ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

A Finite Liouville Dress for c< 1 Boundary Degenerate Matter

P. Furlan, V. B. Petkova, M. Stanishkov

Abstract

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

Keywords: non-critical string, tachyon correlators, boundary conditions.

1 Introduction
Thesimplest exampleof anon-critical string theory is 2dLiouvillegravity inducedby cM < 1matter [1]. It combines
twoVirasoro theories with central charges parametrised by a generically real number b, cM =13−6(b2+1/b2) < 1
and cL = 26 − cM > 25, so that when we add a pair of reparametrisation ghosts of central charge −26 the
total conformal anomaly vanishes. The D-brane dynamics in an open non-critical string is determined by the
boundary correlation functions (numbers) of the physical fields of ghost number one, “massless tachyons”, see
e.g. [2, 3, 4, 5, 6, 7] for more recent discussions.
The full boundary tachyon field factorises into amatter times a Liouville “dressing” vertex operator, producing

a similar factorisation of the full 3-point function. In this work we address our attention to the pure Liouville
factor of it in the case where the matter factor corresponds to degenerate Virasoro representations. The matter
fields are vertex operators of the scaling dimensionΔM(e)= e(e−1/b+b) labelled by degenerate cM < 1Virasoro
representations. This implies that the charges {βi} of the dressing Liouville boundary vertex operators σi Bσiβi , of
scaling dimensions

ΔL(β)= β(Q − β)=1−ΔM(e),

take the values

βi = b + mib −
ni

b
, 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 the
values in (1.1). The range of the boundary parameters σi is generically parametrised by the continuous Liouville
spectrum 2σ − Q – pure imaginary but also admits continuation to real values. These Liouville boundary fields
correspond to the FZZ branes [8].
The matter factor of the 3-point boundary tachyon correlator is a straightforward generalisation of the factor

in the rational b2-case. It is alternatively reproduced by an analytic continuation of a residuum of the integral
Ponsot-Teschner (PT) formula [9] at points corresponding to c > 25degenerateVirasoro representations. The same
analytic continuation applies to fusing matrices, which differ from the boundary field crossing matrices (3-point
boundary correlators) by a renormalisation of the three boundary vertices. Thus the formulae in [9, 10] for the
quantum 3j and 6j symbols, designed generically for the continuous c > 25 spectrum, are in a sense universal, since
we can reproduce from them the Coulomb gas quantities in both c < 1 and c > 25 Virasoro regions. However
this integral formula is not very explicit, and its main characteristics are not immediately visible when applied
to the spectrum of representations (1.1). Another alternative is to solve the pentagon equations recursively. The
final result is a meromorphic expression in the boundary cosmological parameters, the derivation of which we
review here, see [11] for more details. It generalises a special (thermal) case result of [6] and partial results in the
microscopic approach in [5].

2 Boundary 3-point Liouville constant

Thematter fusion rules impose restrictionson the values in (1.1), namely all mkij := mi+mj−mk, n
k
ij = ni+nj−nk

are non-negative integers, so that

2m123 =
3∑

i=1

2mi =0mod 2.

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Acta Polytechnica Vol. 50 No. 3/2010

The 3-point boundary Liouville functions that we are interested in are related to the boundary field crossing
matrices

Cσ2,Q−β3

[
β2 β1

σ3 σ1

]
= 〈σ1Bβ3

σ3Bβ2
σ2Bβ1

σ1〉 = Cσ3,σ2,σ1β3,β2,β1 = S(σ1, β3, σ3)C
σ3,σ2,σ1
Q−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 s

and t channels, lead to an integral pentagon-like equation for the boundary field 3-point functions∫
dβSCσ4,σ3,σ1Q−β3,β2,βsC

σ3,σ2,σ1
Q−βs,β,β1Fβs,βt

[
β2 β

β3 β1

]
= Cσ4,σ2,σ1Q−β3,βt,β1C

σ4,σ3,σ2
Q−βt,β2,β , (2.2)

where Fβs,βt is the fusing matrix computed in [10]. The boundary 3-point functions C
σ3,σ2,σ1
β3,β2,β1

are meromorphic

with respect to the variables β1, β2, β3 [9], while the fusion coeficients Fβs,βt

[
β2 β

β3 β1

]
are meromorphic in all

six variables and invariant under the reflections βi → Q − βi.
When one of the operators corresponds to a degenerate representation, the Fβs,βt and C

σ3,σ2,σ1
β3,β2,β1

coeficients
develop singularities such that the integral in (2.2) gives rise to a finite sum over representations in accordance
with the fusion rules [9]. In particular, for β = −b/2, equation (2.2) becomes (see e.g. [5]):

