Finite Gauss-Sum Modular Kernels: Scalar Gap and a Pure AdS$_3$ Gravity No-Go Theorem

TL;DR

This paper derives explicit modular kernels for non-rational Virasoro CFTs, constructs analytic functionals for spectral bounds, and proves a no-go theorem for pure AdS$_3$ gravity based on spectral gap constraints.

Contribution

It provides closed-form expressions for modular kernels, constructs finite-support functionals for spectral bounds, and establishes a no-go theorem for pure AdS$_3$ gravity.

Findings

Derived explicit $ST^nS$ modular kernels for Virasoro CFTs.

Constructed analytic spectral functionals with finite support.

Proved no pure AdS$_3$ gravity CFT can have a primary gap above $rac{c-1}{12}$.

Abstract

We obtain closed-form expressions for the $ST^nS$ modular kernels of non-rational Virasoro CFTs and use them to construct fully analytic modular-bootstrap functionals. At rational width $\tau$, the Mordell integrals in these kernels reduce to finite quadratic Gauss sums of $\operatorname{sech}/\sec$ profiles with explicit Weil phases, furnishing a canonical finite-dimensional real basis for spectral kernels. From this basis we build finite-support "window" functionals with $\Phi(0)=1$ and $\Phi(p)>0$ on a prescribed low-momentum interval. Applied to the scalar channel of the $ST^1S$ kernel, these functionals yield a rigorous analytic bound on the spinless gap. As a second application we prove an analytic no-go theorem for pure AdS$_3$ gravity: no compact, unitary, Virasoro-only CFT$_2$ can have a primary gap above $\Delta_{\rm BTZ}=(c-1)/12$, because a strictly positive "Mordell surplus" in the odd-spin $ST$ kernel forces an odd-spin primary below $\Delta_{\rm BTZ}$.