Spin foam

Definition

Let $\kappa$ be a finite two-dimensional oriented cell complex, with vertices $v$, edges $e$ and faces $f$, whose boundary $\partial \kappa$ is a graph. A spin foam on $\kappa$ is an assignment of:

1. a nonzero spin $j_f$, labelling an irreducible representation of $SU(2)$ (see representation theory of SU(2)), to each face $f$;
2. an intertwiner $\iota_e$ between the representations of the faces that meet at $e$ to each edge $e$.

The restriction to $\partial \kappa$ is then a spin network. A spin foam model assigns amplitudes $A_f(j_f)$, $A_e(j, \iota)$ and $A_v(j, \iota)$ to faces, edges and vertices. It defines the amplitude of $\kappa$ with a fixed boundary spin network as $$Z(\kappa) = \sum_{j, \iota} \prod_f A_f \prod_e A_e \prod_v A_v.$$ The sum runs over all spin foams on $\kappa$ that restrict to the given boundary spin network.