,(9)where ˜t 1,˜b 1,˜τ1denote the lighter eigenstates.The sfermion mixing angle is given bycos θ˜f =−a f m f(M 2˜f L−m 2˜f 1)2+a 2f m 2f,sin θ˜f =M 2˜f L−m 2˜f1(M 2˜f L−m 2˜f 1)2+a 2f m 2f.(10)The ˜ντappears only in the left–state.Its mass ism 2˜ντ=M 2˜L+1s 4e 2f δij(1−P −P +)−e f c ij δij16s 4W c 4W(v 2e +a 2e)(1−P −P +)−2v e a e (P −−P +) D ZZ,(12)where P −and P +denote the degree of polarization of the e −and e +beams,with the conventionP ±=−1,0,+1for left–polarized,unpolarized,right–polarized e ±beams,respectively.(E.g.,P −=−0.9means that 90%of the electrons are left–polarized and the rest is unpolarized.)v e =4s 2W −1,a e =−1are the vector and axial–vector couplings of the electron to the Z ,s 2W ≡sin 2θW ,c 2W ≡cos 2θW ,and c ijis the Z ˜f i ˜f j coupling (up to a factor 1/cos θW )c ij = T 3f cos 2θ˜f −e f s 2W−12T 3f sin 2θ˜f T 3f sin 2θ˜f −e f s 2W .(13)Furthermore,in Eq.(12)√(s −m 2Z )2+Γ2Z m 2Z,D γZ =s (s −m 2Z )θ˜t =0.98,θ˜b =1.17,and θ˜τ=0.82,respectively.There is a destructive interference between the γand Z –exchange contributions that leads to characteristic minima of the cross sections at specific values of the mixing angles θ˜f ,which according to Eq.(12)depend on √s dependence of the stop and sbottom pair production cross sections for m ˜t 1=220GeV,m ˜t 2=450GeV,cos θ˜t =−0.66,m ˜b 1=284GeV,m ˜b 2=345GeV,cos θ˜b =0.84,and m ˜g =555GeV.Here we have included supersymmetric QCD (i.e.gluon and gluino)corrections [21,22]and initial state radiation (ISR)[23].1The latter typically changes the cross section by ∼15%.The relative importance of the gluon and gluino corrections can be seen in Figs.1c,d where we plot∆σ/σ0for ˜t 1¯˜t 2and ˜b 1¯˜b 1production for the parameters of Fig.1a,b.In addition we also show the leading electroweak corrections in order of Yukawa couplings squared [24]for M =200GeV,µ=800GeV,m A =300GeV,and tan β=4.Let us discuss these corrections in more detail:The standard QCD correction [21](due to virtual gluon exchange and real gluon emission)is proportional to the tree–level cross section:σ=σ0(1+4αss =500GeV in (a)and√s =500GeV for two values of cos θ˜t :cos θ˜t =0.4in (a)andcos θ˜t =0.66in (b).The white windows show the range of polarization of the TESLA design [27].As40060080010001200140001020304050600800100012001400-100102030Figure 1:(a,b)Total cross sections for e +e −→˜ti ¯˜t j ,˜b i ¯˜b j as a function of √Figure 2:Total cross sections of stau and sneutrino pair production as a function of√one can see,one can significantly increase the cross section by using the maximally possible e−and e+ polarization.Here note that the(additional)positron polarization leads to an effective polarization[28] ofP eff=P−−P+σL+σR(16)whereσL:=σ(−|P−|,|P+|)andσR:=σ(|P−|,−|P+|).This observable is sensitive to the amount of mixing of the produced sfermions while kinematical effects only enter at loop level.In Fig.5we show A LR for e+e−→˜t i¯˜t i(i=1,2)as a function of cosθ˜t for90%polarized electrons and unpolarized as well as60%polarized positrons;√s=800GeV we obtainσ(˜t1˜t2)=7.9fb,9.1fb,and14.1fb for (P−,P+)=(0,0),(−0.9,0),and(−0.9,0.6),respectively.The left–right asymmetry,however,hardly varies with cosθ˜t:A LR(˜t1˜t2)≃0.14(0.15)for|P−|=0.9and|P+|=0(0.6),0<|cosθ˜t|<1,and the other parameters as above.We next discuss sbottom production using the scenario of Fig.1b,i.e.m˜b1=284GeV and m˜b2=345GeV.In this case,all three combinations˜b1¯˜b1,˜b1˜b2(≡˜b1¯˜b2+˜b2¯˜b1),and˜b2¯˜b2can be produced at√s=500GeV for unpolarized as well as for polarized e−beams(P+=0).The usefulness of beam polarization to(i)increase the cosθ˜τdependence and(ii)enhance/reduce˜τ1¯˜τ1relative to˜τ2¯˜τ2production is obvious.The left–right asymmetry of˜τi¯˜τi production for the parameters of Fig.9is shown in Fig.10. Here note that,in contrast to˜t and˜b production,A LR(˜τi¯˜τi)is almost zero for maximally mixed staus.-1.-0.50.0.5 1.020*********120140160-1.-0.50.0.5 1.51015202530Figure 3:cos θ˜t dependence of stop pair production cross sections for m ˜t 1=200GeV,m ˜t 2=420GeV,m ˜g =555GeV;√s =1TeV in (b);the label “L”(“R”)denotes P −=−0.9(0.9)with the dashed lines for P +=0,and the dotted lines for |P +|=0.6[sign(P +)=−sign(P −)];the full lineslabeled “U”are for unpolarized beams (P −=P +=0).----Figure 4:Dependence of σ(e +e −→˜t 1¯˜t 1)on degree of electron and positron polarization for √s =1TeV,m ˜t 1=200GeV,m ˜t2=420GeV,and m ˜g =555GeV;the solid lines are for 90%polarized electrons and unpolarized positrons,the dashed lines are for 90%polarized electrons and 60%polarized positrons.-1.-0.50.0.5 1.0510152025-1.-0.50.0.5 1.010203040-1.-0.50.0.5 1.246810Figure 6:cos θ˜b dependence of sbottom pair production cross sections for m ˜b 1=284GeV,m ˜b 2=345GeV,m ˜g =555GeV,√s =800GeV,m ˜b 1=284GeV,m ˜b 2=345GeV,and m ˜g=555GeV.----Figure 8:Dependence of (a)σ(e +e −→˜b 1¯˜b 1)and (b)σ(e +e −→˜b 2¯˜b 2)(in fb)on the degree of electron and positron polarization for√-1.-0.50.0.5 1.20406080-1.-0.50.0.5 1.20406080-1.-0.50.0.5 1.20406080Figure 9:cos θ˜τdependence of stau pair production cross sections for m ˜τ1=156GeV,m ˜τ2=180GeV,√s =500GeV,m ˜τ1=156GeV,m ˜τ2=180GeV,and cos θ˜τ=0.77.------s =500GeV,m ˜τ1=156GeV,m ˜τ2=180GeV,cos θ˜τ=0.77,and m ˜ντ=148GeV.Finally,the dependence of σ(e +e −→˜τi ¯˜τi ),for cos θ˜τ=0.77,and σ(e +e −→˜ντ¯˜ντ)on both the electron and positron polarizations is shown in Fig.11.Notice that one could again substantially increase the cross sections by going beyond 60%e +polarization.3DecaysOwing to the influence of the Yukawa terms and the left–right mixing,the decay patterns of stops,sbottoms,τ-sneutrinos,and staus are in general more complicated than those of the sfermions of the first two generations.As for the sfermions of the first and second generation,there are the decays into neutralinos or charginos (i,j =1,2;k =1,...4):˜t i →t ˜χ0k ,b ˜χ+j ,˜b i →b ˜χ0k ,t ˜χ−j ,(17)˜τi →τ˜χ0k ,ντ˜χ−j ,˜ντ→ντ˜χ0k ,τ˜χ+j .(18)Stops and sbottoms may also decay into gluinos,˜t i →t ˜g ,˜b i →b ˜g (19)and if these decays are kinematically allowed,they are important.If the mass differences |m ˜t i −m ˜b j |and/or |m ˜τi −m ˜ντ|are large enough the transitions ˜t i →˜b j +W +(H +)or ˜b i →˜tj +W −(H −)(20)as well as˜τi →˜ντ+W −(H −)or ˜ντ→˜τi +W +(H +)(21)can occur.Moreover,in case of strong ˜f L −˜f Rmixing the splitting between the two mass eigenstates may be so large that heavier sfermion can decay into the lighter one:˜t2→˜t 1+Z 0(h 0,H 0,A 0),˜b 2→˜b 1+Z 0(h 0,H 0,A 0),˜τ2→˜τ1+Z 0(h 0,H 0,A 0).(22)The SUSY–QCD corrections to the squark decays of Eqs.(17)to (22)have been calculated in [30].TheYukawa coupling corrections to the decay ˜b i →t ˜χ−j have been discussed in [31].All these corrections will be important for precision measurements.The decays of the lighter stop can be still more complicated:If m ˜t 1<m ˜χ01+m t and m ˜t 1<m ˜χ+1+m b the three–body decays [15]˜t 1→W +b ˜χ01,H +b ˜χ01,b ˜l +i νl ,b ˜νl l +(23)can compete with the loop–decay [14]˜t 1→c ˜χ01,2.(24)If also m ˜t 1<m ˜χ01+m b +m W etc.,then four–body decays [16]˜t 1→b f ¯f ′˜χ01(25)have to be taken into account.We have studied numerically the widths and branching ratios of the various sfermion decay modes.In the calculation of the stop and sbottom decay widths we have included the SUSY–QCD correctionsaccording to [30].In Fig.12we show the decay width of ˜t 1→b ˜χ+1as a function of cos θ˜tfor m ˜t 1=200GeV,m ˜t 2=420GeV,tan β=4,M =180GeV,µ=360GeV,(gaugino–like ˜χ+1),and M =360GeV,-1.-0.50.0.5 1.0.0.10.20.30.40.5Figure 12:Decay width of ˜t 1as a function of cos θ˜t for m ˜t 1=200GeV,m ˜t 2=420GeV,tan β=4,{M,µ}={180,360}GeV in (a),and {M,µ}={360,180}GeV in (b);the dashed lines show the results at tree level,the full lines those at O (αs ).µ=180GeV (higgsino–like ˜χ+1).If Γ(˜t 1)<∼200MeV ˜t 1may hadronize before decaying [32].In Fig.12this is the case for a gaugino–like ˜χ+1as well as for a higgsino–like one if cos θ˜t <∼−0.5(or cos θ˜t >∼0.9).In case that all tree–level two–body decay modes are forbidden for ˜t1,higher order decays are impor-tant for its phenomenology.In the following we study examples where three–body decay modes,Eq.(23),are the dominant ones.For fixing the parameters we choose the following procedure:in addition to tan βand µwe use m ˜t 1and cos θ˜t as input parameters in the stop sector.For the sbottom (stau)sector we fix M ˜Q ,M ˜D and A b (M E ,M L ,A τ)as input parameters.(We use this mixed set of parameters in order to avoid unnaturally large values for A b and A τ.)Moreover,we assume for simplicity that the soft SUSY breaking parameters are equal for all generations.Note that,because of SU(2)invariance M ˜Q appears in both the stop and sbottom mass matrices,see Eqs.