Conformal Field Theory in One Dimension

One dimension is the smallest setting in which the conformal bootstrap is already complete enough to be exact: $SL(2,\mathbb R)$ fixes the kinematics, four points leave one cross ratio, blocks are hypergeometric functions, and reflection positivity turns crossing into a positive spectral problem. The simplification is genuine, but so are the global subtleties—points cannot exchange order on a line without collision, and the same algebra can describe an intrinsic CFT, conformal quantum mechanics, or a line defect. The exact one-dimensional crossing and functional framework is developed in Mazáč and Paulos 2019, §§2–3.

Helpful background. Crossing and Positivity supplies the general bootstrap equation. State–Operator Correspondence supplies the relation between primaries and states.

Enter this chapter

Start from the question you need to answer:

If you need to…	Begin with…	You will obtain…
decide what the global group and its cover are	One-Dimensional Conformal Symmetry and SL(2,R)	the line/circle action, primary modules, and positivity qualifications
compare real-line orderings	Primaries, Correlators, and Ordering Sectors	one cross ratio with permutation, statistics, and continuation data
compute an exchanged family	One-Dimensional Conformal Blocks	the Casimir equation, physical hypergeometric block, and shadow distinction
exclude a proposed spectrum	Crossing and Positivity	a positive sum rule and a functional certificate
test a block or crossing implementation	Generalized-Free and Solvable Data	exact bosonic and fermionic spectra, coefficients, and failure tests
interpret where the 1D data came from	Model Handoffs	criteria separating intrinsic, quantum-mechanical, and defect observables

Read the pages in order for a first pass. The cross-ratio and block conventions chosen in the first three pages are held fixed in the last three.

The chapter’s fixed crossing problem

For four identical Hermitian scalar primaries in the ordering $x_1<x_2<x_3<x_4$, use

$$z=\frac{x_{12}x_{34}}{x_{13}x_{24}}, \qquad 0<z<1,$$

and

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle =\frac{\mathcal G(z)}{|x_{12}|^{2\Delta_\phi}|x_{34}|^{2\Delta_\phi}}.$$

The $12\to34$ conformal block and crossing equation are

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

In a reflection-positive identical-scalar problem,

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

Every qualification matters. A different ordering can change a graded sign; a different prefactor changes the displayed crossing vector; a non-Hermitian or nonunitary problem loses scalar nonnegativity; and a continuation outside $0<z<1$ requires a path around the branch points.

One connected chain of reasoning

The six pages form a single derivation:

1. $PSL(2,\mathbb R)$ acts on $\mathbb{RP}^1$, while state representations may require a cover.
2. Fixed cyclic order reduces four-point kinematics to one real interval.
3. The $SL(2,\mathbb R)$ Casimir plus the OPE boundary condition selects one hypergeometric block.
4. Reflection positivity makes its spectral weights nonnegative.
5. A separating functional turns that cone into rigorous exclusions.
6. Generalized-free solutions test every normalization, while the final classification prevents a shared algebra from erasing physical distinctions.

This chain is small enough to reproduce by hand and rich enough to expose the logic used in higher-dimensional numerical and analytic bootstrap.

The general route from Casimir blocks through positive crossing equations to functional exclusions is reviewed in Simmons-Duffin 2017, §§9–10, pp. 43–63, Open PDF; one dimension is the specialization in which every displayed function depends on a single real cross ratio.

Benchmark: generalized-free data

At arbitrary $\Delta_\phi>0$, the ordered generalized-free solutions are

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

They contain towers at $2\Delta_\phi+2n$ and $2\Delta_\phi+2n+1$, respectively. The worked benchmark page gives the exact positive OPE weights and a coefficient-level reconstruction. These solutions anchor three independent checks:

• direct Wick contractions reproduce the reduced correlator;
• the analytic block sum reproduces its series and crossing relation;
• a numerical truncation approaches it with a positive, bounded tail.

None of those checks turns generalized-free data into a local microscopic model. That claim boundary is part of the benchmark.

For positive integer or half-integer $\Delta_\phi$, and under the unitarity and functional-domain hypotheses of the theorem, the fermionic tower saturates the optimal gap bound $\Delta_{\rm gap}\leq2\Delta_\phi+1$ Mazáč 2017, §§4–5, pp. 17–33, Open PDF. This extremality statement is not being extended here to arbitrary $\Delta_\phi$.

Preparation check

You are ready to use the chapter if you can answer these questions:

• Why does the connected coordinate action see $PSL(2,\mathbb R)$ rather than the full cover represented on states?
• Why is $0<z<1$ an ordering statement as well as an OPE convergence domain?
• Which boundary condition removes the shadow solution from a conformal block?
• Which reflection-positive pairing makes $\lambda_{\phi\phi\mathcal O}^2$ nonnegative?
• Why does a functional exclusion fail to prove existence at the boundary?

If the first two are unfamiliar, begin at the global symmetry page. If the last three are unfamiliar, review general crossing and positivity before the block and functional pages.

Review the chapter

Derive the one-dimensional block

Starting from $C_\Delta=\Delta(\Delta-1)$, solve

$$z^2[(1-z)g''(z)-g'(z)]=C_\Delta g(z)$$

with $g(z)\sim z^\Delta$ at the origin.

Solution

Set $g=z^\Delta f$. The equation becomes the hypergeometric equation with parameters $a=b=\Delta$ and $c=2\Delta$. The solution analytic at $z=0$ with unit leading coefficient is $g_\Delta=z^\Delta{}_2F_1(\Delta,\Delta;2\Delta;z)$. The other local solution behaves as $z^{1-\Delta}$ and is the shadow branch.

Diagnose an overclaim

A calculation finds a sharp kink in a finite-derivative bound and a nearby truncated spectrum. What may be concluded?

Solution

The calculation gives a numerical exclusion boundary at the stated derivative order and a candidate extremal spectrum. Identification with a CFT requires stability under higher orders, controlled numerical errors, untruncated crossing, and independent consistency data. The kink alone proves neither existence nor uniqueness.

Classify a line observable

A set of operators transforms under $SL(2,\mathbb R)\times SO(3)$ and includes a dimension-two $SO(3)$ vector. What additional information is needed before calling it a four-dimensional line-defect CFT?

Solution

One needs an ambient four-dimensional CFT, the defect insertion and preserved subgroup, bulk-to-defect data, and the broken-translation Ward identity identifying the vector as the displacement operator. One must also specify the reflection pairing and transverse selection rules. The spectrum alone is compatible with, but does not establish, that interpretation.

Where to continue

• Continue to Two-Dimensional CFT for Virasoro symmetry, rational models, and modular sewing.
• Continue to Boundaries, Defects, and Interfaces for transverse spin, bulk-to-defect OPEs, and displacement observables.
• Continue to Analytic and Lorentzian Bootstrap for complete functional bases, inversion, and Regge-qualified sum rules.
• Reproduce the one-dimensional block, ordering, and functional checks with the conventions fixed in this chapter.

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). arXiv. DOI.
• Simmons-Duffin, D. “TASI Lectures on the Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. World Scientific, 2017. arXiv. DOI.