Conformal Field Theory in One Dimension
A one-dimensional conformal theory offers an explicit setting for the whole bootstrap argument: symmetry fixes the blocks, operator products assemble a correlator, and crossing constrains the spectrum. Ordering along the line and positivity remain essential. The same conformal algebra can describe intrinsic one-dimensional data, quantum mechanics, or a line defect without making those physical systems equivalent.
Enter this chapter
Section titled “Enter this chapter”Begin with One-Dimensional Conformal Symmetry and SL(2,R) to establish the symmetry and its representations.
Helpful background. Crossing and Positivity supplies the general bootstrap equation. State–Operator Correspondence supplies the relation between primaries and states.
Topics
Section titled “Topics”- One-Dimensional Conformal Symmetry and SL(2,R) — Distinguishes the projective action on the compactified line from its action on states.
- Primaries, Correlators, and Ordering Sectors — Tracks operator orderings, statistics, and the single cross ratio.
- One-Dimensional Conformal Blocks and Casimir Equations — Derives the hypergeometric block with its physical short-distance boundary condition.
- Crossing and Positivity in One Dimension — Builds positive crossing equations and explains functional exclusions.
- Generalized-Free and Solvable 1D CFT Data — Tests the equations against exact generalized-free spectra and coefficients.
- Line Defects, Conformal Quantum Mechanics, and One-Dimensional CFTs — Separates intrinsic CFT data, conformal quantum mechanics, and line-defect observables.
How the ideas fit together
Section titled “How the ideas fit together”A conformal block sums one primary and its descendants. In one dimension, a single cross ratio describes each ordering sector, making the block a solution of an ordinary differential equation. Choosing its operator-product asymptotic removes the shadow branch. Crossing compares the resulting expansions; reflection positivity supplies the signs that allow a linear functional to exclude a proposed spectrum. Mazáč and Paulos 2019, §§ 2.1–2.3.
Generalized-free correlators test this entire chain because their contractions and block coefficients can be checked independently. Satisfying this crossing problem does not establish a unique microscopic realization or supply the ambient operators required for a line defect.
Review the chapter
Section titled “Review the chapter”What extra data would turn a consistent one-dimensional spectrum into a specified line defect of a higher-dimensional CFT?
Solution
One must identify the ambient theory and defect insertion, the preserved transverse symmetries, and the bulk-to-defect couplings. Ward identities must connect defect operators to ambient responses, such as displacement of the line. One-dimensional crossing alone supplies none of those choices.
Where to go next
Section titled “Where to go next”Continue to Two-Dimensional CFT: Algebraic and Rational Core for infinite-dimensional chiral symmetry, or to Boundaries, Defects, and Interfaces for ambient–defect consistency.