A note on Liouville theory

An exact differential equation is derived for the evolution of the Liouville effective action with the mass parameter. This derivation is based on properties of the exponential potential and some consequences of the equation are discussed. Liouville theory has been studied for a long time [1] and is currently relevant to two-dimensional quantum gravity or non critical string theory. For a recent review on Liouville formalism, see [2] and references therein. For a recent review on the phenomenological implications, see [3] and references therein. The classical action defining the theory is Sλ[φ̃] = ∫ dξ √ g { 1 2 gab∂ φ̃∂φ̃+Rφ̃+ λme } , (1) where gab is the fixed world sheet metric, g is its determinant and R is the curvature scalar. The dimensionless parameter λ controls the strength of the only dimensionfull parameter m. The field φ̃ is dimensionless, leading to an infinite set of classically marginal interactions. This is specific to the two-dimensional world on which this field lives, which gives rise to the rich conformal properties of the model. 1 The classical Liouville theory is well known, as well as the exact effective potential in a flat background [4]. Still, describing the full quantum theory in a general background remains a difficult task. The aim of this paper is to derive an exact evolution equation for the effective action (the one-particle irreducible graphs generator functional) with λ and to discuss some of its properties. We will see that this evolution equation (13) has a linear structure and is a direct consequence of properties of the exponential potential. The partition function Zλ and the connected graphs generator functional Wλ are defined by Zλ[j] = e −Wλ[j] = ∫