SO(3) ↔ Virasoro: a comparison for conformal blocks

Virasoro conformal blocks are the 2D CFT analogue of partial waves. If you can add angular momenta and use CG/6j symbols, you already know the blueprint: blocks are kinematical, structure constants are dynamical.

Topic	SO(3) / SU(2)	Virasoro
Symmetry	Rotations in 3D	Local conformal symmetry in 2D
Generators	$J_0, J_\pm$ with $[J_0,J_\pm]=\pm J_\pm$, $[J_+,J_-]=2J_0$	$L_n$ with $[L_m,L_n]=(m-n)L_{m+n}+\dfrac{c}{12}m(m^2-1)\delta_{m+n,0}$
Invariant	Quadratic Casimir $\mathbf{J}^2 = j(j+1)$	Central charge $c$; no finite Casimir; reps labeled by highest weight $\Delta$
Reps	Finite‑dimensional spins $j=0,\tfrac12,1,\dots$	Highest‑weight Verma modules $\mathcal{V}_\Delta$ (infinite‑dimensional); nulls at special $\Delta$
Highest‑weight	$\lvert j,m\rangle$ with $J_+\lvert j,j\rangle=0$	$\lvert\Delta\rangle$ with $L_{n>0}\lvert\Delta\rangle=0$, $L_0\lvert\Delta\rangle=\Delta\lvert\Delta\rangle$
Descendants	Ladder by $J_-$: $2j+1$ states	Strings of $L_{-n}$: infinite tower by level $N=\sum n$
Inner product	Positive‑definite; orthonormal $\lvert j,m\rangle$	Shapovalov (Gram) form on descendants; invert $G_N$ by level
Tensor product / OPE	$j_1\otimes j_2=\bigoplus_\ell \ell$	$\phi_1 \times \phi_2 = \sum_{\text{disc.}}\ \text{or}\ \int d\Delta_p\, \mathcal{V}_{\Delta_p}$ (Liouville: continuous)
3‑point kinematics	CG/3j symbols (fixed by symmetry)	Ward identities fix descendant couplings; dynamics in $C_{ijk}$ (DOZZ in Liouville)
Partial waves	Legendre $P_\ell(\cos\theta)$	Conformal blocks $\mathcal{F}(c,\Delta_i,\Delta_p;z)$
4‑pt decomposition	$\mathcal{A}(\theta)=\sum_\ell (2\ell+1) a_\ell\, P_\ell$	$\langle 1234\rangle=\int d\Delta_p\, C_{12p}C_{p34}\, \mathcal{F}(z)\,\overline{\mathcal{F}}(\bar z)$
Recoupling / crossing	Racah/Wigner $6j$ symbols	Fusion kernel (Virasoro “$6j$”, Ponsot–Teschner)
Global subalgebra	Whole algebra	$\mathfrak{sl}_2=\{L_{-1},L_0,L_{+1}\}$
Global block	$P_\ell$ hypergeometric form	$\mathcal{F}_{\text{global}}=z^{\Delta_p-\Delta_1-\Delta_2}\,{}_2F_1(\Delta_p+\Delta_{12},\Delta_p+\Delta_{34};2\Delta_p;z)$
Selection rules	Triangle inequalities	Only with degenerate fields; generic fusion otherwise
Spectrum type	Discrete over $\ell$	Liouville: continuous $\Delta_p=\dfrac{Q^2}{4}+P^2$, $P\in\mathbb{R}_{\ge0}$
Semiclassics	Large $j$ / eikonal	Large $c$ (heavy–light, monodromy); global limit $c\to\infty$

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The dictionary

• Primary ↔ spin highest‑weight. Primary $\phi_\Delta$ corresponds to $\lvert j,j\rangle$. Descendants are $L_{-n}$ ladders (many per level) vs one $J_-$ ladder.
• OPE ↔ tensor product. $\phi_1\times\phi_2$ projects onto intermediate $\Delta_p$, just as $j_1\otimes j_2$ projects onto $\ell$.
• Block ↔ partial wave. For fixed $(c,\Delta_i,\Delta_p)$, $\mathcal{F}$ is universal (symmetry only), like $P_\ell$.
• Crossing ↔ recoupling. Changing channel $(12)(34)\to(13)(24)$ is controlled by a kernel (Virasoro 6j), as SU(2) recoupling uses Wigner 6j.

