Lie group valued moment maps for circle actions were introduced by Dusa McDuff here (§2 the generalized moment map), and then the paper "Lie group valued moment maps" extended this theory to construct the (tame) character varieties of punctured surfaces for nonabelian compact Lie groups, as multiplicative symplectic quotients (so that the monodromy relation becomes a moment map relation), and the quotient naturally gets a symplectic form.

The papers below transpose this theory to handle complex reductive groups, and construct many new examples, e.g. constructing all the wild character varieties of punctured surfaces, as multiplicative symplectic quotients (so the wild monodromy relation becomes a moment map relation). This rests on basic results about the full monodromy data of algebraic connections on smooth open curves, from the Stokes-Birkhoff-Riemann-Hilbert correspondence (the usual local system of analytic solutions is just the most visible piece of the full monodromy data, and is well-known to classify the "regular singular"/tame algebraic connections, as explained here, bottom of p.551).

A new feature is the breaking of structure group at the boundary "fission", and the theorem that the operation taking the formal monodromy is then a group valued moment map.

The prototypical wild monodromy relation appears as equation (2.46) in the 1981 paper of Jimbo-Miwa-Ueno here, closely related to work of Birkhoff. The hope of viewing this relation as a moment map relation was expressed in equation (16) and footnote 1 in the 2001 paper here (this paper also has more background; see also 1999 thesis p.71). The closest sheaf-theoretic way to encode this data is the notion of "Stokes local system", different but equivalent to the original Deligne-Malgrange approach, close to the Martinet-Ramis wild π₁ and Loday-Richaud's Stokes cocycle theorem.

This gives a rigorous algebraic way to construct the topological holomorphic symplectic forms, constructed analytically here and here (using a "straightening trick" to render convergent the Narasimhan/Atiyah-Bott approach, and so get many new holomorphic symplectic forms). It is a TQFT approach involving gluing pieces of surfaces, involving the new "fission" operations (breaking the structure group), complementary to the usual "fusion" product operation.


Quasi-Hamiltonian geometry of meromorphic connections
arXiv 2002, Duke Math. J. 139 (2007) no. 2, 369–405
[Generic connections, any pole order, any complex reductive group]

Through the analytic halo: Fission via irregular singularities
Ann. Inst. Fourier 59, 7 (2009) 2669–2684   (note)
[Simplest fully non-abelian fission spaces, any complex reductive group]

Riemann-Hilbert for tame complex parahoric connections
Transformation groups 16 (2011) no. 1, 27–50
[Tame analogue of fission spaces, any complex reductive group]

Geometry and braiding of Stokes data; Fission and wild character varieties
Annals of Math. 179 (2014) 301–365 (arXiv, pdf)
[All wild log connections on algebraic G-bundles, any complex reductive group]

Twisted wild character varieties (avec D. Yamakawa)
Arxiv 2015
[General setting, including all possible twists, any complex reductive group]

Further Applications:

Global Weyl groups and a new theory of multiplicative quiver varieties
Geometry & Topology 19 (2015) 3467–3536 (pdf)

Simplest examples:

Wild Character Varieties, points on the Riemann sphere and Calabi's examples
Representation Theory, Special Functions and Painlevé Equations — RIMS 2015
Advanced Studies in Pure Mathematics 76 (2018) 67–94 (pdf)


See also many of the slides/videos listed here.