Calabi Yau manifolds are Kähler manifolds with vanishing first Chern class. According to the conjecture of E. Calabi , for a Kähler manifold M , if $c_1 (M) = 0 $ , then M would admit a Ricci-flat Kähler metric . Explicitly, $Ric(g) = -i\partial\bar{\partial}log(det(g))=0 $

Or, $\partial\bar{\partial}log(det(g))=0$. This means that this is a combination of holomorphic and anti-holomorohic fucntions since both holomorphic and anti-holomorphic are present multiplicatively . Then we conclude that $log(det(g))=\bar{f}(x)+f(x)$

$det(g) = \bar{\omega}(x)\omega(x)$ where the Kähler metric is given by $g=i\partial\bar{\partial}K$ , $K$ is the Kähler potential.

Now a Schwarz type topological quantum field theory satisfies the condition that the correlation function of the observables of the theory must be independent of the metric which encodes the geometry of background manifold of the theory i.e. $\frac{\delta}{\delta g^{\mu\nu}}\langle O_{1}....O_{\alpha}\rangle = \frac{\int_{}^{}[D\phi]O_1....O_\alpha e^{-iS[\phi]}}{\int_{}^{}[D\phi]e^{-iS}} = 0 $

If the metric is Hermitian , then its called a Hermitian quantum field theory . Now if we a consider that the metric is Kähler , what additional conditions should we impose on the theory so that the ambient manifold of the theory is Calabi-Yau?

The explicit form of the metric of Calabi-Yau space is still unknown, is that a big restriction against defining such a theory ?