Two-loop string theory and the DVV Vertex

Abstract

We compute the two-loop contributions to the free energy in the null compacti¯cation of perturbative string theory at ¯nite temperature. The cases of bosonic, Type II and heterotic strings are all treated. The calculation exploits an explicit reductive parametrization of the moduli space of in¯nite-momentum frame string worldsheets in terms of branched cover instantons. Various arithmetic and physical properties of the instanton sums are described. Applications to symmetric product orbifold conformal ¯eld theories and to the matrix string theory conjecture are investigated by analyzing the correspondence be- tween the two-loop thermal partition function of DLCQ strings in °at space and the integrated two-point correlator of twist ¯elds in a symmetric product orbifold con- formal ¯eld theory at one-loop order. This is carried out by deriving combinatorial expressions for generic twist ¯eld correlation functions in permutation orbifolds us- ing the covering surface method, by deriving the one-loop modi¯cation of the twist ¯eld interaction vertex, and by relating the two-loop ¯nite temperature DLCQ string theory to the theory of Prym varieties for genus two covers of an elliptic curve. The case of bosonic Z2 orbifolds is worked out explicitly and precise agreement between both amplitudes is found. We use these techniques to derive explicit expressions for Z2 orbifold spin twist ¯eld correlation functions in the Type II and heterotic string theories.