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

by Sachin Grover

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Authors (as registered SciPost users): Sachin Grover
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-01-24 13:05
Submitted by: Grover, Sachin
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We study the even characters of $\widehat{su(2)}$ conformal field theories (CFTs) at admissible fractional levels obtained from the difference of the highest weight characters in the unflavoured limit. We show that admissible even character vectors arise only in three special classes of admissible fractional levels which include the threshold levels, the positive half-odd integer levels, and the isolated level at -$5/4$. Among them, we show that the even characters of the half-odd integer levels map to the difference of characters of $\widehat{su(2)}_{4N+4}$, with $N\in\mathbb{Z}_{>0}$, although we prove that they do not correspond to rational CFTs. The isolated level characters maps to characters of two subsectors with $\widehat{so(5)}_1$ and $\widehat{su(2)}_1$ current algebras. Furthermore, for the $\widehat{su(2)}_1$ subsector of the isolated level, we introduce discrete flavour fugacities. The threshold levels saturate the admissibility bound and their even characters have previously been shown to be proportional to the unflavoured characters of integrable representations in $\widehat{su(2)}_{4N}$ CFTs, where $N\in\mathbb{Z}_{> 0}$ and we reaffirm this result. Except at the three classes of fractional levels, we find special inadmissible characters called quasi-characters which are nice vector valued modular functions but with $q$-series coefficients violating positivity but not integrality.

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Anonymous Report 1 on 2023-3-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2208.09037v2, delivered 2023-03-30, doi: 10.21468/SciPost.Report.6970


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?

  • validity: ok
  • significance: ok
  • originality: good
  • clarity: ok
  • formatting: excellent
  • grammar: good

Author:  Sachin Grover  on 2023-06-06  [id 3711]

(in reply to Report 1 on 2023-03-30)

We thank the referee for raising important points in their report. As per the Editor's guidance, we are awaiting the second referee report before revising the manuscript. In the meantime, we want to address the questions raised in the first report.
We have attached our response, and the proposed corrections are highlighted in colour. We would incorporate the feedback in the revised manuscript.

Please let us know if there are further comments or suggestions. We appreciate it. Thanks.



Author:  Sachin Grover  on 2023-04-11  [id 3578]

(in reply to Report 1 on 2023-03-30)

We thank the referee for the detailed report and for raising important points. In the attachment to this post, we clarify the notion of even characters by demonstrating the cancellation of poles in even characters, first for $\widehat{su(2)}_{-1/2}$ and then at a generic admissible level. Through the process, we clarify why the unflavoured $q$-character should be multiplied by an integer to ensure that their coefficients are integers. We also mention the changes we plan to implement in the manuscript to make the definition of even characters more precise. Let us know if these are okay.

We appreciate the other comments made by the referee and will address them when we submit a formal list of changes. Thanks again.



Anonymous on 2023-04-12  [id 3582]

(in reply to Sachin Grover on 2023-04-11 [id 3578])
answer to question

Thanks for the discussion of even characters and fractional coefficients --- it helps greatly and I do think that adding details like these for the $k=-\frac{1}{2}$ example would be very helpful for the readership.
My confusion arose because there are two separate issues here which are intertwined in the original text. First, there is the identification of the correct convergence regions for the characters. Second, there is the regularisation prescription that results in the even "character" formula with fractional coefficients.
The first is still not quite fixed, in my opinion, but is conceptually not so serious. In the note recently attached, the character formulae are expanded assuming that $|\omega|<1$. However, that is not the correct convergence region as is evidenced by Eqs. (1) and (3) which feature negative coefficients and the wrong powers of $\omega$. The correct region is $1<|\omega|<|q|^{-1}$. The series worked out in (1) and (3) are in fact the characters of the conjugate modules of the irreducible highest weight module of spins $-\frac{1}{4}$ and $-\frac{3}{4}$, respectively.
But, this is not conceptually serious as one may replace all the expansions in $\omega$ by their counterparts in $\omega^{-1}$. Correct formulae now follow.
The second is where the analysis leaves the world of characters behind. Your clear illustration of the process shows what is going on: the subtraction of formal power series is effectively $\frac{1}{1-\omega^{-1}} - \frac{\omega^{-1/2}}{1-\omega^{-1}} = \frac{1}{1+\omega^{-1/2}}$, whence evaluation at $\omega=1$ gives $\frac{1}{2}$. This is of course equivalent to a regularisation of the divergent series $1-1^{-1/2}+1^{-1}-1^{-3/2}+1^{-2}-\cdots = \frac{1}{2}$.
I would appreciate it if a characterisation similar to this, suitably generalised to cover the other admissible levels perhaps, could also be added to the text. My guess is that other readers would appreciate this kind of remark.
Finally, I'd just like to add that I don't think that there is any sense in giving a region of convergence for an even character, except for $|q|<1$. Also, there might be an interpretation of these even "characters" in terms of admissible level parafermion characters which are studied in 1704.05168. However, I haven't tried to make this precise and could be mistaken.
I look forward to the updated manuscript.

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