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Institute for Theoretical Physics

arXiv:hep-th/9111061v1 28 Nov 1991ITP.SB-91-61Nov.’91c = 5/2 Free Fermion Model of WB2 AlgebraChanghyun Ahn∗Institute for Theoretical PhysicsState University of New York at Stony BrookStony Brook, NY 11794-3840We investigate the explicit construction of the WB2 algebra, which is closedand associative for all values of the central charge c, using the Jacobi identityand show the agreement with the results studied previously. Then we illustratea realization of c =52 free fermion model, which is m →∞limit of unitaryminimal series, c(WB2) = 52(1 −12(m+3)(m+4)) based on the cosets ( ˆB2 ⊕ˆB2, ˆB2)at level (1, m).

We confirm by explicit computations that the bosonic currents inthe WB2 algebra are indeed given by the Casimir operators of ˆB2 .∗email: AHN@MAX.PHYSICS.SUNYSB.EDU1

1IntroductionRecently much progress has been made in our understanding the structure oftwo-dimensional rational conformal field theories (RCFTs) for which the spec-trum contains a finite number of irreducible representations of chiral algebra,the operator product algebra (OPA) of all the holomorphic fields in the theory.Cardy [1] was able to show that the sum over irreducible representations of theconformal algebra must be infinite for the central charge c ≥1. Therefore inorder to study RCFTs with c ≥1 , extended conformal algebras which consistof the Virasoro algebra and additional higher spin currents, play an importantrole.

Large classes of extended Virasoro algebras are known: affine Kac-Moodyalgebra [2], superconformal algebra (WB1 ) [3], and a class of W algebras. Someexamples of W algebras have been established: W3 algebra [4], Wn algebra [5],and supersymmetric W algebra [6].Fateev and Lukyanov [7] have considered an infinite dimensional, associativequantum algebra WBn, corresponding to the Lie algebras Bn, consisting of (n+1)fields of spin 4, 6, · · ·, 2n, and a fermion field of spin (n + 1/2) in addition to thespin 2 energy momentum tensor.

The addition of fermion field was necessary forthe quantization of the Hamiltonian structures associated with Lie algebra whosesimple roots have a different length. They conjectured that these (n + 1) fieldsform a closed OPA.

These constructions produce representations of the algebrasover a range of values of c. For n = 1, the WB1 algebra coincides with the N = 1super Virasoro algebra.On the other hand, Watts [8] has shown that W structures may be constructedin coset models of the form ( ˆgn ⊕ˆgn, ˆgn) at level (1, m) with gn one of the finiteLie algebras ABDE and for m sufficiently large.An explicit expression for afermion field which commutes with the diagonal subalgebra was given in terms ofcurrents of ˆBn. The discrete series of c [7] coincide with values obtained from cosetmodels based on ˆBn algebra.

It is expected, but has not been established that hisconstruction are related to those in the free field construction [7]. Recently, heproved that the symmetry of Lie superalgebras of B(0, n) Toda theory was given2

by the classical Poisson bracket analogue of the WBn algebra [9].It wasn’t proven that algebra WB2 are associative for all values of c untilFigueroa-O’Farrill et al. showed that the existence of WB2 explicitly using theperturbative conformal bootstrap [10].

They have been able to show the equiv-alence with the findings conjectured by Fateev and Lukyanov [7] using Coulombgas realization. We make an explicit construction of the Casimir of WB2 from afermion proposed by Watts in [8].This paper is organized as follows.

In section 2, we look at the explicit con-struction of WB2 algebra using the associativity of Laurent expansion operators,which we investigate by considering a graded Jacobi identity for Laurent expan-sion modes. In section 3, we would like to understand how WB2 currents emergefrom fields of ˆB2 algebra and prove a realization of c = 5/2 representation ofsection 2.