(1)–(3).The mass of the heavier stop can thus be calculated from the above set of input parameters as:m 2˜t2=2M 2˜Q +2m 2Z cos 2β 13sin 2θW +2m 2t −m 2˜t 1(1+cos 2θ˜t )tan βM m ˜χ01m ˜χ+2500223M Qm ˜b 23701503420.98M EA τm ˜τ1cos θ˜τ190200m ˜e R220195181Table 1:Parameters and physical quantities used in Fig.13and 14.All masses are given in GeV.size as tan βis small.BR(˜t 1→c ˜χ01)is of order 10−4independent of cos θ˜t and therefore negligible.Near cos θ˜t =−0.3the decay into b W +˜χ01has a branching ratio of ∼100%.Here the ˜t 1b ˜χ+1coupling vanishesleading to the reduction of the decays into sleptons.In Fig.13b the branching ratios for the decays into the different sleptons are shown.As tan βis small the sleptons couple mainly to the gaugino components of ˜χ+1.Therefore,the branching ratios of decays into staus,which are strongly mixed,are reduced.However,the sum of both branching ratios is nearly the same as BR(˜t 1→b νe ˜e +L ).The decays into sneutrinos are preferred by kinematics.The decay ˜t 1→b W +˜χ01is dominated by top quark exchange,followed by chargino contributions.In many cases the interference term between t and ˜χ+1,2is more important than the pure ˜χ+1,2exchange.Moreover,we have found that the contribution from sbottom exchange is in general negligible.In Fig.14we show the branching ratios of ˜t 1decays as a function of tan βfor cos θ˜t =0.6and the other parameters as above.For small tan βthe decay ˜t 1→b W +˜χ01is the most important one.The branchingratios for the decays into sleptons are reduced in the range tan β<∼5because the gaugino component of ˜χ+1decreases and its mass increases.For tan β>∼10the decays into the b τE/final state become more important because of the growing τYukawa coupling and because of kinematics (m ˜τ1decreases with increasing tan β).Here ˜t 1→b ντ˜τ1gives the most important contribution as can be seen in Fig.14b.Even for large tan βthe decay into c ˜χ01is always suppressed.From the requirement that no two–body decays be allowed at tree level follows that m ˜χ+1>m ˜t 1−m b .Therefore,one expects an increase of BR(˜t 1→b W +˜χ01)if m ˜t 1increases,because the decay into b W +˜χ01is dominated by top–quark exchange whereas for the decays into sleptons the ˜χ+1contribution is the dominating one.In general BR(˜t 1→b W +˜χ01)is larger than 80%if m ˜t 1>∼350GeV [15].If the three–body decay modes are kinematically forbidden (or suppressed)four–body decays ˜t 1→b f ¯f ˜χ01come into play.Depending on the MSSM parameter region,these decays can also dominate overthe decay into c ˜χ01.For a discussion,see [16].We now turn to the decays of ˜t 2.Here the bosonic decays of Eqs.20and 22can play an important rˆo le as demonstrated in Figs.15and 16.In Fig.15we show the cos θ˜t dependence of BR(˜t 2)for m ˜t 1=200GeV,m ˜t 2=420GeV,M =180GeV,µ=360GeV,tan β=4,M ˜D =1.1M ˜Q ,A b =−300GeV,and m A =200GeV.As can be seen,the decays into bosons can have branching ratios of several ten percent.The branching ratio of the decay into the gaugino–like ˜χ+1is large if ˜t 2has a rather strong ˜t L component.The decay ˜t 2→˜t 1Z is preferred by strong mixing.The decays into ˜b i W +only occur for |cos θ˜t |>∼0.5because the sbottom masses are related to the stop parameters by our choice M ˜D =1.1M ˜Q .Notice thatBR(˜t 2→˜b i W +)goes to zero for |cos θ˜t |=1as in this case ˜t2=˜t R .We have chosen m A such that decays into all MSSM Higgs bosons be possible.These decays introduce a more complicated dependence on the mixing angle,Eq.(10),because A t =(m 2˜t1−m 2˜t 2)/(2m t )+µcot βdirectly enters the stop–HiggsFigure 13:Branching ratios for ˜t 1decays as a function of cos θ˜t form ˜t 1=220GeV,tan β=3,µ=500GeV,and M =240GeV.The other parameters are given in Table 1.The curves in a)correspond tothe transitions:◦˜t 1→b W +˜χ01,△˜t 1→c ˜χ01,˜t 1→b e +˜νe ,and •˜t 1→b τ+˜ντ.Figure 14:Branching ratios for ˜t 1decays as a function of tan βfor m ˜t 1=220GeV,µ=500GeV,cos θ˜t =0.25and M =240GeV.The other parameters are given in Table 1.The curves in a)correspondto the transitions:◦˜t 1→b W +˜χ01,△˜t 1→c ˜χ01,˜t 1→b e +˜νe ,and •˜t 1→b τ+˜ντ.In gray area m h 0<90GeV.couplings.Here notice also the dependence on the sign of cos θ˜t .Figure 16shows the branching ratios of ˜t 2decays as a function of m ˜t 2for cos θ˜t =−0.66and the other parameters as in Fig.15.Again we compare the fermionic and bosonic decay modes.While for a rather light ˜t 2the decay into b ˜χ+1is the most important one,with increasing mass difference m ˜t 2−m ˜t 1the decays into bosons,especially ˜t2→˜t 1Z ,become dominant.-1.-0.50.0.5 1.01020304050-1.-0.50.0.5 1.1020304050-1-0.500.5100.20.40.60.81-1-0.500.510.20.40.60.81Figure 17:Branching ratios of ˜τ1(a)and ˜τ2(b)decays as afunction of cos θ˜τfor m ˜τ1=156GeV,m ˜τ2=180GeV,M =120GeV,µ=300GeV,and tan β=4.0.0130(b)-----Figure 19:(a)Error bands and 68%CL error ellipse for determining m ˜t 1and cos θ˜t from cross section measurements;the dashed lines are for L =100fb −1and the full lines for L =500fb −1.(b)Error bands for the determination of cos θ˜t from A LR .In both plots m˜t1=200GeV,cos θ˜t=−0.66,√s =800GeV also ˜t2can be produced:σ(˜t 1¯˜t 2+c.c.)=8.75fb for P −=−0.9and P +=0.If this cross section can be measured with a precision of 6%this leads to m ˜t 2=420±8.9GeV (again,we took into account a theoretical uncertainity of 1%).2With tan βand µknown from other measurements this then allows one to determine the soft SUSY breaking parameters of the stop sector.Assuming tan β=4±0.4leads to M ˜Q =298±8GeV andM ˜U =264±7GeV for L =500fb−1.In addition,assuming µ=800±80GeV we get A t =587±35(or −187±35)GeV.The ambiguity in A t exists because the sign of cos θ˜t can hardly be determined from cross section measurements.This may,however,be possible from measuring decay branching ratios or the stop–Higgs couplings.A different method to determine the sfermion mass is to use kinematical distributions.This was studied in [37]for squarks of the 1st and 2nd generation.It was shown that,by fitting the distribution of the minimum kinematically allowed squark mass,it is possible to determine m ˜qwith high precision.To be precise,[37]concluded that at√2Herenote that ˜t 1¯˜t 1is produced at √s =500GeV.One can thusimprove the errors on m ˜t 1,m ˜t 2,and cos θ˜t by combining the information obtained at different energies.However,this is beyond the scope of this study.of <∼0.5%using just 20fb −1of data (assuming that all squarks decay via ˜q →q ˜χ01and m ˜χ01is known).The influence of radiative effects on this method has been studied in [38].Taking into account initial state radiation of photons and gluon radiation in the production and decay processes it turned out that a mass of m ˜q =300GeV could be determined with an accuracy of <∼1%with 50fb −1of data.(Thisresult will still be affected by the error on the assumed m ˜χ01,hadronization effects,and systematic errors.)Although the analysis of [37,38]was performed for squarks of the 1st and 2nd generation,the method is also applicable to the 3rd generation.For the determination of the mixing angle,one can also make use of the left–right asymmetry A LR ,Eq.(16).This quantity is of special interest because kinematic effects and uncertainities in experimental efficiencies largely drop out.At √10σ(pp →t ¯t )for m ˜t 1∼m t and σ(pp →˜t 1¯˜t 1)∼1L =20fb −1.Similar results have been obtained for ˜t1three–body decays into sleptons.The authors of [13]also studied the search for light sbottoms at the Tevatron Run II concentrating on the decay ˜b 1→b ˜χ01within mSUGRA.They conclude that with 2fb −1of data the reach is m ˜b 1<∼200(155)GeV for m ˜χ01≃70(100)GeV.With 20fb −1one is sensitive to m ˜b 1<∼260(200)GeV for m ˜χ01≃70(100)GeV.Moreover,their analysis requires a mass difference of m ˜b 1−m ˜χ01>∼30GeV.The higher reach compared to ˜t 1→c ˜χ01is due to the higher tagging efficiency of b ’s.Similarly,also at the LHC the searchfor sbottoms is,in general,expected to be easier than that for stops.There are,however,cases where the analysis is very difficult,see e.g.[8].The search for staus crucially depends on the possibility of τidentification.At hadron colliders,˜τ’s are produced directly via the Drell–Yan process mediated by γ,Z or W exchange in the s –channel.They can also be produced in decays of charginos or neutralinos originating from squark and gluino cascade decays e.g.,˜q →q ′˜χ±j with ˜χ±j →˜τ±i ντor ˜χ±j →˜νττ±.At Tevatron energies,W pair production is the dominant background,while t ¯tevents,with the b jets being too soft to be detected,are the main background at the LHC.SUSY background mainly comes from ˜χ±˜χ∓production followed by leptonic decays.The Drell–Yan production has a low cross section,and it is practically impossible to extract the signal from the SM background (SUSY background is less important).The situation is different if chargino and neutralino decays into staus have a large branching ratio.As pointed out in [41,42,44]this is the case for large tan βwhere the tau Yukawa coupling becomes important.In [42,43,11]the decays ˜χ±1→˜τ1ντand ˜χ02→˜τ1τ(˜τ1→τ˜χ01)with the τ’s decaying hadronically have been studied.In [44,45],the dilepton mass spectrum of final states with e +e −/µ+µ−/e ±µ∓+E miss T +jets has been used to identify ˜τ1in the decay chain ˜χ02→˜τ1τ→˜χ01τ+τ−with τ→e (µ)+νe (µ)+ντ.It turned out that ˜τ1with m ˜τ1<∼350GeV ought to be discovered at the LHC if m ˜τ1<m ˜χ02and tan β>∼10.From this one can conclude that there exist MSSM parameter regions for which (light)sfermions of the 3rd generation may escape detection at both the Tevatron and the LHC.This is in particular the case for ˜t 1if m ˜t 1<∼250GeV,and for ˜τ1if tan β<∼10(or m ˜τ1>m ˜χ02).In these cases,an e +e −Linear Collider would not only allow for precision measurements but even serve as a discovery machine.6SummaryIn this contribution we discussed the phenomenology of stops,sbottoms,τ–sneutrinos,and staus at an e +e −Linear Collider with√s =500GeV.Also the detection of ˜τ’s is possible atthe LHC only in a quite limited parameter range whereas it should be no problem at the Linear Collider.AcknowledgementsWe thank the organizers of the various ECFA/DESY meetings for creating an intimate and inspiring working atmosphere.We also thank H.U.Martyn and L.Rurua for fruitful discussions and sugges-tions.This work was supported in part by the“Fonds zur F¨o rderung der Wissenschaftlichen Forschung of Austria”,project no.P13139-PHY.W.P.is supported by the Spanish“Ministerio de Educacion y Cultura”under the contract SB97-BU0475382,by DGICYT grant PB98-0693,and by the TMR contract ERBFMRX-CT96-0090.References[1]H.P.Nilles,Phys.Rep.110(1984)1;H.E.Haber,G.L.Kane,Phys.Rep.117(1985)75.