  The role of symmetries in what concerns entanglement entropy has been
extensively explored in the last years and revealed a profound connection with
the quantum field theory's algebraic structure. Recently, it was found that
some universal contributions to the entanglement entropy and mutual information
may be non uniquely defined in theories with generalized symmetries. Here, we
study this issue in detail in the particular case of the entanglement entropy
of the Maxwell theory in $(2+1)$ dimensions for rotationally symmetric regions.
In this setup, the problem can be dimensionally reduced to a half-line. We find
that the only difference between the reduced problem for the Maxwell field and
the reduced scalar free field stems from the Fourier angular $n=0$ mode. This
simplification allows us to check explicitly the many issues that characterize
models with broken global symmetries. Namely, we manifestly show that the
additive algebras break Haag duality, and single out the non-local operators
which are responsible for the failure of this property. More interestingly, we
present concrete lattice realizations that confirm that the logarithmic
"universal" term of the Maxwell entanglement entropy for disks depends on the
details of the algebra assignation. This ambiguity hinders the identification
of possible topological contributions characteristic of models with generalized
symmetries and tarnishes its universal character. We further calculate the
Maxwell mutual information for two nearly complementary concentric disks. We
obtain the expected universal contribution with a log-log dependence and check
that, unlike entropy, this is stable. Accordingly, this supports mutual
information as the appropriate probe to sense additivity-duality breaking and
the consequent universal topological contributions.
