Coupled minimal models revisited II: Constraints from permutation symmetry

Understanding the landscape of 2d CFTs remains an important open problem. In particular, a concrete example of a 2d CFT satisfying the following properties is still unknown: (a) central charge $c > 1$, (b) unitarity, (c) a normalizable vacuum, (d) modular invariance, and (e) absence of extra continuous global symmetries.

In a previous work [arXiv:2211.16503], the authors proposed a construction that could potentially yield such a 2d CFT. The setup begins with a UV CFT given by $N$ copies of a unitary diagonal minimal model: $\left(\mathcal{M}_{m,m+1}\right)^{\otimes N}$, where $N$ is finite and $m$ is taken to be very large. In this regime, the authors identify two weakly relevant operators, denoted $\sigma$ and $\epsilon$, that preserve the $S_N$ global symmetry and do not trigger other relevant operators along the RG flow. By fine tuning their couplings, they discover two new fixed points, ${\rm FP}_{\pm}^*$. These fixed points are expected to satisfy conditions (a)-(d), but whether condition (e) holds remains unclear. The challenge lies in the fact that the UV theory has an extended chiral symmetry given by $Vir^{\otimes N}$, the product of $N$ Virasoro algebras. It is not known whether $Vir^{\otimes N}$ breaks to the diagonal Virasoro algebra $\widehat{Vir}$ in the IR. This is the main question addressed in the present paper.

The main conjecture of this work is as follows. Fix $N > 4$, and let $T$ be any chiral current operator in the UV theory $\left(\mathcal{M}_{m,m+1}\right)^{\otimes N}$ that transforms as a primary under $\widehat{Vir}$. Then, in the IR fixed point ${\rm FP}_{\pm}^*$, $T$ acquires an anomalous dimension of the form
\[
\gamma_T = a_T m^{-2} + O(m^{-3})
\]
with $a_T > 0$. In other words, any conserved current of the UV theory becomes non-conserved in the IR for sufficiently large $m$.

Instead of performing a two-loop computation of $\gamma_T$, the authors invoke the method of weakly broken symmetries: $\bar{\partial}T = \frac{\#}{m} V+O(m^{-2})$, which relates the non-zero leading anomalous dimension to the non-vanishing of the three-point function $\langle T V \sigma\rangle$, where $T$ is the UV current, $V$ is the candidate divergence operator, and $\sigma$ is the relevant deformation operator.

For each spin $J$, there are many chiral currents $T^K$ and candidate divergences $V^L$. The statement that all such currents acquire non-zero anomalous dimensions at order $O(m^{-2})$ is equivalent to that the matrix $M^{LK} = \langle T^L V^K \sigma\rangle$ has maximal rank---equal to the number of independent spin-$J$ currents $T^L$. By first studying the decomposition of the characters of $Vir^{\otimes N}$ into characters of $\widehat{Vir}\times S_N$, the authors showed that for each spin (less than certain truncation) and for each irreducible representation of $S_N$, there are more candidate divergences than currents, making maximal rank possible. They then built up an algorithm for choosing bases of currents and divergences according to $\widehat{Vir}\times S_N$, and verify numerically that maximal rank is indeed achieved for $J \leqslant 10$ and $N = 5,6,7$. This provides a positive test of the conjecture.

This work is a valuable and well-written contribution to the study of irrational 2d CFTs. Based on version 3 of the draft, I have the following suggestions to improve clarity and completeness:

Main Issues
1. Although it is explained in words that the integral in eq.(2.17) is trivial, it would be more reader-friendly to explicitly state the final result. For example, one could write something like:
\begin{equation}
b^K_L = \pi C_{T^K V^L \sigma} + O(1/m),
\end{equation}
where $C_{T^K V^L \sigma}$ is the structure constant of the three-point function. This would also make it more natural to begin section 4.2 with eq.(4.16).

2. What is the technical obstruction to extending the check to $J\leqslant10$ and general (but large) values of $N$? According to table 8, there are only finitely many independent currents with $J \leqslant10$.

3. Since the main technical analysis in section 4.2 is performed numerically, it would be helpful to include a brief summary of the algorithm used, before the end of that section.

Minor Issues
1. In table 8, the entry for $J = 2$ is missing in the list of $d_{J,0}^{(N-1,1)}$.

2. Around (4.1), it would be helpful to remark that "$k$" is the subscript of $n$ instead of $L$ (since both $n$ and $k$ are small). I was personally confused about the placement of the subscripts: $L_{-n_{j,k}}$ vs $L_{-n_j,k}$.

