Generalized-Free and Solvable 1D CFT Data

Generalized-free correlators are exact solutions of one-dimensional crossing whose spectra and OPE coefficients can be written in closed form. They are ideal normalization and convergence tests because Wick contractions, block decomposition, and crossing can be checked independently. They are kinematic CFT data, however—not evidence for a local microscopic action or for a unique interacting theory.

Required background. Crossing and Positivity in One Dimension supplies the positive block sum. Helpful background. Free and Generalized-Free CFT Data compares the same construction in higher dimensions.

Two exact ordered correlators

Normalize an identical primary $\phi$ of dimension $\Delta_\phi>0$ by

$$\langle\phi(x_1)\phi(x_2)\rangle=|x_{12}|^{-2\Delta_\phi}.$$

In the ordering $x_1<x_2<x_3<x_4$, Wick pairings give the reduced generalized-free boson correlator

$$\mathcal G_{\mathrm B}(z) =1+z^{2\Delta_\phi} +\left(\frac{z}{1-z}\right)^{2\Delta_\phi}, \qquad 0<z<1.$$

Changing the sign of the crossed Wick contraction gives the generalized-free fermion solution

$$\mathcal G_{\mathrm F}(z) =1-z^{2\Delta_\phi} +\left(\frac{z}{1-z}\right)^{2\Delta_\phi}.$$

Both obey

$$z^{-2\Delta_\phi}\mathcal G(z) =(1-z)^{-2\Delta_\phi}\mathcal G(1-z)$$

in this ordered convention. The names “boson” and “fermion” refer to the Wick-contraction sign and resulting exchanged tower. A continuation to another ordering must still carry the appropriate graded sign and branch phase.

These correlators also pass the clustering check: as $z\to0$, $\mathcal G(z)\to1$, so the identity is the leading $12$-channel contribution.

Exact spectra and OPE weights

The bosonic and fermionic solutions decompose into the normalized blocks

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

with spectra

$$\Delta_n^{\mathrm B}=2\Delta_\phi+2n, \qquad \Delta_n^{\mathrm F}=2\Delta_\phi+2n+1, \qquad n=0,1,2,\ldots.$$

For either tower, the nonidentity coefficient at an allowed $\Delta$ is

$$a_\Delta^{\mathrm{GFF}} =\frac{2\,\Gamma(\Delta)^2\Gamma(\Delta+2\Delta_\phi-1)} {\Gamma(2\Delta_\phi)^2\Gamma(2\Delta-1) \Gamma(\Delta-2\Delta_\phi+1)}.$$

It is positive for $\Delta_\phi>0$ on both displayed towers. The identity has $a_0=1$ and is not obtained by substituting $\Delta=0$ into this expression. These one-dimensional generalized-free decompositions and their functional duals are worked out in Mazáč and Paulos 2019, §§2–4.

For positive integer or half-integer $\Delta_\phi$, the fermionic spectrum is also the extremal solution that saturates the optimal unitary gap bound, with the theorem’s functional-domain assumptions kept in force Mazáč 2017, §§4–5, pp. 17–33, Open PDF.

As a quick check, set $\Delta_\phi=1$. The bosonic tower begins at $\Delta=2$ with $a_2=2$, followed by $a_4=6/5$. Expanding,

$$\mathcal G_{\mathrm B}(z)=1+2z^2+2z^3+3z^4+O(z^5),$$

while $2g_2+(6/5)g_4$ reproduces every term shown. The fermionic tower begins at $\Delta=3$ with $a_3=2$ and similarly reproduces

$$\mathcal G_{\mathrm F}(z)=1+2z^3+3z^4+O(z^5).$$

These coefficient-level checks catch a missing Wick sign, a different block normalization, or a shift of the double-trace tower immediately.

What makes a strong benchmark

A numerical decomposition should compare more than the final crossing residual. For a truncation at $\Delta\le\Delta_{\max}$, record:

• the exact input correlator and ordered interval;
• the analytic dimensions and OPE weights used;
• the block evaluator and precision;
• the maximum direct-correlator residual on a test grid;
• a bound on the omitted positive tail;
• the residual after $z\leftrightarrow1-z$;
• a failure test with the Wick sign, first dimension, or block normalization deliberately changed.

The figure places this exact fixture in the same ordered crossing pipeline used by analytic and numerical functionals. Inspect the side branch: generalized-free data test the block, ordering, and normalization conventions, but they do not by themselves establish that a functional is positive on a proposed continuous spectrum.

The generalized-free branch supplies exact correlators, block towers, OPE weights, and ordering signs for the one-dimensional crossing pipeline. A functional certificate adds independent domain and positivity conditions; the exact fixture does not turn a truncated search into an existence proof. The diagram is schematic and not to scale.

