Are there really any AdS_2 branes in the euclidean (or not) AdS_3?

We do not find any AdS_2 branes, neither in the H_3^+ WZNW model nor the SL(2,R) WZNW model. We then reexamine the case of the branes that possess a su(2) symmetry: we speculate that they would have to live on the boundary of AdS_3. This cannot be realized in an euclidean spacetime, but in the SL(2,R) WZNW model by analytical continuation.


Introduction
The discussion of maximally symmetric branes in the euclidean (and lorentzian) AdS 3 has received some attention recently [1,2,3,4,5,6,7,8,9,10,11,12]. It seems established that the possible maximally symmetric branes of the model possess either a sl(2, R) symmetry (AdS 2 branes) or a su(2) symmetry (called spherical branes in [12]), according to the different possible gluing conditions for the currents on the boundary of the worldsheet. Using conformal field theory techniques already developped for Liouville field theory in [13,14], the authors of [10,11,12] proposed a microcospic description of these branes.
In [10] were proposed one point functions in the sl(2, R) and su (2) cases, which, as noticed in [12], turned out to be incorrect as the authors had wrong ansatz for its space time dependence. In [11], the one point function in the sl(2, R) case was proposed, in [12], the one point functions in the sl(2, R) (which coincides with the one of [11]), and su(2) cases as well as the boundary reflection amplitude were constructed. The Cardy condition was checked in both cases, leading to a continuous spectrum of boundary fields in the sl(2, R) case, and to a discrete and finite one in the su(2) case. However, these results do not completly lie on solid grounds, as the functional relations satisfied by these one point functions and the boundary reflection amplitude were only partially solved (see the conclusion of [12]). We would like to emphasize that all functional relations must be solved. Thus, we do not consider the respective Cardy conditions as proven, as neither the quantities needed in the closed string channel (the one point function) nor in the open string channel (the boundary reflection amplitude in the sl(2, R) case) are perfectly under control. Some time ago, one of the authors of the present paper (B.P.) has emitted some doubts about the existence of these AdS 2 branes, (see the conclusion of [15]), as it does not seem to be possible to construct coherently the boundary three point function (i.e. the scattering amplitude of open string states). So we decide to reexamine the problem once more. This paper is organized as follows: section one contains basic definitions and notations, as well as the relations taken from [12] that define the AdS 2 and the su(2)-branes. In section two we check in the AdS 2 branes case the variational principle: it is satisfied without the need of adding any boundary action, so we do not find any source that renders the boundary problem interactive. This is problematic, for if the branes are curved, then it should be thanks to some boundary potential. It remains only the case of the straight brane to study (which is not curved by definition, and for which there is no boundary potential), but even in this case we do not manage to check the results of [12] against the second factorization constraint; so we propose to discard the existence of the AdS 2 branes. In section three, we reexamine the validity of the factorization constraints in the su(2) case. They have no solution in an euclidean spacetime, but it might be possible to construct a consistent boundary conformal field theory in the SL(2, R) WZNW model (the results are those of [12], once the analytical continuation from euclidean to lorentzian of the spacetime and worldsheet coordinates is performed). A striking similarity with the boundary conditions that appear in Liouville field theory on the euclidean AdS 2 considered by Zamolodchikov and Zamolodchikov in [14] is discussed.

Preliminaries [12]
The symmetric space H + 3 consists of hermitian 2×2 matrices h with determinant det h = 1 and positive trace. We parametrize this space through coordinates (φ, γ,γ) such that φ is real and γ is a complex coordinate with conjugateγ. The space H + 3 is equipped with the following metric and H-field, We shall consider the following 2-form potential B ′ for H: The B-field is imaginary so the theory is non-unitary. The action functional for closed strings moving on H + 3 then reads 3 : The currents. Let us introduce the following matrices These are matrix representatives of the Lie algebra sl(2, R), i.e. they obey the relations For the chiral currents we use When we expand them according to J(z) = T + J + + T − J − + 2T 0 J 0 , we obtain expressions for the components J + (z) := k γ 2 e 2φ∂ γ −∂γ − 2γ∂φ .
The components of the anti-holomorphic currents are constructed in an analogous way. Both sets of currents are related by complex conjugation (J ± ) * = (J) ∓ and (J 0 ) * = −J 0 .
The AdS 2 branes. They correspond to surfaces which are characterized by the equation where c is a constant. In terms of the coordinates introduced above one gets the equation The currents satisfy the following relations on the boundary of the worldsheet 4 This implies that the current obeys (J ± ) * = −J ± and (J 0 ) * = J 0 at z =z.
Branes that preserve a su(2) symmetry. They are such that with c constant. This equation can be rewritten as: The currents satisfy 5 along the boundary z =z : (2) current algebra on the boundary of the worldsheet.

