Logarithmic CFT and Indecomposable Modules

A logarithmic conformal field theory is characterized by a nondiagonalizable action of dilatations on some space of states or fields. The resulting logarithms are not arbitrary corrections to power laws: conformal Ward identities fix them once the Jordan action and two-point pairing are specified. This page derives the rank-two case and identifies which constants survive changes of logarithmic basis.

Required background. Nonunitary CFTs, effective central charge, and complex data separates loss of positivity from loss of diagonalizability. Highest-weight modules, null states, and the Kac determinant provide the Virasoro modules that can occur as submodules and quotients.

Helpful background. Representations, intertwiners, and invariants introduces exact sequences and invariant maps used to describe extensions.

A rank-two dilatation block

Let $C$ and $D$ be scalar quasiprimary fields of generalized scaling dimension $\Delta$ . In radial quantization, suppose the dilatation generator $\mathcal D=L_0+\bar L_0$ acts on the corresponding states as

\mathcal D|C\rangle=\Delta|C\rangle, \qquad \mathcal D|D\rangle=\Delta|D\rangle+|C\rangle.

Thus $\mathcal D=\Delta\mathbf1+N$ on this two-dimensional generalized eigenspace, with $N^2=0$ but $N\neq0$ . Exponentiating the transformation gives

\begin{aligned} C(e^t x)&=e^{-\Delta t}C(x),\\ D(e^t x)&=e^{-\Delta t}\bigl[D(x)-tC(x)\bigr]. \end{aligned}

The sign of the $tC$ term follows from $e^{-tN}=1-tN$ . It is a useful convention check: reversing it reverses the logarithm below.

The module is reducible because the span of $|C\rangle$ is invariant, but it is indecomposable because no invariant complementary eigenspace exists. Abstractly it fits into a nonsplit exact sequence

0\longrightarrow \mathcal M_C \longrightarrow \mathcal M \longrightarrow \mathcal M_D \longrightarrow0.

The composition factors $\mathcal M_C$ and $\mathcal M_D$ do not determine the extension $\mathcal M$ . Treating their characters as a direct sum discards precisely the information responsible for logarithmic correlators.

Deriving the logarithmic two-point functions

Translation and rotation invariance make the scalar two-point functions depend only on $r=|x|$ . Write

G_{AB}(r)=\langle A(x)B(0)\rangle, \qquad A,B\in\{C,D\}.

The dilatation Ward identity acts on both insertions. With the Jordan action above it gives

\begin{aligned} (r\partial_r+2\Delta)G_{CC}&=0,\\ (r\partial_r+2\Delta)G_{CD}+G_{CC}&=0,\\ (r\partial_r+2\Delta)G_{DD}+G_{CD}+G_{DC}&=0. \end{aligned}

Special conformal invariance for an equal-rank quasiprimary pair forces $G_{CC}=0$ and $G_{CD}=G_{DC}$ . Integrating the remaining equations yields the canonical form first exhibited in logarithmic CFT by Gurarie 1993, §§2–3, pp. 538–543:

\boxed{ \begin{aligned} \langle C(x)C(0)\rangle&=0,\\ \langle C(x)D(0)\rangle&=\frac{b}{|x|^{2\Delta}},\\ \langle D(x)D(0)\rangle&= \frac{d-2b\log(\mu|x|)}{|x|^{2\Delta}}. \end{aligned}}

Here $\mu$ is an arbitrary inverse-length scale inserted to make the logarithm dimensionless. The coefficient $b$ couples the eigenvector to its generalized partner; $d$ is not invariant.

Indeed, the allowed basis change

D\longmapsto D+\alpha C

preserves the Jordan action but sends

d\longmapsto d+2\alpha b.

A change $\mu\mapsto e^s\mu$ similarly shifts $d\mapsto d-2bs$ . Thus a quoted value of $d$ has no meaning without a basis and scale convention. The nonzero nilpotent action $N$ , the existence of the extension, and appropriately normalized logarithmic couplings such as $b$ carry the structural information. Even $b$ rescales if $C$ and $D$ are jointly rescaled, so comparisons must fix a two-point normalization.

An exact $c=-2$ chiral example

In the $c=-2$ model analyzed by Gurarie, a degenerate chiral field $\mu$ has weight $h_\mu=-1/8$ . Its four-point BPZ equation has the two independent solutions

{}_2F_1\!\left(\frac12,\frac12;1;x\right) \quad\text{and}\quad {}_2F_1\!\left(\frac12,\frac12;1;1-x\right).

The second solution has the local form

{}_2F_1\!\left(\frac12,\frac12;1;1-x\right) =A(x)\log x+B(x),

where $A$ and $B$ are regular at $x=0$ and $A(0)\neq0$ . Consequently the $\mu\times\mu$ OPE contains the identity $I$ and a generalized partner $I_1$ satisfying

L_0|I\rangle=0, \qquad L_0|I_1\rangle=|I\rangle.

This is an exact mechanism rather than a formal insertion of logarithms: the collision of the hypergeometric exponents forces the logarithmic solution, and channel consistency prevents simply discarding it. The differential equation, elliptic-integral solution, and Jordan OPE are worked out in Gurarie 1993, §2, pp. 537–541.

