# How do you derive Noether's theorem when the action combines chiral, antichiral, and full superspace?

How do you derive Noether's theorem when the action combines chiral, antichiral, and full superspace?

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The full superspace terms are the "most general ones" but you may convert the chiral and antichiral terms to the full superspace form, too. In particular, $$\int d^2\bar\theta = \int d^2\theta\, d^2 \bar\theta\, \theta_1\theta_2$$ and similarly for its complex conjugate. Sorry if a sign is wrong. Note that $\theta_1\theta_2$ may be written as $\epsilon^{ab}\theta_a \theta_b/2$ up to a sign if you happen to be annoyed by the explicit components.

The fact that chiral superfields only depend on $\theta$ but not $\bar\theta$ doesn't matter: it just means that it is a more constrained field. But you may still imagine that it's a function of the full superspace that just happens to have a special form.

Once you convert the action to an integral over the full superspace, you may proceed just like you would proceed for the full superspace, D-term-like terms. I am actually a bit unfamiliar with this thing - so I would convert the action to the components (no superspace at all) and proceeded just like in non-supersymmetric theories. Of course, the resulting conserved quantities wouldn't be nicely organized in supermultiplets - which they can be in a supersymmetric theory.

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