Finally section 4 contains a few concluding remarks.2WB2 AlgebraAs our starting point, we follow the analysis in [7].The presence of W2current of dimension 2 and W4 current of dimension 4 can be understood fromthe operator product expansion (OPE) of d current of dimension 5/2 with itself:30(2c + 25)d(z)d(w)=1(z −w)525c +1(z −w)32W2(w) +1(z −w)2∂W2(w)+1(z −w)[60(2c + 25)W4(w) + 12∂2W2(w)] + · · · . (2.1)Notice that W4 is not a primary field under the W2(z) = T(z) energy momentumtensor which generates the Virasoro algebra with a central charge c. The aboveOPE can be rewritten as the more familiar form in the following way :U(z)U(w) =1(z −w)525c +1(z −w)32T(w) +1(z −w)2∂T(w)+1(z −w)[ 310∂2T(w) +27(5c + 22)Λ(w) + C45252V (w)] + · · · ,(2.2)3

wheres30(2c + 25)d(z) = U(z), Λ(z) = T 2(z) −310∂2T(z),(2.3)andW4(z) = −(2c + 25)2120(5c + 22)∂2T(z) + 9(2c + 25)20(5c + 22)T 2(z) + (2c + 25)60C45252V (z). (2.4)It is convenient to use V (z), which is a Virasoro primary field of dimension4 , rather than W4(z) for the future developments of WB2 algebra.C45252 isa coupling constant and to be determined later.Extended conformal algebraby a single primary field of 5/2 conformal dimension was found in [4].Hereenlarging the algebra by additional spin 4 field leads to the appearance of V (z)term of eq.

(2.2). We would expect that C45252 should vanish for c = −13/14.On the other hand, it has been proven that one extra spin 4 current algebrawas determined by the closure of Jacobi identity [12, 11].

We are dealing withtwo-dimensional RCFTs, in which there exist conserved currents U(z) of spin 5/2and V (z) of spin 4, and would expect that U(z) field dependence should appearin the OPE V (z)V (w). Recall that U2(z) can be written as the combinations of∂3T(z), ∂Λ(z) and ∂V (z) from eq.

(2.2). We can see that the nontrivial U(z)field dependence has the form of ∂UU(z) ( or U∂U(z) ).

The next step is to workout the OPE V (z)V (w). Through an analysis of [11] with the symmetry underthe interchange of z and w, it leads to the following results:V (z)V (w) =1(z −w)8c4 + A[1(z −w)62T(w) +1(z −w)5∂T(w)−1(z −w)3112∂3T(w) +1(z −w)1120∂5T(w)] + B[1(z −w)42∂2T(w)+1(z −w)3∂3T(w) −1(z −w)112∂5T(w)] + C[1(z −w)22∂4T(w)+1(z −w)∂5T(w)] + D[1(z −w)42Λ(w) +1(z −w)3∂Λ(w)−1(z −w)112∂3Λ(w)] + E[1(z −w)22∂2Λ(w) +1(z −w)∂3Λ(w)]4

+F[1(z −w)22Ξ(w) +1(z −w)∂Ξ(w)] + G[1(z −w)22∆(w)+1(z −w)∂∆(w)] + H[1(z −w)42V (w) +1(z −w)3∂V (w)−1(z −w)112∂3V (w)] + I[1(z −w)22∂2V (w) +1(z −w)∂3V (w)]+J[1(z −w)22Ω(w) +1(z −w)∂Ω(w)]+K[1(z −w)22Γ(w) +1(z −w)∂Γ(w)] + · · · ,(2.5)withΞ(z) = ∂T∂T(z) −370∂4T(z) −317(5c + 22)∂2Λ(z),∆(z) = TΛ(z) −310∂2Λ(z), Ω(z) = TV (z),Γ(z) = ∂UU(z) −518∂T∂T(z). (2.6)Note that the operator Γ(z) defined as above has no central term whichmakes the calculation easier.To guarantee that the complete OPA be asso-ciative, K terms are inevitable.

