Covariant Hodge Laplacian - Imported Knowlpedia Content

Definition

Let $E\to M$ be a vector bundle with bundle metric and compatible connection $A$ over an oriented Riemannian manifold. The covariant Hodge Laplacian on $E$ -valued $k$ -forms is

\Delta_A=d_A d_A^*+d_A^*d_A: \Omega^k(M;E)\longrightarrow\Omega^k(M;E),

where $d_A$ is the exterior covariant derivative and $d_A^*$ is its formal adjoint. It is a formally self-adjoint elliptic operator with scalar principal symbol $|\xi|^2\operatorname{id}$ . Unlike the ordinary de Rham differential, $d_A^2$ need not vanish: curvature acts on the coefficient bundle. The operator preserves form degree and depends on both the Riemannian metric and the connection.

Energy identity and kernel

On a compact manifold without boundary,

\langle\Delta_A\alpha,\alpha\rangle_{L^2} =\|d_A\alpha\|_{L^2}^2+\|d_A^*\alpha\|_{L^2}^2.

Consequently, a smooth form lies in $\ker\Delta_A$ exactly when it is both $d_A$ -closed and $d_A^*$ -closed. This conclusion uses compactness or boundary conditions that eliminate boundary terms.

If $A$ is flat, then $d_A^2=0$ , so $(\Omega^\bullet(M;E),d_A)$ is a complex and the kernel represents cohomology through the corresponding Hodge theorem. For a curved connection, $\ker\Delta_A$ remains analytically meaningful, but it is not generally a cohomological model.

Weitzenböck form

The covariant Hodge Laplacian has a Weitzenböck decomposition

\Delta_A=\nabla_A^*\nabla_A+\mathcal R_g+\mathcal R_{F_A},

where the last two terms are zeroth-order actions of the Riemannian curvature and the curvature of $A$ . Thus the distinction between $\Delta_A$ and the rough connection Laplacian is precisely a curvature correction. This is why “connection Laplacian” is not used here as an unqualified alias.

In gauge theory, the operator and its gauge-fixed variants control infinitesimal deformations and regularity of connections Freed–Uhlenbeck, chapter 2.

References

Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: chapter 2, covariant differential operators, ellipticity, and gauge-theoretic estimates.
Raymond O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. DOI record. Relevant: chapter IV, Hodge Laplacians and elliptic operator theory.