Quantum Anomalies of Tensionless Bosonic Strings

We systematically investigate and compare the worldsheet actions, BRST structures and the quantum anomalies of four different formulations of tensionless ($T = 0$) bosonic string theory: the $(D+2)$-dimensional conformal string \cite{Gustafsson:1994kr}, the $D$-dimensional ILST null string \cite{Isberg:1993av}, the $D$-dimensional Carroll-Weyl gauged string \cite{Sheikh-Jabbari:2026vqh, Sheikh-Jabbari:2026tpf}, and the $D$-dimensional hybrid null string \cite{Chen:2026klv}. By expressing all fields and constraint generators strictly in terms of mode expansions and adopting a unified algebraic framework, we analyze their quantum anomalies under two distinct worldsheet vacua: the induced vacuum and the flipped vacuum. With the BRST-compatible vacuum definition and the symmetric $\alpha=0$ zeta prescription, we show that no critical dimension is inferred from the vanishing of the quantum anomaly in the induced vacuum. In contrast, the flipped highest-weight vacuum leads to non-trivial constraints, reproducing the critical dimension $D=26$ for the ILST null strings, a $\lambda$-dependent critical dimension $D(\lambda)$ for the hybrid null string whose range covers every positive integer $D\geq 4$ (reproduces $D=26$ at $\lambda=1$), and more importantly showing that the conformal string and the Carroll-Weyl gauged string are structurally anomalous with no consistent critical dimension due to the discrepancy of their central charge parameters $\tilde d_i$. Furthermore, the ILST null string model has target-space conformal symmetry $SO(D,2)$, the ghost-completed $SO(D,2)$ charges are closed on the induced-vacuum BRST cohomology in the $\alpha=0$ prescription, whereas the symmetry is quantum mechanically broken in the flipped vacuum.