We derive the current algebra of supersymmetric principal chiral models with
a Wess-Zumino term. At the critical point one obtains two commuting super
Kac-Moody algebra as expected, but in general there are intertwining fields
connecting both right and left sectors, analogously to the bosonic case.
Moreover, in the present supersymmetric extension we have a quadratic algebra,
rather than an affine Lie algebra, due to the mixing between bosonic and
fermionic fields since the purely fermionic sector displays a Lie algebra as
well.Comment: 13 page

Abdalla E

Abdalla M C B

Bernard D

Cutright T L

de Vega H J

Dirac P A M

Drinfeld V G

E Abdalla

Eichenherr H

Faddeev L D

Forger M

Knizhnik V G

L E Saltini

Lepowsky J

M C B Abdalla

Maillet J M

O H G Branco

Polyakov A M

Zamolodchikov A B

English

arXiv

We derive the current algebra of supersymmetric principal chiral models with a Wess-Zumino term. At the critical point one obtains two commuting super-affine Lie algebras as expected, but, in general, them are intertwining fields connecting both right and left sectors, analogously to the bosonic case. Moreover, in the present supersymmetric extension we have a quadratic algebra, rather than an affine Lie algebra, due to the mixing between bosonic and fermionic fields; the purely fermionic sector displays an affine Lie algebra as well

Abdalla, E.

Abdalla, MCB

Branco, OHG

Saltini, L. E.

Acervo Digital da Unesp

CURRENT-ALGEBRA OF SUPER WZNW MODELS

We derive the current algebra of supersymmetric principal chiral models with a Wess-Zumino term. At the critical point one obtains two commuting super Kac-Moody algebra as expected, but in general there are intertwining fields connecting both right and left sectors, analogously to the bosonic case. Moreover, in the present supersymmetric extension we have a quadratic algebra, rather than an affine Lie algebra, due to the mixing between bosonic and fermionic fields since the purely fermionic sector displays a Lie algebra as well

