  We revisit universal features of duality in linear and nonlinear relativistic
scalar and Abelian 1-form theories with single or multiple fields, which
exhibit ordinary or generalized global symmetries. We show that such global
symmetries can be interpreted as generalized Killing isometries on a suitable,
possibly graded, target space of fields or its jet space when the theory
contains higher derivatives. This is realized via a generalized sigma model
perspective motivated from the fact that higher spin particles can be
Nambu-Goldstone bosons of spontaneously broken generalized global symmetries.
We work out in detail the 2D examples of a compact scalar and the massless
Heisenberg pion fireball model and the 4D examples of Maxwell, Born-Infeld and
ModMax electrodynamics. In all cases we identify the 't Hooft anomaly that
obstructs the simultaneous gauging of both global symmetries and confirm the
anomaly matching under duality. These results readily generalize to higher
gauge theories for p-forms. For multifield theories we discuss the
transformation of couplings under duality as two sets of Buscher rules for even
or odd differential forms.
