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It is known that the Dirac index of a $Spin^c$ structure is localized to thecharacteristic submanifold. We introduce the notion of $G^{\pm}(n,s^+,s^-)$structure on a manifold as a common generalization of the $Spin^c$ structureand the $H_n(s)$ structure defined by D.~Freed--M.~Hopkins, and formulate aversion of characteristic submanifold for the $G^{\pm}(n,s^+,s^-)$ structure.We show that the $KO^*(pt)$-valued index associated with the$G^{\pm}(n,s^+,s^-)$ structure is localized to the characteristic submanifold.As an application, we give a topological sufficient condition for the modulispace of $Pin^-(2)$ monopoles to be orientable.

Localization of a $KO^{\ast}(\text{pt})$-valued index and the orientability of the $Pin^-(2)$ monopole moduli space