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Novel Symmetries of Topological Conformal Field theories

arXiv:hep-th/9108008v1 20 Aug 1991TAUP-1898-91August 1991Novel Symmetries of Topological Conformal Field theoriesJ. Sonnenschein and S. Yankielowicz†School of Physics and AstronomyBeverly and Raymond SaclerDepartment of Exact SciencesRamat Aviv Tel-Aviv, 69987, IsraelABSTRACTWe show that various actions of topological conformal theories that were sug-gested recentely are particular cases of a general action.

We prove the invarianceof these models under transformations generated by nilpotent fermionic generatorsof arbitrary conformal dimension, Q(n) and G(n). The later are shown to be thenth covariant derivative with respect to “flat abelian gauge field” of the fermionicfields of those models.

We derive the bosonic counterparts W (n) and R(n) whichtogether with Q(n) and G(n) form a special N = 2 super W∞algebra. The algebraicstructure is discussed and it is shown that it generalizes the so called “topologicalalgebra”.† Work supported in part by the US-Israel Binational Science Fundation and the IsraelAccademy of Sciences

1.IntroductionTwo dimensional gravity and non-critical string theories provide an interestingand useful arena for the study of topological quantum field theories(TQFT’s)[1].In the opposite direction, the general covariant formulation provides an effectivetool to calculate “correlation functions” in the form of powerful recursion rela-tions[2−6]. Several different starting points for topological conformal field theories(TCFT’s) were proposed.

It was realized that “pure gravity”,[7,8] flat two dimen-sional gauge connection[9,10], twisted N = 2 superconformal theories[11,12], the GGconstruction[13]and topological sigma models[15,7,3] were all examples of TCFT’s.In this work we elaborate on the equivalence between the various models andsuggest a general framework to analyze all of them. We show that all these mod-els are invariant under transformations generated by infinitely many bosonic andfermionic generators of arbitrary integer conformal dimension.

The fermionic gen-erators are nilpotent. They have the structure of higher order covariant derivativewith respect to a flat gauge connection of the fermionic fields of the models.

Thissymmetry may be referred to as “N = 2 super W∞” symmetry. We discuss thealgebraic structure of those symmetries, which generalize the minimal “topologicalalgebra” of ref.

[5] and present several useful OPEs’. Infinite towers of bosonicsymmetries as well as supersymmetries were discussed in the past.

[16] Both thefermionic and bosonic symmetry generators discussed here are different thoughpossibly not unrelated to those in the literature.The paper is organized as follows In section 2 we describe the various formu-lations of conformal topological quantum field theories. We start with Einstein’saction of pure gravity, continue with the twisted N = 2 models with zero and non-zero background charge, theories of flat gauge connections, the GG construction andfinely the general BRST invariant( h, 1 −h) systems.

The equivalence of severalof these formulations is discussed in section 3. Section 4 is devoted to the symme-tries of the TCFTs’ generated by a set of infinitely many bosonic W∞as well asfermionic Q∞and G∞generators.

The transformations of the various fields are2

written down. It is shown that the nth generator in each sector can be expressedin terms of an nth covariant derivative with respect to a “flat gauge connection”.The structure of this infinite topological algebra is investigated in section 5.

Insection 6 we summarize the results and comment briefly on the possible implica-tions. Some technical details are presented in the appendices.

In Appendix A weprove the invariance under Q(n), construct the bosonic operator R(n) and exhibitthe corresponding transformation properties of the various fields. In Appendix Bthe calculation of the anomaly term in the OPE Q(n)G(m) is explained.

AppendixC presents the OPE of W (1) with the rest of the operators.2.Actions for Topological Conformal Field TheoriesThe most obvious TQFT in two dimensions is pure Einstein’s gravity ( with nocosmological constant). In refs.

[7, 8] it was realized that this action has in additionto the usual scale and reparametrization invariance a local symmetry referred to asthe “topological symmetry”. The BRST quantized action of this theory was foundout to be[7,8]:ST G =Zd2z[(b¯∂c + β ¯∂γ + cc) + π(∂¯∂ϕ −ˆR(2)) + ˜ψ∂¯∂ψ](1)where (b, c) are the spin (2,-1) reparametrization ghosts, (β, γ) are commutingghosts with the same spins, and (π, ϕ) , ( ˜ψ, ψ) are Grassmann even and odd scalars.This action was derived via two stages of gauge fixing .

In the first the gaugecondition R(2) = ˆR(2) was imposed to fix the “topological symmetry”.In thesecond stage the ordinary reparametrization was fixed ds2 = e2ϕdz ¯dz as well as aghost symmetry. A different derivation of this action was given in ref.

[14] where itwas shown to correspond to a C = −2 matter theory coupled to a Liouville mode.An alternative formulation of pure gravity was written down in terms of flatSL(2, R) gauge connections[9]. This construction was a special case of topologicalflat gauge connection (TFC)[10]associated with the group G. The action of this3

system is described in the “semi-classical limit” which is exact bySFC =Zd2zTr[(A¯∂˜B + Ψ¯∂˜Ψ + c.c) + ∂φ¯∂˜φ + ∂c¯∂˜c](2)where all the fields are in the adjoint representation of the group. A is the gaugeconnection, Ψ is a spin 1 ghost field related to the gauge fixing of the topologicalsymmetry, (˜c, c) are the usual ghost and anti-ghost of the non-Abelian symmetryand ˜φ, φ are Grassmann even “ghost for ghosts”.

A detailed discussion of thismodel can be found in refs. [9, 10].

