Stochastic formulation of the renormalization group: supersymmetric structure and topology of the space of couplings

TL;DR

This paper reformulates the Wilson renormalization group equations as a stochastic process with supersymmetric structure, enabling topological analysis of the space of couplings using Morse theory.

Contribution

It introduces a supersymmetric formulation of the renormalization group as a stochastic process, providing new tools for analyzing the topology of the coupling space.

Findings

Formulation of RG equations as stochastic Langevin equations.

Identification of supersymmetric structure in the space of couplings.

Application of Morse theory to analyze topology of coupling space.

Abstract

The exact or Wilson renormalization group equations can be formulated as a functional Fokker-Planck equation in the infinite-dimensional configuration space of a field theory, suggesting a stochastic process in the space of couplings. Indeed, the ordinary renormalization group differential equations can be supplemented with noise, making them into stochastic Langevin equations. Furthermore, if the renormalization group is a gradient flow, the space of couplings can be endowed with a supersymmetric structure a la Parisi-Sourlas. The formulation of the renormalization group as supersymmetric quantum mechanics is useful for analysing the topology of the space of couplings by means of Morse theory. We present simple examples with one or two couplings.