Anomaly and Mass Spectrum of Tensionless String in Light-cone Gauge

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Recently the method of dualities as the AdS/CFT correspondence has been used in higher spin gauge theories to understand gauge theories, gravities and M theory. Because higher spin gauge theories contain the infinite tower of higher spin fields, their relations with the tensionless limit of string theories have been studied. However the relations are not clear. One of the reasons is probably the poor understanding of the tensionless string theory. Therefore its understanding is very important to investigate the relation between the tensionless string theory and higher spin gauge theories and to consider dualities on the tensionless string theory. Classical string theories have the spacetime symmetry as the global symmetry, as well as point particle theories. The theory of a string with tension, tensionful string theory, has the Poincare symmetry and the tensionless string theory has the spacetime conformal symmetry. As is well known, the quantization of a tensionful string usually causes the anomaly in the Poincare symmetry, the Lorentz anomaly. To avoid this anomaly, the spacetime dimensions and the operator ordering constant are determined. Such anomalies and restrictions are known also in the quantum tensionless string theory. For example, in the light-cone gauge quantization method, the existence of the spacetime conformal anomaly is known in a certain operator order. Because there is only one transverse direction in three dimensional light-cone coordinate, the Lorentz anomaly vanishes trivially for the light-cone string theory in three dimensions. Recently this specialty of three dimensional light-cone gauge has gathered attention and then the mass spectra of some 3-dim. tensionful strings in the light-cone gauge have been investigated in detail. Because there is not any restriction except for D = 3 in the case of the bosonic tensionful string, the operator ordering constant is undetermined. Such an ambiguity is removed for the tensionful superstring. Similarly, the ambiguity in three dimensional theories might be removed by the requirement of a larger symmetry. In this thesis based on [1], we consider a tensionless closed bosonic string in the lightcone gauge in various dimensions and operator orders. The main products of our study are the construction of the non-separable mass eigenfunctions in the Reference order(Rorder), the avoidance of the spacetime conformal anomaly in the Hermitian R-order and the explicit calculation of the spacetime conformal anomaly in the XP-normal order. In three dimensions, we make use of the above specialty of three dimensions to investigate the spacetime conformal symmetry for the tensionless string in the light-cone gauge. In the Hermitian Refernce order, we verify the absence of the spacetime conformal anomaly and the removing of the ambiguity coming from the operator ordering. Furthermore we investigate separable and non-separable mass eigenstates in the theory without the anomaly and then we consider the relations between such eigenstates in terms of the Poincare invariant corresponding to spin. In the XP-normal order, we calculate a dangerous commutator in the spacetime conformal symmetry concretely to find anomalous terms. To avoid the anomaly, we interpret the tensionless string in the XP-normal order as the point-like theory. Furthermore, in D > 3, we investigate the anomaly in the spacetime conformal symmetry in the same way as three dimensional case. In the Hermitian R-order, we find the possibility of avoiding the anomaly. In the XP-normal order, we verify the existence of the anomaly concretely.