2.1 Setup

We consider a unitary, modular invariant 2D CFT with central charge \(c>1\), a (unique) normalizable vacuum and a positive twist gap \(\tau _}>0\) in the spectrum of Virasoro primaries. The torus partition function \(Z(\beta _L,\beta _R)\) of such a CFT is defined by

$$\begin Z (\beta _L,\beta _R) \equiv \text __}} \left( e^ \right) } e^_0 - \frac \right) } \right) . \end$$

(2.1)

where \(\beta _L\) and \(\beta _R\) are the inverse temperatures of the left and right movers, \(L_0\) and \(\bar_0\) are the standard Virasoro algebra generators, and \(\mathcal _}\) is the CFT Hilbert space which is assumed to be the direct sum of Virasoro representations characterized by conformal weights h and \(\bar\)

$$\begin \begin \mathcal _}=\bigoplus _} V_ \otimes V_}. \end \end$$

(2.2)

The twist gap assumption means that \(h,\bar\geqslant \tau _\textrm/2\) for all representations except the vacuum representation (\(h=\bar=0\)). Using Eqs. (2.1) and (2.2), the torus partition function can be written as a sum of Virasoro characters \(\chi _h(\beta _L)\chi (\beta _R)\) over primaries

$$\begin \begin Z(\beta _L,\beta _R)=\sum \limits _}n_}\ \chi _h(\beta _L)\chi _}(\beta _R), \end \end$$

(2.3)

where \(n_}\) counts the degeneracy of the Virasoro primaries with conformal weights h and \(\bar\). For \(c>1\), the characters of Virasoro unitary representations are given by

$$\begin \chi _h (\beta ) \equiv \text _ \left( e^ \right) } \right) = \frac \beta }} \times 1 - e^ &\text h = 0,\\ e^&\text h > 0, \end\right. } \end$$

(2.4)

where the Dedekind eta function \(\eta (\beta )\equiv e^\prod \limits _^(1-e^)\) accounts for the contribution of descendants. Then, we have

$$\begin Z (\beta _L, \beta _R) = \frac (\beta _L, \beta _R)} , \end$$

(2.5)

where the reduced partition function \(\tilde\) is given by

$$\begin \tilde (\beta _L, \beta _R) = e^\left[ (1 - e^) (1 - e^) + \sum _\geqslant T} n_}\,e^}\right] . \end$$

(2.6)

Here, we have denoted \(A \equiv \frac\) and \(T\equiv \tau _}/2\) for convenience. \(n_}\) is the degeneracy of the Virasoro primaries with conformal weights h and \(\bar\). The first term in the square bracket corresponds to the contribution from the vacuum state, while the second term represents the total contribution from Virasoro primaries with twists above the twist gap.

The above formulations assumed a discrete spectrum. The argument below also works for the continuum spectrum.Footnote 4A more uniform way to write Eq. (2.6), applicable to both the a) continuum and b) discrete spectrum is the following

$$\begin \begin\tilde (\beta _L, \beta _R)&= e^\left[ (1 - e^) (1 - e^)\right. \\&\quad \left. +\int _^\textrm h\int _^\textrm\bar\ \rho (h,\bar) e^}\right] . \end \end$$

(2.7)

Here, \(\rho \) is a nonnegative spectral density of Virasoro primaries. In the case of discrete spectrum, \(\rho \) is related to \(n_}\) by

$$\begin \begin \rho (h,\bar)=\sum \limits _'\geqslant T}n_'}\delta (h-h')\delta (\bar-\bar'). \end \end$$

(2.8)

We assume that (a) the partition function Z (or equivalently \(\tilde\)) for a given CFT is finite when \(\beta _L,\beta _R\in (0,\infty )\); (b) Z is modular invariant, i.e., \(Z(\beta _L,\beta _R)\) is invariant under the transformations generated by

$$\begin \begin (\beta _L,\beta _R)\rightarrow&(\beta _L+2\pi i,\beta _R-2\pi i), \\ (\beta _L,\beta _R)\rightarrow&\left( \frac,\frac\right) . \end \end$$

