Non invariant zeta-function regularization in quantum Liouville theory
arXiv:hep-th/0612270v1 26 Dec 2006
nto di Fisica dell’Universit`, Pisa 56100, Italy and a INFN, Sezione di Pisa e-mail: email@example.com
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.
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  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) . In  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  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 , 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  Scl [?B ] = lim
1 1 b2 8π
1 (?a ?B )2 + 8π?b2 e?B 2 + 1 4πi
d2 z z dz d? ? z z ?
dz d? z 2 ? ) + ηn log ε2 n z ? zn z ? zn ? ?
+ log R2
and Sq [?B , χ] =
(?a χ)2 + 4π?e?B (e2bχ ? 1 ? 2bχ) d2 z
+ (2 + b2 ) log R2 +
z dz d? ? z z ?
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 (
4bχ3 8b2 χ4 + + ... 3! 4!
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
? ??B + 8π?b e
2 ?B (z)
ηi δ 2 (z ? zi )
where ηi = bαi and the vertex functions are given by Vα (z) = e2αφ(z) = eη?(z)/b ;
? = 2bφ = ?B + 2bχ.
We recall that the action (1) ascribes to the vertex function Vα (z) the semiclassical dimension ?sc (α) = α(1/b ? α) . In performing the perturbative expansion in b we have to keep η1 , . . . ηn constant . The one loop contribution to the n-point function is given by K? 2 = where
D[χ] e? 2
2 1 D = ? ?z ?z + 4?b2 e?B ≡ ? ? + m2 e?B . (8) ? π 2π The usual invariant zeta-function technique  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) =
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.
Alternatively one can consider the elliptic operator D de?ned in (8) and its determinant generated by the zeta function ζD (s) =
being ?i the eigenvalues Dψi = (? and thus
′ ? log(Det D) = ζD (0).
1 ? + m2 e?B (z) )ψi (z) = ?i ψi (z) 2π
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 (z)?i (z ′ ) λi ψi (z)ψi (z ′ ) ?i
g(z, z ′ ) =
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 )
and ζD (s) =
dt ts?1 Tr(e?tD ).
We recall that given the short time expansion Tr(e?tH ) = we have ζ(0) = c0 and
c?1 + c0 + . . . t
? log(DetH) ≡ ζ ′(0) = γE ζ(0) + Finiteε→0
dt Tr(e?tH ) t
dt Tr(e?tH ) = lim ε→0 t
c?1 dt Tr(e?tH ) ? + c0 log ε t ε
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 ?
? ? ? ?
1 ? + V (z) 2π
V (z) = m2 e?B (z) .
1 ?π |z?z′ |2 2t e 2t
z|e 2π t |z ′ d2 z ′ f (z ′ )
? ′ ? ′
e 2π (t?t ) V e 2π t dt′ + . . .
so that Tr(e
? ?(? 2π +V )t
1 )= 2t
1 d z? 2t
V (z) d2 z + O(t).
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
V (z) d2 z.
? log(DetD) ≡
′ ζD (0)
= γE ζD (0) + Finiteε→0
dt Tr(e?tD ) t
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
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
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
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.