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Paired rank-two Jordan dilations give a single-valued logarithmic coefficient

Example

Let $E_{L} = E_{R} = C e_{0} \oplus C e_{1}$ and $N e_{0} = 0$ , $N e_{1} = e_{0}$ . On $H_{t o y} = E_{L} \otimes_{C} E_{R}$ , define commuting nilpotent dilation parts

$N_{L} = N \otimes I_{E_{R}}, N_{R} = I_{E_{L}} \otimes N, L_{0} = N_{L}, {\overset{ˉ}{L}}_{0} = N_{R} .$

The only declared joint generalized weight is $(0, 0)$ . For $z \in C^{\times}$ , choose a local logarithm $ℓ (z)$ and its conjugate $\overline{ℓ (z)}$ on the Euclidean slice, and define the finite-dimensional dilation operator

$D (z, \overset{ˉ}{z}) = \exp (ℓ (z) L_{0}) \exp (\overline{ℓ (z)} {\overset{ˉ}{L}}_{0}) .$

Set $v = e_{1} \otimes e_{0} + e_{0} \otimes e_{1}$ and let $λ \in H_{t o y}^{*}$ be the coordinate functional with $λ (e_{0} \otimes e_{0}) = 1$ and value zero on the other three tensor basis vectors. The pair $(D, v, λ)$ is an algebraic Jordan-dilation toy that illustrates the logarithmic-sector mechanism in the earlier definition; it is not a full local CFT package and supplies no local fields, Virasoro modes beyond $L_{0}, {\overset{ˉ}{L}}_{0}$ , OPE, or crossing data.

The two nilpotent actions are shown on the relevant vectors:

Each arrow is the action of the indicated left or right nilpotent dilation; the diagram depicts the algebraic data only.

Facts & Assumptions

Given: The vector spaces $E_{L}, E_{R}$ , the map $N$ , the tensor-product operators, the branch $ℓ$ , the vector $v$ , and the functional $λ$ above.

[F1]

Logarithmic 2D CFT data and logarithmic correlators defines a logarithmic sector by a nontrivial left or right nilpotent part of $L_{0}$ or ${\overset{ˉ}{L}}_{0}$ on a joint generalized-weight vector. It distinguishes this sector property from an actual logarithmic correlator.

[F2]

Nilpotent parts on a joint cyclic subspace give finite logarithmic dilation factors gives the finite polynomial formula for commuting nilpotent dilation parts and the branch-continuation rule.

Verification

1.1givenF2algebra

The tensor-factor actions commute. Each squares to zero because $N^{2} = 0$ , and both are nonzero since $N_{L} (e_{1} \otimes e_{0}) = e_{0} \otimes e_{0}$ and $N_{R} (e_{0} \otimes e_{1}) = e_{0} \otimes e_{0}$ . In this toy representation, $v$ satisfies the algebraic generalized-eigenvector relations $L_{0}^{2} v = {\overset{ˉ}{L}}_{0}^{2} v = 0$ with $L_{0} v = N_{L} v$ and ${\overset{ˉ}{L}}_{0} v = N_{R} v$ ; these are exactly the finite-dimensional nilpotence hypotheses of the dilation lemma with $h = \overset{ˉ}{h} = 0$ and $r = s = 2$ . This does not make $v$ a vector in the domain of a full local CFT package.

2.1givenF2step 1.1algebra

Since $N_{L} N_{R} v = 0$ , the lemma's finite expansion reduces to $D (z, \overset{ˉ}{z}) v = v + (ℓ (z) + \overline{ℓ (z)}) e_{0} \otimes e_{0}$ . Applying $λ$ gives $λ (D v) = ℓ (z) + \overline{ℓ (z)} = 2 \ln ∣ z ∣$ , a nonconstant logarithmic coefficient on $C^{\times}$ . This is a matrix coefficient of the toy dilation, not a correlator supplied by a field theory.

3.1givenF2step 2.1algebra∎

Continuing once around $z = 0$ changes $ℓ$ by $2 π i$ and its conjugate by $- 2 π i$ , leaving $λ (D v)$ unchanged. The other coordinate functional that extracts the coefficient of $e_{1} \otimes e_{0}$ instead gives the constant $1$ , so the same toy representation has matrix coefficients with and without logarithms.

Depends on

Dependency tree · 2 levels

Sources

Rongvoram Nivesvivat and Sylvain Ribault, Logarithmic CFT at Generic Central Charge: from Liouville Theory to the Q-state Potts Model, arXiv:2007.04190v1