Toda conformal blocks, quantum groups, and flat connections

TL;DR

This paper explores the deep connections between Toda conformal field theories, quantum groups, and the quantisation of flat connection moduli spaces, introducing new bases and operators for these complex structures.

Contribution

It defines natural bases for Toda conformal blocks using free field representations and expresses monodromy operators as Laurent polynomials of elementary operators, linking to higher rank Fenchel-Nielsen coordinates.

Findings

Monodromy matrix elements are Laurent polynomials of elementary operators.

Elementary operators are quantised higher rank Fenchel-Nielsen coordinates.

The approach provides a new framework for quantising moduli spaces of flat connections.

Abstract

This paper investigates the relations between the Toda conformal field theories, quantum group theory and the quantisation of moduli spaces of flat connections. We use the free field representation of the $\mathcal{W}$-algebras to define natural bases for spaces of conformal blocks of the Toda conformal field theory associated to the Lie algebra ${\mathfrak s}{\mathfrak l}_3$ on the three-punctured sphere with representations of generic type associated to the three punctures. The operator-valued monodromies of degenerate fields can be used to describe the quantisation of the moduli spaces of flat $\mathrm{SL}(3)$-connections. It is shown that the matrix elements of the monodromies can be expressed as Laurent polynomials of more elementary operators which have a simple definition in the free field representation. These operators are identified as quantised counterparts of natural higher rank analogs of the Fenchel-Nielsen coordinates from Teichm\"uller theory. Possible applications to the study of the non-Lagrangian SUSY field theories are briefly outlined.