= First-order action for Abelian worldsheet duality
{title2=$\tfrac12VA_\mu A^\mu+\widetilde z\,\epsilon^{\mu\nu}\partial_\mu A_\nu$}
= First-order worldsheet duality action
{synonym}
Replace the derivative of an isometric coordinate by an independent one-form $A$ and impose $dA=0$ with a <Lagrange multiplier>. In signature $(-,+)$ with $\epsilon^{\tau\sigma}=1$, its equation is $VA^\nu=\epsilon^{\mu\nu}\partial_\mu\widetilde z$. Eliminating $A$ from the full action, including the multiplier term, gives $\tfrac12V^{-1}\partial_\mu\widetilde z\partial^\mu\widetilde z$. Omitting the multiplier term during substitution would give the wrong sign. This derives the block-diagonal metric inversion in the <Buscher rules>.
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