This scenario was invoked also to describeworld-sheet supergravity using the graded Lie group OSP(2|1)[17]. In that casethe gauge connection was decomposed as follows: A = eaJa + ωJ3 + χ±J± wereJa (a = 1, 2), J3 and J± are the generators of OSP(2|1) and ea, ω and χ± whereinterpreted as the zweibein, spin connection and the gravitino respectively.

Similarconstruction for other interesting groups like SL(3, R) were worked out in ref. [12].Yet another formulation of topological theories was found by twisting the N = 2superconformal field theory[11,12].

In the case of the minimal N = 2 models theaction takes the form ofSN=2 =Zd2z∂φ¯∂˜φ + iα√gR(2)φ + (λ¯∂˜λ + c.c)(3)where φ, ˜φ are commuting scalars, λ, ˜λ are (1, 0) anti- commuting ghosts, R(2) isthe world sheet curvature of the background metric and α is a parameter of thetheory.A different description of a TQFT which links it to some group G was found byextrapolating the GH construction to the case of H = G[13]. It is well known that GHcoset models can be described in terms of a WZW model based on a group G where(an anomaly free) subgroup H is gauged[19].

The gauging amounts essentially tosetting the H-currents to zero. Hence for the case H = G only the G-zero modessurvive.

In this case the system is equivalent to three decoupled systems i.e. G-WZW model at level k, G-WZW at negative level −(k + cG) and a free (1,0)-(b,c)4

system in the adjoint representation. Upon bosonization[20](assuming G = SU(N)) one can recast the action into a sum of the following terms.

[21]One term has theform of eqn. (3) where φ and ˜φ are associated with the “hypercharge” currentsof the two SU(N)-WZW models.

(The scalars combine nicely into one complexboson). The ghost system in this term is the (1, 0)(b, c) system associated withthe hypercharge direction.

The rest of the terms are N2 −2 free(1,0)-(b, c) + (β, γ)systems each per one of the extra generators of SU(N).Clearly one can usethe bosonization formulas to recast this form into other equivalent forms. Thisstructure can be modified by introducing appropriate background charges whichdo not change the value of the total central charge ctot = 0.Older members of the family are the topological sigma models (TSM)[15,7,3,5].These models describe a special sector of the maps from two dimensional worldsheet into some target manifolds.

If the target manifold is taken to be flat thenthe expression of the corresponding action is given by:ST SM =Zd2zηµν[∂Xµ ¯∂Xν + (ψµ ¯∂˜ψν + cc)],(4)Xµ is the target space coordinate, ψµ, ˜ψν are world-sheet (1, 0) system. Just as inthe case of TFC, one can deduce this quantum action via a BRST gauge fixing ofa topological symmetry.The later models as well as the previous ones exhibit an important relationto non-critical string theories and their matrix models counterparts once they arecoupled to the topological two dimensional gravity[3,5] of eqn.

(1) . For instancewhen a TSM is coupled to TG to produce a “topological string” model,[9,5] thecorresponding action is the sum of the actions given in eqn.

(1) and (4).An interesting question is to what extent are ordinary critical and non-criticalstring theories, both bosonic and supersymmetric, a special case of TCFTs’. [5]Apartfrom a comment in the last section, we do not consider here string theories asTCFTs’.5

3.A “unified” pictureTwo questions are now in order: (i) can one “unify” the actions of the modelspresented in the last section and (ii) are there other topological models. A straight-forward observation is that for R(2) = 0 all the actions described in the previoussection are special cases of the following general action:S =Zd2zXi[Φ(hi) ¯∂˜Φ(1−hi) + Ψ(hi) ¯∂˜Ψ(1−hi) + cc](5)where Φ(hi), Ψ(hi) are commuting and anti-commuting fields of dimension (hi) andsimilarly for the dimension (1 −hi) fields ˜Φ(1−hi) and ˜Ψ(1−hi).For the termsinvolving a pair of scalars ( commuting or anti-commuting) the passage to theform of the above equation involves a simple redefinition which amounts to rewritethem is a first order form.

For instance for eqn. (3) we rewrite ∂φ¯∂˜φ = W ¯∂˜φwith W = ∂φ.

Since systems which are the same apart from their Grasmanniannature, have conformal anomaly which differ by a sign, it is obvious that the totalconformal anomaly vanishes c = Pi(ci −ci) = 0. One can reformulate (5) asan exact form under fermionic operators Q and G of dimensions zero and onerespectively.S =Zd2zXi[Q(Ψ(hi) ¯∂˜Φ(1−hi)) + cc] =Zd2zXi[G(˜Ψ(1−hi) ¯∂ˇΦ(hi−1)) + cc] (6)where Φ(hi) = ∂ˇΦ(hi−1).

The Q and G transformations of the various fields aregiven byδQΨ(hi) = ǫΦ(hi)δQ ˜Φ(1−hi) = −ǫ˜Ψ(1−hi)δG ˜Ψ(1−hi) = ǫ∂˜Φ(1−hi)δGΦ(hi) = −ǫ∂Ψ(hi)(7)The fact that the action is exact under a zero dimension fermionic symmetry hintsof the possibility to interpret the action as a BRST gauge fixed action. This inter-pretation follows the original TQFTs’ namely, that the “classical” action is zero6

and the “ quantum action” is derived by gauge fixing of a “topological symme-try”. One can take Lclassical(Ψ(hi)) = 0 which is invariant under the “topologicalsymmetry” δΨ(hi) = ǫ(hi)(z, ¯z) or Lclassical(˜Φ(1−hi)) = 0 which is invariant underthe “topological symmetry” δ˜Φ(1−hi) = ǫ(1−hi)(z, ¯z).

