  In this paper we provide two new semantics for proofs in the constructive
modal logics CK and CD. The first semantics is given by extending the syntax of
combinatorial proofs for propositional intuitionistic logic, in which proofs
are factorised in a linear fragment (arena net) and a parallel
weakening-contraction fragment (skew fibration). In particular we provide an
encoding of modal formulas by means of directed graphs (modal arenas), and an
encoding of linear proofs as modal arenas equipped with vertex partitions
satisfying topological criteria. The second semantics is given by means of
winning innocent strategies of a two-player game over modal arenas. This is
given by extending the Heijltjes-Hughes-Stra{\ss}burger correspondence between
intuitionistic combinatorial proofs and winning innocent strategies in a
Hyland-Ong arena. Using our first result, we provide a characterisation of
winning strategies for games on a modal arena corresponding to proofs with
modalities.
