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General Neveu-Schwarz Correlators in Super Liouville Theory

arXiv:hep-th/9111057v1 27 Nov 1991General Neveu-Schwarz Correlators in Super Liouville 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, JapanAbstractIn this paper we compute the N-point correlation functions of the tachyon operatorfrom the Neveu Schwarz sector of super Liouville theory coupled to matter fields (withˆc ≤1) in the super Coulomb gas formulation, on world sheets with spherical topology.We first integrate over the zero mode assuming that the s parameter takes an integervalue, subsequently we continue the parameter to an arbitrary real number. We includedan arbitrary number of screening charges (s.c.) and as a result, after renormalizing thes.c., the external legs and the cosmological constant, the form of the final amplitudesdo not modify.

Remarkably, the result is completely parallel to the bosonic case.Wealso completed a discussion on the calculation of bosonic correlators including arbitraryscreening charges.1

1- IntroductionMatrix models proved to be an efficient means to obtain information about non-criticalstring theory, specially in computations of correlation functions1. However, supersymmet-ric extensions2 are, up to now, not constructed, due to some technical difficulties.On the other hand, in the continuum approach, we have to deal with Liouville theory3.Unfortunately, we still do not understand it very well in spite of much effort.

In particular,it has been difficult to calculate correlation functions in a reliable way because perturba-tion theory does not apply. Recently, however, several authors4−10 succeeded in taming thedifficulties of the Liouville theory and computed exactly correlation functions in the con-tinuum approach to conformal fields coupled to two dimensional gravity.

The technique isbased on the integration over the zero mode of the Liouville field. The resulting amplitudeis a function of a parameter s which depends on the central charge and on the externalmomenta.

The amplitudes can be computed when the above parameter is a non-negativeinteger. Later on, one analytically continues that parameter to real (or complex) values.The results thus obtained agree with the matrix model approach.An advantage of the Liouville approach is that it is very easy to extend it to supersym-metric theories12.

Since matrix models seem to be less powerful (up to now) in this point,it is natural to investigate supersymmetric Liouville theory in order to know somethingabout 2-D supergravity (or off-critical NSR string).Our aim here is to investigate the supersymmetric Liouville theory. We shall computesupersymmetric correlation functions on world sheets with spherical topology in the Neveu-Schwarz sector, where the super-Liouville is coupled to superconformal matter with centralcharge ˆc ≤1, represented as a super Coulomb gas12,13.

The results are remarkable, andvery parallel to the bosonic case; it is equivalent to a redefinition of the cosmologicalconstant, and of the primary superfields, the resulting amplitudes have the same form asthose of the bosonic theory obtained by Di Francesco and Kutasov5. The super Liouvilletheory14 has also been studied along similar lines.

Our present results generalize thosepresented in a recent paper15, as well as others recently obtained in the literature16.This paper is divided as follows: in section 2 we review the computations of bosoniccorrelators, and include the case with an arbitrary number of screening charges. In section 3we calculate the N-point tachyon (Neveu-Schwarz) correlation functions also with screeningcharges.

Results are remarkably similar to the bosonic case5. In section 4 we draw someconclusions.

In Appendix A, we calculate the supersymmetric generalization of equation(B.10) of Dotsenko and Fatteev.2- Bosonic Correlators2.1- The 3-point tachyon amplitudeIn a recent paper Di Francesco and Kutasov6 calculated the N-point tachyon corre-lation functions in Liouville theory on world sheets with spherical topology, coupled to2

c ≤1 conformal matter in a Coulomb gas representation. They worked in the DDK’sframework17 where the total action is given by:S = 12πZd2wpˆgˆgab∂aφ∂bφ −Q4ˆRφ + 2µeαφ + ˆgab∂aX∂bX + iα02ˆRX,(2.1)where φ is the Liouville mode and X represents the matter with c = 1 −12α20.

