Using the fusion principle of Bauer et al. we give explicit expressions for
some null vectors in the highest weight representations of the \bc algebra in
two different forms. These null vectors are the generalization of the Virasoro
ones described by Benoit and Saint-Aubin and analogues of the $W_3$ ones
constructed by Bowcock and Watts. We find connection between quantum Toda
models and the fusion method.Comment: 8 pages, LaTeX, ITP Budapest 50

Bauer

Benoit

Blumenhagen

Bouwknegt

Bowcock

Forgács

Hamada

Kausch

Kent

Z. Bajnok

Physics Letters B

English

arXiv

BAJNOK, Z

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-th/9309014v1  2 Sep 1993ISSN 0133-462X ITP Budapest Report No.503Null vectors of the WBC2 algebraZ. Bajnok∗Institute for Theoretical PhysicsRoland Eötvös UniversityH-1088 Budapest, Puskin u. 5-7, HungaryMarch 20, 2018AbstractUsing the fusion principle of Bauer et al. we give explicit expressions for some nullvectors in the highest weight representations of theWBC2 algebra in two different forms.These null vectors are the generalization of the Virasoro ones described by Benoit andSaint-Aubin and analogues of the W3 ones constructed by Bowcock and Watts. We findconnection between quantum Toda models and the fusion method.∗E-mail: bajnok@ludens.elte.hu1Conformal field theories describe the statistical mechanical systems at their critical point.Their infinite dimensional symmetry algebra, the Virasoro algebra or its extension, playsan important role in analyzing them. In most cases these models correspond to degeneraterepresentations of these algebras. Such representations are crucial since on one hand theypossess a closed operator algebra with only a finite number of primary fields, while on theother hand they contain null vectors. These null vectors lead to differential equations for thecorrelation functions, and in this sense, they can be regarded as equations of motion governingthe dynamics of the theory. Although this application needs explicit expressions for the nullvectors such formulas were not given in general until the last few years.Recently there has been great interest to describe these null vectors [1–6]. First Benoit andSaint Aubin constructed very compact expressions for a subclass of the Virasoro null vectorsthen Bauer et al. rewrote this formula using the fusion method. They showed how to producenull vectors in the generic case by fusing explicitly known null vectors with primary fields.Later Bowcock and Watts extended this method to W algebras. They found that the fusionsof the null vectors and of their descendants should be investigated simultaneously.In this paper we use their method to produce null vectors in the WBC2 algebra. Explicitexpressions for the simplest null vectors are given. In the fusion method we deal with onlythose primary fields which belong to completely degenerate representations and have twoindependent null vectors at level one and two. It turns out that there is a special null vectoramong the descendants ones. This null vector is nothing but the