Abdalla, Elcio

Abdalla, Maria Christina B

Branco, O H G

Saltini, L E

CERN Document Server

Current Algebra of Super WZNW ModelsE. Abdalla 1, M.C.B. Abdalla 2, O.H.G. Branco 3 and L.E. Saltini 4 1,3,4 Instituto de Fsica, Universidade de S~ao Paulo,Cx. Postal 20516, BR-01498-970 S~ao Paulo SP / Brazil2 Instituto de Fsica Teorica, UNESP,Rua Pamplona 145, BR-01405-, S~ao Paulo SP / BrazilAbstractWe derive the current algebra of supersymmetric principal chiralmodels with a Wess-Zumino term. At the critical point one obtainstwo commuting super Kac-Moody algebra as expected, but in generalthere are intertwining fields connecting both right and left sectors,analogously to the bosonic case. Moreover, in the present supersym-metric extension we have a quadratic algebra, rather than an affineLie algebra, due to the mixing between bosonic and fermionic fieldssince the purely fermionic sector displays a Lie algebra as well.Universidade de S~ao PauloIFUSP/P-1026January 1993∗Work supported in part by CNPq1Since the discovery of higher conservation laws for integrable models,algebraic methods have been frequently advocated in order to display thestructure of the dynamics of eld theory. Higher conserved charges implynon trivial constraints for correlation functions, and one often nds sucha strong algebraic machinery that a calculable S-matrix turns out to be aconsequence. In the case of conformally invariant eld theory, the Virasoro(and Kac-Moody) constraints x all correlation functions.Current algebra for integrable non linear sigma models were, up to veryrecently [1] largely unknown. Nevertheless, the Yang-Baxter relations arethe best tools to explore non perturbative properties of integrable models[2]. The starting point of the formulation is very generally the considerationof the Poisson brackets between the spatial part of the Lax pair, leadingto a Lie-Poisson algebra containing an antisymmetric numerical r-matrixwhich obeys the Yang-Baxter relation. The Yang-Baxter relation leads toan (almost) unique S-matrix of the theory [3, 4, 5, 15] . On the other hand,an algebraic strategy has also been eectively and successfully used in thecase of 2-D conformally invariant theories, in order to compute correlationfunctions [6], as well as in the case of two dimensional gravity in the lightcone gauge [7].Recently, the non linear -model with a Wess-Zumino term has beenstudied in this context [8]. The current, which corresponds to a piece ofthe Lax pair of the model [9], is shown to fulll a new ane algebra. Suchalgebras have played an important role in several cases in eld theory (seerefs. [10] and [11]). The hope is that after quantization we could be able toaddress to the problem of computing exact Greens functions, by means ofhigher conserved currents.In the present paper we consider the supersymmetric Wess-Zumino-Wittenmodel [12, 13], which is dened by the actionSsusy(g;  ) =12∫d2x tr@µg−1@µg + (1)+n4∫ 10dr∫d2x tr g−1r @rg g−1r @µgr g−1r @νgr ++142tr∫d2xf  i@= − 14(1− (n24)2)(   )2 + in24g−1@µg  γ5γµ gobtained from the supereld G(x; ) = g(x) +  (x) + 12F (x), satisfyingthe constraint Gy(x; )G(x; ) = 1, after integrating over the Grassman vari-2able  (see ref [12] for details, but noticing change in conventions see belowand ref [15]).The second term in the right hand side of equation (1) is the so calledWess-Zumino term and only depends