Beurling-Selberg Extremization and Modular Bootstrap at High Energies

We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions $[\Delta - \delta,\Delta + \delta]$ at asymptotically large $\Delta$ in 2d unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval $[\Delta - \delta,\Delta + \delta]$ and have Fourier transforms of finite support. The optimization of the bounds over this choice turns out to be exactly the Beurling-Selberg extremization problem, widely known in analytic number theory. We review solutions of this problem and present the corresponding bounds on the number of operators for any $\delta \geq 0$. When $2\delta \in \mathbb Z_{\geq 0}$ the bounds are saturated by known partition functions with integer-spaced spectra. Similar results apply to operators of fixed spin and Virasoro primaries in $c>1$ theories.


Introduction
Crossing equations in Conformal Field Theories (CFT) form a system of infinite number of equations on an infinite number of unknowns: the spectrum and OPE coefficients. Arguably, the simplest way of extracting constraints on conformal data from crossing equations, is to take an (euclidean) OPE limit in one of the channels, call it t-channel. In this limit t-channel is typically dominated by the vacuum or another light state. Next, one finds a density of high-energy states or OPE coefficients in another channel, call it schannel, that reproduces the vacuum contribution in the t-channel in the OPE limit. This density is sometimes called a "crossing kernel". Many examples of this type for various crossing equations have been discussed in the literature [1][2][3][4][5][6][7][8][9][10][11][12][13].
This approximation, when we keep only the lightest state in the t-channel, is extremely useful for obtaining certain coarse-grained features of the high-energy spectrum. Prominently, thermodynamic behavior of high-energy states and aspects of classical geometry of a gravity dual [14] can be understood in this way.
On the other hand, many interesting fine-grained features of the high-energy spectrum are not captured in this approximation. One very basic and important example is discreteness of energy eigenstates. The coarse-grained approximation discussed above typically predicts a continuous spectrum. This is akin to a version of the information paradox in AdS [15] and related to the late-time behavior of correlators [16][17][18][19] and the spectral form-factor [20,21]. Another important example is the Random Matrix Theory (RMT) behavior of energy eigenstates. Just like coarse-grained thermality, RMT is expected to possess a certain degree of universality. One expects that it holds in a broad class of "chaotic" theories. A natural conjecture is that in any CFT without conserved currents except for the stress tensor, the energy eigenstates at asymptotically high energies obey RMT statistics.
In an attempt to derive universal fine-grained features of CFT spectra, such as RMT, one might imagine a two-step strategy. First, we need to understand why the coarsegrained approximation is not enough and quantify its limitations. Second, we would like to make extra assumptions (e.g. absence of conserved currents) that restrict us to chaotic theories. One might expect that these assumptions lift the limitations of the coarsegrained approximation and allow us to probe fine-grained features of the spectrum.
Partial progress on the step one has been made in [22][23][24][25][26][27][28]. The goal of the present work is to report on further progress in this direction. We do not have anything to say about the step two at this time.
In this paper we consider unitary 2d CFTs with a modular invariant partition function At high temperatures it has an asymptotic behavior From this one would like to derive the Cardy formula To do that one is tempted to take the inverse Laplace transform of the vacuum contribution (2) If we kept only the first term in parentheses, i.e. the vacuum contribution, we would indeed get Cardy growth (4) 1 . However, the vacuum contribution dominates only for |Im(β)| 1, while for |Im(β)| 1 we loose control over the integrand. In the latter region contributions of all operators in (6) become of O(1). Naively, one would like to argue that in the limit ∆ → ∞ only the region |Im(β)| 1/∆ gives a significant contribution. Such intuition is usually based on Riemann-Lebesgue lemma. In this case it is not applicable because the function Z( + it) is not integrable. 2 In a theory with discrete spectrum the density of states ρ(∆) is a sum of delta-functions and, therefore, we expect the integral (5) to be very sensitive to the precise value of ∆ even at large ∆: it's either zero or infinite. To compute this fine-grained density one would need to know much more than just the vacuum operator.
