Logarithmic 2D CFT data and logarithmic correlators

Use a full local two-dimensional CFT data package in the exact state-space/OPE presentation of the earlier definition. In particular, the package supplies a complete locally convex state space $\mathcal{H}$, a common invariant dense domain $\mathcal{D}$, commuting left and right Virasoro actions, specified vacuum and local fields, convergent branchwise radial OPEs, bosonic or spin locality, and equality of iterated OPE channels. This is a chosen mathematical package; it does not assert that a named statistical model realizes it.

A joint generalized-weight vector is a vector $v\in \mathcal{D}$ for which there are $h,\bar{h}\in \mathbb{C}$ and positive integers $r,s$ with

$N_L^{r}v=0,N_R^{s}v=0,N_L=L_0-hI,N_R={\bar{L}}_0-\bar{h}I\mathrm{.}$

A logarithmic sector is a declared joint generalized-weight sector containing a vector for which $N_{L}v\mathrm{\ne }0$ or $N_{R}v\mathrm{\ne }0$. Its Jordan-chain lengths and chosen left/right bases are part of the data. On a branch domain, the earlier commuting-Jordan dilation lemma gives the action of $z^{L_0}{\bar{z}}^{{\bar{L}}_0}$ on such a vector explicitly: if $N_L^{r}v=N_R^{s}v=0$, then

$z^{L_0}{\bar{z}}^{{\bar{L}}_0}v=z^h{\bar{z}}^{\bar{h}}{\sum }_{i=0}^{r-1}{\sum }_{j=0}^{s-1}\frac{(\log z{)}^i(\log \bar{z}{)}^j}{i!j!}{N}_L^i{N}_R^{j}v\mathrm{.}$

A logarithmic correlator is a specified weak matrix coefficient of a finite radially ordered product of local fields on one of the package's declared convergence domains. After factoring the declared scalar powers, its chosen branchwise expansion must be supplied as a finite polynomial in the chosen logarithms, with coefficient functions independent of those logarithm branches; at least one positive-degree coefficient must be nonzero. The expansion and its branches are part of the package data, so the criterion refers to the declared expansion. A nonsemisimple sector permits logarithmic terms but does not force every coefficient or correlator to detect them; the declared joint-sector data, local fields, OPE coefficient arrays and correlator functional determine which terms survive. Call the full package logarithmic only when it has both a logarithmic sector and at least one logarithmic correlator. It must give all logarithm branches and field normalizations, and it inherits the joint-sector data, local fields, OPE coefficient arrays, correlator functional and crossed-channel continuation paths required by the full-local-data definition. No tensor-product factorization of left and right sectors is assumed.

The package records the coefficient arrays of its OPEs and correlators in declared Jordan-chain bases as its logarithmic-coupling data. These numbers are not basis independent by default. Under an invertible change of basis on a generalized-weight space, state slots transform by that matrix and dual slots by its inverse; the coefficient arrays transform by the induced tensor action, while the represented fields and correlators remain unchanged. For a linearly independent normalized rank-two chain $N{v}_0=0$, $N{v}_1=v_0$, the chain-preserving changes $v_0^{\mathrm{\prime }}=a{v}_0$, $v_1^{\mathrm{\prime }}=a{v}_1+b{v}_0$ with $a\mathrm{\ne }0$ leave the Jordan matrix fixed but change the field-normalized coefficient arrays. A scalar is called a normalization-independent logarithmic coupling only when its invariance under the declared allowed changes has been proved for that model.

The words rational, unitary, and reflection positive retain the separate flags and exact conditions from the full-local-data definition. A chiral logarithmic module under the earlier module definition is not, by itself, a full local logarithmic CFT: it does not supply the full left/right state space, local fields, convergent OPE, locality, or crossed-channel equality. No general tensor-category or modular property follows from this definition.

Full local two-dimensional CFT data

Logarithmic generalized module

Nilpotent parts on a joint cyclic subspace give finite logarithmic dilation factors

The cited generic-central-charge Virasoro analysis constructs logarithmic representations and conformal blocks and studies model-specific Potts/O(n) structures; its proposed full model applications are not assertions that a full local logarithmic CFT has been constructed. A chiral logarithmic module under the earlier module definition does not itself supply the left/right local fields and OPE package listed above. This item defines that additional full-local data and does not claim that any named loop model meets them.