One-Dimensional Conformal Blocks and Casimir Equations

An $SL(2,\mathbb R)$ conformal block is the contribution of one primary and all of its descendants to an ordered four-point function. In one dimension the quadratic Casimir equation is an ordinary hypergeometric equation, so the physical block, its shadow partner, its branch structure, and its OPE normalization can all be displayed explicitly.

Required background. Primaries, Correlators, and Ordering Sectors fixes $0<z<1$ and the $12\to34$ channel. Blocks and Casimir Equations supplies the general conformal-family construction.

The Casimir eigenvalue problem

Let $L_{-1}=P$, $L_0=D$, and $L_1=K$ with

$$[L_0,L_{\pm1}]=\mp L_{\pm1}, \qquad [L_1,L_{-1}]=2L_0.$$

The quadratic Casimir

$$\mathcal C=L_0^2-\frac12(L_1L_{-1}+L_{-1}L_1)$$

has eigenvalue $C_\Delta=\Delta(\Delta-1)$ on a scalar primary of dimension $\Delta$. Acting with the sum of generators on points 1 and 2 and stripping the identical-scalar prefactor gives

$$\mathcal D_z g_\Delta(z)=\Delta(\Delta-1)g_\Delta(z), \qquad \mathcal D_z=z^2\big[(1-z)\partial_z^2-\partial_z\big].$$

This operator is tied to the cross-ratio and prefactor convention stated above. Changing either can conjugate $\mathcal D_z$ by a power of $z$ or $1-z$ without changing the underlying Casimir.

The conformal-block Casimir construction and the boundary condition that identifies the physical OPE solution are derived in Simmons-Duffin 2017, §9.3, pp. 46–47, Open PDF.

The physical and shadow solutions

The $12$ OPE requires the exchanged primary to contribute $z^\Delta$ as $z\to0^+$. That boundary condition selects

$$g_\Delta(z) =z^\Delta\,{}_2F_1(\Delta,\Delta;2\Delta;z), \qquad 0<z<1.$$

The second local solution has the same Casimir eigenvalue because $C_{1-\Delta}=C_\Delta$:

$$g_{1-\Delta}(z) =z^{1-\Delta}\,{}_2F_1(1-\Delta,1-\Delta;2-2\Delta;z).$$

It is the shadow solution, not a second contribution from the same local conformal family. A conformal partial wave can combine a block and its shadow; an OPE block keeps only the solution with the declared short-distance behavior. At exceptional dimensions the two Frobenius solutions can collide and a logarithmic solution appears, so the generic formulas must be understood by a limiting prescription.

Expanding the hypergeometric function makes the descendant content explicit:

$$g_\Delta(z) =z^\Delta\sum_{n=0}^\infty \frac{(\Delta)_n^2}{(2\Delta)_n\,n!}\,z^n.$$

For $\Delta>0$ every coefficient is positive. This is a property of the block series; positivity of the coefficient multiplying the whole block still requires reflection positivity and Hermitian external operators.

For unequal external dimensions in the same prefactor convention, define $\Delta_{ij}=\Delta_i-\Delta_j$. The corresponding $12\to34$ solution is

$$g_\Delta^{\Delta_{12},\Delta_{34}}(z) =z^\Delta\,{}_2F_1(\Delta-\Delta_{12},\Delta+\Delta_{34};2\Delta;z).$$

Before comparing formulas from different sources, check whether their $\Delta_{12}$ sign and external prefactor agree.

These one-dimensional physical and shadow solutions, with the $z^\Delta$ OPE normalization used here, are developed in Mazáč and Paulos 2019, §§2–3.

Normalization and the OPE decomposition

For identical Hermitian $\phi$ with unit two-point function,

$$\mathcal G(z)=1+\sum_{\mathcal O\ne\mathbf1} a_{\mathcal O}g_{\Delta_{\mathcal O}}(z), \qquad a_{\mathcal O}=\lambda_{\phi\phi\mathcal O}^2\ge0,$$

within $0<z<1$. If several primaries have the same dimension, $a_\Delta$ is the sum of their squared OPE coefficients in this correlator. The block decomposition therefore recovers spectral weight, not a basis inside a degenerate eigenspace.

