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The Bałaban variational problem in the non-linear sigma model

Wojciech Dybalski, Alexander Stottmeister, Yoh Tanimoto

Abstract

The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of Bałaban's approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyze this step for the $O(4)$ non-linear sigma model in two dimensions and demonstrate, in this case, how various ingredients of Bałaban's approach play together. First, using variational calculus on Lie groups, the equation for the critical point is derived. Then, this non-linear equation is solved by the Banach contraction mapping theorem. This step requires detailed control of lattice Green functions and their integral kernels via random walk expansions.

The Bałaban variational problem in the non-linear sigma model

Abstract

The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of Bałaban's approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyze this step for the non-linear sigma model in two dimensions and demonstrate, in this case, how various ingredients of Bałaban's approach play together. First, using variational calculus on Lie groups, the equation for the critical point is derived. Then, this non-linear equation is solved by the Banach contraction mapping theorem. This step requires detailed control of lattice Green functions and their integral kernels via random walk expansions.
Paper Structure (22 sections, 39 theorems, 255 equations, 4 figures)

This paper contains 22 sections, 39 theorems, 255 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The setting and results
  3. Preparations
  4. Linearization of the constraint
  5. Sets of configurations
  6. Geometric considerations
  7. Variational calculus on Lie groups
  8. Tangent space of the constraint manifold
  9. Derivation of the critical point equation
  10. Structure of the expressions $\overrightarrow{W}$
  11. Analytic considerations
  12. Existence of solutions of the critical point equation
  13. Strict positivity of $G^{-1}, G$ and $QG Q^*$
  14. Exponential decay of integral kernels of $G$, $(QGQ^*)^{-1}$
  15. Method of images
  16. ...and 7 more sections

Key Result

Theorem A

Let $\mathrm{G}_0=SU(2)$. Then there exist $0 < \varepsilon, \varepsilon_1\leq 1$ s.t. for $V\in \mathrm{Conf}_{\varepsilon_{1}}(\Omega_1)$ the action $\mathcal{A}$ has a unique critical point over $\mathrm{Conf}_{\varepsilon}(\Omega)$ with the constraint $\mathcal{C}(U)=V$. The parameters $\varepsi

Figures (4)

  • Figure 1: The spanning tree $T(y)$ of the box $B_1(y)$ is indicated in red together with its orientation. The orientation of the bonds of the lattice is fixed by the axes of the coordinate frame.
  • Figure 2: The square containing the origin is the set $\Omega$. The reflections $P_{\mu}$ are defined w.r.t. the dashed lines from the figure. The points $z_j$ of the argument $z$ are as in formula (\ref{['images-test-case-x']}).
  • Figure 3: The centers of $2\tilde{M}$-boxes in $\mathbb{Z}^2$ as blue (thick) dots, with one of the $2\tilde{M}$-boxes indicated.
  • Figure 4: Three families of disjoint boxes in one dimension. Their union is the whole family of the $2\tilde{M}$-boxes centered at $\tilde{M}\mathbb{Z}$ described in Definition \ref{['boxes-def']}.

Theorems & Definitions (50)

  • Theorem A
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 3.1
  • Proposition 3.3
  • Lemma 3.5
  • Definition 3.6
  • ...and 40 more