The purpose of this paper is to define the concept of multi-Dirac structures
and to describe their role in the description of classical field theories. We
begin by outlining a variational principle for field theories, referred to as
the Hamilton-Pontryagin principle, and we show that the resulting field
equations are the Euler-Lagrange equations in implicit form. Secondly, we
introduce multi-Dirac structures as a graded analog of standard Dirac
structures, and we show that the graph of a multisymplectic form determines a
multi-Dirac structure. We then discuss the role of multi-Dirac structures in
field theory by showing that the implicit field equations obtained from the
Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac
structures. Furthermore, we show that any multi-Dirac structure naturally gives
rise to a multi-Poisson bracket. We treat the case of field theories with
nonholonomic constraints, showing that the integrability of the constraints is
equivalent to the integrability of the underlying multi-Dirac structure. We
finish with a number of illustrative examples, including time-dependent
mechanics, nonlinear scalar fields and the electromagnetic field.Comment: 50 pages, v2: correction to prop. 6.1, typographical change