In order to fix the coefficients in the OPE weconsider the condition of associativity. We will use the Jacobi identities for Lau-rent expansion modes.

The commutation relation [Vm, Vn] can be obtained bycontour integral, as usual. For the commutators of newly defined operators of(2.6) with Virasoro operators Lm, see the reference [11].

Also we need to know[Lm, Γn] =HC0(dw/2πi)wn+5 HCw(dz/2πi)zm+1T(z)Γ(w) :[Lm, Γn] = m(m2 −1)5!180[2020(m −2)(m −3) −100(−10c + 27)(m −2)×(m + n + 2) + 2020(m + n + 2)(m + n + 3)]Lm+n+m3!18(5c + 22)[3159(m2 −1) −3(m + 1)(−100c + 775)(m + n + 4)]Λm+n+ m12C45252[13(m2 −1) −15(m + 1)(m + n + 4)]Vm+n + (5m −n)Γm+n. (2.7)5

After a straightforward computation of the following Jacobi identity[Lm, [Vn, Vp]] + [Vp, [Lm, Vn]] + [Vn, [Vp, Lm]] = 0,(2.8)we can have the intermediate results:A = 1, B = 320, C =1168, D =21(5c + 22)E =227(5c + 22) −10136(5c + 22)K,F = −3(19c −524)20(2c −1)(7c + 68) + 7(50c2 + 507c −968)90(2c −1)(7c + 68) KG =12(72c + 13)(2c −1)(7c + 68)(5c + 22) −(734c + 49)(2c −1)(7c + 68)(5c + 22)KH = 3(c + 24)28J + 1928C45252K, I = (5c + 64)336J −5112C45252K. (2.9)We have still now three free parameters, C45252, J, and K. As you can see, theabove results reduce to ones in [11, 12] when K goes to zero.We can checkthat the other Jacobi identity involving Lm, Un, and Up doesn’t give any furtherrestriction on these free parameters.Now we investigate the OPE V (z)U(w) in order to obtain the complete alge-bra.

Then some local operators appearing in the r.h.s. of V (z)U(w) are madeof the products T(w), U(w) and their derivatives.

The most general form of theOPE of V (z)U(w) isV (z)U(w) =a(z −w)4U(w) +b(z −w)3∂U(w)+1(z −w)2[d∂2U(w) + eTU(w)]+1(z −w)[f∂3U(w) + g∂TU(w) + h∂(TU)(w)] + · · · . (2.10)In a similar manner, the unknown structure constants are determined by thefollowing Jacobi identity :[Lm, [Vn, Vp]] + [Up, [Lm, Vn]] + [Vn, [Up, Lm]] = 0.

(2.11)6

Therefore it leads to the following results,C45252a = 15(14c + 13)4(5c + 22) , C45252b = 3(14c + 13)(5c + 22)C45252d = 5(c + 8)(14c + 13)2(5c + 22)(2c + 25), C45252e =45(14c + 13)(5c + 22)(2c + 25)C45252f = (2c −5)(2c + 25), C45252g = −162c + 2025(5c + 22)(2c + 25)C45252h =6(82c + 215)(2c + 25)(5c + 22). (2.12)In order to get C45252, J, and K completely, we should use explicit construction of[Vm, Λn], {Um, (TU)n}, and {Um, (∂TU)n} to satisfy the following Jacobi identity:[Vm, {Un, Up}] −{Up, [Vm, Un]} −{Un, [Vm, Up]} = 0.

