nLab moduli space of flat connections

Idea

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.

Examples

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.

Properties

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

Related concepts

• geometric Langlands correspondence

moduli spaces

• moduli space of complex structures

• moduli space of conformal structures

• moduli space of curves

• moduli space of Riemannian metrics

• moduli stack of elliptic curves

• moduli space of Calabi-Yau spaces

• moduli stack of tori

• moduli stack of formal groups

• moduli space of connections, moduli space of flat connections

• Hitchin moduli space

• moduli space of monopoles

• moduli space of framed manifolds

• moduli space of bundles

• moduli space of (higher) line bundles

• quiver variety

References

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]