We review the derivation of a general formula for the Liouville dressing factor in the boundary 3-point tachyon correlator with c &lt; 1 degenerate matter

Furlan, P.

Petkova, V. B.

Stanishkov, M.

English

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

Acta Polytechnica Vol. 50 No. 3/2010A Finite Liouville Dress for c< 1 Boundary Degenerate MatterP. Furlan, V. B. Petkova, M. StanishkovAbstractWe review the derivation of a general formula for the Liouville dressing factor in the boundary 3-point tachyon correlatorwith c < 1 degenerate matter.Keywords: non-critical string, tachyon correlators, boundary conditions.1 IntroductionThe simplest example of a non-critical string theory is 2d Liouville gravity induced by cM < 1 matter [1]. It combinestwo Virasoro theories with central charges parametrised by a generically real number b, cM = 13−6(b2+1/b2) < 1and cL = 26 − cM > 25, so that when we add a pair of reparametrisation ghosts of central charge −26 thetotal conformal anomaly vanishes. The D-brane dynamics in an open non-critical string is determined by theboundary correlation functions (numbers) of the physical ﬁelds of ghost number one, “massless tachyons”, seee.g. [2, 3, 4, 5, 6, 7] for more recent discussions.The full boundary tachyon ﬁeld factorises into a matter times a Liouville “dressing” vertex operator, producinga similar factorisation of the full 3-point function. In this work we address our attention to the pure Liouvillefactor of it in the case where the matter factor corresponds to degenerate Virasoro representations. The matterﬁelds are vertex operators of the scaling dimension ΔM(e) = e(e−1/b+ b) labelled by degenerate cM < 1 Virasororepresentations. This implies that the charges {βi} of the dressing Liouville boundary vertex operators σiBσiβi , ofscaling dimensionsΔL(β) = β(Q− β) = 1−ΔM (e),take the valuesβi = b +mib− nib, 2mi, 2ni ∈ Z≥0 (1.1)or their reﬂected β → Q− β counterparts (Q = b+ 1/b), so that without loss of generality we shall work with thevalues in (1.1). The range of the boundary parameters σi is generically parametrised by the continuous Liouvillespectrum 2σ − Q – pure imaginary but also admits continuation to real values. These Liouville boundary ﬁeldscorrespond to the FZZ branes [8].The matter factor of the 3-point boundary tachyon correlator is a straightforward generalisation of the factorin the rational b2-case. It is alternatively reproduced by an analytic continuation of a residuum of the integralPonsot-Teschner (PT) formula [9] at points corresponding to c > 25 degenerate Virasoro representations. The sameanalytic continuation applies to fusing matrices, which diﬀer from the boundary ﬁeld crossing matrices (3-pointboundary correlators) by a renormalisation of the three boundary vertices. Thus the formulae in [9, 10] for thequantum 3j and 6j symbols, designed generically for the continuous c > 25 spectrum, are in a sense universal, sincewe