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Toshiya Kawai, Taku Uchino and Sung-Kil Yang

arXiv:hep-th/9112073v1 25 Dec 1991KEK-TH-311KEK preprint 91-190December 1991Higher-Rank SupersymmetryandTopological Field Theory*Toshiya Kawai, Taku Uchino and Sung-Kil YangNational Laboratory for High Energy Physics (KEK)Tsukuba, Ibaraki 305, JapanAbstractThe N = 2 minimal superconformal model can be twisted yielding an example oftopological conformal field theory. In this article we investigate a Lie theoretic extensionof this process.

*To appear in the proceedings of YITP Workshop on “Developments in Strings and FieldTheories” held at Kyoto, Japan, on September 9-12, 1991

1. IntroductionThe twisting of the N = 2 minimal superconformal model [1] gives rise to a funda-mental class of topological field theories [2].

Topological conformal field theories (TCFT)realized as the topological version of certain N = 2 models exhibit remarkable propertiessuch as the existence of the flat coordinate system and the prepotential [3]. It is thus quiteinteresting to ask to what extent one can generalize the relationship between TCFT andN = 2 superconformal theories.

In this contribution we wish to point out that there isa natural Lie theoretic extension of the N = 2 superconformal algebra which makes uspossible to construct a wider class of TCFT.This paper is organized as follows. In sect.2 we introduce the higher-rank generaliza-tion of the N = 2 minimal model.

In sect.3 we show that the original idea of the chiralityin N = 2 theory can be extended in our model. In sect.4, applying the twisting procedure,we discuss topological properties.

In sect.5 we sketch how to evaluate multiple integra-tions in order to demonstrate the BRST exactness of the twisted stress tensor. In sect.6we present some concluding remarks.2.

Higher-Rank SupersymmetryThe close resemblance between the N = 2 minimal model and the SU(2) WZNWmodel has been well recognized. If one considers an arbitrary simple Lie algebra, motivatedby the SU(2) paradigm, the ‘higher-rank’ analog of the N = 2 minimal model appears[4,5].

This is our model based on which we shall construct TCFT. The model is describedby a familiar system (g−parafermion)k × (boson)n with the special values of Coulomb gasparametersα+ =sk + gkg ,α+α−= −1k ,α0 = α+ + α−.

(2.1)Here g is a simple Lie algebra of rank n, the level k is a positive integer, and g is thedual Coxeter number of g. (We follow the convention of [5].) The N = 2 minimal modelcorresponds to the case g = A1.The existence of ‘exotic supersymmetry’ in this system has been expected throughthe analysis of the branching relations associated with the coset model gk ⊕U(1)n/U(1)n[4].

The generators of this symmetry are given by1Gα(z) := ψα(z)eiα+α·ϕ(z),α ∈∆(2.2)1Throughout this paper we suppress normal orderings.1

where ∆is the set of roots of g, ψα(z) are the generating parafermions, and ϕ(z) =(ϕ1(z), . .

., ϕn(z)) are free bosons with their two-point functions ⟨ϕa(z)ϕb(w)⟩= −δablog(z −w). The stress tensor consists of the parafermion piece Tpara(z) and the free bosonpieceT(z) = Tpara(z) −12(∂ϕ(z))2=Xn∈Zz−n−2Ln .

(2.3)The central charge c is given by c =kdk+g with d = dim g. Since ψα(z) has a conformalweight 1−α22k [6,7], the conformal weight of Gα(z) is 1+ α22g . There are also U(1)n currentswhich we normalize byJa(z) =iα+∂ϕa(z) =Xn∈Zz−n−1Jan,a = 1, .

. ., n .

(2.4)The chiral algebra generated by Gα(z), T(z) and Ja(z) is closed and associative by con-struction. We will refer to this symmetry as the higher-rank supersymmetry.

The structureof this algebra is, in general, complicated due to nonlocality as well as nonlinearity, but,as we will see below, the twisting procedure can be studied without much difficulty sincewe do not need the whole algebra2.We now describe the reason why we think of this chiral algebra as the higher-rankextension of the N = 2 algebra. The crucial ingredient in the N = 2 model is the spectralflow [8] and the chiral ring [9].

