Towards a quantization of the double via the enhanced symplectic “category”

This paper considers the enhanced symplectic “category” for purposes of quantizing quasi-Hamiltonian (G)-spaces, where (G) is a compact simple Lie group. Our starting point well-acknowledged analogy between cotangent bundle $$T^*G$$ in Hamiltonian geometry and internally fused double $$D(G)=G\times G$$ geometry. Guillemin Sternberg consider former, studying half-densities phase functions on its so-called character Lagrangians $$\Lambda _{{\mathcal {O}}}\subseteq T^*G$$ . counterpart replaces these with universal centralizers {C}}}\longrightarrow {\mathcal {C}}$$ regular, $$\frac{1}{k}$$ -integral conjugacy classes $${\mathcal {C}}\subseteq We show each centralizer to be “quasi-Hamiltonian Lagrangian” D(G) come equipped half-density function. At same time, we Dehn twist-induced automorphism $$R:D(G)\longrightarrow D(G)$$ that lacks natural analogue. Each Lagrangian $$R(\Lambda {C}}})$$ shown have clean intersection every {C}}'}$$ , function own. leads us possibility well-behaved, notion BKS pairing construct such study properties. facilitated by nice geometric features {C}}})\cap \Lambda reformulation classical pairing. work perhaps first step towards level-k quantization via “category”.