Interactive lab · companion to tutorial 4
Two objects, a 2×2 matrix, and a sum with two terms. Turn the crank yourself until fusion is a shadow of braiding stops being a slogan.
A worksheet, not a result. The mathematics on this page is standard topological quantum computation, roughly 1984–2009. None of it is new. “It was simply new to my hands. I had read it for years and never computed with it, and those turn out to be different states of knowing.” These sheets exist to move a reader from the first state to the second, and they make no claim of their own.
The method is predict–then–check. Every input box, drill and gate here is built for one move: write down what you think the number is before you reveal it. Numbers are sourced to a paper where a paper says them, and marked unchecked where none does — a sheet whose whole method is check it yourself cannot ask you to take its own arithmetic on trust.
If you are coming from an operator formalism, you may want to ask of every quantity here what the declared observable is, and what mapping puts it into the domain before the algebra acts. That is a fair question, and these sheets sit upstream of it — the arithmetic of d² = d + 1 is the same arithmetic whichever way it is answered. They take no position on it. The full note is on the index.
Prerequisites: the Golden Arithmetic lab (φ and the identity φ² = φ + 1) and the Row Times Column lab (rows, columns, eigenvalues).
Fibonacci's modular S-matrix, with 𝒟 = √(1+φ²) = 1.902113. These four numbers are the only input. Everything below is derived from them.
Fusion-matrix eigenvalues, read off the columns as Sτx/S0x: — and —
Pick the three labels, then read the working exactly as you would write it out.
| term | Sax | Sbx | S̄cx | S0x | value |
|---|
With two objects, Nabc has three indices each taking two values — a 2 × 2 × 2 array, eight numbers. Here they are, live.
Read the τ⊗τ row: Nττ1 = Nτττ = 1, everything else 0 or 1 exactly as the fusion rule says. Nothing was put in but four numbers from the braiding.
Press Nudge φ → 1.60. That moves the quantum dimension by about one percent, and the S-matrix entries by less. Watch Nτττ fall to 0.975 — and every other cell come off its integer too.
That is the rigidity Rowell–Stong–Wang exploit. The demand that this formula return non-negative integers is so severe that almost no matrix survives it, which is why a classification problem in physics becomes a Diophantine problem in number theory. The integers are not an output you tune toward; they are a filter almost everything fails.
The Try Yang–Lee button swaps the quantum dimension from φ to the other root −1/φ and renormalises. The eight integers do not change.
That is the Galois conjugate, and it is the whole of tutorials 1 and 3 in one button: the fusion ring cannot tell the two theories apart, because Galois action provably fixes the fusion matrices. Same modulus. Different argument.
Write out Nτττ on paper without this page. Two terms, four factors each. When the φ³ − 1/φ in the numerator turns out to equal the 𝒟² in the denominator, and the answer is exactly 1, you will have derived a fusion rule from nothing but braiding phases — and the slogan will have become a calculation.
Checked against the paper itself: Rowell, Stong & Wang, On classification of modular tensor categories (arXiv:0712.1377) — for the rigidity of the integrality demand, and for the Galois pairing of Fibonacci with Yang–Lee, which they state in their introduction.
Every other number on this page is computed live in your browser from the four S-matrix entries you can see, so there is nothing here to take on trust.