[2]J.Ellis,S.Rudaz,Phys.Lett.B128(1983)248.[3]G.Altarelli,R.R¨u ckl,Phys.Lett.B144(1984)126;I.Bigi,S.Rudaz,Phys.Lett.B153(1985)335.[4]M.Drees,M.M.Nojiri,Nucl.Phys.B369(1992)54.[5]A.Bartl,W.Majerotto,W.Porod,Z.Phys.C64(1994)499.[6]E.Accomando et al.,Phys.Rep.299(1998)1.[7]U.Dydak,diploma thesis,Vienna1996;CMS 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regular matter source for the Kerr spinning particle is discussed. A class of minimal deformations of the Kerr-Newman solution is considered obeying the conditions of regularity and smoothness for the metric and its matter source. It is shown that for charged source corresponding matter forms a rotating bag-like core having (A)dS interior and smooth domain wall boundary. Similarly, the requirement of regularity of the Kerr-Newman electromagnetic field leads to superconducting properties of the core. We further consider the U(I) x U’(I) field model ( which was used by Witten to describe cosmic superconducting strings ), and we show that it can be adapted for description of superconducting bags having a long range external electromagnetic field and another gauge field confined inside the bag. Supersymmetric version of the Witten field model given by Morris is analyzed, and corresponding BPS domain wall solution interpolating between the outer and internal supersymmetric vacua is considered. The charged bag bounded by this BPS domain wall represents an ‘ultra-extreme’ state with a total mass which is lower than BPS bound of the wall. It is also shown that supergravity suggests the AdS vacuum state inside the bag. Peculiarities of this model for the rotating bag-like source of the Kerr-Newman geometry are discussed. PACS: 04.70, 12.60.Jr, 12.39.Ba, 04.65 +e 1

a rXiv:h ep-th/987237v13J ul1998CALT-68-2190SUPERSYMMETRIC GAUGE THEORIES AND GRA VITATIONAL INSTANTONS SERGEY A.CHERKIS California Institute of Technology,Pasadena CA 91125,USA E-mail:cherkis@ Various string theory realizations of three-dimensional gauge theories relate them to gravitational instantons 1,Nahm equations 2and monopoles 3.We use this correspondence to model self-dual gravitational instantons of D k -type as moduli spaces of singular monopoles,find their twistor spaces and metrics.This work provides yet another example of how string theory unites seem-ingly distant physical problems.(See references for detailed results.)The central object considered here is supersymmetric gauge theories in three di-mensions.In particular,we shall be interested in their vacuum structure.The other three problems that turn out to be closely related to these gauge theories are:•Nonabelian monopoles of Prasad and Sommerfield,which are solutions of the Bogomolny equation ∗F =D Φ(where F is the field-strength of a nonabelian connection A =A 1dx 1+A 2dx 2+A 3dx 3and Φis a nonabelian Higgs field).•An integrable system of equations named after Nahm dT i2εijk [T j ,T k ],(1)for T i (s )∈u (n ).These generalize Euler equations for a rotating top.•Solutions of the Euclideanized vacuum Einstein equation called self-dual gravitational instantons ,which are four-dimensional manifolds with self-dual curvature tensorR αβγδ=1The latter provide compactifications of string theory and supergravity that preserve supersymmetry and are of importance in euclidean quantum gravity.The compact examples are delivered by a four-torus and K3.The noncom-pact ones are classified according to their asymptotic behavior and topology.Asymptotically Locally Euclidean (ALE)gravitational instantons asymptoti-cally approach R 4/Γ(Γis a finite subgroup of SU (2)).These were classified by Kronheimer into two infinite (A k and D k )series and three exceptional (E 6,E 7and E 8)cases according to the intersection matrix of their two-cycles.Asymp-totically Locally Flat (ALF)spaces approach the R 3×S 1 /Γmetric.(To be more precise S 1is Hopf fibered over the two-sphere at infinity of R 3.)Sending the radius of the asymptotic S 1to infinity we recover an ALE space of some type,which will determine the type of the initial ALF space.For example,the A k ALF is a (k+1)-centered multi-Taub-NUT space.Here we shall seek to describe the D k ALF space.M theory on an A k ALF space is known to describe (k+1)D6-branes of type IIA string theory.Probing this background with a D2-brane we obtain an N =4U (1)gauge theory with (k+1)electron in the D2-brane worldvolume.As the D2-brane corresponds to an M2-brane in M theory,a vacuum of the above gauge theory corresponds to a position of the M2-brane on the A k ALF space we started with.Thus the moduli space of this gauge theory is the A k ALF.Next,considering M theory on a D k ALF one recovers 4k D6-branes parallel to an orientifold O 6−.On a D2-brane probe this time we find an N =4SU(2)gauge theory with k matter multiplets.Its moduli space is the D k ALF.So far we have related gauge theories and gravitational instantons .D3NS5NS5pp p k D3Figure 1:The brane configuration corresponding to U (2)gauge theory with k matter mul-tiplets on the internal D3-branes.There is another way of realizing these gauge theories.Consider the Chalmers-Hanany-Witten configuration in type IIB string theory (Figure 1).2In the extreme infrared limit the theory in the internal D3-branes will appear to be three-dimensional with N=4supersymmetry.This realizes the gauge theory we are interested in.A vacuum of this theory describes a particular position of the D3-branes.In the U(2)theory on the NS5-branes the internal D3-branes appear as nonabelian monopoles,while every external semiinfinite D3-brane appears as a Dirac monopole in the U(1)of the lower right corner of the U(2).Thus the moduli space of such monopole configurations of non-abelian charge two and with k singularities is also a moduli space of the gauge theory in question.Another way of describing the vacua of the three-dimensional theory on the D3-branes is by considering the reduction of the four-dimensional theory on the interval.For this reduction to respect enough supersymmetry thefields (namely the Higgsfields of the theory on the D3-branes)should depend on the reduced coordinate so that Nahm Equations(1)are satisfied.Thus the Coulomb branch of the three-dimensional gauge theory is described as a moduli space of solutions to Nahm Equations.At this point we have two convenient descriptions of D k ALF space as a moduli space of solutions to Nahm equations and as a moduli space of sin-gular monopoles.It is the latter description that we shall make use of here. Regular monopoles can be described5by considering a scattering problem u·( ∂+ A)−iΦ s=0on every lineγin the three-dimensional space di-rected along u.The space of all lines is a tangent bundle to a sphere T=TP1. Let(ζ,η)be standard coordinates on T,such thatζis a coordinate on the sphere andηon the tangent space.Then the set of lines on which the scat-tering problem has a bound state forms a curve S∈T.S is called a spectral curve and it encodes the monopole data we started with.In case of singular monopoles some of the linesγ∈S will pass through the singular points.These lines define two sets of points Q and P in S,such that Q and P are conjugate to each other with respect to the change of orientation of the lines.Analysis of this situation1shows,that in addition to the spectral curve,we have to consider two sectionsρandξof the line bundles over S with transition functions eµη/ζand e−µη/ζcorrespondingly.Alsoρvanishes at the points of Q andξat those of P.Since we are interested in the case of two monopoles the spectral curve is given byη2+η2(ζ)=0whereη2(ζ)=z+vζ+wζ2−¯vζ3+¯zζ4.z,v and w are the moduli.z and v are complex and w is real.The sectionsρand ξsatisfyρξ= k i=1(η−P i(ζ)),where P i are quadratic inζwith coefficients given by the coordinates of the singularities.The above equations provide the description of the twistor space of the singular monopole moduli space.3Knowing the twistor space one can use the generalized Legendre transform techniques6tofind the auxiliary function F of the moduliF(z,¯z,v,¯v,w)=1ζ3+2 ωr dζ√ζ2−a1ζ2(√−η2−za(ζ)).(2)Imposing the consistency constraint∂F/∂w=0expresses w as a function of z and v.Then the Legendre transform of FK(z,¯z,u,¯u)=F(z,¯z,v,¯v)−uv−¯u¯v,(3) with∂F/∂v=u and∂F/∂¯v=¯u,gives the K¨a hler potential for the D k ALF metric.This agrees with the conjecture of Chalmers7.AcknowledgmentsThe results presented here are obtained in collaboration with Anton Kapustin. This work is partially supported by DOE grant DE-FG03-92-ER40701. References1.S.A.Cherkis and A.Kapustin,“Singular Monopoles and GravitationalInstantons,”hep-th/9711145to appear in Nucl.Phys.B.2.S.A.Cherkis and A.Kapustin,“D k Gravitational Instantons and NahmEquations,”hep-th/9803112.3.S.A.Cherkis and A.Kapustin,“Singular Monopoles and SupersymmetricGauge Theories in Three Dimensions,”hep-th/9711145.4.A.Sen,“A Note on Enhanced Gauge Symmetries in M-and String The-ory,”JHEP09,1(1997)hep-th/9707123.5.M.Atiyah and N.Hitchin,The Geometry and Dynamics of MagneticMonopoles,Princeton Univ.Press,Princeton(1988).6.N.J.Hitchin,A.Karlhede,U.Lindstr¨o m,and M.Roˇc ek,“Hyperk¨a hlerMetrics and Supersymmetry,”Comm.Math.Phys.108535-589(1987), Lindstrom,U.and Roˇc ek,M.“New HyperK¨a hler metrics and New Su-permultiplets,”Comm.Math.Phys115,21(1988).7.G.Chalmers,“The Implicit Metric on a Deformation of the Atiyah-Hitchin Manifold,”hep-th/9709082,“Multi-monopole Moduli Spaces for SU(N)Gauge Group,”hep-th/9605182.4。

