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

Two-dimensional conformal field theory does not cease to be useful when positivity, semisimplicity, compactness, discreteness, or rationality fails. What changes is the mathematical type of the data: a positive Hilbert-space sum may become an indefinite pairing, a diagonal operator may acquire Jordan blocks, a discrete sum may become a direct integral, and a finite modular matrix may become an integral kernel. This chapter develops those replacements without conflating them.

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.

Five independent questions

The adjectives in the chapter title are not synonyms. A theory can fail one condition while satisfying another, so diagnose each condition separately.

Question	Standard discrete unitary answer	Replacement when the answer is no
Is the radial pairing positive definite?	Descendant norms and squared OPE coefficients are nonnegative.	Keep the conformal Ward identities, but use an indefinite or non-Hermitian pairing and do not apply positivity bounds.
Is dilatation diagonalizable?	Local operators decompose into ordinary eigenspaces.	Use indecomposable modules and generalized eigenvectors; correlators can contain logarithms.
Is the target effectively compact?	Zero-mode wavefunctions can be normalized and momenta are discrete.	Use delta-normalized states, direct-integral completeness, and explicit volume normalization.
Is the spectrum discrete?	OPE and torus decompositions are sums.	Specify a measure and contour, and replace sums by integrals plus any discrete residues.
Is the chiral theory rational?	Finitely many characters transform by a finite matrix.	Use modular and fusion kernels, generalized characters, or distribution-valued transforms.

These distinctions are structural. For example, the Yang–Lee minimal model is nonunitary but rational and discrete; a noncompact free boson has a continuous spectrum but an ordinary diagonalizable dilatation operator; a logarithmic theory may have a discrete set of characters while its modules remain nonsemisimple. The finite rational story is reviewed in Di Francesco, Mathieu, and Sénéchal 1997, Chapters 7 and 10, against which these replacements can be compared.

Choose the required structure

Begin with the first observed failure, then check the neighboring possibilities rather than attaching a single label to the whole theory.

If the calculation exhibits…	Read first	The indispensable datum
Negative norms, a lowest conformal weight below zero, or complex dimensions/OPE coefficients	Nonunitary CFTs, effective central charge, and complex data	The pairing, conjugation rule, lowest weight, and the domain in which observables are interpreted
A nondiagonalizable action of $L_0$ or logarithms in correlators	Logarithmic CFT and indecomposable modules	The extension class, Jordan action, and basis-dependent versus invariant logarithmic data
Delta-normalized states, target-volume factors, or continuous OPE labels	Noncompact CFTs and continuous spectra	The spectral measure, zero-mode convention, and contour
A continuous Virasoro spectrum with exponential interactions	Liouville theory and the Virasoro bootstrap	The reflection identification, DOZZ normalization, and pole prescription
A change of OPE channel represented by an integral transform	Irrational CFT, fusion kernels, and crossing	The block normalization, Plancherel measure, fusion contour, and residue terms
Momentum-conserving delta functions or divergent zero-mode volumes	Distributional correlators, zero modes, and normalization	The test-function space, regulator, and whether a volume has been divided out
A torus partition function built from a continuum of characters	Nonrational modular consistency and spectral densities	The modular kernel, spectral density, vacuum/discrete pieces, and sense of equality

The order is intentional. Noncompactness motivates distributional normalization; Liouville theory makes the contour and reflection issues concrete; fusion kernels then separate kinematics from dynamical structure constants; modular consistency tests the resulting spectrum globally.

The scope also has sharp boundaries. Finite semisimple constructions remain in Two-dimensional CFT; general distribution theory remains in Distributions and microlocal analysis; and quantitative torus bounds and thermal consequences continue in Modular and thermal bootstrap. This chapter does not freeze evolving model classifications or numerical bounds: any such statement needs its own dated evidence and assumptions.

What remains universal

Local conformal covariance, Virasoro Ward identities, null-vector equations, OPE associativity, and mapping-class-group consistency remain meaningful beyond the unitary rational setting. Their implementation, however, must be typed correctly.

For an ordinary discrete spectrum one writes schematically

$$\mathbf 1=\sum_i |i\rangle\langle i|, \qquad G(z,\bar z)=\sum_i C_{12i}C_{i34}\,\mathcal F_i(z)\overline{\mathcal F_i(z)}.$$

A continuous diagonal spectrum instead requires

$$\mathbf 1=\int_{\mathcal C} d\mu(\lambda)\, |\lambda\rangle\langle\lambda|, \qquad G(z,\bar z)=\int_{\mathcal C}d\mu(\lambda)\, \rho_{12;34}(\lambda)\, \mathcal F_\lambda(z)\overline{\mathcal F_\lambda(z)},$$

with a declared contour $\mathcal C$, measure $d\mu$, delta normalization, and prescription for poles crossing the contour. An indecomposable spectrum requires generalized eigenvectors as well: the character trace alone can be blind to the nilpotent part of $L_0$.

The usual numerical-bootstrap cone also depends essentially on positivity. Reflection positivity turns identical-scalar OPE coefficients into nonnegative weights; without it, crossing still holds but the separating-functional argument no longer produces the same rigorous exclusions. The role of positivity is stated explicitly in Poland, Rychkov, and Vichi 2019, §III.E.

A common consistency workflow

For any example in this chapter, record the following before manipulating a conformal-block or character expansion:

1. Algebra and central charge. State the chiral algebra, $c$, and any effective central charge. Do not substitute $c_{\mathrm{eff}}$ for $c$ in local Ward identities.
2. States and pairing. State whether states are normalizable, delta-normalized, null, or generalized eigenvectors, and give the conjugation or bilinear pairing.
3. Spectrum and measure. Distinguish discrete multiplicities from a spectral density. Give the contour and every $2\pi$ factor.
4. Zero modes and volumes. State whether a conserved-momentum delta function or target-space volume has been retained, regulated, or divided out.
5. Channel transform. Specify the normalization of blocks or characters and whether the transform is a finite matrix, an integral kernel, or a kernel supplemented by residues.
6. Sense of equality. Say whether crossing or modular covariance holds pointwise, after analytic continuation, or only after pairing with test functions.
7. Failure test. Check a limit that exposes a wrong convention: a negative descendant norm, a Jordan-basis shift, a compact-to-noncompact limit, a contour pole crossing, or a modular Fourier transform.

This workflow prevents two common category errors. A density $\rho(P)$ is not a degeneracy at a single $P$, because changing coordinates on the continuum changes $\rho$ with the Jacobian. Likewise, a finite list of characters does not prove semisimplicity, because a trace need not see the nilpotent operator in a Jordan block.

Review the chapter

You should be able to answer the following questions after working through the leaves.

• Given $c$ and the lowest chiral weight $h_{\min}$, compute $c_{\mathrm{eff}}=c-24h_{\min}$ and explain which asymptotic observable it controls.
• Starting from a rank-two Jordan action for dilatations, derive the logarithm in the two-point function and identify the basis-dependent constant.
• Convert a momentum sum in a compact boson of circumference $L$ into a continuum integral, including the measure and volume factor.
• State the Liouville spectrum, reflection identification, four-point contour, and the condition under which discrete residues must be added.
• Distinguish a fusion kernel from an OPE coefficient and a continuous spectral density from a discrete multiplicity.
• Formulate torus modular consistency as an equality of measures or distributions rather than a finite matrix equation.

An adequate answer includes the relevant normalization, contour, measure, and limiting prescription—not only the formal expression.

References

• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. New York: Springer, 1997. DOI.
• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI.