Leading infrared logarithms for the $\sigma$-model with fields on an arbitrary Riemann manifold

We derive a nonlinear recurrence equation for the infrared leading logarithms (LLs) in the four-dimensional $\sigma$-model with fields on an arbitrary Riemann manifold. The derived equation allows computing the LLs to an essentially unlimited loop order in terms of the geometric characteristics of the Riemann manifold. We reduce solving the $SU(\infty)$ principal chiral field in an arbitrary number of dimensions in the LL approximation to solving a very simple recurrence equation. This result prepares a way to solve the model in an arbitrary number of dimensions as $N\to\infty$.