Cyclic foam topological field theories

This paper proposes an axiomatic form for cyclic foam topological field theories, that is, topological field theories corresponding to string theories where particles are arbitrary graphs. World surfaces in this case are 2-manifolds with one-dimensional singularities. I prove that cyclic foam topological field theories are in one-to-one correspondence with graph-Cardy–Frobenius algebras that are families (A,B⋆,ϕ)(A,B⋆,ϕ) where A={As|s∈S}A={As|s∈S} are families of commutative associative Frobenius algebras, B⋆=⨁σ∈ΣBσB⋆=⨁σ∈ΣBσ is an associative algebra of Frobenius type graduated by graphs, and ϕ={ϕσs:As→End(Bσ)|s∈S,σ∈Σ} is a family of special representations. Examples of cyclic foam topological field theories and graph-Cardy–Frobenius algebras are constructed.