||In the last few years, the notion of symmetry has been enlarged to "generalized symmetry" and these more generalized symmetries have played a critical role in deepening our understanding of QFT, notably IR phases of QFT. In this talk, I will discuss various ways of coupling the axion-Maxwell theory to a sector of topological field theory (TQFT). Contrary to common wisdom, I will show that such topological modifications can lead to direct changes in the local physics with possible observable consequences. This surprise can be realized by a dimensional reduction, namely, a coupling to a TQFT in 4d leads to a non-trivial and local impact on the 2d string world-sheet QFT. I will explain how this happens using the language of anomaly inflow, which provides a way of figuring out the IR-universal observable effect from the TQFT-coupling. There exists another topological modification of the theory, i.e. gauging a discrete subgroup of 0-form shift symmetry, and this time it results in an alteration of the spectrum of cosmic strings. If time permits, I will also discuss generalized symmetries and associated higher-groups of these theories.