Four-Dimensional Spin Foam Perturbation Theory

We define a four-dimensional spin-foam perturbation theory for the ${\rm BF}$-theory with a $B\wedge B$ potential term defined for a compact semi-simple Lie group $G$ on a compact orientable 4-manifold $M$. This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group $U_q (\mathfrak{g})$ where $\mathfrak{g}$ is the Lie algebra of $G$ and $q$ is a root of unity. The Chain–Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners $\Lambda\otimes \Lambda \to A$, where $A$ is the adjoint representation of $\mathfrak{g}$, is 1-dimensional for each irrep $\Lambda$. We calculate the partition function $Z$ in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold $M$. We prove that the first-order perturbative contribution vanishes for finite triangulations, so that we define a dilute-gas limit by using the second-order contribution. We show that $Z$ is an analytic continuation of the Crane–Yetter partition function. Furthermore, we relate $Z$ to the partition function for the $F\wedge F$ theory.