Fusion rules and structure constants of E-series minimal models

Changes according to Jiaxin Qiao's report

1. Our numerical bootstrap results imply that the E-series minimal models' spectra always yield a unique solution to the crossing-symmetry equation. Therefore, this allows us to uniquely factorize the 4-point structure constants into the 3-point structure constants up to some field renormalizations. (This uniqueness is a numerical result, we do not have a proof.)

We have added these details at the beginning of Section 4.1.

1. We have corrected the parity of $s$ in last term of (1.5a) as suggested.

2. The spectrum for the case $q =12$ is block-diagonal w.r.t to the Ising category whereas the spectrum for the case $q =30$ is block-diagonal w.r.t to the Lee-Yang category.

This has been clarified after equation (1.6).

1. To obtain the PSU(n) CFT, we need to assume that the first Kac indices $r$ increases as $q$ is increased.

To make our speculation clearer, we have added the equations (1.11), (1.12), and (1.13), which speculate how to obtain the $PSU(n)$ CFT as the non-rational limit of generalizations of E-series minimal models.

1. We have clarified that the non-chiral fusion rules of $V_{1,2}^d$ are not an assumption.

The non-chiral fusion rules of $V_{1,2}^d$ can be fully determined by using the constraints from the null vectors, the crossing-symmetry equation, and the single-valuedness. Then, the resulting non-chiral fusion rules of $V_{1,2}^d$ produce the primary fields that are allowed by the chiral fusion rules of $R_{1, 2}$.

1. For computing the non-chiral fusion rules, the bootstrap uses the spectrum and the constraints from the null vectors (chiral fusion rules) as the only inputs. Then, we check that the resulting non-chiral fusion rules agree with the extended symmetries and the non-standard constraints in Section 2. For deducing the analytic structure constants, the inputs to the bootstrap are the non-chiral fusion rules, which reduce the number of unknowns in the crossing-symmetry equation and allow us to easily access numerical results at high-precision. All of the above has been clarified at the end of Section 1.4, after equation (1.18).

2. See point 6.

3. We have rewritten 2-point structure constants as simply structure constants since $c_{1, 2, 2}$ and $c_{1, 3, 3}$ on the table (4.9) are the structure constants of the three-point functions $<V_{1,s}V_{2,s'}V_{2,s''} $ and $<V_{1,s}V_{3,s'}V_{3,s''} $.

Changes according to Connor Behan's report

1. We have added the reference for the three-point functions in the A- and D-series in the second point of the 4 items on page, citing as "reference [2] and references therein".

2. On page 3, we have itemized why the E-series minimal models are not fully solved, and we have added reference [3], which discuss how to compute the models' three-point functions on a case-by-case basis.

3. At the end of Section 2.3 we have stated more explicitly that a simple current can be interpreted as an extended $\mathbb{Z}_2$ symmetry, and that Section 2.4 deals with extended symmetries that are not simple currents.

4. We have clarified this point.

5. We have added equations (1.11), (1.12), and (1.13) to clarified our speculation on obtaining the $PSU(n)$ CFT as non-rational limits.

6. On page 9, under the equation (2.7), we have clarified why we choose $r \leq q/2$ except for the case $q = 12$.

We make an exception for the case $q = 12$ because we want all three-point functions to obey our convention for the parity constraints for any $q$: any allowed even coupling also comes with an allowed odd coupling.

For $q = 12$, the case $r = 7$ belongs to the identity sector of the extended symmetry (1.6) whereas $r=5$ belongs to the $\epsilon$. This makes three-point functions for the case $q =12$ agree with our convention for the parity constraints.

1. We have clarified that the convention for the parity $s1 + s2 + s3$ under the equation (2.11). For this parity, we use the notations (1.5) wherein the Kac indices $s_i$ are integers for both diagonal and non-diagonal fields.

2. We have stressed that we write the superscript $e$ in the non-chiral fusion rules when both even and odd parities are allowed.

3. We have explained after (4.3) why the structure constants on the table (4.9) only depends on the second indices through signs.

4. We have corrected these typos.