These properties are in fact realized in our present model.Let us first consider the spectral flow.The existence of U(1)n currents implies an n-parameter spectral flow. We denote the flow operator by U¯η with ¯η being an n-dimensionalflow vector.

One finds an inner automorphism of the algebraU¯η Ln U−1¯η= Ln + ¯η · Jn + |¯η|22α2+δn,0 ,U¯η Jan U−1¯η= Jan + ¯ηaα2+δn,0 ,U¯η Gαr U−1¯η= Gαr+¯η·α ,(2.5)2 See, however, Appendix where we find a connection to the Zamolodchikov-Fateev spin- 4/3current algebra, when g = A2.2

whereGα(z) =Xrz−r−1+ α22gGαr . (2.6)Specializing ¯η =ηg ρ with ρ being half the sum of the positive roots of g yields a one-parameter flowU¯η Ln U−1¯η= Ln + ηg ρ · Jn + c24η2 δn,0 ,U¯η Jan U−1¯η= Jan + cdηρa δn,0 ,U¯η Gαr U−1¯η= Gαr+ ρ·αg η ,(2.7)where the strange formula 12ρ2 = gd has been used.

As in N = 2 theories this flow isimportant when we consider the analogs of the chiral ring and the Ramond ground statesof the present system in the next section.3. Chiral Primary FieldsIn N = 2 superconformal field theory chiral primary fields play a distinguished role[9].

In particular, they become BRST invariant physical observables after twisting [1–3]. To find analogs of chiral primary fields, for which we use the same terminology in thefollowing, let us define the NS sector as the set of fields ANS(z) which are local with respectto the fractional-spin currents Gα(z).

ThenGα(z)ANS(0) =Xr∈Z−α22gz−r−1+ α22g(Gαr ANS) (0) . (3.1)The NS primary states are annihilated by all the positive modes, i.e.

Ln, Jan and Gαn−α22gwith n > 0. Chiral fields Φ are those in the NS sector obeying3I0dz Gαi(z)Φ(0) = 0 ,i = 1, .

. ., n ,(3.2)for each simple root αi of g. Chiral primary fields satisfy both (3.2) and the primarycondition.

These are explicitly obtained asΦΛ(z) = φΛΛ(z)e−iα−Λ·ϕ(z),Λ ∈P k+(3.3)3We note that there are as many similar sets of conditions as the order of the Weyl groupof g corresponding to different choices of the simple root system. Hence for each such choice onecan repeat the whole story in the sequel.3

where φΛΛ is a parafermionic primary field with a conformal weight12(k + g)(Λ + 2ρ) · Λ −Λ22k ,(3.4)and P k+ is the set of dominant weights appearing in the level k WZNW model. Althoughone could consider ΦΛ for any values of the Coulomb gas parameters as a deformation ofthe highest weight primary field of the WZNW model, one special feature arising from(2.1) is that the conformal weight and U(1)n charges of ΦΛ are linearly related:L0|ΦΛ⟩= 1g ρ · J0|ΦΛ⟩,(3.5)withJa0 |ΦΛ⟩= gΛak + g |ΦΛ⟩.

(3.6)Then one can adopt the conventional argument (found e.g. in [9]) and deduce that thereare no short distance singularities between two chiral primary fields:ΦΛ(z)ΦΛ′(w) ∼ΦΛ+Λ′(w) ,(3.7)where we should understand that ΦΛ+Λ′(w) = 0 if Λ + Λ′ ̸∈P k+.

Thus our chiral primaryfields define a finite nilpotent ring.Let us apply the flow (2.7) with η = 1 to the NS sector. The chiral primary states|ΦΛ⟩are then mapped onto the R ground states which have a conformal weight h andU(1)n charges q given byh = c24,q =gk + g (Λ + ρ) −ρ .

(3.8)These states are indeed responsible for the non-vanishing contribution to the ‘generalizedindex’ [4].4. Topological VersionStarting with the higher-rank supersymmetric model introduced in the previous sec-tions let us now construct TCFT.

