Nonunitary, Logarithmic, Noncompact, and Nonrational Two-Dimensional CFT

Beyond the unitary rational setting, two-dimensional CFT requires several distinct generalizations. Negative norms, nondiagonalizable dilatations, continuous spectra, and infinitely many chiral representations change different parts of the theory. This chapter shows which familiar arguments survive and which state expansions, pairings, or integral transforms must replace them.

Enter this chapter

Begin with Nonunitary CFTs, Effective Central Charge, and Complex Data to separate conformal symmetry from positivity. For continuum states, enter through Noncompact CFTs and Continuous Spectra after reviewing completeness.

Helpful background. Highest-weight modules and the Kac determinant supply the Virasoro representation theory used throughout. Completeness and operator bases explain the discrete resolution of the identity that will be generalized to direct integrals. Tempered distributions and Fourier calculus provide the test-function language needed for delta-normalized states and zero modes.

Topics

1. Nonunitary CFTs, Effective Central Charge, and Complex Data — Separates central charge, effective central charge, reality, and positivity.
2. Logarithmic CFT and Indecomposable Modules — Relates Jordan blocks in dilatations to logarithmic correlation functions.
3. Noncompact CFTs and Continuous Spectra — Replaces discrete completeness by a specified spectral measure.
4. Liouville Theory and the Virasoro Bootstrap — Combines reflection relations, structure constants, and continuous Virasoro blocks.
5. Irrational CFT, Fusion Kernels, and Crossing — Treats a change of OPE channel as an integral transform with contour data.
6. Distributional Correlators, Zero Modes, and Normalization — Controls momentum delta functions and target-volume factors using test functions.
7. Nonrational Modular Consistency and Spectral Densities — Formulates modular transformations for continuous characters and spectral densities.

How the ideas fit together

Crossing compares two operator-product decompositions even when positivity is absent. What changes is the inference drawn from it: without a positive pairing, the coefficients no longer form the cone used by unitary functional bounds. The distinction between the crossing equation and its positive coefficients is explicit in Poland, Rychkov, and Vichi 2019, § III.E.

A continuous spectrum creates a different issue. A sum over normalizable eigenstates becomes an integral over delta-normalized states with a measure. This replacement does not itself produce negative norms or Jordan blocks. Liouville theory and the noncompact boson clarify why spectrum, pairing, and diagonalizability must be examined separately.

Review the chapter

Does a continuous spectrum imply that a theory is logarithmic?

Solution

No. Continuity concerns how eigenstates are resolved and integrated. Logarithmic behavior concerns a nondiagonalizable action of dilatations, requiring generalized eigenvectors. A noncompact free boson has continuous momenta while its dilatation operator remains diagonalizable.

Where to go next

Continue to Modular and Thermal Bootstrap to constrain spectral measures and thermal data while retaining their normalization and convergence conditions.

References

• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI.