Formal adjoint of the covariant exterior derivative

Definition

Let $M$ be an oriented $n$ -dimensional Riemannian manifold without boundary, and let $E\to M$ be a vector bundle with bundle metric and compatible connection $A$ . The formal adjoint of the covariant exterior derivative is the operator

d_A^*:\Omega^{r}(M;E)\longrightarrow\Omega^{r-1}(M;E)

characterized by

\langle d_A\alpha,\beta\rangle_{L^2} =\langle\alpha,d_A^*\beta\rangle_{L^2}

for all compactly supported smooth forms of the appropriate degrees. Here $d_A$ is the exterior covariant derivative and the pairing is the $L^2$ pairing. This identity defines a differential expression, not a Hilbert-space adjoint with a specified domain.

Hodge-star formula

Extend the Hodge star to $E$ -valued forms by acting on the form factor. With the convention used here, its restriction to $r$ -forms satisfies

d_A^*=(-1)^{n(r+1)+1}*d_A*.

Equivalent sign formulas occur because authors index the input or output degree differently. The compatibility of $A$ with the fiber metric is what allows integration by parts without an additional derivative of that metric.

For $E$ the trivial real line bundle with its flat connection, $d_A^*$ is the ordinary codifferential. For the gauge-theoretic covariant derivative on an adjoint bundle $\operatorname{ad}P$ , invariance of the Lie-algebra inner product supplies the required bundle metric.

Analytical role

The operator $d_A^*$ enters both gauge fixing and the Yang–Mills equation. A perturbation $a$ is in Coulomb gauge relative to $A$ when $d_A^*a=0$ , while the Yang–Mills equation is $d_A^*F_A=0$ . Together with $d_A$ , it forms the gauge-covariant Hodge Laplacian used in elliptic estimates 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, bundle-valued forms, formal adjoints, and the Yang–Mills equation.
Raymond O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. DOI record. Relevant: chapters III–IV, Hodge-star identities and formal adjoints.