Four-Dimensional $N=1$ Kähler–Hodge Line Data

Let $n\ge 1$ and let $X$ be a nonempty smooth complex manifold of complex dimension $n$. Here this means a nonempty Hausdorff, second-countable space with a holomorphic atlas of charts $z_{\alpha }:U_{\alpha }\to z_{\alpha }(U_{\alpha })\subset {\mathbb{C}}^n$ whose overlap maps are biholomorphic; this atlas also supplies the smooth structure. Choose a cover by such coordinate patches $U_{\alpha }$. On each patch choose a real smooth Kähler potential $K_{\alpha }$ for a positive Hermitian metric $g_{i\bar{j}}={\mathrm{\partial }}_i{\mathrm{\partial }}_{\bar{j}}{K}_{\alpha }\mathrm{.}$ On each overlap $U_{\alpha }\cap U_{\beta }$, require $K_{\alpha }-K_{\beta }=F_{\alpha \beta }+\overline{F_{\alpha \beta }},$ where $F_{\alpha \beta }$ is holomorphic. Fix a positive mass $M_P$. Require $F_{\alpha \alpha }\in 2\pi i{M}_P^2\mathbb{Z}$ on each $U_{\alpha }$, and require $F_{\alpha \beta }+F_{\beta \gamma }+F_{\gamma \alpha }\in 2\pi i{M}_P^2\mathbb{Z}$ on every triple overlap, including repeated indices. Thus $t_{\alpha \beta }:=e^{-F_{\alpha \beta }\mathrm{/}{M}_P^2}$ obeys the normalized bundle cocycle identities $t_{\alpha \alpha }=1$, $t_{\alpha \beta }{t}_{\beta \alpha }=1$, and $t_{\alpha \beta }{t}_{\beta \gamma }=t_{\alpha \gamma }$. A holomorphic line bundle here is a rank-one complex vector bundle with holomorphic local frames and nonzero holomorphic transition functions satisfying these identities. A Kähler–Hodge line datum in this convention is this patch data together with such a line bundle whose local components for a holomorphic section $W$ transform as $W_{\alpha }=e^{-F_{\alpha \beta }\mathrm{/}{M}_P^2}{W}_{\beta }\mathrm{.}$ The displayed transitions are holomorphic and nonzero, and the stated period condition makes their product around every triple overlap equal to $1$. Its Hermitian norm is locally $e^{K_{\alpha }\mathrm{/}{M}_P^2}\mathrm{\mid }{W}_{\alpha }{\mathrm{\mid }}^2$, which is independent of patch. In four-dimensional units use $[z^i]=[M_P]=1$, $[K_{\alpha }]=[F_{\alpha \beta }]=2$, and $[W]=3$. Define the Kähler-covariant derivative by $D_{i}W:={\mathrm{\partial }}_{i}W+\frac{K_i}{M_P^2}W,K_i:={\mathrm{\partial }}_i{K}_{\alpha }\mathrm{.}$ The definition is local on the stated patches. It does not assert that arbitrary Kähler potentials admit the required integral cocycle or line bundle.

Kähler coordinate-patch data for a rigid $N=1$ chiral model

Complex manifolds, holomorphic vector bundles and Hermitian metrics

Hohl, §5 opening discussion, equations (5.1)–(5.6)

Louis, §9.1, equations (9.3)–(9.4)