AdS branes
The star conditions for the currents give the following boundary conditions for the fields at z =z: If one sets: z = τ + iσ,z = τ − iσ , ∂ z = 1 2 (∂ τ − i∂ σ ) , ∂z = 1 2 (∂ τ + i∂ σ ) , then one can rewrite the boundary conditions as Then, using these conditions, the variational principle states that at σ = 0, The variational principle is thus satisfied without the need of adding any boundary term in the action. This absence of boundary potential leads to some problems with respect to the analysis of [12]: it suggests that the observables depend on the bulk cosmological constant only (called λ b in [12]), whereas the one point function and boundary two point function proposed in [12] behaves like λ α b f (cos πb 2 (2ρ + 1)) where ρ is the boundary condition and α some exponent. This scaling is actually what we would have expected had we found a boundary potential of the form √ λ b cos πb 2 (2ρ + 1) R dxB(x) (the real axis is the boundary of the worldsheet). Of course this argument is not sufficient to exclude the particular case of the straight brane for which c = 0 (cos πb 2 (2ρ + 1) ≡ 0 in this case): the 5 Same as above.
scaling of [12] matches the expected scaling; however, the computation in this particular case shows that 6 the one point function (amongst others) proposed in [11,12] does not satisfy the factorization constraint arising when one considers the degenerate field with spin 1/2b 2 . We do not see how to construct a coherent conformal field theory in this case.

su(2)-branes
Let us remind that to construct the one point function in this case, one first starts with a bulk two point function, where one the fields has a spin 1/2, the other a spin j (see [14] for such a discussion). There are two equivalent ways of expressing this two point function: either one first merges the two operators in the bulk (this is given by a special case of the three point function structure constant), then one approaches the resulting field to the boundary, which is given by the one point function; in the other channel, the two fields in the bulk do approach the boundary, which gives a product of two one point functions U ρ (1/2)U ρ (j) (ρ is the boundary label). It is not clear whether the validity condition for such a factorization has been properly discussed in [10,12]: one requires a two point function to factorize into a product of two one-point function, but, as mentionned in [14], this can be the case only when the fields are very far apart in the spacetime when they approach the boundary; to have the geodesical distance between the fields become infinite as they approach the boundary of the worldsheet, the fields should be at the boundary of H + 3 , i.e. at φ = +∞, and consequently γγ + 1 = 0 for the relation written in (10) to be satisfied 7 . This equation does not have any solution in the euclidean spacetime, so it seems that these su(2)-branes do not exist in H + point out that the relation e φ ∼ c 1 + ab , valid at 1 + ab = 0 , φ = +∞, is very reminiscent of what was found for Liouville field theory on the pseudosphere (euclidean AdS 2 ) [14]. In this case the metric on the Lobachevski plane is where φ L is the Liouville field, and R is interpreted here as the radius of the pseudosphere. It was shown in [14] that the boundary fields live on the boundary of the surface parametrized by zz = 1 (called the absolute), and that the possible boundary conditions are in one to one correspondence with the degenerate representations of the Virasoro algebra. We believe it is no accident if the boundary three point function of SL(2, R) model can be written in terms of the boundary three point function in Liouville field theory 10 (we discard on both sides the worldsheet and space-time dependence, and consider only the structure constants).
where C stands for the boundary three point function in the SL(2, R) WZNW model, D is the boundary three point function in Liouville field theory; a boundary operator is labelled by its spin j, and its left and right boundary conditions ρ 1 and ρ 2 . The labels here are integers submitted to the conditions 11 ρ 2 = ρ 1 + j 1 − n, j 3 = j 2 + j 1 − m, n, m ∈ N We do not infer that these two models are equivalent: it was shown in [14] in the pseudosphere case, the possible boundary conditions are parametrized by two positive integers (s, t), and if s > 1, the one point function does not have any usual classical limit. In the WZNW model considered here, the boundary conditions are parametrized by one positive integer only, and the one point function does have a smooth classical limit. The situation is very analogous to Liouville theory on the pseudosphere with s = 1.