a r X i v :h e p -t h /0002134v 1 16 F eb 2000ITP–UH–02/00February 2000NBI–HE–00–05hep-th/0002134Manifestly N=3Supersymmetric Euler-Heisenberg Action in Light-Cone SuperspaceThomas B¨o ttner,Sergei V.Ketov 1and Thomas Lau Institut f¨u r Theoretische Physik,Universit¨a t HannoverAppelstraße 2,Hannover 30167,Germany boettner,ketov,lau@itp.uni-hannover.deAbstractWe find a manifestly N=3supersymmetric generalization of the four-dimensional Euler-Heisenberg (four-derivative,or F 4)part of the Born-Infeld action in light-cone gauge,by using N=3light-cone superspace.1IntroductionThe Born-Infeld (BI)action in flat spacetime,2S BI =1−det(ηµν+bF µν),(1.1)is the particular non-linear generalization of Maxwell theory,F µν=∂µA ν−∂νA µ.The action (1.1)was initially introduced to regularize both the electric field and the self-energy of a point-like charge in electrodynamics [1].Much later,the BI action was recognized as the leading contribution to the effective action of open strings in an abelian background with constant field strength F [2],and as the essential part of the D3-brane action as well [3],with b =2πα′.The action (1.1)has many remarkable properties,e.g.,causal propagation and electric-magnetic duality [4,5].The BI Lagrangian can be rewritten to the formL =−1b 21−4p µνp µν,Q =i2εµνρσp ρσ,(1.4)have been introduced.Eliminating p µνfrom eq.(1.2)results in the equivalent La-grangian L =11+b 216(F ˜F )2,(1.5)where we have defined F 2=F µνF µν,˜Fµν=12We use ηµν=diag(+,−,−,−)and ¯h =c =1.supersymmetric generalizations of the four-dimensional bosonic BI action(1.1)are not known in any form.Supersymmetry apparently prefers the parametrization of the BI action in terms of the Maxwell term L2=−132 (F2)2+(F˜F)2 =12 Fµν±i˜Fµν .(1.6) This term is known as the Euler-Heisenberg(EH)Lagrangian[11].The EH action alsoappears as the bosonic part of the one-loop effective action in N=1supersymmetric√scalar electrodynamics with the parameter b−1=2b2 1−Lorentz invariance.The light-cone formulation is,therefore,very suitable for an off-shell formulation of N-extended supersymmetric gaugefield theories with manifest supersymmetry in N-extended light-cone superspace[17,18,19].We define light-cone coordinates in Minkowski spacetime asx+=12 x0+x3,x−=12 x0−x3 ,x=12 x1+ix2,¯x=12 x1−ix2 ,(2.1)and similarly for the gauge vectorfield,Aµ→(A+,A−,A,¯A).The real coordinate x+is going to be considered as‘light-cone time’.The linear transformation(2.1)of spacetime coordinates is obviously non-singular(with the Jacobian equal to i),while it does not preserve the Minkowski metric(i.e.it is not a Lorentz-transformation).The light-cone gauge readsA+=0.(2.2) In this(physical)gauge the A−component of the gaugefield Aµis supposed to be eliminated via its(non-dynamical)equation of motion,whereas the transverse components(A,¯A)are supposed to represent the physical propagatingfields.It is easy to solve the equation of motion for A−in the Maxwell theory,where it takes the form of a linear equation in the light-cone gauge(cf.refs.[17,18,19]). It becomes,however,a highly non-trivial problem in the BI or EH theory,where it takes the form of a non-linear partial differential equation.The equations of motion amount to the conservation law for the p-tensor,∂µpµν=0,(2.3) while the pµνin the BI theory is given bypµν=b2Fµν−b4 2F2−b42∂µFµ−F2+14Fµ−∂µF2+b4 −116∂µFµ−(F˜F)2+132Fµ−∂µ(F˜F)2 .(2.5)We use a perturbative Ansatz,in powers of the small parameter b2,for a solution to eq.(2.5),A−(x)=∞ n=0b2n A−(2n)(x).(2.6) As regards the leading and sub-leading terms,wefindA−(0)=1(∂+)2 −14˜Fµ−∂µ(F˜F)+14F2+b22∂+A−∂+¯A∂22δm n P+,m,n=1,2,3,(3.1)where the supersymmetry charges Q r transform in the fundamental representation of SU(3).A natural representation of the algebra(3.1)in N=3light-cone superspace Z=(xµ,θm,¯θn)is given byQ m=−∂√∂θn−i2¯θn∂+.(3.2)The covariant derivatives in N=3light-cone superspace areD m=−∂√∂θn +i2¯θn∂+.(3.3)They anticommute with the supersymmetry charges(3.2)and obey the same alge-bra(3.1).The irreducible off-shell representations of N=3light-cone supersymmetry are easily obtained by imposing the covariant chirality condition on N=3light-cone superfieldsφ(Z),D mφ(Z)=0.(3.4)A solution to eq.(3.4)in components is just given by an arbitrary complex function φ(x+,x−+i2θm¯θm,x,¯x;θn)≡φ(y;θ).Its expansion in the chiral superspace readsφ(y;θ)=1∂+θm¯χm(y)+i3!εmnpθmθnθpψ(y).(3.5)The light-cone N=3supersymmetry transformation laws for the components areδA=iεn¯χn,δ¯χm=√2εpqr¯εq¯χr−iεpψ,δψ=−√where (D )3=εmnp D m D n D p and similarly for (¯D)3.After some trials and errors,we find36(−i √∂+¯φ+2b 2 1∂+3∂+2φ¯∂2φ ∂+2¯φ∂2¯φ+14∂+3 ∂+2φ2φ ∂+2¯φ2¯φ−12∂+3 ∂+2φ2φ¯∂2φ ∂+2¯φ−12∂+3¯∂∂+φ¯∂∂+φ2φ ∂+2¯φ−12∂+A −∂+¯A ∂22C p 2¯Cp −2b 222∂+2∂+2C p ¯∂2A +¯∂2C p ∂+2A∂+2¯Cp ∂2¯A +∂2¯C p ∂+2¯A+14Cp2∂∂+¯Cp ∂∂+¯A 2¯A +2¯C p ∂∂+¯A∂∂+¯A(3.9)−14∂+2∂+2C p2A ¯∂2A +¯∂2C p ∂+2A 2A +2C p ¯∂2A∂+2A∂+2¯Cp +h .c ..One of the obvious features of both eqs.(3.8)and (3.9)is the apparent presence ofhigher derivatives,as may have been expected from the experience with the manifestly N=2supersymmetric generalization of the BI action in the covariant N=2superspace [9].The expected correspondence to the component D3-brane effective action having non-linearly realized extended supersymmetry and no higher derivatives implies the existence of a field redefinition that would eliminate the higher-derivative terms in our action and make its N=3supersymmetry to be non-linearly realised (i.e non-manifest)[6].We also note the absence of quartic (C 4)scalar terms and the on-shell (2A =2C =0)invariance of our action under constant shifts,C p (x )→C p (x )+c p ,which are supposed to be related to the possible interpretation of the C p fields as the Goldstone scalars associated with spontaneoulsy broken translations in the full N=3BI action.4ConclusionOur main results are given by eqs.(2.8),(3.8)and(3.9).Our initial motivation was to construct an N=4supersymmetric generalization of the EH action in the light-cone gauge.The N=4light-cone supersymmetry algebra is given by eq.(3.1),where the indices now take four values.Equations(3.2),(3.3)and(3.4)are still valid in N=4 light-cone superspace,where they have to supplemented by an extra(generalized reality)condition[17],D m D n¯φ=148∂+2εmnpq¯D m¯D n¯D p¯D qφ.(4.1)The restricted chiral N=4light-cone superfieldφis equivalent to the chiral N=3 superfield in eq.(3.5).Our efforts to construct an N=4generalization of eq.(2.8) along the similar lines(sect.3)unexpectedly failed,while eq.(4.1)was the main obstruction.We conclude that even a manifestly N=4supersymmetric generalization of the EH action in the light-cone gauge seems to be highly non-trivial,if any,not to mention an even more ambitious(manifest)N=4supersymmetrization of the BI action.AcknowledgementsWe are grateful to Norbert Dragon,Gordon Chalmers,Olaf Lechtenfeld and Daniela Zanon for useful discussions.References[1]M.Born,Proc.Roy.Soc.A143(1934)410;M.Born and L.Infeld,Proc.Roy.Soc.A144(1934)425;M.Born,Ann.Inst.Poincar´e7(1939)155[2]E.S.Fradkin and A.A.Tseytlin,Phys.Lett.163B(1985)123;A.Abouelsaood,C.G.Callan,C.R.Nappi and S.A.Yost,Nucl.Phys.B280(1987)599[3]C.G.Callan,and J.Maldacena,Nucl.Phys.B513(1998)198;G.W.Gibbons,Nucl.Phys.B514(1998)603;R.G.Leigh,Mod.Phys.Lett.A4(1989)2767[4]J.Plebanski.Lectures on Non-Linear Electrodynamics,Nordita(1968);S.Deser,J.McCarthy and O.Sarioglu,Class.and Quantum Grav.16(1999) 841[5]M.K.Gaillard and B.Zumino,Nucl.Phys.B193(1981)221;G.W.Gibbons and D.A.Rasheed,Nucl.Phys.B454(1995)185[6]A.A.Tseytlin,Born-Infeld Action,Supersymmetry and String Theory,in“YuriGol’fand Memorial Volume”,edited by M.Shifman,World Scientific,2000;hep-th/9908105[7]S.Cecotti and S.Ferrara,Phys.Lett.187B(1987)335[8]S.Deser and R.Puzalowski,J.Phys.A13(1980)2501[9]S.V.Ketov,Mod.Phys.Lett.A14(1999)501;Nucl.Phys.B553(1999)250[10]S.M.Kuzenko and S.Theisen,Supersymmetric Duality Rotations,M¨u nchenpreprint;hep-th/0001068[11]W.Heisenberg and H.Euler,Z.Phys.98(1936)714[12]M.Roˇc ek and A.A.Tseytlin,Phys.Rev.D59(1999)106001[13]A.De Giovanni, A.Santambrogio and D.Zanon,(α′)4Corrections to theN=2Supersymmetric Born-Infeld Action,Milano and Leuven preprint,hep-th/9907214[14]H.Liu and A.A.Tseytlin,JHEP9910(1999)003[15]F.Gonzalez-Rey and M.Roˇc ek,Phys.Lett.434B(1998)303[16]G.G.Harwell and P.S.Howe,Class.and Quantum Grav.12(1995)1823;P.S.Howe and P.C.West,Phys.Lett.389B(1996)273[17]L.Brink,O.Lindgren and B.E.W.Nilsson,Phys.Lett.123B(1983)323andNucl.Phys.B212(1983)401[18]S.Mandelstam,Nucl.Phys.B213(1983)149[19]S.V.Ketov,Theor.Math.Phys.63(1985)470.。

WU-AP/46/95
arXiv:hep-th/9508029v1 7 Aug 1995

Evaporation and Fate of Dilatonic Black Holes
Jun-ichirou Koga(a) and Kei-ichi Maeda(b) Department of Physics, Waseda University, Shinjuku-ku, Tokyo 169, Japan
