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The zipper condition for $4$-tensors in two-dimensional topological order and the higher relative commutants of a subfactor arising from a commuting square

Yasuyuki Kawahigashi

TL;DR

This work identifies the $4$-tensors used in tensor-network models of 2D topological order with bi-unitary connections from Jones’ subfactor theory and shows that the zipper condition on these tensors corresponds to flat fields of strings in the associated higher relative commutants. By treating the $4$-tensors as bi-unitary connections, the authors establish that the zipper/flatness properties are equivalent to intertwining conditions within the string-algebra framework, without requiring finite depth or flatness a priori. They further generalize the construction to allow four index sets to differ and introduce a half-version of the zipper condition that suffices for the central morphism properties. The results bridge tensor-network descriptions of quantum symmetries with subfactor theory and clarify the exact algebraic conditions under which these correspondences hold, offering precise normalization and morphism criteria for practical computations.

Abstract

They recently study two-dimensional topological order in condensed matter physics in terms of tensor networks involving certain 3- and 4-tensors. Their 3-tensors satisfying the ``zipper condition'' play an important role there. We identify their 4-tensors with bi-unitary connections in Jones' subfactor theory in operator algebras with precise normalization constants. Then we prove that their tensors satisfying the zipper condition are the same as flat fields of strings in subfactor theory which correspond to elements in the higher relative commutants of the subfactor arising from the bi-unitary connection. This is what we expect, since the zipper condition is a kind of pentagon relations, but we clarify what conditions are exactly needed for this -- we do not need the flatness or the finite depth condition for the bi-unitary connection. We actually generalize their 4-tensors so that the four index sets of the 4-tensors can be all different and work on a ``half-version'' of the zipper condition.

The zipper condition for $4$-tensors in two-dimensional topological order and the higher relative commutants of a subfactor arising from a commuting square

TL;DR

This work identifies the -tensors used in tensor-network models of 2D topological order with bi-unitary connections from Jones’ subfactor theory and shows that the zipper condition on these tensors corresponds to flat fields of strings in the associated higher relative commutants. By treating the -tensors as bi-unitary connections, the authors establish that the zipper/flatness properties are equivalent to intertwining conditions within the string-algebra framework, without requiring finite depth or flatness a priori. They further generalize the construction to allow four index sets to differ and introduce a half-version of the zipper condition that suffices for the central morphism properties. The results bridge tensor-network descriptions of quantum symmetries with subfactor theory and clarify the exact algebraic conditions under which these correspondences hold, offering precise normalization and morphism criteria for practical computations.

Abstract

They recently study two-dimensional topological order in condensed matter physics in terms of tensor networks involving certain 3- and 4-tensors. Their 3-tensors satisfying the ``zipper condition'' play an important role there. We identify their 4-tensors with bi-unitary connections in Jones' subfactor theory in operator algebras with precise normalization constants. Then we prove that their tensors satisfying the zipper condition are the same as flat fields of strings in subfactor theory which correspond to elements in the higher relative commutants of the subfactor arising from the bi-unitary connection. This is what we expect, since the zipper condition is a kind of pentagon relations, but we clarify what conditions are exactly needed for this -- we do not need the flatness or the finite depth condition for the bi-unitary connection. We actually generalize their 4-tensors so that the four index sets of the 4-tensors can be all different and work on a ``half-version'' of the zipper condition.

Paper Structure

This paper contains 4 sections, 2 theorems, 3 equations, 35 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Bi-unitary connections
  3. $4$-tensors
  4. Flatness and the zipper condition

Key Result

Proposition 2.1

The above action of flat fields of strings gives a self-intertwiner of $X^W$ commuting with the left and right actions of hyperfinite II$_1$ factors arising from the string algebras on $\mathcal{G}_0$ and $\mathcal{G}_2$, and all self-intertwiners of $X^W$ arise in this way.

Figures (35)

  • Figure 1: Four graphs
  • Figure 2: Example 1:How four graphs are connected
  • Figure 3: Example 2:How four graphs are connected
  • Figure 4: A cell
  • Figure 5: A connection value
  • ...and 30 more figures

Theorems & Definitions (3)

  • Proposition 2.1
  • Theorem 4.1
  • proof