FollowingDDK, the constant Q is determined by imposing a vanishing total central charge,Q =r25 −c3= 2q2 + α20,(2.2)and α is determined by requiring eαφ to be a (1,1) conformal operator,i.e., −12α(α+Q) = 1.We define the solutions to this equation by:α± = −Q2 ± |α0|,α+α−= 2;(2.3)the semiclassical limit (c →∞) fixes α = α+.In the following we calculate gravitationally-dressed tachyon amplitudes:⟨Tk1 · · ·TkN ⟩=* NYj=1Zd2zjeikjX(zj)+β(kj)φ(zj)+(2.4)the dressing parameter β is fixed imposing eikjX+βjφ to be a (1,1) conformal operator:βj = β(kj) = −Q2 + |kj −α0|. (2.5)An important ingredient in the calculation of ⟨Tk1 · · ·Tkn⟩is the integration over thematter (X0) and the Liouville (φ0) zero modes.

We make the following split4,5φ = φ0 + ˜φX = X0 + ˜X(2.6)where the fields ˜φ and ˜X obey the conditionZd2w ˜φ =Zd2w ˜X = 0. (2.7)The integration over the zero modes X0 and φ0 gives the following results;Z ∞−∞DX0eiX0PNi=1 ki−2α0= 2πδ(NXi=1ki −2α0),(2.8a)3

Z ∞−∞Dφ0eiφ0PNj=1 βj+Q−eα+φ0µπRd2weα+ ˜φ= Γ(−s)−α+µπZd2weα+ ˜φs,(2.8b)wheres = −1α+NXj=1βj + Q. (2.9)And we have used that on the sphere18πRd2w√ˆg ˆR = 1.

We thus obtain⟨Tk1 · · · TkN ⟩= 2πδNXj=1kj −2α0AN(k1 · · · kN)(2.10a)AN(k1 · · · kN) = Γ(−s)−α+µπs* NYj=1Zd2zjeikj+βjφ(zj)Zd2weα+φs+0(2.10b)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:1)⟨X(w)X(z)⟩0 = ⟨φ(w)φ(z)⟩0 = ln |w −z|−2(2.11)and fixing the residual SL(2C) invariance of the conformal gauge on the sphere by choosing(z1 = 0 , z2 = 1 , z3 = ∞), we have in the case of the 3-point function:A3(k1, k2, k3) = Γ(−s)−α+µπs ZsYj=1d2wj|wj|2α|1 −wj|2βsYi

The above integral hasbeen calculated by Dotsenko and Fatteev12 ( see formula (B.9), of the second paper). Usingtheir result Di Francesco and Kutasov obtained:A3(k1, k2, k3) = Γ(−s)−α+µπsΓ(s + 1)[∆(1 −ρ)]ssYi=1∆(iρ)×s−1Yi=0∆(1 + α + iρ)∆(1 + β + iρ)∆(−1 −α −β −(s −1 + i)ρ)(2.13)where ∆(x) = Γ(x)/Γ(1 −x).

Choosing the kinematics2) k1, k3 ≥α0 , k2 < α0 we caneliminate β using (2.6), (2.9) and the momentum conservation: P3i=1 ki = 2α0:,β =ρ(1 −s),α0 > 0−1 −ρ,α0 < 0(2.14)1) Hereafter we drop the tilde in the fields defined by (2.9), since no confusion can occur.2) Notice that our notation differs from Ref. [6] by the exchange of k2 and k3.4

Back in (2.13) it is easy to see that for α0 > 0 there appears a factor Γ(0) in the denominatorof A3 and the amplitude vanishes identically. For α0 < 0, using12(β21 −k21) = ρ −α,(2.15a)12(β22 −k22) = 1 + α(s −1)ρ,(2.15b)12(β23 −k23) = −s,(2.15c)we can write the 3-point amplitude in a rather compact formA3 = [µ∆(−ρ)]s3Yj=1∆12(β2j −k2j).

