moduli space of connections in nLab

Contents

Contents

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 Σ\Sigma is a compact smooth manifold, then the moduli space of flat connections over Σ\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 GG be a compact Lie group. Assume either that GG is simply connected or is a torus (what we really need below is that any two commuting elements in GG sit jointly in one maximal torus).

The moduli space of GG flat connections on a 2-dimensional torus A≃S 1×S 1A \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

[Π(A),BG] ≃[B[A,S 1],BG] ≃Hom Grp(ℤ×ℤ,G)// adG \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 GG, and GG acts on that by conjugation. Now any two commuting elements can be taken to sit in a maximal torus T↪GT \hookrightarrow G, and up to conjugation we can take this to be one fixed maximal torus. This means that the moduli space is actually

π 0(Hom Grp([A,S 1],G)// adG)≃Hom Grp([A,S 1],T)/W, \pi_0 \left( Hom_{Grp}([A,S^1], G)//_{ad} G \right) \simeq Hom_{Grp}([A, S^1], T)/W \,,

where WW is the Weyl group W=N G(T)/TW = N_G(T)/T.

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

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

where now [T,S 1][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).

• Donaldson-Uhlenbeck-Yau theorem

• Tamagawa numbers

• Hitchin connection

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