Of course, what is really meant by (4) is the coarse-grained density, i.e. the density of states averaged over a small window around ∆. One may consider different ways to average, for example The question then remains: how does one derive (7)? This was explained in [25], where a formula analogous to (5) was derived for the averaged density of states, but with an insertion of certain kernels φ ± (t). The role of φ ± (t) is to cutoff the troubling region t → ∞, where one looses control over the integrand. This effectively localizes the integral to the region of sufficiently small t, where the integral is controlled by the vacuum contribution. The price to pay for this modification is that instead of equality we get upper/lower bounds (hence "±" in φ ± (t)) on the averaged density of states. This will be reviewed is section 2.
It was shown in [25] that a simple sufficient choice of φ ± (t) is to require that they have finite support |t| < 2π. In this paper we find the optimal functions φ ± (t) of this type. The problem boils down to finding functions φ ± (x) that majorize/minorize a characteristic function of an interval θ [−δ,δ] (x) and minimize L 1 norm ||φ ± − θ [−δ,δ] || with a constraint that Fourier transformations φ ± (t) have finite support |t| < 2π. This turns out to be a classic Beurling-Selberg problem [29,30], widely known in analytic number theory.
In particular, this allows us to derive a simple bound where ρ 0 is defined in (18). These bounds hold for any δ ≥ 0. It turns out that they are optimal, among those obtained from bandlimited functions φ ± (x), only when 2δ ∈ Z.
In this case they are also saturated by c = 4k, k ∈ Z >0 partition functions with integer spaced spectra, for example, Klein's j-invariant. We derive these results for 2δ ∈ Z in section (3). In section 4 we derive (8) for any δ ≥ 0. The optimal bounds for 2δ / ∈ Z is derived in section 5.
Note the following two simple consequences of (8). The lower bound implies that: 1)In any window of size 2δ > 1 at asymptotically high energies there is non-zero number of operators.
This result was previously established in [25,26]. If we have Virasoro symmetry this is trivial due to descendants. 3 But the whole analysis can be repeated almost verbatim for Virasoro primaries (when c > 1) and the result remains the same.
The upper bound in the limit δ → 0 becomes ρ 0 (∆). Therefore: 2)The maximum degeneracy of an individual operator with dimension ∆ is ρ 0 (∆) up to additive corrections suppressed at asymptotically high energies. This is again saturated by partition functions with integer-spaced spectra for c = 4k, k ∈ Z >0 , as we will discuss in section 3.3.
In section 6 we generalize our results to counting operators of fixed spin J. Similar results also hold for counting Virasoro primaries when c > 1. The functions φ ± (x) considered in this paper can be applied in that case as well. The only thing that is different is the definition of ρ 0 . See [25] for details.
The main results of this paper are the upper and lower bounds (8), (17), (79), (80), and similar formulas for fixed spin operators (101), (106), (107), (108). These bounds are optimal among those that can be obtained using bandlimited functions φ ± (x). We do not know whether the bounds can be saturated for central charges c = 4k, k ∈ Z >0 or for 2δ / ∈ Z.

Review
In this section we review the setup of [25] and make a few additional comments. We start with defining two continuous integrable functions φ ± (∆ ) that bound the indicator function of an interval Here we have been careful with the ends of the interval. The function θ (∆−δ,∆+δ) vanishes at the ends, while θ [∆−δ,∆+δ] at the ends is 1. Since the functions φ ± are continuous, we can think of φ + as a bound from above where we include the edges, while φ − is a bound from below where we do not include the edges, see the figure 4 1. This will be important later, when we check optimality of our bounds. It will correspond to whether we include or not the states at the edges. Until then we will not distinguish the two θ's for the sake of brevity. Multiplying (9) by exponential factors 5 e ∆±δ−∆ , integrating with the density of states and taking the Fourier transform of φ ± , one finds the bounds on the number of states [25] e β(∆−δ−c/12) The integrals in (10) are reminiscent of the inverse Laplace transform (5) in the sense that we integrate the partition function over imaginary inverse temperatures. However, in comparison wtih (5), here we have much more control over the integrand. In particular, we can choose the functions φ ± (∆ ) in such a way that the region t 1, where we lose control over the integrand, does not contribute. This leads us to consider φ ± (∆ ) such that their Fourier φ ± (t) have finite support |t| < Λ. Functions of this type are called bandlimited functions.
