---

N = 2 String as a Topological Conformal Theory

arXiv:hep-th/9111059v1 28 Nov 1991N = 2 String as a Topological Conformal TheoryJoaquim Gomis∗and Hiroshi Suzuki†Uji Research Center, Yukawa Institute forTheoretical Physics, Kyoto University, Uji 611, JapanABSTRACTWe prove that critical and subcritical N = 2 string theory gives a realization ofan N = 2 superfield extension of the topological conformal algebra. The essentialobservation is the vanishing of the background ghost charge.The N = 2superstring introduced by Ademollo et.

al. [1,2,3,4,5] has the critical dimension2, there are no transverse degrees of freedom, the physical spectrum contains afinite number of particles, all massless and bosonic.

There is a general belief thatthis is topological quantum theory. In this note we will prove that critical andsubcritical N = 2 strings are a topological field theories [6,7] in the sense that thereparametrization BRST current algebra gives a realization of an N = 2 superfieldextension of the topological conformal algebra [8,6].

The key ingredient for thisproof is that N = 2 string has no total background ghost charge and therefore noghost number anomaly [9].∗Permanent address: Departament d’Estructura i Constituents de la Mat`eria, Universitatde Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia, Spain.Electronic address:quim@ebubecm1.bitnet† JSPS Junior Scientist Fellow.Also at Department of Physics, Hiroshima University,Higashi-Hiroshima 724, Japan. Electronic address: suzuki@jpnrifp.bitnet

Let us first recall some basic facts of N = 2 string in superfield formalism,∗inN = 2 superspace is described in terms of bosonic (z, ¯z) and fermionic (θ±, ¯θ±)coordinates. We can define covariant derivativesD± =∂∂θ∓+ θ±∂,¯D± =∂∂¯θ∓+ ¯θ± ¯∂.

(1)The action can be written in terms of two superfields Sµ(z, ¯z, θ+, ¯θ+, θ−, ¯θ−) andSµ∗(z, ¯z, θ+, ¯θ+, θ−, ¯θ−) satisfying two constraintsD−Sµ = ¯D−Sµ = 0(2)andD+Sµ∗= ¯D+Sµ∗= 0. (3)The action is [1]A =Zdzd¯zZdθ+d¯θ+dθ−d¯θ−Sµ∗Sµ.

(4)The solution of the equation of motionsD+ ¯D+Sµ = 0,¯D−D−Sµ∗= 0(5)can be written asSµ = Sµ1 + Sµ2(6)whereD−Sµ1 = ¯D−Sµ1 = ¯D+Sµ1 = 0,D−Sµ2 = ¯D−Sµ2 = D+Sµ2 = 0. (7)A real superfield Xµ is constructed viaXµ z, θ+, θ−= Sµ1z + θ−θ+, θ−+ Sµ∗1z + θ+θ−, θ+.

(8)∗We follow the notation of ref. [4].2

The components of Xµ(Z) areXµ(Z) = Xµ(z) + θ−ψ+µ(z) + θ+ψ−µ(z) + iθ−θ+∂Y µ(z)(9)where Xµ(z) and Y µ(z) are free bosonic fields and ψ±µ(z) are free fermions.∗We have the following operator product expansionXµ(Za)Xν(Zb) ∼ηµν ln Zab(10)where Zab isZab = za −zb −θ+a θ−b + θ−a θ+b. (11)N = 2 primary conformal superfields ψhq (Z) are characterized by a weight h and acharge q.

They have the following OPE with the energy momentum tensor T(Z)T(Za)ψhq (Zb) ∼h θ−abθ+abZ2abψhq (Zb) −q2Zabψhq (Zb)+12Zabθ−abD+b −θ+abD−bψhq (Zb) + θ−abθ+abZab∂zbψhq (Zb)(12)where θ±ab = θ±a −θ±b . The contribution to the energy momentum tensor from XµisT X(Z) = 12D−XµD+Xµ(Z).

