Hi,


I have a superspace spanned by 4 commuting coordinates + 2 anti-commuting ones $\{x^\mu,\theta^\alpha\}$, I have to do the change of coordinates $dx^\mu\to dy^\mu= dx^\mu+d\theta^\alpha \eta\_\alpha^{\;\:\mu}$ where $\eta$ have to be a local function i.e: $\eta\equiv\eta(x)$, and leave the $d\theta$'s unchanged, so how can I translate this change on coordinates themselves? In particular I need to express $\partial\_\mu$ in the new coordinate system.


If the $\eta$ fcts where global it would be simply $x^\mu\to y^\mu= x^\mu+\theta^\alpha \eta\_\alpha^{\;\:\mu}$, so what if $\eta$'s where not global?



Change of coordinates introduced through dx
First of all, I would not call $dx^\mu \mapsto dy^\mu$ a change of coordinates. Not every transformation on 1-forms is going to be the pullback by a smooth map. In fact, your question is whether the transformation $(dx^\mu,d\theta^\alpha) \mapsto (dy^\mu,d\theta^\alpha)$ comes from a change of coordinates.


Pullbacks commute with exterior derivative, so that if $dy^\mu$ is a pullback then applying $d$ to your $dy^\mu$ should give zero. If you do that you find that it is not unless $\eta$ is constant.


