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

Toda conformal blocks, quantum groups, and flat connections