CFT: Virasoro Minimal Models and WZW SU(2)$_k$

QAtlas exposes two parametric families of two-dimensional rational conformal field theories as concrete dispatch tags:

• MinimalModel(p, p_prime) — unitary and non-unitary Virasoro minimal models $\mathcal{M}(p, p^\prime)$.
• WZWSU2(k) — Wess–Zumino–Witten models with affine $\widehat{\mathfrak{su}}(2)_k$ symmetry.

Both expose their CFT data as exact Rational{Int} values via fetch on CentralCharge, ConformalWeights, and (minimal models only) PrimaryFields.

Virasoro Minimal Models $\mathcal{M}(p, p^\prime)$

Following Belavin, Polyakov, Zamolodchikov (BPZ, 1984) the central charge of the minimal model labelled by coprime integers $p > p^\prime \geq 2$ is

\[c(p, p^\prime) = 1 - \frac{6 (p - p^\prime)^2}{p \, p^\prime}.\]

Primary fields are labelled by $(r, s)$ with $1 \leq r \leq p^\prime - 1$, $1 \leq s \leq p - 1$, with conformal weight given by the Kac formula:

\[h_{r, s}(p, p^\prime) = \frac{(p \, r - p^\prime s)^2 - (p - p^\prime)^2}{4 \, p \, p^\prime}.\]

The Kac table has the symmetry $h_{r, s} = h_{p^\prime - r, p - s}$, so the number of distinct primaries is $(p - 1)(p^\prime - 1) / 2$.

Special cases

Usage

using QAtlas

m = MinimalModel(4, 3)
fetch(m, CentralCharge())                  # 1//2
fetch(m, ConformalWeights(); r=1, s=2)     # 1//16  (σ)
fetch(m, ConformalWeights(); r=2, s=1)     # 1//2   (ε)
fetch(m, PrimaryFields())                  # 3 NamedTuples (r, s, h)

Cross-check with Universality(:Ising)

The c = 1//2 stored on Universality(:Ising)'s exponent NamedTuple at $d = 2$ agrees identically with fetch(MinimalModel(4, 3), CentralCharge()):

fetch(MinimalModel(4, 3), CentralCharge()) ==
    QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=2).c
# true

This is a sanity check that the CFT-side parametric data and the universality-class table agree as Rational{Int} values.

Validation

MinimalModel(p, p_prime) validates its arguments at construction time and throws DomainError for any of:

• \[p^\prime < 2\]

  ,

• \[p \leq p^\prime\]

  ,

• \[\gcd(p, p^\prime) \neq 1\]

  (non-coprime).

fetch(::MinimalModel, ConformalWeights(); r, s) throws DomainError for $(r, s)$ outside the fundamental rectangle $1 \leq r \leq p^\prime - 1$, $1 \leq s \leq p - 1$.

WZW SU(2)$_k$

Following Knizhnik–Zamolodchikov (1984) the Sugawara central charge of the level-$k$ WZW model on the affine algebra $\widehat{\mathfrak{su}}(2)_k$ is

\[c(k) = \frac{3 k}{k + 2}, \qquad k = 1, 2, 3, \dots\]

The primary fields are labelled by SU(2) spin $j \in \{0, 1/2, 1, 3/2, \dots, k/2\}$ with conformal weight

\[h_j = \frac{j (j + 1)}{k + 2}.\]

Special cases

Usage

fetch(WZWSU2(1), CentralCharge())                    # 1//1
fetch(WZWSU2(1), ConformalWeights(); j=1//2)         # 1//4
fetch(WZWSU2(2), ConformalWeights(); j=1)            # 1//2

Validation

WZWSU2(k) requires $k \geq 1$ and throws DomainError otherwise.

fetch(::WZWSU2, ConformalWeights(); j) requires j to be a non-negative half-integer (Integer or Rational{Int} with $2 j \in \mathbb{Z}_{\geq 0}$) with $0 \leq j \leq k/2$. Floats and non-half-integer rationals (e.g. 1//3) raise DomainError.

References

• A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241, 333 (1984).
• D. Friedan, Z. Qiu, S. Shenker, Conformal invariance, unitarity, and critical exponents in two dimensions, Phys. Rev. Lett. 52, 1575 (1984).
• V. G. Knizhnik, A. B. Zamolodchikov, Current algebra and Wess–Zumino model in two dimensions, Nucl. Phys. B 247, 83 (1984).
• E. Witten, Non-abelian bosonization in two dimensions, Comm. Math. Phys. 92, 455 (1984).
• P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory (Springer, 1997), Ch. 7 (minimal models), Ch. 15 (WZW).
• I. Affleck, Quantum spin chains and the Haldane gap, J. Phys.: Condens. Matter 1, 3047 (1989).