String theory in covariant gauge

The path integral representing the propagators of open and closed bosonic strings are evaluated in manifestly covariant proper-time gauges. For the covariant classical action in the path integrals, we follow Polyakov; however, the measures employed in the path integrals are canonical ones which are suitable for the BRST (Becchi-Rouet-Stora-Tyutin) method, rather than those conventionally used for the Polyakov strings. We can understand some aspects of the connection between the geometrical approach of Polyakov which reveals some elegant mathematical properties of the path integral such as the modular invariance and the algebraic approach of Siegel which puts emphasis on the BRST invariance in constructing the covariant second quantized string theories. The Neveu-Schwarz-Ramond (NSR) super-string is also discussed as an extension of the bosonic string theories. We provide explanations in detail for the appearance of the modular and the super-modular parameters as a consequence of covariant gauge fixing as well as for their roles in the path integrals. Depending on the spin structure, the NSR super-string has two different sectors for the open string and four sectors for the closed string. We perform explicitly the evaluation of the path integral representing the propagator for each sector by use of the BRST formalism