Superconformal Field Theory with Boundary Fermionic Model
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a rXiv:h ep-th/98653v16J u n1998Superconformal Field Theory with Boundary:Fermionic Model S.A.Apikyan a,b †,D.A.Sahakyan b ‡a )Yerevan Physics Institute Alikhanian 2,Yerevan,375036Armenia.b )Yerevan State University,Al.Manoogian 1,Yerevan,375049Armenia.February 7,2008Abstract Fermionic model of Superconformal field theory with boundary is considered.There were written the ”boundary”Ward Identity for this theory and also constructed boundary states for fermionic and spin models.For this model were derived ”bootstrap”equations for boundary structure constants.1IntroductionSuperconformal invariant theories with boundary undoubtedly have a great interest themselves as well as in their connection with open superstring the-ories [1].There are other reasons for studying Superconformal Field Theory on manifolds with boundary.They relate to the connection with statistical mechanics [2].In introduction we will give only a brief review of necessary facts from superconformal field theory.Background on superconformal field theory can be found in refs.[3],[4].In complex coordinates ˆz =(z,θ)two-dimensional superconformal trans-formations are:δθ=ε+12¯θ¯u ¯z δz =u +θε;δ¯z =¯u +¯θ¯ε(1)and these supertransformations are generated by W (z,θ)=12¯S (¯z )+¯θ¯T (¯z ).Currents T (z )and S (z )are generators of holo-morphic conformal and supersymmetric transformations respectively.Coef-ficients of the Laurent expansion of T (z )and S (z )form the superconformal algebra:[L m ,L n ]=(m −n )L m +n +c2(r 2−12−r )S m +r (2)where r runs over half-integer values (Neveu-Schwarz–NS )or integer values (Ramond–R ).One can construct irreducible representation of this algebra in each sector from primary states.To each highest weight state of NS -algebra:L n |h =0n >0S r |h =0r >0L 0|h =h |h(3)corresponds primary superfield V h (z,θ)=φ(z )+θψ(z ):|h =φ(0)|0 ;S −1/2|h =ψ(0)|0 ;(4)In the Ramond sector superconformal current has zero mode,which form two dimensional Clifford Algebra with the Fermion Number Operator Γ=(−)F ,commuting with the L 0.As a result,we have double degeneration of the2ground state[3].In this space we can choose the following ortogonal basis |h+ =R h+(0)|0 ,|h− =R h−(0)|0 (where R±-Ramond spinfields):|h− =S0|h+ (5) where|h+ and|h− are eigenvectors of operator(−)F with eigenvalues+1 and−1respectively having the same conformal weight ing commutation relations(2)we can obtain:S0|h− =S20|h+ =(L0−c16)|h+ (6)Thus,if one normalizes|h+ as, h+|h+ =1,then from(6)it follows,that h−|h− =h−c16,it can be chosen basis|h′± such, that S0|h′± = 16|h′∓ and which is ortonormal.In further we will use the basis(5).Let us note,that if h=c2πi C dzdθv(ˆz)W(ˆz)Φ(ˆz)=1where v(ˆz)=u(z)+2θε(z)is infinitesimal parameter.A superfieldΦ(ˆz)is a primary superconformalfield if it obeysδvΦ=v∂zΦ+1∂θ+θ∂(z−z i)d(z−z i)2] Φ1,...,Φn(9)whereˆ∆=∆Φ+1∂θandS(z)Φ1,...,Φn =[−2 ∆iθi z−z i(∂∂z i)] Φ1,...,Φn (10)In the same way one canfind anti–holomorphic Ward Identity.It is easy to see,that the requirement of preservation of the geometry gives strong limitations on parameters of superconformal transformationεand u. One can see that the coefficients of expansion u andεmust be real.Therefore holomorphical and anti–holomorphical transformations are not independent, and in relation(7)we should considerδv+δ¯v.For let us make analytical continuation of T and S on to lower half plane.T(z)=¯T(z)S(z)=¯S(z)for Imz<0(11)It means that now we have only one algebra(2)in opposite to”bulk”theory, there were two,holomorphical and anti–holomorphical algebras,which is consistent with the fact,that in theory with boundary we have only one set of coefficient in expansion of parametersε,ing(7)and(11)we get: (δv+δ¯v)Φ(ˆz,¯z)=12πi C+∪C−(εS+uT)Φdz(12) where contour C+∪C−contains all points(z1¯z1,...,z n¯z n).