The criteria for the existence of a glass transition in a planar vortex array with quenched disorder are studied. Applying a replica Bethe ansatz, we obtain for self-avoiding vortices the exact quenched average free energy and effective stiffness which is found to be in excellent agreement with recent numerical results for the related random bond dimer model [C. Zeng, P. L. Leath, and T. Hwa, Phys. Rev. Lett. 83, 4860 (1999)] Including a repulsive vortex interaction and a finite vortex persistence length xi, we find that for xi-->0 the system is at all temperatures in a glassy phase; a glass transition exists only for finite xi. Our results indicate that planar vortex arrays in superconducting films are glassy at presumably all temperatures.
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