We would like to bound the number of states in (10) at large ∆. In this case we imagine β to be small. Later we will see that to optimize the bounds β should be related to ∆ by the standard thermodynamic relation.
The quantity |Z(β + it)| 2 is called the spectral form-factor [20,21]. Its typical behavior is that it is large at t = 0 and decays exponentially at small times t due to oscillating phases. Early times are controlled by the vacuum in the dual channel [21]. For a large enough t = t rec phases can come into sync again and a recurrence happens, when the form-factor is again large. See figure 2. The recurrence time and the value of the formfactor at the recurrence time depend on the particular theory we study. Therefore, we expect that the integrand in (10) is controlled by the vacuum in the dual channel only for t t rec . This suggests that we should take Λ t rec . Figure 2: Spectral formfactor in 2d Ising (β = 1). It represents a typical behavior: it is large at early times and the initial decay is controlled by the vacuum in the S-dual channel. After a certain time a recurrence, generically only partial, happens. In chaotic theories the recurrence time is typically very long.
For partition functions with integer-spaced spectra, such as Klein's j-invariant, the recurrence time is t rec = 2π. In fact, as was shown in [25] and will be reviewed below, in any 2d CFT for Λ ≤ 2π the integrals in (10) are dominated by the vacuum in the S-dual channel. Therefore, t rec = 2π is the shortest recurrence time among modular invariant partition functions.
Using modular invariance and splitting the partition function into light (below c/12) and heavy (above c/12) operators, we find the upper bound in the limit ∆ → ∞, β → 0 (lower bound is similar) [25] ∆+δ Now we show that the first term here is dominated by t = 0, while the second term is dominated by t = Λ. We estimate heavy operators by dropping the phases In the last estimate we again used modular invariance and that the integral over t is dominated by t ∼ Λ ∼ 1, while β → 0. This is true because the contribution of an individual heavy operator ∆ ≥ c is continuous and has support |t| < Λ, we can estimate near Λ that The first integral in the RHS of (12) gets contributions from a finite number of light operators below c/12. It is dominated by the vacuum and gives Here the integral is dominated by the saddle t = 0 and the prefactor β 3/2 comes from integrating over fluctuations. Putting together the estimates we get from (12) ∆+δ Now it is clear that Λ = 2π is the biggest value for which heavy operators are suppressed as we take β → 0. From now on we will make this choice for Λ.
Minimization of the first term in (16) over β leads to the thermodynamic relation β = π c/3∆.
Lower bound is similar. Finally, we have bounds at asymptotically large ∆

Beurling-Selberg problem and Paley-Wiener theorem
At this point finding a bound on the density of states boils down to finding the functions φ ± (x) with the desired properties. Namely, we would like to solve the following problem.
Equivalently, we would like to minimize the area between φ ± and θ with the constraints that φ ± are bandlimited and bound θ from above/below. This type of problem was first considered by Beurling [29] and Selberg [30] and has various applications in analytic number theory [31,32] and signal processing [33].
The Beurling-Selberg problem formulated above was solved for 2δ ∈ Z in [30] and for 2δ / ∈ Z in [33,34] 6 Our task will be to simply use their results in the bounds (17). We describe the construction of [30] in section 3 and [34] in section 5. The following classic result will be very useful in solving the Beurling-Selberg problem. Theorem (Paley-Wiener): Suppose φ ∈ L 2 (R). Then φ can be extended to the complex plane as an entire function with |φ(z)| ≤ Be Λ|z| for some B > 0, if and only if Fourier φ(t) is supported on |t| < Λ.
