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arXiv:hep-th/9202089v1 26 Feb 1992æ11

Correlation Functions in Non Critical (Super) String TheoryE. Abdalla1, M.C.B.

Abdalla2,D. Dalmazi2, Koji Harada31Instituto 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, Brazil3Department of Physics, Kyushu University, Fukuoka 812, JapanAbstractWe consider the correlation functions of the tachyon vertex operator of the superLiouville theory coupled to matter fields in the super Coulomb gas formulation, on worldsheets with spherical topology. After integrating over the zero mode and assuming that thes parameter takes an integer value, we subsequently continue it to an arbitrary real numberand compute the correlators in a closed form.

We also included an arbitrary number ofscreening charges and, as a result, after renormalizing them, as well as the external legsand the cosmological constant, the form of the final amplitudes do not modify. The resultis remarkably parallel to the bosonic case.

For completeness, we discussed the calculationof bosonic correlators including arbitrary screening charges.2

1- IntroductionTwo dimensional gravity is not only a toy model for the theory of gravitation, but alsodescribes phenomena such as random surfaces and string theory away from criticality1.The discretized counterpart, namely matrix models, proved to be an efficient means toobtain information, especially about non critical string theory, while computing correlationfunctions2.In the continuum approach4, in the conformal gauge, we have to face Liouville theory5.However, although several important developments have been achieved6, we still lack someimportant points, in spite of much effort which has been spent.In particular, it is difficult to calculate correlation functions in a reliable way becauseperturbation theory does not apply. Recently, however, several authors7−14 succeeded intaming the difficulties of the Liouville theory and computed exactly correlation functionsin the continuum approach to conformal fields coupled to two dimensional gravity.

Thetechnique is based on the integration over the zero mode of the Liouville field. The resultingamplitude is a function of a parameter s which depends on the central charge and on theexternal momenta.

The amplitudes can be computed when the above parameter is a non-negative integer. Later on, one analytically continues that parameter to real (or complex)values.

The results for the correlation functions of the tachyon operator thus obtainedagree with the matrix model approach.2 General correlation functions including arbitraryscreening charges are also computed, and may be useful for the purpose of studying fusionrules of minimal models coupled to two dimensional gravity.25Opposite to the bosonic case supersymmetric matrix models are up to now scarcelyknown. There are few papers in the literature concerning this approach3,25, and it is desir-able to have results from an alternative approach for comparison.

For this reason, severalgroups are studying the continuum approach to two dimensional supergravity17−21. Ouraim here is to investigate the supersymmetric Liouville theory.

We shall compute supersym-metric correlation functions on world sheets with spherical topology in the Neveu-Schwarzsector, where the super-Liouville is coupled to superconformal matter with central chargeˆc ≤1, represented as a super Coulomb gas23,17. The results are remarkable, and very paral-lel to the bosonic case; after a redefinition of the cosmological constant, and of the primarysuperfields, the resulting amplitudes have the same form as those of the bosonic theoryobtained by Di Francesco and Kutasov9.

Our present results generalize those presented ina recent paper19, as well as others recently obtained in the literature20,21,26.The paper is organized as follows. In section 2 we review some computations of thebosonic correlators and generalize them to include an arbitrary number of s.c. .

In section3 we calculate the N-point correlator of the Neveu-Schwarz vertex operator in the n = 1two dimensional supergravity coupled to N = 1 supermatter including an arbitrary numbers.c..Our results include the limiting cases c = 1 (ˆc = 1) and in those situations somephysical conclusions can be drawn in the section 4: the amplitude factorizes, and theexpected intermediate poles have zero residue, due to strong kinematic constraints. Allresults are at least consistent with the matrix model approach, with possible exceptionabout the inclusion of screening charges, in the 3-point correlation function.

In the latter3

case, fusion rules deserve careful study.2- Bosonic Correlation FunctionsSome N-point tachyon correlation function in Liouville theory coupled to c ≤1 con-formal matter were recently calculated by Di Francesco and Kutasov9. They worked inthe DDK’s framework22 where the total action is given by:S = 12πZd2wpˆgˆgab∂aφ∂bφ −Q4ˆRφ + 2µeαφ + ˆgab∂aX∂bX + iα02ˆRX,(2.1)here φ represents the Liouville mode and X is the matter field with the central chargegiven by c = 1 −12α20.

