Four-Dimensional Spin Foam Perturbation Theory

We define a four-dimensional spin-foam perturbation theory for the BF-theorywith a B ∧ B potential term defined for a compact semi-simple Lie group G on a compactorientable 4-manifold M . This is done by using the formal spin foam perturbative seriescoming from the spin-foam generating functional. We then regularize the terms in theperturbative series by passing to the category of representations of the quantum group Uq (g)where g is the Lie algebra of G and q is a root of unity. The Chain–Mail formalism can beused to calculate the perturbative terms when the vector space of intertwiners Λ ⊗ Λ → A,where A is the adjoint representation of g, is 1-dimensional for each irrep Λ. We calculate thepartition function Z in the dilute-gas limit for a special class of triangulations of restrictedlocal 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 continuationof the Crane–Yetter partition function. Furthermore, we relate Z to the partition functionfor the F ∧ F theory.