(13)N = 2 superstring action is invariant under several local gauge transformations.We will work in the superconformal gauge. Gauge fixing generates a Faddeev–Popov determinant expressible as a superfield action using N = 2 superfield ghostC and antighost BC ≡c + iθ+γ−−iθ−γ+ + iθ−θ+ξ,B ≡−iη −iθ+β−−iθ−β+ + θ−θ+b.

(14)The ghosts c and b are for the τ-σ reparametrization invariances, γ± and β± are thesuper ghosts for the two local supersymmetry transformations and ξ and η are the∗We consider only the holomorphic part.3

ghosts associated with the local U(1) symmetry. Their Lagrangians are the firstorder systems with background charge Q [10] and statistics ǫ of (Q, ǫ) = (−3, +),(2, −) and (−1, +) respectively.

Notice that the total background ghost chargevanishes. The ghost action in terms of superfield is given byAgh = 1πZd2zdθ+dθ−B ¯∂C + (c.

c.). (15)The non-zero fundamental operator product expansion for the holomorphic part isC(Za)B(Zb) ∼θ−abθ+abZab∼B(Za)C(Zb).

(16)The ghost energy momentum tensor isT gh(Z) = ∂(CB)(Z) −12D+CD−B(Z) −12D−CD+B(Z). (17)The superfields B(Z) and C(Z) are q = 0 conformal fields with h = +1 andh = −1.If we consider the total energy momentum tensorT = T X + T gh(18)the OPE of T with itself becomesT(Za)T(Zb) ∼D −24Z2ab+ θ−abθ+abZ2abT(Zb) +12Zabθ−abD+b −θ+abD−bT(Zb)+ θ−abθ+abZab∂T(Zb)(19)where D is the dimension of the target space.

For D = 2,∗T is a q = 0, h = 0conformal superfield.∗A first attempt to interpret this theory as a four dimensional (2, 2) theory was done byD’Adda and Lizzi in ref. [2].4

The N = 2 BRST charge QB [3] isQB =IdZ CT X + 12T gh(Z)(20)whereIdZ =Idz2πiZdθ+dθ−(21)and when D = 2 it satisfiesQ2B = 0(22)since the total energy momentum tensor has no anomalies. Furthermore one cancheck thatT(Z) = {QB, B(Z)} .

(23)Now we can construct the ghost number current jghostjghost(Z) = −BC(Z). (24)Since the total background ghost charge is zero and this current is not anomalous,the ghost current is a primary field with q = 0 and h = 0.

The previous facts arecharacteristic of N = 2 string, for N = 0 and N = 1 due to the non-vanishing ofthe background ghost charge, the ghost current is anomalous and the ghost currentis not a primary field [10,11].Now let us construct the BRST current jB, we will use a relationjB(Z) = −QB, jghost(Z)(25)in such a way that jB will be BRST invariant. Explicitly one findsjB(Z) = C(Z)T X + 12T gh+ 14D−CD+CB+ 14D+ CD−CB(26)5

where the total derivative pieces∗ensure that jB(Z) is a primary superfield withq = 0 and h = 0.At this point we can define the N = 2 superfield extension of the topologicalconformal algebra. The generators areT(Z) ≡T(Z),G(Z) ≡jB(Z),¯G(Z) ≡B(Z),J(Z) ≡jghost(Z).

(27)The relevant operator product expansions areT(Za)Ψ(Zb) ∼h θ−abθ+abZ2abΨ(Zb) +12Zabθ−abD+b −θ+abD−bΨ(Zb)+ θ−abθ+abZab∂zbΨ(Zb)(28)for Ψ = T, G, ¯G, and J with h = 1, 0, 1, and 0 respectively andG(Za) ¯G(Zb) ∼12Zabθ−abD+b −θ+abD−bJ(Zb) + θ−abθ+abZabT(Zb),G(Za)G(Zb) ∼0,¯G(Za) ¯G(Zb) ∼0,J(Za)J(Zb) ∼0,J(Za)G(Zb) ∼θ−abθ+abZabG(Zb),J(Za) ¯G(Zb) ∼−θ−abθ+abZab¯G(Zb). (29)Summing up we should conclude that the critical N = 2 string is a topologicalfield theory in the sense that the reparametrization BRST current algebra gives∗The sign of the last term in eq.

