Remarks Concerning Polyakov's Conjecture for the 3D Ising Model and the Hierarchical Approximation

TL;DR

This paper explores the use of hierarchical approximations and new variable formulations to analyze the continuum limit of the 3D Ising model, building on Polyakov's conjecture.

Contribution

It introduces novel hierarchical model formulations with dual and fermionic variables and discusses their implications for the renormalization group analysis.

Findings

New hierarchical formulations with dual and fermionic variables

Insights into the renormalization group transformation

Connections to string model reformulations

Abstract

We consider the possibility of using the hierarchical approximation to understand the continuum limit of a reformulation of the 3D Ising model initiated by Polyakov. We introduce several new formulations of the hierarchical model using dual or fermionic variables. We discuss several aspects of the renormalization group transformation in terms of these new variables. We mention a reformulation of the model closely related to string models proposed by Zabrodin.