An entire function φ bounded by |φ(z)| < Be Λ|z| in the complex plane is usually called a function of exponential type Λ. Therefore, Paley-Wiener theorem can be stated as equivalence between functions of exponential type Λ and functions whose Fourier transform has finite support [−Λ, Λ]. The practical convenience of Paley-Wiener theorem is that one can determine the support of φ(t) simply by looking at the growth of φ(z) and vice versa.
For the full proof we refer the reader to, for example, [35]. It's easy to see that finite support of φ leads to the boundedness of φ The proof in the other direction proceeds in two steps. First, using Phragmén-Lindelöf principle one shows that boundedness condition of the theorem actually implies a stronger bound |φ(x + iy)| ≤ Ce Λ|y| . Second, one can deform the contour parallel to the real line and estimate using the boundedness of φ Taking y → ∞ we get φ(t) = 0 for t > Λ. Similarly, shifting the contour to the lower half-plane gives φ(t) = 0 for t < −Λ. See [35] for details or [36] for a more pedagogical discussion.

Extremal functions for 2δ ∈ Z
In this section we assume 2δ ∈ Z. The construction of [30] proceeds in two steps. First, one derives bounds on φ ± (0) from Poisson summation formula. Second, one constructs functions that saturate these bounds, thus showing their optimality.

A bound from Poisson summation
One easy way to get estimates on the allowed φ ± (0) is to use the Poisson resummation formula Applying this to φ ± and taking into account that φ ± (t) have support |t| < 2π, only one term in the LHS survives 2π φ ± (0) = n∈Z φ ± (n + r) .
Since φ ± (x) bound θ [−δ,δ] (x) from above and below, we can estimate the zero mode as where we also optimized over r ∈ [0, 1) since φ ± (0) doesn't depend on it. The RHS in (23) is simply the maximum number of integer-spaced numbers n+r, that can be put into the interval [−δ, δ]. Similarly, the RHS of (24) is the minimum number of integer-spaced numbers n + r, that can be put into the interval (−δ, δ).
While the bounds (23), (24) are true for any δ, they are not always optimal. Later in this section we will construct functions φ ± that saturate (23), (24) when 2δ ∈ Z. On the other hand, if 2δ / ∈ Z the bounds (23), (24) are not optimal and functions φ ± saturating them do not exist. We will discuss optimal bounds for 2δ / ∈ Z in section 5. When 2δ ∈ Z the bounds (23), (24) are simply To obtain these bounds we must choose r in (23), (24) as follows. In the bound for φ + it is clear that to attain the maximum number of integer-spaced numbers n + r inside of the interval [−δ, δ], we need to put one of the numbers n + r at the edge of the interval, as should be clear from the figure 1. Therefore, if δ ∈ Z, then n + r must be integers and r = 0. If δ ∈ Z + 1 2 , then n + r must be half-integers and r = 1 2 . In the bound for φ − we need to attain the minimum number of integer-spaced numbers inside of (−δ, δ). The choice of r is the same as for φ + , i.e. we need to put one of the numbers n + r at the edge of the interval, see figure 1. To summarize, for 2δ ∈ Z we have two cases

Extremal functions
Now let's construct functions φ ± that saturate (25), (26). From the derivation of these bounds it's clear that saturation happens if where the last condition on the derivative comes from the fact that φ + (x) ≥ θ [−δ,δ] (x) and therefore φ + should be touching θ at the points n + r, see figure 1. Similarly, for φ − we have The functions φ ± (x) are essentially fixed by these properties. For δ ∈ Z we consdier One can think of these functions as follows. Let's discuss φ + . We start with sin 2 (πx), which vanishes at all integer x. These are the second order zeros that we want outside of [−δ, δ]. Then the role of the terms in parentheses in (34) is to cancel zeroes of sin 2 (πx) inside of the interval [−δ, δ]. The residues of the 2nd order poles in parentheses are fixed by φ + (n) = 1, |n| ≤ δ. The first order poles are allowed only at the ends x = ±δ. This is because we want x = n, |n| < δ to be minimums of φ + in order for φ + to bound θ from above. Finally, the residues of 1st order poles at x = ±δ must be related in order to have The constant term is not allowed for the same reason. Similar comments apply to φ − .
