Cyclic Foam Topological Field Theories

TL;DR

This paper introduces an axiomatic framework for Cyclic Foam Topological Field theories, linking them to graph-Cardy-Frobenius algebras and providing concrete examples of such theories.

Contribution

It establishes a one-to-one correspondence between Cyclic Foam Topological Field theories and graph-Cardy-Frobenius algebras, expanding the mathematical understanding of string-like topological models.

Findings

Established the axiomatic foundation for Cyclic Foam Topological Field theories.

Proved the correspondence with graph-Cardy-Frobenius algebras.

Constructed explicit examples of these theories and algebras.

Abstract

This paper proposes an axiomatic 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 two-manifolds with one-dimensional singularities. We proved that Cyclic Foam Topological Field theories one-to-one correspond to graph-Cardy-Frobenius algebras, that are families $(A,B_\star,\phi)$, where $A=\{A^s|s\in S\}$ are families of commutative associative Frobenius algebras, $B_\star = \bigoplus_{\sigma\in\Sigma} B_\sigma$ is an graduated by graphes, associative algebras of Frobenius type and $\phi=\{\phi_\sigma^s: A^s\to (B_\sigma)|s\in S,\sigma\in \Sigma\}$ is a family of special representations. There are constructed examples of Cyclic Foam Topological Field theories and its graph-Cardy-Frobenius algebras