Beurling-Selberg Extremization and Modular Bootstrap at High Energies

1- establishes a useful connection between contemporary physics and classic mathematics literatures

Cardy's formula for the asymptotic growth of states in a 2d CFT is a classic result, but the first detailed rigorous discussion was given in 2019 in [25]. This discussion is important as it shows to what extent Cardy's formula is supposed to be a general result, rather than a generic expectation. Ref [25] introduced a method to bound CFT spectral density integrated over a finite interval. The method needed certain functions which bound the characteristic function of the interval above and below, and are bandlimited (have Fourier transform supported in $[-2\pi,2\pi]$). In [25] some reasonable such functions were used, giving rise to bounds with non-optimal O(1) constants. In the paper under review, the authors describe how to choose best-possible such bandlimited functions giving rise to best-possible O(1) constants. It turns out that the problem of finding such functions is the classic Beurling-Selberg problem, and the authors borrow its ingenious solution from mathematics literature. I recommend the paper for publication, however I have several revision requests.

1- I found formulation of the main results of the paper somewhat vague. What does it mean that (8) is true for $\Delta\to\infty$? Does it mean that if I relax the constants in the r.h.s. and l.h.s. by an $\epsilon>0$ the bounds will be true for all $\Delta≥ \Delta_0(\epsilon)$? Do the authors's methods allow to determine how $\Delta_0$ depend on $\epsilon$, and if not why not?

2- It would be nice if the authors reduced the dependence of this work on [25]. As it stands, the reader not only should be familiar with the reasoning of [25], but also with the notation (e.g. $Z_H$ is not explicitly defined here). Furthermore, the reasoning seems somewhat different, e.g. the HKS bound does not seem to be mentioned in the paper under review; is it needed or used here, perhaps implicitly? If not why not? Several times the authors say: "See [25]", and a more pointed reference would be welcome.

3- At several places (p.4, p.20), the authors state that their arguments apply also to Virasoro primaries, with the same results. I think it would be nice to at least state those results (even if they have a very similar formulation, as this will facilitate future referencing), and provide a short explanation (at least in words if not in equations) what are the changes needed and why they don't invalidate the arguments. As it stands, the readers not perfectly familiar with [25] may be left incredulous.

More minor remarks

4- p. 3 looses -> loses

5 - p.17, Eq. (89), second equation, should $\tau$ be $\bar\tau$?

1. Places rigorous bounds on interesting quantities in CFT
2. Written clearly and concisely, and defines the problem carefully
3. Formalizes various intuitive expectations people had about CFT
4. Makes interesting connections to previous results in the math literature

1. Mostly is a result on rational CFT. It is expected (but not proved) that for irrational CFTs the bounds can be vastly improved
2. Mostly a followup of previous work in the literature (including previous work by both authors)

This paper analyzes universal properties of 2d CFTs by looking at the modular bootstrap. It shows that any compact, unitary 2d CFT must have operators with scaling dimension $\Delta$ separated by at most $1$ in the limit of large energy. Although this result by itself is trivial (since the Virasoro descendants provide examples of such operators), the authors claim a simple generalization of their argument applies to Virasoro primaries if $c>1$. This bound of integer-separated operators can be saturated by theories that holomorphically factorize, i.e. RCFTs with one character. An example of a modular-invariant that saturates this bound is Klein's $j$-invariant (although there are CFTs that realize this partition function, the arguments in this paper are insensitive to that). They also describe a simple generalization and find a similar result for operators at fixed spin, namely that at any fixed spin the operators are at most integer separated at asymptotically large energies. The authors are also able to show that at asymptotically large energies, the number of operators of energy $\Delta$ must be at most a certain explicit quantity, and this bound is again saturated by holomorphically-factorized partition functions.

The main technique this paper uses is by bounding a Heaviside theta function of width $2\delta$ centered at $\Delta$ by two smooth functions with Fourier transforms of finite support. By bounding the expressions for the density of states integrated against the lower and upper smooth functions, the authors are able to place rigorous upper and lower bounds on the number of operators in a certain window of size $2\delta$ centered on $\Delta$. It turns out that this problem of bounding such a theta function was previously explored by mathematicians and is known as the Beurling-Selberg extremization problem. By applying this technique to the CFT problem, the authors are able to provide stronger bounds on the number of operators at asymptotically large energy for any fixed width $\delta$. For example if the width $2\delta$ is less than $1$, then there is no lower bound of operators, which is reflected in the fact that there exist modular-invariant functions with support only at integer values of energy. In the limit as $\delta\rightarrow 0$, the authors find a nontrivial upper bound, which provides an interesting universal upper bound on the density of states at a particular energy $\Delta$.

This paper is clearly written and provides interesting results in the study of rational conformal field theory, and interesting connections to optimization problems in analytic number theory. I recommend that it should be published, with very minor modifications below.

1. Very minor typos. On p. 3, "loose" $\rightarrow$ "lose" (twice)
On p. 10, "consdier" $\rightarrow$ "consider"
On p. 24, "degeneraices" $\rightarrow$ "degeneracies"
2. When discussing the saturation by partition functions of $c=4k$ (e.g. in the Introduction and in Sec $3.3$), it seems to me that there are not necessarily $SL(2, \mathbb Z)$ invariant partition functions that saturate these bounds if $c = 4 ~\text{mod}~ 8$. Rather it seems that there are $S$-invariant functions that saturate these bounds (that do not necessarily consistently "split" amongst spins into an $SL(2, \mathbb Z)$ invariant function). Is this correct? If so, then since for the majority of the paper the authors only use $S$-invariance, it would be good to clarify this point.
3. In (7) should we expect $\delta$ dependence on the RHS, i.e. should the RHS be something like $2\delta e^{2\pi \sqrt{\frac{c\Delta}3}}$?
4. Most of the paper is about the case of states rather than Virasoro primaries, where the result is technically trivial. The authors write that a minor modification of the argument will go through for the case of Virasoro primaries. If it is not too difficult it may be useful to expand on this point in more detail. It seems to me that two qualitative features may change when one considers Virasoro primaries if $c>1$: (1) The modular crossing equation will pick up a "weight" coming from the Virasoro character transforming as a modular form of weight $-\frac12$. (2) There are null states in representations of the Virasoro vacuum character which will effect vacuum states and chiral operators. Although intuitively one might expect that these two changes will not affect the main result, it would be good to explain these more rigorously if possible, since the main result of this paper is a careful rigorous derivation of expected-results in CFT.