Entrywise conjugation does not match the raw ADHM real moment map

Let $V={\mathbb{C}}^k$ and $W={\mathbb{C}}^N$ with their standard Hermitian products, and let ${\alpha }_1,{\alpha }_2\in {\mathrm{End}}_{\mathbb{C}}(V)$, $P:W\to V$, and $Q:V\to W$. Identify the conjugate spaces with these standard-coordinate spaces, so the bars below mean entrywise complex conjugation. Suppose the source-convention equations are

$[{\alpha }_1,{\alpha }_2]+PQ=0,[{\alpha }_1,{\alpha }_1^{†}]+[{\alpha }_2,{\alpha }_2^{†}]-P{P}^{†}+Q^{†}Q=0.$

For $B_i=\overline{{\alpha }_i}$, $I=\bar{P}$, and $J=\bar{Q}$, use the raw moment maps

${\mu }_{\mathbb{C}}=[B_1,B_2]+IJ,{\mu }_{\mathbb{R}}=[B_1,B_1^{†}]+[B_2,B_2^{†}]+I{I}^{†}-J^{†}J\mathrm{.}$

Under this proposed substitution,

$B_i=\overline{{\alpha }_i},I=\bar{P},J=\bar{Q},$

the raw complex moment map vanishes, but the raw real moment map is

${\mu }_{\mathbb{R}}=\bar{C}+\bar{X}-\bar{Y},$

where $C=[{\alpha }_1,{\alpha }_1^{†}]+[{\alpha }_2,{\alpha }_2^{†}]$, $X=P{P}^{†}$, and $Y=Q^{†}Q$. The conjugated source real equation gives $\bar{C}-\bar{X}+\bar{Y}=0$, so on source solutions the proposed raw real moment map equals $2\bar{C}$; the source equations alone therefore do not establish its vanishing. This failure occurs for the explicit $k=N=2$ solution of the two displayed source equations in the proof below; no source full-rank condition is asserted for that witness.

Donaldson's ADHM matrix equations in the source convention

ADHM linear data, stability, costability and FI levels

Donaldson, §6, printed pp. 10–11