Given a space $X$ and a group $G$, a $G$-flat connection or $G$-local system is a map $X \to \flat \mathbf{B}G$, or equivalently a map $\Pi(X) \to \mathbf{B}G$. A *moduli space of flat connections* is a moduli space/moduli stack of such structure.

The moduli space of flat connections for suitable Lie groups over Riemann surfaces appears as the phase space of $G$-Chern-Simons theory over these surfaces. It carries itself a projectively flat connection, the *Knizhnik-Zamolodchikov connection* or *Hitchin connection*.

The Narasimhan–Seshadri theorem asserts that the moduli space of flat connections on a Riemann surface is naturally a complex manifold.

For more see the references at

moduli space of connections.

- Jörg Teschner,
*Quantization of moduli spaces of flat connections and Liouville theory*, proceedings of the International Congress of Mathematics 2014 (arXiv:1405.0359)

Introducing Fock-Goncharov coordinates on moduli spaces of flat connections:

- Vladimir V. Fock, Alexander B. Goncharov,
*Moduli spaces of local systems and higher Teichmüller theory*, Publ. Math. Inst. Hautes Études Sci.**103**, 1-211 (2006) [arXiv:math/0311149, doi:10.1007/s10240-006-0039-4]

Last revised on April 14, 2024 at 05:16:08. See the history of this page for a list of all contributions to it.