By Elcio Abdalla
This publication encompasses a survey of using the Liouville (and super-Liouville) equation in (super)string concept outdoors the serious size, and of the complementary process in line with the discretized space-time, often called the matrix version procedure. Supersymmetry is given specific realization, either within the continuum formula, via the Liouville equation, and during the honour of the super-eigenvalue challenge. The tools provided listed here are vital in a number of advanced difficulties, e.g. random surfaces, 2-D gravity and large-N quantum chromodynamics. The comparability of other equipment within the research of such difficulties allows a cross-evaluation of the consequences whilst either equipment are acceptable and new predictions while just one of the tools can be utilized.
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Additional info for 2D-Gravity in Non-Critical Strings: Discrete and Continuum Approaches (Planetology)
Is guaranteed if x. 92) jj' - p(m - 1) + 2n N 2 - a'0 - f3'0 - 2 LJ) " p't - naJ - ma') 1=4 (,<1) The above symmetries still do not permit to calculate Fir. It is very important to know its asymptotic behaviour. 93) are not enough to compute f;,m. 54) simplifies considerably, after the elimination of f3 as functions of sand p, using kinematic relations. In the present case we can eliminate most of the parameters of f/r as a function of the independent variables a, f3, p, s and p) (for 4 ::; j ::; N - 1).
The total action is the same as in the previous section (S L + skt). '\~n. ,\~ f( -8) IT f d 2z, IT f d 2t) IT f d2rk m. 89) (Q p,). =1 )=4 n N II It. <) m x II 2P ,<)=1 n X :, s 2 ,=1 X 4P Iz, - z) 1 ,<)=4 s X II ,<)=1 m Ir. 76), we have also a) = k 1 k) - PIP), ii) = d+k) , p) = k 2k) -P2P)l jj) = d_k)l p;) = t(klk) - PIP)), p) = -a+p)l with 4 :::; j,l :::; N. 91) is very difficult to be calculated in general (even in the case n = m = 0). However we notice that it possesses a set 49 of symmetries: It is invariant under X, --+ 1 - X, once we exchange variabl(~s as a +-t f3 , a~ +-t f3; , a +-t jj and a' +-t jj'.
32) where A r , = Or, r: e,B,q'>(z,) , a = -a+(31, (3 = -a+(32, P = -at/2. Fortunately the s interrelated integrals have been calculated by Dotsenko and Fatteev 30 (see Appendix C, Eq. 33) where Ll(x) = r(x)/r(l - x). 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.
2D-Gravity in Non-Critical Strings: Discrete and Continuum Approaches (Planetology) by Elcio Abdalla