A bstract We study reductions of 6d theories on a d -dimensional manifold M _( d ), focusing on the interplay between symmetries, anomalies, and dynamics of the resulting (6 ? d )-dimensional theory T [ M _( d )]. We refine and generalize the notion of “polarization” to polarization on M _( d ), which serves to fix the spectrum of local and extended operators in T [ M _( d )]. Another important feature of theories T [ M _( d )] is that they often possess higher-group symmetries, such as 2-group and 3-group symmetries. We study the origin of such symmetries as well as physical implications including symmetry breaking and symmetry enhancement in the renormalization group flow. To better probe the IR physics, we also investigate the ’t Hooft anomaly of 5d Chern-Simons matter theories. The present paper focuses on developing the general framework as well as the special case of d = 0 and 1, while an upcoming paper will discuss the case of d = 2, 3 and 4.
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