Deformation complex of a flat connection - Imported Knowlpedia Content

Definition

Let $A$ be a flat connection on a principal $G$ -bundle $P\to M$ . The deformation complex of $A$ is the twisted de Rham complex

0\longrightarrow\Omega^0(M;\operatorname{ad}P) \xrightarrow{\,d_A\,}\Omega^1(M;\operatorname{ad}P) \xrightarrow{\,d_A\,}\Omega^2(M;\operatorname{ad}P) \xrightarrow{\,d_A\,}\cdots ,

where $d_A$ is the covariant exterior derivative. It is a complex because $d_A^2\alpha=[F_A\wedge\alpha]$ and flatness gives $F_A=0$ . The first three terms govern deformations of the flatness equation modulo gauge transformations; the full complex computes the cohomology of the flat adjoint bundle. Its cohomology groups are denoted $H_A^j$ or $H^j(M;\operatorname{ad}P_A)$ .

Infinitesimal meaning

For a one-parameter family $A+ta+O(t^2)$ , the linearized curvature is $d_Aa$ , so infinitesimal flat deformations are the closed one-forms $\ker(d_A:\Omega^1\to\Omega^2)$ . An infinitesimal gauge transformation generated by $\xi$ changes $A$ by $d_A\xi$ , up to the action convention. Hence

H_A^1 = \frac{\ker(d_A:\Omega^1\to\Omega^2)} {\operatorname{im}(d_A:\Omega^0\to\Omega^1)}

is the formal tangent space to the flat moduli space. Similarly, $H_A^0$ is the Lie algebra of the stabilizer, and $H_A^2$ is the primary obstruction space.

Nonlinear deformation equation

Writing a nearby connection as $A+a$ , flatness becomes the Maurer–Cartan equation

d_Aa+\tfrac12[a\wedge a]=0.

The linear term gives the complex, while the quadratic term explains why a class in $H_A^1$ need not integrate to an actual family. In representation-theoretic settings the associated differential graded Lie algebra controls the local deformation germ; the compact Kähler case and its quadraticity properties are developed in Goldman–Millson, §§1–3.

Ellipticity and examples

On a compact Riemannian manifold, the twisted de Rham complex is elliptic. Hodge theory then gives finite-dimensional harmonic representatives of its cohomology. If $G$ is abelian and $P$ is trivial, the adjoint action is trivial and the complex is the ordinary de Rham complex tensored with the Lie algebra of $G$ .

References

William M. Goldman and John 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, differential graded Lie algebras, deformation functors, and local representation germs.