Cσ3,β2−t b2

⎡
⎣ β2 −b2

σ4 σ2

⎤
⎦Cσ2=σ3± b2 ,β3

⎡
⎣ β2 − t b2 β1

σ4 σ1

⎤
⎦ =

F+t

⎡
⎣ β2 −b2

β3 β1

⎤
⎦ Cσ3± b2 β1− b2

⎡
⎣ −b2 β1

σ3 σ1

⎤
⎦Cσ3,β3

⎡
⎣ β2 β1 − b2

σ4 σ1

⎤
⎦ + (2.3)

F−t

⎡
⎣ β2 −b2

β3 β1

⎤
⎦ Cσ3± b2 β1+ b2

⎡
⎣ −b2 β1

σ3 σ1

⎤
⎦Cσ3,β3

⎡
⎣ β2 β1 + b2

σ4 σ1

⎤
⎦ ,

t = ±1

where C and F are the appropriate residues of C and F. For t =1 it becomes

Cσ3,σ2±
b
2 ,σ1

β3,β2,β1
=

Γ(1−2β2b)Γ((2β1 − b)b)
Γ(1− b(β2 + β3 − β1))Γ(b(β1 + β3 − β2 − b))

· Cσ3,σ2,σ1
β3,β2+

b
2 ,β1−

b
2
+ (2.4)

b2
√

λLΓ(1−2β2b)Γ(1−2β1b)g ∓ (σ2, β1, σ1)
2sinπb(2β1 − Q)Γ(1− b(β1 + β2 + β3 − Q))Γ(1− b(β1 + β2 − β3))

· Cσ3,σ2,σ1
β3,β2+

b
2 ,β1+

b
2
,

where λL = πμγ(b
2) is the (normalised) cosmological constant and

g−(σ2, β1, σ1) = g+(σ2, Q − β1, σ1)= −4sinπb
(

β1 − σ1 − σ2 +
b

2

)
sinπb

(
β1 − σ2 + σ1 −

b

2

)
. (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. The
derivation of (2.4) is standard and the coeficients in front of the correlators are given by the products of fusing
matrix elements and 3-point boundary functions containing a fundamental field. The latter are computed by
free field Coulomb gas methods [8], assuming that for degenerate representations the Cardy multiplicity coincides
with the Verlinde multiplicity. In the case above, this means that the two boundaries of the field σ2Bσ1− b2

satisfy

σ2 = σ1 ± b/2.

2.1 The simplest correlator

We start with the derivation of the simplest correlator with three identical charges equal to b, i.e., the correlator
of three cosmological operators, or boundary Liouville screening charges. It is reproduced by the second term on

the r.h.s. of the equality (2.4) choosing β1 =
b

2
= β2, β3 = b. For this choice the equation needs regularisation

since the coeficient in front of the correlator becomes divergent. The remaining two correlators are represented

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Acta Polytechnica Vol. 50 No. 3/2010

as reflections (2.1) with respect to β3 (the l.h.s.) and β2 (the first term on the r.h.s.) of correlators which also
diverge, if we assume that they are given by the integral PT formula. Indeed they satisfy the charge conservation
conditions (Q − β3)+β2+β1 = Q and β3+(Q − β2 − b/2)+(β1 − b/2)= Q, respectively, and their residua equal
1/2π (to agree with the normalisation in [9]). Thus, in a proper regularisation of (2.4), these two correlators are
replaced by the corresponding reflection amplitudes, which appear as the initial data in the equation. We recall
their general expression computed in [8],

S(σ2, β, σ1) =
2π

bΓ(1+ 1
b
(Q −2β))Γ(b(Q −2β))

G2(σ2, β, σ1), (2.6)

G2(σ2, β, σ1) =
λ
1
2b(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)= 2sinπb(α − b)Sb(α − b) and Γb(x) is the double gamma function; Sb(b)= b. In
the case under consideration here β = b and inserting in (2.4) we reproduce the cyclically symmetric expression
proposed in the microscopic approach [5],

C
σ3,σ2,σ1
b,b,b =

2π
√

λl
−1

(Γ(1− b2))2Γ( 1
b2
−1)

·
G2(σ3, b, σ1)− G2(σ3, b, σ2)

g−(σ1,
b
2, σ2)

= (2.7)

2πλ
Q−3b
2b

L

Sb(
2
b
)(Γ(1− b2))2Γ( 1

b2
−1)

·
(c̃1(c2 − c3)+ c̃2(c3 − c1)+ c̃3(c1 − c2)

(c2 − c1)(c1 − c3)(c3 − c2)

where the boundary cosmological constants ∼ ci and their dual appear,

ci =2cosπb(b −2σi), c̃i =2cosπ
1
b

(
1
b
−2σi

)
. (2.8)

Similar regularisedversions of (2.4) arise for other values of the charges corresponding to reflections ofCoulomb
gas correlators.

