Anomaly of Tensionless String in Light-cone Gauge

TL;DR

This paper investigates the spacetime conformal anomaly in the quantum tensionless string theory within the light-cone gauge, highlighting differences between three and higher dimensions and the impact of operator ordering on anomalies.

Contribution

It analyzes the conformal anomaly in the light-cone gauge for tensionless strings, considering mode expansion and operator order effects in different spacetime dimensions.

Findings

Conformal anomaly depends on spacetime dimension and operator order.

Anomalies appear differently in D=3 and D>3 cases.

Operator ordering influences the presence of dangerous commutators.

Abstract

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.