Four-Dimensional Spin Foam Perturbation Theory

• Main Authors: João Faria Martins, Aleksandar Mikovic
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2011-10-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: spin foam models; BF-theory; spin networks; dilute-gas limit; Crane-Yetter invariant; spin-foam perturbation theory
• Online Access: http://dx.doi.org/10.3842/SIGMA.2011.094

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.