In the geometric quantization approach to quantum mechanics, the Hilbert space of a $2n$-dimensional Kähler manifold is constructed inside $L^2(\mu)$, where $\mu=\omega\wedge\ldots\wedge\omega$ is the $n$-fold exterior product of the Kähler 2-form $\omega$. Which measure takes its place when the Kähler manifold is infinite-dimensional (so that the quantized problem is presumably a field theory)?

I'd like to think that for 2-dimensional QFT (or at least for 2-dimensional conformal field theory) there should be results in this direction.