2D-Gravity in Non-Critical Strings: Discrete and Continuum by E. Abdalla, M.C.B. Abdalla, D. Dalmazi, A. Zadra

By E. Abdalla, M.C.B. Abdalla, D. Dalmazi, A. Zadra

This e-book encompasses a survey of using the Liouville (and super-Liouville) equation in (super)string conception outdoors the serious measurement, and of the complementary strategy in accordance with the discretized space-time, referred to as the matrix version procedure. Supersymmetry is given specific cognizance, either within the continuum formula, via the Liouville equation, and during the honor of the super-eigenvalue challenge. The tools provided listed below are very important in a number of complicated difficulties, e.g. random surfaces, 2-D gravity and large-N quantum chromodynamics. The comparability of other tools within the examine of such difficulties allows a cross-evaluation of the consequences whilst either equipment are acceptable and new predictions whilst just one of the tools can be utilized.

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Therefore, it is possible that string theory has a phase 27 transition at this point, where the string degenerates to a branched polymer or is in a crumpled phase 51 • Therefore at this point one is led to the study of random surfaces, and related critical phenomena52 . The propagation of strings along random surfaces is a rather general problem in physics. It describes low energy QCD 53 in the large N limit. In fact we may argue that the large N limit is related to two-dimensional gravity (Liouville theory)54.

Otherwise a correction arises, and we have C-I-24~-y(l-c)(25-c-24~) h A h · . al ·bl al 'Yh=O = 12(1 ~) were ... assumes t e IDlIDm POSSI e vue. 29 2. 1. Introduction and General Overview Two-dimensional gravity describes important phenomena ll ,43, such as random surfaces and string theories away from criticality. Up to recently, the most efficient means of extracting results from the theory of two-dimensional gravity was the matrix model approach 56 ,70, to be discussed later. In this theory, one obtains a series expansion in the genus with determined coefficients.

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. Moreover, the plethora of information on Liouville theory, both at classical as well as in the quantum level, together with its role in the geometry of Riemann surfaces, practically force us to face a comparison of the results obtained in the light cone gauge with further discussion in the conformal gauge, and compels us to re-obtain all relevant results directly in the Liouville theory.

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