  In this work, we investigate the role of multivalued fields in the
formulation of monopole operators and their connection with topological states
of matter. In quantum field theory it is known that certain states describe
collective modes of the fundamental fields and are created by operators that
are often non-local, being defined over lines or higher-dimensional surfaces.
For this reason, they may be sensitive to global, topological, properties of
the system and depend on nonperturbative data. Such operators are generally
known as monopole operators. Sometimes they act as disorder operators because
their nonzero expectation values define a disordered vacuum associated with a
condensate of the collective modes, also known as defects. In this work we
investigate the definition of these operators and their relation to the
multivalued properties of the fundamental fields. We study several examples of
scalar field theories and generalize the discussion to $p$-forms, with the main
purpose of studying new field configurations that may be related to topological
states of matter. We specifically investigate the so-called chiral vortex
configurations in topological superconductors. We highlight an important aspect
of this formalism, which is the splitting of the fields in their regular and
singular parts that identifies an ambiguity that can be explored, much like
gauge symmetry, in order to define observables.