From the literature17 we know that the constant Q is determinedby imposing a vanishing total central charge, and is given byQ = 2q2 + α20.The value of α is determined by requiring eαφ to be a (1,1) conformal operator, yieldingthe equation −α2 (α + Q) = 1, whose solutions are labeled by α±α± = −Q2 ± |α0|,α+α−= 2(2.2)the semiclassical limit (c →−∞) fixes α = α+.The gravitationally-dressed tachyon amplitudes are the objects we are interested in:⟨Tk1 · · ·TkN ⟩=* NYj=1Zd2zjeikjX(zj)+β(kj)φ(zj)+(2.3)where we fix the dressing parameter β imposing eikjX+βjφ to be a (1,1) conformal operatorand supposing the space-time energy to be positive:E = β(k) + Q2 = |kj −α0|. (2.4)In the calculation of the amplitudes ⟨Tk1 · · · TkN ⟩the main ingredient is the integrationover the matter (X0) and the Liouville (φ0) zero modes.

This is the so called zero modetechnique: one splits4,5 both, the matter and the Liouville fields as a sum of the zeromode (X0), (φ0) plus fluctuations ( ˜X), (˜φ), where the fluctuations are orthogonal to thezero mode. After such splitting we are left with the following integrals:Z ∞−∞DX0eiX0PNi=1 ki−2α0= 2πδ NXi=1ki −2α0!,Z ∞−∞Dφ0eiφ0PNj=1 βj+Q−eα+φ0µπRd2weα+ ˜φ= Γ(−s)−α+µπZd2weα+ ˜φs,(2.5)4

where we have used that on the sphere18πRd2w√ˆg ˆR = 1 ands = −1α+NXj=1βj + Q. (2.6)We thus obtain for the amplitude⟨Tk1 · · · TkN ⟩= 2πδNXj=1kj −2α0Γ(−s)−α+µπs×* NYj=1Zd2zjeikj+βjφ(zj)Zd2weα+φs+0(2.7)where ⟨· · ·⟩0 means that now the correlation functions are calculated as in the free theory(µ = 0).

The strategy to obtain AN is to assume first that s is a non-negative integer andto continue the result to any real s at the end. Thus, using free propagators⟨X(w)X(z)⟩0 = ⟨φ(w)φ(z)⟩0 = ln |w −z|−2(2.8)and fixing the residual SL(2, C) invariance of the conformal gauge on the sphere by choosing(z1 = 0 , z2 = 1 , z3 = ∞), the 3-point function is written as:A3(k1, k2, k3) = Γ(−s)−α+µπs ZsYj=1d2wj|wj|2α|1 −wj|2βsYi

Choosing the kinematicsk1, k3 ≥α0 , k2 < α0 ≤0 (notice that our notation differs from Ref. [9] by the exchangeof k2 and k3.) we can eliminate β using (2.4), (2.6) and the momentum conservation, onecan write the 3-point amplitude in a rather compact formA3 = [µ∆(−ρ)]s3Yj=1∆12(β2j −k2j),(2.10)where ∆(x) = Γ(x)/Γ(1 −x).

After redefinitions of the cosmological constant and of theexternal fields asµ →µ∆(−ρ),Tkj →Tkj∆ 12(β2j −k2j ),(2.11)Di Francesco and Kutasov9 obtained for the three-point functionA3 = µs,(2.12)5

which is also obtained in the matrix model approach. In the following we shall see that asimilar expression holds for general N-point tachyon amplitudes with an arbitrary numberof screening charges.The screening charges are introduced in the form of n operators eid+X and m operatorseid−X, with d± solutions of:12d(d −2α0) = 1 , (d+d−= −α+α−= −2).

Integrating overthe zero-modes again we get:*Tk1Tk2Tk3 1n!nYi=1Zd2tieid+X(ti)! 1m!mYi=1Zd2rieid−X(ri)!+= 2πδ(3Xi=1ki + nd+ + md−−2α0)Anm3(k1, k2, k3)(2.13)where the amplitude Anm3(k1, k2, k3) is given by the expressionAnm3(k1, k2, k3) = Γ(−s)−α+µπsnYi=1Zd2ti|ti|2˜α|1 −ti|2 ˜βnYi

(2.14)The parameters α, β and ρ are defined as before, and the remaining parameters are˜α = d+k1,˜β = d+k2,˜ρ = 12d2+˜α′ = d−k1,˜β′ = d−k2,˜ρ′ = 12d2−. (2.15)Notice that the gravitational part of the amplitude (integrals over zi) is the same asin the case without screening charges.

