Toward Shifted symplectic Chiralization: Deformation of Courant Algebroids

Abstract

We study the chiralization of Poisson--Lie theory through graded symplectic geometry. We compare Courant deformations and classical $r$- and $R$-matrix constructions, obtaining integrable hierarchies and free field realizations. We give a variational description of the obstruction to Lie conformal bialgebra compatibility and recover Liberati's obstruction in the Virasoro case. We also introduce a vertex bialgebra analogue of quantized universal enveloping algebras and construct explicit quantizations to all orders of Cartan type Lie conformal bialgebra structures on $\operatorname{Cur}(\mathfrak{sl}_2)$. These results contribute to the first author's broader programme of developing shifted symplectic chiralization as a geometric method for constructing chiral analogues of Poisson-Lie structures via $n$-shifted symplectic geometry, and exploring its connections with AKSZ theory and quantization.