(2.16)Thus, after the redefinitions of the cosmological constant and of the external fields asµ →µ∆(−ρ),Tkj →Tkj∆ 12(β2j −k2j ),(2.17)Di Francesco and Kutasov6 obtained for the three-point functionA3 = µs,(2.18)which is also obtained in the matrix model approach. In the next sub-sections we shallsee that this expression holds for general N-point tachyon amplitudes with an arbitrarynumber of s.c..2.2- The 3-point tachyon amplitude with an arbitrary number of screeningchargesNow we show explicitly how one generalizes the previous calculation to the case whichincludes an arbitrary number of screening charges in the matter sector.

we introduce noperators eid+X and m operators eid−X, with d± solutions of:12d(d −2α0) = 1 , (d+d−=−α+α−= −2). Integrating over the zero-modes again we get:*Tk1Tk2Tk3 1n!nYi=1Zd2tieid+X(ti)!

1m!mYi=1Zd2rieid−X(ri)!+=2πδ(Xi=1ki + nd+ + md−−2α0)Anm3(k1, k2, k3)(2.19)5

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 01 −m −(s + n)ρ.α0 < 0It is easy to see, assuming s ≥m + 2, that for α0 > 0 the amplitude vanishes again due toa factor Γ(−n) in the denominator in the gravitational part of the amplitude.

Thereforewe concentrate now on the α0 < 0 case where we have˜α = α −2ρ,˜α′ = −2 + ˜ρα˜β = m −1 + (s + n)ρ,˜β′ = s + n + ρ−1(m −1)˜ρ = −ρ,˜ρ′ = −ρ−1(2.23)6

Substituting in (2.22) we obtain:Anm3=µπsΓ(−s)Γ(s + 1)πs+n+m(˜ρ)−4nm(CM+GDMEM+G)(2.24)whereCM+G =∆(1 + ρ−1)m [∆(1 + ρ)]nmYi=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)(2.25a)DM =m−1Yi=0∆(−1 −n + ρ−1α −iρ−1)∆(1 −s −ρ−1α + iρ−1)(2.25b)EM+G =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)(2.25c)To get a simple expression for Anm3we look for ∆(ρ−α)∆(ρ(s−n+1)+α−m+1)∆(−mρ−1−(s+n)) which corresponds to Q3i=1 ∆( 12(β2i −k2i )). We expect that these terms show upin the result.

For example:EM+G =n+1Yi=2∆(1+α−iρ)s−1Yi=0∆(1+α+iρ)s−1Yi=s−n∆(m−α−iρ)s−n−2Yi=−(n+1)∆(m−α−iρ) (2.25d)Using ∆(x)∆(1 −x) = 1, and∆(x + 1) = −x2∆(x), we easily get:EM+G = (−)m(s+n+1)∆(1 −m + α + (s −n + 1)ρ)∆(ρ −α)×n+1Y1−s(m −1 −α + iρ)2(m −2 −α + iρ) · · ·(−α + iρ)2(2.26a)Analogously, we also arrive atDM = (−)m(n+s+1)ρ2m(n+s+1)n+1Y1−s Γ(−α + iρ)Γ(m −α + iρ)2(2.26b)CM+G = (−)sρ−2(s+n)+2m(n−s)Γ(−s)Γ(s + 1)∆(1 + ρ−1)m [∆(1 + ρ)]n [∆(1 −ρ)]s(2.26c)7

Now substituting (2.26) into (2.24) we haveAnm3= [µ∆(−ρ)]s −π∆(ρ−1)m [−π∆(ρ)]n3Yi=1(−π)∆12(β2i −k2i ). (2.27)Therefore redefining the screening operators aseid+X →1∆(ρ)eid+X(2.29a)eid−X →1∆(ρ−1)eid−X,(2.29b)the operators Tki and the cosmological constant µ as before (see (2.17)), we get the verysimple result:Anm3= µs,(2.29)which should be compared to (2.18).