The first step is to twist the model through the redefi-nition of the stress tensorT(z) →ˆT(z) = T(z) + 1g ρ · J(z) . (4.1)4

The new stress tensor ˆT(z) satisfies the Virasoro algebra with a vanishing central charge.Among the supercurrents Gα(z) with α ∈∆we see that the currents Gαj(z) with simpleroots αj acquire conformal weights 1 with respect to the new stress tensor. Hence BRSToperators in the topological version will be constructed from Gαj(z).

The BRST structureof the theory is governed by a directed graph essentially determined by the affine Weylgroup. One associates a BRST charge to each arrow, the explicit form of which is givenby a multiple integral of Gαj and depends on the location of the arrow in the graph.In TCFT the stress tensor is expressed as a BRST commutatorˆT(z) = {Q, G(z)} .

(4.2)For G(z) we take the current G−θ(z) whose conformal weight is 2 with respect to ˆT(z)and the explicit expression of the BRST commutator was proposed in [10]:ˆT(z) = (const.) Ts(z)Ts(z) :=Z· · ·ZΓnYi=1aiYj=1dz(j)iGαi(z(j)i)G−θ(z),(4.3)where θ = Pni=1 aiαi is the highest root of g. The contour Γ of the integrals must bechosen appropriately [11].

The evaluation of Ts(z) will be studied in the next section.The physical states in our topological theory are realized as the non-vanishing BRSTcohomology. Though the analysis of the BRST complex is quite complicated, the indexcalculation based on the branching relation shows that the chiral primary fields ΦΛ(z),Λ ∈P k+ turn out to be the basic physical observables.

In fact the BRST invariance ofΦΛ(z) immediately follows from the chirality condition (3.2) since the action of BRSTcharges on ΦΛ(z) is given by0 = {Q(j), ΦΛ(z)} =Izdw Gαj(w)ΦΛ(z),j = 1, . .

., n.(4.4)All these states have zero conformal weights with respect to ˆT(z) by virtue of (3.5). Thusthe physical states are labeled by the U(1)n charge (3.6).Let us take a look at correlation functions.

Since the stress tensor takes the BRSTexact form (4.3), correlators of the basic physical operators are independent of their worldsheet positions, and hence there is no notion of distance in the theory. This is a familiarphenomenon observed in twisted N = 2 theories [1–3].

We also note that upon twisting wehave coupled the system to a ‘background charge at infinity’. The selection rule of U(1)ncharges arising from this features our topological field theory.

We intend to further discussthe properties of correlation functions, including perturbed behaviors, in a subsequentpaper [12].5

5. Multiple IntegralsIn this section we briefly comment on the validity of (4.3).

For simplicity, considerthe case g = An and k = 1. By a standard calculation we obtainTs(z) =ZΓnYj=1dzjn−1Yj=1(zj −zj+1)(z1 −z)(zn −z)−α2+× expiα+nXj=1αj · (ϕ(zj) −ϕ(z)),(5.1)where α2+ = n+2n+1 and we have used the fact θ = Pnj=1 αj.

If we make a change of variableszj →z+Qjl=1 ul, j = 1, . .

., n and integrate over u1, we find, after standard manipulationsof contours, thatTs(z) ∝nYj=2Iduj u(jn+1 −1)−1j(1 −uj)−1n+1 −1×"−12 α1 · ∂ϕ(z) +nXj=2(u2 · · ·uj)αj · ∂ϕ(z)!2+ i2(n + 1)α0 α1 · ∂2ϕ(z) +nXj=2(u2 · · · uj)2 αj · ∂2ϕ(z)!#. (5.2)By repeatedly applying the recursion property of the beta function,B(a, b) =Idt ta−1(1 −t)b−1,B(a + 1, b) =aa + bB(a, b) ,(5.3)it is easy to see thatTs(z) ∝−12nXi,j=1(ωi · ωj)(αi · ∂ϕ)(αj · ∂ϕ)(z) + iα0ρ · ∂2ϕ(z) ,(5.4)where ω1, .

. ., ωn are the fundamental weights of An andωi · ωj = min(i, j) −ijn + 1 ,2ρ =nXi=1i(n + 1 −i)αi .

