In their classification of subfactors of index less than 5, Izumi, Jones, Morrison, and Snyder introduced the diagrammatic branch matrix UU — the canonical gauge representative of a flat biunitary connection at the initial branch point of a candidate principal graph pair — and proved (Theorem 1.7) that the eigenvalues of UtUU^t U are two 1's together with the eigenvalues of the planar-algebra two-click rotation on the non-Temperley–Lieb box space one past the initial branch point. Because rotational eigenvalues are roots of unity, this converts continuous gauge-parameter families of candidate connections into discrete constraints that confine flatness to finitely many parameter values; the authors used it to prove uniqueness of the 3311 subfactor and remarked that the relationship "should be applicable more generally". Later work applied the branch-matrix machinery case by case — to triple-point obstructions (Snyder), to a candidate graph in the index interval (5,3+5)(5,3+\sqrt{5}) (Morrison–Peters), to the 2222 graph at index 5 (Izumi–Morrison–Penneys–Peters–Snyder), to the 4442 graph at index 3+53+\sqrt{5} (Morrison–Penneys), and within the classification up to index 5145\tfrac{1}{4} (Afzaly–Morrison–Penneys) — but no general theory removing the theorem's hypotheses (initial branch points with multiple edges; connection data beyond the initial branch matrix) and no systematic, implemented procedure exist. The problem is to develop the correspondence into a general, hypothesis-minimal tool for constraining flatness of biunitary connections on arbitrary candidate principal graph pairs, and to demonstrate it on settings beyond the published case-by-case treatments.

Background

A subfactor is an inclusion N⊂MN\subset M of von Neumann algebras with trivial centers; its Jones index [M:N]≥1[M:N]\geq 1 measures the size of the inclusion. The standard invariant of a finite-index subfactor — the tower of higher relative commutants — is encoded combinatorially by the principal graphs, a pair of bipartite graphs whose vertices record the irreducible bimodules appearing in tensor powers of the basic NN-MM bimodule and whose adjacency encodes fusion. A subfactor has finite depth when these graphs are finite, and classification of small-index subfactors proceeds in two stages: enumerate all candidate principal graphs compatible with the index bound and known restrictions, then eliminate each candidate or construct a subfactor realizing it.

Two equivalent languages describe the standard invariant. A planar algebra is a graded family of box spaces acted on by planar tangles; the Temperley–Lieb subalgebra is the part spanned by non-crossing diagrams, and a subfactor planar algebra is nn-supertransitive when its box spaces agree with Temperley–Lieb up to depth nn, so the first non-Temperley–Lieb elements appear one step past the initial branch point of the principal graph. The one-click rotation turns a box by π\pi and the two-click rotation ρ\rho turns it by 2π2\pi; in the finite-depth setting the eigenvalues of ρ\rho on the Temperley–Lieb complement are roots of unity — the starting point of Jones' quadratic tangles analysis — so rotational spectra carry strong arithmetic constraints. A biunitary connection is the same data written differently: a system of unitary matrices indexed by pairs of edges of the candidate graph pair, defined only up to gauge (unit complex rescalings), subject to normalization and unitarity axioms; the connection comes from a subfactor exactly when it is flat, a local associativity condition, so classification reduces to deciding which members of a typically continuous family of connections are flat.

The series "Subfactors of index less than 5" carried this program through the index range (4,5)(4,5): Part 1 enumerated the candidates, leaving two infinite families of candidate graph pairs with 4-valent initial vertices (the weeds Q\mathcal{Q} and Q′\mathcal{Q}' — a weed is a graph pair representing an infinite family of candidate graph pairs obtained by translation) plus finitely many vines (an explicit finite list of candidate graph pairs, each again considered with all its translations); Part 2 eliminated the triple-point weeds; Part 4 reduced the infinitely many vine translations to finitely many cases by a cyclotomicity argument (a number-theoretic integrality test on graph norms); Part 3 (Izumi–Jones–Morrison–Snyder) eliminated the two quadruple-point weeds and proved uniqueness, up to duality, of the 3311 Goodman–de la Harpe–Jones subfactor — the subfactor of index 3+33+\sqrt{3} whose principal graph begins with a 4-valent branch point, arising from the Coxeter-graph construction in the book of Goodman, de la Harpe, and Jones.