>From(12)fol-lows,that,in contrast to”bulk”theory where T(z)and S(z)acts only on (z1,...,z n),in theory with boundary the action of T(z)and S(z)is extended4to(z1,¯z1,...,z n,¯z n)and therefore in the relations of the type(9),(10)the doubling of terms on the right hand sides takes place due to terms with z′i=¯z i.3Boundary StatesFurther we will consider boundary state problem in superconformalfield theory with boundary.We will deal with theories defined on the upper half plane or strip,which one can also interpretate as a world sheet of an open superstring.Mapping of the upper half plane on to strip is given by the conformal trasformation z=eτ+iσ,where(τ,σ)are coordinates on strip.In general superconformalfield theory with boundary,the unique require-ment on boundary condition is the superconformal invariance: T(z=e t)=¯T(¯z=e t)T(z=e t+iπ)=¯T(¯z=e t−iπ)S(z=e t)=¯S(¯z=e t)S(z=e t+iπ)=¯S(¯z=e t−iπ)NS−sector S(z=e t)=¯S(¯z=e t)S(z=e t+iπ)=−¯S(¯z=e t−iπ)R−sector(13) It is well known that there is an isomorphism between the space of conformal invariant boundary conditions and the space of boundary states.Indeed, if one compactifies t by mod2πImτ(τis purely imaginary)(in this way we obtain the theory defined on cylinder with radius Imτ),then partition function with boundary conditionsαandβat the ends of cylinder can be written as follows:Z NS αβ=T re2πiτH openαβ(14)>From the other side,the same partition function can be considered as a propagation of closed superstring onσdirection between states α|,|β , Zαβ= α|e−πH cyl|β = α|e−πwhereζ=e−i(t+iσ).One can rewrite conditions(16),intheform:(L n−¯L−n)|B =0(S r+i¯S−r)|B =0(17) where r∈Z or r∈Z+1|h,N (20) where U NS is an anti-unitary operator,satisfying the following condition:L n U NS=U NS L nU NS S r=−iS r U NS(−)FU NS|h =|h(21) From(17)one can derive analytical expression for UU NS|h,N =1−ii|⊗ j|,where i|⊗ j|(S r+iS−r|h =0 andi|⊗ j|(S r+iS−r)|n ⊗Ui|U i|S−ri|US−r(−)Fj|¯S r|U+i =0(23)6By the same way we can show,that thefirst equation in(17)is also satisfied. It is more interesting to study Ramond sector.At the beginning let us consider the case h=c|h±,N (24)where U R is anti-unitary operator,satisfying to condition:L n U R=U R L n(25)U R S r=−iS r U R(−)FSince the ground state is now non–trivial,we have freedom in a definition of the action U R on this space.And we have the only one restriction on U R:(U R S0+iS0U R(−)F)|h± =0(26) In representation,where|h+ = 10 and|h− = 0 16S0and(−)F can be represented asS0= 16σx;(−)F=σz(27) whereσx andσz are Pauli ing(26)and representation(27),we get:U R= a−ic c−ia (28) where a and c satisfy anti-unitary condition:aa∗+cc∗=1and ac∗+a∗c=0. Thus,we get,that in opposite to Neveu-Schwarz sector in Ramond sector U R is not determined uniqely.It is interesting to note,that if h=cwhereχNS(q)=q−c/24T rq L0is the character of the superconformal algebras in NS-sector.By non-negative integer n hαβdenoted the number of times thatrepresentation h occurs in the spectrum of H openαβ.The character formulas forthe NS and R algebra have been derived by Goddard,Kent and Olive[8]and by Kac and Wakimoto[9]and under the modular transformationτ→−1/τthe character for the fermionic model transform linearly[10],χNS h(q)= S h′hχNS h′(˜q)(30) which leads toZ NSαβ= n hαβS h′hχNS h′(˜q)(31) where˜q=e−2πi/τ.In order to have complete set of boundary states de-fined by equation(20),we have to consider diagonal bulk theory.Following to Kastor[10]there are different superconformal theories corresponding to different modular invariant combination