Localization of a $K{O}^{\ast }(\text{pt})$-valued index and the orientability of the $Pi{n}^{-}(2)$ monopole moduli space

Abstract

It is known that the Dirac index of a $Spi{n}^c$ structure is localized to the characteristic submanifold. We introduce the notion of $G^{\pm }(n,s^{+},s^{-})$ structure on a manifold as a common generalization of the $Spi{n}^c$ structure and the $H_n(s)$ structure defined by D.~Freed--M.~Hopkins, and formulate a version of characteristic submanifold for the $G^{\pm }(n,s^{+},s^{-})$ structure. We show that the $K{O}^{\ast }(pt)$-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 moduli space of $Pi{n}^{-}(2)$ monopoles to be orientable.

27 pages

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