A comprehensive survey of the use of the Liouville (and super-Liouville) equation in (super)string theory outside the critical dimension, and of the complementary approach based on the discretized space-time - known as the matrix model approach. The authors pay particular attention to supersymmetry, both in the continuum formulation and through the consideration of the super-eigenvalue problem. The methods presented here are important in a large number of complex problems, e.g. random surfaces, 2-D gravity and large-N quantum chromodynamics, and this comparitive study of the different methods permits a cross-evaluation of the results when both methods are valid, combined with new predictions when only one of the methods may be applied.
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This is a survey of the use of Liouville (and super-Liouville) equations in (super)string theory outside the critical dimension, and of the complementary approach based on the discretized space-time, known as the matrix model approach. Supersymmetry is given particular attention, both in the continuum formulation, by means of the Liouville equation, and through the consideration of the super-eigenvalue problem. The methods presented here are important in a large number of complex problems, such as random surfaces, 2-D gravity and large-N quantum chromodynamics. The comparison of different methods in the study of such problems permits a cross-evaluation of the results when both methods are applicable and new predictions when only one of the methods may be used.
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