Finite Gauss-Sum Modular Kernels: Scalar Gap and a Pure AdS$_3$ Gravity No-Go Theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915644825403392 |
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| author | Tierz, Miguel |
| author_facet | Tierz, Miguel |
| contents | 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 $Δ_{\rm BTZ}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_00361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite Gauss-Sum Modular Kernels: Scalar Gap and a Pure AdS$_3$ Gravity No-Go Theorem Tierz, Miguel High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics 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 $Δ_{\rm BTZ}$. |
| title | Finite Gauss-Sum Modular Kernels: Scalar Gap and a Pure AdS$_3$ Gravity No-Go Theorem |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2512.00361 |