By Elcio Abdalla

This booklet incorporates a survey of using the Liouville (and super-Liouville) equation in (super)string thought outdoor the serious measurement, and of the complementary procedure in line with the discretized space-time, referred to as the matrix version process. Supersymmetry is given specific realization, either within the continuum formula, via the Liouville equation, and during the respect of the super-eigenvalue challenge. The tools offered listed below are very important in numerous complicated difficulties, e.g. random surfaces, 2-D gravity and large-N quantum chromodynamics. The comparability of alternative equipment within the learn of such difficulties allows a cross-evaluation of the implications while either equipment are appropriate and new predictions whilst just one of the tools can be used.

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**Example text**

1); iii) each four legs vertex implies a factor A1 PI = -t(1 + ta:"), and finally iv) the momentum is conserved through any vertex. We have for the five point function, the following elements: for the first diagram in Fig. 121) hold in any kinematic region and do not need to be further examined in order to be extended. The last step now is to calculate the irreducible part A~PI. We first notice that the cuts at k, + kJ = 00 only appear due to zero energy intermediate particles (remember p = ±ilk, + k J - 001 = -iE), thus they can only correspond to the massless "tachyons" Tk.

Later, we shall see that the results are in complete agreement with the ones obtained via matrix models. The main ingredient in this technique is the integration over the zero mode in the Liouville field (~o). The importance of this zero mode integration had already been noticed previously by Gupta, Trivedi and Wise 72 . They have used it to obtain non-perturbative partition functions for fixed area (Z(A)) in conformal theories coupled to two-dimensional gravity, only for certain values of the central charge.

Integrating out the zero mode cPo, Dotsenko obtained, besides the usual cosmological term to the power of s, a second term to the power s. 47b) 40 Although such a trick might a priori not be justified, the final result for the ratios of correlation functions does not seem to depend on the details of the continuation in s or s. 48) with the dressed tachyonic operator given by T k for the Liouville and matter fields written as = J d 2 zyge'kX +M and the action As before we have Q = 2}2 + a6; a± = -~ ± laol; moreover, a+a_ = 2.