Tutorial 4 · Verlinde's formula, from scratch

Where the
Integers Come From

Counting how anyons fuse is combinatorics — small non-negative integers. How a theory behaves on a torus is analysis. In 1988 Erik Verlinde noticed they are the same question, and wrote down a formula that turns one into the other.

What this is

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 φ² = φ + 1 — Section 3 leans on it hard) and Derivation 2, the modular group and why S exists at all, from the paper course. The Verlinde by Hand lab turns the same crank with the numbers filled in live. No Berry phase.

Start from a number. Put the four entries of the Fibonacci S-matrix into Verlinde's formula and out comes exactly 1 — twice. Not 0.99999; exactly, because two irrational expressions turn out to be the same number. This tutorial is how a matrix of braiding phases produces the integers that count fusion channels, and Section 3 is the twenty minutes of pencil work that makes it stick.

Section 1Two structures that look unrelated

Fix a topological order with finitely many anyon types a, b, c, …, one of which is the vacuum, written 0. There are two quite different things you can ask about it.

The counting question

Bring a next to b. What can they become, and in how many distinguishable ways? The answer is a table of non-negative integers:

a ⊗ b = Σ_c N_ab^c · cN is a count of fusion channels. Nothing here is continuous. Nothing here involves a phase.

Collect these into matrices — one per anyon type — by fixing the first index:

(N_a)_b^c = N_ab^cFor Fibonacci: N_τ = [[0,1],[1,1]], the matrix whose Perron–Frobenius eigenvalue is φ.

The torus question

Now put the theory on a torus instead of a plane. A torus can be described by a shape parameter τ, but the same torus has many descriptions — you can relabel which cycle you call "space" and which "time." The relabellings form the modular group SL(2,ℤ), generated by two moves:

T : τ → τ + 1 — shear. Acts diagonally, carrying the twists θ_a.
S : τ → −1/τ — swap the two cycles. This is the interesting one.

Fig. 1 — why there is an S at all

A torus has two independent loops and nothing intrinsically distinguishes them. So there must be an operation that exchanges them — and that operation is S. The matrix you have been dividing by 𝒟 is this geometric move, written down. T shears one loop along the other; together they generate SL(2,ℤ). ⚠ The τ in S : τ → −1/τ is the torus modulus, not the Fibonacci anyon — the collision is real and lives in the literature.

The theory's states on the torus are labelled by anyon type. Swapping the cycles mixes those labels, and the mixing is a matrix — the S-matrix, S_ab. Physically it is the amplitude for the worldline of a to link the worldline of b. It is braiding data. It is phases.

Hold the strangeness for a second

The first question is how many ways can these two things combine. The second is what happens when I redescribe a doughnut. There is no obvious reason these should be related at all — and the relation is the content of the theorem.

Section 2The formula

N_ab^c = Σ_x ( S_ax S_bx S̄_cx ) / S_0xSum over all anyon types x. Bar denotes complex conjugate. 0 is the vacuum.

On the left: integers, counting. On the right: nothing but entries of the modular S-matrix. The fusion rules are not independent data. They are a consequence of the braiding.

The cleaner way to say it

The formula is equivalent to a statement that is easier to hold in your head:

N_a = S Λ_a S^† , where (Λ_a)_xx = S_ax / S_0xS simultaneously diagonalises every fusion matrix, and the eigenvalues are ratios of S entries.

So: one matrix, S, diagonalises all of them at once. The fusion matrices commute — they must, since fusion is associative and commutative — and commuting matrices share eigenvectors. Verlinde identifies those shared eigenvectors as the columns of the modular S-matrix, and that identification is the whole theorem.

The eigenvalues λ_a(x) = S_ax/S_0x are the characters of the fusion ring — the one-dimensional representations. A rank-2 ring has two of them. Remember that.

Section 3Do it by hand — Fibonacci

This is worth twenty minutes of your Saturday, because the punchline lands only if you turn the crank yourself. Two objects, 1 and τ. Set 𝒟 = √(1 + φ²), and note φ² = φ + 1 so 𝒟² = 2 + φ ≈ 3.618.

S = (1/𝒟) · [[ 1 , φ ] , [ φ , −1 ]]Rows and columns indexed (0, τ). Check for yourself that S² = identity.

Step 1 — read off the eigenvalues

The eigenvalues of N_τ should be S_τx/S_0x, one per column:

column x = 0: (φ/𝒟) / (1/𝒟) = φ
column x = τ: (−1/𝒟) / (φ/𝒟) = −1/φ

And independently, N_τ = [[0,1],[1,1]] has characteristic polynomial λ² − λ − 1 = 0, whose roots are φ and −1/φ. They match.