Note that the growth of (34), (35) in the complex plane is bounded by |φ ± (z)| ≤ Be 2π|z| . By Paley-Wiener theorem the Fourier transforms φ ± (t) have finite support |t| < 2π, just what we wanted our functions to satisfy.
Selberg's functions (34)- (38) indeed give the best possible bounds for 2δ ∈ Z that can be obtained from (17). However, we are not guaranteed that there is a fully S-invariant partition function Z(β) that saturates (41), since using bandlimited functions in (10) could be too crude in the first place. Here, we show that there is a zoo of S-invaraint partition functions for c = 4k, k ∈ Z >0 that saturate the bounds (41). Let us consider two nice examples of S-invariant partition functions at c = 4, 12. They are (τ = i β 2π ) given by where j(τ ) is Klein's j-function. The dimensions of operators in both partition functions are given by non-negative integers ∆ k = k ∈ Z ≥0 . The degeneracy d k of an operator ∆ k at large k is 7 where ρ 0 is defined in (18) and corrections are suppressed at k → ∞. Now it is easy to check that the bounds (41) are saturated. First, recall that the upper bound always bounds the number of operators in the window [∆ − δ, ∆ + δ] where we include the "edge states" at ∆ ± δ. While the lower bound holds for the number of operators in (∆ − δ, ∆ + δ) where we do not include the edge states. See the discussion below (9).
For the upper bound, we are counting the number of states in the interval [∆−δ, ∆+δ] of size 2δ ∈ Z. There can be at most 2δ + 1 integers in this interval, each corresponding to an operator with degeneracy ρ 0 (∆). Thus we have (2δ + 1)ρ 0 (∆) operators.
More generally, the bounds (10) would be saturated 8 if the functions φ ± take the following values on the physical spectrum ∆ ph 7 This follows from the classic results of Rademacher and Zuckerman [37][38][39]. One can also derive this using Ingham's theorem [40], which we explain in the appendix B. 8 We also lose precision when we multiply (9) by e β(∆−∆ ±δ) . However, this factor is unimportant at large ∆, where we take β = π c 3∆ → 0 and ∆ ≈ ∆. This is indeed the case for c = 4, 12 partition functions (42), (43) and Selberg's functions (34)- (38). The Selberg's functions vanish at the physical spectrum ∆ ph = k ∈ Z outside and are one inside of the corresponding interval.
With the understanding of (45) it is clear that any partition function with an integerspaced spectrum saturates the bounds (41), because Selberg's functions (34)-(38) satisfy (45) in this case. A large class of such partition functions is given by where P k (x) = x k + . . . is a monic (to ensure that the vacuum is unique) polynomial such that q-expansion coefficients of the partition function are non-negative. These partition functions correspond to the central charges c = 4a + 12k = 4, 8, 12, 16, . . . . The functions φ ± (∆) are reminiscent of the conformal bootstrap extremal functionals (e.g. [41][42][43][44][45][46][47]): outside of a certain interval they are non-negative/non-positive and vanish at the dimensions of physical operators.

A simple non-optimal bound for 2δ / ∈ Z
In this section we will show that the bounds (41) in fact hold for any δ ≥ 0. They turn out to be sub-optimal when 2δ / ∈ Z in the sense that they do not minimize/maximize φ ± (0) with the constraints described in section 2.1. We will discuss the optimal functions for 2δ / ∈ Z in the next section. However, the optimal functions in the general case are more complicated and we would like to start with a simpler sub-optimal bound.
