Full Two-Dimensional CFT from Chiral Nets

A left–right tensor product of chiral nets is already a local net on two-dimensional double cones, but a full CFT generally contains additional charged fields. For completely rational chiral data, a commutative Q-system in the product of the left sector category with the reverse-braided right category constructs such a local extension. A modular-invariant multiplicity matrix is necessary data, not by itself sufficient data.

Required background. Conformal Nets and Covariance Axioms supplies the chiral local algebras, Extensions, Orbifolds, Cosets, and Alpha-Induction supplies extension criteria, and Chiral Blocks, Sewing, and Modular Invariance supplies the physical left–right decomposition. Helpful background. Conformal Boundaries and Defects explains why full centers also organize boundary conditions.

Double-cone nets from two chiral halves

In two-dimensional Minkowski space with signature $(+,-)$ , use light-ray coordinates $x^\pm=t\pm x$ . A double cone is $O=I\times J$ , where $I$ is an interval on the left light ray and $J$ one on the right. Given chiral nets $\mathcal A_L$ and $\mathcal A_R$ , the product subnet is

\mathcal A_2(O)= \mathcal A_L(I)\,\overline\otimes\,\mathcal A_R(J).

If two double cones are spacelike separated, their left intervals occur in one order and their right intervals in the opposite order. This reversal is why the relevant category is

\operatorname{DHR}(\mathcal A_L) \boxtimes \operatorname{DHR}(\mathcal A_R)^{\mathrm{rev}}.

The product net inherits isotony, locality, vacuum, positive energy, and Möbius or diffeomorphism covariance factorwise. A full net $\mathcal B_2$ with the chosen chiral symmetry is a local covariant extension $\mathcal A_2(O)\subset\mathcal B_2(O)$ , consistently for every double cone. The common Hilbert space, unique vacuum, and finite-index inclusion are part of the hypotheses; a torus partition function alone supplies none of them.

Full centers as the sufficiency theorem

Let the chiral nets be completely rational. Finite local extensions of $\mathcal A_2$ correspond to commutative Q-systems in the product category. If $A$ is a chiral Q-system, its full center $Z(A)$ is a canonical commutative Q-system in the left–right product; Morita-equivalent chiral Q-systems have equivalent full centers. Bischoff, Kawahigashi, and Longo prove that the generalized Longo–Rehren construction agrees with this categorical full center and classify maximal full nets by the resulting Morita classes Bischoff, Kawahigashi, and Longo 2015, §§4.1 and 6.1, pp. 1153–1160 and 1171–1173.

The mechanism is concrete. The Q-system multiplication defines products of charged generators, associativity makes the local algebra well defined, C*-positivity gives a Hilbert-space representation, and commutativity with the product braiding proves spacelike locality. Its sector decomposition determines a nonnegative integer coupling matrix $Z_{ij}$ with $Z_{00}=1$ . Alpha-induction then implies $ZS=SZ$ and $ZT=TZ$ . The arrows do not reverse: modular commutation does not supply the multiplication or positivity.

For a maximal full extension, several equivalent conclusions hold: maximal index, trivial DHR sector theory for the full net, and modular-invariant coupling. Proposition 6.6 of Bischoff, Kawahigashi, and Longo 2015, pp. 1174–1175 states the equivalence under complete rationality and finite irreducible inclusion. It is not a theorem about nonrational chiral theories or arbitrary integer matrices.

Diagonal Ising full CFT

Take identical left and right Ising nets. In $\mathcal C_{\mathrm{Ising}}\boxtimes\mathcal C_{\mathrm{Ising}}^{\mathrm{rev}}$ , the diagonal full-center object is

\Theta=(\mathbf1\boxtimes\mathbf1) \oplus(\sigma\boxtimes\sigma) \oplus(\varepsilon\boxtimes\varepsilon).

Reverse braiding on the right cancels the left monodromy, so the canonical multiplication is commutative. The vacuum summand occurs once, giving $Z_{00}=1$ , and the coupling matrix is the diagonal Ising invariant,

Z=\lvert\chi_{\mathbf1}\rvert^2+ \lvert\chi_\sigma\rvert^2+ \lvert\chi_\varepsilon\rvert^2.

This completes the exact local-net construction associated with Chiral Blocks, Sewing, and Modular Invariance: the blocks identify the sector pairing, while the full-center Q-system proves operator-algebraic locality and vacuum multiplicity.

There is an independent index check. Each Ising chiral net has $\mu=4$ , the extension index is $d(\Theta)=4$ , and the full-net index formula gives

\mu_{\mathcal B_2}= \frac{\mu_L\mu_R}{[\mathcal B_2:\mathcal A_2]^2} =\frac{4\cdot4}{4^2}=1.

Thus the diagonal full center is maximal and has no nontrivial DHR sectors, consistently with Proposition 6.6.

Locality beyond the partition function

The double-cone locality check uses more structure than the symmetry of $Z$ . Charged generators in spacelike-separated double cones exchange with the left braiding and the inverse right braiding; commutativity of the full-center multiplication cancels their product. Associativity then makes this exchange compatible with triple products, while the C*-relations supply adjoints and positivity. A torus partition function records only multiplicities of left–right sectors and cannot test any of these operator identities. It also does not determine the embeddings $\mathcal B_2(O_1)\subset\mathcal B_2(O_2)$ or the vacuum representation. These are why a commutative Q-system is sufficient in the stated rational finite-index regime and a modular matrix is not.

Adversarial multiplicity matrix

In the Ising sector order $(\mathbf1,\sigma,\varepsilon)$ , let $M=\operatorname{diag}(1,0,1)$ . It has nonnegative integer entries, $M_{00}=1$ , and commutes with the diagonal $T$ matrix. But it does not commute with the displayed Ising $S$ matrix: for example $(MS)_{\mathbf1\sigma}=1/\sqrt2$ while $(SM)_{\mathbf1\sigma}=0$ . It is therefore rejected even as a modular invariant. More generally, a matrix commuting with both generators can still fail to arise from a positive local Q-system. The surviving conclusion is a numerical consistency candidate, not a full CFT.

Exercises

Compute $d(\Theta)$ for the diagonal Ising object and use the full-net index formula to test maximality.

Solution

The summand dimensions are $1$ , $d_\sigma^2=2$ , and $1$ , hence $d(\Theta)=4$ . With $\mu_L=\mu_R=4$ , the formula gives $\mu_{\mathcal B_2}=16/16=1$ . The full extension is maximal within the stated completely rational finite-index regime.