---

Correlation functions in super Liouville theory

arXiv:hep-th/9108025v2 16 Sep 1991Correlation functions in super Liouville theoryE. Abdalla1, M.C.B.

Abdalla2,D. Dalmazi2, Koji Harada1(a)1Instituto de F´ısica, Univ.

S˜ao Paulo, CP 20516, S˜ao Paulo, Brazil2Instituto de F´ısica Te´orica, UNESP, Rua Pamplona 145,CEP 01405, S˜ao Paulo, BrazilWe calculate three- and four-point functions in super Liouville theory coupled tosuper Coulomb gas on world sheets with spherical topology. We first integrate overthe zero mode and assume that a parameter takes an integer value.

After calculatingthe amplitudes, we formally continue the parameter to an arbitrary real number.Remarkably the result is completely parallel to the bosonic case, the amplitudesbeing of the same form as those of the bosonic case.Typeset Using REVTEX1

The matrix model definition of 2D-gravity has been proving to be very powerful incalculating correlation functions [1], although it seems difficult to generalize the results tosupersymmetric theories. On the other hand, in the continuum approach (Liouville theory)[2–5] it is difficult to calculate correlation functions, while its supersymmetric generalization(super Liouville theory) [6–8] is well known.

Recently, however, several authors [9–13] haveexactly calculated correlation functions in the continuum approach to conformal matterfields coupled to 2D-gravity. (See also Ref.

[14]). They have used a technique based on theintegration over the Liouville zero mode, and their results agree with those obtained earlierin the discrete approach (matrix models).

It is thus very urgent to extend the continuummethod [15] to the supersymmetric case, i.e., superconformal matter fields coupled to 2D-supergravity.The aim of this Letter is to calculate the three- and four-point functions in super Liouvilletheory coupled to superconformal matter with the central charge ˆc < 1, represented as superCoulomb gas [16]. Our approach is close to that of Di Francesco and Kutasov [10].

Theresult is remarkable and is very parallel to the bosonic case; it amounts to a redefinition ofthe cosmological constant and of the primary superfields, resulting the same amplitudes asthose of the bosonic theory.The relevant framework has been given by Distler, Hlousek and Kawai [8].With atranslation invariant measure, the total action is given by S = SSL + SM,SSL = 14πZd2z ˆE12ˆDαΦSL ˆDαΦSL −QˆY ΦSL −4iµeα+ΦSL,(1)SM = 14πZd2z ˆE(12ˆDαΦM ˆDαΦM + 2iα0 ˆY ΦM),(2)where ΦSL, ΦM, are super Liouville and matter superfields respectively. (See Refs.

[8,17]).The matter sector has the central charge ˆcm = 1 −8α20. The parameters Q and α± are givenbyQ = 2q1 + α20,α± = −Q2 ± 12qQ2 −4 = −Q2 ± |α0|.

(3)The (gravitationally dressed) primary superfield eΨNS is given by2

eΨNS(z, k) = d2z ˆEeikΦM(z)eβ(k)ΦSL(z),(4)β(k) = −12Q + |k −α0|. (5)Note that the expressions for α± and β(k) are the same in terms of Q and α0 as those ofthe bosonic theory.Screening charges in the matter sector are of the form d2zeid±ΦM (z), where d± are thetwo solutions of 12d(d −2α0) = 12.

In this Letter, however, we will concentrate on the casewithout screening charges. The case with screening charges, N(≥4)- point functions andthe inclusion of the Ramond sector will be discussed elsewhere.We shall calculate three-point functions of the primary field eΨNS on world sheets withspherical topology (without screening charges), that is,* 3Yi=1ZeΨNS(zi, ki)+≡Z[D ˆEΦSL][D ˆEΦM]3Yi=1eΨNS(zi, ki)e−S.

(6)Our first step is to integrate over the zero modes,* 3Yi=1ZeΨNS(zi, ki)+≡2πδ 3Xi=1ki −2α0!A(k1, k2, k3),A(k1, k2, k3)=Γ(−s)(−π2 )3(iµπ )s*Z3Yi=1d2˜zieikiΦM (˜zi)eβiΦSL(˜zi)Zd2zeα+ΦSL(z)s+0,(7)where ⟨· · ·⟩0 denotes the expectation value evaluated in the free theory (µ = 0) and wehave absorbed the factor [α+(−π/2)3]−1 into the normalization of the path integral. Theparameter s is defined ass = −1α+"Q +3Xi=1βi#.

(8)In general, s can take any real value and there is no obvious way of calculating thepath-integral. However, if we assume that s is a non-negative integer [8–13], this is justa free-field correlator.

Under this assumption, we evaluate the path-integral, and formallycontinue s to non-integer values. For s non-negative integer,3

A(k1, k2, k3) = Γ(−s)(−π2 )3iµπs×Z3Yi=1d2˜zisYi=1d2zi3Yi

[18]. Alternatively,using ΦSL = φ + θψ + ¯θ ¯ψ, we can writeA(k1, k2, k3)= Γ(−s)(−π2 )3 iα2+µπ!sβ21ZsYi=1d2zisYi=1|zi|−2α+β1|1 −zi|−2α+β2sYi

(10)This is non-vanishing only for s odd; we thus write s ≡2m+1. One may evaluate ⟨ψ · · · ψ⟩0and ⟨ψ · · · ψ⟩0 independently.

Since the rest of the integrand is symmetric, one may writethe result in a simple form by relabelling coordinates:A(k1, k2, k3) = −i(−π2 )3Γ(−s)Γ(s + 1) 1α2+ α2+µπ!sIm(α, β; ρ)(11)whereIm(α, β; ρ) =α22mm!Zd2wmYi=1d2ζid2ηi|w|2α−2|1−w|2βmYi=1|w−ζi|4ρ|w−ηi|4ρ×mYi=1|ζi|2α|ηi|2α|1 −ζi|2β|1 −ηi|2βmYi,j|ζi −ηj|4ρmYi

It is easy to check that when m = 0. Thelarge-α and large-β behaviors are consistent with it, and it is physically natural because4

the amplitude should be symmetric under the exchange of two external momenta. ThusIm(α, β; ρ) exhibits the following symmetryIm(α, β; ρ) = Im(−1 −α −β −mρ, β; ρ)(13)Thus we may write Im(α, β; ρ) in the following wayIm(α, β; ρ) = Cm(α, β; ρ)mYi=0∆(1+α+2iρ)∆(1+β+2iρ)∆(−α−β+(2i−4m)ρ)×mYi=1∆(12 + α + (2i −1)ρ)∆(12 + β + (2i −1)ρ)∆(−12 −α −β + (2i −4m −1)ρ)(14)where Cm(α, β; ρ) has the same symmetries as Im(α, β; ρ), and where ∆(x) ≡Γ(x)/Γ(1−x).By looking at the large-α behavior:Im(α, β; ρ) ∼α−2m−2(2m+1)β−4ρm(2m+1),(15)one can confirm that Cm(α, β; ρ) is, as a function of α, bounded as |α| →∞and analytic.This means that Cm(α, β; ρ) is independent of α, and by symmetry, of β as well; Cm = Cm(ρ).It is hard to calculate Cm(ρ).For this purpose, it is useful to consider the simplerintegral:Jm(α, β; γ; ρ) =ZmYi=1d2ζid2ηimYi=1|ζi|2α|ηi|2α|1 −ζi|2β|1 −ηi|2βmYi,j|ζi −ηj|4ρ×mYi

(16)By using similar arguments, one may obtainJm(α, β; γ; ρ)= eCm(γ; ρ)m−1Yi=0∆(1 + α + 2iρ)∆(1 + β + 2iρ)∆(−1 −α −β −2γ + (2i −4m + 2)ρ)×mYi=1∆(1+α+γ+(2i−1)ρ)∆(1+β+γ+ (2i−1)ρ)∆(−1−α−β−γ+ (2i−4m+2)ρ). (17)5

Again, it is very difficult to calculateeCm(γ; ρ).Unfortunately we could not get it in arigorous way. A series of trials and errors, however, led us to the following form;eCm(γ; ρ) = π2m2m m!

[∆(−(γ + ρ))]2mmYi=1∆(1 + 2(γ + iρ)) ∆(1 + γ + (2i −1)ρ) . (18)This is consistent with eC1(γ; ρ) = π22 [∆(−(γ + ρ))]2 ∆(1 + γ + ρ) ∆(1 + 2(γ + ρ)) and thetwo other (calculable) cases ρ = 0 and γ = 0 (up to symmetry factors).

