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We show how to regularize and renormalize the QED at the one-loop order by means of the intrinsic loop regularization method proposed by the authors. All the results are the same as those derived by means of other regularization methods.

We consider the classical Liouville field theory on the Riemann surface with h>1 genus. In terms of the uniformization theorem of the Riemann surface, we show explicitly the classical exchange algebra (CEA) for the chiral components of the Liouville field. We find that this classical exchange algebra is just the same as that on a Riemann sphere with 2h punctures.

We find that there exist certain intrinsic relations between the divergent diagrams and the convergent ones at the same loop order in some renormalizable quantum field theories. Whereupon we propose a new method for regularization and renormalization of those divergent diagrams. We name it the intrinsic loop regularization. In this paper, we take the renormalized 4 theory up to the second order of the coupling constant as an example to present this method.

Starting from the classical Liouville theory, we study its quantum theory through canonical quantization. We find that if the Poisson bracket relations between two vectors are dominated by the classical r-matrix in the classical case, their quantum analogue is replaced by the exchange relations dominated by the quantum R-matrix. The quantum group structure in the quantum Liouville theory is studied and the central charge of the quantum Liouville theory is also obtained.

We introduce the Beltrami gauge symmetry related to the Beltrami parametrization of the metrics and the Beltrami equations. We explore its role in 2d induced gravity and show that Polyakov's-SL(2R) and KPZ's residual symmetries are subsymmetries of the Beltrami gauge symmetry in the light-cone gauge. We also find that the 2d induced gravity with or without matter can be reformulated as Beltrami-Liouville field theories on Riemann surfaces of higher genus.