Flat Fields in Subfactor Theory

• Flat fields of strings in subfactor theory are defined by precise algebraic and diagrammatic conditions that enforce commutation properties in tensor networks.
• Bi-unitary connections and 4-tensors are linked via zipper conditions, establishing a correspondence between subfactor invariants and topological sectors in condensed matter models.
• The analysis of flat connections and higher relative commutants offers a robust framework for classifying small-index subfactors and understanding anyon models in two-dimensional topological phases.

Flat fields of strings in subfactor theory encapsulate the precise algebraic and diagrammatic conditions required for certain higher-degree tensor objects to realize commutation properties fundamental to both operator algebraic subfactor theory and their manifestation in tensor-network models of two-dimensional topological order. The emergence of the “zipper condition” for 4-tensors in tensor network models has led to the identification of these structures with specific flatness constraints on fields of strings in the context of the planar algebra associated to a subfactor, establishing a detailed correspondence between topological sectors in condensed matter models and the higher relative commutants of subfactors.

1. Bi-Unitary Connections and the Structure of Commuting Squares

A bi-unitary connection arises in Jones’s theory as a function $W(\xi_0,\xi_1,\xi_2,\xi_3)$ assigning complex numbers to cells—quadruples of edges traversing four finite, bipartite, oriented graphs $_0,_1,_2,_3$ under strict source and range conditions: $$s(\xi_0) = s(\xi_1) = x_0, \quad r(\xi_0) = s(\xi_3) = x_3, \quad r(\xi_1) = s(\xi_2) = x_1, \quad r(\xi_2) = r(\xi_3) = x_2.$$ Normalization crucially involves a positive Perron–Frobenius vector $\mu$ and eigenvalues $\beta_0,\beta_1 > 1$, enforcing

$$\sum_{x} \Delta_{i,xy} \mu(x) = \beta_{\lfloor i/2 \rfloor} \mu(y), \quad \sum_{y} \Delta_{i,xy} \mu(y) = \beta_{\lfloor i/2 \rfloor} \mu(x).$$

Unitarity consists of two sets of “horizontal” and “vertical” equations, ensuring that $W$ and its renormalized $\pi/2$-rotations $W',...$ satisfy contraction identities represented diagrammatically by “jellyfish” and cap-cup diagrams. The bi-unitarity condition means these relations hold for all three $90^\circ$-rotations, and two connections are unitarily equivalent if related by unitaries $U, V$ acting on the relevant edge sets.

2. 4-Tensor / Bi-Unitary Connection Correspondence

Tensor-network approaches to 2D topological phases introduce 4-tensors $a_{\xi;\eta}^{\rho,\sigma}$, with “physical” edge indices $\xi,\eta$ and “virtual” edges/lines $\rho,\sigma$. The key normalization

$$W_a(\xi, \rho, \eta, \sigma) = a^{\rho, \sigma}_{\xi; \eta} \cdot \sqrt[4]{\frac{\mu(s(\xi))\,\mu(r(\eta))}{\mu(r(\xi))\,\mu(s(\eta))}}$$

ensures that $W_a$ is a bi-unitary connection if and only if $a$ satisfies pentagon-type (fusion unitarity) equations. The rotation symmetry in these tensors aligns with planar isotopy in the graphical calculus, corresponding to performing $\pi/2$-rotations of the tensor legs.

This correspondence is central: every normalized 4-tensor in the physical literature with the required symmetry properties may be interpreted as a bi-unitary connection in subfactor theory. The explicit normalization makes precise the translation between tensor-network symmetries and algebraic constraints in operator algebras.

3. Flat Fields of Strings and Higher Relative Commutants

Given an irreducible bi-unitary connection $W$, one constructs a tower of finite-dimensional “string algebras” $A_{jk}$ via Jones’s basic construction; for example, $A_{00} = \mathbb{C}^{|V_0|}$, $$A_{10} \simeq \mathrm{End}(\bigoplus_{\xi} \text{paths of length 1 in }\,{_0} \cup {_2})$$, and so forth. These algebras are equipped with orthogonal Jones projections ($e_n, f_n$), which generate commuting squares.

