# Vanishing of field strength in gauged WZW model

Consider gWZW action $S_{gWZW}(g,A)=S_{WZW}(g)+S_{gauge}(g,A)$, where $S_{WZW}$ is usual WZW action being sum of sigma model and WZ terms for field $g$ taking values in group $G$ and $S_{gauge}=\int d^2\sigma\text{Tr}(A_+\partial _-gg^{-1}-A_-g^{-1}\partial _+g-g^{-1}A_+gA_-+A_+A_-)$ is gauge action which depends on field $g$ and gauge fields $A_\pm$, the later take values in subalgebra of group $G$ algebra which we want to gauge.

According to equations of motion following from gWZW action field strength $F_{+-}=\partial _+A_--\partial _-A_+ +[A_+,A_-]$ vanishes on-shell. To show it one shall project equation of motion for field $g$ on gauge subalgebra and substitute constraints for gauge fields $A_\pm$.

My question is: are we allowed to use $F_{+-}=0$ equation of motion while studying symmetries of action - that is are we allowed to substitute $F_{+-}=0$ in variation of action justifying it by claim that fields $A_\pm$ are non-dynamical?

I have natural doubts that this is allowed because as described above equation $F_{+-}=0$ is obtained not just with the help of equations following from variation of action by non-dynamical $A_\pm$ fields but also using $g$ field equations. This is topical issue e.g. when one deals with sort of supersymmetric extension of gWZW model (some non-ordinary or potentially-deformed kinds of them - because there's no necessity to set $F_{+-}=0$ in usual supersymmetric gWZW model). If so to obtain $F_{+-}=0$ equation one also needs to use fermionic fields equations of motion.

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