Abstract We study both spherically symmetric and rotating black holes with dilaton coupling, and discuss the evaporation of these black holes via Hawking’s quantum radiation and their fates. We find that the dilaton coupling constant α drastically affects the emission rates, and therefore the fates of the black holes. When the charge is conserved, the emission rate from the non-rotating hole is drastically changed beyond α = 1 (a superstring theory) and diverges in the extreme limit. In the rotating cases, we analyze the slowly rotating black hole solution with arbitrary α √as well as three exact solutions, the Kerr–Newman (α = 0), and Kaluza–Klein (α = 3), and Sen black hole (α = 1 and with axion field). Beyond the same critical value of α ∼ 1, the emission rate becomes very large near the maximally charged limit, while for α < 1 it remains finite. The black hole with α > 1 may evolve into a naked singularity due to its large emission rate. We also consider the effects of a discharge process by investigating superradiance for the non-rotating dilatonic black hole.

a r X i v :h e p -t h /0307063v 1 6 J u l 2003hep-th/0307063Notes on supersymmetric Sp (N )theories with an antisymmetric tensorFreddy Cachazo School of Natural Sciences,Institute for Advanced Study,Princeton NJ 08540USA We study supersymmetric Sp (N )gauge theories with an antisymmetric tensor and degree n +1tree level superpotential.The generalized Konishi anomaly equations derived in hep-th/0304119and hep-th/0304138are used to compute the low energy superpotential of the theory.This is done by imposing a certain integrality condition on the periods of a meromorphic one form.Explicit computations for Sp (2),Sp (4),Sp (6)and Sp (8)with cubic superpotential are done and full agreement with the results of the dynamically gen-erated superpotential approach is found.As a byproduct,we find a very precise map fromSp (N )to a U (N +2n )theory with one adjoint and a degree n +1tree level superpotential.July,20031.IntroductionN=1supersymmetric gauge theories have provided a very rich arena to explore nonperturbative quantum effects.Holomorphy has been the key tool in controlling the strong coupling dynamics.In the90’s,these techniques were extensively used to determine the structure of holomorphic quantities in the infrared of asymptotically free theories(for a review,see e.g.[1,2]).For pure N=1SU(N)a massive vacuum is generated with low energy superpotentialW low=N(Λ3N)1/N.(1.1) This has the same low energy physics as the proposal of Veneziano and Yankielowicz[3]. They introduced a single scalar superfield S which is a fundamentalfield in the infrared but it is composite in the microscopic theory.In the latter,S is given by the glueball superfield −1S N +N .(1.2)At the extremum of(1.2),the fundamentalfield S acquires an expectation value S N=Λ3N and(1.2)reduces to(1.1).Clearly,the only contribution to this vev is nonpertubative and in perturbation theory we can write S N p.t.=0.It was proven in[4]that in the microscopic theory,S,defined as the glueball superfield,satisfies the identity S N≃0,where≃meansvalid in the chiral ring of the theory.In[4],it was also suggested that the exchange of the identity S N≃0in the microscopic theory by thefield equation S N p.t.=0in the effective theory is a sign of dual descriptions.If this is the case,no information of the constraint S N p.t=0should be visible in the off-shell low energy effective superpotential for S.In general,for any pure N=1gauge theory,with a simple Lie group G,the Veneziano-Yankielowicz superpotential is given byW VY=S ln Λ3hRecently,a new way of computing low energy holomorphic physics of a large class of N=1gauge theories using bosonic matrix models was conjectured[6,7,8].This con-jecture motivated the development of superfield techniques[9]and of generalized Konishi anomalies[4]to prove it.Using the superfield techniques[10],and using the generalized Konishi anomalies [11,12],the low energy superpotential for Sp(N)gauge theories with one matterfield in the antisymmetric representation and tree level superpotential was computed.These theories were also studied in the90’s using holomorphy to determine the dynamically generated superpotentials[13,14].In the far IR,after allfields have been integrated out,the two approaches should give the same answer.By this we mean a single function W low,the low energy superpotential,that depends on the dynamically generated scaleΛand tree level superpotential parameters.However,a discrepancy was found in[10]for several examples,namely,Sp(4),Sp(6) and Sp(8)in the unbroken classical vacuum with a cubic superpotential.The difference always sets in at orderΛ3h where h=N/2+1is the dual Coxeter number of Sp(N).It was then suggested that perhaps S satisfies relations coming from its glueball origin.This explanation,however,would contradict the dual picture mentioned above.In[15],a possible explanation to the discrepancy was suggested.An ambiguity in the UV behavior of a theory was discovered if one allows for supergroups.A prescription to“F-term complete”a theory was given by thinking about the gauge group G(N)as embedded in a supergroup G(N+k|k)with k→∞.Moreover,the matrix model,glueball superpoten-tial,or Konishi anomaly computations,were claimed to be computing the superpotential of the F-completion G(N+k|k).Therefore,the analysis of[10,11]and[12],which was compared to a standard Sp(N)field theory calculation[13,14],had actually being done for Sp(N)×Sp(0).Here Sp(0)is a remnant of the F-completion Sp(N+k1+k2|k1+k2)broken down to Sp(N+k1|k1)×Sp(k2|k2).This completion produces residual instantons that are not present in the standard UV completion of Sp(N).In these notes we show that by using the generalized Konishi anomaly equations derived in[11]and[12]and imposing that the periods of the generating function of chiral operators,T(z)= Tr12πi A1T(z)dz=N,1we get full agreement with the dynamically generated superpotential approach[13,14],i.e, no discrepancy was observed at any order computed in the examples.In particular,these contain the examples considered in[10].The A k’s in(1.4)refer to the A-cycles of a genus g=1Riemann surfacey2=W′(z)2+f(z)(1.5)where W′(z)=0is the classical F-termfield equation.The IR dynamics is taken into account by imposing that the period of T(z)dz over the B-cycle of(1.5)be an integer[16,17].Wefind that the cut A2does not close up on-shell.This is very surprising atfirst due to the second equation in(1.4).The fact that the vanishing of the period of T(z)dz through a given cut implies that the cut disappears on-shell has been usually assumed for U(N)theories with excellent results so far(see e.g.[18,4]).Here we show that this assumption is not valid in general and it is the reason for the discrepancy observed in [10,11,12].The main difference between Sp(N)with antisymmetric tensor and U(N)with adjoint is that in the former the branch points of(1.5)are simple poles of T(z)dz while in the latter they are regular points.We also consider general degree n+1superpotentials and generic classical vacua.In this case,the IR dynamics is also taken into account by imposing that the periods over the B-cycles of(1.5),which in general has genus n−1,be integers[16,17].The actual computation is carried out by mapping the problem to a U(N+2n)theory. This is a very precise map which might suggest a new kind of duality.To give an example,Sp(N)with one antisymmetric tensor1corresponds to U(N+2n) with one adjoint.If Sp(N)is classically unbroken,then U(N+2n)is classically broken to U(N+2)×U(2)n−1.Quantum mechanically,the vacua of Sp(N)correspond to U(N+2n) vacua with confinement index t=2(defined in[19]).Given that t=2,the computation is effectively carried out in U(N/2+n)classically broken to U(N/2+1)×U(1)n−1.In particular,for a cubic superpotential n=2and U(N/2+2)is broken to U(N/2+1)×U(1). The U(1)factor enters in the low energy superpotential starting at orderΛ3(N/2+1).This explains why the discrepancy found in[10]started at different orders for different values of N.The presence of the U (1)factors is reminiscent of the Sp (0)factors in [15].However,itisimportant to note that these U (1)factors are forced upon us and are crucial to the agreement with the standard Sp (N )dynamically generated superpotential results of[13,14].This paper is organized as follows:In section 2we consider Sp (N )theories with one antisymmetric tensor in detail.The anomaly equations are solved for a general degree n +1superpotential.The mapping to U (N +2n )is shown and used to compute W low .In section 3,the low energy superpotential for Sp (4),Sp (6)and Sp (8)is computed.Also,as a consistency check,Sp (0)and Sp (2)are considered.In section 4,we end with some conclusions and open questions.In appendix A,we review and compute relevant U (N +2n )results.Finally,in appendix B,we review the dynamically generated superpotential computation of W low for Sp (4),Sp (6)and Sp (8).2.Sp (N )with antisymmetric matterWe consider N =1supersymmetric Sp (N )gauge theories with a chiral superfield in the antisymmetric representation Φ≡AJ .A is a N ×N antisymmetric matrix and J is the invariant antisymmetric tensor of Sp (N ),i.e.,1N/2×N/2×iσ2.The theory is deformed by a degree n +1tree level superpotential W tree =n k =0g kz −Φ,R (z )=−1z −Φ.(2.2)Following [4],a set of equations which constraint the generating functions (2.2)was derived in [11,12].These equations are obtained by studying a generalization of the Konishi anomaly [20,21]and they can be used to determine T (z )and R (z )up to 2n complex parameters.2.1.Anomaly equations and solutionThe equations are given by[11,12],[W′(z)R(z)]−=1dz R(z).(2.3)Thefirst equation gives R(z)in terms of W′(z)= n k=0g k z k and a degree n−1 polynomial f(z)≡−2[W′(z)R(z)]+as follows,R(z)=W′(z)−y(z)+2W′′(z)y(z).