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Four‑point factorization: global vs full Virasoro

Global (SL(2)) block — the Legendre look‑alike

With $\Delta_{12}=\Delta_1-\Delta_2$ and $\Delta_{34}=\Delta_3-\Delta_4$,

$$\mathcal{F}_{\text{global}}(\Delta_i,\Delta_p;z) = z^{\Delta_p-\Delta_1-\Delta_2}\; {}_2F_{1}\!\big(\Delta_p+\Delta_{12},\ \Delta_p+\Delta_{34};\ 2\Delta_p;\ z\big).$$

This is the $c\to\infty$ limit of Virasoro blocks with fixed dimensions (only $L_{\pm1,0}$ descendants contribute).

Full Virasoro block — summing all descendants

At level $N$, descendants are labeled by partitions $Y$ of $N$ (e.g. $Y=(2,1,1)$ stands for $L_{-2}L_{-1}^2$). Using Ward identities,

$$\mathcal{F}(c,\Delta_i,\Delta_p; z) = z^{\Delta_p-\Delta_1-\Delta_2}\sum_{N=0}^\infty z^N \sum_{\lvert Y\rvert=\lvert Y'\rvert=N} \frac{\langle V_1 V_2\, \mathcal{L}_{-Y}\lvert\Delta_p\rangle\, (G_N^{-1})_{Y,Y'}\, \langle \Delta_p\rvert \mathcal{L}_{Y'} V_3 V_4\rangle} {\langle V_1V_2\lvert\Delta_p\rangle\,\langle \Delta_p\rvert V_3V_4\rangle}.$$

• Level 0: $1$.
• Level 1: basis $\{L_{-1}\lvert\Delta_p\rangle\}$; $G_1=2\Delta_p$, $$\mathcal{F}=z^{\Delta_p-\Delta_1-\Delta_2}\Big[1+\frac{(\Delta_p+\Delta_{12})(\Delta_p+\Delta_{34})}{2\Delta_p}\,z+\cdots\Big].$$ (Matches the global block at order $z^1$.)
• Level 2: basis $\{L_{-2}\lvert\Delta_p\rangle,\ L_{-1}^2\lvert\Delta_p\rangle\}$, $$G_2=\begin{pmatrix} 4\Delta_p+\tfrac{c}{2} & 6\Delta_p\\[2pt] 6\Delta_p & 4\Delta_p(2\Delta_p+1) \end{pmatrix},$$ where $L_{-2}$ produces the first genuine Virasoro correction.

Key idea. Global vs Virasoro differs precisely by the tower of $L_{-n\ge2}$ descendants organized by the Gram matrix.

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Liouville specifics (continuous spectrum)

In Liouville, the intermediate primary is labeled by $P\in\mathbb{R}_{\ge0}$ with

$$\Delta_p=\frac{Q^2}{4}+P^2,\qquad Q=b+\frac1b,\qquad c=1+6Q^2.$$

The sphere four‑point correlator is a partial‑wave integral

$$\langle V_{\alpha_1}(\infty)V_{\alpha_2}(1)V_{\alpha_3}(z,\bar z)V_{\alpha_4}(0)\rangle = \int_0^\infty \! dP\; C(\alpha_1,\alpha_2,\tfrac{Q}{2}+iP)\, C(\alpha_3,\alpha_4,\tfrac{Q}{2}-iP)\, \big|\mathcal{F}(c,\Delta_i,\Delta_p; z)\big|^2,$$

where blocks are kinematics and DOZZ structure constants $C$ encode dynamics.

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Four efficient ways to compute blocks

1) Low‑level expansion (by hand or CAS)

• Build $G_N$ and its inverse for $N=1,2,3$.
• Use Ward identities for descendant numerators.
• Ideal for checks, identical external weights, and intuition.

2) Zamolodchikov’s elliptic $q$‑recursion (fast & robust)

Introduce the elliptic nome

$$q(z)=\exp\!\Big(-\pi\,\frac{K(1-z)}{K(z)}\Big),\qquad K=\text{complete elliptic integral}.$$

Then

$$\mathcal{F}(c,\Delta_i,\Delta_p; z)= \Lambda(c,\Delta_i,\Delta_p; z)\, H(c,\Delta_i,\Delta_p; q),$$

with a simple prefactor $\Lambda$ and a meromorphic series

$$H=1+\sum_{m,n\ge1} \frac{(16q)^{mn}\, R_{m,n}(\Delta_i,c)}{\Delta_p-\Delta_{m,n}(c)}\, H\big(c,\Delta_i,\Delta_{m,n}(c)+mn;\ q\big),$$

where $\Delta_{m,n}(c)=\dfrac{Q^2-(mb+n/b)^2}{4}$ and residues $R_{m,n}$ come from degenerate fusion. Why it’s good: $\lvert q(z)\rvert\ll1$ on most of the $z$‑plane ⇒ rapid convergence.

