The emergence of (3+1)-dimensional expanding space-time in the Lorentzian type IIB matrix model is an intriguing phenomenon which was observed in Monte Carlo studies of this model. In particular, this may be taken as a support to the conjecture that the model is a nonperturbative formulation of superstring theory in (9+1) dimensions. In this paper we investigate the space-time structure of the matrices generated by simulating this model and its simplified versions, and find that the expanding part of the space is described essentially by the Pauli matrices. We argue that this is due to an approximation used in the simulation to avoid the sign problem, which actually amounts to replacing e iS b by e βS b (β > 0) in the partition function, where S b is the bosonic part of the action. We also discuss the possibility of obtaining a regular space-time with the (3+1)-dimensional expanding behavior in the original model with the correct e iS b factor.

Monte Carlo simulation of gauge theories with a θ term is known to be extremely difficult due to the sign problem. Recently there has been major progress in solving this problem based on the idea of complexifying dynamical variables. Here we consider the complex Langevin method (CLM), which is a promising approach for its low computational cost. The drawback of this method, however, is the existence of a condition that has to be met in order for the results to be correct. As a first step, we apply the method to 2D U(1) gauge theory on a torus with a θ term, which can be solved analytically. We find that a naive implementation of the method fails because of the topological nature of the θ term. In order to circumvent this problem, we simulate the same theory on a punctured torus, which is equivalent to the original model in the infinite volume limit for |θ| < π. Rather surprisingly, we find that the CLM works and reproduces the exact results for a punctured torus even at large θ, where the link variables near the puncture become very far from being unitary.

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The tensor renormalization group method is a promising approach to lattice field theories, which is free from the sign problem unlike standard Monte Carlo methods. One of the remaining issues is the application to gauge theories, which is so far limited to U(1) and SU(2) gauge groups. In the case of higher rank, it becomes highly nontrivial to restrict the number of representations in the character expansion to be used in constructing the fundamental tensor. We propose a practical strategy to accomplish this and demonstrate it in 2D U(N) and SU(N) gauge theories, which are exactly solvable. Using this strategy, we obtain the singular-value spectrum of the fundamental tensor, which turns out to have a definite profile in the large-N limit. For the U(N) case, in particular, we show that the large-N behavior of the singular-value spectrum changes qualitatively at the critical coupling of the Gross-Witten-Wadia phase transition. As an interesting consequence, we find a new type of volume independence in the large-N limit of the 2D U(N) gauge theory with the θ term in the strong coupling phase, which goes beyond the Eguchi-Kawai reduction.

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