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Type IIB string theory is chiral. Do there exist non-chiral $\mathcal{N}=2$ II SUGRA theories? The answer is apparently yes. Additionally how do we understand the fact that Type IIB SUGRA is chiral and $\mathcal{N}=2$ SUSY gauge theory is (famously) non-chiral? What is the relation or what should I think about the non-chirality of extended susy and chirality of extended SUGRA?

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In fact the answer is "yes", the non-chiral type $II$ sugra thoery is called type $II$ A. You can obtain it by dimensional reduction from the $M$-theory sugra in $d=11$. The (massless) spectrum of type $II$ B contains spinor representations of just one chirality (which one is matter of convention), while type $II$ A contains representations of both chiralities.

The fact you say about the $\mathcal{N}=2$ supergauge theory is true in $d=4$, but not in general.

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