(2.6)Note that the combination c(z)=c(z)+2W′′(z)is again a polynomial of degree n−1. In addition,note that the last term in(2.6)can be written as a logarithmic derivative. Combining these observations(2.6)becomesT(z)= c(z)dz log W′(z)2+f(z) .(2.7) From(2.7)we learn that T(z)dz is a meromorphic differential on the Riemann surface (2.5).It has simple poles at z=∞on thefirst and second sheets,and in all branch points. T(z)dz depends on2n complex parameters given by the coefficients of f(z)and c(z).The n coefficients in c (z )can be fixed by imposing12πi B i T (z )dz =b i for i =1,...,n −1.(2.9)The n th parameter in f (z )is related to the scale of the theory,Λ,and depends on the regularization scheme.The parameters b i ’s are determined by the IR dynamics.In [17],itwas shown thatfor a U ( N)theory with adjoint and fundamental matter,solving the field equations of the low energy effective superpotential is equivalent to imposing the integrality of the periodsb i ’s in (2.9).Here we assume that the IR dynamics of the Sp (N )theory is also determined by imposing the integrality of b i .This surprisingly simple condition seems to be a generic feature of theories where a hyperelliptic Riemann surface emerges.It would be interesting to explore this in more detail.We comment more on this issue in section 4.At this point we have all the ingredients to compute T (z )and W low .However,it proves more convenient to map this problem to a known one.As a byproduct we find a surprisingly precise relation to a U (N +2n )theory.2.2.Mapping of Sp (N )to U (N +2n )Consider U ( N)with a single adjoint chiral superfield ΦU and tree level superpotential W tree(U)=n k =0h k 32π2Tr W UαW αUz −ΦU(2.12)are given in terms of two degree n −1polynomials f U (z )=−4[W ′U (z )R U (z )]+and c U (z )=[W ′U (z )T (z )]+as followsT U (z )=c U (z )2(W ′U (z )−y U (z )).(2.13)These are meromorphic functions on the Riemann surface y 2U =W ′U (z )2+f U (z ).As mentioned before,the IR dynamics of this theory is determined by simply imposing the integrality of all the periods of T U (z )dz .Motivated by the form of the equations (2.3)and (2.11)weproposethe following map,W ′U (z )=2W ′(z ),R U (z )=R (z ).(2.14)From these equations it is easy to get thatf U (z )=4f (z ),y U (z )=2y (z ).(2.15)The key observation is that T (z )for Sp (N )given by (2.7)has the same form as T U (z )in (2.13)except for a logarithmic derivative of a meromorphic function on (2.5).Therefore,the extra term does not affect the dynamics since it has integer periods automatically.This leads us to identifyT U (z )=T (z )+dz −Φ = Tr 1dzlog W ′(z )2+f (z ) .(2.17)Note that the first n orders in (2.17),i.e.,up to 1/z n are equivalent to[W ′(z )T (z )]++2W ′′(z )=[W ′(z )T U (z )]+.(2.18)Therefore,we do not get either extra information or an inconsistency by studying c U(z)= 2 c(z).Of particular interest is the leading order in(2.17),Tr1=Tr1U−2n.(2.19) This gives N=N+2n as claimed in the introduction.Let us also mention a relation which follows from(2.14)and will play a key role in the sequelS U=S.(2.20)Classical breaking patternIn order to complete the map we have to identify the correct classical breaking pattern.A U( N)theory with adjointΦU and tree level superpotential(2.10)generically has classical vacua with unbroken U( N1)×...×U( N n).The relation between N i and N i can be found by studying the periods of T U(z)through the A-cycles.In the classical vacuum where Sp(N)is broken to Sp(N1)×...×Sp(N n)we have (2.8),1dzlog W′(z)2+f(z) satisfies12πi A i T U(z)dz=12πi A iψ(z)dz=N i+2for i=1,...,n.(2.23)Note that this is consistent with N= n i=1 N i.In the Sp(N)unbroken vacuum,U(N+2n)is broken to U(N+2)×U(2)n−1. Vacua identificationThe last step in defining the associated U( N)problem is the identification of vacua in the quantum theory.Recall that in our normalization of Sp(N),N is an even number. Therefore,the set of numbers N i=N i+2with i=1,...,n has2as common divisor.Equivalently,if m is the maximum common divisor of the set N i,i=1,...,n,then m is even.The number of vacua for U(N+2n)broken to n i=1U(N i+2)is n i=1(N i+2).As shown in[19],these vacua fall into separate phases distinguished by the“confinement index”t(defined in[19]).t takes values in the set of integer factors of m.As mentioned above,m is always even.Therefore,at least two possibilities are present in this theory: t=1and t=2.A very useful fact about vacua with confinement index t>1is the following.T U(z) evaluated at vacua of U(N+2n)with classical breaking U(N1+2)×...×U(N n+2)and confinement index t can be effectively computed in a U((N+2n)/t)theory with classical breaking U((N1+2)/t)×...×U((N n+2)/t)and same tree level superpotential.Let us denote by T u(z)the generating function of chiral operators of such a U((N+2n)/t)theory. The relation between the two is very simple[19],T U(z)=t T u(z).(2.24) Consider for example the unbroken Sp(N)vacuum.This theory is maximally confin-ing.Clearly m=2since N i=0for i=2,...,n.Note also that U(N+2n)vacua with t=1are in a non confining phase.Therefore t must be equal to2.Motivated by this result,we propose that in general t=2.This is consistent with the fact that the center of Sp(N)is Z Z2.Let us check this proposal by counting the number of vacua in the general case.Pure supersymmetric Sp(N i)is expected to have h i=N i/2+1vacua distinguished by thesolutions to the effective superpotential(1.3),i.e.,S h i i=Λ3h ii.Therefore,when Sp(N)is classically broken to n i=1Sp(N i)wefind n i=1h i vacua.In the U(N+2n)theory we have n i=1(N i+2)which is2n times larger.However,the number of vacua with t=2is less and it is computed in U(N/2+n)broken to n i=1U(N i/2+1).Therefore,the number of vacua with t=2is n i=1(N i/2+1)which is the desired answer.However,this counting is not completely correct.The reason is that for each vacuum of U(N/2+n)there are t=2vacua in U(N+2n)[19].This would lead to twice the expected number.The way this happens is through the relation between the scales of the theories[22,19].If we denote byΛu andΛU the scales of U(N/2+n)and U(N+2n) respectively,thenΛN+2n u =ηΛN+2nUwithηt=1.(2.25)As we will see next,in the case of a cubic superpotential and unbroken Sp(N)classical vacuum this puzzle is resolved because the Sp(N)low energy superpotential only depends onΛ2(N+2n)u and therefore it is invariant under the Z Z2actionη→−η.We believe that this is also true in the general case.We leave the proof of this for future work.2.3.Low energy superpotentialThe low energy superpotential,W low,is a single function of the dynamically generated scale of the theoryΛand the tree level superpotential parameters.W low is defined so that the expectation value of the chiral operators TrΦk and S=−1∂g k =1∂logΛH =(N+4)S.(2.26)The factor(N+4)in the last equation is the coefficient of the holomorphic beta function of Sp(N)withΦan antisymmetric tensor.We denote byΛH the scale of the high energy theory beforeΦis integrated out.The equations(2.26)can be used tofind W low up to an irrelevant constant independent of the couplings in the superpotential andΛH.For simplicity,and also because it is the case used in the examples,let us consider acubic superpotentialW tree=g2TrΦ2+λTrΦ.(2.27)Using(2.17)we getTr1 = Tr1U −4, TrΦ = TrΦU +2mg2−2λg−m3g2 .(2.28)These equations,together with(2.26),determine W low once the corresponding vev’s of U(N+4)are known.An important consistency check is the integrability of(2.26).We assume that the U(N+4)problem has been solved,i.e.,W low(U)is known.More explicitly,we assume that the system∂W low(U)k+1 TrΦk+1U ,∂W low(U)has been integrated.Recall that h2=2g,h1=2m,and h0=2λare the couplings of W tree(U)(2.10).Let us propose an ansatz for W low,W low(ΛH,g,m,λ)=13m3g.(2.30)The ansatz forΛU=ΛU(g,ΛH)implies,by dimensional analysis,thatΛU is proportional toΛH up to a g dependent factor.This can be shown by taking a derivative with respect to logΛH of(2.30).This leads to∂W lowΛU∂ΛU∂ΛU ∂ΛUg.(2.32)Using(2.20)and the second equation in(2.29)we get,ΛU(ΛH,g)=g−2superpotential of U(N/2+2)classically broken to U(N/2+1)×U(1).The scales of the theories are related by[22,19]ΛN+4 u =ηΛN+4U.(2.35)withη2=1.This is the source of the doubling of vacua discussed at the end of section2.2. However,note that(2.33)and(2.34)imply thatΛ3,which is the expansion parameter in the Sp(N)theory,is invariant under the Z Z2actionη→−η.Effective superpotential for SFinally,in the case of the unbroken Sp(N)vacuum,an effective superpotential for S can be easily computed by integrating it in.This is done by performing a Legendre transform in two steps.First,we introduce the intermediate superpotentialW int(S,ΛH,C,g,m,λ)=W low(C,g,m,λ)+(N+4)S logΛHS(N/2+1) +(N/2+1) +∞ k=2a k S k.(2.37)3.ExamplesIn this section we consider the six cases studied in[10].Namely,Sp(4),Sp(6)and Sp(8)withΦin the antisymmetric traceful representation and withΦin the antisymmetric traceless representation.The theory is deformed by a cubic tree level superpotential and studied around the unbroken classical vacuum.As a consistency check of our approach we also include at the end of this section the analysis of Sp(0)and Sp(2).The tree level superpotential is given by(2.27),W tree=g2TrΦ2+λTrΦ.(3.1)In the traceful case we setλ=0and for the traceless case we use it as a Lagrange multiplier imposing the vanishing of TrΦ .The strategy to compute the low energy superpotential for Sp(4),Sp(6)and Sp(8)is based on the integrability of(2.26),which was proven in the section2.3.W low is obtained by first computing1Then(2.24)is usedto get12 TrΦ2 is obtained fromthe third equation in(2.28).Finally,1g2 gΛU2 N/2+1.