(2.9)

The invariance under the first transformation implies that the spin \(J:=\left|h-\bar\right|\) of any Virasoro primary state must be an integer. The invariance condition under the second transformation (which is called S modular transformation),

$$\begin Z (\beta _L, \beta _R) = Z_} \left( \frac, \frac \right) , \end$$

(2.10)

can be formulated in terms of reduced partition function \(\tilde\) as follows. By (a) and the positivity of the spectral density, the convergence domain of \(Z(\beta _L,\beta _R)\) (or equivalently \(\tilde(\beta _L,\beta _R)\)) can be extended to the complex domain of \((\beta _L,\beta _R)\) withFootnote 5

$$\begin \begin \textrm(\beta _L),\textrm(\beta _R)\in (0,\infty ). \end \end$$

(2.11)

Since under S modular transformation, \(\eta \) behaves as \(\eta (\beta )=\sqrt}\eta (\frac)\), Eqs. (2.5) and (2.10) imply that \(\tilde\) transforms as

$$\begin \tilde (\beta _L, \beta _R) = \sqrt} \tilde \left( \frac, \frac \right) . \end$$

(2.12)

Notice that the complex domain (2.11) is preserved by the S modular transformation. Therefore, we have two convergent expansions of \(\tilde(\beta _L,\beta _R)\) for \((\beta _L,\beta _R)\) in the domain (2.11):

Direct channel: expanding l.h.s. of (2.12) in terms of (2.7).

Dual channel: expanding r.h.s. of (2.12) in terms of (2.7) (with \(\beta _L,\beta _R\) replaced by \(\frac,\frac\)).

2.2 Review of the Twist Accumulation Point

Under the above setup, one can show that in the theory, there is at least one family of Virasoro primaries \(\mathcal _i\) with \(h_i\rightarrow A\) and \(\bar_i\rightarrow \infty \) [28, 29, 32, 33]. In other words, \((h=A,\bar=\infty )\) is an accumulation point in the spectrum of Virasoro primaries. The same is true with h and \(\bar\) interchanged. Here, let us briefly explain why it is true. For more technical details, see [28], section 3.

We consider the reduced partition function \(\tilde(\beta _L,\beta _R)\) for real and positive \((\beta _L,\beta _R)\). We take the double lightcone (DLC) limit, defined byFootnote 6

$$\begin \begin \text \quad \beta _L \rightarrow \infty , \quad \beta _R \rightarrow 0, \quad \mathfrak (\beta _L,\beta _R):=\frac -\beta _L-\frac\log (\beta _L)\rightarrow \infty . \end \end$$

(2.13)

By the \([\ldots ]\) factor in the r.h.s. of (2.6), the limit \(\beta _\rightarrow \infty \) favors the contribution from the lowest h (i.e., the vacuum term), while the limit \(\beta _\rightarrow 0\) favors the accumulative contribution from high \(\bar\). The important feature of the DLC limit (2.13) is that \(\beta _\) approaches 0 much faster than \(\beta _L\) approaches \(\infty \) (where the introduction of the logarithmic term in \(\mathfrak (\beta _L,\beta _R)\) is just for technical reason), so the high-\(\bar\) contribution wins. However, if we look at same limit from the dual-channel point of view, i.e., the expansion of the r.h.s. of (2.12) in terms of \(e^}\) and \(e^}\), the limit \(\frac}\rightarrow \infty \) is much faster than the limit \(\frac\rightarrow 0\), so the vacuum term wins in the dual channel. Based on this argument, one can show that in the DLC limit, the partition function \(\tilde(\beta _L,\beta _R)\) is dominated by the vacuum term (the first term in Eq. (2.6)) in the dual channel, i.e.,

$$\begin \underset}\ \frac (\beta _L,\beta _R)}\beta _R^} e^}} = 1. \end$$

(2.14)

