Character expansion developed in real space renormalization group (RSRG) approach is applied to U(1) lattice gauge theory with θ-term in 2 dimensions. Topological charge distribution P (Q) is shown to be of Gaussian form at any β(inverse coupling constant). The partition function Z(θ) at large volume is shown to be given by the elliptic theta function. It provides the information of the zeros of partition function as an analytic function of ζ = e iθ (θ = theta parameter). These partition function zeros lead to the phase transition at θ = π. Analytical results will be compared with the MC simulation results. In MC simulation, we adopt (i)"set method" and (ii)"trial function method".

The two dimensional CP' model with a B term is simulated. We compute the topological charge distribution P(Q) by employing the "set method" and "trial function method", which are effective in calculations for a very wide range of Q and large volume. The distribution P(Q) exhibits Gaussian behavior in the small /3 (inverse coupling constant) region and deviates from this in the large (3 region. The free energy and its moment are calculated as a function of B. For small /3, the partition function is given by the elliptic theta function, and the distribution of its zeroes on the complex B plane leads to a first order phase transition at B = TC. In the large (3 region, on the other hand, this first order phase transition disappears, but definite conclusion concerning the transition is not reached due to large errors. § 1. Introduction 175The two dimensional cpN-I model is a suitable laboratory to study dynamics of QCD. Topology is expected to play an important role in the non-perturbative nature of the dynamics of such theories. Numerical studies of the topological aspects of the cpN-l model have made much progress.l)-3 ) However, a full understanding of the dynamics of the model requires study of an additional contribution of the imaginary part of the action, i.e., the 8 term. The degeneracy of the different topological sectors is resolved into a unique vacuum labeled by the parameter 8. As shown by certain analytic studies, various models with the 8 term, in general, exhibit a rich phase structure. 4 H) It is then worthwhile to study effects of the e term on the cpN-I model. 8 > From a realistic point of view also, it is significant to clarify the matter of strong CP violation in QCD.Introduction of the 8 term precludes ordinary simulations because of the complex Boltzmann factor. An idea to circumvent this problem is to introduce the constrained updating of the fields, in which the topological charge, being a functional of the dynamical fields, is constrained to take a given value Q. Thus the phase factor eieQ is factored out so that the partition function is given by the summation of the probability distribution P( Q) weighted by e; 8 Q over all possible values of the topological charge Q. This algorithm was adopted in simulating the two dimensional U(l) gauge model. 4 > So far, topological aspects of the CP' model have been studied considerably, both theoretically and numerically. Most works, however, are limited to theories without the 8 term. This is one of our motives for studying the effects of the 8 term on the model by means of Monte Carlo simulations. We present here the results for P( Q) and the free energy F( 8) and its moments as a function of 0 by surveying a compara-

A U(2) lattice gauge theory with 8-term in 2 space-time dimensions is investigated. It has a non-Abelian real action and Abelian (U(l) type) imaginary action. The imaginary action is defined as the standard 8-term. As the effect of a renormalization group (RG) transformation, the non-Abelian imaginary action is induced. After many steps of the RG transformation, the non-Abelian part dies away. After several steps of the RG transformations, the renormalized action approaches the so-called heat kernel action. A phase transition is found only at 8 = 7r. §1. Introduction 791 Much progress has been made in lattice gauge theory approach in the past twenty years. The systems studied, however, are limited mostly to those without topological terms (=O-term). This is due mainly to the difficulty in treating the O-term in numerical calculations in Euclidean space-time. Numerical calculations are performed with the use of a probability weight defined by the Euclidean action. When the type of system considered is limited to those without a O-term, the probability weight is given by a positive quantity. But when a O-term is added to the Euclidean action, we obtain a complex weight due to the fact that the O-term is a purely imaginary quantity in Euclidean space-time. The O-term, however, is expected to lead to physically interesting effects.In 4 space-time dimensions, the O-term leads to strong CP violation which is severely suppressed in the real world, although it is not excluded from the theoretical framework. A system with a O-term leads to oblique confinement and rich phase structures. 1) -3)In 3 space-time dimensions, the O-term is called a Chern-Simons term and is related to new physical effects, e.g., fractional statistics. In compact U(l) gauge theory, the system is in the confinement phase, as discussed by Polyakov, due to the monopole excitation. It is conjectured that the system undergoes a phase transition. 4 ) The system is in a "deconfined phase" when the O-term is included, since existence of O-term has the effect of washing out magnetic monopoles, in which case the system cannot be in a confined phase for arbitrary non-zero (J. In two space-time dimensions, the compact U(l) system is in the confinement phase. The existence of the O-term, however, leads to a deconfinement phase transiat