First, note that it is not completely trivial to generalize (34), (35), (37), (38) to noninteger 2δ because of the sum in parenthesis. However, one can use the following trick due to Selberg [30]. Let's consider the upper bound. We would like to bound from above. Therefore, we can first take the function (34) and construct a function that bounds sgn(x) as Note that the only admissible choice (36) as δ → ∞ is λ ± = 1. Now we consider a new function φ + that bounds θ for any δ ≥ 0 Similarly, for the lower bound The functions B ± (x) were first considered by Beurling [29]. One can show that the functions (50), (53) give the same result (39), (40) for the zero mode φ ± (0) and also reduce to (34), (35), (37), (38) for 2δ ∈ Z. See appendix C for details. Thus, we have a bound for any δ Two comments are in order. First, the lower bound shows that in any window of size 2δ > 1 there is a non-zero number of operators, the result previously established in [25,26]. Second, as we take δ → 0 the upper bound becomes ρ 0 (∆). This implies that the maximum degeneracy of an individual operator is at most ρ 0 (∆) up to additive error terms suppressed at asymptotically large ∆. In fact, partition functions c = 4, 12, 16, . . . considered in section 3.3 saturate this bound on degeneracy, see section 3.3.

Extremal functions for 2δ / ∈ Z
When the size of the window is not integer 2δ / ∈ Z the functions (50), (53) we constructed previously turn out not to be optimal. The optimal functions in non-integer case were found by Littmann in [34]. In this section we describe his results and their implications for our bounds.

Generalized Poisson summation
In the integer case it was very useful, for both deriving functions φ ± and proving their optimality, to use Poisson summation formula (22) for functions with finite Fourier support. A generalization that is useful in non-integer case was found by Littman [34]. Suppose a continuous integrable function φ(x) has Fourier of support 9 |t| < 2π. Then where γ > 0, r ∈ R and x n , n ∈ Z are the roots B(x n ) = 0 of In particular, if we take γ → ∞ in (55), the roots become integer spaced x n = n +r and we recover the Poisson formula (22). But (55) is true for any γ > 0, r ∈ R. 10 To bound θ [−δ,δ] (x) we would like ±δ to be among the nodes x n , similarly to the integer case. This determines γ, r that we take. The analytic expressions for γ, r depend on the 9 Both (22) and (55) can be easily generalized to functions of support |t| < Λ by a rescaling of variables. 10 Littmann's formula (55) has a beautiful interpretation in terms of de Branges reproducing kernel Hilbert spaces of entire function [48]. Roughly speaking, if φ(z) = f (z)f (z), where φ is of exponential type 2π (i.e. Fourier of support 2π) and f is of exponential type π, then the LHS of the formula (55) can be interpreted as the norm of f in a certain Hilbert space. And the RHS is the same norm written as an expansion over an orthonormal basis in this Hilbert space. We prefer not to delve into details about this interpretation, as this would take us too far from our goals. We refer the interested reader to [34].
The function B(x) then takes the form Similarly to (23,24), applying (55) to φ ± we find These bounds can be saturated and we construct the corresponding functions φ ± next.

Extremal functions
The idea of the construction is analogous to section 3. We start with φ + . To saturate (61) we must have The last condition on the derivative comes from the fact that φ + (x) ≥ θ [−δ,δ] (x), so the nodes x n = ±δ must be local minimums. We take the following ansatz Note that by Paley-Wiener theorem the Fourier transform φ(t) has support |t| < 2π. The factor B(x) 2 ensures that (63, 65) are satisfied for |x n | > δ, i.e. we have 2nd order zeros outside of the interval. A simple calculation shows that the conditions (64, 65) for Now the only undetermined coefficients are b(±δ). We fix them by requiring that φ + is integrable The cancellation of order x, 1, 1 x terms in x → ∞ expansion leads to To solve these constraints, note that if x n is a root of B(x), then −x n is also a root, see The function φ − (x) is constructed in a similar manner. We require and we find The analysis at x n = ±δ is the same and therefore the coefficients a(x n ), b(x n ) are given by (67, 68).
Finally, one needs to check that φ ± (x) in (66, 77) indeed bound θ [−δ,δ] (x) from above and below. This was proved in [34]. 11 To summarize, we constructed optimal functions (66, 77) that saturate (61,62) where x n are roots of B(x) defined in (56) and γ, r are given by (58, 59). In general, the nodes x n are some transcendental numbers that we can't write down explicitly. However, the difference φ + (0) − φ − (0) gets a contribution only from x n = ±δ and takes a simple form This already demonstrates that (79, 80) give better bounds than (54), where the difference was 2.