Here, the denominator is the asymptotic behavior of the vacuum term in the dual channel:

$$\begin \begin \sqrt}\tilde_\left( \frac,\frac\right) \equiv&\sqrt}e^+\frac\right) }\left( 1-e^}\right) \left( 1-e^}\right) \\ \sim&\frac\beta _R^} e^}\quad (\beta _\rightarrow \infty ,\ \beta _\rightarrow 0). \\ \end \end$$

(2.15)

Now, we consider the direct channel (i.e., the l.h.s. of Eq. (2.12)) and ask which part of the spectrum in the direct channel contributes to the asymptotic behavior (2.14). Let \(\Omega \) denote a set of \((h, \bar)\) pairs, subject to the condition that \(\Omega \) excludes the vacuum state represented by the pair \((0,0)\). We define \(\tilde_\) to be the partial sum of Eq. (2.6) with \((h,\bar)\in \Omega \):

$$\begin \begin \tilde_(\beta _L,\beta _R):=\sum _)\in \Omega } n_}\,e^} \end \end$$

(2.16)

In what follows, we will only state the conditions of \(\Omega \), e.g., \(\tilde_\) is the same as \(\tilde_\) with \(\Omega =[A+\varepsilon ,\infty )\times (0,\infty )\).

The claim is that in the DLC limit, the direct channel is dominated by the sum over \(h\in (A-\varepsilon ,A+\varepsilon )\) and \(\bar\geqslant \bar_*\):

$$\begin \begin \underset} \frac_\geqslant \bar_*} \left( \beta _L, \beta _R \right) }\beta _R^} e^}} = 1. \end \end$$

(2.17)

Here, \(\varepsilon >0\) can be arbitrarily small and \(\bar_*\) can be arbitrarily large. (But they are fixed when we take the DLC limit.) To prove this claim [28], demonstrated that in the direct channel, the total contribution from other \((h,\bar)\) pairs is suppressed compared to the dual-channel vacuum term. The main idea of the proof involves decomposing the contributions into three distinct parts::

$$\begin & (1)\ h = \bar = 0\ (\text ),\quad (2)\ T \leqslant h \leqslant A - \varepsilon \ (\text ),\quad \\ & (3)\ h \geqslant A + \varepsilon \ (\text ). \end$$

In the DLC limit, a direct computation reveals that the contribution from part (1) is subleading in comparison with Eq. (2.15). Similarly, the contribution from part (3) is also subleading. This is primarily due to the fact that \( e^} \leqslant e^} \), which decays exponentially as \( \beta _ \rightarrow \infty \). The subleading nature of the contribution from part (2), however, is not immediately obvious. This is because \( e^} \geqslant e^} \), which grows exponentially as \( \beta _ \rightarrow \infty \). Moreover, as \( \beta _ \rightarrow 0 \), the contribution from part (2) becomes increasingly significant, accruing a greater high-\(\bar\) contribution. Nonetheless, it remains subleading due to the modular invariance condition (2.12). This condition effectively restricts the density of the high-\(\bar\) spectrum, thereby preventing the emergence of an \( e^}} \) behavior.

As a consequence of (2.17), \((h=A,\bar=\infty )\) must be an accumulation point in the spectrum. Otherwise one can find sufficiently small \(\varepsilon \) and sufficiently large \(\bar_*\) such that \(\tilde_\geqslant \bar_*}=0\), contradicting Eq. (2.17). By interchanging the roles of \(\beta _L\) and \(\beta _\) in the above argument, we can show that \((h=\infty ,\bar=A)\) is also an accumulation point in the spectrum.

