Exactly solvable 2d CFTs

Spectrums and loop CFTs

See Sections 2.1 and 2.4 of Ref.

Course questions

1. In the Ising minimal model, what are the OPE spectrums? Write the channel spectrums of all inequivalent 4-point functions.
2. In a generalized minimal model, write the 3 channel spectrums of $⟨V_{⟨3,3⟩}^d{V}_{⟨2,4⟩}^d{V}_{⟨1,4⟩}^d{V}_{⟨2,5⟩}^d⟩$.
3. What are the largest possible non-diagonal spectrums for a CFT with 2 degenerate fields, such that all spins are integer?

ESDI: Conformal dimensions of non-diagonal fields

We consider the non-diagonal primary fields with integer conformal spins, in a CFT that has a degenerate field $V_{⟨1,3⟩}^d$. We assume $\mathrm{\Re }{\beta }^2>0$.

1. Which non-diagonal fields $V_{(r,s)}$ can exist?
2. Show that the set of their total conformal dimensions $\mathrm{\Delta }+\overline{\mathrm{\Delta }}$ is discrete and bounded from below.
3. Show that there are finitely many fields whose total dimension is below any given bound.
4. Assuming ${\beta }^2\in ℝ$, sketch the set of non-diagonal fields on a plot whose axes are $\mathrm{\Delta }-\overline{\mathrm{\Delta }}$ and $\mathrm{\Delta }+\overline{\mathrm{\Delta }}$.

Liouville theory and non-diagonal minimal models

Liouville theory

See Section 2.2.2 of Ref.

D-series minimal models

See Section 2.3.1 of Ref.

Course questions

1. From OPEs in generalized minimal models, deduce the OPEs in Liouville theory with $c\le 1$. In particular, show that the measure of integration over momentums is the uniform measure $dP$.
2. What is the intersection of the spectrums of the A-series and D-series minimal models at the same central charge? Is this the spectrum of a consistent CFT?
3. Show that the spectrum of the D-series minimal model DMM${}_{6,5}$ is a subset of the spectrum of the 3-state Potts CFT. Explain why the diagonal primary field $V_{P_{(0,\frac{1}{2})}}$ appears in the non-diagonal sector of DMM${}_{6,5}$. Admitting that $V_{P_{(0,\frac{1}{2})}}$ transforms in the standard representation of the symmetric group $S_3$, explain why it appears twice in DMM${}_{6,5}$.

ESES: E-series minimal models

1. Find the spectrums of E-series minimal models in Wikipedia.
2. Which diagonal degenerate fields exist in E-series minimal models?
3. Compare the spectrums of the A-series, D-series and E-series minimal models with the same central charge.
4. In the case $q=12$, rewrite the spectrum in terms of non-diagonal fields $V_{(r,s)}$, and compare with loop models.