A field of strings $f$ on ${_0}$ is an assignment of matrices $f_{\rho, \rho'}$ to strings of length-2 paths in ${_2} \circ {_3}$, subject to flatness constraints:

• Half-flatness: $$\sum_{\rho_1, \rho_2} f_{\rho_1, \rho_2} W_a(\ldots; \rho_1, \rho_2) = \delta_{\ldots} \,\tilde f_{\ldots}$$
• Full flatness: $\sum f\,W\,W' = \delta\,f$ Equivalently, $f$ lies in $A_{10} \cap A_{02}$, and, iteratively, in the intersection of all $A_{10} \cap A_{0k}$ (the higher relative commutants of $A_{\infty, 0} \subset A_{\infty, 2}$).

This formalizes the notion that such $f$ represent operators invariant under the full top algebra action, concretely realizing the notion of “topological defects” or “anyon projectors” in the physical context.

4. Zipper and Half-Zipper Conditions: Algebraic and Diagrammatic Realization

The zipper and half-zipper conditions originate from tensor-network constructions of MPO symmetries and projectors. Their diagrammatic realization involves composing 2-tensors $F$ and $\tilde F$ with the 4-tensor $a$ by “pulling a zipper” through the tensor:

• Half-zipper: $F \circ a = a \circ \tilde F$
• Full zipper: $F \circ a = a \circ F$ Both conditions expand into families of linear pentagon-type equations involving normalization constants and summations over the virtual indices.

The equivalence theorem demonstrates that:

• $a$ satisfying the half-zipper (respectively, zipper) condition is equivalent (under a precise identification of indices and normalizations) to the string field $f$ satisfying half-flatness (respectively, full flatness), and thus to $f$ being an element of the higher relative commutants.
• The MPO tensors $F, \tilde F$ are realized as intertwiners arising from $f$ in the algebraic context of the commutants.

Graphical proofs involve inserting $f$ into the “jellyfish” diagrams, then applying bi-unitarity (including the cap-cup identity) to demonstrate the needed intertwining or commutation relations.

5. Consequences for Two-Dimensional Topological Order and Anyon Models

In the framework of PEPS and MPOs for anyon models, the zipper condition is imposed to guarantee that the MPO acts as a projector (central idempotent) in the double-layer. The identification with flat fields of strings shows:

• The physical projectors underlying virtual topological symmetry correspond directly to elements of the higher relative commutants in subfactor theory.
• The entire planar algebra of the subfactor acts on the virtual indices; all anyon labels, modular $S$ and $T$ matrices, and related modular data emerge from the higher commutant structure.
• The finite-depth condition in physical models coincides with the flatness of the bi-unitary connection (Ocneanu flatness), ensuring a genuine finite tensor category and precluding infinite multiplicities.
• The “half-zipper” perspective accounts for the existence of chiral (left- or right-) MPO intertwiners in models where full flatness is not available, indicating partial symmetry.

This creates a structured dictionary between algebraic data from subfactor theory and topological invariants classifying two-dimensional phases, situating the paper of flatness and commutants as fundamental to both operator-algebraic and condensed matter descriptions.

6. Flat Connections and Uniqueness in Subfactor Classification

The classification of small-index subfactors in the interval $(5, 3+\sqrt{5})$ is directly governed by the existence and uniqueness of flat bi-unitary connections on 4-partite (principal) graphs. In this regime:

• The bi-unitary condition is enforced by vertical and horizontal unitarity, with normalization taken via the Perron-Frobenius vector and renormalization identities.
• Flatness is the requirement that, for $K$ a biunitary connection, there exist 2-box elements $x, y$ such that $K y K^* = K x K^*$.
• Flat connections correspond canonically to subfactor planar algebras. Morrison and Peters demonstrate, via classification (using the FusionAtlas “odometer” and dimension bounds), that in this interval only two quantum-group subfactors arise, each uniquely determined by their flat connection (Morrison et al., 2012).

This analysis underscores the central role of flatness and the higher relative commutants—not only as a structural feature in tensor-network models, but also as a rigidity mechanism for the classification of subfactors within a given index range.

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A plausible implication is that the bridging of conditions from tensor-network symmetries to flatness in subfactor theory provides a robust framework for transferring classification results and structural invariants between condensed matter and operator algebraic approaches, enhancing both the understanding of topological phases and the algebraic classification of subfactors.