(5.5)6

Finally using the formulanXi,j=1(ωi · ωj)(αi)a(αj)b = δab ,(5.6)which follows from Pna=1(αi)a(ωj)a = δij and Pni=1(αi)a(ωi)b = δab we arrive atTs(z) ∝−12(∂ϕ(z))2 + iα0ρ · ∂2ϕ(z) . (5.7)For other Lie algebras we encounter more complicated integrals.

For instance, in thecase g = Dn we have to deal with lower moments with respect to a two-variable Selbergdensity to derive the desired result. Further details of the calculation will be reportedelsewhere [12].6.

Concluding RemarksWe have seen that our higher-rank supersymmetric model is a fairly natural exten-sion of the N = 2 minimal model and its topological version possesses all the propertiescharacteristic to TCFT.Let us point out some interesting issues which remain to be properly understood. Onemay notice that the modified stress tensor ˆT(z) is that describing the gk ⊕g0/gk ≃gk/gkcoset theory.

According to recent works [13,14] it has been established that correlationfunctions in topological gk/gk model yield the fusion algebra of the WZNW model gk.Furthermore, this result can be understood in terms of the deformed chiral ring of topo-logical N = 2 theory [15,16]. Thus it will be significant to clarify whether the presentchiral primary ring, after certain deformation, has any relevance to the gk fusion algebra[12].It should also be mentioned that the higher-rank supersymmetric models are real-ized in the critical limit of solvable vertex-type lattice models [17].

Quite recently, Saleur[18] has observed an interesting structure of solvable lattice models whose critical behav-iors are described by the N = 2 minimal model. It seems important to seek a similarcorrespondence in our higher-rank setting.Finally we note that the parallelism with N = 2 supersymmetry is yet to be completed.What is crucially missing in higher-rank supersymmetry is its possible connection to certaingeometry if there is any.

It would be very exciting if one can find a geometric interpretationwhich could be an analog of the deep relation between Ricci-flat K¨ahler manifolds andN = 2 superconformal field theories [19].7

AcknowledgementsWe thank E. Date, T. Eguchi and Y. Yamada for helpful discussions.Appendix.The Zamolodchikov-Fateev spin-4/3 algebra [20] is generated by two spin 4/3 currentsψ(z) and ψ∗(z) which satisfy the operator product expansionsψ(z)ψ(w) ∼λ(z −w)4/3ψ∗(w) + 12(z −w)∂ψ∗(w) + · · ·,ψ∗(z)ψ∗(w) ∼λ(z −w)4/3ψ(w) + 12(z −w)∂ψ(w) + · · ·,ψ(z)ψ∗(w) ∼1(z −w)8/3I + 83c(z −w)2 T(w) + · · ·. (A.1)The associativity gives a constraint [20]9λ2c = 4(8 −c) ,(A.2)which can be parametrized asc = 21 −12v(v + 4),λ2 = 13(v + 2)2(v −2)(v + 6) .

(A.3)The minimal unitary series corresponds to v = 2, 3, . .

. and is realized by the GKO coset(A1)4 ⊕(A1)v−2/(A1)v+2.We present here another realization of this algebra.

Consider Gα(z), α ∈∆, definedin (2.2) for g = A2. Note that all these currents have spin 4/3.

The level-k A2-parafermioncurrents ψα(z) generate the algebra [6,7]ψα(z)ψβ(w) ∼Kα,β(z −w)1+α·β/kψα+β(w) + 12(z −w)∂ψα+β(w) + · · ·,ψα(z)ψ−α(w) ∼1(z −w)2−α2/kI + k + 33k (z −w)2 Tpara(w) + · · ·,(A.4)8

where Kα,β = Kβ,α, Kα1,α2 = Kα1,−θ = Kα2,−θ = 1/√k and otherwise zero. It is thenstraightforward to check that4ψ(z) =1√3Xα=α1,α2,−θGα(z) ,ψ∗(z) =1√3Xα=α1,α2,−θG−α(z) ,T(z) = Tpara(z) −12(∂ϕ(z))2 ,(A.5)satisfy (A.1) withc =8kk + 3,λ =2√3k.

(A.6)The associativity is clear by construction. This construction is rather reminiscent of makingan N = 1 superconformal generator out of two N = 2 superconformal generators.4For k = 1, this realization coincides with that considered in [21].9

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