The integrals over ti and rj (matter contributions)have been calculated by Dotsenko and Fatteev16 (See their formula (B.10) in the secondpaper of [16]); the result turns out to beAnm3=µπsΓ(−s)Γ(s+1)πs+n+m˜ρ−4nm[∆(1−˜ρ)]n[∆(1−˜ρ′]mmYi=1∆(i˜ρ′−n)nYi=1∆(i˜ρ)×m−1Yi=0∆(1 −n + ˜α′ + i˜ρ′)∆(1 −n + ˜β′ + i˜ρ′)∆(−1 + n −˜α′ −˜β′ −(n −1 + i)˜ρ′)×n−1Yi=0∆(1 + ˜α + i˜ρ)∆(1 + ˜β + i˜ρ)∆(−1 + 2m −˜α −˜β −(n −1 + i)˜ρ)× [∆(1 −ρ)]ssYi=1∆(iρ)s−1Yi=0∆(1 + α + iρ)∆(1 + β + iρ)∆(−1−α−β−(s−1+i)ρ)(2.16)6

Assuming α0 < 0, and the same kinematics (k1 , k3 ≥α0 , k2 < α0) we can eliminateβ again, using (2.4), (2.6) and momentum conservation. In this way, we have˜α = α −2ρ,˜α′ = −2 + ˜ραβ = −1 −m −(s + n)ρ,˜β = m −1 + (s + n)ρ,˜β′ = s + n + ρ−1(m −1)˜ρ = −ρ,˜ρ′ = −ρ−1.

(2.17)Substituting in (2.16) we obtain the three point function with arbitrary screeningsAnm3=µπsΓ(−s)Γ(s + 1)πs+n+m(˜ρ)−4nm ∆(1 + ρ−1)m [∆(1 + ρ)]n×mYi=1∆(iρ−1 −n)nYi=1∆(−iρ) [∆(1 −ρ)]ssYi=1∆(iρ)×n−1Yi=0∆(m + (s + n −i)ρ)s−1Yi=0∆(−m −(s + n −i)ρ)×m−1Yi=0∆(1 + s + (m −1 −i)ρ−1)×m−1Yi=0∆(−1 −n + ρ−1α −iρ−1)∆(1 −s −ρ−1α + iρ−1)×n−1Yi=0∆(1 + α −(i + 2)ρ)∆(m −α −(s −1 −i)ρ)×s−1Yi=0∆(1 + α + iρ)∆(m −α + (n + 1 −i)ρ). (2.18)To get a simpler expression for Anm3we look for the term ∆(ρ−α)∆(ρ(s−n+1)+α−m+1)×∆(−mρ−1−(s+n)) which corresponds to Q3i=1 ∆( 12(β2i −k2i )).

After algebraic manipulationswe getAnm3= [µ∆(−ρ)]s −π∆(ρ−1)m [−π∆(ρ)]n3Yi=1(−π)∆12(β2i −k2i ). (2.19)This result has been also obtained by Di Francesco and Kutasov9,20, as well as Aokiand D’Hoker14.

Note that the factors ∆(ρ) and ∆(ρ−1) can be easily understood; thescreening operators are renormalized like the tachyon vertex operators Tk with vanishingdressing β(k).Remembering the momentum conservation law:3Xi=1ki = (1 −n)d+ + (1 −m)d−,(2.20)7

we can use (2.19) with n = m = 1 and the formula below to obtain the partition functionZ:∂3Z∂µ3 = A113 (ki →0)(2.21)thus,Z = ∆(ρ)∆(ρ−1)ρ3[µ∆(−ρ)]1−ρ−1(ρ −1)(ρ + 1). (2.22)We can also get the two-point function by taking, e.g., k3 →0 in (2.19) and (2.21):A˜n ˜m3(k1, k2, k3 →0) = ∂∂µA˜n ˜m2(k1, k2),(2.23)where (˜n, ˜m) are fixed by (2.20) with k3 = 0.

Thus we finally arrive atA˜n ˜m2(k1, k2) = [µ∆(−ρ)]ρ−1 α+2P2i=1 βi+ρ−1ρ−1α+2P2i=1 βi + ρ −1∆(ρ−1) ˜m [∆(ρ)]˜n2Yi=1∆(β2i −k2i )2(2.24)We are able now to calculate ratios of correlation functions to compare with otherresults in the literature. We have (in a generic kinematic region):R =Anm3(k1, k2k3)ZA˜n1, ˜m12(k1, k1)A˜n2, ˜m22(k2, k2)A˜n3, ˜m32(k3, k3)=Q3i=1 α+|ki −α0|(ρ −1)(1 + ρ)(2.25)In the case of minimal models the momenta assume the following form:krir′i = (1 −ri)2d+ + (1 −r′i)2d−.

