Exactly solvable 2d CFTs

## Spectrums and loop CFTs

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See Sections 2.1 and 2.4 of Ref.[1]

### Course questions

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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 ⟩ {\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?

### ESDI: Conformal dimensions of non-diagonal fields

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

1. Which non-diagonal fields  V ( r , s ) {\displaystyle V_{(r,s)}} can exist?
2. Show that the set of their total conformal dimensions  Δ + Δ ¯ {\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  β 2 ∈ R {\displaystyle \beta ^{2}\in \mathbb {R} } , sketch the set of non-diagonal fields on a plot whose axes are  Δ − Δ ¯ {\displaystyle \Delta -{\bar {\Delta }}} and  Δ + Δ ¯ {\displaystyle \Delta +{\bar {\Delta }}} .

## Liouville theory and non-diagonal minimal models

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### Liouville theory

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See Section 2.2.2 of Ref.[1]

### D-series minimal models

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See Section 2.3.1 of Ref.[1]

### Course questions

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

### ESES: E-series minimal models

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

## References

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1. ↑ 1.0 1.1 1.2 Ribault, Sylvain (2024-11-26). "Exactly solvable conformal field theories". arXiv.org. Retrieved 2024-11-27.