This result has been also obtained by Di Francescoand Kutasov6,16 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).2.3- N-Point tachyon amplitude (N ≥4) with an arbitrary number of screeningchargesRepeating the zero-mode trick in the most general case of an N-point function witharbitrary screening charges we have3):AnmN= (−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,(2.30)3) We have absorbed a factor π3/α+ in the measure of the path integral.8

where s = −1α+ (PNi=1 βi + Q). Fixing the SL(2, C) symmetry we get:AnmN= (−π)3 µπsΓ(−s)InmN,InmN=ZNYj=4d2zj|zj|2α′j|1 −zj|2βjNYi

(2.32)The integral InmNfor the case n = m = 0 has been calculated by Di Francesco andKutasov6.We shall use the same technique for arbitrary n, m.First we notice thattranslation invariance (wi →1 −wi , zi →1 −zi , ti →1 −ti , ri →1 −ri) implies(α ↔β , α′j ↔β′j , ˜α ↔˜β , ˜α′ ↔˜β′) so after the elimination of the remaining parametersas 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). (2.33)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)(2.34)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 Fatteev12 in the case of contourintegrals. Take for instance the simple case:I(α, β) =Zd2w|w|2α|1 −w|2β = π∆(1 + α)∆(1 + β)∆(−1 −α −β)(2.35)9

by making a change of variables w →e−˜wαw∗→e−˜w∗αwe have :I(α →∞, β) ≈α−2−2β ˜I(β). (2.36)This large-α behaviour can be checked by using Stirling’s formula (Γ(α+B) ∼αBΓ(α))on the r.h.s.

of equation (2.35). Applying this technique to InmNwe get:InmN≈α2β+2ρ(s−N−n+3)+2P −2m(2.37)where we have used the kinematics:k1, k2, · · ·, kN−1 ≥α0 , kN < α0(2.38)and assumed α0 < 0, which permits us to eliminate the remaining parameters in terms ofα, β, pj and ρ as follows:pN = −1 −m −ρ(N + s + n −3)β′j = β + pj −2ρβ′N = m −1 + (ρ −β)(N + s + n −3) −mρ−1βρ′jl = 12(pj + pl) −ρρ′jN = m −12+ (ρ −pj)2(N + s + n −3) −mρ−12pj,˜α = α −2ρ,˜α′ = αρ−1 −2,˜ρ = −ρ,˜β = β −2ρ,˜β′ = βρ−1 −2,˜ρ′ = −ρ−1˜αj = pj −2ρ,˜α′j = ρ−1pj −2,˜αN = m −1 + ρ(N + s + n −3),˜α′N = N + s + n −3 + ρ−1(m −1).

(2.39)where 4 ≤j, l ≤N −1. Notice that eliminating pN the symmetry under inversion implies:InmN (α, β, pj, ρ) = InmN (m −1 −P −α −β + ρ(N + n −1 −s), β, pj, ρ)(2.40)It is not difficult to check (using Stirling’s formula) that the following Ansatz is consistentwith (2.33), (2.37) and (2.40):AnmN= f nmN (ρ, pj)∆(ρ−α)∆(ρ−β)∆(1−m+P +α+β+ρ(s+2−N −n))(2.41a)AnmN= f nmN (ρ, pj)3Yj=1∆12(β2j −k2j)(2.41b)Following Di Francesco and Kutasov6, we can fix f nmN (ρ, pj) by using the 3-pointfunction Anm3through4):AnmN (k1, k2, kj →0, kN) = (−π)N−3 ∂∂µN−3Anm3(k1, k2, kN),3 ≤j ≤N −1(2.42)4) Notice that limkj→0 βj = α+, thus s = −1α+ NPj=1βj +Q→˜s+3−N where ˜s = −1α+Pj=1,2,Nβj +Q.10

Now using the result for Anm3(formula (2.27)) we get:f nmN (ρ, pj)=[−π∆(ρ−1)]m[−π∆(ρ)]n∂N−3∂µµs+N−3[∆(−ρ)]sNYj=4(−π)∆(12(β2j −k2j )). (2.43)Back in (2.41) we haveAnmN= (s + N −3)(s + N −4) · · ·(s + 1) [µ∆(−ρ)]s−π∆(ρ−1)m [−π∆(ρ)]nNYj=1(−π)∆(12(β2j −k2j)),(2.44)therefore, redefining the screening operators, Tkj and µ as before, we have:AnmN= ∂N−3∂µ µs+N−3(2.45)which is a remarkable result.

