# Non-hermitian connections in Super-Yang Mills

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If you have a lagrangian for chiral superfields in 4 dimensions which is also invariant under SU(N) and you try to gauge the symmetry , You introduce a connection $\Gamma$ so as to make $\Phi^{\dagger}\Gamma \Phi$ gauge invariant under super gauge transformation. If $\Gamma$ is hermitian , you have introduced  vector superfields into the lagrangian coupled with the chiral superfields. Now why don't we consider cases in which $\Gamma$ is not hermitian ?

Hopefully this comment isn't nonsense and/or unrelated.  There is a theory called $\mathcal{N}=1^{*}$ which is a massive deformation of $\mathcal{N}=4$ SYM in four-dimensions where you give a mass $M$ to the three chiral superfields $\Phi_{i}$ with $i=1,2,3$.  And indeed, you consider terms in the Lagrangian like $M \Phi^{\dagger} \Phi$.  You really want $M \in \mathbb{C}$ in general as opposed to $M \in \mathbb{R}$.  And in some contexts, this inspires one to consider not merely Hermitian matrix models, but actually holomorphic matrix models.  So I think you probably would want to consider more general $\Gamma$ in your case.  Sorry this isn't too concrete; just a thought.

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