Four-Dimensional Spin Foam Perturbation Theory

[摘要] We define a four-dimensional spin-foam perturbation theory for the BF-theory with a B ∧ 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 ( g ) where 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 Λ⊗Λ→ A , where A is the adjoint representation of g , is 1-dimensional for each irrep Λ. 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 ∧ F theory.

[关键词] spin foam models;BF-theory;spin networks;dilute-gas limit;Crane–Yetter invariant;spin-foam perturbation theory