Coherent Quantization using Coloured Surfaces

TL;DR

This paper extends the framework of quantizing Lie bialgebras to moduli spaces of flat connections, demonstrating how quilted surface embeddings induce morphisms and applying this to quantize specific geometric varieties.

Contribution

It introduces a general approach to quantize moduli spaces via quilted surface embeddings, linking Lie bialgebra quantization with geometric structures.

Findings

Quantization of the variety of Lagrangian subalgebras.

Quantization of the de-Cocini Procesi wonderful compactification.

Compatibility with the action of quantized Poisson Lie groups.

Abstract

In this note, we revisit the quantization of Lie bialgebras described by the second author, placing it in the more general framework of the quantization of moduli spaces developed in our previous work. In particular, we show that embeddings of quilted surfaces (which are compatible with the choice of skeleton) induce morphisms between the corresponding quantized moduli spaces of flat connections. As an application, we describe quantizations of both the variety of Lagrangian subalgebras and the de-Cocini Procesi wonderful compactification, which are compatible with the action of the (quantized) Poisson Lie group.