Replacing the parameters oftransformations with ghost fields Φ(hi) for the first formulation and ˜Ψ(1−hi) forthe second, imposing holomorphicity of the original fields as the gauge condition¯∂Ψ(hi) = 0 or ¯∂˜Φ(hi) = 0, and using the “BRST” transformations of eqn. (7) onegets the action (5) .

Notice that this prescription is different from the BRST gaugefixing that was applied for the cases of pure gravity[7], TFC[9]and TSM[7]. It is thusapparent that various different starting points for TCFT lead to the same theory.This point will be further discussed in the last section.So far we considered only the case of R(2) = 0.

For the twisted N = 2 actionit is equivalent to taking α = 0 which is the semiclassical limit of this action sincek = ( 1α2 −2) →∞. As we show in the next section, for pure imaginary α, namely,negative k, one can generalize the construction by redefining ˜φ →ˆφ = ˜φ −iαφ.4.The symmetriesBy definition, all the “physical observables” of a TQFT are invariant under anarbitrary variation of the metric of the underlying manifold.

(The notion of physicalobservables refers to correlation functions of products of operators which are scalarsand gauge invariant with respect to any local symmetry in the system. )Thisimplies that the energy momentum tensor can be expressed as an exact operatorunder a nilpotent fermionic symmetryTαβ = {Q, Gαβ}.

(8)It is straightforward to check that eqn. (8) guarantees the metric independence.

[15]Moreover, it is easy to see that in fact the TQFT actions given in the previoussection are all exact under the fermionic symmetry. This is obviously the situationfor the TG, TFC and TSM models since the quantum action by construction isBRST exact as well as for any other model following eqn.

(6) .7

A topological conformal field theory (TCFT) is characterized by the fact thatthe trace of the classical energy momentum vanishes. All the TQFT models pre-sented in the previous section share this property.

In these cases Tαβ as well asGαβ, the BRST current Qα and the ghost number current Jα can be split intotheir holomorphic and antiholomorphic parts. Hence one gets the following rela-tions[5]which reflect the BRST multiplet structureT(z) = {Q, G(z)}Q(z) = −[Q, J(z)](9)By Laurant expansion of these operators one finds using Jacobi identities the TCFTalgebra.

This algebra together with its generalization will be presented in the nextsection.Next we analyze the symmetries of the TCFT. Let us first discuss the sym-metries generated by J, T, Q,and G. To simplify the notation we choose todemonstrate all the features in the twisted N = 2 model eqn.

(3) with α = 0 orR(2) = 0. Later we explain how to generalize it to the case of non-flat world sheetand to the other models included in the general form of the action given in eqn.

(5) .The following transformations of the fields leave the action invariant.δJλ = −ǫλδJ ˜λ = ǫ˜λδJφ = −ǫδJ ˜φ = ǫδQλ = ǫ∂φδQ ˜φ = −ǫ˜λδG˜λ = −ǫ∂˜φδGφ = ǫλδTλ = (∂ǫλ + ǫ∂λ)δT ˜λ = ǫ∂˜λδT φ = ǫ∂φδT ˜φ = ǫ∂˜φ(10)The parameters of transformation ǫ are holomorphic function ǫ = ǫ(z). For theJ and Q transformations ǫ has dimension zero, for T and G dimension one andfor Q and G they are Grassmanian variables.

Obviously the action is invariantalso under similar transformations generated by the anti-holomorphic counterparts,¯J, ¯T, ¯Q, ¯G. From here on we discuss only the holomorphic transformations.

Notice8

that unlike usual BRST transformations where the parameter of transformation ǫis a global parameter, here ǫ = ǫ(z) also for the fermionic symmetries generatedby Q and G. Hence they generate an infinite dimensional algebra. Using the OPEof the basic fieldsφ(z)˜φ(ω) = −log(z −ω)λ(z)˜λ(ω) =1z −ω(11)it is straightforward to extract the currents that generate the above transforma-tions:J = −(λ˜λ + a∂φ −˜a∂˜φ)T = −(∂φ∂˜φ + λ∂˜λ + a∂2φ)Q = ˜λ∂φ + ˜a∂˜λG = −(λ∂˜φ + a∂λ)(12)Note that the terms proportional to a and ˜a are total derivatives so they do notcontribute to the corresponding charges and therefore cannot be determined fromthe classical transformations alone.

Even for parameters of transformations whichare not global but rather are holomorphic functions, in which case the total deriva-tive terms do contribute to the transformations, they cannot be determined. Henceone can generally multiply each of them with an arbitrary parameter .

However,imposing the relations of eqn. (9) reduces the number of parameters from five totwo a and ˜a as stated in eqn.

(12) . These parameters will play a role in thecorresponding algebra as will be discussed in the next section.

In fact there aresome additional relations among the symmetry generators˜T(z) = {G, Q(z)}G(z) = −[G, ˜J(z)](13)which are all summarized in the following diagram:˜T,TQG−J, −˜J(14)where A −−B →C denotes acting with a charge B on a current A(z) to generatea current C(z). The currents ˜T, T and ˜J, J correspond to the same symmetrytransformation and are related to one another by φ ↔−˜φ and λ ↔˜λ.9

We wish now to address the question of whether the transformations of eqn. (10) exhaust the symmetries of the TCFT models.

In what follows we consideronly compact Riemann surfaces so the invariance of the action will be checkedalways up to total derivatives. The answer to this question is definitely no.

Thearsenal of symmetries is much richer. There are in fact three types of symmetrytransformations: (i) bosonic or fermionic transformations which involve only thecommuting or the anticommuting parts of the action like δJ for a = ˜a = 0 (ii)bosonic symmetries acting on both sectors like δT and (iii) fermionic symmetriesmixing the two sectors like δQ and δG.

