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IFUP-TH/2006-41

Non invariant zeta-function regularization in quantum Liouville theory

1

arXiv:hep-th/0612270v1 26 Dec 2006

Pietro Menotti

Dipartimento di Fisica dell’Universit`, Pisa 56100, Italy and a INFN, Sezione di Pisa e-mail: menotti@df.unipi.it

Abstract We consider two possible zeta-function regularization schemes of quantum Liouville theory. One refers to the Laplace-Beltrami operator covariant under conformal transformations, the other to the naive non invariant operator. The ?rst produces an invariant regularization which however does not give rise to a theory invariant under the full conformal group. The other is equivalent to the regularization proposed by Zamolodchikov and Zamolodchikov and gives rise to a theory invariant under the full conformal group.

1

Work supported in part by M.I.U.R.

Quantum Liouville theory has been the subject of intense study following di?erent lines of attack. While the bootstrap [1, 2, 3, 4, 5] starts from the requirement of obtaining a theory invariant under the full in?nite dimensional conformal group, the more conventional ?eld theory techniques like the hamiltonian and the functional approaches depend in a critical way on the regularization scheme adopted. In the hamiltonian treatment [6] for the theory compacti?ed on a circle the normal ordering regularization gives rise to a theory invariant under the full in?nite dimensional conformal group. It came somewhat of a surprise that in the functional approach the regularization which realizes the full conformal invariance is the non invariant regularization introduced by Zamolodchikov and Zamolodchikov (ZZ) [4]. In [7] it was shown that such a regularization provides the correct quantum dimensions to the vertex functions on the sphere at least to two loops while in [8] it was shown that such a result holds true to all order perturbation theory on the pseudosphere. Here we consider the approach in which the determinant of a non covariant operator is computed in the framework of the zeta function regularization and show that this procedure is equivalent to the non invariant regularization of the Green function at coincident points proposed by ZZ [4], and extensively used in [7, 8, 9, 10]. For de?niteness we shall refer to the case of sphere topology. The complete action is given by SL [?B , χ] = Scl [?B ] + Sq [?B , χ] where [7] Scl [?B ] = lim

N

εn →0

R→∞

1 1 b2 8π

Γ

1 (?a ?B )2 + 8π?b2 e?B 2 + 1 4πi

d2 z z dz d? ? z z ?

(1)

?

n=1

ηn

1 4πi

?B (

?Γn

dz d? z 2 ? ) + ηn log ε2 n z ? zn z ? zn ? ?

?B

?ΓR

+ log R2

and Sq [?B , χ] =

εn →0

lim

R→∞

1 4π

(?a χ)2 + 4π?e?B (e2bχ ? 1 ? 2bχ) d2 z

Γ

+ (2 + b2 ) log R2 +

1 4πi

?B

?ΓR

z dz d? ? z z ?

+

b 2πi

χ

?ΓR

z dz d? ? z z ?

( . 2)

In eq.(2) Γ is a disk of radius R from which disks of radius εn around the singularities have been removed. We recall that Scl is O(1/b2 ) while the ?rst integral appearing in the quantum action (2) can be expanded as 1 4π (?a χ)2 + 8π?b2 e?B χ2 + 8π?b2 e?B (

Γ

4bχ3 8b2 χ4 + + ... 3! 4!

d2 z.

(3)

The quantum n-point function is given by Vα1 (z1 )Vα2 (z2 ) . . . Vαn (zn ) = e?Scl [?B ] 1 D[χ] e?Sq (4)

where the ?B appearing in the classical and quantum actions is the solution of the classical Liouville equation in presence of n sources

n

? ??B + 8π?b e

2 ?B (z)

= 8π

i=1

ηi δ 2 (z ? zi )

(5)

where ηi = bαi and the vertex functions are given by Vα (z) = e2αφ(z) = eη?(z)/b ;

2

? = 2bφ = ?B + 2bχ.

(6)

We recall that the action (1) ascribes to the vertex function Vα (z) the semiclassical dimension ?sc (α) = α(1/b ? α) [11]. In performing the perturbative expansion in b we have to keep η1 , . . . ηn constant [3]. The one loop contribution to the n-point function is given by K? 2 = where

1

D[χ] e? 2

1

R

χ(z)Dχ(z)d2 z

(7)

2 1 D = ? ?z ?z + 4?b2 e?B ≡ ? ? + m2 e?B . (8) ? π 2π The usual invariant zeta-function technique [12] for the computation of the functional determinant K consists in writing χ(z)Dχ(z) d2 z = χ(z) ? 1 ?LB + m2 χ(z) dρ(z) 2π (9)

being dρ(z) = e?B (z) d2 z the conformal invariant measure and ?LB = e??B ? (10)

the covariant Laplace-Beltrami operator on the background ?B (z) generated by the n