2.2 One parameter correlators, cyclic symmetry

We shall use Eq. (2.4) as a recursion relation, starting from the explicit expression (2.7). Let us first introduce
some general notation:

G(−)(σ2, β, σ1) := Sb(−β + σ2 + σ1)Sb(Q − β + σ2 − σ1)= g−
(

σ2, β +
b

2
, σ1

)
G(−)(σ2, β + b, σ1). (2.9)

For a non-negative integer k and an integer n of parity p(n) denote

B(σ2, σ1)
(k;p(n)) :=

G(−)(σ2, −kb2 −
n
2b , σ1)

G(−)(σ2, b +
kb
2 −

n
2b , 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 �=0

g−

(
σ2,

b

2
− k

b

2
+

n

2b
, σ1

)
g−

(
σ2,

b

2
+ k

b

2
+

n

2b
, σ1

)
= c21 + c

2
2 − c1c2(−1)

n2cosπkb2 − (2sinπkb2)2 (2.11)

while B(σ2, σ1)
(0;p(n)) =(−1)nc2 − c1. Similarly, we define the dual B̃(σ2, σ1)(n;p(k))

B̃(σ2, σ1)
(n;p(k)) :=

G(−)(σ2, − n2b −
kb
2 , σ1)

G(−)(σ2,
1
b
+ n2b −

kb
2 , σ1)

= (−1)(k+1)(n+1)B̃(σ1, σ2)(n;p(k)) (2.12)

so that the reflection amplitude is expressed as the ratio of polynomials

λ
2β2−Q
2b

L G2(σ2, β2 = b + m2b −
n2
b

, σ1)

Sb(2β2 − Q)
=

G(−)(σ2, β2, σ1)
G(−)(σ2, Q − β2, σ1)

=
B̃(σ2, σ1)

(2n2;p(2m2))

B(σ2, σ1)(2m2;p(2n2))
. (2.13)

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Acta Polytechnica Vol. 50 No. 3/2010

Finally we introduce

P2 ≡ P σ3,σ2,σ1β3,β2,β1 := (−1)
m312+2m2λ

−
m3
12
2

L

Sb((2m1 +1)b)Sb((2m2 +1)b)
Sb(b)

·

m312∑
p=0

Sb((m
3
12 +1)b)

Sb((p +1)b)Sb((m312 +1− p)b)
×

G2(σ2 + p
b
2, β2 − p

b
2, σ3)

G2(σ2, β2, σ3)
(2.14)

G2(σ2 − (m312 − p)b2, β1 − (m
3
12 − p)b2, σ1)

G2(σ2, β1, σ1)

andsimilarly P1 and P3,whichareobtained from(2.14)bycyclicpermutations. Thefinite sum(2.14) isproportional
to 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 which

the first term on the r.h.s. of (2.4) reduces to a 2-point function in order to obtain recursively the most general
correlator with m213 = 0. Then using the analog of the general equation (2.4) for shifts of the pair (β3, β2), we
obtain

C
σ3,σ2,σ1
β3,β2,β1

= −
λ

Q−β123
2b

L

∏
(β3, β2, β1)

B(σ1, σ2)(2m1;0)B(σ2, σ3)(2m2;0)B(σ3, σ1)(2m3;0)
F

σ3,σ2,σ1
β3,β2,β1

,

F
σ3,σ2,σ1
β3,β2,β1

= (−1)2m1((−1)2m2c̃2 − c̃3)B(σ3, σ1)(2m3;0)P σ3,σ2,σ1β3,β2,β1 −
(−1)2m2((−1)2m3c̃3 − c̃1)B(σ2, σ3)(2m2;0)P σ2,σ1,σ3β2,β1,β3 = (2.15)

−
(
c̃1B(σ3, σ2)

(2m2;0)P
σ2,σ1,σ3
β2,β1,β3

+ c̃2B(σ1, σ3)
(2m3;0)P

σ3,σ2,σ1
β3,β2,β1

+ c̃3B(σ2, σ1)
(2m1;0)P

σ1,σ3,σ2
β1,β3,β2

)
where ∏

(β3, β2, β1)=
be0(Q−β123)Γb(2q − β123)Γb(Q − β123)Γb(Q − β213)Γb(Q − β312)

Sb(
1
b
)Sb(

2
b
)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)P

σ3,σ2,σ1
β3,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 the
fact 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 duality

transformation b → 1/b (changing notation mi → ni) gives the correlator in the other thermal case, when all
mi =0 in (1.1). In this case the product of B

(0;p(2ni)) replaces the denominator in (2.15) and the formula confirms
the structure suggested in themicroscopic approach of [5]. The dual polynomial P̃ 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 one
reproduces the formula in [6] for the case {mi = 0, ni – integers}. The expression in [6] is however not explicitly
symmetric under cyclic permutations, rather this symmetry is checked to hold on examples.