Before dwelling into the second and thirdtypes let us write the most general invariance of each of the sectors separately.Let us look for instance on the bosonic sector. This part of the action is invariantunder the separate transformation of φ and ˜φ as follows:δφ = ǫ∂˜W ˜F( ˜W)δ ˜φ = ǫ∂W F(W)(15)where W = ∂φ, F(W) is a general function of W, ∂WF(W) is its derivative withrespect to W and similarly for the fields with ˜φ.

In particular any polynomialsof W and ˜W for F and ˜F will do the job. Symmetries which leave the fermionicsector invariant are for example those which are generated by λ (or ˜λ) δ˜λ = ǫ(δλ = ǫ).Next we want to check whether there are generalizations of the fermionic sym-metries generated by Q and G. One finds that the following currents generate suchsymmetries.Q(n) = Dn˜λG(n) = ˜Dnλ(16)where the covariant derivatives are D = ∂+W = ∂+∂φ and ˜D = ∂−˜W = ∂−∂˜φand Dn is the nth power of D. The fermionic generators Q and G are the special ofn = 1, Q = Q(1) and G = G(1) with ˜a = −a = 1.

From the corresponding OPEs’one gets that ˜a = −a. One can generalize the covariant derivative given above toincorporate a ̸= 1 as in eqn.

(12) and still maintain the structure of eqn. (16) by10

the following redefinitions λ →λ′ = ˜aλW →W ′ = 1˜aW and the same for ˜λ and˜W. Alternatively one can only redefine λ and ˜λ and take 1˜a as the charge in thecovariant derivative.

One can view this covariant derivative as if its source is anabelian gauge field which is taken to be a pure gauge, namely, zero field strength orflat gauge connection. We denote the set of infinitely many symmetry generatorsQ(n) as Q∞and G(n) as G∞.

It is obvious from eqn. (16) that the G∞and thecorresponding transformation laws are related to those of Q∞by the replacementφ →−˜φ and ˜λ →λ.

We thus describe here only the Q∞symmetries. Under thelater only λ and ˜φ transform as follows:δQ(n)λ = D(n)−ǫδQ(n) ˜φ = −n−1Xi=0Di−ǫD(n−1−i)˜λ(17)where D−= −∂+ W such that (DA)B −A(D−B) = ∂(AB).

The parameter oftransformation ǫ has conformal dimensions −(n −1). To be specific here are theglobal transformations generated by the first three lowest generators (omitting theparameter of transformations)δQ(1)λ = WδQ(1) ˜φ = −˜λδQ(2)λ = W 2 −∂WδQ(2) ˜φ = −(∂˜λ + 2W ˜λ)δQ(3)λ = W 3 −3W∂W + ∂2WδQ(3) ˜φ = −(∂2˜λ + 3W∂˜λ + 3W 2˜λ).

(18)It is straightforward to check that these transformations leave the action in-variant. In appendix A we show that the general transformations eqn.

(16) areindeed symmetry transformations.So far have we discussed the fermionic symmetries, now to complete the gener-alization of the relation given by eqn. (14) we define a double set of infinite bosonicoperators as followsW (n)(ω) =12πiIωdzQ(z)G(n−1)(ω)˜W (n)(ω) =12πiIωdzG(z)Q(n−1)(ω)(19)11

Using the OPE’s of eqn. (11) we find for the W∞W (n) = ˜D(n−1) ˆW (1) + G(n−1)˜λ(20)where ˆW (1) = W −λ˜λ and ˜DW = (∂−˜W)W,˜D˜λ = ∂˜λ .

For some applicationsit is convenient to express W (n) as follows:W (n) = ˜D(n−2)W (2) −n−2Xk=0(n−1k)G(k)∂n−1−k˜λ(21)where G(0) = λ.For ˜W (n) we interchange W with −˜W and λ with ˜λ.Theexpressions for the lowest W (n) areW (1) = W + ˜W −λ˜λW (2) = ∂W −W ˜W −λ∂˜λW (3) = ∂2W −∂(W ˜W) −˜W∂W + ˜W 2W −λ∂2˜λ −2∂λ∂˜λ + 2 ˜Wλ∂˜λ(22)W (1) is not determined by eqn. (19) .Its form is dictated by the algebra ofthe generators.

Again it is easy to check that W (1) = −J and W (2) = T with˜a = −a = 1. The invariance under the W (n) transformations follows from that ofG(n):δW (n)S = [ǫW (n), S] = [ǫ{Q, G(n)}, S] = {Q, [G(n), S]} + {ǫG(n)[Q, S]} = 0(23)since [Q, S] = [G(n)S] = 0.

In the same manner Q(n) invariance implies that of˜W (n). For completeness we write now the transformations under the W∞symmetryof the various fieldsδW (n)λ = ( ˜D(n−1)−ǫ)λ −ǫ ˜D(n−1)λδW (n)˜λ = ˜D(n−1)−(ǫ˜λ) −( ˜D(n−1)−ǫ)λδW (n) ˜φ = −˜D(n−1)−ǫδW (n)φ = −n−2Xi=0˜Di−ǫD(n−2−i) ˆW (1) −˜Di−(ǫ˜λ) ˜D(n−2−i)λ(24)where ˜D−ǫ = −(∂+ ˜W)ǫ.

How can we generalize the diagram of eqn. (14)?

One is tempted to think that there is the same structure also for the nth level.12

By definition W (n) and ˜W (n) are created by applying Q and G on G(n) and Q(n)respectively, however Q(n) and G(n) are not derivable from W (n−1) and ˜W (n−1) byacting with Q and G. One has to modify W (n) and ˜W (n) in the following way togenerate from them Q(n) and G(n). First note that12πiIωdzQ(z) ˜W (n+1)(ω) = [{Q, G}, Q(n)] + [{Q, Q(n)}, G] = [W (2), Q(n)] = ∂Q(n).