can reproduce from them the Coulomb gas quantities in both c < 1 and c > 25 Virasoro regions. Howeverthis integral formula is not very explicit, and its main characteristics are not immediately visible when appliedto the spectrum of representations (1.1). Another alternative is to solve the pentagon equations recursively. Theﬁnal result is a meromorphic expression in the boundary cosmological parameters, the derivation of which wereview here, see [11] for more details. It generalises a special (thermal) case result of [6] and partial results in themicroscopic approach in [5].2 Boundary 3-point Liouville constantThe matter fusion rules impose restrictions on the values in (1.1), namely allmkij := mi+mj−mk, nkij = ni+nj−nkare non-negative integers, so that2m123 =3∑i=12mi = 0 mod 2.84Acta Polytechnica Vol. 50 No. 3/2010The 3-point boundary Liouville functions that we are interested in are related to the boundary ﬁeld crossingmatricesCσ2,Q−β3[β2 β1σ3 σ1]= 〈σ1Bβ3σ3Bβ2σ2Bβ1σ1〉 = Cσ3,σ2,σ1β3,β2,β1 = S(σ1, β3, σ3)Cσ3,σ2,σ1Q−β3,β2,β1 , (2.1)where S(σ1, β3, σ3) is the reﬂection amplitude [8].The associativity condition for OPE of boundary ﬁelds, together with the fusion transformation relating the sand t channels, lead to an integral pentagon-like equation for the boundary ﬁeld 3-point functions∫dβSCσ4,σ3,σ1Q−β3,β2,βsCσ3,σ2,σ1Q−βs,β,β1Fβs,βt[β2 ββ3 β1]= Cσ4,σ2,σ1Q−β3,βt,β1Cσ4,σ3,σ2Q−βt,β2,β , (2.2)where Fβs,βt is the fusing matrix computed in [10]. The boundary 3-point functions Cσ3,σ2,σ1β3,β2,β1 are meromorphicwith respect to the variables β1, β2, β3 [9], while the fusion coeﬁcients Fβs,βt[β2 ββ3 β1]are meromorphic in allsix variables and invariant under the reﬂections βi → Q− βi.When one of the operators corresponds to a degenerate representation, the Fβs,βt and Cσ3,σ2,σ1β3,β2,β1 coeﬁcientsdevelop singularities such that the integral in (2.2) gives rise to a ﬁnite sum over representations in accordancewith the fusion rules [9]. In particular, for β = −b/2, equation (2.2) becomes (see e.g. [5]):Cσ3,β2−t b2⎡⎣ β2 − b2σ4 σ2⎤⎦ Cσ2=σ3± b2 ,β3⎡⎣ β2 − t b2 β1σ4 σ1⎤⎦ =F+t⎡⎣ β2 − b2β3 β1⎤⎦Cσ3± b2β1− b2⎡⎣ − b2 β1σ3 σ1⎤⎦ Cσ3,β3⎡⎣ β2 β1 − b2σ4 σ1⎤⎦+ (2.3)F−t⎡⎣ β2 − b2β3 β1⎤⎦Cσ3± b2β1+ b2⎡⎣ − b2 β1σ3 σ1⎤⎦ Cσ3,β3⎡⎣ β2 β1 + b2σ4 σ1⎤⎦ ,t = ±1where C and F are the appropriate residues of C and F . For t = 1 it becomesCσ3,σ2± b2 ,σ1β3,β2,β1 =Γ(1− 2β2b)Γ((2β1 − b)b)Γ(1− b(β2 + β3 − β1))Γ(b(β1 + β3 − β2 − b)) · Cσ3,σ2,σ1β3,β2+ b2 ,β1− b2+ (2.4)b2√λLΓ(1− 2β2b)Γ(1− 2β1b)g ∓ (σ2, β1, σ1)2 sinπb(2β1 −Q)Γ(1− b(β1 + β2 + β3 −Q))Γ(1− b(β1 + β2 − β3)) · Cσ3,σ2,σ1β3,β2+ b2 ,β1+b2,where λL = πμγ(b2) is the (normalised) cosmological constant andg−(σ2, β1, σ1) = g+(σ2, Q− β1, σ1) = − 4 sinπb(β1 − σ1 − σ2 + b2)sinπb(β1 − σ2 + σ1 − b2). (2.5)There is a second equation with a shift β2 − b/2 on the r.h.s. as well as two dual equations for 1/b→ b/2. Thederivation of (2.4) is standard and the coeﬁcients in front of