(2.26)Plugging back in (2.25) we find for the ratio R the resultR =⟨Tr1r′1Tr2r′2Tr3r′3⟩2Z⟨Tr1r′1Tr1r′1⟩⟨Tr2r′2Tr2r′2⟩⟨Tr3r′3Tr3r′3⟩=α+23 Q3i=1 |rid+ + r′id−|(ρ −1)(ρ + 1)(2.27)That is exactly the same result obtained by Dotsenko10 (in a given kinematic region)which reduces to the result of Goulian and Li for ri = r′i; all these results agree with thoseobtained by the matrix model approach.At this point the following remark is in order.After renormalizing the screeningcharges:eid+X →eid−X∆(ρ),eid−X →eid−X∆(ρ−1),(2.28)and the tachyon operators Tki as well as the cosmological constant µ as before (see (2.11))we get the renormalized amplitude.Anm3= µs. (2.29)8

Using the above result we would be able to exactly reproduce the ratio in (2.27).Thus, the comparison of those ratios with matrix models is not a precise test and onlymeasures the scalling of the amplitudes w.r.t. the cosmological constant.

All singularitiescontained in the ∆functions cancel out in such ratios. If on the other hand compare,the 3-point function (2.19) directly with the matrix model result we would find a preciseagreement only for c = 1 (α0 = 0) where the amplitude can be written as a function of therenormalized cosmological constant as:Anm3∼(˜µ)s3Yi=1Γ(1 −√2|ki|).

(2.30)For c < 1 the amplitude obtained via matrix model are finite and the singularitiescontained in (2.19) are not observed. Neither do the fusion rules for minimal models (seediscussion in [27,28]) appear in (2.19), although they can be seen via matrix models.We now generalize the cases known in the literature for N > 3 including screen-ing charges.

Repeating the zero-mode technique in the most general case of an N-pointfunction with arbitrary screening charges we haveAnmN= (−π)3 µπsΓ(−s)NYi=1Zd2zinYj=1Z d2tjn!mYk=1Z d2rkm!×sYl=1Zd2wlDeikiX(zi)eid+X(tj)eid−X(rn)E0Deβiφ(zi)eα+φ(wl)E0,(2.31)where s = −1α+ (PNi=1 βi + Q) and a factor π3/α+ has been absorbed in the measure ofthe path integral. Fixing the SL(2, C) symmetry we get:AnmN= (−π)3 µπsΓ(−s)InmN,InmN=ZNYj=4d2zj|zj|2αj|1 −zj|2βjNYi

whereαj = k1kj −β1βj,˜αj = d+kjβj = k2kj −β2βj,˜αj = d−kjρlj = 12(klkj −βlβj), pj = −α+βj,4 ≤j, l ≤N. (2.33)The integral above has been calculated by Di Francesco and Kutasov9, for the casen = m = 0.

For arbitrary n, m we shall use the same technique. Notice that translationinvariancewi →1 −wi , zi →1 −zi , ti →1 −ti , ri →1 −riimplies the symmetry relationsα ↔β , αj ↔βj , ˜α ↔˜β , ˜α′ ↔˜β′so that after the elimination of the remaining parameters as a function of α, β, pj and ρ(j = 4, 5, · · ·, N −1), InmNexhibits an α-β symmetryInmN (α, β, pj, ρ) = InmN (β, α, pj, ρ).

(2.34)Similarly by the inversion of all variables wi, zi, ti, ri we have:InmN (α, β, pj, ρ) = InmN (−2 −α −β −2ρ(s −1) −pN −P, β, pj, ρ)(2.35)where P = PN−1j=4 pj. Further information about InmNcan be obtained in the limit α →∞(or β →∞), by using a technique applied by Dotsenko and Fatteev16 in the case of contourintegrals, we foundInmN≈α2β+2ρ(s−N−n+3)+2P −2m(2.36)where we have used the kinematics: k1, k2, · · ·, kN−1 ≥α0 , kN < α0 and assumed α0 < 0To eliminate most of the parameters as a function of α, β, pj and ρ we use (2.4), (2.6) andmomentum conservation.

After such elimination the symmetry (2.25) becomes:InmN (α, β, pj, ρ) = InmN (m −1 −P −α −β + ρ(N + n −1 −s), β, pj, ρ)(2.37)Using Stirling’s formula, it is not difficult to check that the following Ansatz is consistentwith (2.34), (2.36) and (2.37):AnmN= f nmN (ρ, pj)∆(ρ−α)∆(ρ−β)∆(1−m+P +α+β+ρ(s+2−N −n))AnmN= f nmN (ρ, pj)3Yj=1∆12(β2j −k2j ). (2.38)Now we can fix f nmN (ρ, pj) by using the 3-point function Anm3.AnmN (k1, k2, kj →0, kN) = (−π)N−3 ∂∂µN−3Anm3(k1, k2, kN),3 ≤j ≤N −1.

