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)C((λ_c-λ)N^m/(m+1/2)) in the limit in which N^-1 goes to zero and λ goes to a critical value λ_c we have determined the universal non-linear differential equation satisfied by the function C. From this equation it is found that a phase transition takes place for some finite value of the scaling parameter (λ_c-λ)N^m/(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.

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