the correlators are given by the products of fusingmatrix elements and 3-point boundary functions containing a fundamental ﬁeld. The latter are computed byfree ﬁeld Coulomb gas methods [8], assuming that for degenerate representations the Cardy multiplicity coincideswith the Verlinde multiplicity. In the case above, this means that the two boundaries of the ﬁeld σ2Bσ1− b2satisfyσ2 = σ1 ± b/2.2.1 The simplest correlatorWe start with the derivation of the simplest correlator with three identical charges equal to b, i.e., the correlatorof three cosmological operators, or boundary Liouville screening charges. It is reproduced by the second term onthe r.h.s. of the equality (2.4) choosing β1 =b2= β2, β3 = b. For this choice the equation needs regularisationsince the coeﬁcient in front of the correlator becomes divergent. The remaining two correlators are represented85Acta Polytechnica Vol. 50 No. 3/2010as reﬂections (2.1) with respect to β3 (the l.h.s.) and β2 (the ﬁrst term on the r.h.s.) of correlators which alsodiverge, if we assume that they are given by the integral PT formula. Indeed they satisfy the charge conservationconditions (Q− β3) + β2 + β1 = Q and β3 + (Q− β2 − b/2) + (β1 − b/2) = Q, respectively, and their residua equal1/2π (to agree with the normalisation in [9]). Thus, in a proper regularisation of (2.4), these two correlators arereplaced by the corresponding reﬂection amplitudes, which appear as the initial data in the equation. We recalltheir general expression computed in [8],S(σ2, β, σ1) =2πbΓ(1 + 1b (Q− 2β))Γ(b(Q− 2β))G2(σ2, β, σ1), (2.6)G2(σ2, β, σ1) =λ12b (Q−2β)L Sb(2β −Q)∏s=± Sb(β + s(σ2 + σ1 −Q))Sb(β + s(σ2 − σ1)),where Sb(α) = Γb(α)/Γb(Q− α) = 2 sinπb(α− b)Sb(α− b) and Γb(x) is the double gamma function; Sb(b) = b. Inthe case under consideration here β = b and inserting in (2.4) we reproduce the cyclically symmetric expressionproposed in the microscopic approach [5],Cσ3,σ2,σ1b,b,b =2π√λl−1(Γ(1 − b2))2Γ( 1b2 − 1)· G2(σ3, b, σ1)−G2(σ3, b, σ2)g−(σ1, b2 , σ2)= (2.7)2πλQ−3b2bLSb(2b )(Γ(1 − b2))2Γ( 1b2 − 1)· (c˜1(c2 − c3) + c˜2(c3 − c1) + c˜3(c1 − c2)(c2 − c1)(c1 − c3)(c3 − c2)where the boundary cosmological constants ∼ ci and their dual appear,ci = 2 cosπb(b− 2σi), c˜i = 2 cosπ 1b(1b− 2σi). (2.8)Similar regularised versions of (2.4) arise for other values of the charges corresponding to reﬂections of Coulombgas correlators.2.2 One parameter correlators, cyclic symmetryWe shall use Eq. (2.4) as a recursion relation, starting from the explicit expression (2.7). Let us ﬁrst introducesome general notation:G(−)(σ2, β, σ1) := Sb(−β + σ2 + σ1)Sb(Q− β + σ2 − σ1) = g−(σ2, β +b2, σ1)G(−)(σ2, β + b, σ1). (2.9)For a non-negative integer k and an integer n of parity p(n) denoteB(σ2, σ1)(k;p(n)) :=G(−)(σ2,−kb2 − n2b , σ1)G(−)(σ2, b+ kb2 − n2b , σ1)= (−1)(k+1)(n+1)B(σ1, σ2)(k;p(n)). (2.10)Applying (2.9), the ratio (2.10) is expressed as a k + 1 order polynomial in {ci} using