quantum equation of motionof the appropriate Toda model and in order to produce null vectors one has to study the fusionof this descendant null vector. We start with the representation theory of the WBC2 algebra.Then we describe how to handle the action of a spin j chiral current in the fusion procedure.Finally, using this result we study two different cases namely the case of the B2 and C2 Todamodels, respectively.The WBC2 algebra is one of the simplest W algebras. It is an extension of the Virasoroalgebra generated by the Virasoro field, L(z), and a spin 4 chiral current, W (z) [7]. The highestweight (hw.) representation of the algebra contains a hw. vector, |hw〉, which satisfies:Ln|hw〉 = δn,0h|hw〉 ; Wn|hw〉 = δn,0w|hw〉 ; n ≥ 0 (1)In order to analyze the determinant formula [8] we use the following reparametrisation of theW weights :h(x, y) = Q4(x2 + 2xy + 2y2)− 14(5Q+ 10Q− 14)w(x, y) = B̄(4Q(Q− 3)(27Q− 32)y3(y + 2x)−Q(3Q− 2)(16Q− 27)x3(x+ 4y)+ (Q2−2)Q(14Q(x2 + 2xy + 2y2 − 6Qx2y2) +Q− 6 + 2Q))(2)whereB̄−1 =32Q2√(4Q− 5)(5Q− 8)(7Q− 6)(3Q− 7)(3Q− 2)(Q− 3)(75Q2 − 226Q+ 150)Here we used Q instead of the central charge (C). They are related to each other as2C = 86− 60/Q− 30Q. Now the determinant formula dictates that ifx = a− c2Qor y = b− d1Q(3)then the representation is degenerate and there is a null vector level ac or bd. If both conditionshold at the same time then the representation is completely degenerate and contains twoindependent null vectors. We restrict ourselves to only such representations. The hw. vectorin this case is denoted by |ab; cd〉 and the two independent null vectors by Oa;c|ab; cd〉 andOb;d|ab; cd〉, respectively.To produce null vectors we follow the method of Bauer et al.. They considered the fusionof a primary field, φh0w0(z), belonging to a degenerate representation, with another primaryfield, |hw〉 = φhw(0). Setting the null vector equal to zero and demanding its consistencyimplies recursive equations for the descendant fields appearing in the OPE. Solving theseequations iteratively explicit expression can be obtained for the null states in the correspondingrepresentations. Everything one needs to know is the action of generators after the fusion.Using contour deformation technique for an arbitrary spin j generator we have:{W−nφh0w0(z)}|hw〉 ={(−1)n+jzn[(n− 1n− j)w0 −j−1∑k=1zkfk,n,j(1)W−k]φh0w0(z)}|hw〉+{(−1)n+jznj−1∑k=1zkfk,n,j(q)W−k +∞∑l=0(n+ l − jl)(pz)lW−n−l}φh0w0(z)|hw〉 (4)where we have used fk,n,j(q) =∑ki=1(n−i−1n−j)(j−i−1k−i)qk−i. The fusion point, the argument ofthe resulting field, is qz and p = 1 − q. For j = 2 the formula yields the result of [2] if weuse L−1φ(z) = ∂zφ(z). Unfortunately in the WBC2 , j = 4, case we have unknown modes:W−1 ;W−2 ;W−3. However following the idea of Bowcock and Watts [3] we are able to solvethis problem. In the WA2 case they could eliminate this kind of modes considering not onlythe two independent null vectors but also their descendants simultaneously.For simplicity we deal with the following case: completely degenerate representationshaving null vectors at