More precisely, the defect spin content in the IR is a subset of the UV one due to possible massive decouplings.

Note that this implies the closure of this category under fusion.

E.g., $\mathcal{M}(3,4)\to \mathcal{M}(2,5)$, or $\mathcal{M}(4,5)\to \mathcal{M}(3,4)$. See Appendix B in Supplemental Material [33] for more examples.

The known flows [40, 41] $\mathcal{M}(2k+1,2)\to \mathcal{M}(2k+1,2)$ are not displayed in the figure given that these degenerate cases where all the topological lines are broken are not determined by our method.

We find it worth mentioning that these kernels were derived from an educated guess. We asked their Fourier transform to have the following properties: (1) be a ratio of trigonometric functions; (2) be nonvanishing at $\omega =0$ (as this value enters the expressions of $c_{\mathrm{eff}}^{\mathrm{UV}/\mathrm{IR}}$ and $h_{(r,s)}$); (3) be decaying as $\omega \to \pm \infty$; (4) present poles at $\omega =\pm \mathrm{i}$ (which produce the bulk term in the UV expansion of $f(r)$); and (6) present two free parameters (that, along with the twist $\alpha$, allow to tune the UV and IR central charges and the UV conformal dimension at will). Asking additionally for a minimal number of trigonometric functions, we arrived at the expressions (9).

The limit case $\kappa =2$ yields the ${\varphi }_{(1,3)}$ massless flows [26, 27, 28]. This limit is nontrivial and is discussed in details in Supplemental Material [33].

Specifically, $z_{(r,s)}=1+[(p(r-1)-q(s-1))(\text{mod}2)]$.

The case ${\varphi }_{(2,1)}$ in Eq. (3.9) in [31] is recovered from (13) by setting $\mu =1/2$, redefining $p=2P-J$, and $I=J/2$ and swapping the indices of the minimal models and of the primary fields.

It is perhaps worth remarking that we did not ask for the known integrable cases to be included in the educated guess that yielded (9). This is a post hoc result that reveals our kernels to be a one-parameter deformation of the known integrable ones.