(26) is different from the expression of the BRST currentin ref. [4], which is primary however not BRST invariant.6

a representation of a N = 2 superfield extension of the topological conformalalgebra.∗Actually Ooguri and Vafa [5] have shown an close relation between N = 2string and self-dual gravity and self-dual Yang–Mills, it will be interesting to studythe implications of our results from this type of theories.With respect to N = 2 subcritical strings or N = 2 super Liouville theorywe can also show that these theories are topological. For analyzing these theoriesone might do some hypothesis about a local ansatz for the Jacobian between anon-trivial measure and the free measure [13].

Or we might construct an effectivetheory [14] based on anomalous identities associated to the superconformal, theBRST and the ghost number symmetries. In the latter, the renormalization ofthe coupling constant (see below) is considered as the one-loop order effect of theBRST invariant measure.

We will follow the second procedure with the vanishingcosmological constant, generalizing the results of the N = 0 and N = 1 case [14] toN = 2 super Liouville theory. To construct the relevant operators for the effectivetheory [15] we should introduce a Liouville superfieldΦ(Z) = φ(z) + θ−φ+(z) + θ+φ−(z) + iθ−θ+∂ρ(z)(30)with the operator productΦ(Za)Φ(Zb) = ln Zab.

(31)The form of the energy momentum tensor isT = T X + T gh + T Liouville(32)where T X and T gh are given by eqs. (13) and (17).

To construct T Liouville wemake an observation that apart from an inhomogenious piece, Liouville superfield∗Our algebra in eqs. (28) and (29), in terms of the component fields, is not same as thetwisted N = 4 superconformal algebra by Nojiri [12].7

behaves as Xµ. Then we haveT Liouville = 12D−ΦD+Φ + κ∂Φ(33)where κ is the renormalized coupling constant of the effective theory and alge-braically it can be calculated imposing that the total energy momentum tensor Tbehaves as a superfield with q = 0 and h = 0.

This condition impliesκ2 = 1 −D4. (34)For the ghost number current as we have described before there is no ghostnumber anomaly, thus the current is given byjghost(Z) = −BC(Z)(35)as for the critical string case, jghost is a superfield with q = 0 and h = 0.The BRST charge associated to the BRST symmetry is given byQB =ZDZ CT X + T Liouville + 12T gh(36)and it satisfies Q2B = 0 for any D. Furthermore one can also checkT = {QBRST, B} .

(37)Then we determine the total divergent pieces of the BRST current by using eq.(25). Explicitly one findsjB(Z) = C(Z)T X + T Liouville + 12T gh+ 14D−CD+CB+ 14D+ CD−CB(38)where the total derivative pieces ensures that jB(Z) is a primary N = 2 superfieldwith q = 0 and h = 0.8

At this point it is interesting to comment about the differences of the analo-gous calculation for the N = 0 and N = 1 cases [14]. The two main differenceswith respect to those cases are i) the appearance of a Liouville term in the ghostnumber current ii) the appearance of a divergent terms with a Liouville field in theexpression of BRST current.

The presence of these terms implies that the previoustheories are not topological for arbitrary values of the dimension [16]. The mainreason for that is the anomalous behavior of the ghost number current.As in the critical string case T, jB, jgh and B give a representation of theN = 2 superfield extension of the topological conformal algebra eqs.

(28) and (29).It is also useful to see how the critical string can be considered as a subcriticalstring in dimension 1 plus the Liouville superfield, in fact for this situation, κ = 0and all the operators of the effective theory coincide with the ones of the criticalstring, since in this case there is no restriction for the possible values of D. Distler,Hlousek and Kawai [13] notice already that their local ansatz for the Jacobianworks for every D. It will be interesting to find the dimension D in which one canestablish the equivalence with the recent proposed N = 2 topological supergravity[17,18,19] with gauge group Osp(2|2).In summary in this note we have shown that the N = 2 string theory is atopological field theory in any dimension D in the sense that the reparametrizationBRST current algebra realizes an N = 2 superfield extension of the topologicalconformal algebra. This result is peculiar to the N = 2 string and it is due to thefact that the background charge vanishes and therefore there is no ghost numberanomaly.Acknowledgements: We acknowledge Prof. K. Fujikawa for discussions and a crit-ical reading of the manuscript.