In terms of scaling dimension \(\Delta =h+\bar\) and spin \(J=|h-\bar|\), the above argument implies that the theory must include a family of Virasoro primary operators \(\mathcal _\) with

$$\begin \begin \Delta ,J\rightarrow \infty ,\quad \Delta -J\rightarrow 2A\left( \equiv \frac\right) . \end \end$$

(2.18)

Given that \(\Delta - J\) is conventionally defined as the “twist” of the operator in CFT literature, we refer to the point where \(h = A\) and \(\bar = \infty \) as a “twist accumulation point.” For general CFTs, (2.18) is slightly weaker than the existence of both \((h\rightarrow \infty ,\bar\rightarrow A)\) and \((h\rightarrow A,\bar\rightarrow \infty )\) families. In a CFT with conserved parity, these two statements are the equivalent.

Before concluding this subsection, we would like to point out the roles of the properties of the coefficients \( n_} \) (or the spectral density \( \rho (h,\bar) \)). Throughout the above argument, the crucial requirement was the absolute convergence of Eq. (2.6) (or Eq. (2.7)) within the range \( \beta _, \beta _ \in (0, +\infty ) \). This absolute convergence is a consequence of the positivity of the coefficients and their convergence property in the same regime. However, it is noteworthy that our conclusion regarding the twist accumulation point remains valid even if the coefficients become negative or complex, provided that absolute convergence is maintained. Additionally, it is not necessary for the \( n_} \) values to be integers, this is a key factor in the applicability of our argument to the continuum spectrum. Furthermore, the above argument does not require that \( n_} \) or \( \rho (h,\bar) \) be supported only at points where \( h - \bar \in \mathbb \).

In summary, any modular invariant function \( \tilde(\beta _,\beta _) \) that possesses an absolutely convergent expansion as defined in (2.6) or (2.7) will exhibit a twist accumulation point (\(h = A,\ \bar = \infty \)).

However, for the specific results of this paper, the positivity of \( n_} \) and the constraint of integer-spin \( h - \bar \in \mathbb \) are significant in our analysis. It is important to note, though, that the \( n_} \) values are not necessarily required to be integers.

2.3 Main Theorem

In the previous subsection, we reviewed that in the double lightcone limit, the dominant contribution to the reduced partition function \(\tilde(\beta _L,\beta _R)\) comes from the spectrum with high spin and twist near 2A in the direct channel, while the vacuum state (\(h=\bar=0\)) dominates in the dual channel. This observation implies a connection between the spectral density \(\rho (h,\bar)\) (as given by Eq. (2.7)) near the accumulation point \((h=A,\bar=\infty )\) and the vacuum term of the partition function in the dual channel. In fact, by conducting a more thorough analysis of the arguments presented in [28], we can not only establish the existence of an infinite number of operators near the accumulation point \((h=A, \bar=\infty )\) but also estimate how many such operators there are. To achieve this quantitative understanding, we will employ Tauberian theory [25], building upon similar reasoning presented in [23, 50], and integrate it with the arguments put forth in [28]. This combined approach will be the focus of the remaining sections in this paper.

The object we are going to study is the total number of Virasoro primaries with h in the range \((A-\varepsilon , A+\varepsilon )\) and \(\bar=h+J\) where the spin J is fixed. This quantity is denoted as \(\mathcal _J(\varepsilon )\) and can be expressed as the sum of the degeneracies \(n_\) of Virasoro primaries over the specified range of h:

$$\begin \mathcal _J(\varepsilon ) := \sum \limits _ n_. \end$$

(2.19)

Our goal is to derive non-trivial asymptotic two-sided bounds on \(\mathcal _J(\varepsilon )\) in the limit \(J\rightarrow \infty \) and \(\varepsilon \rightarrow 0\), under specific constraints between \(\varepsilon \) and J. However, due to technical limitations, a direct estimate of \(\mathcal _J(\varepsilon )\) is not feasible.Footnote 7 To overcome this, we introduce another quantity \(\mathcal _J(\beta _L,\varepsilon )\) by assigning a \(\beta _\)-dependent weight to each degeneracy \(n_}\) of Virasoro primaries:

$$\begin \begin \mathcal _J(\beta _L,\varepsilon ):=\sum \limits _n_e^. \end \end$$

