Refined Dijkgraaf-Witten invariant of spin 3-manifold

We give a construction of a state sum invariant of a closed spin 3-manifold based on a super 3-cocycle $(\tilde{\alpha },\omega )$ and a combinatorial representation of a spin 3-manifold, where $\omega$ is a ${\mathbb{Z}}_2$-valued cocycle and $\tilde{\alpha }$ is a 3-cochain satisfying a 3-cocycle condition with a sign coming from the 2-cocycle $\omega$. The definition of the invariant is similar to the state sum construction of the Dijkgraaf-Witten invariant, except it uses the spin structure to take care of the sign in the 3-cocycle condition. We also give an example of the invariant and see that it is sensitive to the spin structure.