We generalize this technique to the NS sector of the super-symmetric theory in the next section.3- Supersymmetric Correlators3.1- The 3-point fermionic NS correlatorIn a recent paper15 we have calculated the 3- and 4-point NS correlations functionsusing DHK formulation13 of super Liouville theory coupled to superconformal matter onthe sphere without screening charge. The total action S is given by S = SSL + SM whereSSL = 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 matter sectorhas the central charge ˆcm = 1 −8α20. Analogous to the bosonic case the parameters Q andα± are given by (compare with (2.2))Q = 2q1 + α20,α± = −Q2 ± 12pQ2 −4 = −Q2 ± |α0|,α+α−= 1.

(3.2)The (gravitationally dressed) primary superfields ˜ΨNS are given by˜ΨNS(zi, ki) = d2z ˆEeikΦM (z)eβ(k)ΦSL(z)11

whereβ(k) = −Q2 + |k −α0|. (3.3)In what follows we review the calculation of the three-point function of the primarysuperfield ˜ΨNS, that is:* 3Yi=1Z˜ΨNS(zi, ki)+≡Z[D ˆEΦSL][D ˆEΦM]3Yi=1˜ΨNS(zi, ki)e−S.

(3.4)The method will closely parallel the bosonic case. After integrating over the bosoniczero modes we get* 3Yi=1Z˜ΨNS(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(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 (2.9)).For s non-negative integer, we haveA(k1, k2, k3) = Γ(−s)(−π2 )3(iµπ )s×Z3Yi=1d2˜zisYi=1d2zi3Yi

Indeed, the generators ofthe superconformal transformations on the (z, θ) variablesL0 = z∂z + 12θ∂θ −jL1 = ∂zL−1 = z2∂z + zθ∂θ −2jzQ1/2 = ir12(∂θ + θ∂z)Q−1/2 = ir12(z∂θ + zθ∂z −2jθ)(3.7)12

imply that we can fix z1 = 0 , z2 = 1 , z3 = ∞, θ2 = θ3 = 0 , θ1 = θ. The integral isthe supersymmetric generalization of (B.9) of Ref.[12].

Alternatively, it is expressed incomponents (ΦSL = φ + θψ + ¯θ ¯ψ):A(k1, k2, k3) = Γ(−s)(−π2 )3(iα2+µπ)sβ21ZsYi=1d2zisYi=1|zi|−2α+β1|1−zi|−2α+β2sYi

One may evaluate⟨ψ · · · ψ⟩0 and ⟨ψ · · · ψ⟩0 independently. Since the rest of the integrand is symmetric, onemay write the result in a simple form by relabelling coordinates: (compare with A3 bosonicformula (2.12)).A(k1, k2, k3) = Γ(−s)(−π2 )3 1α2+(iα2+µπ)sα2(−1)s+12 s!

!×ZsYi=1d2zisYi=1|zi|2α|1 −zi|2βsYi

[15] we calculated Il in detail byusing the symmetries Il(α, β; ρ) = Il(β, α; ρ), Il(α, β; ρ) = Il(−1 −α −β −4lρ, β; ρ) and13

looking at its large α behavior (see Ref. [15]) we obtained:Il(α, β; ρ) = −π2l+122l∆12 −ρ2l+1lYi=1∆(2 + ρ)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)(3.11)We can choose, without loss of generality k1, k3 ≥α0, k2 ≤α0.