level one and two. First we analyze the u = |21; 11〉 field. The h, w ofthis field are ∆, ω, respectively:∆ =14(5Q− 6)ω = −Q8∆√√√√√(4Q− 5)( 8Q− 5)( 2Q− 3)( 6Q− 7)(226− 75Q− 150Q)(Q− 3)(3Q− 7)(5)Analyzing the C2 Toda model in WZNW framework we obtained [9] that the only relevantoperator, u, has a null state at level four. Studying the covariance of this quantum equationof motion we concluded that u must belong to a completely degenerate representation and soit has the following null states. At level one the null vector is:O−1u = W−1u+ β1L−1u = 0 (6)3whereβ1 = −2(4Q− 5)(8/Q− 5)(7Q− 6)(3Q− 2)NandN =ω∆1(4Q− 5)(8/Q− 5)(7Q− 6)(3Q− 2)At level two this completely degenerate representation has an independent null vector. How-ever it is more convenient to use the following null state:O−2u = W−2u+ β2L−2u+ β11L2−1u = 0 (7)β2 = 2(23− 10Q)(7Q− 6)(3Q− 2)N ; β11 = 4(13Q− 25)(7Q− 6)(3Q− 2)NWe use the following descendant null vector of the two independent null vectors at level three:O−3u = W−3u+ β3L−3u+ β12L−1L−2u+ β111L3−1u = 0 (8)whereβ111 =16Q(75Q2 − 226Q+ 150)Nβ3 = −6(5Q− 22)(4Q− 5)(Q− 2)N ; β12 = −24(34Q2 − 113Q+ 82)NFinally u satisfies the quantum Toda equation of motion, which is again a descendant nullvector:O−4u = W−4u+ β4L−4u+ β22L2−2u+ β13L−1L−3u+ β112L2−1L−2u+ β1111L4−1u = 0 (9)whereβ4 = −2Q(3Q− 2)(10Q3 − 73Q2 + 312Q− 360)Nβ13 = −8(27Q3 − 263Q2 + 548Q− 300)N ; β22 = 4(3Q− 2)(16Q− 27)Nβ1111 = Qβ111 ; β1112 = −β111The first three null vectors carry enough information to eliminate the unknownW modes whilewe can use the Toda equation to fuse with another primary field. This equation indicates whichrepresentations can occur after the fusion. We did this analysis in [9] and found that the nonvanishing matrix elements of u are:|21; 11〉 × |ab; cd〉 →|a+ 1 b ; c d〉|a− 1 b ; c d〉|a+ 1 b− 1; c d〉|a− 1 b+ 1; c d〉(10)For producing null vectors we consider the following fusion:|21; 11〉 × |0r; ss′〉 → |1r; ss′〉 (11)4Using the method of [2] we expand the two OPE asu(z)|0r; ss′〉 =∑n≥0zn−yfn ; {O−4u(z)}|0r; ss′〉 =∑n≥0zn−y−4f̃n (12)where y = ∆+ h0r;ss′ − h1r;ss′ . Now using (4) and choosing the fusion point at p = 1 ; q = 0we get the following recursive equation:f̃n = αnfn +∑j≥1{W−j + γn,jL−j + β22Λ−j}(13)whereαn = N(75Q2 − 226Q+ 150)16Q2n(n− s)(n+ rQ− s− s′)(n + rQ− 2s− s′)γn,j = β112(n− y − 2)(n− y − 3) + β13(n− y − 3)(j − 2) + β4(j − 2)(j − 3)/2+3(β2 + β12(n− y − 2) + β3(j − 2)) (14)and Λ−j =j−1∑i≥1L−iL−j+i. The vanishing of αn at n = 0 and n = s explicitly shows that thefusion is allowed and that there is a null state at level s, respectively. Solving these equationsiteratively we find for the null state at level s:O1;s|1r; ss′〉 =∑ni:∑pi=1ni=s∏pi=1(W−ni + γNi,niL−ni + β22Λ−ni)∏p−1i=1 αNi|1r; ss′〉 (15)where Ni =∑ij=1 nj . Here and from now on we use the ordering from left to the right.Alternative expressions can be obtained by choosing the fusion point in a different way. Theresult is very elegant if we use q = 1. In this case :O1;s|1r; ss′〉 =∑ni=1,2,3,4:∑pi=1ni=s∏pi=1 ΓNini∏p−1i=1 αNi|1r; ss′〉 (16)whereΓn1 = O−1 +((n− y − 1)(6β11 + 9β111(n− y − 2) + 4β1111(n− y − 2)(n− y − 3))+ 3β21(h+ y + 2− n) + 2β112(n− y − 3)(h+ y − n)− β13(2h+ y + 1− n))L−1Γn2 = 3O−2 +((n− y − 2)(3β12 + β112(n− y − 3)) + 2β22(h + y + 2− n))L−2+(9β111(n− y − 2) + 6β1111(n− y − 2)(n− y − 3) + β112(h + y + 2− n))L2−1Γn3 = 3O−3 + (n− y − 3)(β13L−3 + 2β112L−1L−2 + 4β1111L3−1)Γn4 = O−4 (17)5We remark that the computations for the |11; 12〉 state proceed in the same way as in the|21; 11〉 case; the final result can be obtained form eq. (15)-(17) by the Q → 2Qsubstitution.The null vector produced in this manner is in the |ss′; r1〉 representation at level s.There are two other cases when the representation contains two independent null vectorsat level one and two. The first can be described by ũ = |12; 11〉 while the other one can beobtained from this one by the Q → 2Qsubstitution. Both correspond to the Toda field of theB2 Toda model. The h, w weights of ũ are:∆̃ = 2Q− 2ω̃ =12∆̃√√√√√(4Q− 5)( 2Q− 3)( 6Q− 7)(Q− 3)(226− 75Q− 150Q)( 8Q− 5)(3Q− 7)(18)As before, we analyze the leading null vectors of ũ and their descendants simultaneously. Theũ field satisfies equations entirely analogous to eq.(6)-(8) the only difference is that we haveto substitute every β with the following β̃: At level one we have:β̃1 = −2(4Q− 5)(Q− 3)(7Q− 6)(3Q− 2)(Q+ 1)ÑwhereÑ =ω̃∆̃1(4Q− 5)(Q− 3)(7Q− 6)(3Q− 2)(Q+ 1)At level two we use again instead of the independent null states their convenient combinationwhose coefficients are:β̃2 = 4Q(Q−1)(16Q−27)(Q+1)(3Q−2)Ñ ; β̃11 = −(59Q2−199Q+50)(Q+1)(3Q−2)ÑThe coefficients of the null states at level three are: β̃12 = 6Q(41Q2 − 113Q+ 68)(Q+ 1)Ñβ̃3 = 6Q(7Q− 6)(Q− 3)2(Q+ 1)Ñ ; β̃111 =Q(Q+ 1)8β111However, contrary to the previous case, this representation has no null states of the form of(9) at level four. Analyzing the B2 Toda model we learn that ũ has to satisfy the followingequation, which shows the existence of a descendant null state at level five:O−5ũ = β̃14L−1L−4u−2(Q+1)β̃L−1W−4u+ β̃W−5u+ β̃122L−1L2−2u+ β̃113L2−1L−3u+β̃1112L3−1L−2u+ β̃11111L5−1u+ β̃23L−2L−3u+ β̃5L−5u = 0 (19)whereβ̃5 = 4(Q+ 1)(Q− 3)(3Q3 − 2Q2 + 48Q− 60)Ñ ; β̃1112 = −Q4β111β̃14 = −6(Q− 3)(3Q3 − 2Q2 + 48Q− 60)Ñ ; β̃113 = −6(Q− 3)(7Q2 − 54Q+ 50)Ñβ̃122 = 2Q(Q−3)(27Q−32)Ñ ; β̃23 = −2Q(Q−3)(Q+1)(27Q−32)Ñ ; β̃11111 =18β1116Eq.(19) is the quantum equation of motion of the B2 Toda model. From this equation weconclude that the possible fusions of ũ are the following:|12; 11〉 × |a b; c d〉 →|a b+ 1; c d〉|a− 2 b+ 1; c d〉|a b; c d〉|a b− 1; c d〉|a+ 2 b− 1; c d〉(20)Considering the|12; 11〉 × |r0; ss′〉 → |r1; ss′〉 (21)fusion and using the same technique as before: i.e. expanding the two OPE asu(z)|r0; ss′〉 =∑n≥0zn−yfn ; {O−5u(z)}|r0; ss′〉 =∑n≥0zn−y−5f̃n (22)with y = ∆̃ + hr0;ss′ − hr1;ss′ and using (4) we obtainf̃n = α̃nfn+∑j≥1{Ñ(j−4−2(n− y − 4)Q+ 1)W−j + γ̃n,jL−j + β̃122(n− y−4)Λ−j + β̃23Λ′−j}(23)whereα̃n = Ñ2Q(75Q2−226Q+150)n(n−s′)(n+r + 12Q−s−s′)(n+rQ−2s−s′)(n+rQ−2s−2s′)andγ̃n,j = (n− y − 4)(β̃114(j−22)+ (n− y − 3)(β̃1112(n− y − 4) + β̃113(j − 2)))−β̃5(j−23)+ 2β̃122(h + y + 1− n) + β̃23((h+ 3 + y − n)(j − 2) + n− y − 