ASD deformation complex - Imported Knowlpedia Content

Definition

Let $A$ be an anti-self-dual connection on a principal $G$ -bundle $P\to X$ over an oriented Riemannian four-manifold. The ASD deformation complex at $A$ is

0\longrightarrow\Omega^0(X;\operatorname{ad}P) \xrightarrow{\,d_A\,} \Omega^1(X;\operatorname{ad}P) \xrightarrow{\,d_A^+\,} \Omega^{2,+}(X;\operatorname{ad}P) \longrightarrow0,

where $d_A$ is the covariant exterior derivative and $d_A^+=\operatorname{pr}_+\circ d_A$ . The identity $d_A^+d_A\phi=[F_A^+,\phi]=0$ makes this a complex. Its cohomology groups $H_A^0,H_A^1,H_A^2$ record, respectively, infinitesimal stabilizers, infinitesimal ASD deformations modulo infinitesimal gauge transformations, and obstructions to extending infinitesimal deformations. It is precisely the linear complex obtained from the ASD equation and the infinitesimal gauge action at the connection.

Ellipticity

The symbol sequence is exact away from the zero section, so the complex is elliptic. On a closed manifold its Sobolev completions are Fredholm and all three cohomology groups are finite-dimensional. Combining the last differential with the formal adjoint of the first gives the elliptic operator

d_A^*\oplus d_A^+: \Omega^1(X;\operatorname{ad}P)\longrightarrow \Omega^0(X;\operatorname{ad}P)\oplus\Omega^{2,+}(X;\operatorname{ad}P).

Its kernel is a canonical harmonic model for $H_A^1$ .

Interpretation of the cohomology

The Lie algebra of the stabilizer of $A$ is $H_A^0=\ker d_A$ . The Zariski tangent space to the ASD moduli space is $H_A^1=\ker d_A^+/\operatorname{im}d_A$ . Finally,

H_A^2=\Omega^{2,+}(X;\operatorname{ad}P)/\operatorname{im}d_A^+

is the cokernel of the linearized ASD equation after gauge fixing. Vanishing of $H_A^2$ is the regularity condition used in the implicit-function theorem.

Index and conventions

The index $\dim H_A^1-\dim H_A^0-\dim H_A^2$ is computed topologically by the Atiyah–Singer index theorem. Some sources study self-dual connections instead; they then replace $\Omega^{2,+}$ by $\Omega^{2,-}$ . Reversing the orientation exchanges the two complexes.

References

Michael F. Atiyah, Nigel J. Hitchin, and Isadore M. Singer, “Self-Duality in Four-Dimensional Riemannian Geometry,” Proceedings of the Royal Society A 362 (1978), 425–461. DOI record. Relevant: the elliptic deformation complex and its index.
Simon K. Donaldson and Peter B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990. DOI record. Relevant: §§4.2–4.3, infinitesimal deformations and obstruction spaces.