pp-wave Green-Schwarz superstring, polygon divergent structure, and conformal field theory

In the semi-light-cone gauge g_ab=e^2φδ_ab, γ¯⁺θ=0, we evaluate the φ-dependent effective action for the pp-wave Green-Schwarz (GS) superstring in both harmonic and group coordinates. When we compute the fermionic φ-dependent effective action in harmonic coordinates, we find a new triangular one-loop Feynman diagram. We show that the bosonic φ-dependent effective action cancels with the fermionic one, indicating that the pp-wave GS superstring is a conformal field theory. We introduce the group coordinates preserving SO(4)×SO(4) and conformal symmetry. Group coordinates are interesting because vertex operators take simple forms in them. The new feature of the group coordinates is that there are logarithmic divergences from n-gons, so that the divergent structure is more complicated than in harmonic coordinates. After summing over all contributions from n-gons, we show that in group coordinates the GS superstring on a pp-wave RR background is still a conformal field theory.