We propose a method to define a d + 1 dimensional geometry from a d dimensional quantum field theory in the 1/N expansion. We first construct a d + 1 dimensional field theory from the d dimensional one via the gradient flow equation, whose flow time t represents the energy scale of the system such that t → 0 corresponds to the ultra-violet (UV) while t → ∞ to the infra-red (IR). We then define the induced metric from d + 1 dimensional field operators. We show that the metric defined in this way becomes classical in the large N limit, in a sense that quantum fluctuations of the metric are suppressed as 1/N due to the large N factorization property. As a concrete example, we apply our method to the O(N) non-linear σ model in two dimensions. We calculate the three dimensional induced metric, which is shown to describe an AdS space in the massless limit. We finally discuss several open issues in future studies.

We generalize the gradient flow equation for field theories with nonlinearly realized symmetry. Applying the formalism to super Yang-Mills theory, we construct a supersymmetric extension of the gradient flow equation. It can be shown that the super gauge symmetry is preserved in the gradient flow. Furthermore, choosing an appropriate modification term to damp the gauge degrees of freedom, we obtain a gradient flow equation which is closed within the Wess-Zumino gauge.

We revisit the globally coupled map lattice (GCML). We show that in the so called turbulent regime various periodic cluster attractor states are formed even though the coupling between the maps are very small relative to the nonlinearity in the element maps. Most outstanding is a maximally symmetric three cluster attractor in period three motion (MSCA) due to the foliation of the period three window of the element logistic maps. An analytic approach is proposed which explains successfully the systematics of various periodicity manifestations in the turbulent regime. The linear stability of the period three cluster attractors is investigated. 05.45.+b,05.90.+m

We study the gradient flow equation for the O(N) nonlinear sigma model in two dimensions at large N . We parameterize solution of the field at flow time t in powers of bare fields by introducing the coefficient function X n for the n-th power term (n = 1, 3, · · · ). Reducing the flow equation by keeping only the contributions at leading order in large N , we obtain a set of equations for X n 's, which can be solved iteratively starting from n = 1. For n = 1 case, we find an explicit form of the exact solution. Using this solution, we show that the two point function at finite flow time t is finite. As an application, we obtain the non-perturbative running coupling defined from the energy density. We also discuss the solution for n = 3 case.

We propose a supersymmetric gradient flow equation in the four-dimensional Wess-Zumino model. The flow is constructed in two ways. One is based on the off-shell component fields and the other is based on the superfield formalism, in which the same result is provided. The obtained flow is supersymmetric because the flow time derivative and the supersymmetry transformation commute with each other. Solving the equation, we find that it has a damping oscillation with the flow time for nonzero mass, which is different from the Yang-Mills flow. The on-shell flow equation is also discussed.