Exactly solvable field theories of closed strings

Abstract

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 ) ⨍((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 ⨍. 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.