(25)Now it is easy to see that if one adds λQ(n) to ˜W (n+1) one gets (see Appendix A):12πiIωdzQ(z)[ ˜W (n+1)+λQ(n)](ω) =12πiIωdzQ(z) ˜R(n+1)(ω) = ∂Q(n)+WQ(n) = Q(n+1). (26)and similarly for G(n+1).

The explicit expressions for R(n) and ˜R(n) areR(n) = ˜D(n−1) ˆW (1)˜R(n) = D(n−1)˜ˆW (1). (27)The diagram of eqn.

(14) takes now the split form˜W (n), W (n)Q(n−1)G(n−1)Q(n−1)G(n−1)˜R(n−1), R(n−1)(28)Are the R(n) and ˜R(n) generators of symmetries? It is easy to see that δR(n)Sis closed under Q.

[Q(n), S] = [[Q, R(n)], S] = [Q, [R(n), S]] −[R(n), [Q, S]] = 0. (29)It turns out, as we show in Appendix A, that δR(n)S =Rdz ¯∂(ǫλDn˜λ) = 0.

Thetransformations of the various fields under R(n) are also written down in the ap-pendix.13

The next task is to show that all these fermionic and bosonic symmetries arein fact shared by all the models of the previous section. To prove this we firsttreat the general case of eqn.

(9) and then we consider the case of eqn. (3) forR(2) ̸= 0 and α ̸= 0.

The action (9) is clearly a sum of decoupled actions, ( as longas it is not coupled to TG) so we can separate the symmetry generators for eachseparate part Q(n)i, G(n)iand W (n)i. To construct the generators in a form similarto eqns.

(16) and (21) we need dimension one fields as connections in the covariantderivatives. For this purpose one can “bosonize”[18]the bosonic system in eqn.

(9)in the following wayZd2zXi[Φ(hi) ¯∂˜Φ(1−hi)] =12Zd2zXi[∂ρi ¯∂ρi −14Qi√gR(2)ρi + 2ηi ¯∂ξi](30)where Φ(hi)(z) = eρi(z)∂ξi and ˜Φ(1−hi) = e−ρi(z)ηi and Qi = −(1 −2hi). Settingnow R(2) = 0 we take ∂ρi as the connection of the following covariant derivativesDi = ∂+ ∂ρi and ˜Di = ∂−∂ρi.

The expression for the symmetry generators arethusQ(n)i= Dni ˜Ψ(1−hi)G(n)i= ˜Dni Ψ(hi)W (n)i= ˜D(n−1)iˆW (1)i+ G(n−1) ˜Ψ(1−hi)= ˜D(n−2)iW (2)i−n−1Xk=1(n−1k)G(k)i∂n−1−k ˜Ψ(1−hi)(31)where now ˆW (1)i= ∂ρi +Ψ(hi) ˜Ψ(hi−1) and W (2)i= −[(∂ρi)2 −∂2ρi +Ψ(hi)∂˜Ψ(hi−1)]Notice that unlike the discussion above here there is only one scalar for each sys-tem (ρi) rather than two (φ, ˜φ). Nonetheless, the transformations of the fieldsΨ(hi), ˜Ψ(1−hi) and ρi, which are given by eqn.

(17) and (24) with some obviousrenaming, leave the action of eqn. (30) invariant due to the factor half in front of∂ρi ¯∂ρi.We want to consider now the case of a non-flat world-sheet.

Once we turnon the curvature, the parameters α in the model of (9) as well as Q of the above14

discussion play an important role. In the case of the twisted N = 2 theory thelevel k = ( 1α2 −2) determines the dimension of the (moduli) space on which all“physical” correlators are cohomologies[12,5].

Do we loose the symmetry structuregenerated by the infinitely many generators of eqn. (31) ?

It turns out that thoseinvariances persist also in the R(2) ̸= 0 case. To realize this phenomena we useagain the conformal metric ds2 = eϕdzd¯z.

In this picture the action of eqn. (3)takes the formS(N=2) =Zd2z∂φ¯∂˜φ + iα∂¯∂ϕφ + (λ∂˜λ + c.c)=Zd2z∂φ¯∂ˆφ + (λ∂˜λ + c.c)ˆφ = ˜φ −iαϕ(32)This redefinition make sense if α is pure imaginary namely for negative k. Thelater are natural if the starting point of the twisted N = 2 is SL(2, R) WZWmodel rather than an SU(2) model.

Following this redefinition the form of all thegenerators remains the same apart form the fact that ˆφ is replacing ˜φ.Two remarks are in order: (i) Since a fixed world sheet metric was introducedit is clear that the ghost sectors of the pure gravity theory have to be invoked andhence the whole action of eqn. (1) has to be added.

The other remark refers tothe form of T. Building it from G we get nowIωdzQ(z)G(ω) = T(ω) = ∂φ(∂˜φ + iα∂ϕ) + λ¯∂˜λ −∂2φ(33)This may look an unfamiliar expression but in fact this is exactly what has to beachieved for this metric. [12]15

5.The algebraic structureAn algebra for the TCFT’s was written down in ref. [5].

This algebra canbe deduced from the OPEs’ of the various pairs of operators made out of J, T, Qand G, using Jacobi identities. The OPEs’ follow from the “topological condition”given in eqn.

(8) . The algebra is characterized by the three anomalous terms inthe Kac- Moddy algebra of J, in [L, J] and in {G, Q}, which are all determined byone parameter denoted in ref.

