Exactly solvable 2d CFTs

See Sections 2.1 and 2.4 of Ref.^[1]

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 ${\displaystyle \left\langle V_{\langle 3,3\rangle }^{d}V_{\langle 2,4\rangle }^{d}V_{\langle 1,4\rangle }^{d}V_{\langle 2,5\rangle }^{d}\right\rangle }$.
3. What are the largest possible non-diagonal spectrums for a CFT with 2 degenerate fields, such that all spins are integer?

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

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

See Section 2.2.2 of Ref.^[1]

See Section 2.3.1 of Ref.^[1]

1. From OPEs in generalized minimal models, deduce the OPEs in Liouville theory with ${\displaystyle c\leq 1}$. In particular, show that the measure of integration over momentums is the uniform measure ${\displaystyle dP}$.
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${\displaystyle _{6,5}}$ is a subset of the spectrum of the 3-state Potts CFT. Explain why the diagonal primary field ${\displaystyle V_{P_{(0,{\frac {1}{2}})}}}$ appears in the non-diagonal sector of DMM${\displaystyle _{6,5}}$. Admitting that ${\displaystyle V_{P_{(0,{\frac {1}{2}})}}}$ transforms in the standard representation of the symmetric group ${\displaystyle S_{3}}$, explain why it appears twice in DMM${\displaystyle _{6,5}}$.

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 ${\displaystyle q=12}$, rewrite the spectrum in terms of non-diagonal fields ${\displaystyle V_{(r,s)}}$, and compare with loop models.