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\bea\label{V1} G_{00}+G_{11}=(\alpha^2+\beta^2)g_{MN}\frac{d\tilde{X}^M}{d\xi}\frac{d\tilde{X}^N}{d\xi} +2\beta\Lambda^\mu g_{\mu N} \frac{d\tilde{X}^N}{d\xi} +\Lambda^\mu\Lambda^\nu g_{\mu \nu} =0,\eea \bea\label{V2} G_{01}&=&\alpha\beta g_{MN}\frac{d\tilde{X}^M}{d\xi}\frac{d\tilde{X}^N}{d\xi}+\alpha\Lambda^\mu g_{\mu ...

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\bea\nn &&-(\alpha^2-\beta^2)\left[g_{LK}\frac{d^2\tilde{X}^K}{d\xi^2}+ \Gamma_{L,MN}\frac{d\tilde{X}^M}{d\xi}\frac{d\tilde{X}^N}{d\xi}\right] +2\beta\Lambda^\mu\Gamma_{L,\mu N}\frac{d\tilde{X}^N}{d\xi} +\Lambda^\mu\Lambda^\nu \Gamma_{L,\mu\nu} \\ \label{EM} &&= \alpha\Lambda^\mu H_{L\mu N}\frac{d\tilde{X}^N}{d\xi},\eea \bea\label{Q} ...

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Take a section of \(\mathbf{T\mathbb{C} \otimes T^*\mathcal{O}}\) (i.e. a vector field on the conifold tensored with a 1-form on the orbifold) and apply the **exterior covariant derivative only on the orbifold side**, then contract or antisymmetrize in a way that produces genuine curvature or rotation. $$ \mathbf{R_\xi ...

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