Modular Group Action and Topological Order Degeneracy via Torus Moduli — E8 Intelligence Research

FINDING: The modular group SL(2,Z) acts on the moduli space of tori, and its representation theory encodes ground state degeneracy of 2+1D topological orders; the chiral central charge c− relates to this degeneracy via a complex vector bundle over the moduli space. | MATH: SL(2,Z) = { [[a,b],[c,d]] | a,b,c,d ∈ Z, ad−bc=1 }; torus moduli space M_τ = H/PSL(2,Z) (H = upper half-plane); ground state degeneracy D(Σ²) = |det(S)|² for anyon model with modular S-matrix; chiral central charge c− appears in the phase of the modular T-matrix: T = e^{−2πi c−/24} · diag(e^{2πi h_a}); the relation (arXiv:2004.11904) shows the degeneracy on Σ^d forms a vector bundle over M_{Σ^d} with flat connection (projectively) — the monodromy is exactly the SL(2,Z) representation. | CONNECTION: The modular group's generators S and T satisfy (ST)³ = S² = I (projectively), and the fixed points of SL(2,Z) on the upper half-plane are τ = i (order 4, S) and τ = e^{2πi/3} (order 6, ST) — these are the elliptic points w Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com