$\newcommand{\g}{\mathfrak g}$ $\newcommand{\h}{\mathfrak h}$ In short my question is :
Has someone worked out the flat connection that one should get from the gauged WZW model in genus 0 ?
Some motivation : One of my favourite things in life is the KZ connection. This is a flat connection on the configuration space of points in the plane which can be defined from every pair of a Lie algebra $\g$ and a non degenerate element $t\in S^2(\g)^{\g}$. This construction is at the heart of the relation between low dimensional topology and deformation quantization: on the one hand its monodromy gives knots invariants which are specialization of the Kontsevich integral. This invariant can be understood as a certain ribbon structure on the category of $\g[[\hbar]]$-modules which in turn can be used to show the existence of a deformation-quantization of various interesting structures, e.g. if $\g$ is simple and $t$ is the inverse of the Killing form this recover the category of modules over the associated quantum group. More generally this is the main ingredient of the formality of the little 2-discs operad and the ancestor of the theory of shifted symplectic structure and their quantization.
The origin of this connection is as follow: for $\g$ simple there is a general construction of a certain conformal field theory related to the representation theory of the affinization of $\g$ at a given level $k$. This general formalism provide a canonical projectively flat connection on the moduli spaces of marked curves $M_{g,n}$. Knizhnik-Zamolodchikov have computed this connection in genus $g=0$ and observed that it comes in fact from an actual flat connection on the configuration space of point of which the moduli space $M_{0,n}$ is an appropriate quotient. Then taking a perturbative expansion as $k\rightarrow \infty$ you get something which depends really on $\g$ and not on its affinization. Then tou can realize that you can replace $\g$ by any Lie algebra equipped with an appropriate invariant pairing. I must confess that to me most of this is pure black magic !
All of this is one of the starting point of Drinfeld's theory of (quasi-) Lie bialgebra and (quasi-)-quantum envelopping algebras. Indeed, if $\h$ is a Lagrangian Lie subalgebra in $\g$ with respect to the pairing induced by $t$ satisfying some extra conditions then it turns $\h$ into a so-called Lie bialgebra and realize $\g$ as the so-called Drinfeld double of $\h$.
It turns out that the same kind of data (a Lie algebra equipped with a pairing and a Lagrangian subalgebra) can be used to define another conformal field theory known as the gauged WZW model. Needless to say that I've no doubt that Drinfeld was very well aware of that ! Indeed there is a lot of paper dealing with gauged WZW models which contains the word Lie bialgebra or Poisson Lie groups which is morally the same thing.
Yet, it seems to me that the general formalism of conformal field theories should also provide a flat connection on moduli spaces, maybe with some kind of boundary condition whatever this means exactly. It's temptating to think that its genus 0 version should be related to quantization of Lie bialgebras somehow and to various related topological objects. However I've been unable to find any reference dealing with this question, let alone compute it explicitely which I find surprizing considering the usefulness of the original KZ equation !
Hence I'd be happy if someone could either point to a reference or explain to me what is known and what is not on this problem.
edit To answer Pavel's question here is what was explained to me by José Figueroa-O'Farrill. You need to pick a subgroup $H \subset G \times G$ and the condition for it to be gauged is as follow: the corresponding Lie algebra embedding is characterized by two morphisms $l,r:\h \rightarrow \g$. Then the condition is $$\forall x,y \in \h, \langle l(x),l(y) \rangle = \langle r(x),r(y) \rangle$$ As Pavel point out if $\h$ is a Casimir subalgebra of $\g$ then the diagonal embedding does the trick.
Another way of realizing this condition is by taking a Lagrangian (or even just isotropic I guess) subalgebra $\h \subset \g$ and to define $l$ to be this embedding and $r$ to be simply 0.