Step 2 — recover the fusion rule

Now run the full formula. Take a = b = τ and ask for c = 0. Since S is real here, S̄_0x/S_0x = 1 and the sum collapses:

N_ττ⁰ = Σ_x S_τx² = (φ/𝒟)² + (−1/𝒟)² = (φ² + 1)/𝒟² = 𝒟²/𝒟² = 1

Now c = τ:

N_ττ^τ = Σ_x S_τx³ / S_0x = φ³/𝒟² − 1/(𝒟²φ) = (φ³ − 1/φ) / 𝒟²φ³ = 2φ + 1 and 1/φ = φ − 1, so the numerator is (2φ+1) − (φ−1) = φ + 2. And 𝒟² = φ + 2.

N_ττ^τ = (φ + 2)/(φ + 2) = 1

Fig. 2 — two terms, one integer

The two contributions to N_ττ^τ. Neither is a nice number on its own — φ³/𝒟² and −1/(φ𝒟²), full of irrationals. Their sum is exactly 1, because φ³ − 1/φ = φ + 2 and 𝒟² = 2 + φ are the same number. The integers are not approached; they are hit.

What you just did

You started from a matrix of braiding phases and, with no other input, produced the integers 1 and 1 — which is exactly τ ⊗ τ = 1 ⊕ τ. The fusion rule was never independent information. It was recoverable from S all along.

Section 4The punchline

Look again at the two eigenvalues you read off the columns of S:

λ(0) = φ λ(τ) = −1/φThe two characters of the Fibonacci fusion ring. Both are roots of d² = d + 1.

These are the same two numbers that separate Fibonacci from Yang–Lee in tutorial 1's table. And now you can see what they are: not two rival choices of dimension imposed from outside, but the two characters of one fusion ring, sitting as two columns of one matrix.

The quantum dimension is defined as d_a = S_0a/S₀₀ — a ratio against whichever column you designate the vacuum. So:

Fig. 3 — which column you call the vacuum

The columns of S are the characters of the fusion ring — its two one-dimensional representations. The quantum dimension is a ratio taken against whichever column you designate the vacuum. Pick the all-positive one (the Perron–Frobenius character) and you get d = φ, positive dimensions, a universal quantum computer. Pick the other and you get −1/φ, a negative dimension, and the Yang–Lee edge singularity. Unitarity is exactly the condition that the vacuum sits on the Perron–Frobenius character.

The distinction, stated exactly

Fibonacci and Yang–Lee differ by which character of the fusion ring is designated the vacuum.

Choose the Perron–Frobenius character — the one whose entries are all positive — and you get d_τ = φ, positive dimensions, a unitary theory, a universal quantum computer. Choose the other, and you get d_τ = −1/φ, a negative dimension, non-unitarity, and the Yang–Lee edge singularity.

Unitarity is exactly the condition that the vacuum sits on the Perron–Frobenius character.

That is as sharp as "fusion carries the magnitude, braiding carries the phase" gets. The ring hands you a finite set of characters. Which one you call the ground state is extra data the ring does not contain — and it is the whole difference between a computer and a critical point.

Section 5Why this is the engine of the classification

Here is the part that connects Verlinde to Rowell–Stong–Wang, and it is the reason number theory can classify physical theories at all.

The left side of Verlinde's formula must be a non-negative integer. The right side is built from entries of S. That is an extremely rigid demand: most matrices, plugged into that expression, produce irrational numbers or negative ones and are thereby disqualified.

The move

Verlinde converts "is this a consistent topological order?" — a question about physics — into "does this matrix yield non-negative integers?" — a question about arithmetic.

That is why RSW can classify by solving Diophantine equations. They are not being clever with an unrelated tool. They are using the only tool the problem admits, because Verlinde already made it an arithmetic problem.

And it explains the finiteness you meet in tutorial 3: the fibre over a fusion ring is small because integrality is so restrictive that only a handful of S-matrices survive.

Section 6Provenance, and how to cite it

Worth getting right, because the attribution is not a single paper:

Verlinde (1988), Nucl. Phys. B300, 360, stated the relation — largely as an observation and conjecture, from the structure of rational conformal field theory.
Moore & Seiberg (1989) supplied the derivation in the CFT setting, from the consistency of the theory on higher-genus surfaces.
In the axiomatic modular-tensor-category setting it is a theorem, provable from the MTC axioms directly.

So "Verlinde's formula" names the result, but "Verlinde proved" would be wrong. Say conjectured by Verlinde, derived by Moore–Seiberg, theorem in the categorical setting — three different epistemic statuses, and they are worth keeping apart.

Sources

Checked against the paper itself: Rowell, Stong & Wang (arXiv:0712.1377) for the Diophantine classification described in Section 5.

Not consulted for this page: Verlinde (1988), Nucl. Phys. B300, 360, and Moore & Seiberg (1989). Both are behind publisher paywalls and neither was opened while writing this. The formula is reproduced in essentially every set of lecture notes on rational conformal field theory or topological order, many of them free — which is a practical route for reading, though not for citing.

Everything in Section 3 was computed and checked while writing. Everything in Sections 4–5 is standard but reconstructed rather than read from a source — in particular, check Section 4's vacuum-designation statement against Rowell–Stong–Wang before it carries any weight.