An Exact Solution to O(26) Sigma Model Coupled to 2-D Quantum Gravity

By a mapping to the bosonic string theory, we present an exact solution to the O(26) sigma model coupled to 2-D quantum gravity. In particular, we obtain the exact gravitational dressing to the various matter operators classified by the irreducible representations of O(26). We also derive the exact form of the gravitationally modified beta function for the original coupling constant e2. The relation between our exact solution and the asymptotic solution given in ref[3] is discussed in various aspects. ⋆ work supported by NSERC postdoctoral fellowship e-mail: [email protected] Consider the following two dimensional O(N) nonlinear sigma model[1] I = 1 e2 ∫ dσ∂n∂ni. (1) This model is well known to exhibit such features as the asymptotic freedom at the UV limit, and dimensional transmutation leading to massive modes as low-lying excitations, etc. An easy way to visualize these properties is through the large N expansion, in which the mass of the vector-like excitation is related to the bare coupling constant by m2 = Λ2 exp(− 4π Ne ). An exact solution to this model is also available by the means of Bethe Ansatz[2]. We may ask what would happen if this model is coupled to the 2-D gravity. This is a problem of the gravitational dressing to a renormalizable but not conformally invariant 2-D field theory. This problem has recently attracted some attention, see Ref.[3,4]. In particular, a modification to the beta functions in the presence of gravity has been derived to the one-loop level[3]. The modification, suspected to be universal, is verified successfully in several cases including the gravitational dressing to the O(N) sigma model (1). It is important to extend these results to higher loops, and if possible, to obtain exact solutions to this problem. In analogy to the 2-D conformal matters coupled to gravity, there are the light-cone and conformal gauge approaches available to this problem[5,6]. The conformal gauge approach, known as the DDK approach in the case of conformal matter coupling to gravity, requires to introduce a fiducial world-sheet metric ĝ, with respect to which the gauge fixed theory is exactly conformal invariant. As usual, the Liouville mode φ serves as the only dynamic degree of freedom left from gravity, and its coupling to the matter is encoded in a nonlinear sigma model with both φ and the matter fields as coordinates. The conformal invariance can be implemented by requiring the beta functions of this sigma model to vanish exactly. This procedure is reduced to the DDK approach when the bare matter is conformally invariant. When the bare matter is not conformally invariant, one must in principle solve the equation of the exact beta functions. Up to the one-loop level, these beta functions read [7] β ij = Rij −∇i∇jΦ β = 26−N 3 −∇Φ− (∇Φ) (2) where Gij is the target space metric and Φ is the dilaton background. In principle, the beta functions at higher orders can be derived by the means of background expansion. Unfortu-