of charactersZ NS,R= nm,kl F nm,kl N nm,klχnm(q)¯χkl(¯q)(32)here the factor F is equal to2for the nonsupersymmetric R highest weight states,which one twofold degenerated,and is equal to1otherwise.N nm,kl is the number of highest weight states(h nm,¯h kl)in the theory which one obeys to following sum rule for NS,N nm,kl sinπnn′p+2sinπkk′p+2=1and comparing with the(31)we geth′S h h′n h′αβ= α|h h|β (36) This equation for fermionic model is the same which one found Cardy[6]for conformal theory.As result of this equation boundary states|˜h can be read|˜h = h′S h′hthen we can write short distance expansion forφ(z,¯z)near boundary asφ(z,¯z)= i(z−¯z)∆Bφi−∆φC BφφB i[φB i(x)](40)Ψ(z,¯z)= r(z−¯z)∆BΨr−∆ΨC BΨΨB r[ΨB r(x)](41) here[φB(x)],[ΨB(x)]–are conformal class ofφB,ΨB boundary vertex opera-tors and C BφφB ,C BΨΨB–are boundary structure constants of theory.Now weare interested to obtain these boundary structure constants.First of all let’s note that for identity boundary operator corresponding structure constant is equal to constant factor of one point boundary correlation function.One point boundary correlation(with boundary condition labelled by B)of NS superfield with corresponding to superconformal invariance and boundary Ward identity can be writtenΦ(ˆz,¯z) B=A BΦ(z−¯z)∆; F(z,¯z) B=∆A BΦ0|B(45)and using the superconformal physical boundary states(37)wefindA hφ=(S00)1/2(Sφ0)1/2(46)To determine the boundary structure constants C BφφB ,C BΨΨBwe need somedynamical principle.Associativity of the boundary operator algebra imposes10global constraints on correlation function as usual.For this,let’s consider2-point functions,φi(z1,¯z1)φj(z2,¯z2) B; Ψr(z1,¯z1)Ψσ(z2,¯z2) B(47) in two channels.We can evaluate these correlation functions using bulk OPE taking z1→z2,¯z1→¯z2and can alternatively be evaluated using boundaryOPE by taking z1→¯z1,z2→¯z2.Associativity of the operator algebra implies that correlation function of these two channels should give the sameresult(crossing symmetry),k C BφiφB k C BφjφB k F k ij(1−η)= m C ijm A Bφm F m ij(η)(48)ρC BΨrΨBρC BΨσΨBρFρrσ(1−η)= m C rσm A Bφm F m rσ(η)(49)hereη=|z1−z2|2/|z1−¯z2|2is cross-ratios,F k ij(η),C ijm are conformalblocks and bulk structure constants respectively..The conformal blocks are solutions of differential equations.According to different basis of differential equations the solutions are expressed by each other linearly[11],F k ij(η)= αk,pl ij,m F m pl(1−η)(50) Inserting to the equation(48-49)we getC B φiφBk C BφjφBk= m C ijm A Bφmαk,ij ij,m(51)C BΨrΨBρC BΨσΨBρ= m C rσm A Bφmαρ,rσrσ,m(52)In second”bootstrap”equation we see that right hand side is nonzero due to A Bφ.So,all boundary structure constants are expressed via well known bulk quantities.Finally,we note that an examination for”spin model”will be given elsewhere.This work was partially supported by the INTAS foundation under grant 96-0482.We thank the ICTP(Trieste)for its hospitality where this work was completed.11References[1]J.Polchinski,hep-th/9611050[2]I.Affleck,A.Ludwig,Nucl.Phys.B360(1991)641.[3]D.Friedan,Z.Qiu,S.Shenker.Phys.Lett.B151(1985)37[4]M.Bershadsky,V.Knizhnik and M.Teitelman,Phys.Lett.151B(1985)31.[5]N.Ishibashi,University of Tokyo preprint UT-530(1988)[6]J.Cardy.Nucl.Phys.B324(1989)581.[7]J.Cardy.D.Lewellen,Phys.Lett.B259,(1991)274.[8]P.Goddard,A.Kent and D.Olive,Comm.Math.Phys.103(1986)105.[9]V.Kac,M.Wakimoto,in Proc.Symp.on Conformal groups and struc-tures,Clausthal,Lecture Notes in Physics(Springer,1985).[10]A.Cappelli,Phys.Lett.B185(1987)82, D.Kastor,Nucl.Phys.B280[FS18](1987)304,J.Cohn and D.Friedan Nucl.Phys.B296(1988)779.[11]Y.Kitazawa,N.Ishibashi, A.Kato,K.Kobayashi,Y.Matsuo,S.Odake,University of Tokyo,preprint UT-522(1987).12。