1 charges. The determinant of the elliptic operator ? 2π ?LB + m2 is de?ned through the

zeta function ζ(s) =

∞

λ?s i

i=1

(11)

being λi the eigenvalues of the operator H H?i = λi ?i where H=? 1 ?LB + m2 . 2π (12)

For an elliptic operator the sum (11) converges for Re s su?ciently large and positive and the determinant is de?ned by analytic continuation as ? log(DetH) = ζ ′(0). 2 (13)

Such a value is usually computed by the heat kernel technique [13, 14], which we shall also employ in the following. The great advantage of the zeta-function regularization is to provide an invariant regularization scheme as the eigenvalues λi are invariant under conformal transformations (SL(2C) for the sphere, U(1, 1) for the pseudosphere).

1 Associated to the operator H = ? 2π ?LB + m2 we can consider the Green function

HG(z, z ′ ) = δ 2 (z ? z ′ )e??B (z ) ≡ δI (z, z ′ ) where δI (z, z ′ ) is the invariant delta function.

′

(14)

Alternatively one can consider the elliptic operator D de?ned in (8) and its determinant generated by the zeta function ζD (s) =

i=1 ∞

??s i

(15)

being ?i the eigenvalues Dψi = (? and thus

′ ? log(Det D) = ζD (0).

1 ? + m2 e?B (z) )ψi (z) = ?i ψi (z) 2π

(16)

(17)

The Green function for the operator D is de?ned by Dg(z, z ′ ) = δ(z ? z ′ ) (18)

and multiplying by e??B (z) we see that G(z, z ′ ) = g(z, z ′ ). The determinant (17), being D non covariant is not an invariant under conformal transformations. By using the spectral representation we can also write G(z, z ′ ) =

i

?i (z)?i (z ′ ) λi ψi (z)ψi (z ′ ) ?i

(19)

g(z, z ′ ) =

i

(20)

where the ?i (z) and ψi (z) are normalized by |?i (z)|2 dρ(z) = 1; and |ψi (z)|2 d2 z = 1 (21)

being dρ(z) the invariant measure e?B (z)d2 z. We shall exploit in both cases the heat kernel technique. We have ζ(s) = 1 Γ(s)

∞

dt ts?1 Tr(e?tH )

0

(22)

3

and ζD (s) =

1 Γ(s)

∞

dt ts?1 Tr(e?tD ).

0

(23)

We recall that given the short time expansion Tr(e?tH ) = we have ζ(0) = c0 and

∞

c?1 + c0 + . . . t

(24)

? log(DetH) ≡ ζ ′(0) = γE ζ(0) + Finiteε→0

ε

dt Tr(e?tH ) t

(25)

where Finiteε→0

ε

∞

dt Tr(e?tH ) = lim ε→0 t

∞ ε

c?1 dt Tr(e?tH ) ? + c0 log ε t ε

(26)

which gives the value of the determinant of the operator H and similarly for D. We examine ?rst (23) writing D=? We have z|e 2π t |z ′ = where (e 2π t f )(z) = and e?(? 2π +V )t = e 2π t ?

0

? ? ? ?

1 ? + V (z) 2π

with

V (z) = m2 e?B (z) .

(27)

1 ?π |z?z′ |2 2t e 2t

?

(28)

z|e 2π t |z ′ d2 z ′ f (z ′ )

t

? ′ ? ′

(29)

e 2π (t?t ) V e 2π t dt′ + . . .

t

(30)

so that Tr(e

? ?(? 2π +V )t

1 )= 2t

1 d z? 2t

2

dt′

0

V (z) d2 z + O(t).

(31)

The coe?cient of the ?rst term is the in?nite volume term; as such term is proportional to 1/t it does not contribute to the ?nite part. In addition we see that c0 = ζD (0) = ? We compute

∞

1 2

V (z) d2 z.