(2.13)But we refrain from giving the explicit expressions for the above ( anti) commu-tators, which are rather complicated and not illuminating. Finally, withC45252 =vuut6(14c + 13)(5c + 22) , J =4√6(7c −115)(2c + 25)q(5c + 22)(14c + 13)K =30(5c + 22)(2c + 25)(14c + 13),(2.14)putting all together, all the coefficients can be determined in terms of only theVirasoro central charge c:a =15q(14c + 13)4q6(5c + 22), b = 3√14c + 13q6(5c + 22)d =5(c + 8)√14c + 132(2c + 25)q6(5c + 22), e =45√14c + 13(2c + 25)q6(5c + 22)f =(2c −5)√5c + 22(2c + 25)q6(14c + 13), g = −162c + 2025(2c + 25)q6(5c + 22)(14c + 13)7

h =6(82c + 215)(2c + 25)q6(5c + 22)(14c + 13),E =3696c2 + 31957c −3487042(2c + 25)(5c + 22)(14c + 13), F =2158c + 2130560(2c + 25)(14c + 13)G =54(32c −5)(2c + 25)(5c + 22)(14c + 13), H =3√6(2c2 + 83c −490)(2c + 25)q(5c + 22)(14c + 13)I =√6(10c2 −197c −2810)24(2c + 25)q(5c + 22)(14c + 13). (2.15)Note that WB2 algebra is a subalgebra of Zamolodchikov’s spin 5/2 algebra[4] for c = −13/14 because C45252 vanishes for this value of c. These results arein agreement with the findings of [10].† The one thing which we would like tostress is the fact that all the remaining Jacobi identities, [Um, [Vn, Vp]] + cycl.

=0, [Um, {Un, Up}]+cycl. = 0, and [Vm, [Vn, Vp]]+cycl.

= 0, are consistent with theabove results after a long calculation with MathematicaTM [13].3The Five Free Fermion ModelThe coset models [8] are defined in terms of the currents Eab(1)(z), and Eab(2)(z),of level 1 and m, respectively, which generate the algebra g = ˆB2 ⊕ˆB2. Thegenerator of the diagonal subalgebra g′ = ˆB2, which has level m′ = 1 + m, isgiven byE′ab(z) = Eab(1)(z) + Eab(2)(z).

(3.1)The coset Virasoro algebra is generated by the difference T(1)(z)+T(2)(z)−T ′(z),where T(1)(z) and T(2)(z) are Sugawara stress energy tensors:˜T(z) = −116Eab(1)Eab(1)(z) −14(m + 3)Eab(2)Eab(2)(z) +14(m + 4)E′abE′ab(z). (3.2)† In this way we discovered that there is a misprint of eq.

(13) of ref.[10]. The factor 5 in thedenominator should be in the numerator.8

Of course, ˜T(z) commutes with E′ab(z). The coset central charge of the unitaryminimal models for WB2 is˜c = c(WB2) = 52 + 10mm + 3 −10(m + 1)m + 4= 52(1 −12(m + 3)(m + 4))(3.3)where m = 1, 2, · · ·.In this section we focus on the limit m →∞, which gives us a model ofc = 5/2 that is invariant under the affine Lie algebra ˆB2 at level 1.

This modelcan be represented by 5 free fermions ψa of dimension 1/2, where the index atakes values in the adjoint representation of B2 and a = 1, · · · , 5. We will showhow the currents of WB2 can be constructed from these free fermion fields orbasic fields Eab of ˆB2 and consider their OPA.The defining OPE of the basic fermion fields is given by, as usual,ψa(z)ψb(w) =1(z −w)δab + · · · .

(3.4)We can define dimension 1 currents Eab(z) as composites of the free fermionsEab(z) = ψaψb(z)(3.5)which satisfy, at level 1, the usual ˆB2 OPEEab(z)Ecd(w) =1(z −w)2(δbcδad −δacδbd)+1(z −w)[δbcEad(w) + δadEbc(w) −δacEbd(w) −δbdEac(w)] + · · ·. (3.6)Watts [8] has pointed out that U(z) of dimension 5/2, which is invariant underthe horizontal subalgebra, can be expressed as follows using B2 invariant ǫabcdetensor.U(z) =1120ǫabcdeψaEbcEde(z).‡(3.7)‡Multiple composite operators are always regularized from the right to left, unless otherwisestated.

