First-order action for Abelian worldsheet duality ($\frac{1}{2}V{A}_{\mu }{A}^{\mu }+\tilde{z}{ϵ}^{\mu \nu }{\partial }_{\mu }{A}_{\nu }$)

Replace the derivative of an isometric coordinate by an independent one-form $A$ and impose $dA=0$ with a Lagrange multiplier. In signature $(-,+)$ with ${ϵ}^{\tau \sigma }=1$, its equation is $V{A}^{\nu }={ϵ}^{\mu \nu }{\partial }_{\mu }\tilde{z}$. Eliminating $A$ from the full action, including the multiplier term, gives $\frac{1}{2}{V}^{-1}{\partial }_{\mu }\tilde{z}{\partial }^{\mu }\tilde{z}$. Omitting the multiplier term during substitution would give the wrong sign. This derives the block-diagonal metric inversion in the Buscher rules.