Covariant quantization of the Green-Schwarz superstring in a Calabi-Yau background

After adding a scalar chiral boson to the usual superspace variables, the four-dimensional Green-Schwarz superstring is quantized in a manifestly SO(3,1) super-Poincaré covariant manner. The constraints are all first class and form an N = 2 superconformal algebra with c = -3. Since the Calabi-Yau degrees of freedom are described by an N = 2 superconformal field theory with c = 9, the combined Green-Schwarz and Calabi-Yau systems form the c = 6 matter sector of a critical N = 2 string. Using the standard N = 2 super-Virasoro ghosts, a nilpotent BRST charge is defined and vertex operators for the massless supermultiplets are constructed. Four-dimensional superstring amplitudes can be calculated with manifest SO(3,1) super-Poincaré invariance by evaluating correlation functions of these BRST-invariant vertex operators on N = 2 super-Riemann surfaces.