(3.5)Thefinal result of using(3.5)in the superpotentials of appendix A are listed below.Note that W eff(S)can easily be computed using(2.36)but we will not do it here. Sp(4)low energy superpotential:Traceful case:W low=3Λ3−1m3−1m6−187m9−1235m12+O(Λ18).(3.6) Traceless case:W low=3Λ3+O(Λ18).(3.7)Sp(6)low energy superpotential:Traceful case:W low=4Λ3−3m3−47m6−75m9−7437m12+O(Λ18).(3.8)13Traceless case:W low=4Λ3−1m3−7m6−5m9−221m12+O(Λ18).(3.9)Sp(8)low energy superpotential:Traceful case:W low=5Λ3−5m3−13m6−65m9−2147m12+O(Λ18).(3.10) Traceless case:W low=5Λ3−1m3−1m6−7m9−1m12+O(Λ18).(3.11)Comparing these results with the ones obtained by using the dynamically generated superpotentials reviewed in appendix B,wefind complete agreement to all orders com-puted.Sp(0)and Sp(2)low energy superpotentials.A consistency check:Sp(0)and Sp(2)are special.The former should clearly give a trivial result.The latter is interesting since in our convention Sp(2)≃SU(2)and the antisymmetric matter is actually a singlet of the gauge group.This means that it does not participate in the strong coupling dynamics of the theory.This is why Sp(2)is a non trivial consistency check of our formalism.According to the general analysis of section2,we are instructed to consider U(4)→U(2)×U(2)and U(6)→U(4)×U(2)pute W low(U)in the vacua with t=2and use it in(2.30).Since t=2we only have to consider U(2)→U(1)×U(1)and U(3)→U(2)×U(1)respectively.In the case of U(2)→U(1)×U(1)it is known that S u=0[16]and therefore the low energy superpotential does not depend onΛu.This implies that W low(u)can be evaluated at anyΛu in particular,atΛu=0to get the classical answer,W low(u)(Λu,g u,m u,λu)=1g2u−m uλuUsing this in(2.30)wefind2W low=0.(3.13)This is indeed the correct result.In the case U(3)→U(2)×U(1),the low energy superpotential is given by[24,22],W low(u)(Λu,g u,m u,λu=0)=1g2u+2g uΛ3u.(3.14)Using this in(2.30)together with the matching relation(3.5)we get,W low=2Λ3.(3.15) This is the correct answer for pure N=1SU(2)gauge theory as can be seen from(1.1).4.Conclusions and open questionsThe low energy superpotential of N=1Sp(N)theories with matter in the anti-symmetric tensor representation and tree level superpotential was computed using the generalized Konishi anomaly equations found in[11]and[12].A key role was played by the generating function of chiral operators T(z).Quantum mechanically,T(z)dz becomes a meromorphic differential on a Riemann surface.A remarkably simple condition on T(z)dz was imposed which accounts for the full IR dynamics.The condition is the integrality of its A-and B-periods.Based on all the examples in the literature and the ones presented here,it is reasonable to propose that this is always the case in theories where a hyperelliptic Riemann surface emerges.One possible explanation is the following:In such theories a string theory realization with a dual involvingfluxes might be available.On the dual side,thefluxes through compact cycles are quantized and are given as the periods of a closed real form.In the projection to a Riemann surface3this form gives rise to a one form with integer periods.However, the one form is real instead of meromorphic.It is only on-shell that this real one form becomes meromorphic and agrees with T(z)dz.On the other hand,T(z)dz is meromorphic by definition but the integrality of its periods is valid only on-shell.It would be interesting to explore the full range of validity of this dual description.In the process of imposing the integrality condition of the periods of T(z)dz in the Sp(N)theory,we found a very precise map to a U(N+2n)theory.The map was given for a general degree n+1tree level superpotential.In the vacuum where Sp(N)is classically broken to n i=1Sp(N i),U(N+2n)is broken to n i=1U(N i+2).However,one point we did not discuss is the following:In the IR,U(N+2n)has low energy group U(1)n.Their couplings are frozen and depend on the parameters in the superpotential andΛU.What is the interpretation of this on the Sp(N)side?In section2we found a very simple expression(2.30)for the low energy superpotential W low of Sp(N)with a cubic tree level superpotential.It would be interesting to generalize it to tree level superpotentials of arbitrary degree.Generalizations of the approach presented in this work to Sp(N)and SO(N)with symmetric/antisymmetric tensors and fundamentals is surely possible and interesting.We leave it for future work.As mentioned in the introduction we have shown that the vanishing of the period of T(z)dz through a given cut does not imply that the cut closes up on-shell.Also surprising is the opposite case,namely,the period of T(z)dz is non zero but the cut closes up on-shell. This does not arise in the cases studied in this work but we believe it will show up for SO(N)with a symmetric tensor.This is currently under investigation.We also would like to comment on the role of the term dAppendix A.U(N+4)resultsIn this appendix we review and derive the relevant U(N+4)field theory results needed to compute the low energy superpotential W low of the Sp(N)theory discussed in section 3.The theory has a cubic tree level superpotentialW tree(U)=g u2TrΦ2U+λu TrΦU,(A.1)and it is classically broken to U(N+2)×U(2).Here we use the notation g u,m u and λu for both the U(N+4)and the U(N/2+2)theories as they share the same tree level superpotential.There are two possible ways to proceed.We will briefly discuss thefirst one,which is very powerful and can be applied to arbitrary high values of N.However,this is not what we use since the three cases considered in section3map to three cases already considered in[19]using a strong coupling analysis.The reader can skip thefirst one unless interested in larger values of N for which the strong coupling analysis is complicated.Matrix model effective superpotentialThefirst method is to use the matrix model[6,7,8]to compute an effective superpo-tential for U(N+4)around the classical vacuum U(N+2)×U(2).This superpotential is a function of S1=−14πi A2y U(z)dz given by W eff(S1,S2)=−1ΛU +2πi(b1S1+b2S2),(A.2)where B r i are regularized non-compact cycles,Λ0is a cut off,which should be taken to infinity at the end of the computation and b i’s are integers(for more details see e.g.[17]).At the extremum of(A.2)we get W low(U)=W eff(<S1>,<S2>).This result can be used in(2.30)to get the Sp(N)low energy superpotential W low.Strong coupling approachThe second method is to use a strong coupling analysis where the tree level super-potential is thought of as a deformation of a N=2U(N+4)theory.Since we are only17interested in vacua with confinement index t=2we can consider a N=2U(N/2+2)the-ory.The only N=1supersymmetric vacua are located at points in the N=2Coulomb moduli space that satisfyP2 N(z)−4Λ2 N u=112 TrΦu 2−.(A.5)v Clearly,in the semiclassical limitΛu→0,a→0and P4(z)→z3(z+v),showing that here U(4)is broken to U(3)×U(1).From(A.3)and(A.5),wefind that1,determineδ,v and a as functions of m u/g u,λu/g u andΛu.vThese equations can be solved in a power expansion inΛu aroundΛu=0.We also choose to expand around the classical solutionv cl= g u 2−4λu2 TrΦ2U .Using this last result in(2.28)weget1g2u 2T+1481T4+153140m u 8/3.(A.9) Finally,note that only the ratio m u/g u appears and can be replaced by m/g.Recall that the Sp(N)tree level superpotential is given by(3.1)and it is proportional to(A.1). W low can then be obtained by integrating(A.8)with respect to m,W Traceful low(Sp)=m327T4−4940m 8/3.(A.11)For the traceless case we have to useλto set the second equation in(2.28)to zero, i.e.,TrΦU +2m ug u=0.(A.13)The solution to(A.13)isλug2uT+O(T6).(A.14) Using this and following the same procedure as before we compute1g2u T+O(T6) .(A.15)19Integrating with respect to m we findW Traceless low(Sp)=m 3c 3.(A.17)In the semiclassical limit Λu →0,a →0and P 5(z )→z 4(z +c ),showing that here U (5)is broken to U (4)×U (1).From(A.3)and (A.17),we find that1m ug u.(A.19)In the traceful case with λu =0we get1g 2u4T +12T 2+14132T 5+O (T 6)(A.20)with T =(g u Λu /m u )5/2.Integrating with respect to m after replacing m u /g u by m/g we get,W Tracefullow(Sp)=m 33T 3−75T 4−7437g u=m 2u3T −418T 3−357776T 5+O (T 6).(A.22)Using this we compute1g 2u4T +46T 3+352592T 5+O (T 6).(A.23)。

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multi valley 34 Multi valley calculus . . . . . . . . . . . . . . . . . . . . . . . 34 ¯ valley and II ¯ valley revisited . . . . . . . . . . . . . . . . . 38 II 40
August, 1998
Abstract The elucidation of the properties of the instantons in the topologically trivial sector has been a long-standing puzzle. Here we claim that the properties can be summarized in terms of the geometrical structure in the configuration space, the valley. The evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbative calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A ‘proof’ of Bogomolny’s “trick” is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of the perturbation theory is confirmed by explicit calculations, in some cases to the 478-th order. A new type of supersymmetry is found as a by-product, and our result is shown to be consistent with the non-renormalization theorem. The prediction of the energy levels is confirmed with numerical solutions of the Schr¨ odinger equation.

