On zigzag-invariant strings

We propose a world-sheet realization of the zigzag-invariant strings as a perturbed WZNW model at large negative level. We argue that the gravitational dressing produces additional fixed points of the dressed renormalization beta function. One of these new critical points can be interpreted as a zigzag-invariant string model. ∗e-mail: [email protected] †e-mail: [email protected] A world-sheet description of the QCD string has not as yet been discovered. At the same time, the general feeling is that we are getting very close to unveiling the one of the most intriguing enigma of theoretical physics. Recently, a major breakthrough has been achieved [1],[2],[3]. It has been shown that a Yang-Mills theory in D-dimensions can be obtained from a (super)gravity defined in a D + 1-dimensional space-time. The open problem is to understand the concrete world-sheet realisation of these ideas. Here we would like to attempt a formulation of a two-dimensional model which may have properties similar to the ones required for the QCD string. According to Polyakov, the latter has to be described by the following world-sheet action [1] SP = ∫ dξ [ (∂φ) + a(φ)(∂x) + Φ(φ)R √ g ] , (1) where we omit possible antisymmetric fields, like Bμν or Ramond-Ramond fields in the supersymmetric case. Here φ is the Liouville field of 2D gravity, x (μ runs from 1 to D) are coordinates of the confining string, Φ is the dilaton field, R is the curvature of the world-sheet and a(φ) is the running string tension. The zigzag symmetry requires the existence of a certain value of the Liouville field φ∗ such that [1] a(φ∗) = 0. (2) A concrete world-sheet realisation of Polyakov’s ansatz appears to be fairly intricate. Our first step is to consider a certain non-conformal model with the running coupling constant in front of the kinetic term. Such a theory has been discussed in [4]. It is a (non-unitary) WZNW model perturbed by its kinetic term. The corresponding action is written as follows S( ) = SWZNW (G, k) − ∫ dz O(z, z̄). (3) Here is a small constant, SWZNW (G, k) is the WZNW model on the group manifold G at level k and O(z, z̄) = 1 cV (G) JJ̄ φ, (4) where J ≡ Jt = −k 2 ∂gg−1, J̄ ≡ J̄t = −k 2 g−1∂̄g, φ = Tr(g−1tagtb), cV (G) = −fac d f bd c ηab dim G . (5)