Analytic Modular-Bootstrap Functionals

• Analytic modular-bootstrap functionals are rigorously constructed linear operators on 2d CFT spectral data that exploit modular invariance.
• They employ Beurling-Selberg extremization, finite Gauss-sum kernels, and Mordell integral techniques to yield sharp bounds on spectral densities and operator gaps.
• They provide closed-form analytic gap bounds and no-go theorems for pure AdS₃ gravity without relying on numerical semidefinite programming.

Analytic modular-bootstrap functionals are rigorously constructed linear functionals on the spectral data of two-dimensional unitary, modular-invariant conformal field theories (2d CFTs). They provide sharp, sometimes optimal, analytic bounds on spectral densities, operator gaps, and averaged OPE coefficients without recourse to semidefinite programming. Central to this methodology are explicitly constructed test functions that exploit the modular transformation properties and permit Beurling-Selberg-type extremization, Gaussian sums, and Mordell integral techniques, making use of deep analytic machinery from both number theory and function theory. Analytic modular-bootstrap functionals have led to new, closed-form, rigorous bounds on gaps and have resolved outstanding questions such as the impossible realization of pure AdS$_3$ gravity as a compact, unitary, Virasoro-only CFT$_2$.

1. Modular Bootstrap and Linear Functionals

The modular bootstrap investigates constraints on operator data imposed by modular invariance of the torus partition function $Z(\beta)$ and related observables. Mathematically, $Z(\beta)$ must obey $Z(\beta) = Z(4\pi^2/\beta)$, arising from the $S$ transformation $\tau\to -1/\tau$ on the modular parameter $\tau = i\beta/2\pi$.

Analytic modular-bootstrap functionals are linear operations $F[\,\cdot\,]$ designed to act on the spectrum via the modular crossing relation to either annihilate vacuum contributions, extract asymptotic data (via inverse Laplace transforms), or directly produce rigorous inequalities by virtue of spectrum positivity. Distinct analytic techniques emerge at extreme and intermediate temperatures:

• In the high/low-temperature regime, inverse Laplace transformations conjugate modular crossing to relate asymptotic spectral data to vacuum contributions, often recoverable via saddle point or residue methods.
• Around fixed modular points (e.g., $\beta=2\pi$), differential operators odd under modular duality yield functionals $F$ that extract gap information or enforce positivity constraints on OPE coefficients or spectral densities (Brehm et al., 2019).

2. Beurling-Selberg Extremization in the Modular Bootstrap

A prominent analytical tool in this context is Beurling-Selberg extremization, originally developed in analytic number theory for bounding indicator functions by majorants and minorants with prescribed Fourier support. In modular bootstrap applications, it yields optimal, band-limited test functions $\phi^\pm(x)$ majorizing and minorizing the indicator function ${\bf 1}_{[-\delta,\delta]}(x)$, constructed so that their Fourier transforms vanish outside a fixed interval.

These test functions lead to upper and lower bounds ($\alpha^+[Z], \alpha^-[Z]$) on the number of primaries in a window $[\Delta-\delta, \Delta+\delta]$: $$\alpha^{-}[Z]\ \leq\ \int_{\Delta-\delta}^{\Delta+\delta} \rho(\Delta')\,d\Delta'\ \leq\ \alpha^+[Z]$$ In the case $2\delta\in \mathbb{Z}$, explicit formulas for $\phi^\pm$ exist in terms of Dirichlet kernels and their Fourier transforms, yielding exact expressions for these bounds. For general $\delta$, extremality is achieved via de Branges-type function theory, and the interpolation conditions arising from Poisson or Littmann's theorem completely determine the extremal functionals (Mukhametzhanov et al., 2020).

The functionals are optimal amongst all band-limited (i.e., Fourier support $|t|\leq 2\pi$) linear functionals: $$\alpha^-[Z] = 2\pi \rho_0(\Delta) \widehat{\phi}_-(0), \quad \alpha^+[Z] = 2\pi \rho_0(\Delta) \widehat{\phi}_+(0)$$ with $\rho_0$ the Cardy prefactor. This construction leads to the sharpest known analytic window bounds, matching integer-spaced spectra in extremal cases.

3. Finite Gauss-Sum Kernels and Mordell Integrals

Recent developments exploit closed-form modular kernels expressed as Mordell integrals, which, on rational slices $\tau=n$ ($n\in\mathbb{Z}_{>0}$), reduce to finite quadratic Gauss sums with explicit Weil phases. With $h(\tau,z)$ the Mordell integral: $h(n,z) = \frac{1}{\sqrt{n}}\sum_{r=0}^{n-1} W_n(r) \sech\left(\frac{\pi}{\sqrt n}(z + i(r + \tfrac{1}{2}))\right), \quad W_n(r) = \exp\left(\frac{\pi i}{n} r(r+1)\right)$ For real spectral parameter $p$, this expansion translates $\sech$ to $\sec$ profiles.