that for k = 0g−(σ2,b2− k b2+n2b, σ1)g−(σ2,b2+ kb2+n2b, σ1)= c21 + c22 − c1c2(−1)n2 cosπkb2 − (2 sinπkb2)2 (2.11)while B(σ2, σ1)(0;p(n)) = (−1)nc2 − c1. Similarly, we deﬁne the dual B˜(σ2, σ1)(n;p(k))B˜(σ2, σ1)(n;p(k)) :=G(−)(σ2,− n2b − kb2 , σ1)G(−)(σ2, 1b +n2b − kb2 , σ1)= (−1)(k+1)(n+1)B˜(σ1, σ2)(n;p(k)) (2.12)so that the reﬂection amplitude is expressed as the ratio of polynomialsλ2β2−Q2bL G2(σ2, β2 = b+m2b− n2b , σ1)Sb(2β2 −Q) =G(−)(σ2, β2, σ1)G(−)(σ2, Q− β2, σ1) =B˜(σ2, σ1)(2n2;p(2m2))B(σ2, σ1)(2m2;p(2n2)). (2.13)86Acta Polytechnica Vol. 50 No. 3/2010Finally we introduceP2 ≡ P σ3,σ2,σ1β3,β2,β1 := (−1)m312+2m2λ−m3122LSb((2m1 + 1)b)Sb((2m2 + 1)b)Sb(b)·m312∑p=0Sb((m312 + 1)b)Sb((p+ 1)b)Sb((m312 + 1− p)b)× G2(σ2 + pb2 , β2 − p b2 , σ3)G2(σ2, β2, σ3)(2.14)G2(σ2 − (m312 − p) b2 , β1 − (m312 − p) b2 , σ1)G2(σ2, β1, σ1)and similarly P1 and P3, which are obtained from (2.14) by cyclic permutations. The ﬁnite sum (2.14) is proportionalto a truncated basic hypergeometric function 4φ3(. . . ; q, q). It can be expanded as a polynomial in the variables{ci} (a special case of Askey-Wilson polynomials).We begin with the “thermal” case with all ni = 0 in (1.1). We ﬁrst use such a regularised equation in whichthe ﬁrst term on the r.h.s. of (2.4) reduces to a 2-point function in order to obtain recursively the most generalcorrelator with m213 = 0. Then using the analog of the general equation (2.4) for shifts of the pair (β3, β2), weobtainCσ3,σ2,σ1β3,β2,β1 = −λQ−β1232bL∏(β3, β2, β1)B(σ1, σ2)(2m1;0)B(σ2, σ3)(2m2;0)B(σ3, σ1)(2m3;0)F σ3,σ2,σ1β3,β2,β1 ,F σ3,σ2,σ1β3,β2,β1 = (−1)2m1((−1)2m2 c˜2 − c˜3)B(σ3, σ1)(2m3;0)Pσ3,σ2,σ1β3,β2,β1−(−1)2m2((−1)2m3 c˜3 − c˜1)B(σ2, σ3)(2m2;0)P σ2,σ1,σ3β2,β1,β3 = (2.15)−(c˜1B(σ3, σ2)(2m2;0)Pσ2,σ1,σ3β2,β1,β3+ c˜2B(σ1, σ3)(2m3;0)Pσ3,σ2,σ1β3,β2,β1+ c˜3B(σ2, σ1)(2m1;0)Pσ1,σ3,σ2β1,β3,β2)where ∏(β3, β2, β1) =be0(Q−β123)Γb(2q − β123)Γb(Q− β123)Γb(Q− β213)Γb(Q− β312)Sb(1b )Sb(2b )Γb(Q)Γb(Q− 2β1)Γb(Q− 2β2)Γb(Q− 2β3). (2.16)In the last equality of (2.15) we have exploited (2.10) and the relation.B(σ3, σ1)(2m3;0)Pσ3,σ2,σ1β3,β2,β1+ cyclic permutations = 0 (2.17)which is equivalent to the cyclic symmetry of the correlator, now explicit in (2.15). Symmetry is ensured by thefact that the expression given by the ﬁrst equality satisﬁes all the equations related by cyclic permutations.The composition of the reﬂection of all three ﬁelds with the reﬂection amplitude as in (2.1) and the dualitytransformation b → 1/b (changing notation mi → ni) gives the correlator in the other thermal case, when allmi = 0 in (1.1). In this case the product of B(0;p(2ni)) replaces the denominator in (2.15) and the formula conﬁrmsthe structure suggested in the microscopic approach of [5]. The dual polynomial P˜ σ3,σ2,σ1β3,β2,β1 is deﬁned by changing in(2.14) βi → Q−βi, b→ 1/b, mi → ni. With the help of some identities for the basic hypergeometric functions onereproduces the formula in [6] for the case {mi = 0, ni – integers}. The expression in [6] is however not explicitlysymmetric under cyclic permutations, rather this symmetry is checked to hold on examples.2.3 The general correlatorTo obtain the Liouville correlator deﬁned for general values (1.1), we can either use the dual pentagon equations,or we can start from the correlator with all mi = 0. In one of the steps, the pentagon equation (2.4) is regularisedagain so that the second term on the r.h.s. is given by G2 times a non-trivial Coulomb gas Liouville correlator.The ﬁnal result is an expression generalising the ﬁrst line in (2.15),Cσ3,σ2,σ1β3,β2,β1 =λQ−β1232bL∏′(β3, β2, β1)B(σ1, σ2)(2m1;p(2n1))B(σ2, σ3)(2m2;p(2n2))B(σ3, σ1)(2m3;p(2n3))×(−1)2m22n1((−1)2m1+2n2B˜(σ2, σ3)(2n2;p(2m2))P˜ σ2,σ1,σ3β2,β1,β3 B(σ3, σ1)(2m3;p(2n3))Pσ3,σ2,σ1β3,β2,β1− (2.18)(−1)2m2+2n1B˜(σ3, σ1)(2n3;p(2m3))P˜ σ3,σ2,σ1β3,β2,β1 B(σ2, σ3)(2m2;p(2n2))Pσ2,σ1,σ3β2,β1,β3)with the prefactor ∏′(β3, β2, β1) =(−1)m123n123∏(β3, β2, β1)S3b (1b )Sb(2b − b)Sb(n312+1b )Sb(n123+1b )Sb(n213+1b )Sb(n123+2b − b). (2.19)87Acta Polytechnica Vol. 50 No. 3/2010Here, say, the polynomial P2 is given by the ﬁrst formula (2.14), where now all βi are given by (1.1), with onlythe sign in front of (2.14) modiﬁed to (−1)m312(1+2n3)+2m32n3+2m2 = (−1)m123(1+2n3)+2m1 . Let us also write downthe expression for one of the dual polynomialsP˜1 ≡ P˜ σ2,σ1,σ3β2,β1,β3(−1)n123(1+2m2)+2n3λ−n213/2L Sb(2n1+1b )Sb(2n3+1b )Sb(1b )n213∑u=0Sb(n213+1b )Sb(1+ub )Sb(n213+1−ub )× (2.20)G2(σ1 + u2b , Q− β1 − u2b , σ2)G2(σ1, Q− β1, σ2)G2(σ1 − n213−u2b , Q− β3 − n213−u2b σ3)G2(σ1, Q− β3, σ3) .The cyclic symmetry of the full correlator is ensured by construction and is equivalent to a relation generalising(2.17),(−1)2n2(2m2+1)B(σ3, σ1)(2m3;p(2n3))P2 + cyclic permutations = 0 (2.21)and its dual with the dual polynomials and mi ↔ ni. In particular, when all mi = 0 the dual relation reproducesthe cyclic identity satisﬁed by the ﬁrst order dual polynomials B˜(σ2, σ3)(0;p(2m2)) = (−1)2m2 c˜2 − c˜3, etc., whichappear in the numerator in (2.15). The composition of the duality transformation b→ 1/b, mi ↔ ni with reﬂectionof all three ﬁelds keeps (2.18) invariant.3 Summary and discussionWe have obtained the general Liouville dressing factor in the tachyon 3-point boundary correlator with degeneratec < 1 representations. Formula (2.18) represents the Liouville correlator as a ratio of polynomials of the boundarycosmological parameters ci, c˜i generalising the partial results in [5, 6]. This solution of the Liouville pentagonequations extends to the minimal gravity theory with rational b2, in which case there may appear further truncationsof the sums. The general 3-point boundary tachyon correlator is a product of (2.18) and the matter 3-point boundarycorrelator, satisfying a 4-term equation, see [11] for an explicit formula and further discussion.A possible extension of our result would allow us to describe also