Symmetries and Critical Dimensions of Tensionless Branes

In this work, we investigate the worldsheet symmetry of bosonic brane theories and its quantum consistency in the tensionless limit. We find that the residual worldsheet symmetry after specific gauge fixing is generated by a novel algebra, denoted as $g_{\lambda }^{(p)}$. To achieve full quantization of the tensionless brane, we introduce a $bc$ ghost system and derive the overall BRST charge. Moreover, we calculate the quantum anomaly of the $g_{\lambda }^{(p)}$ algebra for general parameters $p$ and $\lambda$ in the framework of canonical quantization. After demanding that this quantum anomaly vanishes, we successfully derive the critical dimensions of the bosonic brane theories. Especially, we obtain nontrivial solutions: $p=3$ in $D=4$ spacetime dimensions when $\lambda =-3$ and $p=6$ in $D=7$ spacetime dimensions when $\lambda =3$.