The Yang-Mills heat flow on the moduli space of framed bundles on a surface

We study the analog of the Yang-Mills heat flow on the moduli space of framed bundles on a cut surface. Existence and convergence of the heat flow give a stratification of Morse type invariant under the action of the loop group. We use the stratification to prove versions of Kähler quantization commutes with reduction and Kirwan surjectivity.