The leading normalization $g_\Delta(z)=z^\Delta(1+O(z))$ is essential. Multiplying every block by a dimension-dependent factor simply divides $a_\Delta$ by that factor; crossing is unchanged, but quoted OPE coefficients are not.

The Euler representation

$$g_\Delta(z) =\frac{\Gamma(2\Delta)}{\Gamma(\Delta)^2} z^\Delta\int_0^1 \frac{[t(1-t)]^{\Delta-1}}{(1-zt)^\Delta}\,dt, \qquad \operatorname{Re}\Delta>0,$$

is useful for sign checks and stable numerical evaluation. It also makes clear that the displayed integral is a first-sheet formula with $z$ away from the cut $[1,\infty)$.

Radial convergence and numerical checks

The one-dimensional radial coordinate

$$\rho=\frac{z}{(1+\sqrt{1-z})^2}, \qquad z=\frac{4\rho}{(1+\rho)^2},$$

maps the twice-cut $z$ plane to the unit disk. For $0<z<1$, one has $0<\rho<1$; at the crossing-symmetric point $z=1/2$, $\rho=3-2\sqrt2\approx0.1716$. A radial expansion therefore converges much faster there than the raw $z$ series. The general convergence argument follows from nested state-preparation spheres Hogervorst and Rychkov 2013, §§2–3.

A trustworthy block implementation should pass four independent tests:

1. the Casimir residual $\mathcal D_zg_\Delta-\Delta(\Delta-1)g_\Delta$ vanishes;
2. $z^{-\Delta}g_\Delta(z)\to1$ as $z\to0^+$;
3. the $z$ series, Euler integral, and hypergeometric form agree in their common domain;
4. analytic continuation uses a declared side of the $[1,\infty)$ cut and reproduces its predicted phase or discontinuity.

Near $z=1$, direct hypergeometric evaluation can lose precision through cancellation. Transform to a basis adapted to $1-z$, increase precision, and compare overlapping representations rather than trusting a single black-box call.

Common pitfalls

Keeping both Casimir solutions in an OPE. The Casimir equation alone cannot select a conformal block. The $z\to0$ OPE boundary condition removes the shadow solution.

Reading block-series positivity as theory unitarity. Positive Taylor coefficients of $g_\Delta$ do not prove $a_\Delta\ge0$. The latter comes from the reflection-positive state decomposition.

Evaluating across a cut without a sheet. Hypergeometric software chooses a branch convention. A Lorentzian correlator must instead specify the continuation path that produces the required boundary value.

Exercises

Verify the first two nontrivial coefficients in the block series directly from the Casimir equation.

Solution

Write $g=z^\Delta(1+b_1z+b_2z^2+\cdots)$ and substitute into the equation. Matching powers gives $b_1=\Delta/2$ and $b_2=\Delta(\Delta+1)^2/[4(2\Delta+1)]$. These agree with $(\Delta)_n^2/[(2\Delta)_n n!]$ for $n=1,2$.

At $\Delta=1$, simplify the block and check its OPE behavior.

Solution

The identity ${}_2F_1(1,1;2;z)=-\log(1-z)/z$ gives $g_1(z)=-\log(1-z)$. Its small-$z$ expansion is $z+z^2/2+\cdots$, so the leading coefficient is one as required.

References

• Hogervorst, M., and Rychkov, S. “Radial Coordinates for Conformal Blocks.” Physical Review D 87, 106004 (2013). arXiv. DOI.
• Mazáč, D., and Paulos, M. F. “The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices.” Journal of High Energy Physics 2019, 162 (2019), §§2–3. arXiv. DOI.
• Simmons-Duffin, D. “TASI Lectures on the Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. World Scientific, 2017, §9. arXiv. DOI.