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

Abstract

In the semi-light-cone gauge ${g}_{\mathrm{ab}}{=e}^{2\ensuremath{\varphi}}{\ensuremath{\delta}}_{\mathrm{ab}},$ ${\overline{\ensuremath{\gamma}}}^{+}\ensuremath{\theta}=0,$ we evaluate the \ensuremath{\varphi}-dependent effective action for the pp-wave Green-Schwarz (GS) superstring in both harmonic and group coordinates. When we compute the fermionic \ensuremath{\varphi}-dependent effective action in harmonic coordinates, we find a new triangular one-loop Feynman diagram. We show that the bosonic \ensuremath{\varphi}-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 $\mathrm{SO}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(4)$ 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.