Higher topological structures in parametrized quantum many-body wavefunctions

In this talk, we study the parametrized family of ground states for one dimensional gapped quantum many-body systems. This family of systems are characterized by a parameter space $X$ and a symmetry group $G$. We find that there is a higher topological structure underlying this parametrized family of wave functions. When there is no global symmetry, this higher topological structure is characterized by the so-called Dixmier-Douady class in $H^3(X,Z)$; when there is a global symmetry $G$, it is characterized by the equivariant Dimmer-Douady class in $H^3_G(X,Z)$. The corresponding topological invariants organize how locally defined wave functions glue across parameter patches and diagnose nontrivial "twists" that persis under adiabatic evolution.