A class of exact pp-wave string models with interacting light-cone gauge   actions
  We find a general class of pp-wave string solutions with NS-NS $H_3$ or R-R $F_3$ field strengths, which are analogous to solutions with non-constant $F_5$ recently considered by Maldacena and Maoz (hep-th/0207284). We show that: (i) all pp-wave solutions supported by non-constant $H_3$ or $F_p$ fields are exact type II superstring solutions to all orders in $\a'$; (ii) the corresponding light-cone gauge Green-Schwarz actions are non-linear in bosons but always quadratic in fermions, and describe UV finite 2-d theories; (iii) the pp-wave backgrounds supported by non-constant $F_3$ field do not have, in contrast to their $F_5$-field counterparts, ``supernumerary'' supersymmetries and thus the associated light-cone GS actions do not possess 2-d supersymmetry. We consider a specific example where the pp-wave $F_3$ background is parametrized by an arbitrary holomorphic function of one complex bosonic coordinate. The corresponding GS action has the same bosonic part, similar Yukawa terms but twice as many interacting world-sheet fermions as the (2,2) supersymmetric model originating from the analogous $F_5$ background. We also discuss the structure of massless scalar vertex operators in the models related to N=2 super sine-Gordon and N=2 super Liouville theories. 
J.G. Russo and A.A. Tseytlin