A moduli space of connections on bundles over some prescribed space.
Often one considers flat connections only, see at moduli space of flat connections.
If $\Sigma$ is a compact smooth manifold, then the moduli space of flat connections over $\Sigma$ is compact.
Over a complex manifold/complex variety, the Koszul-Malgrange theorem identifies holomorphic flat connections on complex vector bundles with holomorphic vector bundles. This identifies the moduli space of flat connections as a complex manifold with (a non-abelian version of) the first Griffiths intermediate Jacobian. See at intermediate Jacobian – Examples – k=0.
More specifically over a Riemann surface Narasimhan–Seshadri theorem identifies the moduli spaces of flat connections with that of certain complex spaces of stable holomorphic vector bundles. This space appears as the phase space for Chern-Simons theory over that surface. See there for more.
More generally, the Donaldson-Uhlenbeck-Yau theorem similarly gives a Kähler structure on the moduli space of flat connections also over higher dimensional Kähler manifolds (Scheinost-Schottenloher 96, corollary 1.16).
Let $G$ be a compact Lie group. Assume either that $G$ is simply connected or is a torus (what we really need below is that any two commuting elements in $G$ sit jointly in one maximal torus).
The moduli space of $G$ flat connections on a 2-dimensional torus $A \simeq S^1 \times S^1$ (e.g. underlying a complex elliptic curve) has the following description:
first, the moduli stack of flat connections is
(see also the discussion at characters and fundamental groups of tori).
Here a single flat connection is just a choice of pair of two commuting elements in $G$, and $G$ acts on that by conjugation. Now any two commuting elements can be taken to sit in a maximal torus $T \hookrightarrow G$, and up to conjugation we can take this to be one fixed maximal torus. This means that the moduli space is actually
where $W$ is the Weyl group $W = N_G(T)/T$.
Moreover, by Pontryagin duality this may be re-expressed as
where now $[T,S^1]$ is the character group of the maximal torus.
In this form the moduli space of flat connections appears prominently for instance in the discussion of equivariant elliptic cohomology. But beware that the above interpretation in algebraic geometry is at least more subtle, see (Lurie 15).
Original articles include
Michael Atiyah, Raoul Bott, The Yang-Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (jstor, lighning summary)
Nan-Kuo Ho, Chiu-Chu Melissa Liu, On the Connectedness of Moduli Spaces of Flat Connections over Compact Surfaces, Canad. J. Math. 56(2004), 1228-1236 (publisher)
Reviews and lecture notes include, for the case of flat connections
Remi Janner, Notes on the moduli space of flat connections, 2005 (pdf)
Daan Michiels, Moduli spaces of flat connections, Master Thesis Leuven 2013 (pdf)
Jörg Teschner, Quantization of moduli spaces of flat connections and Liouville theory, proceedings of the International Congress of Mathematics 2014 (arXiv:1405.0359)
Vladimir Fock, Alexander Goncharov, Symplectic double for moduli spaces of G-local systems on surfaces (arXiv:1410.3526)
and for the case of general and logarithmic connections
Detailed discussion of moduli space of flat connections also on higher dimensional base spaces is in
For more references see at Hitchin connection.
Discussion in algebraic geometry includes
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