(32)

? log(DetD) ≡

′ ζD (0)

= γE ζD (0) + Finiteε→0

ε

dt Tr(e?tD ) t

(33)

by taking a variation of the background ?B as due e.g. to a change of the strength of the sources. 4

We have ? δ log(Det D) = γE δζD (0) ? Finiteε→0

ε

∞

dt Tr(δV e?tD ) =

= γE δζD (0) ? Finitet→0 Tr(δV D?1 e?tD ) = = γE δζD (0) ? Finitet→0 d2 z δV (z) g(z, z ′ )d2 z ′ z ′ |e?tD |z (34)

being g(z, z ′ ) the Green function de?ned in (18). The ?nite part appearing in the above equation can be computed by exploiting again the short time expansion (30). We have ? d2 z δV (z) g(z, z ′ ) d2 z ′ z ′ |e?tD |z = ? d2 z δV (z) g(z, z ′ ) d2 z ′ 1 ?π |z?z′ |2 2t e + O(V ). 2t (35)

We shall show in the following that the O(V ) term does not contribute to the ?nite part as it goes to zero for t → 0. With regard to the ?rst term it equals ?

|z?z ′ |2 1 1 d2 z δV (z) ? log |z ? z ′ |2 + gF (z) + o(|z ? z ′ | d2 z ′ e?π 2t 2 2t

(36)

which for t → 0 goes over to ? δV (z) gF (z) d2 z + d2 z δV (z)

|z?z ′ |2 1 1 log |z ? z ′ |2 d2 z ′ e?π 2t 2 2t

(37)

and we have written 1 g(z, z ′ ) = ? log |z ? z ′ |2 + gF (z) + o(|z ? z ′ |). 2 with the result 1 π Finitet→0 (log t ? log ? γE ) 2 2 1 π d2 z δV (z) = ? (log + γE ) 2 2 d2 z δV (z). (39) (38)

Thus we have to compute the ?nite part of the second term in eq.(37). This is easily done

We must now examine in eq.(35) the O(V ) term

t

IV =

d z δV (z)g(z, z )d z

2

′

2 ′ 0

′′ ′ 2 ?z|2 1 ′′ ? π|z ?z ′ )| 2 ′′ 1 ? π|z 2t′ 2(t?t e dt′ . V (z ) d z ′ e 2(t ? t′ ) 2t

′′

(40)

We have that δV (z) is integrable and also d2 z |δV (z) log |z ? z ′ |2 | < ∞. (41)

At this stage were it V (z) bounded i.e. |V (z)| < VM it would follow immediately that |IV | < t VM gM + c + 1 π log( ) 2 2t 5 δV (z)d2 z (42)

being gM = sup |g(z, z ′ ) + log |z ? z ′ | | + sup |g(z, z ′ )|

|z?z ′ |<1 |z?z ′ |>1

(43)

and c a numerical constant. The r.h.s. of eq.(42) vanishes for t → 0. In the case at hand V (z) = m2 e?B with ?B satisfying eq.(5) is not bounded but has locally integrable singularities. Such a situation can be dealt with as follows. Isolate the singularities with disks of given radius around them. For the contribution outside these disks the above reasoning applies. For the disk contributions we can use the following bound. Let the singularity of e?B (z) be located in zero and given by const |z|?2γ where due to the local integrability of the area γ < 1.

t

′′

dt

0

′

′′ ′ 2 |z ?z|2 1 ′′ ′′ 1 ?π |z ?z | e 2(t?t′ ) |z |?2γ d2 z ′ e?π 2t′ = 2(t ? t′ ) 2t

tγ?1 = 4

t 0

dt′ t′ (t ? t′ )γ = tγ?1 4

e

t 0

?π |z

′ ?z|2 2t

e

?u 2

2

z z′ + ′) u + 2( ′ t?t t e?π

|z ′ ?z|2 2t

t′ (t ? t′ ) t

?2γ

d2 u <

dt′ t′ (t ? t′ )γ

e?

u2 2

|u|?2γ d2 u < (44)

< 2γ?2 π γ+1/2

Γ2 (1 ? γ) ?γ ?π |z′ ?z|2 2t t e . Γ(3/2 ? γ)

As γ < 1 repeating the argument which leads to eq.(42) we have that the integral (40) goes to zero for t → 0. Putting all contributions together we ?nd

′ ? δζD (0) = δ log(DetD) =

d2 z δV (z) gF (z) +

1 π log + γE . 2 2

(45)

Thus we have for the variation of the determinant the same result which is obtained formally from δ log(DetD)? 2 = ?

1

1 2

d2 z δV (z)g(z, z)

