Covariant quantization of the string in dimensionsD≤26using a Becchi-Rouet-Stora formulation

Abstract

The conventional open string theory is cast into the Becchi-Rouet-Stora (BRS) formulation of Fradkin and Vilkovisky. Upon quantizing this theory, the nilpotency condition ${\hat{Q}}^2=0\mathrm{}$ of the BRS charge requires $D=26\mathrm{}$ and ${\alpha }_0=1\mathrm{}$ for the space-time dimension $D$ and intercept parameter ${\alpha }_0$. Unitarity is proved by relating this theory to the old covariant quantization. The string theory is then generalized as suggested by Polyakov, taking into account the conformal or trace anomaly by $\mathcal{L}={\mathcal{L}}_{\mathrm{string}}+C{\mathcal{L}}_1$, where ${\mathcal{L}}_1$ yields the Liouville equation in the orthonormal gauge. Under the assumption that the exact quantization of Liouville's equation does not yield any additional anomalies, we show that the condition ${\hat{Q}}^2=0\mathrm{}$ implies $C=\frac{\mathrm{}(D-26\mathrm{})\mathrm{}}{48\pi }$, in agreement with Polyakov's result, and the "intercept parameter" $\beta =\frac{\mathrm{}(D-2\mathrm{})\mathrm{}}{24}$.