[5] as d = dJJ = dQG = −dT J.Next we want to analyze the algebraic structure of the set W (n), Q(n) and G(n).We first wish to confirm that the OPEs’ which led to the algebra of ref. [5] arethose of the symmetry generators for n = 1 and W (2).

Using the definitions ofeqns. (16) and (21) and the basic OPEs’ (11) it is straightforward to check thatthe resulting OPE’s :W (1)(z)Q(1)(ω) = Q(1)(ω)(z −ω)W (1)(z)G(1)(ω) = −G(1)(ω)(z −ω)W (1)(z)W (1)(ω) = −1(z −ω)2W (1)(z)W (2)(ω) =−1(z −ω)3 −W (1)(ω)(z −ω)2W (2)(z)Q(1)(ω) = Q(1)(ω)(z −ω)2 + ∂Q(1)(ω)(z −ω)W (2)(z)G(1)(ω) = 2G(1)(ω)(z −ω)2 + ∂G(1)(ω)(z −ω)W (2)(z)W (2)(ω) = 2W (2)(ω)(z −ω)2 + ∂W (2)(ω)(z −ω)Q(1)(z)G(1)(ω) =−1(z −ω)3 + W (1)(ω)(z −ω)2 + W (2)(ω)(z −ω)(34)is identical to those of ref.

[5]. Since when acting on Q(n) with W (1) it is in factonly ˆW (1) which operates, one can use the later as the “ghost number current”when acting on Q(n).

Similarly one can useˆ˜W (1) when applied on G(n). We nowreturn to the more general form of the symmetry generators namely those with˜a = −a ̸= 1 given in eqn.

(12) and in the discussion following eqn. (16) .

It isstraightforward to check that for this case one derives the same OPEs’ apart fromthe fact that now d = dJJ = dQG = −dT J = 2a˜a + 1 For the parametrization of16

ref. [12] one thus gets d =kk+2.

Switching on R(2) introduces, as was explained insection 3, the redefinition of ˜W →˜W + iα∂ϕ. It is easy to check that the OPEs’of eqn.

(34) stay in tack under the this modification.We proceed now to the operators beyond the “minimal topological algebra”[5].First we examine the OPE of W (1) and W (2) with the rest of the operators. InAppendix C it is shown in the context of model (3) thatW (1)(z)Q(n)(ω) = Q(n)(ω)(z −ω)W (1)(z)G(n)(ω) = −G(n)(ω)(z −ω)(35)12πiIωdzW (1)(z)W (n)(ω) = 0(36)It is thus clear that W (1) plays the role of the ghost number current and that theQ(n) and G(n) have ghost number 1, -1 respectively.

It is shown in Appendix Cthat the term proportional to1(z−ω) in W (1)(z)W (n)(ω) vanishes which leads to eqn. (36) .

Hence, as expected from the its definition, W (n) has a zero ghost number.Similarly it is not surprising to notice that W (2) is the energy momentum tensorand Q(n), G(n) and W (n) all carry dimension equal to n. and n + 1 respectively.W (2)(z)Q(n)(ω) = ...n(n −1)Q(n−1)(ω)(z −ω)3+ nQ(n)(ω)(z −ω)2 + ∂Q(n)(ω)(z −ω)(37)and similarly for W (2)G(n).The next question of interest is whether the OPEs’ and the corresponding com-mutation relations are linear or whether products of generators and their derivativesshow up in them. It turns out that the algebra is not linear.

We demonstrate itnow in the following two examples:W (1)(z)W (3)(ω) =−2(z −ω)4 + 2W (1)(ω)(z −ω)3 −[2W (2)+ : ˜R(1) ˜R(1) : +∂˜R(1)](ω)(z −ω)2(38)where the : : denotes normal ordering as explained in Appendix C. A similarstructure show up in17

Q(2)(z)G(1)(ω) =−4(z −ω)4−2W (1)(ω)(z −ω)3 −[2W (2)+ : ˜R(1) ˜R(1) : +∂˜R(1)](ω)(z −ω)2+∆(3)(ω)(z −ω)(39)Where ∆(3) = −W 2 ˜W + ∂(W 2) + ∂W ˜W + ∂2˜λλ + 2(∂W)˜λλ + 2W∂˜λλ.Another obvious property of Q(n) and G(n) is nillpotency . This is a specialcase of the anticommuting relations{Q(n)Q(m)} = 0{G(n), G(m)} = 0.

(40)The derivation of Q(n)(z)G(m)(ω) is straightforward though tedious. In Ap-pendix B we present the calculation of the anomalous term.6.

Summary and DiscussionSince the original path-integral approach to TQFT’s it was known that a basicproperty of all the TQFTs’ is the fact that all the non-zero modes are canceledout from the “physical observables”. This characteristic feature should manifestitself in terms of a large set of symmetry constraints on physical states.

In thisnote we have investigated the symmetry structure of topological theories.Weshowed that the TCFTs’ are in fact invariant under transformations generatedby nilpotent pairs of fermionic operators of arbitrary conformal dimension. Aninteresting feature of these generators is that they are in fact the nth covariantderivative on the basic fermions of the theory.

The covariant derivative is withrespect to a “flat abelian gauge connection”. We showed that the generic model canbe derived as a BRST gauge fixed action of a theory with a “topological symmetry”in which holomophicity condition was imposed.

It is thus plausible that the laterconstruction and the infinite tower of symmetries are related. In this case, it isnot hard to envision, that all the TCFTs’ models that we considered are describedby cohomologies on moduli spaces of flat connections and their generalization to18

higher spin fields.The bosonic counterparts of the fermionic symmetries W (n)and R(n) where also expressed as covariant derivatives. The complete algebraicstructure was not extracted in the present work.