We proceed now as inthe bosonic case and we have (compare with (2.14))β = ρ(1 −s) (α0 > 0)−12 −ρs (α0 < 0). (3.12)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 ](3.13)By redefining the cosmological constant and the primary superfield ˜ΨNSµ →2∆ 12 −ρµ,˜ΨNS(kj) →1(−i2π)∆ 12[1 + β2j −k2j] ˜ΨNS(kj),(3.14)we getA(k1, k2, k3) = µs. (3.15)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(3.14) with (2.17). 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, as it should.3.2- The 3-point NS correlator with arbitrary s.c. (Anm3)Now we shall 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 , where14

d± 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,(3.16)Integrating over the Grasmann variables and fixing thedSL(2) symmetry as before(˜z1 = 0 , ˜z2 = ∞, ˜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

(−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(3.18)whereInmM (˜α, ˜β; ˜ρ) =nYi=1Zd2ti|ti|2˜α|1 −ti|2 ˜βnYi

×mYi=1Zd2ri|ri|2˜α′|1 −ri|2 ˜β′ mYi

The integral JnmMdiffers from InmMby the introduction of a factor ξξ(0)and JsG can be obtained from IsG by dropping ψψ(0). Henceforth we assume, for simplicity,n + m = even, s = odd.

We will work out explicitly only the case n, m even. However, thefinal result for the amplitude does not depend on which case we choose.

In the AppendixA we calculate InmMfor n and m even and we get:InmM (˜α, ˜β; ˜ρ) = (−)n+m2πn+m2n+m n!m!−˜ρ2−2nm ∆12 −˜ρ2n ∆12 −˜ρ′2m×n2Y1∆(i˜ρ)∆12 + ˜ρi −12m2Y1∆(i˜ρ′ −n2 )∆12 −n2 −˜ρ′i −12×n2 −1Yi=0∆(1 + ˜α + i˜ρ)∆(1 + ˜β + i˜ρ)∆(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 + i˜ρ′)∆(1 −n2 + ˜β′ + i˜ρ′)∆(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))(3.21)In the case where s = 2l + 1 the gravitational contribution to Anm(k1, k2, k3), i.e, IsG isjust the same as in the case without screening charges, thus from the last section we have16

the supersymmetric generalization of (B.9) of Ref. [12]: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 + 2i˜ρ)∆(−˜α −˜β + 2˜ρ(i −s + 1))×s−12Yi=0∆(12 + ˜α + (2i −1)˜ρ)∆(12 + ˜β + (2i −1)˜ρ)∆(−12 −˜α −˜β + ˜ρ(2i −2s + 1))(3.22)To obtain Anm(k1, k2, k3) (see (3.18)) we have to calculate Inm × IsG.

Using the samekinematics as in the case without screening charges, it is easy to deduce:β = n2 + ρ(m −s + 1), α0 > 0−12 −m2 −(n + s)ρ, α0 < 0(3.23)As before, in the case α0 > 0 the amplitude vanishes if we assume5) s ≥m + 2because there appears Γ(−n) in the denominator of IsG after the substitution of β aboveand consequently IsG vanishes. Therefore we have to look at the α0 < 0 case to have a non-trivial amplitude.

In this case we have (remember that d+ = −α+ , d−= α−, if α0 < 0):˜α = α −2ρ,˜α′ = −1 + ρ−1α2˜ρ = −2ρ,˜ρ′ = −ρ−12˜β = −β −1 = (n + 1)ρ + m2 −12˜β′ = (n + s)2+ ρ−14 (m −1). (3.24)5) Actually the kinematics chosen here is self only if s >> n, m.17

Substituting in (3.21) and (3.22) and using (3.18) we have a very involved expression: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=1δ(−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))(3.25)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 and 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 )(3.26)Therefore after redefining the cosmological constant, the NS operators and the screeningchargeseid+ΦM(ti) →−iπ2 ∆12 + ρ−1eid+ΦM(ti)(3.27a)eid−ΦM(ti) →−iπ2 ∆12 + ρ−14−1eid−ΦM (ti)(3.27b)ΨNS →−iπ2 ∆12 + 12(β2i −k2i )−1ΨNS(3.27c)µ →12∆12 −ρ−1µ(3.27d)we obtain the very simple result:Anm3(k1, k2, k3) = µs(3.28)In view of the complexity of (3.25), the simplicity of the result is remarkable.3.3- N-point (N ≥4) supersymmetric correlation functions with an arbitrarynumber of screening chargesIn this subsection we show that it is possible to obtain a simple result for the mostgeneral case of a N-point amplitude with an arbitrary number of screening charges (AnmN ).In that general case we have to calculate the following integral6)AnmN (k1, · · · , kN) = Γ(−s)−α+iµπs * NYi=1Zd2˜zieikiΦM(˜zi)+βiΦSL(˜zi)×nYi=1Zd2tieidΦM (˜ti)mYj=1Zd2rjeid−ΦM (˜rj)sYj=1Zd2zjeiα+ΦSL(˜zi)+0(4.1)6) We are computing 1PI amplitudes; see in this respect ref. [6]19