3. The last paragraph of section 4 states that the numerical check was done for $J < 10$, but all the tables seem to include preparations for $J = 10$ as well (also the introduction says $J\leqslant10$). Has the case $J = 10$ actually been checked?

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Understanding the space of CFTs is a fundamental problem of mathematical physics, but CFTs are vastly more complicated than Lie algebras, and classifying all of them is far from achievable. However, one can start with smaller spaces of CFTs. First we can focus on 2d CFTs, which are more tractable thanks to their infinite-dimensional symmetry algebra. Then we can restrict to compact, unitary, modular-invariant CFTs, a physically well-motivated class which is expected to be sparse in the space of all CFTs. For simplicity we can further assume that there is a twist gap, which not only implies that there is no extended chiral symmetry, but also (together with modular invariance) leads to strong constraints on the spectrum.

The space of CFTs that obey all these assumptions is still expected to be vast, but there are no confirmed examples beyond minimal models. It is reasonably easy to construct candidate CFTs by perturbing products of minimal models and following the resulting RG flows, but it is much harder to prove that these candidates have a twist gap. The weaker statement that there is no extended chiral symmetry is very plausible in many cases, if only because extended symmetries typically do not arise by accident. Proving this statement remains technically challenging, and this is the challenge that the present article addresses.

So, we start from a product $\mathcal{P}_N$ of $N$ unitary diagonal minimal models. This product CFT has a large extended chiral symmetry: the product of the $N$ Virasoro algebras. It also has a permutation symmetry, described by the symmetric group $S_N$. Then we apply a perturbation that preserves permutation symmetry, and breaks chiral symmetry. But how do we know that no chiral symmetry is restored (or appears) at the fixed point, beyond conformal symmetry? The present article's approach is to focus on short RG flows, generated by nearly marginal operators. Such flows are accurately described by perturbation theory around the original product CFT $\mathcal{P}_N$. Unitary minimal models come with an integer parameter $m$, and they can be described perturbatively in $\frac{1}{m}$ near $m=\infty$: we therefore consider perturbing operators of dimension $1-O(\frac{1}{m})$, leading to fixed points at critical couplings $O(\frac{1}{m})$.

Consider chiral currents in the product CFT $\mathcal{P}_N$, i.e. primary fields $T^K$ of left and right conformal dimensions $(\Delta,\bar\Delta)=(J,0)$, which obey $\bar\partial T^K=0$. The breaking of the corresponding extended symmetries occurs if $\bar\partial T^K$ becomes nonzero along the flow. As explained in Sections 2.2 and 4.2, this is equivalent to the non-vanishing in $\mathcal{P}_N$ of 3-point functions of the type $\left<T^KV^L\sigma\right>$, where $\sigma$ is the perturbing (nearly marginal) operator and $V^L$ are primary fields of conformal dimensions $(\Delta,\bar\Delta)=(J,1)$. This non-vanishing of 3-point functions in a product of minimal models is now a tractable technical question, which the authors answer up to $J=10$ using computer calculations. Their result is that indeed, there are no extended symmetries.

How confident can we be that this result is true? This is hard to say, for two main reasons. First, we do no know for which values of the parameters $m$ and $N$ the result is supposed to hold. We know that $m$ should be large but finite, which must mean larger than some $N$-dependent bound. (Does the bound also depends on $J$?) The calculations described in Section 4.2 seem $N$-dependent, since they are about a matrix whose size $d_{J,0}$ depends on $N$, but it is not stated which values of $N$ were investigated. Second, the result cannot in principle be checked because it heavily relies on computer code that is not publicly accessible. And since the result is a negative statement, it would be hard to check by other means.

Nevertheless, this article is valuable for identifying large families of CFTs that are most probably unitary, compact, without extended chiral symmetry, and for introducing methods that could allow us to tame these CFTs. These methods include the analysis of the action of $S_N$ in Section 3. Although this analysis seems to play no role in the main result, it uncovers fundamental structures in the spectrum and its dependence on $N$. Somewhat paradoxically, while this work aims to explore CFT way beyond the well-known minimal models, it reaffirms the crucial importance of exactly solved models, since the CFTs under consideration are built from minimal models.

Main issues:

1. What exactly is proved in this article, and what is conjectured? First there is the statement from Section 4.2 that symmetries are broken by the perturbation. This does not seem to depend much on $m$, as explained in Section 3.3. But for which values of $J,N$ has this statement been established? The statement that the currents "appear to lift" in Section 4 is too vague. Then there is the statement that RG flows are short enough that the fixed point has no extended symmetry. This requires $m\geq m_0$, what can we say on $m_0$? How does $m_0$ depend on $J,N$? Can $m_0$ be estimated?

2. In Sections 3.2 and 4.1, the counting of currents and divergences by $S_N$ irrep is done in a pedestrian way, by constructing them explicitly. Such explicit constructions are needed in Section 4.2. Nevertheless, it would be interesting to have more general, analytic results on the decomposition of the spectrum into $S_N$ irreps. States are counted using partition functions: there are also twisted partition functions, which account for the action of $S_N$, and decompose into characters of the Virasoro algebra and of $S_N$. This works for determining the spectrum of the $q$-state Potts model, see https://arxiv.org/abs/2208.14298 . Could this work here?

3. Starting from the title, the article gives the impression that $S_N$ symmetry is crucial. But then there is the apparent admission in footnote 14 page 21 that $S_N$ irreps are not used to derive the main result. It would be good to state earlier and more explicitly which role $S_N$ symmetry plays.

Some other questions, and suggestions for improving the text:

1. Typos: encoutners, Verma modula, memoized, eachother, vaccua (twice), vaccuum (twice).

2. In the introduction, the questions Q1, Q2 about spaces of CFTs could be complemented by a Q3 about CFTs with a twist gap, instead of postponing the subject to the outlook.

3. The notation $\frac{3}{f}-\frac12$ for the dimension of $\epsilon^i$ in Eq. (1.1) differs from the rest of the article, where the same number would be called $h_{(1,2)}$. How is Eq. (1.1) related to Figure 1? Do they have the same range of values of $q$? Does the perturbation become marginal for $q=2$ and $q=4$?

4. Is it for technical simplicity that only diagonal minimal models are used, or is there another reason for not using non-diagonal minimal models?

5. The integer $m$ is called the rank of a minimal model, is this standard terminology?

6. On page 3, the statement that irrationality follows from modular invariance and $c>1$ could be clarified.

7. The discussion in Section 2.1 about recovering minimal models from generalized minimal models is nice, but is it relevant? At the end we focus on $\phi_{(1,2)}$ and $\phi_{(1,3)}$, which obey $r_i, s_i\ll m$.

8. In Eq. (2.7), parentheses would be welcome.

9. After Eq. (2.12), a factor $\sqrt{N}$ seems to be missing in $g^*_\epsilon$ for the fully stable fixed point.

10. In the legend of Figure 2, $FP_\pm^*$ could be called by their names, rather than the "fixed points in the middle".

11. Page 8, the explanation why checking primaries is sufficient could be clarified.

12. In Section 2.2, it would be nice to carry out the reasoning until the end, and clearly state the outcome: that we only need to compute $\left<T^KV^L\sigma\right>$ in the unperturbed CFT. Instead, this statement is postponed to Eq. (4.14), leaving us with Eq. (2.17) which looks not so easy to evaluate. Moreover, it would be useful to clearly define divergences and divergence candidates.

13. In Section 2.3, the reasoning that ends by invoking Hecke operators would deserve a conclusion.

14. In Eq. (3.1) should there be $N$?

15. In Section 3.1, most of the material is standard and covered in the Wikipedia article on Representation theory of the symmetric group. Does this material need to be repeated in each article that uses $S_N$ representations?

16. When writing $S_N$ representations, it might be simpler to remove the first line of the Young diagrams, and to write $(1)$ instead of $(N-1,1)$.

17. At the very end of Section 3.2, it would be nice to conclude on the meaning of the single case where there are fewer divergences than currents. Are you conjecturing that recombination occurs, even though the sufficient (but not necessary) condition of having enough divergences in the right $S_N$ irrep is not obeyed?

18. Why is there Eq. (3.19)? What should we deduce from it?

19. In Table 5, it would be good to state more explicitly what $n$ is, and to demonstrate the agreement with Table 2 in an example, for example $J=4$.

20. On page 20 the explanation that starts with "To remove redundant degrees of freedom" is obscure. Such technical details could either be explained more carefully (with examples) or omitted altogether.

21. The expression "Making this parallel" is awkward, it should be said more explicitly that this is about code and not physics.

22. In Section 4.2, instead of the statement "this is the last step which is sensitive...", it would be good to state more precisely how the calculations are sensitive to $N$ and to the index structure.

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As a matter of principle I do not make recommendations on whether to publish articles or not. Editorial decisions belong to editors. Since the computer system makes it necessary to have a recommendation, I picked the first one in the list.
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Publish (surpasses expectations and criteria for this Journal; among top 10%)