3) BPZ differential equations (degenerate insertion)

If one external is degenerate (e.g. $\alpha=-\tfrac{b}{2}$ or $-\tfrac{1}{2b}$), the chiral piece satisfies a second‑order Fuchs ODE. Solutions are hypergeometric in the simplest case; monodromy picks the internal $\Delta_p$. This is the closest analogue to strong SU(2) selection rules.

4) AGT / Nekrasov (instanton counting)

AGT equates the Virasoro chiral block with the instanton partition function of 4d $\mathcal{N}=2$ SU(2) with $N_f=4$ in an $\Omega$‑background. A practical dictionary (one consistent choice):

$$b=\sqrt{\frac{\varepsilon_1}{\varepsilon_2}},\quad Q=b+\frac1b,\quad \alpha=\frac{Q}{2}+\frac{a}{\sqrt{\varepsilon_1\varepsilon_2}},\quad \alpha_i=\frac{Q}{2}+\frac{m_i}{\sqrt{\varepsilon_1\varepsilon_2}},\quad q=z.$$

• Compute the Nekrasov instanton series $Z_{\text{inst}}(a,m_i,q\mid\varepsilon_{1,2})$ by summing over pairs of Young diagrams.
• Remove the decoupled $U(1)$ if you start from $U(2)$ expressions.
• Assemble the block with a minimal prefactor so that $$\mathcal{F}(z)=z^{\Delta_p-\Delta_1-\Delta_2}\Big(1+\sum_{K\ge1}c_K z^K\Big),$$ where $c_K$ are the instanton coefficients. Use cases: fast numerics, high‑order series, cross‑checks for recursion/monodromy.

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Crossing as recoupling: the Virasoro “$6j$”

Changing channel is implemented by the fusion kernel

$$\mathcal{F}_s(\Delta_p; z)=\int d\Delta_{p'}\ \mathbf{F}\!\left[ \begin{smallmatrix} \Delta_1\ \Delta_2\\[-2pt] \Delta_3\ \Delta_4 \end{smallmatrix} \ \Big|\ \Delta_p\!\to\!\Delta_{p'}\right]\, \mathcal{F}_t(\Delta_{p'}; 1-z),$$

the Virasoro analogue of Wigner–Racah $6j$. In Liouville this kernel is known (Ponsot–Teschner) and enforces crossing symmetry.

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Worked checkpoint: identical external weights

Let $\Delta_i=\Delta$. Then $\Delta_{12}=\Delta_{34}=0$.

• Global $$\mathcal{F}_{\text{global}}=z^{\Delta_p-2\Delta}\,{}_2F_1(\Delta_p,\Delta_p;2\Delta_p;z) = z^{\Delta_p-2\Delta}\left[1+\frac{\Delta_p}{2}z+\frac{\Delta_p(\Delta_p+1)}{4(2\Delta_p+1)}z^2+\cdots\right].$$
• Virasoro (level 2)
  The $z^2$ coefficient differs from the global value by $c$‑dependent terms via $L_{-2}$. As $c\to\infty$ this difference disappears, recovering the global block.

Use this as a unit test for any implementation.

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Remarks

• Kinematics vs dynamics. Blocks depend only on $(c,\Delta_i,\Delta_p)$, not on DOZZ normalizations.
• Channel confusion. $s$‑ and $t$‑channel blocks are related by a kernel, not equality.
• Global vs Virasoro. They coincide at level 1; differences start at level 2 via $L_{-2}$.
• Selection rules. Generic Virasoro fusion has no SU(2)‑like triangle rule; selection arises only with degenerate fields.
• Liouville spectrum. Integrate over $P$ (continuous), not a sum over discrete $\ell$.
• Normalize three‑point factors so the block starts as $z^{\Delta_p-\Delta_1-\Delta_2}(1+\cdots)$.

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