nLab moduli space of connections

Idea

A moduli space of connections on bundles over some prescribed space.

Often one considers flat connections (local systems) only, see at moduli space of flat connections.

Properties

Compactness

If $\Sigma$ is a compact smooth manifold, then the moduli space of flat connections over $\Sigma$ is compact.

Over complex manifolds / complex varieties

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).

Examples

Flat connections over a torus

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

$$\begin{aligned} [\Pi(A), \mathbf{B}G] & \simeq [B [A, S^1], \mathbf{B}G] \\ & \simeq Hom_{Grp}(\mathbb{Z} \times \mathbb{Z}, G)//_{ad} G \end{aligned}$$

(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

$$\pi_0 \left( Hom_{Grp}([A,S^1], G)//_{ad} G \right) \simeq Hom_{Grp}([A, S^1], T)/W \,,$$

where $W$ is the Weyl group $W = N_G(T)/T$.

Moreover, by Pontryagin duality this may be re-expressed as

$$\cdots \simeq Hom_{Grp}([T,S^1], A)/W \,.$$

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).

Related concepts

• Donaldson-Uhlenbeck-Yau theorem

• Tamagawa numbers

• Hitchin connection

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

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)

• Nigel Hitchin, Flat connections and geometric quantization, Comm.Math.Phys., 131 (1990) 347-380

• 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 doi

Review and lecture notes, mostly 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 V. Fock, Alexander G. Goncharov?, Symplectic double for moduli spaces of G-local systems on surfaces (arXiv:1410.3526)

and for the case of general and logarithmic connections

• Indranil Biswas, V.Munoz, Moduli spaces of connections on a Riemann surface (pdf)

Introcuction of 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]

Detailed discussion of moduli space of flat connections also on higher dimensional base spaces is in

• Peter Scheinost, Martin Schottenloher, pp. 154 (11 of 76) of Metaplectic quantization of the moduli spaces of flat and parabolic bundles, J. reine angew. Mathematik, 466 (1996) (web)

For more references see at Hitchin connection.

Discussion in algebraic geometry includes

• Jacob Lurie, MO comment 2015