the 3-point boundary tachyon correlators cor-responding to the ZZ branes. For this purpose, the roles of the matter and Liouville spectra and the correspondingcorrelators are essentially inverted: the Coulomb gas Liouville correlator for degenerate c > 25 representationsdescribing both the charges and the boundaries should be combined with a matter factor obtained by analytic con-tinuation of the solution (2.18). Note that the corresponding discrete c < 1 spectrum parametrises the irreduciblerepresentations embedded as submodules of the reducible Virasoro modules. The analogous characteristics of thec > 25 spectrum (1.1) have been exploited in the construction of the 4-point bulk tachyon correlators [12].AcknowledgementP. Furlan acknowledges support from the Italian Ministry of Education, Universities and Research (MIUR).V. B. Petkova acknowledges hospitality from the Service de Physique The`orique, CEA-Saclay, France, and ICTPand INFN, Italy. This research has received some support from the French-Bulgarian RILA project, contract3/8-2006.References[1] Ginsparg, P., Moore, G.: Lectures on 2D gravity and 2D string theory (TASI 1992), hep-th/9304011.[2] Martinec, E. J.: The annular report on non-critical string theory, hep-th/0305148.[3] Seiberg, N., Shih, D.: JHEP 0402 (2004) 021, hep-th/0312170.[4] Kostov, I. K.: Nucl. Phys. B 689, 3 (2004), hep-th/0312301.[5] Kostov, I. K., Ponsot, B., Serban, D.: Nucl. Phys. B 683, 309 (2004), hep-th/0307189.[6] Alexandrov, S. Y., Imeroni, E.: Nucl.Phys. B 731 (2005) 242, hep-th/0504199.[7] Basu, A., Martinec, E. J.: Phys. Rev. D 72, 106007 (2005), hep-th/0509142.[8] Fateev, V., Zamolodchikov, A., Zamolodchikov, Al.: Boundary Liouville ﬁeld theory. I: Boundary state andboundary two-point function, hep-th/0001012.88Acta Polytechnica Vol. 50 No. 3/2010[9] Ponsot, B., Teschner, J.: Nucl. Phys. B 622 (2002) 309, hep-th/0110244.[10] Ponsot, B., Teschner, J.: Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/99111110; Comm. Math. Phys. 224, 3 (2001) 613, math.QA/0007097.[11] Furlan, P., Petkova, V. B., Stanishkov, M.: Non-critical string pentagon equations and their solutions, toappear In: J. Phys. A, arXiv:0805.0134.[12] Belavin, A., Zamolodchikov, Al.: Theor. Math. Phys. 147 (2006) 729, hep-th/0510214.P. FurlanDipartimento di Fisica Teorica dell’Universita` di Trieste, ItalyIstituto Nazionale di Fisica Nucleare (INFN)Sezione di Trieste, ItalyV. B. PetkovaInstitute for Nuclear Research and Nuclear Energy (INRNE), BulgarianAcademy of Sciences (BAS), BulgariaM. StanishkovInstitute for Nuclear Research and Nuclear Energy (INRNE), BulgarianAcademy of Sciences (BAS), Bulgaria89

A Finite Liouville Dress for c

We review the derivation of a general formula for the Liouville dressing factor in the boundary 3-point tachyon correlator with c &lt; 1 degenerate matter.</p

A Finite Liouville Dress for c

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