(2.20)

Importantly, \(\mathcal _J(\varepsilon )\) and \(\mathcal _J(\beta _L,\varepsilon )\) are related by the following inequality:

$$\begin \begin e^}\mathcal _J(\beta _,\varepsilon )\leqslant \mathcal _J(\varepsilon )\leqslant e^}\mathcal _J(\beta _,\varepsilon ). \end \end$$

(2.21)

This inequality provides an upper and lower bound for \(\mathcal _J(\varepsilon )\) in terms of \(\mathcal _J(\beta _L,\varepsilon )\), with a dependence on the parameter \(\varepsilon \) and the inverse temperature \(\beta _L\). So our approach involves two main steps. First, we will derive asymptotic two-sided bounds for \(\mathcal _J(\beta _L,\varepsilon )\). Then, we will use Eq. (2.21) to obtain corresponding bounds for \(\mathcal _J(\varepsilon )\).

To estimate \(\mathcal _J(\beta _L,\varepsilon )\), we introduce the DLC\(_w\) (double lightcone) limit defined as follows:

$$\begin \begin \textrm_w\ \textrm:&\quad \beta _L,\ J\rightarrow \infty ,\quad \frac\sqrt}-\beta _L\rightarrow \infty \,,\\ &\quad \beta _L^\log \rightarrow 0\,. \end \end$$

(2.22)

The reason we still refer to it as the “DLC” limit, similar to (2.13), will become clearer later. For now, a brief explanation is that by introducing the additional identification

$$\begin \beta _=2\pi \sqrt}, \end$$

(2.22) becomes a slightly stronger form of (2.13). We will revisit this point later around (2.41).

With the aforementioned setup, we present our main theorem as follows:

Theorem 2.1

Take any unitary, modular invariant 2D CFT with central charge \(c>1\) (i.e., \(A\equiv \frac>0\)), a unique normalizable vacuum and a twist gap \(\tau _}\equiv 2T>0\) in the spectrum of non-trivial Virasoro primaries.

Then for any \(w\in \left( \frac,1\right) \) fixed, and \(\varepsilon \) within the range

$$\begin \begin&\varepsilon _\textrm(\beta _,J)\leqslant \varepsilon \leqslant 1-\frac, \\ \varepsilon _}(\beta _,J)&:=\max \left\}\frac}},\ \frac}}+\frac}}\right\} , \end \end$$

(2.23)

the quantity \(\mathcal _J\), defined in (2.20), satisfies the following asymptotic two-sided bounds in the DLC\(_w\) limit (2.22):

$$\begin \begin \frac\frac}\lesssim \frac_J(\beta _L,\varepsilon )}\beta _L^J^e^}} \lesssim \frac\frac}, \end \end$$

(2.24)

which is uniform in \(\varepsilon \). Here and throughout this paper, by \(a\lesssim b\) we mean

$$\begin \begin \limsup \frac\leqslant 1 \end \end$$

(2.25)

in the considered limit.

Let’s make some remarks in Theorem 2.1:

Remark 2.2 (a)

In Eq. (2.24), both upper and lower bounds are strictly positive quantities for the assumed ranges of w and \(\varepsilon \). The upper bound is always greater than the lower bound. This is because the upper bound is consistently larger than \(\frac\), while the lower bound monotonically decreases with \(\varepsilon \) in the interval \(\varepsilon \in \left( 0,1-\frac\right) \). Notably, when \(\varepsilon =0\), the lower bound is less than or equal to \(\frac\). This observation provides a consistency check for the validity of the two-sided bounds.

(b)

The gap between the upper and lower bounds in Eq. (2.24) decreases as we increase w (i.e., when a stronger DLC\(_w\) limit is imposed) and decrease \(\varepsilon \). In the limit \(\varepsilon \rightarrow 0\) and \(w\rightarrow 1\), both the upper and lower bounds converge to 1.

