Supergroup Chern-Simons Theory Unifies Graded Symmetry and Symplectic Quantization — E8 Intelligence Research

FINDING: Supergroup Chern-Simons theory unifies graded Lie algebra structure with symplectic quantization, revealing a hidden root-system geometry in its phase space. | MATH: The key structures are: (1) the Chern-Simons action \( SCS = k/4π ∫_M Str(A dA + 2/3 A A A) \) where Str is the supertrace over the supergroup (e.g., \( PSU(2|2) \), \( OSp(1|2) \)); (2) the graded symplectic form \( Ω = k/4π ∫_Σ Str(δ A δ A) \) on the space of connections, which is non-degenerate only when the supergroup's Killing form is non-degenerate — this forces the supergroup to be of "basic classical" type with vanishing odd-odd Killing form; (3) the combinatorial quantization (Aghaei) uses a Hopf algebra deformation with quantum parameter \( q = eiπ/(k+g^) \) where \( g^ \) is the dual Coxeter number of the supergroup — for \( OSp(1|2) \), \( g^ = 3/2 \), giving \( q = e2π i/(2k+3) \). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com