The modular kernels $K^{ST^nS}(p)$ relevant for continuous partition function transformations in non-rational $c > 1$ theories are then explicitly decomposable as: $$K^{ST^nS}(p) = \sum_{r=0}^{n-1} B_{n,r} g_{n,r}(p) + C_n \Xi_n(p) + A_n \frac{2}{\cosh(\pi p)}$$ where $$g_{n,r}(p) = 2 \sec\left(\frac{\pi}{\sqrt n}(p + r + \tfrac{1}{2})\right)$$ and $\Xi_n(p)$ encodes metaplectic structure. This provides a canonical finite real basis for analytic spectral kernels in rational-width modular problems (Tierz, 29 Nov 2025).

4. Construction and Properties of Window Functionals

Given this explicit real basis, analytic window functionals $\Phi(p)$ are constructed as finite linear combinations: $$\Phi(p) = \sum \alpha_{n,r} g_{n,r}(p) + \sum \beta_n \Xi_n(p)$$ A positive window functional on $[0, P_{\max}]$ satisfies $\Phi(0)=1$ and $\Phi(p)>0$ for all $p\in[0, P_{\max}]$. Theorem 3.1 (Tierz, 29 Nov 2025) asserts the existence of such functionals for all $P_{\max} \in (0,2]$. In simple cases, a single $g_{n,r}$ column suffices if its poles fall outside $[0,2]$. General constructions utilize sums of basis elements, with positivity verifiable via explicit grid checks (Grid-to-interval Lemma 4.1).

No semidefinite programming is required; all positivity statements and functionals are constructively analytic. These functionals allow for rigorous, closed-form, analytic window bounds and gap theorems, sharpened beyond the reach of previous numerical approaches.

5. Analytic Gap Bounds and No-Go Theorems

Employing the $ST^1S$ kernel, a phase-matched real functional $\Phi_1(p)$ leads to the currently sharpest purely analytic scalar gap bound: $\Delta_1 \leq \frac{c-1}{12} + 0.2282370622$ where the precise additive term is obtained by finding the unique positive root of a transcendental equation for $p_\star$ such that $E(p_\star) = 0$, with $$E(p) = 2\cosh(\pi p) - 2\min\{1,4e^{-\pi p}\} - 2/\cosh(\pi p)$$ (Tierz, 29 Nov 2025).

An analytic no-go theorem for pure AdS$_3$ gravity arises by projecting the modular crossing at $\tau=\rho=e^{2\pi i/3}$ onto the odd-spin sector and utilizing the positivity of the "Mordell surplus" in the corresponding kernel: $\Delta_{\rm BTZ} = \frac{c-1}{12}$ No compact, unitary, Virasoro-only CFT$_2$ can avoid having a primary below this threshold, as the total evaluation of an odd-spin window functional yields a strictly positive contribution from the Mordell tail, impossible to cancel with vacuum and even-spin terms (Tierz, 29 Nov 2025).

6. Relation to Other Analytic Bootstrap Methods

The S-modular analytic bootstrap organizes functionals via inversion formulas, odd derivatives at fixed points, and the construction of S-invariant combinations. The Laplace inversion technique yields universal Cardy-like asymptotics, while the fixed-point method yields rigorous, sometimes optimal, gap bounds and constraints on OPE coefficients. The explicit analytic control—most sharply realized in the Gauss-sum/Mordell framework—circumvents both numerical limitations and loss of spectral resolution inherent in broader numerical approaches (Brehm et al., 2019).

Extremal functionals, both in the Beurling-Selberg (window) and finite basis (Gauss-sum) settings, define the points where the analytic bootstrap establishes its most stringent bounds. The major technical advance of recent work is the explicit realization of these extremal functionals in closed form, including precise positivity domains and spectral supports.

7. Summary Table: Core Analytic Structures

Structure	Formula / Description	Reference
Beurling-Selberg extremals $\phi^\pm$	Entire functions, Fourier support $|t|<2\pi$, interpolate indicator functions; bounds on $\int \rho(\Delta)\,d\Delta$	(Mukhametzhanov et al., 2020)
Gauss-sum basis $g_{n,r}(p),\,\Xi_n(p)$	Canonical real basis for $ST^nS$ continuous kernels at rational $\tau=n$, explicit pole structure	(Tierz, 29 Nov 2025)
Window functional $\Phi(p)$	Positive linear combination, support $[0,P_{\max}]$, normalized at $p=0$	(Tierz, 29 Nov 2025)
Odd modular differential functionals	$F[\cdot]=\lim_{\beta\to2\pi}(D_\beta)[\cdot]$, with $D_\beta=-D_{4\pi^2/\beta}$	(Brehm et al., 2019)
Analytic gap bound (scalar)	$\Delta_1 \leq (c-1)/12 + 0.228...$ via $ST^1S$ phase-matched window	(Tierz, 29 Nov 2025)
AdS$_3$ gravity no-go	Mordell surplus in odd-spin sector forces a state below $\Delta_{BTZ}$	(Tierz, 29 Nov 2025)

Each analytic structure above forms a crucial component in the derivation of exact, closed-form bounds and provides a platform for further conceptual and technical advances in CFT modular bootstrap.