(2.39)10

Now using the result for Anm3we get:f nmN (ρ, pj)=[−π∆(ρ−1)]m[−π∆(ρ)]n∂N−3∂µµs+N−3[∆(−ρ)]sNYj=4(−π)∆(12(β2j −k2j )). (2.40)We finally return to (2.38) and obtainAnmN= (s + N −3)(s + N −4) · · ·(s + 1) [µ∆(−ρ)]s−π∆(ρ−1)m [−π∆(ρ)]nNYj=1(−π)∆(12(β2j −k2j)),(2.41)therefore, redefining the screening operators, Tkj and µ as before, we have:AnmN= ∂N−3∂µ µs+N−3(2.42)which is a remarkable result; however, it is valid only in the kinematic region alreadymentioned.

In order to extend for general k, we can use the same technique as used in[20]. Notice that the amplitude (2.41) factorizes as in the case without screening charges.3- Supersymmetric CorrelatorsIn a recent paper19 we have calculated the 3- and 4-point NS correlations functionsusing DHK formulation17 of super Liouville theory coupled to superconformal matter onthe sphere without screening charges.

The total action S is given by the sum of the superLiouville action SSL and the matter piece SM,SSL = 14πZd2z ˆE12ˆDαΦSL ˆDαΦSL −Q ˆY ΦSL −4iµeα+ΦSL,SM = 14πZd2z ˆE(12ˆDαΦM ˆDαΦM + 2iα0 ˆY ΦM),(3.1)where ΦSL, ΦM are super Liouville and matter superfields respectively. The central chargeof the matter sector is c =32ˆc , (ˆc = 1 −8α20).Analogously to the bosonic case theparameters Q and α± are given by (compare with (2.2))Q = 2q1 + α20,α± = −Q2 ± 12pQ2 −4 = −Q2 ± |α0|,α+α−= 1.

(3.2)We shall call ˜ΨNS the gravitationally dressed primary superfields, whose form is given by˜ΨNS(zi, ki) = d2z ˆEeikΦM (z)eβ(k)ΦSL(z), whereβ(k) = −Q2 + |k −α0|. (3.3)11

The calculation of the three-point function of the primary superfield ˜ΨNS, involvesthe expression:* 3Yi=1Z˜ΨNS(zi, ki)+≡Z[D ˆEΦSL][D ˆEΦM]3Yi=1˜ΨNS(zi, ki)e−S. (3.4)We closely follow the method already used in the bosonic case.

After integrating overthe bosonic zero modes we get* 3Yi=1Z˜ΨNS(zi, ki)+≡2πδ 3Xi=1ki −2α0!A3(k1, k2, k3),A(k1, k2, k3) = Γ(−s)(−π2 )3(iµπ )s*Z3Yi=1d2˜zieikiΦM (˜zi)eβiΦSL(˜zi)Zd2zeα+ΦSL(z)s+0(3.5)where ⟨· · ·⟩0 denotes again the expectation value evaluated in the free theory (µ = 0) andwe have absorbed the factor [α+(−π/2)3]−1 into the normalization of the path integral.the parameter s is defined as in the bosonic case (see (2.6)).For s non-negative integer, after fixing the cSL2 gauge,30 ˜z1 = 0 , ˜z2 = 1 , ˜z3 = ∞, ˜θ2 =˜θ3 = 0 , ˜θ1 = θ, in components (ΦSL = φ + θψ + ¯θ ¯ψ) (the integral above is the supersym-metric generalization of (B.9) of Ref. [16]) we haveA(k1, k2, k3) = Γ(−s)(−π2 )3(iα2+µπ)sβ21×ZsYi=1d2zisYi=1|zi|−2α+β1|1−zi|−2α+β2sYi

(3.6)Observe that this is non-vanishing only for s odd (s = 2l+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:A3(k1, k2, k3) = Γ(−s)(−π2 )3 1α2+(iα2+µπ)sα2(−1)s+12 s!

!×ZsYi=1d2zisYi=1|zi|2α|1 −zi|2βsYi

whereIl(α, β; ρ) =12ll!α2Zd2wlYi=1d2ζid2ηi|w|2α−2|1 −w|2βlYi=1|w −ζi|4ρ|w −ηi|4ρ×lYi=1|ζi|2α|1 −ζi|2β|1 −ηi|2βlYi,j|ζi −ηj|4ρlYi

[19] we calculated Il in detail by using the sym-metries Il(α, β; ρ) = Il(β, α; ρ), Il(α, β; ρ) = Il(−1 −α −β −4lρ, β; ρ) and looking at itslarge α behavior we obtained:Il(α, β; ρ) = −π2l+122l∆12 −ρ2l+1lYi=1∆(2iρ)lYi=1∆12 + (2i + 1)ρ×lYi=0∆(1 + α + 2iρ)∆(1 + β + 2iρ)∆(−α −β + (2i −4l)ρ)×lYi=1∆(12 + α + (2i −1)ρ)∆(12 +β+(2i−1)ρ)∆(−12 −α−β+(2i−4l−1)ρ)(3.9)We can choose, k1, k3 ≥α0, k2 ≤α0. We proceed now as in the bosonic case, obtainingfor the parameter β,β = ρ(1 −s) (α0 > 0)−12 −ρs (α0 < 0).

