The renormalization group describes the change in a physical system as the energy scale decreases from high to low. In quantum field theory, the exact renormalization group (ERG)—formulated by Wilson and Polchinski—follows a cutoff-dependent Wilson action by integrating high-energy degrees of freedom.

Earlier ERG studies primarily used differential flow equations to describe infinitesimal changes in the cutoff scale. However, these equations are local differential equations on the configuration space the space of all field configurations—and thus cannot adequately describing the global topology. This is particularly crucial when the configuration space is multiply connected with a nontrivial fundamental group , as such topologies are closely related to topological terms in field theory.

In this study, we focus on a functional integral kernel that describes a finite ERG transformation, which we term the ERG kernel. The basic observation is that ERG transformations in field theory exhibit a structure corresponding to time evolution in quantum mechanics. In quantum mechanics, infinitesimal time evolution is described by the Schrödinger equation, whereas finite evolution is described by an integral kernel. Similarly, infinitesimal ERG transformations are described by flow equations, whereas finite ERG transformations are represented by ERG kernels.

For quantum mechanics on a multiply connected configuration space, the time-evolution kernel can be constructed by lifting the problem to the universal covering space and summing over the elements of the fundamental group with weight factors . This construction yields a natural formulation of finite ERG transformations that incorporates the global topology of the configuration space.

Based on the algebraic properties required of the ERG kernel, we obtain two important results. First, the composition law of ERG transformations dictates that the weights  are independent of the cutoff scale. This establishes a nonrenormalization theorem for topological terms originating from . Second, the normalization condition that preserves the partition function results in a quantization condition. In nonlinear sigma models, this condition is equivalent to the level quantization of the Wess–Zumino–Witten term.

Results show that the level quantization and nonrenormalization of topological terms follow nonperturbatively from the algebraic and geometric structures of finite ERG transformations. Additionally, the formulation is expected to be applicable to other field theories with topological terms, including the Chern–Simons term in the three-dimensional Yang–Mills theory.

(written by Taichi Tanaka on behalf of all authors.)

Share this topic

Fields