(46)

where one replaces the divergent quantity g(z, z) by the regularized value [4] g(z, z) = lim (g(z, z ′ ) + ′

z →z

1 log |z ? z ′ |2 ) + C = gF (z) + C 2

(47)

with C =

1 2

log(π/2) + γE . The contribution (45) changes the dimensions of the vertex

?eld Vα (z) from the semiclassical value ?sc (α) = α(1/b ? α) to the quantum dimension ?(α) = α(1/b + b ? α) [4, 7, 8, 9, 10]. At the perturbative level on the pseudosphere ZZ [4] chose C = 0. Such a constant can be absorbed in a constant shift in the Liouville ?eld. This can be shown for the contribution of the quantum determinant to all order 6

in ηj = bαj for the n-point function of eq.(4). In fact the background ?eld solves eq.(5); taking the derivative w.r.t. ηj and integrating over the whole plane we have d2 z ?(?b2 e?B ) 2 d z=1 ?ηj (48)

insofar due to the behavior of ?(?b2 e?B )/?ηj at in?nity only the gradient of the ?eld ?B around the charges contributes to the boundary integral and the integral (48) converges. As a result integrating back in ηj , (DetD)?1/2 gets multiplied by e?2Cηj = e?2Cbαj corresponding to the shift in the ?eld φ → φ ? bC. The fact that a change of the regularized value of the Green function at coincident points is equivalent to the above mentioned shift in the ?eld φ can be shown on the pseudosphere to all order perturbation theory by exploiting the identity which relates the tadpole graph and the simple loop [4] ? 4b2 ? g (z, z ′ )dρ(z ′ )?(z ′ , z) + g (z, z) = ? g ? 1 2 (49)

which holds for the Green function on the background of the pseudosphere [4, 16] g (z, z ′ ) = ? ? 1 2 1+ω log ω + 2 1?ω (50)

with ω = |(z ? z ′ )/(1 ? z ′ z )|2 and extending a combinatorial argument developed in [8] in ? connection with the quantum dimensions of the vertex function on the pseudosphere to all order perturbation theory. We want now to compare the result (45) with the computation of the determinant of the covariant operator H eq.(12). We recall that in the eigenvalue equation (12) in presence of conical singularities it may occur that both behaviors of the solution ?n at the singularity are square integrable in the invariant metric dρ(z) [15] and this fact gives rise to the problem of the self-adjoint extension of the operator (12). A standard way for doing that is to regularize the singularities and then take the limit when the regularization is removed. The main point however is that the so de?ned operator has an invariant spectrum and as such through the zeta- function procedure gives rise to an invariant value for the functional determinant. The derivative w.r.t. ηj of the change of log(DetH)? 2 under dilatations vanishes and the same happens for the O(b0 ) boundary terms of eq.(2). Thus at one loop the correction to the semiclassical dimensions ?sc (α) = α(1/b ? α) vanishes. In particular at one loop the cosmological term e2bφ maintains the weights (?(b), ?(b)) = (1?b2 , 1?b2 ). Higher order corrections cannot bring such weights to the value (1, 1), because the two loop correction is of the form f (η) b2 [7] and as we must have for the dimension, ?(0) = 0 we have f (0) = 0. But then at two loop we have ?(b) = 1 ? b2 + O(b4 ) which cannot be identically 1 in b. 7

1

For the sake of comparison we shall compute the determinant of the covariant operator H given in eq.(12) in the regularized case. The related zeta-function is given by eq.(22) where the trace is Tr(e?tH ) =

n

dρ(z)?n (z)(e?tH ?n )(z) = e?tλn

n

(51)

dρ(z)?n (z) z|e?tH |z ′ d2 z ′ ?n (z ′ ) = and we used the following de?nition for z|e?tH |z ′ (e?tH f )(z) ≡ and thus it follows Tr(e?tH ) = d2 z z|e?tH |z . z|e?tH |z ′ d2 z ′ f (z ′ )

(52)

(53)

The determinant of H is given by eq.(25). The value of ζ(0) in eq.(25) is found by standard techniques [14] to be ζ(0) = c0 = ? m2 2 e?B d2 z + 1 24π R(z) e?B d2 z (54)

where R(z) is the curvature related to the metric e?B idz ∧ d?/2. The ?rst integral apart z the multiplicative constant is the volume while the second is simply the Gauss-Bonnet term and as such a topological invariant. The change of ζ ′ (0) under a small change of ?B is given by

∞

δζ (0) = γE δζ(0) ? Finiteε→0

ε

′

dt d2 z δ?B (z)e??B (z) z|

1 ?e?tH |z = 2π

??B ? +m2 ) 2π

(55) |z .

γE δζ(0) ? Using

1 Finitet→0 2π

d2 z δ?B (z)e??B (z) ?G(z, z ′ )dρ(z ′ ) z ′ |e?t(?e

e??B (z) ? ?G(z, z ′ ) = δI (z, z ′ ) ? m2 G(z, z ′ ) 2π we obtain δζ ′(0) = γE δζ(0) + Finitet→0 m2 Finitet→0 d2 z δ?B (z) z|e?tH |z ?

(56)

(57)

d2 z δ?B (z) G(z, z ′ ) dρ(z ′ ) z ′ |e?tH |z .