The normalization of U(z) is chosen such that ǫabcdeǫabcde = 120.9

After a tedious calculation, using the rearragement lemmas [14] , we arrive at thefollowing result for the OPE of U(z) with U(w) §,U(z)U(w)=1(z −w)5 −1(z −w)3ψa∂ψa(w) −1(z −w)212ψa∂2ψa(w)+1(z −w)[12ψa∂ψaψb∂ψb(w) −16ψa∂3ψa(w)] + · · · . (3.8)During this calculation, we used the fact that ǫabcdeǫafghi((EbcEde)(EghEij))(z) =864ψa∂ψaψb∂ψb(z) −384ψa∂3ψa(z).Comparing this with eq.

(2.2), one canreadily see thatV (z)=18√69[7ψa∂ψaψb∂ψb(z) −6∂ψa∂2ψa(z) + 23ψa∂3ψa(z)]=s23192[2823T 2(z) + 3323∂2T(z) + ψa∂3ψa(z)]=−140√69EabEcdEacEbd(z)(3.9)which is the unique ( up to a normalization ) dimension 4 primary field underthe energy momentum tensor, T(z) = −12ψa∂ψa(z) = −116EabEab(z) which is theform of the second order Casimir. For B2 algebra, the number of independentCasimirs equals the rank of B2 (=2).

Therefore we have in addition to the secondCasimir only a fourth order Casimir given by (3.9). The fact that the fourth orderCasimir operator is generated in the OPE U(z)U(w) confirms Casimir algebrasconsisting of T(z) and V (z) are not the usual spin-4 algebras [11, 12].In order to find the complete structure of WB2, one has to take the OPEU(z)V (w) :s19223 U(z)V (w) =1(z −w)4120231120ǫabcdeψaEbcEde(w)+1(z −w)324231120ǫabcde∂(ψaEbcEde)(w)§A product of two of ǫ tensors can be expressed as a determinant in which the entries areδ’s.10

+1(z −w)21120[4823154 ǫabcdeψaψbψcψd∂2ψe(w) −823ǫabcde∂2(ψaEbcEde)(w)]+1(z −w)1120[−823154 ǫabcde∂(ψaψbψcψd∂2ψe)(w)+5423{23ǫabcde∂3(ψaEbcEde)(w) + 5ǫabcde∂3ψaEbcEde(w)−53ǫabcdeψa∂3(EbcEde)(w) + 20ǫabcdeψaψbψc∂ψd∂2ψe(w)}]=1(z −w)412023 U(w) +1(z −w)32423∂U(w) +1(z −w)2[4823TU(w)−823∂2U(w)] +1(z −w)[−823∂(TU)(w) + 5423∂TU(w)] + · · ·. (3.10)Basically, it agrees with the expressions given in (2.10) and (2.15) for c = 52.

Weare ready to consider the OPE of V (z)V (w). We explicitly computed V (z)V (w),obtained from eq.

(3.9), which is given byV (z)V (w) =1(z −w)858 +1(z −w)62T(w) +1(z −w)5∂T(w)+1(z −w)42[ 320∂2T(w) + 1423Λ(w) −274√69V (w)]+1(z −w)3[ 115∂3T(w) + 1423∂Λ(w) −274√69∂V (w)]+1(z −w)22[ 1168∂4T(w) +9083278208∂2Λ(w) + 15184∆(w)+ 89288Ξ(w) −94√69∂2V (w) −132√69Ω(w) + 2332Γ(w)]+1(z −w)[ 1560∂5T(w) −5029278208∂3Λ(w) + 15184∂∆(w)+ 89288∂Ξ(w) −2716√69∂3V (w) −132√69∂Ω(w) + 2332∂Γ(w)](3.11)where Λ(z), ∆(z), Ξ(z), Ω(z) and Γ(z) which are expressed in terms of ψa(z)’s11

