We have absorbed the usual coefficient on the RHS into the quadratic form Tr. Note that the low-energy effective description of type I string theory contains an additional $R^2$ term, necessary for the anomaly cancellation.

The bracket with sections of $L$ descends to a flat $L$-connection on the complex smooth vector bundle $L^{\perp }/L$. Since $L\cong T^{0,1}M$ this is equivalent to a Dolbeault operator on ${\mathrm{\Omega }}^{0,•}(L^{\perp }/L)$, which we have denoted ${\overline{\partial }}_{\mathcal{E}}$.

Note that due to the lack of R-symmetry which could be used for regrading, the twist of the $\mathcal{N}=1$ theory in ten dimensions is expected to only carry a ${\mathbb{Z}}_2$-grading (instead of a full $\mathbb{Z}$-grading).

More explicitly, the natural map $G\to G_{\mathbb{C}}$ allows us to construct $P_{\mathbb{C}}$ as the associated bundle $P{\times }_G{G}_{\mathbb{C}}$. This (still regarded as a smooth object) inherits a connection, denoted ${\nabla }^{\mathbb{C}}$. In particular the tangent space at each point in $P^{\mathbb{C}}$ can be identified with the direct sum of a tangent space on the base with the Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$. Since both of these spaces carry a complex structure, we may attempt to combine them into a complex structure on $P^{\mathbb{C}}$. This indeed works—the integrability is ensured by the condition $F^{0,2}=0$.

Here Calabi–Yau manifold is understood as a complex manifold with a holomorphic volume form.

Using the explicit expressions for ${\mathcal{E}}^{•}$ and the identification ${\mathrm{\Omega }}^{0,•}(T^{1,0}M)\cong {\mathrm{\Omega }}^{4,•}$ provided by the volume form, the equality $(\ker ({\mathrm{div}}_{\mathrm{\Omega }}){)}^{\perp }=\mathrm{im}\mathcal{D}$ reduces to $[\ker (\partial :{\mathrm{\Omega }}^{4,•}\to {\mathrm{\Omega }}^{5,•}){]}^{\perp }=\partial {\mathrm{\Omega }}^{0,•}$, where $\perp$ on the LHS is taken w.r.t. the wedge-and-integrate pairing. Choosing now any auxiliary Hermitian metric, the LHS becomes $[\ker ({\partial }^{†}:{\mathrm{\Omega }}^{1,•}\to {\mathrm{\Omega }}^{0,•}){]}^{\perp }$, with $\perp$ taken w.r.t. the metric. Using the Hodge decomposition we then conclude that $(\ker {\partial }^{†}{)}^{\perp }=\mathrm{im}\partial$.