2.3 The general correlator

To 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 regularised
again 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),

C
σ3,σ2,σ1
β3,β2,β1

=
λ

Q−β123
2b

L

∏′
(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+2n2B̃(σ2, σ3)(2n2;p(2m2))P̃ σ2,σ1,σ3β2,β1,β3 B(σ3, σ1)

(2m3;p(2n3))P
σ3,σ2,σ1
β3,β2,β1

− (2.18)

(−1)2m2+2n1B̃(σ3, σ1)(2n3;p(2m3))P̃ σ3,σ2,σ1β3,β2,β1 B(σ2, σ3)
(2m2;p(2n2))P

σ2,σ1,σ3
β2,β1,β3

)
with the prefactor ∏′

(β3, β2, β1)=
(−1)m123n123

∏
(β3, β2, β1)S

3
b(
1
b
)Sb(

2
b
− b)

Sb(
n312+1

b
)Sb(

n123+1
b
)Sb(

n213+1
b
)Sb(

n123+2
b

− b)
. (2.19)

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Acta Polytechnica Vol. 50 No. 3/2010

Here, say, the polynomial P2 is given by the first formula (2.14), where now all βi are given by (1.1), with only
the sign in front of (2.14) modified to (−1)m

3
12(1+2n3)+2m32n3+2m2 = (−1)m123(1+2n3)+2m1. Let us also write down

the expression for one of the dual polynomials

P̃1 ≡ P̃ σ2,σ1,σ3β2,β1,β3
(−1)n123(1+2m2)+2n3λ−n

2
13/2

L Sb(
2n1+1

b
)Sb(

2n3+1
b
)

Sb(
1
b
)

n213∑
u=0

Sb(
n213+1

b
)

Sb(
1+u

b
)Sb(

n213+1−u
b
)
× (2.20)

G2(σ1 +
u
2b , Q − β1 −

u
2b , σ2)

G2(σ1, Q − β1, σ2)
G2(σ1 −

n213−u
2b , Q − β3 −

n213−u
2b 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 reproduces
the cyclic identity satisfied by the first order dual polynomials B̃(σ2, σ3)

(0;p(2m2)) = (−1)2m2c̃2 − c̃3, etc., which
appear in the numerator in (2.15). The composition of the duality transformation b → 1/b, mi ↔ ni with reflection
of all three fields keeps (2.18) invariant.

3 Summary and discussion

We have obtained the general Liouville dressing factor in the tachyon 3-point boundary correlatorwith degenerate
c < 1 representations. Formula (2.18) represents the Liouville correlator as a ratio of polynomials of the boundary
cosmological parameters ci, c̃i generalising the partial results in [5, 6]. This solution of the Liouville pentagon
equations extends to theminimalgravity theorywith rational b2, inwhichcase theremayappear further truncations
of the sums. Thegeneral3-pointboundarytachyoncorrelator is aproductof (2.18)andthematter3-pointboundary
correlator, 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 andLiouville spectra and the corresponding
correlators are essentially inverted: the Coulomb gas Liouville correlator for degenerate c > 25 representations
describing both the charges and the boundaries should be combinedwith amatter factor obtained by analytic con-
tinuation of the solution (2.18). Note that the corresponding discrete c < 1 spectrum parametrises the irreducible
representations embedded as submodules of the reducible Virasoro modules. The analogous characteristics of the
c > 25 spectrum (1.1) have been exploited in the construction of the 4-point bulk tachyon correlators [12].

Acknowledgement

P. Furlan acknowledges support from the Italian Ministry of Education, Universities and Research (MIUR).
V. B. Petkova acknowledges hospitality from the Service de Physique Thèorique, CEA-Saclay, France, and ICTP
and INFN, Italy. This research has received some support from the French-Bulgarian RILA project, contract
3/8-2006.

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 and
boundary two-point function, hep-th/0001012.

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[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, to
appear In: J. Phys. A, arXiv:0805.0134.

[12] Belavin, A., Zamolodchikov, Al.: Theor. Math. Phys. 147 (2006) 729, hep-th/0510214.

P. Furlan
Dipartimento di Fisica Teorica dell’Università di Trieste, Italy
Istituto Nazionale di Fisica Nucleare (INFN)
Sezione di Trieste, Italy

V. B. Petkova
Institute for Nuclear Research and Nuclear Energy (INRNE), Bulgarian
Academy of Sciences (BAS), Bulgaria

M. Stanishkov
Institute for Nuclear Research and Nuclear Energy (INRNE), Bulgarian
Academy of Sciences (BAS), Bulgaria

89