where s = −1α+ (PNi=1 βi + Q) and PNi=1 ki + nd+ + md−= 2α0. After fixing the gSL2symmetry 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

with α, β, ρ, ˜α, ˜β, ˜ρ, ˜α′, ˜β′, ˜ρ′ defined as before andα′j = k1kj −β1βj,˜αj = d+kj,β′j = k2kj −β2βj,˜α′j = d−kj,ρ′jl = 12(kjkl −βjβl),pj = −α+βj,4 ≤j, l ≤N(4.4)Using the kinematics: k1, k2, · · ·, kN−1 ≥α0 , kN < α0 it is possible to eliminate allparameters in terms of α, β, ρ and pj(4 ≤j ≤N −1):pN = −(m + 1)2−ρ(N + s + n −3)α′j = α + pj −2ρ,β′j = β + pj −2ρ,4 ≤j ≤N −1α′N = (m −1)2+ (ρ −α)(N + s + n −3) −mρ−1β2ρ′jn = (m −1)4−mρ−14pj + (ρ −pj)2(N + s + n −3)ρ′jl = 12(pl + pj) −ρ˜α = α −2ρ,˜β = β −2ρ,˜α′ = −1 + ρ−12 α,˜β′ = −1β2 ρ−1(4.5)where 4 ≤j, l ≤N −1. Using the symmetries:AnmN (α, β, ρ, p1, p2, · · ·, pN−1) = AnmN (β, α, ρ, p1, p2, · · ·, pN−1)(4.6)AnmN (α, β, ρ, p1, · · ·, pN−1)=AnmN (−α−β+(m−1)2−P+ρ(N+n−s−1), β, ρ, p1, · · ·, pN−1)(4.7)(with P =N−1Pj=4pj), and the large-α behaviour:AnmN (α →∞) ∼α1−m+2β+2ρ(s−N−n+3)+2P(4.8)we have the ansatz:AnmN =f nmN (ρ, p1, · · ·, pN−1)∆(12+ρ−α)∆(12+ρ−β)∆(1−m2 +P+α+β+ρ(2+s−n−N)) (4.9)By sending ki (3 ≤i ≤N −1) to zero, which implies pj →2ρ (4 ≤j ≤N −1), we candetermine f nmN (ρ, p1, · · ·pN−1) using:AnmN (α, β, ρ, ki →0) =−iπ2N−3 ∂N−3∂µAnm3(k1, k2, kN)(4.10)21

and the result for Anm3(see (3.26)). We getfN(ρ, p1, · · ·, pN−1) =−iπ2 ∆(12 + ρ)n −iπ2 ∆(12 + ρ−14 )m −iπ2N×∆(12 −ρ)s N−1Yi=4∆(12 +ρ−pj)!∆−(s+n+N −4)2−m4 ρ−1× ∂∂µN−3 hµ2is+N−3.