We also thank Prof. M. Ninomiya for a readingof the manuscript. One of us (J. G.) would like to thank Yukawa Institute forTheoretical Physics for hospitality.9

REFERENCES1. M. Ademollo, L. Brink, A. D’Adda, R. D’Auria, E. Napolitano, S. Sciuto,E.

Del Giudice, P. Di Vecchia, S. Ferrara, F. Gliozzi, R. Musto and R.Pettorino, Phys. Lett.

B62 (1976) 105;M. Ademollo, L. Brink, A. D’Adda, R. D’Auria, E. Napolitano, S. Sciuto,E. Del Giudice, P. Di Vecchia, S. Ferrara, F. Gliozzi, R. Musto, R. Pettorinoand J. H. Schwarz, Nucl.

Phys. B111 (1976) 77.2.

D. J. Bruth, D. B. Fairle and R. G. Yates, Nucl. Phys.

108 (1976) 310;L. Brink and J. H. Schwarz, Nucl. Phys.

B121 (1977) 285;E. S. Fradkin and A. A. Tseytlin, Phys.

Lett. B106 (1981) 63;P. Bouwknegt and P. van Nieuwenhuizen, Class.

Quantum Grav. 3 (1986)207;A. D’Adda and F. Lizzi, Phys.

Lett. B191 (1987) 85;S. D. Mathur ans S. Mukhi, Nucl.

Phys. B302 (1988) 130;N. Ohta and S. Osabe, Phys.

Rev. D39 (1989) 1641;M. Corvi, V. A. Kosteleck´y and P. Moxhay, Phys.

Rev. D39 (1989) 1611.3.

A. Bilal, Phys. Lett.

B180 (1986) 255;A. R. Bogojevic and Z. Hlousek, Phys. Lett.

B179 (1986) 69;S. D. Mathur ans S. Mukhi, Phys. Rev.

D36 (1987) 465.4. J. Gomis, Phys.

Rev. D40 (1989) 408.5.

H. Ooguri and C. Vafa, Nucl. Phys.

B361 (1991) 469.6. E. Witten, Commun.

Math. Phys.

117 (1988) 353.7. E. Witten, Commun.

Math. Phys.

118 (1988) 411; Nucl. Phys.

B340 (1990)281.8. T. Eguchi and S.-K. Yang, Mod.

Phys. Lett.

A5 (1990) 1693.9. K. Fujikawa, Phys.

Rev. D25 (1982) 2584.10.

D. Friedan, E. Martinec and S. Shenker, Nucl. Phys.

B271 (1986) 93.10

11. S. P. Martin, Phys.

Lett. B191 (1987) 81.12.

S. Nojiri, Phys. Lett.

B262 (1991) 419; Phys. Lett.

B264 (1991) 57.13. J. Distler, Z. Hlousek and H. Kawai, Int.

J. Mod. Phys.

A5 (1990) 391;I. Antoniadis, C. Bachas and C. Kounnas, Phys. Lett.

B242 (1990) 185.14. K. Fujikawa, T. Inagaki and H. Suzuki, Phys.

Lett. B213 (1988) 279; Nucl.Phys.

B332 (1990) 499.15. J. Gomis and H. Suzuki, to appear.16.

K. Fujikawa and H. Suzuki, Nucl. Phys.

B361 (1991) 539.17. K. Li, Nucl.

Phys. B346 (1990) 329.18.

J. Gomis and J. Roca, Phys. Lett.

B268 (1991) 197.19. S. Govindarajan, P. Nelson and S.-J.

Rey, Nucl. Phys.

B365 (1991) 633.11

---

출처: arXiv:9111.059 • 원문 보기