A finite scaling check

The formulas can be checked without differentiating. Under $x\mapsto e^t x$ ,

\frac{d-2b\log(\mu e^t|x|)}{|e^t x|^{2\Delta}} =e^{-2\Delta t} \frac{d-2bt-2b\log(\mu|x|)}{|x|^{2\Delta}}.

The extra $-2bt$ is exactly the sum of the two mixed correlators generated by transforming both $D$ insertions. If it is absent, either the finite Jordan transformation or the two-point function has been normalized inconsistently.

Null states, quotients, and fusion

Indecomposability must be checked at the module level, not inferred from a logarithm seen in one perturbative expression. A practical analysis has four stages:

Find singular and subsingular vectors in the candidate module.
State which null submodules are quotiented and which zero-norm states remain paired with generalized partners.
Compute the action of $L_0$ and enough other modes on representatives to decide whether the exact sequence splits.
State the fusion category and boundary conditions under which the proposed indecomposable module is produced.

Quotienting every zero-norm state can destroy a logarithmic theory: $|C\rangle$ itself has zero self-pairing but nonzero pairing with $|D\rangle$ . Conversely, a degenerate Gram matrix does not by itself prove a Jordan block; an ordinary null vector can be consistently removed from a diagonalizable module.

Fusion is also assumption-sensitive. In a nonsemisimple category, the fusion product of irreducible modules can be reducible but indecomposable. Writing only the list of composition factors omits how one factor is glued to another, and therefore cannot determine logarithmic OPEs.

Why characters are insufficient

On a rank-two generalized eigenspace,

q^{L_0}=q^h e^{(\log q)N} =q^h\bigl(\mathbf1+(\log q)N\bigr).

Because $\operatorname{tr}N=0$ , the ordinary trace sees

\operatorname{tr}q^{L_0}=2q^h,

exactly as it would for two diagonal states of weight $h$ . An ordinary character can therefore reproduce the correct graded dimension while missing the extension and the logarithmic coupling. Modular closure may require pseudo-traces, torus amplitudes with additional insertions, or other generalized characters; the correct choice depends on the category and pairing, not just on the list of weights.

Locating the logarithmic branch

The figure below should be read at the central fork. The logarithmic branch changes the spectral decomposition of dilatations even when the set of generalized eigenvalues is discrete.

A diagnostic map separating an indefinite pairing, a Jordan block with logarithmic correlators, and a delta-normalized continuum, with the Jordan branch highlighted conceptually — Loss of semisimplicity is diagnosed by a nilpotent part of dilatation and requires extension data in addition to eigenvalues and characters. The diagram is schematic and not to scale.

The diagram’s content has the following structured equivalent:

Test	Ordinary diagonal module	Rank-two logarithmic module
Minimal polynomial of $\mathcal D$ at $\Delta$	$x-\Delta$	$(x-\Delta)^2$
Two-point dependence	$\lvert x\rvert^{-2\Delta}$	$\lvert x\rvert^{-2\Delta}$ and $\lvert x\rvert^{-2\Delta}\log(\mu\lvert x\rvert)$
Basis data	Eigenvector normalization	Eigenvector normalization plus generalized-partner shift
Ordinary character	Records multiplicity	Records multiplicity but can miss $N$ and $b$
Required extra structure	None beyond the pairing	Nonsplit extension, logarithmic coupling, generalized torus amplitudes when needed

A bounded calculation can exponentiate finite Jordan matrices and test basis changes. A finite matrix example verifies the formulas above; it does not determine the module category or fusion rules of a full CFT.

For the independent replacement of sums by direct integrals, continue to Noncompact CFTs and continuous spectra.

Common pitfalls

A logarithm is not automatically a Jordan block. Perturbative logarithms can arise from expanding a family of ordinary powers in a parameter. A logarithmic CFT claim requires a limiting operator basis and a demonstrably nondiagonalizable dilatation action.

A zero norm is not automatically a null state to quotient. In the pair above, $C$ has zero self-pairing but $\langle C D\rangle\neq0$ . Removing it would make the generalized partner and its correlators ill-defined.

Characters do not classify indecomposable modules. They count generalized eigenspaces with signs or multiplicities determined by the trace; they need not record extension classes or logarithmic couplings.

Exercises

Solve the Ward identities

Assume $G_{CC}=0$ , $G_{CD}=G_{DC}=b r^{-2\Delta}$ , and the rank-two Ward identities above. Derive $G_{DD}$ .

Solution

Write $G_{DD}=r^{-2\Delta}f(r)$ . The last Ward identity becomes

r^{-2\Delta}r f'(r)+2b r^{-2\Delta}=0,

so $r f'(r)=-2b$ . Hence $f(r)=d-2b\log(\mu r)$ , where a change of $\mu$ is absorbed into $d$ .

Detect what the trace misses

Let $L_0=h\mathbf1+N$ on an $r$ -dimensional Jordan block with nilpotent $N$ . Show that the ordinary character contribution is $r q^h$ , independent of the off-diagonal entries of $N$ .