It is very difficult toget anything else consistent with these constraints. And a posteriori it seems to be correctsince it gives a physically reasonable result.

Let us assume that (18) is correct and see itsconsequences.The two integrals are related byIm(ǫ, β; ρ) = −π2mm!∆(1 + ǫ) ∆(1 + β) ∆(−ǫ −β) Jm(2ρ, β; −1/2; ρ) . (19)Therefore Cm(ρ) = −(π/2mm!) eCm(−1/2, ρ)∆( 12 −ρ)∆( 12 + (2m+ 1)ρ).

If we substitute (18)we getCm(ρ) = −π2m+122m∆12 −ρ2m+1 mYi=1∆(2iρ)mYi=0∆12 + (2i + 1)ρ. (20)Now we are ready to write down the amplitude.

Without loss of generality, we can choosek1, k3 ≥α0, k2 ≤α0. By using (5), (8) andP3i=1 ki = 2α0, one getsβ =ρ(1 −s) (α0 > 0)−12 −ρs (α0 < 0).It is easily seen that, for α0 > 0, A = 0 identically, as in the bosonic theory.

For α0 < 0,there are many cancellations, leading toIm(α, β, ρ)= Cm(ρ)mYi=0∆12 −(2i + 1)ρ mYi=1∆(−2iρ) ∆(1 + α + 2mρ) ∆12 −α + ρ= (−1)m+1π2m+1(m! )2∆12 −ρ2m+1(4ρ)−2m∆(1 + α + 2mρ) ∆12 −α + ρ.

(21)We finally obtain the three-point function:6

A(k1, k2, k3) = (−iπ2 )3µ2 ∆12 −ρs∆12 −s2∆(1 + α + 2mρ) ∆12 −α + ρ=µ2 ∆12 −ρs3Yj=1(−iπ2 )∆12[1 + β2j −k2j]. (22)By redefining the cosmological constant and the primary superfield eΨNS asµ →2∆12 −ρµ,eΨNS(kj) →1(−i2π)∆12[1 + β2j −k2j] eΨNS(kj),(23)we get our main resultA(k1, k2, k3) = µs.

(24)Remarkably, this amplitude is of the same form as the bosonic one [10].It is natural to expect that this feature continues to be true for N(≥4)- point functions.In fact, for k1, k2, k3 ≥α0, k4 ≤α0 < 0 (and without screening charges), the four pointfunction turns out to beA(k1, k2, k3, k4) = (s + 1)µs ,(25)with the same redefinition of the cosmological constant and the primary superfields. In orderto get the amplitude for general ki, one may argue, as in Ref.

[10], that non-analyticity comesentirely from massless intermediate states and one may calculate the amplitude by using theanalyticity of the one particle irreducible (1PI) correlators. After setting µ = 1, we obtainthe four-point function for all ki:A(k1, k2, k3, k4) = −α−[|k1+k2−α0|+|k1+k3−α0|+|k1+k4−α0|]+A1P I,(26)with A1P I = −12(1 + α2−).

Compare with Eq. (37) in Ref.

[10]. The analogy to the bosoniccase is obvious.

A detailed account will appear elsewhere.The close connection to the bosonic amplitudes was suggested from super-KP systems[19], and more recently, from supermatrix models [20]. We think, however, that our demon-stration is more direct.In conclusion, we calculated the three- and four-point functions of super Liouville theorycoupled to super Coulomb gas (without screening charges) on a sphere and found that they7

are essentially the same as those of the usual Liouville theory, obtained in Ref. [11].

As a by-product we get the supersymmetric generalization of (B.9) formula of Ref. [18] (s = 2m+1),1s!ZsYi=1d2zid2˜θsYi=1|zi + ˜θθi|2αsYi=1|1 −zi|2βsYi

would like to thank the members of Instituto de F´ısica Te´orica, UNESP, for theirhospitality extended to him and FAPESP (# 90/1799-9) for the financial support.Healso acknowledges communications on 2D gravity with Y.Tanii and with N.Ishibashi. D.Dthanks FAPESP (# 90/2246-3) for financial support.

