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

Miguel Tierz

TL;DR

The paper develops an analytic, kernel-based approach to the modular bootstrap for non-rational Virasoro CFTs, making the ST^nS kernels explicit via Mordell integrals and a finite Gauss-sum basis. It constructs positive window functionals from this basis, yielding a rigorous analytic bound on the spinless scalar gap and a no-go theorem ruling out pure Virasoro AdS$_3$ gravity with a BTZ gap. A key mechanism is the Mordell surplus at the elliptic point, which cannot be canceled by any discrete spectrum, ensuring odd-spin primaries below the BTZ threshold. The framework connects elliptic-point modular bootstrap with Mordell–Gauss–Weil structures and has implications for ensemble holography and the role of stringy or higher-spin degrees of freedom in holographic theories.

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 $τ$, 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 $Φ(0)=1$ and $Φ(p)>0$ 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 $Δ_{\rm BTZ}=(c-1)/12$, because a strictly positive "Mordell surplus" in the odd-spin $ST$ kernel forces an odd-spin primary below $Δ_{\rm BTZ}$.

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

TL;DR

The paper develops an analytic, kernel-based approach to the modular bootstrap for non-rational Virasoro CFTs, making the ST^nS kernels explicit via Mordell integrals and a finite Gauss-sum basis. It constructs positive window functionals from this basis, yielding a rigorous analytic bound on the spinless scalar gap and a no-go theorem ruling out pure Virasoro AdS gravity with a BTZ gap. A key mechanism is the Mordell surplus at the elliptic point, which cannot be canceled by any discrete spectrum, ensuring odd-spin primaries below the BTZ threshold. The framework connects elliptic-point modular bootstrap with Mordell–Gauss–Weil structures and has implications for ensemble holography and the role of stringy or higher-spin degrees of freedom in holographic theories.

Abstract

We obtain closed-form expressions for the modular kernels of non-rational Virasoro CFTs and use them to construct fully analytic modular-bootstrap functionals. At rational width , the Mordell integrals in these kernels reduce to finite quadratic Gauss sums of 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 and on a prescribed low-momentum interval. Applied to the scalar channel of the 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 gravity: no compact, unitary, Virasoro-only CFT can have a primary gap above , because a strictly positive "Mordell surplus" in the odd-spin kernel forces an odd-spin primary below .

Paper Structure

This paper contains 43 sections, 18 theorems, 153 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Relation to elliptic-point modular bootstrap.
  3. Preliminaries
  4. Modular kernels
  5. Spinless parametrization.
  6. Cusp expansion of Mordell integrals
  7. Arithmetic Gauss sums
  8. Pole structure
  9. Finite Gauss--sum basis
  10. Main theorems
  11. Window functionals
  12. Positive functional viewpoint.
  13. Phase-matched Mordell bound at $\tau=1$
  14. Scalar gap
  15. No–go for pure Virasoro AdS$_3$ gravity
  16. ...and 28 more sections

Key Result

Lemma 2.1

For $n\in\mathbb{N}$, where $W_n(r)=\exp[\pi i r(r+1)/n]$ are the quadratic Weil phases. From the perspective of the Weil representation Weil1964, they implement the action of $SL(2,\mathbb{Z})$ on the space of half–integral weight theta functions at width $n$. In particular, the finite sums $\sum_r W_n(r)\,\mathop{\math

Theorems & Definitions (41)

  • Lemma 2.1: Finite cusp expansion at width $n$
  • Lemma 2.2: Quadratic Gauss sum
  • proof
  • Lemma 2.3: Poles and residues
  • Proposition 2.4: Finite Gauss--sum basis
  • proof
  • Theorem 3.1: Existence of positive window functionals
  • Remark 3.2: Explicit examples
  • Definition 3.3: Positive above a threshold and vacuum--negative
  • Lemma 3.4: Phase-matched Mordell bound at $\tau=1$
  • ...and 31 more