(c)

In the DLC\(_w\) limit, the lower bound \(\varepsilon _\textrm(\beta _L,J)\) for \(\varepsilon \) approaches zero. We note that our choice of \(\varepsilon _}(\beta _L,J)\) is not optimal with respect to the method that we use, in the sense that the coefficients of the logarithms in (2.23) can be further improved. But we expect that the current form of \(\varepsilon _}\) already captures its essential behavior in the double lightcone limit, namely \(\varepsilon _\textrm=O(J^\log J)\).

Using Theorem 2.1, we can obtain an estimate for \(\mathcal _J(\varepsilon )\). Let us consider the following constraints, which are compatible with the DLC\(_w\) limit (2.22) (when \(J\rightarrow \infty \)):

$$\begin \begin \beta _=3\kappa ^ J^,\quad \varepsilon =\kappa J^\log \quad \left( \kappa ^<\frac}\ \textrm\right) . \end \end$$

(2.26)

Substituting these values into Eq. (2.21) and Theorem 2.1, and choosing, e.g., \(w^2=\frac\), we obtain the following result:

Corollary 2.3

Given any fixed \(\kappa \in \left( \frac},\infty \right) \), we have

$$\begin \begin \mathcal _J(\varepsilon \equiv \kappa J^\log J)=J^e^+f_(J)}, \end \end$$

(2.27)

where the error term \(f_(J)\) satisfies the bound

$$\begin \begin \left|f_(J)\right|\leqslant 3\log (J+1)+C(\kappa ), \end \end$$

(2.28)

with \(C(\kappa )\) being a finite constant.

Before going to the proof, we have three remarks.

Remark 2.4 (1)

Recall the twist accumulation point is given by \(\tau =2A\equiv \frac\), Corollary 2.3 tells us that at large spin J, the number of states that are very closed to the twist accumulation point grows exponentially as \(e^J}}\), with additional slow-growth factors that are bounded by powers of J. This implies that the average spacing between adjacent states in this regime is approximately given by \(e^J}}\). However, we cannot at present rule out the possibility of having all the states piling up near the end points of the interval. Therefore, the rigorous upper bound on spacing is given by the size of window, i.e., \(J^\log J\).

(2)

In Corollary 2.3, it is crucial to note that the lower bound of \(\kappa \) is proportional to \(T^\). This dependence clearly indicates that our analysis will not be valid if the theory does not have a twist gap. Also, our choice of the lower bound of \(\kappa \) here is not optimal. It is possible to improve it, e.g., by choosing other w or by further improving our analysis in the paper.

(3)

If we further assume that the theory has some critical spin \(J_*\), above which there are no Virasoro primaries with twist strictly below \(2A\equiv \frac\), then all the Virasoro primaries have h greater than or equal to A when \(\bar\geqslant h+J_*\). Consequently, considering the exponential term \(e^}\leqslant 1\), we find that the number of Virasoro primaries with h in the window \(\left[ A,A+\kappa J^\log J\right) \) cannot be smaller than \(\mathcal _J\). This leads to a more precise lower bound on \(\mathcal _J\), given by:

$$\begin \begin \mathcal _J(\varepsilon \equiv \kappa J^\log J)\geqslant \textrm (J+1)^e^}, \end \end$$

(2.29)

where the constant prefactor is strictly positive. Here, the power index \(-5/4\) is obtained by choosing \(\beta _\sim J^\) in (2.24).