When δ is sufficiently small, the only roots in the range |x n | ≤ δ are x n = 0, ±δ and the expressions (79, 80) take a simple form. Namely For larger δ one can find roots of (56) numerically. We plot (79,80) as functions of δ in the figure 3.
The dashed red lines are from the bounds (54), which are optimal only for 2δ ∈ Z. The blue and red lines touch when 2δ ∈ Z.

Fixed spin
Now we generalize the discussion of previous sections to operators of fixed spin J. We turn on an angular potential Ω and consider the grand canonical partition function The dimension ∆ and spin J are given by h +h and |h −h| ∈ Z respectively. The main technical difference with the previous discussion will be the double sum over h,h instead of a single sum over ∆. This will lead to a different splitting to light and heavy operators and to a different value Λ of the support of φ ± (t).
First, we project onto operators of fixed spin J Our goal is to derive bounds for the density of states ρ J (∆) at large ∆ and fixed spin J.
The analog of (10) is where f ± (x) are functions that bound θ [−δ,δ] (x) from above/below and have Fourier f ± (t) of support |t| < Λ. We changed notation for these functions to distinguish from the previous sections. To use (87) we would like to estimate the integrals To do that we go to the dual channel τ → −1/τ,τ → −1/τ and split into light and heavy operators where primes denote the dual channel (90), (91). Light (L) and heavy (H) operators are defined by 12 The convenience of this splitting is that there are only finite number of light operators and thus the first term in (92) will be dominated by the vacuum in the limit β → 0.
In the light sector the integral is dominated by t = 0 and Ω = 0. Using saddle-point approximation we have For the heavy part, dropping the phases, i.e. retaining only the first term in each of (90), (91), we can estimate First, we estimate as follows Here the expressions in the exponentials under the sums are always positive. Therefore, the integrals over t, Ω are dominated by the regions, where the effective inverse temperatures β ± = 4π 2 β β 2 +(Ω±t) 2 are the smallest. In particular, as we will verify below, there are such Ω, t, that β ± → 0. In this case we estimate (97) by the vacuum in the S-dual channel Using this estimate we have 12 Note that our definition of heavy/light operators is different from [49].
where we also used that f ± (t) ∼ Λ ∓ t is vanishing near the end of the support ±Λ. The integrals were computed by expanding near t = ±Λ, Ω = ±π.
Collecting the results we have, for example, for the upper bound For heavy operators to be suppressed in β → 0 limit, we set Λ = π − β 1/4 . Note, that for this Λ and Ω = π, t = Λ the effective inverse temperatures in (97) are small β ± → 0. Namely, So that the approximation (98) is justified. 13 Similar comments apply to the lower bound.
As before, we set β = π c/3∆. In the ∆ → ∞ limit, we obtain bounds where ρ 0 J (∆) is given by Now we turn our attention to finding optimal f ± (0). The results obtained in the previous sections are almost readily applicable upon proper scaling. We go through the generalized version of the salient equations appearing in § 3, 4, 5 below. The only difference is that we need functions with Fourier of support Λ = π − β 1/4 instead of 2π. This is achieved by considering where φ ± (x) are functions with Fourier of support 2π considered in § 3, 4, 5. The rescaling of δ is needed so that f ± (x) bound θ [−δ,δ] (x) from above/below.
• Beurling-Selberg § 3, 4 : Taking functions (50),(53) that satisfy (39), (40) and setting Λ = π in (104) we obtain Therefore, the bounds (101) become When 2δ · Λ 2π = δ ∈ Z, these bounds are optimal among those that can be obtained with bandlimited functions f ± . They are saturated if the spacing between operators of spin J is ∆ n+1 − ∆ n = 2. We can construct such examples from the partition functions considered in § 3.3. We consider the partition functions (46) as c L = 8k chiral partition functions. We tensor each of them with their anti-chiral copies to obtain a partition function with c L = c R = 8k. These have integer spaced spectrum h,h ∈ Z. For a fixed J = |h −h|, the spacing between operators ∆ = J + 2 min(h,h) is 2 and the asymptotic degeneracy is exactly given by ρ 0 J (∆), hence these partition functions saturate the bound. The factors δ ± 1 come from counting how many numbers spaced by 2 can be fit in the interval of length 2δ ∈ 2Z.