(3.10)Now we are ready to write down the amplitude. For α0 < 0 we have the non-trivialamplitude:A(k1, k2, k3) = (−iπ2 )3µ2 ∆12 −ρs∆12 −s2∆(1 + α −(s −1)ρ) ∆12 −α + ρ=µ2 ∆12 −ρs3Yj=1(−iπ2 )∆12[1 + β2j −k2j ](3.11)In the case ˆc = 1, we obtain for the external legs, renormalization factors of theform ∆(1 −|ki|), which should be compared to the bosonic case (2.30); it permits as wellcomparison to super matrix model as well, whenever those are available.By redefining the cosmological constant and the primary superfield ˜ΨNSµ →2∆ 12 −ρµ,˜ΨNS(kj) →1(−i2π)∆ 12[1 + β2j −k2j] ˜ΨNS(kj),(3.12)we getA3(k1, k2, k3) = µs.

(3.13)13

As in the bosonic case we have a remarkably simple result.The only differences withrespect to the bosonic case are in the details of the renormalization factors. Compare(3.12) with (2.11).

Note that the singular point at the renormalization of the cosmologicalconstant is ρ = −1 in the bosonic case, which corresponds to c = 1, and ρ = −12 in thesupersymmetric case, corresponding to ˆc = 1 or c = 3/2.We shall now generalize the above result to the case which includes screening chargesin the supermatter sector. We consider n charges eid+ΦM and m charges eid−ΦM , whered± are solutions of the equation 12d(d −2α0) = 12.

After integrating over the matter andLiouville zero modes we get* 3Yi=1Z˜ΨNS(˜zi, ki)nYi=1Z d2tin! eid+ΦM(ti)mYi=1Z d2rim!

eid−ΦM(ri)+≡2πδ 3Xi=1ki + nd+ + md−−2α0!Anm3(k1, k2, k3)Anm3(k1, k2, k3) = Γ(−s)(−π2 )3(iµπ )s* nYi=1Z d2tin! eid+ΦM (ti)mYi=1Z d2rim!

eid−ΦM(ri)×Z3Yi=1d2˜zieikiΦM (˜zi)eβiΦSL(˜zi)Zd2zeα+ΦSL(z)s+0,(3.14)Integrating over the Grasmann variables and fixing thedSL(2) symmetry as before(˜z1 = 0 , ˜z2 = 1 , ˜z3 = ∞, ˜θ1 = θ , ˜θ2 = ˜θ3 = 0) we obtain (using d+d−= −α+α−= −1)Anm3(k1, k2, k3) = Γ(−s)−π23 iµα2+πs (−d2+)nn! (−d2−)mm!×nYi=1Zd2ti|ti|−2d+k1|1 −ti|−d+k2nYi

(3.15)Since the vacuum expectation value of an odd number of ψψ (or ξξ ) operators is zerowe have only two non-trivial cases: in the first case n + m = odd, s = even and in the14

second one n + m = even, s = odd. Thus we have (see also [21] for comparison)Anm3(k1, k2, k3) = Γ(−s)−π23 iµα2+πs (−d2+)nn!

(−d2−)mm!×InmM (˜α, ˜β; ρ) × IsG(α, β; ρ) , n + m = even , s = oddJnmM (˜α, ˜β; ρ) × JsG(α, β; ρ) , n + m = odd , s = even(3.16)whereInmM (˜α, ˜β; ρ) =nYi=1Zd2ti|ti|2˜α|1 −ti|2 ˜βnYi

Henceforth we assume, for simplicity, n + m = even, s = odd. We willwork out explicitly only the case n, m even.

However, the final result for the amplitudedoes not depend on which case we choose. In ref.