Work of E.A. and M.C.B.A.

is partiallysupported by CNPq. (a) Address after October 1991, Department of Physics, Kyushu University, Fukuoka812, Japan.8

REFERENCES[1] E. Br´ezin and V. A. Kazakov, Phys. Lett.

B236,144 (1990); M. R. Douglas and S. H.Shenker, Nucl. Phys.

B335, 635 (1990); D. J. Gross and A.

A. Migdal, Phys. Rev.

Lett.64, 127 (1990). [2] A. M. Polyakov, Phys.

Lett. B103, 207 (1981).

[3] T. L. Curtright and C. B. Thorn, Phys. Rev.

Lett. 48, 1309 (1982); E. Braaten, T. L.Curtright and C. B. Thorn, Phys.

Lett. B118, 115 (1982); Ann.

Phys. 147, 365 (1983);E. Braaten, T. L. Curtright, G. Gandour and C. B. Thorn, Phys.

Rev. Lett.

51, 19(1983); Ann. Phys.

153, 147 (1984); J. L. Gervais and A. Neveu, Nucl. Phys.

B199, 50(1982); B209, 125 (1982); B224, 329 (1983); B238, 123,396 (1984); E. D’Hoker andR. Jackiw, Phys.

Rev. D26, 3517 (1982); T. Yoneya, Phys.

Lett. B148, 111 (1984).

[4] F. David, Mod. Phys.

Lett. A3, 1651 (1988); J. Distler and H. Kawai, Nucl.

Phys.B321, 509 (1989). [5] N. Seiberg, Lecture at 1990 Yukawa Int.

Sem. Common Trends in Math.

and QuantumField Theory, and Cargese meeting Random Surfaces, Quantum Gravity and Strings,May 27, June 2, 1990; J. Polchinski, Strings ’90 Conference, College Station, TX, Mar12-17, 1990, Nucl. Phys.

B357 241 (1991). [6] A. M. Polyakov Phys.

Lett. B103 , 211 (1981).

[7] J. F. Arvis, Nucl. Phys.

B212, 151 (1983); B218, 309 (1983); O. Babelon, Nucl. Phys.B258, 680 (1985); Phys.

Lett. 141B, 353 (1984); T. Curtright and G. Ghandour, Phys.Lett.

B136, 50 (1984). [8] J. Distler, Z. Hlousek and H. Kawai, Int.

J. Mod. Phys.

A5, 391 (1990). [9] M. Goulian and M. Li, Phys.

Rev. Lett.

66, 2051 (1991). [10] P. Di Francesco and D. Kutasov, Phys.

Lett. B261, 385 (1991).9

[11] Vl. S. Dotsenko, Paris VI preprint PAR-LPTHE 91-18 (1991).

[12] Y. Kitazawa, Harvard Univ. preprint HUTP-91/A013 (1991).

[13] N. Sakai and Y. Tanii, Tokyo Institute of Technology preprint TIT/HEP-169 (STUPP-91-116) (1991). [14] A. Gupta, S. P. Trivedi and M. B.

Wise, Nucl. Phys.

B340, 475 (1990). [15] (See also) M. Bershadsky and I. R. Klebanov, Phys.

Rev. Lett.

65, 3088 (1990); HarvardUniv. preprint HUTP-91/A002 (PUPT-1236) (1991) ; A. M. Polyakov, Mod.

Phys. Lett.A6, 635 (1991).

[16] M. A. Bershadsky, V. G. Knizhnik and M. G. Teitelman, Phys. Lett.

B151, 131 (1985);D. Friedan, Z. Qiu and S. Shenker, Phys. Lett.

B151, 37 (1985). [17] E. Martinec, Phys.

Rev. D28, 2604 (1983).

[18] Vl. S. Dotsenko and V. Fateev, Nucl.

Phys. B240, 312 (1984); B251, 691 (1985).

[19] P. Di Francesco, J. Distler and D. Kutasov, Mod. Phys.

Lett. A5, 2135 (1990).

[20] L. Alvarez-Gaum´e and J. L. Ma˜nez, Mod. Phys.

Lett. A6, 2039 (1991).10

---

출처: arXiv:9108.025 • 원문 보기