The tetrahedron graph in Liouville theory on the pseudosphere 1 Pietro Menotti

We compute analytically the tetrahedron graph in Liouville theory on the pseudosphere. The result allows to extend the check of the bootstrap formula of Zamolodchikov and Zamolodchikov to third order perturbation theory of the coefficient G3. We obtain complete agreement. A.B. Zamolodchikov and Al.B. Zamolodchikov [1] proposed an exact non perturbative formula for the one-point function 〈 Va(z1) 〉 = 〈 e φ(z1) 〉 in Liouville theory on the PoincaréLobachevskiy pseudosphere, being φ the Liouville field. The exact quantum conformal dimensions of the vertex operator Va(z1) are a (Q − a). The background charge Q is related to the coupling constant b by Q = b + b, which is related to the central charge of the theory c by c = 1 + 6Q. Thus, the form of the one-point function is 〈Va(z1) 〉 = U(a) ( 1− z1z̄1 )2a(Q−a) (1) where the proposed formula for the structure constant U(a) is [1] U(a) = [ πμγ(b) ] − a b Γ(bQ) Γ(Q/b) Q Γ(b(Q− 2a)) Γ(b−1(Q− 2a)) (Q− 2a) . (2) A similar conjecture has been put forward for the structure constant of the three point function on the sphere [2] and in this case it has been verified up to order b in perturbation theory [5], within the hamiltonian approach [3, 4]. It is useful to consider [1] instead of 〈 e1 〉 the expansion in powers of a of the logarithm of (1) log 〈 e1 〉