Turbulent Two-Dimensional Magnetohydrodynamics and Conformal Field Theory

We show that an infinite number of non-unitary minimal models may describe two dimensional turbulent magnetohydrodynamics (MHD), both in the presence and absence of the Alf 'ven effect. We argue that the existence of a critical dynamical index results in the Alf 'ven effect or equivalently the equipartition of energy. We show that there are an infinite number of conserved quantities in 2D-MHDturbulent systems both in the limit of vanishing the viscocities and in force free case. In the force free case, using the non-unitary minimal modelM_{2, 7}we derive the correlation functions for the velocity stream function and magnetic flux function. Generalising this simple model we find the exponents of the energy spectrum in the inertial range for a class of conformal field theories.