(4.11)So, the final result for the general N-point function with arbitrary screening chargescan be written in a simple form:AnmN= (s + N −3)(s + N −4) · · ·(s + 1)sµ∆(12 −ρ)s×−iπ2 ∆(12 + ρ)n −iπ2 ∆(12 + ρ)m NYi=1−iπ2N∆(12(1 + β2i −k2i )) (4.12)Redefining ΨNS , µ and the screening charges we have our final result:AnmN= ∂∂µN−3µs+N−3(4.13)4- ConclusionWe have computed exactly N-point correlators in the NS sector of super Liouvilletheory conformally coupled to c ≤3/2 supermatter in a supercoulomb gas representationincluding an arbitrary number of screening charges in the matter sector. We also gen-eralized previous results for bosonic amplitudes to the case including arbitrary s.c.. Wehave learned that in all those cases the final N-point amplitude with n, m s.c. has thesame (rather simple) form given by (4.13) above in terms of the renormalized cosmologicalconstant µ and the parameter s which is a function of the matter central charge and theexternal momenta.

This confirms suspicions18,19 in that direction using a proposal of thematrix model approach. The close connection to the bosonic amplitude was suggestedfrom super KP systems as well19.

This demonstration is however more direct (see also [15]and [16]). The similarity is striking, and the hope is that a full treatment of the theoryby the super matrix model approach could be checked against our results.

Furthermore,the 4-point NS amplitude at s = 0 recently obtained by Di Francesco and Kutasov16 canbe also compared with our result (in that limit) and the check is positive. It should bestressed, however, that our results for arbitrary s permits to visualize the role of the bar-rier at c = 1 (in the bosonic case), since the renormalization µ →µ/∆(−ρ) is infinity at22

c = 1 (where ρ = −1). In the supersymmetric case, the barrier occurs at c = 3/2, therenormalization µ →µ/∆(1/2 −ρ) is infinity at c = 3/2 (where ρ = −1/2).

It would beinteresting to see whether this barrier indeed disappears for N=2 supersymmetry, and wehope to obtain also the N=2 super correlators. The next interesting question concerns theRamond sector.

Work in this direction is in progress. There are, however, new difficultiesin that case.23

Appendix AIn this appendix we calculate (for n and m even) the following integral:InmM (α, β; ρ) =nYi=1Zd2ti|ti|2α|1 −ti|2βnYi

In orderto obtain Inm we first notice that by making translations (ti →1 −ti , ri →1 −ri) andinversions (ri →1/ri , ti →1/ti) we have the symmetries, respectively:Imn(α, β, ρ) = Imn(β, α; ρ)(A.2)Imn(α, β; ρ) = Imn(m −1 −α −β −ρ(n −1), β; ρ). (A.3)Changing variables we can analyse the asymptotic behaviour of Imn for large α:Imn(α →∞, β; ρ) = α(2nm−n−m−2nβ−2mβ′−ρ(n−1)n−ρ′(m−1)m).

(A.4)Another information can be used, namely that for ρ = ρ′ = −1 (α′ = α , β′ = β) theintegral must be a function of n and m through the combination n+m. It is not difficult tocheck (using Stirling’s formula Γ(α + c) ∼αcΓ(α), for large-α), that the following Ansatzis consistent with all requirements given above:Imn(α, β; ρ) = Cmn(ρ)×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))(A.5)24

The coefficient Cmn(ρ) can be basically obtained by noticing that the integral Imnreduces to a known integral when m or n vanishes:I0n = (−)n/2(n!! )Jn/2(α, β, γ = −1/2, ρ/2)(A.6)Im0 = (−)m/2(m!!

)Jm/2(α′, β′, γ = −1/2, ρ′/2)(A.7)We have calculated the integral Jm(α, β; ρ) in Ref. [15].

For the reader’s convenience wegive the result:Jm(α, β; γ; ρ) =π2m2m m! [∆(−(γ + ρ))]2mmYi=1∆(1 + 2(γ + iρ)) ∆(1 + γ + (2i −1)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)(A.8)Using the above result in eq.

(A.6) and (A.7) with the Ansatz (A.5) we obtain Cmn(ρ):Cnm(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(A.9)which determines Imn(α, β; ρ) completely.AcknowledgmentsThe work of K.H. (contract # 90/1799-9) and D.D.

(contract # 90/2246-3) was sup-ported by FAPESP while the work of E.A. and M.C.B.A.

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