Quasi-Characters in $\widehat{su(2)}$ Current Algebra at Fractional Levels

This paper uses the known modular properties of certain fractional level $SU(2)$ WZW models to construct other representations of the modular group and identifies these representations with other conformal models, generally unitary rational ones. The results are that one can do this when $k+2$ is $2/2N+1$, $2N+3/2$ or $3/4$. In each case, the author explores the other modular representations in detail and offers many speculative connections.

The procedure appears to be as follows:

1. Start with the irreducible highest-weight characters and partition them into those which converge when the Jacobi variable (here denoted by $z$) is sent to $0$ and those which diverge.
2. The divergent characters come in pairs related by a $Z_2$ symmetry that preserves the conformal dimension. The difference of such pairs (here called an even character) is stated to converge at $z=0$.
3. The set of convergent characters and differences of pairs carries a representation of the modular group.
4. The differences of pairs usually need to be multiplied by an integer so that their $q$-series expansions have integer coefficients.
5. If the results have positive integer coefficients, which is quite rare, try to identify them as the characters of some unitary rational CFT.

The motivation behind this is stated to be 4d-2d duality, specifically the well known observation that the Schur index of the 4d theory is the $q$-character (here called an unflavoured character) of the vacuum module of some 2d (usually non-rational, always non-unitary) CFT. The link to even characters appears to be that the S-transform of the vacuum character is a linear combination of them. Moreover, this transformation has a physical interpretation in the 4d theories.

I have to say that I find this link and the preceding motivation to be rather tenuous. Nevertheless, the paper represents a detailed exploration of the results of the procedure outlined above, independent of what it may be useful for.

There are several problems that I think need to be addressed before this paper can be considered for publication. The most important is that the notion of the even characters is not properly defined in the text. As stated in (2.19), one simply takes the difference of two characters. However, the $SU(2)$ weights (here $J^0_0$-eigenvalues) of the two representations do not differ by integers, so there is no cancellation. With the definition presented in the paper, it appears that the difference still diverges at $z=0$. To put it another way, the residues of the characters at $z=0$ are not the same.

This is easier to see with an example. For $k=-1/2$, the $Z_2$ symmetry relates the irreducible highest weight modules with $SU(2)$ weights $-1/4$ and $-3/4$. It goes without saying that these weights differ by a non-integer so subtracting their characters just gives infinitely many powers of $\omega = e^{2\pi iz}$ with positive coefficients, and infinitely many with negative coefficients, at each conformal grade.

It is also concerning that even characters apparently need to be multiplied by an integer before their coefficients are integers. The difference of two characters always has integer coefficients before specialising to $z=0$ and so also after (assuming specialising is possible). Something more subtle than the naive "difference of characters" is being used here.

A much more precise definition is therefore needed. I'll mention that one lesson learned in [17] is that the region of convergence of the characters is essential to any fractional level study because choosing different regions in which to expand gives characters of inequivalent representations! Perhaps the definition of even character being used here is using different convergence regions for the two characters? As the author notes that the sum of these characters (here the odd characters) are related to Virasoro minimal models, I'll mention is that this was carefully reformulated in [11,13] by saying that the sum of one of these characters and the conjugate of the other (ie, $z \to z^{-1}$) gives the character of a relaxed representation which turns out to be proportional to that of an irreducible module for the corresponding Virasoro minimal model.

In our $SU(2)$ example above, taking the highest weight representation $-1/4$ means that the conjugate of the $-3/4$ is $3/4$ (but no longer highest weight) which now differs from $-1/4$ by an integer. There is no cancellation in their sum of course, but the fact that their weights differ by $1$ is crucial for the existence of the relaxed representation. I suggest that there must be a similar interpretation for the stated cancellation in the even character case.

Besides this problem, I have a few more comments and corrections. Please let me know if I have misunderstood anything.

1. The end of the second paragraph suggests that CFTs with a finite number of primaries are rational. This is false and the original counterexample is the triplet model, see hep-th/9604026.

2. The word "representation" appears to be used frequently to mean "irreducible highest-weight representation" or something similar (but perhaps not always). Perhaps it could be clarified that this is the case unless otherwise noted?

3. Footnote 3 claims that "admissible" will be defined in the next section. But, I didn't see it defined there (or anywhere else). Is it the Kac-Wakimoto definition from their 1988 paper, so only applicable to irreducible highest-weight representations? Or does it mean _any_ representation of the vertex operator algebra underlying the CFT under consideration? Or something else?

4. The term "big category of admissible modules" after (1.1) is perplexing, especially as reference is made to [17] where the term is never used. Please be more precise here.

5. There appears to be a systematic misunderstanding of the term "integrable" in the paper. As far as I can tell, it is universally understood (in this context) to mean that a representation decomposes as a direct sum of finite dimensional $su(2)$ representations, for _any choice_ of $su(2)$ subalgebra. As such, $SU(2)_k$ has no (non-zero) integrable highest weight representations unless $k$ is a non-negative integer. However, the author mentions them in many places for fractional levels. Perhaps they are thinking of representations whose characters converge at $z=0$, ie. finite-dimensionality for _one special choice_ of $su(2)$ subalgebra?

6. In the second paragraph of Sec. 1.1, what does "admissibility of the even characters" mean? These objects don't correspond to representations in general. Do you mean both the representations whose characters are being subtracted are admissible?

7. After (2.4), the RCFT discussion suggests that we add the constraint of positivity of fusion coefficients if the S-matrix is unitary. But, the S-matrix of a RCFT is always unitary...

8. Immediately after, I didn't understand what was meant by "reduced S and T-modular matrices".

9. I also didn't understand "a unitary S-matrix implies a 1-1 correspondence of the unflavoured characters with the modules". What could unitarity possibly have to do with the linear independence of $q$-characters?

10. After (2.7), it is claimed that the spectrum is invariant under $j \to -1-j$. But, this is false as (2.9) notes.

11. The last sentence in Sec. 2.1 is baffling, especially as it references [17]. Shouldn't a $q$-expansion (about $q=0$) always give the degeneracies (multiplicities) of weights in a module? The subtle discussion in [17] concerns expansions in $\omega$ (or $z$) which is much trickier.

12. Is there a typo in (2.20)?

13. I guess (2.23) only holds for $k \ne 0$?

14. I also dislike the reference in the first paragraph of Sec. 2.3 to [17], seemingly in support of something "manifest" that I honestly find incomprehensible.

15. In Sec. 3.1, it may be worth noting that the term "threshold level" also has the name "boundary level" in the literature. I believe it was introduced by Kac-Roan-Wakimoto in their 2003 article, but it could have been earlier.

16. I'll add that it is somewhat dangerous to identify a unitary CFT from its modular data. There are many known examples where different RCFTs give the same representation of the modular group. I would not be surprised if there were also examples of inequivalent RCFTs whose $q$-characters matched!

17. In the conclusion, it is claimed that even characters are expanded in the region $|z|<1$ and $|q|<1$ and that the radius of convergence is the same as that for a RCFT. However, this can't be correct as there can be poles in $|z|<1$ depending on $\tau$, right?