linearly on the time derivative of theeld g. Hence it proves useful to rewrite it in terms of A(g), introducedin order to permit the canonical quantization of the theory [14, 15], beendened in such a way that∫ 10dr∫d2x"µν tr g−1r @rgr g−1r @µgr g−1r @νgr ∫d2x tr A(g)@0g (2)and@Aij@glk− @Akl@gji= @1g−1il g−1kj − g−1il @1g−1kj : (3)We shall use the following notation:  = nλ24pi, "01 = 1, γ0 =(0 −ii 0),γ1 =(0 ii 0), and γ5 = γ0γ1 . When  = 1 we have a super conformallyinvariant theory. Canonical quantization is straightforward, on account ofref [8] and [14, 15]); we have the following canonically conjugated momenta:gij =@L@(@0gij)=12@0g−1ji −n4Aji(g) +i42 jkγ1 kmg−1mi (4)andψij =@L@(@0 ij)=i42 jiγ0 =i42 yji ; (5)were the rst term in the right side at the equation (4), is the momentumcanonically conjugated to the eld g in the principal chiral model only:gij =12@0g−1ji (6)with the following Poisson algebra:fgij(x); gkl(y)g = 0 ; (7)fgij(x); gkl(y)g = 0 ;fgij(x); gkl(y)g = ikjl(x1 − y1) ;3f ij(x);  kl(y)g+ = 0 ;fψij(x);ψkl(y)g+ = 0 ;f ij(x);ψkl(y)g+ = ikjl(x1 − y1) :However, as we have already mentioned this model has been obtained viaa supereld formulation where we have imposed the constraintGy(x; )G(x; ) = 1 (8)which leads, for the fermion component, to the relation yij = −g−1ik  klg−1lj : (9)In the phase space we have the constraintΩil = ψli +i42g−1im mkg−1kl (10)which must be implemented using the Dirac method [16]. The basic elementof the method is the dirac matrixQ = fΩij(x);Ωkl(y)g = i22gyilgykj(x1 − y1) (11)satisfying∫dzfΩij(x);Ωmn(z)gfΩmn(z);Ωkl(y)g−1 = ikjl(x1 − y1) : (12)From Q we can obtain the Dirac brackets (notice that as we havefgij(x);Ωkl(y)g = 0 the Dirac bracket of gij with any other eld is thePoisson bracket itself). Writing the equation (4) for the eld gij instead theeld gij , they read:fgij(x); gkl(y)gD = 0 ; (13)4fgij(x);  kl(y)gD = 0 ;fgij(x);ψkl(y)gD = 0 ;fgij(x);gkl(y)gD = ikjl(x1 − y1) ;fgij(x);gkl(y)gD = f12(1−2)[gyjkψml mngyni − gyliψmj mngynk] +2Fji,lkg(x1−y1) ;fgij(x);  kl(y)gD = −12fik[gyjm ml−gyjmγ5 ml] + jl[ kmgymi+ kmγ5gymi]g(x1−y1) ;fgij(x);ψkl(y)gD =12fgyliψkj + ψilgyjk − [ψilγ5gyjk − gyliγ5ψkj]g(x1 − y1) ;f ij(x);  kl(y)gD = −22igilgkj(x1 − y1) ;f ij(x);ψkl(y)gD =12ikjl(x1 − y1) ;fψij(x);ψkl(y)gD =−i82gyligyjk(x1 − y1) ;whereFij,kl = (gyil)0(x)gykj(x)− gyil(x)(gykj)0(x) :We are now ready to study the relevant algebraic properties of the super-symetric WZW theory. In analogy with the bosonic case [8], we consider theconserved Noether currents, which in the supersymetric case are given by:JR ,a(x) = −12trf(1 )[g( _(gy) (gy)0) + i4 (1 γ5) y]tag(x) (14)5andJL ,a(x) = −12trf(1 )[(− _(gy) (gy)0)g + i4 y(1 γ5) ]tag(x) : (15)Here we introduced the notation with the indices of a basis ta of theLie algebra G were the elds g are dened, with the structure constants f cabdened as:[ta; tb] = fcab tcand any eld JRor L,a is dened as:JRor L,a = (J; tRor La ) = −tr(JRor L ta) :In addition, it is important in the case of theories containing fermions tointroduce the fermionic currents:iR,a(x) = −i44tr[ (1 γ5) yta](x) (16)andiL,a(x) = −i44tr[ y(1 γ5) ta](x) : (17)Since we know, from the case of the study of supersymetric non localcharges, that the above purely fermionic objects do also show up indepen-dently [15, 17, 18] of the currents [10]. Moreover, there are also intertwiningoperators already in the bosonic