For the 3311 uniqueness step the authors introduced the branch matrix: the (k+2)×(k+2)(k+2)\times(k+2) unitary matrix of connection values on loops through the initial branch point of the two graphs. They showed that every flat biunitary connection has a canonical gauge representative — the diagrammatic branch matrix, characterized within its gauge class by a real top-left entry of sign (−1)n+1(-1)^{n+1} and positive real remaining first-row and first-column entries — and proved (Theorem 1.7) that for an nn-supertransitive graph pair whose initial branch point is (k+2)(k+2)-valent without multiple edges, the eigenvalues of UtUU^t U are two 1's together with the eigenvalues of ρ\rho on the Temperley–Lieb complement one past the initial branch point; in particular tr⁡(UtU)−2\operatorname{tr}(U^t U)-2 equals the sum of those rotational eigenvalues. Since rotational eigenvalues are roots of unity, this identity converts the continuous gauge-parameter family of connections into discrete constraints strong enough to pin flatness to finitely many parameter values — for 3311 it forces tr⁡(UtU)=2\operatorname{tr}(U^t U)=2 and yields uniqueness. The theorem is already stated for arbitrary valence k+2k+2; what its hypotheses exclude are initial branch points with multiple edges, and it constrains only the single matrix at the initial branch point, not the deeper connection data. The authors remark that this relationship between rotational eigenvalues and connections "should be applicable more generally"; subsequent papers applied it case by case (see previous progress), but the general theory and systematic tool sketched by that remark remain to be built.

Problem Statement

Develop the branch-matrix / rotation-eigenvalue correspondence of Izumi-Jones-Morrison-Snyder (Theorem 1.7 of Subfactors of index less than 5, part 3) into a general, systematic tool for constraining and constructing flat biunitary connections on candidate principal graph pairs. Concretely, beyond the case-by-case treatments available in the 2012-2015 literature:

(1) Prove the correspondence in the settings its current hypotheses exclude: initial branch points with multiple edges, where the 1-dimensional intertwiner spaces underlying the diagrammatic gauge normalization can fail, and extension from the single matrix at the initial branch point to connection data at vertices deeper in the graphs, yielding explicit necessary eigenvalue or trace conditions on that deeper data for flatness.

(2) Give a systematic, reproducible procedure (ideally implemented) that, given an arbitrary candidate principal graph pair with its continuous family of biunitary connections modulo gauge, extracts the discrete constraints arising from the root-of-unity property of the rotational eigenvalues together with the identity of Theorem 1.7, so that flatness is confined to finitely many parameter values or excluded outright.

(3) Demonstrate the resulting theory on at least one setting not settled by the existing case-by-case arguments, for example candidate graphs in an index range above 5145\tfrac{1}{4}, graphs with multiple edges at the initial branch point, or a new construction of subfactors or of unitary fusion categories (the semisimple C*-tensor categories abstracting the bimodule fusion encoded by the principal graphs).

The common scientific objective is a single general correspondence between connection gauge data and the planar-algebraic rotation spectrum. The three components jointly support that objective: (1) supplies the hypothesis-minimal statement, (2) makes it a tool rather than a collection of ad hoc arguments, and (3) is the non-vacuity check that the general theory genuinely extends the published instances (the 3311, 2222, and 4442 graph pairs and the triple-point obstructions) rather than repackaging them. The objective is the correspondence itself rather than an advance of the subfactor classification frontier, and the correspondence should be general enough to decide at least one case that the existing case-by-case arguments did not.

References

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  2. Noah Snyder. A rotational approach to triple point obstructions. Analysis & PDE 6 (2013) 1923-1928. DOI: 10.2140/apde.2013.6.1923. https://arxiv.org/abs/1207.5090
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