Tachyon Correlators in C < 1 Non-Rational Liouville Gravity

Tachyon correlators in cM < 1 non-rational Liouville gravity

I. Kostov and V. Petkova Nucl.Phys. B769, B770 (2007)

Two approaches to cM < 1 non-critical string - compared - world-sheet CFT and matrix model (target space CFT) 2bφ I. Continuum - conformal gauge gab = e ˆgab , ˆgab = δab locally

eﬀective action - 2d CFT: cM < 1 -“matter” , cL > 25 - Liouville

free 1 2 2 2 ghosts A = d x (∂aχ) +(∂aφ) +(Qφ + ie0χ) ˆgRˆ + A (b, c) 4π Z h p i background charges = + 1 = 1 - real Q b b , e0 b − b, b

tot vanishing conformal anomaly c = cM + cL + cghosts = 0 interaction - matter and Liouville screening charges -

A = µ e2bφ + µ e−2ibχ = λ T + + λ T + , int Z L M L b M 0 or/and

2 φ 2 iχ − − ˜ b b ˜ ˜ Aint = µ˜L e +µ ˜M e = λL T + λM T0 , Z 1/b 2 2 ˜ 1/b2 ˜ −1/b2 λL = πγ(b ) µL, λM = πγ(−b ) µM , λL = λL , λM = λM - examples of ”massless tachyons” - BRST -invariant operators of ghost number 1 associated with vertex operators of dim (1, 1),

ǫ 2ieχ 2αφ Vα ∼ e e , △M (e)+ △L(α) = 1 , ”mass-shell” condition parametrized by the target space momentum P 1 ε α = 2(Q − εP )= εe + b , and chirality ε = ±1 ,

± 2 ± • integrated (1,1)-forms Tα ≡ d xVα R ± c¯c ± • QBRST-closed (0,0)-forms: Wα ≡ Vα

Problem: tachyon correlation ”functions”(numbers) on the sphere

(ǫ) = ε1 ε2 ε3 ε4 εn Gn Wα1 Wα2 Wα3 Tα4 ··· Tαn

= ··· h iM h iL R R

1 1 (Q− αi) ( ei−e0) (ǫ) b i b i = λL λ Gn (P1,P2,...Pn) P M P • Ground ring – OPE of ghost number 0, dim 0 physical operators [Witten ‘91]

k Oi Oj = nij Ok mod QBRST − exact terms β Oi Wα = ciα Wβ tachyons - modules of the ring,

• (free ﬁeld) generators of the ground ring a±(x)= a±(z) a±(¯z) ,

( )= : bc 1 ( + ) −b(φ−iχ) : a− z − b ∂z φ iχ e − 1 (φ+iχ) a+(z)= : (bc − b ∂z(φ − iχ)) e b : both matter and Liouville degenerate vertex operators (unlike the tachyons)

• Functional relations for the tachyon correlators (on the sphere) [‘91- Kutasov-Martinec-Seiberg, ] [Bershadsky- Kutasov ],[’03- Seiberg-Shih, Kostov] [’91 Di Francesco-Kutasov ] old work - trivial matter - now OPE deformed by Liouville and matter interactions 1 1 (Q− αi) ( ei−e0) b i b i • 3-point tachyon correlator G3(α1 , α2 , α3)= N(α1,α2,α3)λL λ P M P

bρ bρ N(α1 ± 2 ,α2,α3)= N(α1,α2 ± 2 ,α3) , ρ = ±1 X± X±

• simplest solution - for generic momenta - N(α1,α2,α3)=1, obtained also from 3-point matter -Liouville factorization

Liou Matt C (α1,α2,α3)C (e1,e2,e3) W ε1 W ε2 W ε3 = α1 α2 α3 3 −ǫj 2 2 j=1 b γ(αj − ej ) MattQ with the generic matter constant C (e1,e2,e3) computed similarly as the DOZZ Liouville constant

• non-trivial solutions for momenta corresponding to degenerate matter

Pi = e0 − 2ei = ni/b − mib, ni, mi ∈ N (or Liouville) reps

N(P1,P2,P3)= Nm1,m2,m3 Nn1,n2,n3 tensor-product decomposition multiplicities of irreps of su(2) of dimensions mk - ground ring – the representation ring of su(2) × su(2) ǫ ǫ • n ≥ 4− point functs - free ﬁeld OPE further deformed by the tachyons Tα = Vα in the correlator serving as ”screening charges”R

a W + T + (T −T − )k ∼ W + , k = 0, 1,... − α2 α3 0 1/b b n α2+α3− + 2 b OPE coeﬀs C(ǫ) α - 4-point functions computed by Coulomb gas −b/2 α2 α3 ” inhomogeneous associativity eqs” - string analog of the locality eqs

(ǫ) (ǫ) α (ǫ) G (α1 ± b/2,α2,α3,α4)+ C G (α,α2,α4) 4 −b/2 α1α3 3 X± Xα (ǫ) (ǫ) α (ǫ) = G (α1,α2 ± b/2,α3,α4)+ C G (α1,α,α4) 4 −b/2 α2α3 3 X± Xα

ﬁnite number of contact terms if

• 1. one of the ﬁelds corresponds to a degenerate Vir representation, or

• 2. the four momenta are restricted by a (matter) charge conservation condition 4 = 2 2 + 2n Z i=1 Pi e0 − mb b , m,n ∈ ≥0 , P • General form of the solutions

P = + Σ + permutations P

• Ground ring relations derived for the ﬁxed chirality {ǫi ,i = 1, 2, 3, 4} correlators - unphysical set - partially symmetric ”locality” requirement - symmetrised correlators, consistent with the matter fusion rules, formally P replaced by |P |

NP1,P2,P |P | NP,P3,P4 XP • the local correlators satisfy complicated diﬀerence eqs, depending on the momenta

• for the case of (one) degenerate ﬁeld - diﬀerent, more constructive method [A. Belavin, Al. Zamolodchikov] II. Microscopic, discrete realisation of 2d gravity - generalization of rational ADE string models [Kostov ‘91]

- target space - graph, x = 1, 2, 3,...

An •••• ... • • → A∞ SOS (height) model on ﬂuctuating lattice loop gas expansion ,

→ dual formulation as a matrix chain model Mx, N → ∞ (scaling) limit

→ collective field formulation in terms of a c = 1 chiral field Ψx(z) on a Riemann surface - infinite branch cover of the spectral plane

operator solution of Virasoro constraints - finite diagram technique for evaluation of n - loop amplitudes → local field correlators • However different interpretation of the matter interaction - only ”order operators” P = ±m(1/b − b), closed under fusion

no underlying ”pure matter” CFT

⇒ introduce unconventional “diagonal” perturbation of Liouville gravity

matter screening charges replaced by tachyons of matter charge e0 = 1/b − b

2 2 −2biχ iχ 2e0iχ+2bφ 2e0iχ+ φ e , e b → e , e b adapt the ground ring structure: perturbed ground ring operator A = a−a+ projects to shifts of momenta by ±e0

• 4-point tachyon correlators coincide with the expressions obtained by the Feynman diagram technique of the matrix model