case [8, 1], which are described by the eldsjab(x) =12tr[gytagtb](x) : (18)In the case of supersymetric theories we have also the fermionic partnerjψR,ab(x) =12tr[gy iR tagtb](x) (19)andjψL,ab(x) =12tr[gyta giL tb](x) : (20)We shall see that after coupling bosons with fermions, an innite numberof elds will be present, as a consequence of a quadratic algebra, which is6absent in the purely bosonic, as well as in the purely fermionic models. WeintroduceKR ,abc(x) = −12tr[gytaiRtbgtc](x) ; (21)KL,abc(x) = −12tr[gytagtbiLtc](x) ; (22)Y R ,abcd(x) =12tr[gytagtbgyiRtcgtd ; ](x) (23)Y L ,abcd(x) =12tr[gytagtbgytcgiLtd](x) : (24)We are now in position to write down the full algebra. The purely rightsector is very simple. For the purely left sector one substitutes (R! L) and! −. We have:fJR ,a(x); JR ,b(y)g = −12(1 )f cabf(3 )JR ,c − (1 )JR ,c+ (25)+22(1)[(32−2)iR,c−(52+2)iR,c]g(x1−y1)2(1)2ab0(x1−y1) ;fJR ,a(x); JR ,b(y)g = −12f cabf(1− )2JR− ,c + (1 + )2JR+ ,c+ (26)−(1− 2)222[iR−,c + iR+,c]g(x1 − y1) ;fiR,a(x); JR ,b(y)g = −12(1 )2f cabiR,c(x1 − y1) ; (27)fiR,a(x); JR ,b(y)g =12(1 )2f cabiR,c(x1 − y1) ; (28)fiR,a(x); iR,b(y)g =12f cabiR,c(x1 − y1) ; (29)fiR,a(x); iR,b(y)g = 0 ; (30)7whereab = (ta; tb) = − 12tr(tatb)Notice that the intertwining operators did not appear at all up to thispoint, The mixed sector is more involved. We have:fJR ,a(x); JL ,b(y)g = (1−2)[(1)jab(x)+(1)jab(y)]0(x1−y1)+ (31)−(1−2)24[(1−2)(jψR−ab+jψR+ab−jψL−ab−jψL+ab)+8(jψRab−jψLab)](x1−y1)fJR ,a(x); JL ,b(y)g = (1− 2)[(1 )j0ab(x)+ (32)−24(1 )(3 )[jψR−ab + jψR+ab− jψL−ab − jψL+ab]](x1 − y1)fiR,a(x); JL ,b(y)g =12(1− 2)[jψLab − jψRab](x1 − y1) ; (33)fiR,a(x); JL ,b(y)g =12(1 )(3 )[jψLab − jψRab](x1 − y1) ; (34)fiL,a(x); JR ,b(y)g = −12(1− 2)[jψLba − jψRba](x1 − y1) ; (35)fiL,a(x); JR ,b(y)g = −12(1 )(3 )[jψLba − jψRba](x1 − y1) ; (36)fiR,a(x); iL,b(y)g =12[jψLab − jψRab](x1 − y1) ; (37)fiR,a(x); iL,b(y)g = 0 : (38)We see now in equations (31, 32) the explicit appearance of the bosonicintertwiners as well the fermionic one in equations (31, 32, 33, 34, 35, 36,37). Notice also the simplicity of the purely fermionic components brackets.In the critical case, ! +1 we see that a whole sector decouples completely,8the same being true for  ! −1. We have in that case a super Kac-Moodyalgebra, as expected, and the model is completely soluble. In fact, the con-served charges can be used in order to provide a complete solution of theGreen functions, the only missing ingredient with respect to our computa-tion being the super Virasoro generators. We nally write down the part ofthe algebra involving the intertwiners. We have:fJL ,a(x); jbc(y)g = −(1 )f dacjbd(x1 − y1) ; (39)fJR ,a(x); jbc(y)g = −(1 )fdabjdc(x1 − y1) ; (40)fjab(x); jcd(y)g = 0 ; (41)fjab(x); iL,c(y)g = 0 ; (42)fjab(x); iR,c(y)g = 0 ; (43)fjab(x); jψR,cd(y)g = 0 ; (44)fjψR,ab(x); JL ,c(y)g = −12(1)[2jψRac,b−(1)jψRab,c−(1)KL,abc](x1−y1) ;(45)fjψR,ab(x); JL ,c(y)g = −12(1)[2jψRac,b+(1)jψRab,c−(3)KL,abc](x1−y1) ;(46)fjψR,ab(x); JR ,c(y)g =12(1)[2jψRac,b−(1)jψRca,b−(1)KR cab](x1−y1) ;(47)fjψR,ab(x); JR ,c(y)g =12(1)[2jψRac,b+(1)jψRca,b−(3)KR cab](x1−y1) ;(48)9fjψR,ab(x); iL,c(y)g = −12[jψRab,c −KLabc](x1 − y1) ; (49)fjψR,ab(x); iL,c(y)g = 0 ; (50)fjψR,ab(x); iR,c(y)g =12[jψRca,b −KR cab](x1 − y1) ; (51)fjψR,ab(x); iR,c(y)g = 0 ; (52)fjψR,ab(x); jψL,cd(y)g =12[KLcab,d −KRcab,d](x1 − y1) ; (53)fjψR,ab(x); jψL,cd(y)g = 0 ; (54)fjψR,ab(x); jψR,cd(y)g =12[Y R abcd − Y R cdab](x1 − y1) ; (55)fjψR,ab(x); jψR,cd(y)g = 0 (56)where(jψRab)ij =12(gyiRtagtb)ij(jψRab)ij =12(gyiRtatbg)ij(KRabc)ij = −12(gytaiRtbgtc)ij(KRabc)ij = −12(gytatbiRtcg)ij(jψLab)ij =12(gytagiLtb)ij10(jψLab)ij =12(gytatbgiL)ij(KLabc)ij = −12(gytagtbiLtc)ij(KLabc)ij = −12(gytatbgtciL)ijIt is clear now that the algebra is quadratic. Therefore, a new structurearises in the case of symmetric theories involving the WZW term. Indeed,integrability of the bosonic model [9, 19] is not clear in the supersymetriccase [12]. However it is certainly true that for  = 0 the theory must beintegrable. Indeed, there is a considerable simplication, but the algebraobtained is still quadratic.Acknowledgement: the authors wish to thank prof. Michael Forger forseveral discussions and an initial colaboration, as well as a partial suport ofFAPESP.References[1] M. Forger, J. Laartz and U. Scha¨per, Commun. Math Phys.146(1992)397.[2] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theoryof Solitons, Spring Verlag, Berlin, 1987.[3] A. B. Zamolodchikov, Al. B. Zamolodchikov, Annals of Phys.120(1979)253.A. B. Zamolodchikov, Al. B. Zamolodchikov, Nucl. Phys.B133(1978)525.[4] H. J. de Vega, H. Eichenherr and J. M. Maillet, Commun. Math. Phys.,92 (1984)507.11[5] O. Babelon and C. M. Viallet, Phys. Lett. 237B(1990)411.J. M. Maillet, Phys. Lett. 162B(1985)137; 167B(1986)401; Nucl. Phys.B269(1986)54.[6] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. PhysB241(1984)333.[7] A.M. Polyakov, Mod. Phys. Lett. A2(1987)893.V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov, Mod. Phys.Lett. A3(1988)819.E. Abdalla, M. C. B. Abdalla and A. Zadra, Mod. Phys. Lett.A4(1989)849.[8] E. Abdalla and M. Forger, Mod. Phys. Lett. A7(1992)2437-2447.[9] M. C. B. Abdalla, Phys. Lett. B152(1985)215.[10] H.Halpern, Commun.Math.Phys. 25(1972)253.[11] Vertex Operators in Mathematics and Physics, Proceedings edited by J.Lepowsky, S. Mandelstam and I. M. Singer, Springer Ferlag, 1983.[12] E. Abdalla and M. C. B. Abdalla, Phys. Lett. 152B(1985)59.[13] P. di Vecchia, V. G. Knizhnik, J. L. Petersen and P. Rosso, Nucl. Phys.B253(1985)701.[14] E. Abdalla and K. Rothe, Phys. Rev. D36(1987)3170.[15] Non Perturbative Methods in Two Dimensional Quantum Field Theory,E. Abdalla, M. C. B. Abdalla and K. 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Current algebra of super WZNW models

Acervo Digital da Universidade Estadual Paulista

Journal of Physics A General Physics

We derive the current algebra of supersymmetric principal chiral models with a Wess-Zumino term. At the critical point one obtains two commuting super-affine Lie algebras as expected, but, in general, them are intertwining fields connecting both right and left sectors, analogously to the bosonic case. Moreover, in the present supersymmetric extension we have a quadratic algebra, rather than an affine Lie algebra, due to the mixing between bosonic and fermionic fields; the purely fermionic sector displays an affine Lie algebra as well.UNIV SAO PAULO,INST FIS,BR-01498970 SAO PAULO,BRAZILUNESP,INST FIS TEOR,BR-01405 SAO PAULO,BRAZILUNESP,INST FIS TEOR,BR-01405 SAO PAULO,BRAZI

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Current Algebra of Super WZNW Models

Current Algebra of Super WZNW Models

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