Tangent space to flat-connection moduli - Imported Knowlpedia Content

Statement

Let $A$ be a flat connection on a principal $G$ -bundle $P\to M$ , and let $H_A^j$ denote the cohomology of its deformation complex. The deformation-theoretic tangent space to flat connections modulo gauge at $[A]$ is

T^{\mathrm{def}}_{[A]}\mathcal M_{\mathrm{flat}} \cong H_A^1 = \frac{\ker(d_A:\Omega^1(M;\operatorname{ad}P)\to\Omega^2(M;\operatorname{ad}P))} {\operatorname{im}(d_A:\Omega^0(M;\operatorname{ad}P)\to\Omega^1(M;\operatorname{ad}P))}.

If $A$ is irreducible and unobstructed so that the gauge quotient is smooth near $[A]$ , this is its ordinary tangent space. Moreover, $H_A^0$ is the infinitesimal stabilizer and $H_A^2$ is the primary obstruction space.

Linearization

Write a nearby connection as $A+a$ . Its curvature is

F_{A+a}=F_A+d_Aa+\tfrac12[a\wedge a].

Since $F_A=0$ , the linearized flatness equation is $d_Aa=0$ . An infinitesimal gauge transformation generated by $\xi\in\Omega^0(M;\operatorname{ad}P)$ changes $a$ by $d_A\xi$ , up to the sign convention for the gauge action. Taking solutions modulo these tangent gauge directions gives $H_A^1$ .

The quadratic term defines the first obstruction map $H_A^1\to H_A^2$ . Hence a class in $H_A^1$ need not integrate to a curve of flat connections.

Holonomy description

Through the holonomy correspondence, the same cohomology is group cohomology with local coefficients:

H_A^1\cong H^1\!\left(\pi_1(M);\mathfrak g_{\operatorname{Ad}\rho_A}\right)

under the standard comparison hypotheses. It is therefore also the infinitesimal deformation space of the holonomy representation $\rho_A:\pi_1(M)\to G$ modulo infinitesimal conjugation. Goldman and Millson formulate the local germ using the controlling differential graded Lie algebra Goldman–Millson, §§1–3.

Smooth and singular points

If $H_A^2=0$ , standard elliptic slice models on a compact base give unobstructedness. If in addition the stabilizer has been removed or is only the unavoidable center, the quotient is locally a smooth space modeled on $H_A^1$ . Nonzero $H_A^2$ permits obstructions but does not prove that the obstruction map is nonzero.

At a reducible connection, the coarse quotient can be singular because the stabilizer acts on $H_A^1$ . In that case $H_A^1$ remains the degree-one tangent cohomology of the moduli problem, but it should not automatically be called the Zariski tangent space of the coarse orbit space.

References

M. F. Atiyah and R. Bott, “The Yang–Mills Equations over Riemann Surfaces,” Philosophical Transactions of the Royal Society of London A 308 (1983), 523–615. DOI record. Relevant: §§6–7, the gauge quotient and elliptic deformation complex.
W. M. Goldman and J. J. Millson, “The Deformation Theory of Representations of Fundamental Groups of Compact Kähler Manifolds,” Publications Mathématiques de l’IHÉS 67 (1988), 43–96. DOI record. Relevant: §§1–3, tangent cohomology, differential graded Lie algebras, and obstruction germs.