a r X i v :h e p -t h /9507046v 1 7 J u l 1995NIKHEF/95-032DAMTP-95/36hep-th/9507046NEW SUPERSYMMETRY OF THE MONOPOLEF.De Jonghe,A.J.Macfarlane ∗,K.Peeters and J.W.van HoltenNIKHEF,Postbus 41882,1009DB Amsterdam,The NetherlandsJune 1995Abstract The non-relativistic dynamics of a spin-1/2particle in a monopole field possesses a rich supersymmetry structure.One supersymmetry,uncovered by d’Hoker and Vinet,is of the standard type:it squares to the Hamiltonian.In this paper we show the presence of another supersymmetry which squares to the Casimir invariant of the full rotation group.The geometrical origin of this supersymmetry is traced,and its relationship with the constrained dynamics of a spinning particle on a sphere centered at the monopole is described.∗On leave of absence from DAMTP,University of Cambridge,Silver Street,Cambridge CB39EW,UK,and St.John’s College,Cambridge.1.This paper discusses spin-1( p−e A)2−e B. S,(1)2where all vectors are3-dimensional.In particular S= σ/2, B=g r/r3,and the vector potential A is to be defined patchwise in the well-known way.It is more than ten years now since it was observed that this model posesses a hidden supersymmetry[1].This supersymmetry is of the so-called N=12,in terms of Pauli matrices,which gives rise to the description[5]of the spin of the monopole.In the rest of this paper,we will work at the level of the quantum theory,i.e.after the transition from Poisson brackets to canonical(anti)commutators.However, we will not use the specific representation of the Grassmann coordinates in terms of Pauli matrices that was mentioned above,since this can obscure the difference between Grassmann even and Grassmann odd operators.In this paper,we show that the dynamics of a spin-1[ J2−e2g2+12.(4)rThe operator˜Q—which coincides up to a constant shift with the operator A in[1]—plays an important role in the determination of the spectrum of the model.To see the equivalence between˜Q and A,one has to use the representation of the Grassmann coordinates by Pauli matrices,and,in this way,one loses the insight that A is in fact a Grassmann odd operator.In the original analysis,A was introduced only on algebraic grounds, and its geometrical interpretation remained unclear.Here we explain that it corresponds to a new type of supersymmetry of the problem,and our way of obtaining this extra supercharge shows the geometry of the monopole configuration to be considerably richer than anticipated.In particular,since the new supercharge˜Q squares to essentially J2,our analysis shows that the full supersymmetry algebra of the monopole is in fact a non-linear algebra.2.Our discussion is based,firstly,on a general method[6]which uses Killing-Yano tensors[8]to generate additional supersymmetries in a theory which already possesses some known supersymmetries of a standard type,and,secondly,on the realisation that vital theoretical information can stem from the employment,where appropriate,of fermionic phase spaces of odd dimension.We have seen that the last circumstance does apply to the monopole.Also,it is true that theflat background of the monopole admits the simplest available non-trivial example of a Killing-Yano tensor,viz.f ij=εijk x k.(5)1This satisfies trivially the conditions generally obeyed[7,6]by such a tensor,namely,f ij=−f ji,f ij,k+f ik,j=0.(6) Further,we can use it,as in[6],to define a hermitian supercharge˜Q=˜Q†,which obeys(2)and(3).We develop these matters below starting from the superfield formulation.Our treatment may be compared with the recent work of[9],which added electromagnetic interactions to the curved space analysis of spinning particles in d dimensions in[6].Although[9]does not address the special properties and simplifications that occur in the specific case of d=3,our work evidently has links with the general results found there.We use scalar superfieldsΦi=x i+iθψi,i=(1,2,3),(7) involving one real Grassmann variable such thatθ2=0,θ=θ∗,a real coordinate x i and Majorana fermions ψi.The most general Lagrangian we can build is of the formS= dt dθ i6kεijk DΦi DΦj DΦk ,(8)where the operator D=∂θ−iθ∂t.This contains kinetic,electromagnetic and torsion terms,but no pure potential term can be built without the use of spinor superfields(see e.g.[4]for more details on this point). However,we do not need the latter here and will simplify(8)by taking the coupling constant for the torsion term to be zero atfirst.Writing the action out in componentfields one obtainsL=12iψi˙ψi+e˙x i A i(x)−12εijk F jk.As usual[1],p i=˙x i+eA i,[x i,p j]=iδij,{ψi,ψj}=δij,(10) and H=Q2for the superchargeQ=(p i−eA i(x))ψi.(11) Following the procedure of[6],we seek a new supercharge˜Q,which obeys(2)and hence describes a new supersymmetry of H,in the form˜Q=(pi−eA i)f ijψj+ibwhere S i=−i.(14)r3The condition on B here is a special case of the more general conditionfλ[µFν]λ=0(15) found in[9].It remains to compute and interpret the conserved quantity K that one obtains by squaring this new supercharge.A direct computation gives the answer(3)announced in the introduction,where the total angular momentum J can also be written asJ= r×( p−e A)+ S−eg1r.(16)rThefirst term L in the last formula,shows that the part of K quadratic in the momenta:12 J2−e2g2+1Such a structure in which the Killing-Yano tensoris related to a square root of total angular momentumis familiar from other examples[6,13].It represents the particular kind of non-linearity familiar also from finite-dimensional W-algebras[14].3.Recently it was shown by Rietdijk and one of us[15],that there exists a certain duality which relates two theories in which the role of the Killing-Yano tensor and the vierbein and the role of the supercharges Q and˜Q are parison of(11)and(12)indicates that similar ideas are relevant in the theory studied here.Thus one is led to consider not only our original theory but also the one in which K and˜Q are Hamiltonian and‘first’supersymmetry.One knows the Hamiltonian of the latter theory and the canonical equations for the variables that occur in it,so that its dynamical content can be completely determined.In fact we can show that it describes a particle of spin12 x2i−ρ2 −iλψi x i.(21)The full set of constraints(primary and secondary)then becomesx2i=ρ2,x iψi=0,x i(p i−eA i)=0.(22) After introducing Dirac-brackets,the constraints can be used to write the classical Hamiltonian asH∗=12ρ2−eB· S,(24)where M i=−f ij(p j−eA j)=εijk x j(p k−eA k).Clearly,this supersymmetric model describes a subclass of the solutions of the original model(those with spherical symmetry),and involves precisely the Stackel-Killing tensor as its metric and the Killing-Yano tensor as the dreibein.Interestingly,the modification introduced here changes the new Killing-Yano-based supersymmetry of the original model into a standard supersymmetry of the constrained Hamiltonian.In the quantum theory,tofind the corresponding result,we rescale the new supercharge(12)by a factorρ˜Q=12ρ−iεijkσi x j D k+12 −D i δij−x i x j4ρ2,(26)As a result,the supercharge(25)is naturally identified with the proper restriction of˜Q/ρof the original model to the sphere with radiusρ;however,restricted to the sphere it plays the role of the ordinary supercharge Q:4it is a square root of the(constrained)Hamiltonian.Thus we see,how the new non-standard supercharge of one model is related to the standard supersymmetry of another model.Moreover,the second model is here a constrained version of the original model.We remark on the appearance of(25)in the constrained model.In the constrained model,the representation of the fermionsψi and of the momentum p i requires an extra projection operator K ij in comparison with the representation in the unconstrained model.However,the extra terms do not contribute to the representation of˜Q in the constrained model,basically owing to the zero mode of the Killing-Yano tensor f ij. AcknowledgementsThe research of FDJ and JWvH is supported by the Human Capital and Mobility Program through the network on Constrained Dynamical Systems.The research of KP is supported by Stichting FOM.AJM thanks JWvH for the hospitality extended to him at NIKHEF at the time when much of the present research was performed. References[1]E.d’Hoker and L.Vinet,Phys.Lett.137B(1984)72.[2]M.Sakimoto,Phys.Lett.B151(1985)115.[3]R.H.Rietdijk,Ph.D.thesis Amsterdam1992,Applications of supersymmetric quantum mechanics.[4]A.J.Macfarlane,Nucl.Phys.B438(1995)455.[5]J.L.Martin,Proc.Roy.Soc.251(1959)536and ibid.543.[6]G.W.Gibbons,R.H.Rietdijk and J.W.van Holten,Nucl.Phys.B404(1993)42.[7]B.Carter and R.G.McLenaghan,Phys.Rev.D19(1979),1093.[8]K.Yano,Ann.Math.55(1952),328.[9]M.Tanimoto,TIT/HEP-277/COSMO-50,gr-qc/9501006.[10]J.W.van Holten and R.H.Rietdijk,Class.Quantum Grav.11(1993)559.[11]R.A.Coles and G.Papadopoulos,Class.Quantum Grav.7(1990)427.[12]R.Jackiw,Ann.Phys.129(1980)183.[13]J.W.van Holten,Phys.Lett.B342(1995)47.[14]J.de Boer,F.Harmsze and T.Tjin,preprint ITP-SB-95-07and ITFA-03-2/95.[15]R.Rietdijk and J.W.van Holten,Durham/NIKHEF preprint(july1995).5。