A few simple consequences of (106) are: 1. The lower bound implies that in any window of size 2δ > 2 there is a non-zero number of operators of given spin J. This is again not impressive if the theory has Virasoro symmetry. However, the analysis can be easily generalized to counting Virasoro primaries for c > 1 with the same result.
2. The upper bound, in the δ → 0 limit, implies that the maximum degeneracy of an individual operator with dimension ∆ and spin J is ρ 0 J (∆).
3. In any unitary 2d CFT operators of all spins J ∈ Z must be present. Again, this is trivial if Virasoro symmetry is present. However, the analysis can be easily generalized to counting Virasoro primaries for c > 1 with the same result.
• Littmann § 5: When δ / ∈ Z the bounds (106) are not optimal. The optimal bounds are obtained from the functions (66), (77). Taking the equations (79), (80) and setting Λ = π in (104), we get for the zero modes where x n are solutions of the equation andγ, r are defined by Appendix A: Majorization and Minorization by φ ± In this appendix, we show that φ ± (x) majorize and minorize the indicator function of the interval [−δ, δ] for 2δ ∈ Z + , given the inequality (36) is satisfied.
In order to treat δ ∈ Z and δ ∈ Z + 1 2 cases simultaneously, we define functions F ± (y) = φ ± (y − δ) We want to show that if the inequalities (36) are satisfied, we have We will be explicitly doing the analysis for F − (y) below. The analysis for F + (y) is similar. There are two regions of interest. For y ∈ (0, 2δ), we want F − (y) − 1 ≤ 0 and for y / ∈ (0, 2δ), we want F − (y) ≤ 0. We will see that requiring the former gives the upper bound on λ − , while requiring the latter provides us with the lower bound on λ − .
• y ∈ (0, 2δ): Using the identity Thus, we require The quantity in the brackets is minimized for y = δ, as we will show below. Therefore Finally, to show that the RHS of (114) is indeed minimized for y = δ, following [30], we argue as follows. We wish to show that for y ∈ (0, 2δ) We multiply the L.H.S by 2δ and write it as Now the idea is to put a lower bound on term I and an upper bound on term II, such that the lower bound on term I is still bigger than the upper bound on term II, resulting in L ≥ 0, which we want to prove.
Term I: Since the function g has positive second derivative, the trapezoidal rule always overestimates the integral below and we have g(n) + g(n + 1) 2 ≥ n+1 n dx g(x) , which, upon summing over n, gives us ∞ n=0 g(n) ≥ 1 2 Term II: We note that where we have used the Jensen's inequality g(k) ≤ k+1/2 k−1/2 dn g(n) as g is a positive, decreasing function of n with positive second derivative.

Appendix B: Degeneracy of states for integer spaced spectra
In this section we use Ingham's theorem [3,25,40] to derive an asymptotic formula for the degeneracy of states d k in CFTs with integer spectra ∆ k = k ∈ Z. We assume that degeneracies of operators are non-decreasing d k+1 ≥ d k and consider a modular invariant partition function Let us introduce an auxiliary function [40] F (β) = (1 − e −β )Z(β) = e The Ingham's theorem [3,25,40] implies that where ρ 0 is defined in eq. (18).
To apply (123) to partition functions considered in § 3.3 we need to check that d k+1 ≥ d k is satisfied. First, it's easy to check the following statement. Consider two series expansions, one with non-decreasing and one with non-negative degeneracies A(q) = ∞ n=0 a n q n , a n+1 ≥ a n ≥ 0 , Then the product AB gives rise to non-decreasing degeneracies c N Indeed, one simply compares term by term Now let's show that j(τ ) 1/3 has non-decreasing expansion coefficients. We recall that j(τ ) 1/3 = 1 + 240 n σ 3 (n)q n η(τ ) 8 , q = e 2πiτ .