2h)−2(n−y−4)Q+1(3β̃2 + β̃12(x− 2) + 3β̃3(j − 2))− (8β̃2 + β̃12(x− 2) + 6β̃3(j − 2)); (24)Λ−j is the same as before and Λ′−j =∑j−1i≥1 (i− 2)L−iL−j+i. One can prove that in the |r1; ss′〉representation there is a null state at level s′. An explicit expression for this null vector canbe obtained by solving the recursion:O1;s′ |r1; ss′〉 ==∑ni:∑pi=1ni=s∏pi=1{Ñ(j − 4− 2(n−y−4)Q+1)W−j + γ̃n,jL−j + β̃122(n− y − 4)Λ−j + β̃23Λ′−j}∏p−1i=1 α̃Ni|r1; ss′〉where Ni =∑ij=1 nj . An alternative expression can be obtained by choosing the fusion pointat q = 1, however this straightforward formula looks complicated so we omit it here.Summarizing we find that the fusion rules of the Toda fields –u, ũ – are the same asthe fusion rules of the (10), (01) representations of the corresponding Lie algebra. However7the fusions of the various Toda fields are determined by the quantum equations of motionof the corresponding Toda models and these quantum equations of motion take the form ofparticular descendant null vectors. So we conclude that for producing null vectors via thefusion method one has to consider these descendant null vectors. Furthermore we conjecturehow to generate null vectors in the most general case, i.e. in the highest weight representationof the WBC2 algebra corresponding to the |rr′, ss′〉 hw. vector. One has to consider theWZNW model associated not to the defining but to a higher dimensional representation,labelled by (r, 0), of the appropriate group and use the Drinfeld-Sokolov reduction [10] toobtain the corresponding Toda model. Fusing its |r1, 11〉 type special descendant null vectorwith a |0r′, ss′〉 primary field a null vector of the above mentioned representation is hoped tobe obtained.I would like to thank L. Palla and P. Bántay for the many useful discussions and commentsand L. P. for the careful reading of the manuscript.References[1] L. Benoit and Y. Saint Aubin Phys. Lett. 215B (1988) 517[2] M. Bauer, P. di Francesco,C. Itzykson and J.-B. Zuber Nucl. Phys. B362 (1991) 515 Phys. Lett.260B (1991) 323[3] P. Bowcock and G.M.T. Watts Phys. Lett. 297B (1992) 282[4] G.M.T. Watts Null Vectors of the Superconformal Algebra: The Ramond Sector CambridgeUniversity Preprint DAMPT 93-14, hep-th 9306034[5] L. Benoit and Y. Saint Aubin Int. J. Mod. Phys. A13 (1992) 3023, Letters in Math. Phys. 23(1992) 117[6] Adrian Kent Phys. Lett. 273B (1991) 56, Phys. Lett. 278B (1992) 443[7] Bouwknegt Phys. Lett. 207B (1988) 295 K.-J. Hamada and M. Takao Phys. Lett. 209B (1988)247 R. Blumenhagen, M. Flohr, A. Kliem, W. Nahm, A. Recknagel and R. Varnhagen Nucl.Phys. B361 (1991) 255[8] H.G.Kausch and G.M.T.Watts Int. J. Mod. Phys. A7 (1992) 4175, Duality in Quantum Todatheory and W -algebras Cambridge University Preprint DAMPT 92-07[9] Z. Bajnok C2 Toda Theory in Reduced WZNW Framework ITP Budapest Report No.501 hep-th9304019 submitted to Int.J.Mod.Phys. A[10] P. Forgács, A. Wipf, J. Balog, L. Fehér and L. O’Raifeartaigh: Phys. Lett. B227 (1989) 214;B244 (1990) 435; Ann. Phys. (N. Y.) 203 (1990) 76.8

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