The index of \(-5/4\) in \(\mathcal _J(\varepsilon )\) can be understood by considering the contribution from the vacuum character in the dual channel. This can be naively reproduced by only taking into account this part of the contribution. To see this, we rewrite the dual vacuum character in terms of the Laplace transform of the modular crossing kernel:

$$\begin \begin \sqrt}e^}\left( 1-e^}\right)&=\int _A^\infty \,\textrm h\,\sqrt}\left[ \cosh (4\pi \sqrt)\right. \\ &\quad \left. -\cosh \left( 4\pi \sqrt\right) \right] e^. \\ \end \end$$

(2.30)

Therefore, a naive computation of the “vacuum character” contribution to \(\mathcal _J(\varepsilon )\) is as follows:

$$\begin \begin \left[ \mathcal _J(\varepsilon )\right] _\textrm=&\int _A^d h\int _^\sqrt-A)}} \\&\times \left[ \cosh (4\pi \sqrt)-\cosh \left( 4\pi \sqrt\right) \right] \\&\times \left[ \cosh (4\pi \sqrt-A)})-\cosh \left( 4\pi \sqrt-A)}\right) \right] \\ \sim&\textrm\,\varepsilon ^J^e^}\quad (\varepsilon \ll 1,\ J\gg 1). \\ \end \end$$

(2.31)

By choosing \(\varepsilon =\kappa J^\log J\), we obtain the correct index of \(-5/4\) in (2.29). We expect that this is the optimal power index of J for the lower bound of \(\mathcal _J(\varepsilon \equiv \kappa J^\log J)\), in the sense that the index cannot be larger. One possible approach to verify the optimality is to examine explicit examples of torus partition functions, e.g., the one presented in [56].

2.4 Sketch of the Proof

To derive the two-sided asymptotic bounds (2.24) for \(\mathcal _J(\beta _L,\varepsilon )\) in the DLC\(_w\) limit, we introduce several tricks as follows.

Fig. 1

Illustration of the idea behind Eq. (2.32). The blue lines represent the allowed positions of the spectrum, constrained by \(h-\bar\in \mathbb \). We aim to count the spectrum around the pink line (\(h=A\)). We choose two windows (shown in red) with the same width in h but different widths in \(\bar\). Due to the integer-spin constraint, the spectrum inside the two windows is the same, as long as the windows intersect with only one of the blue lines

The first trick relies on the fact that only integer spins are allowed. (Here, we only consider bosonic CFTs.) This implies that the spectrum is empty for values of \(h-\bar\) that are non-integers. Using this property, we can express \(\mathcal _J\) in a different form as follows (see Fig. 1 for a clearer visual representation):

$$\begin \begin \mathcal _(\beta _L,\varepsilon )=\mathcal (\beta _L,\bar,\varepsilon ,\delta )\quad \forall \delta \in (\varepsilon ,1-\varepsilon ), \end \end$$

(2.32)

where

$$\begin \begin \bar\equiv A+J \end \end$$

(2.33)

and \(\mathcal (\beta _L,\bar,\varepsilon ,\delta )\) is defined as

$$\begin \begin \mathcal (\beta _L,\bar,\varepsilon ,\delta ):=&\int _^\textrm h\int _-\delta }^+\delta } \textrm\bar\,\rho (h,\bar)e^, \\ \end \end$$

(2.34)

where \(\rho (h,\bar)\) represents the spectral density of Virasoro primaries in the continuum-spectrum version of \(\tilde\) given by the integral in Eq. (2.7).

Now, the problem is reduced to obtaining the upper and lower bounds for \(\mathcal (\beta _L,\bar,\varepsilon ,\delta )\) in the DLC\(_w\) limit. To achieve this, we express the DLC\(_w\) limit (2.22) in terms of \(\beta _L\) and \(\bar\), taking into account that \(\bar=A+J\):

$$\begin \begin \textrm_w\ \textrm:\quad&\beta _L\rightarrow \infty ,\quad \bar\rightarrow \infty ,\quad \frac\sqrt-A}}-\beta _L\rightarrow \infty , \\&\beta _L^\log \bar\rightarrow 0, \\ \end \end$$

(2.35)

where w is the same parameter introduced in (2.22).