Therefore it is not clear to whatextent the algebra of the bosonic generators is related to various W∞which werediscussed in the literature.[16]. The implications of this very rich algebraic structureon the Hilbert space of physical states is under investigation.

We believe that it isthis algebraic structure which is responsible for the decoupling of all the non-zeromodes from the physical observables.We did not discuss in this work the application of the “minimal toplogicalalgebra” to string theories. It was realized[5,22] that the set of J, Q, G and T donot close the algebra and one has to introduce additional symmetry generators.It was also found out that the non-critical string theory of c = 1 share a “highersymmetry”.

[23]The role of the new symmetries presented in this work in the realmof string theories is under current investigation.Acknowledgements:J.S wants to thank R.D. Peccei and the HET group of UCLAfor the worm hospitality during his stay in UCLA where part of this work wasdone.

He also wants to thank R. Brooks for his comments on the manuscript. S.Ywants to thank the theory group of CERN.

Part of this work was done during hisstay in CERN.19

REFERENCES1. E. Witten, Comm.

Math. Phys.

117 (1988) 353.2. E. Witten, Nucl.

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R. Dijkgraaf, E. Witten, Nucl. Phys.

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B352 (1991) 59;“Notes On Topological String THeory And 2D Quantum Gravity,” Princetonpreprint PUPT-1217 (1990).6. K. Aoki, D. Montano, J. Sonnenschein,Phys.

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72513. M. Spiegelglas and S. Yankielowicz, “G/G Topological Field Theory byCosetting,” “Fusion Rules As Amplitudes in G/G Theories,” preprints, toappear.14.

J. Distler, “2-D Quantum Gravity, Topological Field Theory, and the Mul-ticritical Matrix Models,” Princeton preprint PUPTHY-1161 (1990).15. E. Witten Comm.

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Pope, L. J. Romans, E. Sezgin, X. Shen, and S. StellePhys. Lett.

243B (1990) 350, 245 (1990) 447, 256 (1990) 350; “w∞Gravityand Super w∞CERN-TH 5743/90; E. Bergshoeff, M. Vasiliev and B. de WitPhys. Lett.

256B (1991) 199 and references therein.20

17. K. Aoki, D. Montano, J. Sonnenschein “The role of Contact Algebra in theMulti Matrix Models” Caltech preprint CALT-68-1676, to appear in Mod.Phys.

Lett.18. D. Friedan, E. Martinec and S. Shenkar Nucl.

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Schnitzer Nucl. Phys.

B329 (1990)649D. Karabali Gauged WZW Models and the Coset Construction of ConformalField Theories”, talk at the NATO ARW 18th Int.

Conf. on DifferentialGeometric Methods in Theoretical Physics, Tahoe CA, 1989 H.J.

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J. Distler and Z. Qiu, Nucl. Phys.B336(1990)533 D. Nemeschansky, Phys.Lett.B224(1989)121; Gerasimov, A. Marshakov, A. Morozov, A. Olshanet-skii, S. Shatashvilli Int.

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A5(1990)249521. D. Nemeschansky and S. Yankielowicz 1990 unpublished see also M. Spigel-glas and S. Yankielowicz ref.

[13] in which the U(1) case is discussed indetails.22. R. Budzynski and M. Spalinski, ”on the chiral algebra of topological confor-mal field theories”, TUM-TH-121/91 preprint.23.

R. Brooks “Residual Symmetries in c=1 noncritical string theory” MIT-CTP1957 (1991).21

APPENDIX AWe want to show now that the Q∞, G∞and W∞transformations leave theTCFT models invariant. Again we present the explicit proof for the α = 0 case ofeqn.

(12) and later we explain how to extend the proof to the rest of the cases.Obviously only λ and ˜φ transform by Q(n). Recall eqn.

(17)δQ(n)λ = Dn−ǫδQ(n) ˜φ = −n−1Xi=0Di−ǫD(n−1−i)˜λ. (A.1)The action thus transforms intoδSN=2 =Zd2z[¯∂Wn−1Xi=0Di˜λD(n−1−i)−ǫ + Dn−ǫ¯∂˜λ].

(A.2)Now this last expression is in fact a total derivative of the form ¯∂(Dn−ǫ˜λ). In orderto prove that we have to show that¯∂Wn−1Xi=0DiλD(n−1−i)−ǫ = ¯∂(Dn−ǫ)˜λ(A.3)Let us expand the term on the right of the last equation:¯∂(Dnǫ)˜λ = ¯∂(−∂+ W)(Dn−1ǫ)˜λ = ¯∂W(Dn−1ǫ)˜λ + ¯∂(Dn−1ǫ)D˜λ(A.4)The term on the right is the first term in the sum of eqn.

(A.1) . Further iterationof expanding the term to the right generates exactly all the terms in the sum ofeqn.

(A.4) .The invariance of the action under G(n) follows from an identical proof withthe obvious replacements λ →˜λ W ↔−˜WNow the generalization to the rest of the TCFT models is very straightforward.For the R(2) ̸= 0 case one can again pass to the modified field ˆφ. As for the generalcase of eqn.

(13). The same reasoning of above leads to the conclusion that thevariation of the action under for example Q(n)iis δS =Rd2z ¯∂(Dni ǫ˜Ψ(1−hi).22

Next we want to explain the relations of diagram (14) . We use again theexample of (12) for R(2) = 0.