The figure’s relationships can be checked without the image:

Stage	Generalized-free datum	Exact check	Failure exposed
Ordering	$x_1<x_2<x_3<x_4$, hence $0<z<1$	Wick contractions use the declared graded order	Missing permutation or statistics sign
Block basis	$g_\Delta(z)=z^\Delta{}_2F_1(\Delta,\Delta;2\Delta;z)$	Casimir equation and $z^\Delta$ OPE limit	Shadow or normalization substituted for the block
Spectrum	$2\Delta_\phi+2n$ or $2\Delta_\phi+2n+1$	First nonidentity power and every analytic coefficient	Bosonic tower paired with the fermionic sign
Crossing	Exact equality under $z\leftrightarrow1-z$	Direct correlator and block sum agree with a bounded tail	Small sampled residual mistaken for an exact sum
Functional use	The exact tower is a test input	Every asserted sign is checked on its stated spectral set	Benchmark agreement promoted to an optimal bound or model identity

A reproducible calculation should use the same $z$, block, and ordering conventions as this chapter. The equations above provide the analytic benchmark against which its outputs should be checked.

Deformations and contact ambiguities

One may perturb the generalized-free solution,

$$\Delta_n=\Delta_n^{(0)}+g\,\gamma_n+O(g^2), \qquad a_n=a_n^{(0)}+g\,a_n^{(1)}+O(g^2),$$

and solve crossing order by order. Differentiating a block produces logarithms through

$$g_{\Delta_n}(z) =g_{\Delta_n^{(0)}}(z) +g\,\gamma_n\,\partial_\Delta g_\Delta(z) \big|_{\Delta=\Delta_n^{(0)}}+O(g^2).$$

Crossing alone can leave homogeneous, contact-type solutions. Their admissibility depends on endpoint and Regge bounds, and perturbative CFT data need not resum to an exact unitary theory. In the $SL(2)$ functional basis, the contact-term ambiguity is fixed in Mazáč and Paulos 2019, §3.4, pp. 21–23, Open PDF. The broader analytic classification belongs to Analytic Functionals and Polyakov Blocks, while the Mellin interpretation belongs to Large-N Crossing and Contact Ambiguities.

Comparing low-dimensional benchmarks

The following structured comparison is the chapter’s portion of the low-dimensional model table. It keeps properties that are often conflated in separate columns.

Benchmark	Spectrum in this channel	Dilatation action	Reflection positivity	Exact evidence	Important limitation
Generalized-free boson	Discrete $2\Delta_\phi+2n$	Diagonal	Yes for $\Delta_\phi>0$ in the stated sector	Wick correlator, positive coefficients, exact block sum	Does not supply a local microscopic action
Generalized-free fermion	Discrete $2\Delta_\phi+2n+1$	Diagonal	Yes in the compatible graded ordering	Wick correlator, positive coefficients, exact block sum	Statistics signs must follow the ordering convention
Finite block truncation	Finite approximation	Diagonal by construction	Only if every retained weight is nonnegative	Residual and tail estimate	Is not an exact crossing solution unless the tail vanishes
Perturbative contact solution	Deformed double-trace data	Diagonal order by order	Must be checked order by order	Linearized crossing and endpoint bounds	May not resum to a complete CFT

Exact rational models, nonunitary theories, logarithmic modules, and noncompact spectra occupy different rows on the two-dimensional core and nonunitary/logarithmic chapter. A label such as “solvable” never substitutes for specifying spectrum type, diagonalizability, and positivity.

Common pitfalls

Calling generalized free data a free local field theory. The correlators solve conformal kinematics and crossing. Local equations of motion, a stress tensor, and a microscopic action are extra claims.

Using the bosonic tower with the fermionic Wick sign. The sign removes alternating low-dimension contributions and shifts the exchanged tower by one. Check the first nonidentity power before fitting coefficients.

Treating a small truncated residual as exact data. A residual has meaning only together with precision, sampling domain, and an omitted-tail bound.

Exercises

Verify crossing for $\mathcal G_{\mathrm B}$ and $\mathcal G_{\mathrm F}$ directly.

Solution

Multiply $\mathcal G(1-z)$ by $(z/(1-z))^{2\Delta_\phi}$. The identity and crossed terms exchange, while the middle term retains its plus sign for $\mathcal G_{\mathrm B}$ and its minus sign for $\mathcal G_{\mathrm F}$. The result is $\mathcal G(z)$ in both cases.

Use the gamma-function formula to compute $a_2$ and $a_4$ for the bosonic solution at $\Delta_\phi=1$.

Solution

At $\Delta=2$, the formula gives $2\Gamma(2)^2\Gamma(3)/[\Gamma(2)^2\Gamma(3)\Gamma(1)]=2$. At $\Delta=4$, it gives $2\Gamma(4)^2\Gamma(5)/[\Gamma(2)^2\Gamma(7)\Gamma(3)]=6/5$.

References

• Mazáč, D. “Analytic Bounds and Emergence of $\mathrm{AdS}_2$ Physics from the Conformal Bootstrap.” Journal of High Energy Physics 2017, 146 (2017). 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), §§2–4. arXiv. DOI.
• Mazáč, D., and Paulos, M. F. “The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation.” Journal of High Energy Physics 2019, 163 (2019). arXiv. DOI.