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Ising Cardy annulus multiplicities

Example

Use the diagonal Ising VOA $V$ of central charge $c = \frac{1}{2}$ and its three irreducible ordinary modules $M_{0} = V$ , $M_{ϵ}$ , and $M_{σ}$ , with conformal weights $0, \frac{1}{2}, \frac{1}{16}$ and character labels $0, ϵ, σ$ in that order; take the source modular matrix $S = \frac{1}{2} (\begin{array}{ccc} 1 & 1 & \sqrt{2} \\ 1 & 1 & - \sqrt{2} \\ \sqrt{2} & - \sqrt{2} & 0 \end{array}) .$ Each Ising module is self-contragredient; fix module isomorphisms $J_{i} : M_{i}^{\lor} \to M_{i}$ and use the normalized weight-block coevaluation Ishibashi vectors from Diagonal rational annulus and Cardy-state conventions. With positive square roots in that convention, the Cardy coefficients are $∣ 0 ⟩ = 2^{- 1 / 2} ∣ 0 ⟩ ⟩ + 2^{- 1 / 2} ∣ ϵ ⟩ ⟩ + 2^{- 1 / 4} ∣ σ ⟩ ⟩, ∣ ϵ ⟩ = 2^{- 1 / 2} ∣ 0 ⟩ ⟩ + 2^{- 1 / 2} ∣ ϵ ⟩ ⟩ - 2^{- 1 / 4} ∣ σ ⟩ ⟩, ∣ σ ⟩ = ∣ 0 ⟩ ⟩ - ∣ ϵ ⟩ ⟩ .$ For $δ > 0$ , the open and closed names are $q_{o} = e^{- π δ}$ and $q_{c} = e^{- 4 π / δ}$ . The open-channel character sums for boundary labels $a, b \in {0, ϵ, σ}$ are $\begin{array}{cccc} Z_{a b}^{o p e n} & b = 0 & b = ϵ & b = σ \\ a = 0 & χ_{0} (q_{o}) & χ_{ϵ} (q_{o}) & χ_{σ} (q_{o}) \\ a = ϵ & χ_{ϵ} (q_{o}) & χ_{0} (q_{o}) & χ_{σ} (q_{o}) \\ a = σ & χ_{σ} (q_{o}) & χ_{σ} (q_{o}) & χ_{0} (q_{o}) + χ_{ϵ} (q_{o}) \end{array}$ In particular the $σ$ - $σ$ annulus has two open sectors, $0$ and $ϵ$ , each with multiplicity one.

Example references

Diagonal rational annulus and Cardy-state conventions

Facts & Assumptions

Given: The diagonal Ising character system, the displayed character $S$ transformation, and the Cardy-state construction in the source annulus conventions.

[F1]

Diagonal rational annulus and Cardy-state conventions: the vacuum label is fixed by $M_{0} = V$ . Since all Ising labels are self-dual, the closed overlap is $\sum_{i} S_{a i} S_{b i} χ_{i} (q_{c}) / S_{0 i}$ ; the nome parameters satisfy $τ_{c} = - 1 / τ_{o}$ .

[A1]

Cardy's diagonal Ising example supplies the three-label character transformation matrix $S$ above and the stated Cardy-state prescription. Ginsparg's lecture notes independently give the Ising character families and fusion products. The example is a matrix and character calculation conditional on those source data; it does not prove existence of a full local CFT or a path-integral realization.

Proof

Given: The diagonal Ising character system, the displayed character $S$ transformation, and the Cardy-state construction in the source annulus conventions.

1.1givenF1A1algebra

The rows of $S$ are $(\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}})$ , $(\frac{1}{2}, \frac{1}{2}, - \frac{1}{\sqrt{2}})$ , and $(\frac{1}{\sqrt{2}}, - \frac{1}{\sqrt{2}}, 0)$ . Their squared norms are $1$ and their pairwise inner products are $0$ , so $S S^{T} = I$ . The vacuum-row entries are $(\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}})$ , hence the positive roots are $d_{0} = d_{ϵ} = 2^{- 1 / 2}$ and $d_{σ} = 2^{- 1 / 4}$ . Substitution into the Cardy coefficient formula gives the three displayed states.

2.1givenF1A1step 1.1algebra

For fixed $b$ , the coefficient of $χ_{j} (q_{o})$ after modularly transforming the closed overlap is $m_{a b}^{j} = \sum_{i} \frac{S_{a i} S_{b i} S_{i j}}{S_{0 i}} = {(S diag ⁣ (\frac{S_{b i}}{S_{0 i}}) S^{T})}_{a j},$ using symmetry of the displayed matrix and $S_{0 i} \neq 0$ . The three diagonal ratio matrices, for $b = 0, ϵ, σ$ , have entries respectively $(1, 1, 1)$ , $(1, 1, - 1)$ , and $(\sqrt{2}, - \sqrt{2}, 0)$ .

3.1givenA1step 2.1algebra

For $b = 0$ , the ratio matrix is the identity, so $m_{\cdot, 0}^{\cdot} = S S^{T} = I$ . For $b = ϵ$ , direct multiplication with $diag (1, 1, - 1)$ gives the second matrix below; for $b = σ$ , multiplication with $diag (\sqrt{2}, - \sqrt{2}, 0)$ gives the third: $m_{\cdot, 0}^{\cdot} = I, m_{\cdot, ϵ}^{\cdot} = (\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}), m_{\cdot, σ}^{\cdot} = (\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \end{array}) .$ with rows indexed by $a$ and columns by $j$ in the order $0, ϵ, σ$ . For the potentially nontrivial last row, the formula in step 2.1 gives $m_{σ σ}^{0} = \frac{(1 / \sqrt{2})^{2} (1 / 2)}{1 / 2} + \frac{(- 1 / \sqrt{2})^{2} (1 / 2)}{1 / 2} = 1, m_{σ σ}^{ϵ} = \frac{(1 / \sqrt{2})^{2} (1 / 2)}{1 / 2} + \frac{(- 1 / \sqrt{2})^{2} (1 / 2)}{1 / 2} = 1, m_{σ σ}^{σ} = \frac{(1 / \sqrt{2})^{2} (1 / \sqrt{2})}{1 / 2} + \frac{(- 1 / \sqrt{2})^{2} (- 1 / \sqrt{2})}{1 / 2} = 0.$ Thus every coefficient is $0$ or $1$ , and the $σ$ - $σ$ row has one unit coefficient in each of columns $0, ϵ$ .

4.1givenF1A1step 2.1step 3.1algebra∎

Reading the matrices row by row gives the displayed open-channel table. The coefficient formula in step 2.1 came from applying the character $S$ transform to the closed overlap, so each table entry equals that closed-channel expression after $τ_{c} = - 1 / τ_{o}$ . These are nonnegative integer multiplicities for all nine boundary pairs. The calculation verifies the annulus character identity for this supplied Ising Cardy datum only.

Proof status

The finite character-matrix calculation is complete under [A1]. It treats the diagonal Ising Cardy states and supplies no claim about arbitrary Virasoro-gluing states or a full local CFT construction.

Remarks

Ginsparg's lecture notes independently give $σ \times σ = 0 + ϵ$ , $ϵ \times ϵ = 0$ , and $ϵ \times σ = σ$ . The annulus multiplicity matrices above are checked directly from the displayed $S$ matrix, so the example does not rely on a categorical Hopf-link matrix.

Depends on

Diagonal rational annulus and Cardy-state conventions