Let us show first thatQ(n) =IωQ(z) ˜W (n)(ω) =IW ˜λ(z)[−Dn−1 ˜W −Dn−1(˜λλ) + Dn−1˜λ)λ](A.5)The first term in the integral gives Dn−1(∂˜λ). Plugging the OPE of λ˜λ into theother two terms, recalling that Dλ = ∂λ one gets for the second term Dn−1(W ˜λ)so that altogether we get for the first two terms Dn−1(∂˜λ + W ˜λ) = Dn˜λ = Q(n).It is thus clear that omitting the last term Q(n−1)λ produces R(n).

Under theinterchange of λ with ˜λ and W with −˜W we find in a complete analogy the sameresults for G(n).Our next task to examine whether R(n) generate symmetry transformations,namely, we want to check if δR(n)S =12πiHω dz[ǫR(n)](z)S = 0. Since we know that˜R(n+1) = ˜W (n+1) + λQ(n) and since we know that ˜W (n) are symmetry generatorsit is enough to show that λQ(n) leaves the action invariant.

It is straightforwardto realize that the later holds. The transformation of the various fields are foundto beδR(n)λ = −( ˜D(n−1)−ǫ)λδR(n)˜λ = ˜D(n−1)−(ǫ)˜λδR(n) ˜φ = −˜D(n−1)−ǫδR(n)φ = −n−2Xi=0˜Di−ǫD(n−2−i) ˆW (1) −˜Di−(ǫ) ˜D(n−2−i)(λ˜λ)(A.6)23

APPENDIX BWe compute the anomaly term in the OPE of Q(m)G(n). The notion of anomalyrefers here to the term proportional to1(z−ω)m+n+1 which obviously is a number.One gets this term by performing a complete contraction of all the fields.Forn > m the general form of a term in the expansion which can contribute to theanomalous term has the form[W i∂jW∂m−(i+j)−1˜λ](z)[ ˜W i∂l ˜W∂n−(i+l)−1λ](ω)(B.1)where i = 0, ...m −1, j = 1, ...m −(i + 1) and l = 1, ....n −(i + 1).

In addition onecan have the case with no derivatives on W and ˜W. The contribution of a termof the form of (B.1) is found by performing all possible contractions between thefields.

One gets(−1)m[m + n −(2i + j + l)]!i! (j + 1)!

(l + 1)! + (l + j + 1)!

](B.2)The contribution of the terms with no W and ˜W derivatives are (−1)m[m+n−2i]!i!.What is left over to do is to figure out the multiplicity factors Bi and Dijl of eachof the terms and then perform the summation, namely:Anom =(−1)mmXi=0Bi[(m + n) −2i]!i! (−1)mmXi=0−1m−i−1Xj=1Dijl[(m + n) −2(i + 1) −(j + l)]!i!

[(j + 1)! (l + 1)!

+ (l + j + 1)! ](B.3)It is easy to check that Bi = (mi )(ni ) and similarly one can get an expression forDijl24

APPENDIX CAs in the previous sections we work here in the context of the flat world sheetof eqn. (3) .

Thus following eqn. (21) W (1) = (W + ˜W −λ˜λ).

When acting onQ(n) = Dn˜λ obviously only the second and the third terms in W (1) can contribute.Let us first look on the residue, namely, the1(z−ω) terms. Since following eqn.

(11)the OPE ˜W(z)W(ω) =1(z−ω)2 and when ˜W is applied on Q(n)(ω) there are no termsat z to expand, the only contributions can come from λ˜λ(z)Q(n)(ω). Denoting ageneric term in Q(n) as CkFk(W, ∂W)∂n−k˜λ where Ck is some numerical coefficientand Fk(W∂W) is a dim k polynomial of W and derivatives of W, thanλ˜λ(z)CkFk∂n−k˜λ(ω) = −CkFk˜λ(z)[n −k]!

(z −ω)n+1−k = ...−1(z −ω)CkFk(W, ∂W)∂n−k˜λ(C.1)It is thus clear that the residue is really −Q(n).We want to show now that all the terms multiplying1(z−ω)l for l > 1 vanish.Terms proportional to1(z−ω)2 are generated by contraction between the λ˜λ and Q(n)and between ˜W and powers of W in fk. Rewriting the later as Ckfk = Ci,kW igk−iwe get a contribution of −Pi,k Ci,kW i(n−k)∂n−k˜λ where as the ˜WW contractionslead Pi,k Ci,kiW i−1(n−k)∂n−k˜λ.

Now since (i+1)Ci+1,k = (n−k)Ci,k for k ̸= i+1and (i + 1)Ci+1,k = (n −i)Ci,k for k = i + 1 the two contributions cancel eachother.Next we compute the terms multiplying1(z−ω)j for j = 1, 2, 3. Following thesame steps as for W (1) one can realize that from contraction the λ∂˜λ(z) term oneget −CkFk(n−k)!l!(z−ω)n+1−k−l∂l+1˜λ.

For l = n −k one can exactly the action of thederivative on ˜λ in ∂Q(n). When ˜WW is contracted with powers of W in fk onegets the action of the derivative of this part and same applies for ∂jW factors infk.

So altogether applying the chain rule one gets the term ∂Q(n)(z−ω). Repeating theanalysis now for j = 2, 3 one derives the eqn.

(37) .25

We present here the explicit calculation of W (1)W (3) The terms multipying thevarious powers of1(z−ω) are1(z −ω)3 : 4 ˜W + 2W −2λ˜λ −2 ˜W = 2W (1)1(z −ω)2 : 2∂W −λ∂˜λ + ∂˜W −2W ˜W −˜W 2 + 2˜λ∂λ −2˜λW ˜λ−[W (2)+ : R(1) ˜R(1) : +∂˜R(1)](C.2)where the normal order product : R(1) ˜R(1) : is given by W 2 + 2Wλ˜λ + ∂λ˜λ + ∂˜λλ.26

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