EXACTLY SOLVABLE FIELD-THEORIES OF CLOSED STRINGS

Field theories of closed strings are shown to be exactly solvable for a central charge of matter fields c=1-6/m(m+1),m=1,2, 3, .... The two-point function χ(λ,N), in which λ is the cosmological constant and N-1 is the string coupling constant, obeys a scaling law χ(λ,N=N-(m+ 1 2){cauchy integral}((λc-λ)Nm/(m+ 1 2)) in the limit in which N-1 goes to zero and λ goes to a critical value λc; we have determined the universal non-linear differential equation satisfied by the function {cauchy integral}. From this equation it is found that a phase transition takes place for some finite value of the scaling parameter (λc-λ)Nm/(m+ 1 2); this transition is a "condensation of handles" on the world sheet, characterized by a divergence of the averaged genus of the world sheets. The cases m=2,3 are elaborated in more details, and the case m=1, which corresponds to the embedding of a bosonic string in -2 dimensions, is reduced to explicit quadratures. © 1990.