[24] we have calculated InmMfor n and15

m even and we get:InmM (˜α, ˜β; ρ) = (−)n+m2πn+m2n+m n!m!−ρ2−2nm ∆12 −ρ2n ∆12 −ρ′2m×n2Y1∆(iρ)∆12 + ρi −12m2Y1∆(iρ′ −n2 )∆12 −n2 −ρ′i −12×n2 −1Yi=0∆(1 + ˜α + iρ)∆(1 + ˜β + iρ)∆(m −˜α −˜β + ρ(i −n + 1))×n2Yi=1∆(12 +˜α+(i−12)ρ)∆(12 + ˜β+(i−12)ρ)∆(−12 −˜α+m−˜β+ρ(i−n+ 12))×m2 −1Yi=0∆(1 + ˜α′ −n2 + iρ′)∆(1 −n2 + ˜β′ + iρ′)∆(n2 −˜α′ −˜β′ + ρ′(i −m + 1))×m2Yi=1∆(12 −n2 +˜α′+(i−12)ρ′)∆(12 −n2 + ˜β′+(i−12)ρ′)∆(−12 + n2 −˜α′−˜β′+ρ′(i−m+ 12))(3.19)In the case where s = 2l + 1 the gravitational contribution to Anm3(k1, k2, k3), i.e, IsG isjust the same as in the case without screening charges, thus from the last section we havethe supersymmetric generalization of (B.9) of ref. [16]:IsG = (−)s−12πs2s−1 s!∆12 −ρss−12Yi=1∆(2iρ)s−12Yi=0∆(12 + (2i + 1)ρ)×s−12Yi=0∆(1 + ˜α + 2i˜ρ)∆(1 + ˜β + 2iρ)∆(−˜α −˜β + 2ρ(i −s + 1))×s−12Yi=0∆(12 + ˜α + (2i −1)ρ)∆(12 + ˜β + (2i −1)ρ)∆(−12 −˜α −˜β + ρ(2i −2s + 1))(3.20)To obtain Anm3(k1, k2, k3) (see (3.16)) we have to calculate Inm × IsG.

Using the samekinematics as in the case without screening charges, and considering that α0 < 0 it is easy16

to deduce:˜α = α −2ρ,˜α′ = −1 + ρ−1α2ρ = −2ρ,ρ′ = −ρ−12β = −12 −m2 −(n + s)ρ˜β = −β −1 = (n + 1)ρ + m2 −12˜β′ = (n + s)2+ ρ−14 (m −1). (3.21)Substituting in (3.19) and (3.20) and using (3.16) we obtain a very involved expression17

(see also [26,21]):Anm3(k1, k2, k3)= Γ(−s)−π23 iµπsα2(s−1)+(2ρ)n−m(−)n+m+s−12πs+n+m2m+n+s−1 ρ−2mns!×∆(12 + ρ)n ∆(12 + ρ−14 )m×n2Yi=1∆(−2iρ)∆(12 + ρ(1 −2i))m2Yi=1∆(−n2 −iρ−12)∆12 −n2 +14 −i2ρ−1×n2 −1Yi=0∆12 + m2 + ρ(n + s −2i)n2Yi=1∆m2 + ρ(n + s + 1 −2i)×m2 −1Yi=0∆1 + s2 −ρ−12 (i + 12 −n2 )m2Yi=1∆12 + s2 −ρ−12 (i −m2 )×∆(12 −ρ)ss−12Yi=1∆(2iρ)s−12Yi=0∆12 + (2i + 1)ρ×s−12Yi=0∆(12 −m2 + (2i −n −s)ρ)s−12Yi=1∆(−m2 + (2i −1 −n −s)ρ)×n2Yi=1∆(12 + α −ρ(2i + 1))∆(12 + m2 −α −ρ(s −n −2 + 2i))×s−12Yi=0∆(m2 + 12 −α + (2i −s + n + 2)ρ)s−12Yi=1∆(12 + α + (2i −1)ρ)×n2Yi=1∆(m2 −α −ρ(2i −n + s −1))∆(1 + α −2iρ)×s−12Yi=0∆(1 + α + 2iρ)s−12Yi=1∆(m2 −α + (2i −s + n + 1)ρ)×m2Yi=1∆(12 −s2 −ρ−12 (i + α −m2 ))∆(−n2 −ρ−12 (i −1 −α))×m2Yi=1∆(−n2 −12 −ρ−12 (i −α −12))∆(1 −s2 −ρ−12 (i −(m + 1)2+ α)). (3.22)In order to obtain a simple expression for the amplitude we have to combine in eachterm the matter and the gravitational parts as in the bosonic case.

The calculation is more18

complicate now, but we finally getAnm3(k1, k2, k3) =−π23 µ2 ∆(12 −ρ)s −iπ2 ∆(12 + ρ−14 )m −iπ2 ∆(12 + ρ)n× ∆ρ −α + 12∆12 −n + s2−mρ−14∆(1 −m2 + α + (s −n −1)ρ)=µ2 ∆12 −ρs −iπ2 ∆12 + ρ−14m −iπ2 ∆12 + ρn×3Yi=1−iπ2∆12 + 12(β2i −k2i ). (3.23)Therefore after redefining the cosmological constant, the NS operators and the screeningchargeseid+ΦM(ti) →−iπ2 ∆12 + ρ−1eid+ΦM(ti)(3.24a)eid−ΦM(ti) →−iπ2 ∆12 + ρ−14−1eid−ΦM (ti)(3.24b)ΨNS →−iπ2 ∆12 + 12(β2i −k2i )−1ΨNS(3.24c)µ →12∆12 −ρ−1µ.

