6 v 2 5 O ct 1 99 1 On the Solution of Topological Landau - Ginzburg

The solution is given for the c = 3 topological matter model whose underlying conformal theory has Landau-Ginzburg model W = − 4 (x + y)+αxy. While consistency conditions are used to solve it, this model is probably at the limit of such techniques. By using the flatness of the metric of the space of coupling constants I rederive the differential equation that relates the parameter α to the flat coordinate t. This simpler method is also applied to the x + y-model. USC-91/023 August 1991 1 Solution of the x + y-model I first use the methods described in [1, 2] to compute the perturbed topological correlation functions and their prepotential for the superconformal field theory that can be described by the following Landau-Ginzburg potential: W = − 1 4 (x + y) + αxy (1) where the parameter α is complex. When α vanishes the model reduces to the tensor product of two minimal models with two commuting U(1) currents. Define φi, i = 0, ..., 8 to be the chiral primary fields such that for α vanishing we have: φ0 = 1, φ1 = x, φ2 = y, φ3 = x , φ4 = xy, φ5 = y , φ6 = x y, φ7 = xy , φ8 = x y. Let ti be the coupling constants corresponding to the chiral primary fields φi, such that ti = 0 for i = 0 to 8 corresponds to the unperturbed model with α = 0. The perturbed three-point functions are defined by: Cijk = 〈φiφjφk exp[ ∑ l tl ∫ dz G− − 1 2 G̃− − 1 2 φl]〉 , (2) from which one can construct a prepotential [1, 3] F such that Cijk = ∂F ∂ti∂tj∂tk . (3) From the expansion of the exponential in (2) and the use of the symmetries of the unperturbed potential one determines the non-vanishing correlators. Integrating (3) one readily gets the general form of the prepotential F : F(~t )= 1 2 tt0 + t(t1t7 + t2t6 + t3t5) + 1 2 tt4 + t1t2t4f0 + 2 (t1t3 + t 2 2t5)f1 + 1 2 (t1t5 + t 2 2t3)f2 + 1 4 (t1t 2 7 + t 2 2t 2 6)f3 + 4 (t1t 2 6 + t 2 2t 2 7)f4 + t1t2t6t7f5 + 1 2 (t1t 2 3t7 + t2t 2 5t6)f6 +t3t5(t1t7 + t2t6)f7 + t4(t1t3t6 + t2t5t7)f8 + 1 2 t4(t1t7 + t2t6)f9 + 2 (t1t 2 5t7 + t2t 2 3t6)f10 + t4(t1t5t6 + t2t3t7)f11 + 1 6 t3t5(t 2 3 + t 2 5)f12 + 1 24 (t3 + t 4 5)f13 + 1 4 t4(t 2 3 + t 2 5)f14 + 1 4 t3t 2 5f15 + 2 t3t 2 4t5f17 + 1 24 t4f19 + (higher order terms) (4) where fn ≡ fn(t). There are fifty four unknown functions to be determined in the complete expression for F ! The fn’s explicitly appearing in equation (4) correspond to all the three and four-point functions of the relevant perturbations. The remaining fn’s, which determine the higher order perturbed functions, can all be easily found in terms of the three and four-point functions. The functions fn all have the following form: fn(t) = ∑ ∞ m=0 amt 2m+n where n ≡ n mod 2 and t = t8. These functions are determined by solving the highly redundant set of equations obtained from requiring that the Cijk be the structure constants of an associative algebra Cijp g Cklq = Cikp g Cjlq (5)