By Orlando Alvarez, Enzo Marinari, Paul Windey
The Cargese Workshop Random Surfaces and Quantum Gravity was once held from might 27 to June 2, 1990. Little used to be identified approximately string thought within the non-perturbative regime sooner than Oetober 1989 while non-perturbative equations for the string partition capabilities have been came upon by utilizing tools in accordance with the random triangulations of surfaees. This set of tools seasoned vides a deseription of non-eritical string idea or equivalently of the coupling of topic fields to quantum gravity in dimensions. The Cargese assembly was once very winning in that it supplied the 1st chance to collect many of the energetic staff within the box for a fuH week of lectures and wide casual discussions approximately those exeiting new advancements. the most effects have been reviewed, contemporary advances have been defined, new effects and conjectures (which seem for the 1st time in those lawsuits) have been offered and mentioned. one of the most vital issues mentioned on the workshop have been: The relation of KdV thought to loop equations and the Virasoro algebra, new ends up in Liouville box conception, potent (1 + 1) dimensional concept for two - D quantum gravity coupled to c = 1 subject and its fermionization, notion for a brand new geometrical interpretation of the string equation and attainable definition of quantum Riemann surfaces, dialogue of the string equation for the multi-matrix types, hyperlinks with topological box theories of gravity, matters in utilizing objective area supersymmetry to outline solid theories, definition of the partition functionality through analytic continuation, new versions of random surfaces
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Additional info for Random Surfaces and Quantum Gravity
T. 6). 2). 21) In the foHowing we will eaH order parameters the q - 1 first operators 4>1,' .. 3) in analogy with the lattice model 40 description . In the topological approach , these fields are the gravitational primaries, and playaspecial role. The picture so far deserves a few comments. An important issue is the p -+ q symmetry. Though in the matrix model P and Q appear on a different footing, the (p,q) and (q,p) conformal theories are known to be equivalent. 3) one could as well have taken P = LP, Q = I:j:I T/j(Lj)+.
The sealing limit of loop equations involving more loop operators ean be obtained in a similar way. 7) Those equations ean be used to eompute reeursively (in the topologieal expansion) eorrelation functions in the sealing limit (see  The interest of the loop equations is not merely ealeulational. 11) , and from the fact that the partition function is the so-ealled T-function of the eorresponding KdV hierarehy. Moreover those reeursion relations ean also be obtained from the formulation of 2-d gravity as a topologieal fiels theory  .
10)) , so that what this really means is that the three point function vanishes if 1+ m + r ~ 2n + 1. But this is precisely the missing fusion rule we were after. Similarly one can easily calculate the three point functions when they don't vanish by using convenient m, l, r. 8d). 5) (see ), but the results involve complicated sums over binomial coefficients. The present derivation avoids evaluating these sums. To calculate correlation functions of o/m m > n, it is convenient to use the recursiori relations for gravitational descendants in terms of primaries, which originate in topological field theory  and therefore hold in the KdV framework as weil.
Random Surfaces and Quantum Gravity by Orlando Alvarez, Enzo Marinari, Paul Windey