(1) Let us first consider principal $U(1)$ bundle on $4$-dimensional manifold $X_4$ with $A$ the connection and $F(A)$ the curvature and the following index: \begin{eqnarray} \int_{X_4}\text{ch}_2(A)=\frac{1}{8\pi^2}\int_{X_4}F(A)\wedge F(A), \end{eqnarray} which is supposed to take integral number on the manifolds which support a spin structure (so-called spin manifolds).
However, on a general manifold, \begin{eqnarray} \int_{X_4}\text{ch}_2(A)\in\frac{\mathbb{Z}}{2}, \end{eqnarray} the minimum of which can be realized on $X_4=CP^2$. My question is whether we could realize, e.g. $\int_{X_4}\text{ch}_2(A)=1/2$, on (non-orientable) pin manifolds, e.g. $X_4=RP^4$.
(2) In addition, what about the 2-dimensional $X_2$ manifold cases? We can also consider \begin{eqnarray} \int_{X_2}\text{ch}_2(A)=\frac{i}{2\pi}\int_{X_2}F(A), \end{eqnarray} and whether there is a 2D's analog that $\int_{X_2}\text{ch}_2(A)=1/2$ with some $U(1)$ bundle over a special chosen $X_2$, e.g. $X_2$ non-orientable such as $X_2=RP^2$?