according toΛ(z) = 14ψa∂ψaψb∂ψb(z) + 2140∂ψa∂2ψa(z) −7120ψa∂3ψa(z)∆(z) = −18ψa∂ψaψb∂ψbψc∂ψc(z) −320ψa∂2ψaψb∂2ψb(z)−6380ψa∂ψa∂ψb∂2ψb(z) + 780ψa∂ψaψb∂3ψb(z) −161400∂2ψa∂3ψa(z)+ 491600∂ψa∂4ψa(z) +71600ψa∂5ψa(z)Ξ(z) = −31483ψa∂ψa∂ψb∂2ψb(z) −31483ψa∂ψaψb∂3ψb(z)+ 3591932ψa∂2ψaψb∂2ψb(z) + 10847245∂2ψa∂3ψa(z)−327245ψa∂5ψa(z) +867245∂ψa∂4ψa(z)Ω(z) =s23192[−746ψa∂ψaψb∂ψbψc∂ψc(z) −1546ψa∂ψa∂ψb∂2ψb(z)+1146ψa∂ψaψb∂3ψb(z) + 323∂2ψa∂3ψa(z)−792∂ψa∂4ψa(z) +3460ψa∂5ψa(z)]Γ(z) = −16ψa∂ψaψb∂ψbψc∂ψc(z) + 16ψa∂ψaψb∂3ψb(z)+ 118ψa∂2ψaψb∂2ψb(z) +11080ψa∂5ψa(z)−5108∂2ψa∂3ψa(z),(3.12)which agree with the formulas (2.6).Crucial point to arrive at this result was to reexpress ∂UU(z) appearing in1/(z −w)2 of V (z)V (w) in terms of 7 independent fields, consisting of compositesof ψa and their derivatives, and recombine with ∂2V (z), TV (z), and T(z) descen-dants. The OPE of two Virasoro primary fields can be represented as the sum12

of Virasoro conformal families, i.e., Virasoro descendants and Virasoro primaryfields [15]. Then we can identify unique Virasoro primary spin 6 field [10] withΦ(z) =1576(5ψa∂ψaψb∂ψbψc∂ψc(z) −174 ψa∂ψa∂ψb∂2ψb(z)−15ψa∂2ψaψb∂2ψb(z) + 1320ψa∂ψaψb∂3ψb(z) + 13∂2ψa∂3ψa(z)−112∂ψa∂4ψa(z) +1300ψa∂5ψa(z)).

(3.13)Of course, Φ(z) is a descendant w.r.t. the full WB2 algebra.

Eq. (3.11) and (3.12)agree with the expression for the WB2 algebra as given in eq.

(2.5) and (2.15) forc = 5/2. The results obtained so far can be summarized as follows.

In c = 5/2free fermion model, a consistent OPA can be made out of energy momentumtensor T(z) and additional currents U(z) of dimension 5/2 and V (z)of dimension4, corresponding to a fourth order Casimir of B2.4ConclusionThe remaining problem is to construct the WB2 algebra in the coset modelsbased on ( ˆB2 ⊕ˆB2, ˆB2) at level (1, m), which can be viewed as perturbations ofthe m →∞model discussed before. Then the dimension 5/2 coset field ˜U(z) wasgiven in [8] where ˜c is as in eq.

(3.3). We can do calculate the OPE ˜U(z) ˜U(w).Then the dimension 4 coset field ˜V (z) can be obtained from the singular part ofOPE ˜U(z) ˜U(w).

We would like to show explicitly that the algebras, consisting of˜T(z), ˜U(z) and ˜V (z), closes in the coset model. Then,this construction will leadto an explicit realization of the c < 5/2 unitary representations of WB2 algebra.We leave it further investigation [16].It is a pleasure to thank M. Rocek and K. Schoutens for reading the manuscriptand discussions, A. Sevrin for drawing my attention to ref.

[7]. This work wassupported in part by grant NSF-91-08054.13

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