Anomaly of Tensionless String in Light-cone Gauge

The classical tensionless string theory has the spacetime conformal symmetry. We expect and require that the quantum tensionless string theory has it too. In the BRST quantization method, the theory has no spacetime conformal anomaly in two dimensions. On the other hand, in the light-cone gauge quantization without the mode expansion, the theory in $D>3$ has the spacetime conformal anomaly in the traceless part of $[\mathcal{J}^{-I}, \mathcal{K}^{J}]$ in some operator order. In this paper, we consider a tensionless closed bosonic string in the light-cone gauge and investigate the spacetime conformal anomaly in the theory with the mode expansion. The appearance of the spacetime conformal anomaly in the light-cone gauge is different between the case of $D>3$ and the case of $D=3$ and depends on the choice of the operator order. Therefore we must consider dangerous commutators in the spacetime conformal symmetry of $D>3$ and $D=3$ in each operator order separately. Specifically we calculate dangerous commutators, $[\mathcal{J}^{-I},\mathcal{K}^{K}]$ in $D>3$ and $\tilde{\mathcal{K}}^{-}\equiv -i[\mathcal{J}^{-}, \tilde{\mathcal{K}}^{-}]$ and $[\mathcal{J}^{-}, \tilde{\mathcal{K}}^{-}]$ in $D=3$, in two types of the operator order.