Modular SL(2,Z) Action and Apéry Interpolation in Supergroup Chern-Simons — E8 Intelligence Research

FINDING: Quantum group SL(2,Z) action on Chern-Simons level quantization is implicit in the modular data of supergroup Chern-Simons theories, with critical L-values of modular forms (weight 4) interpolating Apéry-like sequences. | MATH: Chern-Simons level \(k\) quantizes via \(k + h^\) (shift by dual Coxeter number); SL(2,Z) acts on the Hilbert space of the theory via modular transformations \(τ ↦ (aτ+b)/(cτ+d)\), \(ad-bc=1\). The interpolated Apéry numbers \(A_n⁽ᵗ⁾\) satisfy \(A_n⁽ᵗ⁾ = ∑ₖ₌₀^n {n}{k}^2 {n+k}{k}^2 t^k\), and Zagier's evaluation gives \(A_n⁽ᵗ⁾ ∝ L(f, 2)\) for a weight-4 modular form \(f\). | CONNECTION: The critical L-value \(L(f,2)\) for weight 4 modular forms sits at the central point \(s = k/2 = 2\), echoing the self-dual point of the modular group. The level shift \(k+h^\) for supergroups (e.g., \(PSL(2|2)\)) yields rational levels \(k = p/q\), and the quantum dimension ratios at these levels produce algebraic n Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com