By Elcio Abdalla

This booklet encompasses a survey of using the Liouville (and super-Liouville) equation in (super)string thought outdoors the serious measurement, and of the complementary procedure according to the discretized space-time, often called the matrix version method. Supersymmetry is given specific consciousness, either within the continuum formula, via the Liouville equation, and during the dignity of the super-eigenvalue challenge. The tools offered listed here are vital in a good number of complicated difficulties, e.g. random surfaces, 2-D gravity and large-N quantum chromodynamics. The comparability of alternative equipment within the research of such difficulties allows a cross-evaluation of the consequences whilst either tools are appropriate and new predictions while just one of the tools can be used.

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**Example text**

The full Virasoro algebra can be obtained by computing the expectation value of the product of the energy momentum tensors at nearby points. 150) In order to balance the central charges, we have to consider the following set of fields i) matter fields, with central charge c; ii) diffeomorphism ghosts, with central charge -15; iii) superconformal ghosts, with central charge -~; iv) the Sugawara piece of the supergravity multiplet contributing ,,~!! j 2 v) a linear term contributing -611: to the central charge.

Notice that in the case of the N-point function with N > 3 we still have N - 3 integrals over d 2 z, (i = 4" .. ,N) and such integrals together with the s integrals over d 2 w, will result into s + N - 3 coupled integrals which in principle are very involved. For this reason let us restrict for a moment to the case N = 3. 27), a positive integer. It is useful to extend it to arbitrary real values of s. 35) Now using Q:+ 8 = -/1fI and the definition of 1 1 Q:+ 2q 8 = --(~),. 35) for arbitrary values of 8).

The V± operators can be introduced directly in the matter sector of the action, defining When we calculate the 3-point function using the above action we expand e-s~ in A_ and A+ and the integration over the matter zero mode leads us to a new conservation law, which is satisfied only 3) by the term with n V+ charges and m V_ charges from the above expression. (Tk,Tk2Tkg)SdS~ = A+A~ \ Tk,Tk2Tkg x (~! (~! fj! )) + il! : k, + nd+ + md- - ,=1 3) 2ao = °. 75) One admIts that d_ and d+ are incommensurable, in order that nd+ be not cancelled by md_.