Using the same technique as in deriving eq.(45) we ?nd δζ ′(0) =

8

γE ? 1 24π

m2 δ 2

e?B (z) d2 z +

1 δ 24π

R(z) e?B d2 z

?

m2 2

dρ(z)δ?B (z)+

(58)

δ?B (z)R(z) dρ(z) ? m2

1 1 π e?B (z) d2 z δ?B (z)(gF (z) + ?B (z) + (log + γE ) = 2 2 2

m2 R(z) ? ) = ? dρ(z) δ?B (z)( 2 24π 1 π 1 ?m2 dρ(z) δ?B (z)(gF (z) + ?B (z) + log( ) + γE ). 2 2 2 In the ?rst term one recognizes the variation of the conformal anomaly in presence of the “mass” m while the second term is the same as the result (45) except for a di?erent regularization of the Green function. The additional contribution ?B (z)/2 can be understood as an invariant regularization of the Green function at coincident points obtained

1 by subtracting the divergence ? 2 (log 2σ(z, z ′ )) being 2σ(z, z ′ ) the square of the invariant

distance which for small z ? z ′ reduces to e?B (z) |z ? z ′ |2 . Invariant regularization of this kind for the Green function has been considered in the context of Liouville theory e.g. in [16]; the resulting theory is invariant only under the group SO(2, 1). In conclusion, the zeta-function regularization [12] and the related heat kernel technique [13] has been introduced in the literature with the aim of providing an invariant regulator. Here it has been shown that in quantum Liouville theory, due to the non invariance of the total action SL [?B , χ] = Scl [?B ] + Sq [?B , χ] a non invariant regularization is necessary and that this is provided by the zeta-function regularization of a non covariant operator. Such a situation may well occur in other quantum ?eld theory models.

Acknowledgments

I am grateful to Damiano Anselmi for useful discussions.

References

[1] M. Goulian, Miao Li, Phys. Rev. Lett. 66 (1991) 2051; H. Dorn and H.J. Otto, Nucl. Phys. B429 (1994) 375. [2] J. Teschner, Phys. Lett. B363 (1995) 65; Class. Quant. Grav. 18 (2001) R153; Int. J. Mod. Phys. A19 S2 (2004) 436; Nucl.Phys.B622 (2002) 309; From Liouville Theory to the Quantum Geometry of the Riemann Surfaces, hep-th/0308031; [3] A.B. Zamolodchikov and Al.B. Zamolodchikov, Nucl. Phys. B477 (1996) 577.

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[4] A.B. Zamolodchikov and Al.B. Zamolodchikov, Liouville Field Theory on a Pseudosphere, hep-th/0101152. [5] V. Fateev, A.B. Zamolodchikov and Al.B. Zamolodchikov, Boundary Liouville Field Theory I. Boundary State and Boundary Two-point Function, hep-th/0001012; J. Teschner, Remarks on Liouville theory with boundary, hep-th/0009138. [6] T.L. Curtright and C.B. Thorn, Phys. Rev. Lett. 48 (1982) 1309; E. Braaten, T.L. Curtright and C.B. Thorn, Phys. Rev. Lett. 51 (1983) 19; Ann. Phys. 147 (1983) 365; G. Jorjadze and G. Weigt, Phys. Lett. B581 (2004) 133. [7] P. Menotti and G. Vajente, Nucl. Phys. B709 (2005) 465. [8] P. Menotti and E. Tonni, Phys. Lett. B633 (2006) 404; JHEP 0606:020,2006 [9] P. Menotti and E. Tonni, Phys. Lett. B586 (2004) 425; Nucl. Phys. B707 (2005) 321. [10] P. Menotti and E. Tonni, JHEP 0606:022, 2006 [11] L.A. Takhtajan, Mod. Phys. Lett. A11 (1996) 93. [12] J.S. Dowker and R. Critchley, Phys. Rev. D 13 (1976) 3224; S. Hawking, Comm. Math. Phys. 55 (1977) 133. [13] B.S. DeWitt, Dynamical Theory of Groups and Fields, Gordon and Breach, New York, 1965. [14] R. Balian and C. Bloch, Ann. Phys.64 (1971) 271; O. Alvarez, Nucl. Phys. B216 (1983) 125; A.O. Barvinski and G.A. Vilkoviski, Phys. Rep. 119 (1985) 1. [15] P. Menotti and P.P. Peirano, Phys. Lett. B353 (1995) 444; Nucl. Phys. B473 (1996) 426; Nucl.Phys.Proc.Suppl. 57 (1997) 82. [16] E. D’Hoker, D.Z. Freedman and R. Jackiw, Phys. Rev. D28 (1983) 2583.

10

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