(3.24d)we obtain the very simple result:Anm3(k1, k2, k3) = µs. (3.25)In view of the complexity of (3.22), the simplicity of the result is remarkable.As in the bosonic case we can calculate ratios of correlation functions either using(3.23) or (3.25).

We obtain in a generic kinematic region:R =⟨ΨNS(kr1r′1)ΨNS(kr2r′2)ΨNS(kr3r′3)⟩2ZQ3j=1⟨ΨNS(krjr′j)ΨNS(krjr′j)⟩(3.26)R = (2α+)3Q3i=1 |rid−+ r′id+|(2ρ −1)(2ρ + 1). (3.27)Compare with (2.27).The above result agrees with other results26,21 simultaneously obtained in the litera-ture.

Although the the continuations to non integer values of s used in [26] and [21] arenot the same and, in principle, do not correspond to the procedure used here, the physicalresults seem to be independent of such details.19

We now show that it is possible to obtain a simple result for the most general case of aN-point amplitude with an arbitrary number of screening charges (AnmN ). In that generalcase to compute the amplitudes we have to calculate the following integralAnmN (k1, · · ·, kN) = Γ(−s)−α+iµπs * NYi=1Zd2˜zieikiΦM(˜zi)+βiΦSL(˜zi)×nYi=1Zd2tieid∨ΦM(˜ti)mYj=1Zd2rjeid−ΦM(˜rj)sYj=1Zd2zjeα+ΦSL(˜zi)+0(3.28)where s = −1α+ (PNi=1 βi + Q) and PNi=1 ki + nd+ + md−= 2α0.

After fixing the dSL2symmetry as before and integrating over the Grassmann variables the amplitude becomesAnmN= Γ(−s)−π23 iµα2+πs(−d2+)n(−d2−)m×NYj=4Zd2˜zjnYi=1Zd2timYi=1Zd2risYi=1d2wi|wi|−2α+β1|1 −wi|−2α+β2×Yi

×nYi=1|ti|2˜α|1 −ti|2 ˜βnYi

We arrive at the resultAnmN= (s + N −3)(s + N −4) · · ·(s + 1)µ∆(12 −ρ)s×−iπ2 ∆(12 + ρ)n−iπ2 ∆(12 + ρ−14 )m NYi=1−iπ2N∆(12(1 + β2i −k2i ))(3.34)Redefining ΨNS , µ and the screening charges we have our final result:AnmN= ∂∂µN−3µs+N−3,(3.35)which has the same functional form as the bosonic amplitude (2.42). As in that case theabove result is correct only in the kinematic region used to calculate it and has to becontinued outside this region.21

4- ConclusionIn the first part of this paper we have generalized previous results9 for the N-pointtachyon correlator in Liouville theory coupled to conformal matter (on the sphere) withc ≤1 to the case which includes screening charges in the matter sector.The resultsmight be useful in understanding the issue of fusion rules in the calculation of the 3-pointcorrelator (see discussion in [20]).In the second part we have obtained the N-point NS-correlators in super Liouvillecoupled to ˆc ≤1 matter (also on the sphere), including screening charges thus generalizingthe results of [21,26] to the limit case ˆc = 1, (N > 3), and the results of [20], obtained fors = 0, to any value of s.We have obtained, explicit formulae for the corresponding 2D-integrals involved andthe final form of the amplitudes factorizes in the N-external legs factors, confirming theresults obtained in [20] (for s = 0) through a detailed analysis of the pole structure of theintegrals.In our calculations it was possible to see the singularity in the renormalization of thecosmological constant (µ →˜µ/∆( 12 −ρ)) at the point ˆc = 1 (ρ = −1/2). This is similarto the bosonic case where µ →˜µ/∆(−ρ) and the c = 1 , (ρ = −1) point is also singular.The final (renormalized) N-point amplitude has the same form of the bosonic one.

Thesimilarity with the bosonic case has been found before in the discrete approach [20,25].Finally we should stress that our results must be continued to other kinematic regions,this is likely to be very similar to the bosonic case (without s.c.) worked out in [20].As a further development it would be interesting to carry out analogous calculationsin the case with N = 2 supersymmetry and see, among other aspects, whether the barrierat c = 3 indeed disappears. The inclusion of the Ramond sector (for s ̸= 0) is also ofinterest.AcknowledgmentsThe work of K.H.

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