A bstract We consider the SU( N ) Yang-Mills theory, whose topological sectors are restricted to the instanton number with integer multiples of p . We can formulate such a quantum field theory maintaining locality and unitarity, and the model contains both 2 π -periodic scalar and 3-form gauge fields. This can be interpreted as coupling a topological theory to Yang-Mills theory, so the local dynamics becomes identical with that of pure Yang-Mills theory. The theory has not only ℤ N 1-form symmetry but also ℤ p 3-form symmetry, and we study the global nature of this theory from the recent ’t Hooft anomaly matching. The computation of ’t Hooft anomaly incorporates an intriguing higher-group structure. We also carefully examine that how such kinematical constraint is realized in the dynamics by using the large- N and also the reliable semiclassics on ℝ 3 × S 1 , and we find that the topological susceptibility plays a role of the order parameter for the ℤ p 3-form symmetry. Introducing a fermion in the fundamental or adjoint representation, we find that the chiral symmetry becomes larger than the usual case by ℤ p , and it leads to the extra p vacua by discrete chiral symmetry breaking. No dynamical domain wall can interpolate those extra vacua since such objects must be charged under the 3-form symmetry in order to match the ’t Hooft anomaly.