By E. Abdalla, M.C.B. Abdalla, D. Dalmazi, A. Zadra
This e-book incorporates a survey of using the Liouville (and super-Liouville) equation in (super)string concept outdoors the severe size, and of the complementary technique according to the discretized space-time, often called the matrix version procedure. Supersymmetry is given specific recognition, either within the continuum formula, by way of the Liouville equation, and during the distinction of the super-eigenvalue challenge. The tools provided listed below are very important in a great number of complicated difficulties, e.g. random surfaces, 2-D gravity and large-N quantum chromodynamics. The comparability of other tools within the research of such difficulties allows a cross-evaluation of the consequences while either tools are appropriate and new predictions while just one of the tools can be utilized.
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This booklet includes a survey of using the Liouville (and super-Liouville) equation in (super)string conception outdoor the serious size, and of the complementary strategy according to the discretized space-time, often called the matrix version method. Supersymmetry is given specific awareness, either within the continuum formula, via the Liouville equation, and during the honour of the super-eigenvalue challenge.
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Extra info for 2D-Gravity in Non-Critical Strings: Discrete and Continuum Approaches
121)). 152) Notice that il = ~ is a fixed point (in the same way as il = 1 was a fixed point in the bosonic theory). Conclusions similar to the bosonic case may be drawn. A new series of critical indices emerges, and may be compared with the usual results. The previous discussion of quantum gravity gives us hope that the full problem may be understood by the methods of conformal field theory. Nevertheless, the quantization in the light-cone gauge poses several restrictions. The first concerns the role played by the cosmological constant, which has been discussed in the framework of this gauge in [38,39], but for the computation of correlation functions, it plays a rather unclear role.
113) can be depicted as in Fig. 1 below + Fig. 1: Contributions to the ~ amplitude. 54 An analogous picture holds for N > 4. 100b) - L (n, ... TkNTk=O) = [-2Ia N N Ik, - aol- (2 -1)(1 ,=1 +1 2a :")] (Tk,'" T kN ) . 119) For N = 4 this expression suggests the diagrammatic picture of Fig. 2. The contribution Ag PI is a contact term. We complete now As, which is known in the given kinematic region as before. + + Fig. 2: Contributions to the five point function. The strategy to extend As to other kinematic regions is first to compute the reducible diagrams in Fig.
A more elegant way of understanding the cosmological term in the c = 1 theory, is by inspection using the space-time interpretation of the theory. 69) with -ko(k o + Q) + k 2 = 2. 72) is the presence of poles at V2lkJI = 1,2,3, .... These states are called discrete states and are remnants 96 of the massive states of string in higher dimensions. In our case of a two-dimensional embedding space these discrete states do not have particle interpretation since there is no room for transverse excitations and they only appear for specific values of the momentum 97 • This problem is discussed in more detail in section 5 ofthis Chapter.
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