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.