Topological real-space responses and stable integer invariants from dimensional reduction

Dimensional reduction of topological response is a powerful tool for understanding topological field theory of insulating states at various spatial dimensions. Thus far this procedure has been implemented by Qi et al. for time-reversal symmetric, spin-orbit coupled insulators (symplectic, class AII). In this work, we show that the dimensional reduction of four-dimensional quantum Hall effect can be performed for all three Wigner Dyson classes to understand stable topological states at three and two spatial dimensions, which are not predicted by the ten-fold classification scheme. We accomplish this goal by using magnetic monopoles and flux tubes as non-perturbative, real-space probes. We show that many topological phases of matter, which are often called ''fragile topological insulators" actually support stable invariants and quantized response.