To proceed, we introduce the next trick which was used in [23, 50]. Let us consider two functions \(\phi _(x)\) satisfying the inequality

$$\begin \begin \phi _-(x)\leqslant \theta _\delta (x)\leqslant \phi _+(x),\quad \theta _\delta (x):=\,\theta (x\in [-\delta ,\delta ]). \end \end$$

(2.36)

In addition, for technical reasons, we require that \(\phi _\) are band-limited functions, meaning that their Fourier transforms \(\hat_\pm \) have compact support:

$$\begin \begin \phi _(x)=&\int \textrm t\ \hat_\pm (t)e^, \\ \textrm(\hat_\pm )\subset&[-\Lambda ,\Lambda ]\quad \mathrm \Lambda <2\pi w. \\ \end \end$$

(2.37)

The functions satisfying these conditions exist [57]. Later, for the specific range of w we are interested, we will give explicit expression for \(\phi _\), see (2.96) for \(\Lambda =2\pi \) and (B.1) for any \(\Lambda \).

Here again, w corresponds to the parameter in Eq. (2.35). The choice of \(\Lambda <2\pi w\) will be clarified at the end of Sect. 2.5.2. By substituting Eqs. (2.34) and (2.36) into the definition of \(\mathcal \), we obtain an upper bound for \(\mathcal \) given by:

$$\begin \begin \mathcal (\beta _L,\bar,\varepsilon ,\delta )\leqslant&\int _^\textrm h\int _0^\infty \textrm\bar\,\rho (h,\bar)e^+\delta -\bar)\beta _R}\theta _\delta (\bar-\bar) \\ \leqslant&\int _^\textrm h\int _0^\infty \textrm\bar\,\rho (h,\bar)e^+\delta -\bar)\beta _R}\phi _+(\bar-\bar) \\ =&e^+\delta )\beta _R}\int _^\textrm h\int _0^\infty \textrm\bar\,\rho (h,\bar)e^\beta _R}\int \textrm t \hat_+(t)e^-\bar)t} \\ =&e^+\delta -A)\beta _R}\int \textrm t\ \tilde_(\beta _L,\beta _R+i t)\hat_+(t)e^-A)t}. \\ \end \end$$

(2.38)

In the first line, we used \(e^+\delta -\bar)\beta _R}\geqslant 1\) in the support of \(\theta _\delta (\bar-\bar)\). In the second line, we bounded \(\theta _\delta \) by \(\phi _+\). In the third line, we rewrote \(\phi _+\) as the Fourier transform of \(\hat_+\). Finally, in the last line we used the definition of \(\tilde_\).

Similarly, we have the following lower bound for \(\mathcal \):

$$\begin \begin \mathcal (\beta _L,\bar,\varepsilon ,\delta )\geqslant e^-\delta -A)\beta _R}\int \textrm t\ \tilde_(\beta _L,\beta _R+i t)\hat_-(t)e^-A)t}, \end \end$$

(2.39)

where \(\hat_\) is the Fourier transform of \(\phi _\). It is worth noting that although the bounds depend on \(\beta _R\), the quantity \(\mathcal \) itself does not. The final result, given by Eq. (2.24), will be obtained by selecting an appropriate value for \(\beta _R\). Here, we choose \(\beta _\) to beFootnote 8

$$\begin \begin \beta _R=2\pi \sqrt-A}}. \end \end$$

(2.40)

With this choice, the limit (2.35) can be expressed as:

$$\begin \begin \text _w \text \quad&\beta _L \rightarrow \infty , \quad \beta _R \rightarrow 0, \quad \mathfrak _w(\beta _L,\beta _R):=\frac -\beta _L\rightarrow \infty , \\&\beta _L^\log \beta _\rightarrow 0. \\ \end \end$$

(2.41)

We observe that (2.41) is slightly stronger than (2.13): The inclusion of the \(w^2\) term in the third equation of (2.41) is sufficient to eliminate the logarithmic term \(\log (\beta _)\) present in the third equation of (2.13). The last equation in (2.41) is introduced for technical reasons.

From now on, we will always assume (2.33) and (2.40) by default. Consequently, the three formulations of the DLC\(_w\) limit, namely (2.22), (