Geometric spin foams, Yang-Mills theory and background-independent models
Florian Conrady
TL;DR
The paper develops a geometric, non-perturbative framework linking lattice gauge theory to spin foam sums by grouping plaquettes into single-colored surfaces and treating spin foams as branched, lattice-embedded surfaces with area-weighted amplitudes. It derives a simple dual spin foam model for SU$(N)$ Yang–Mills theory in $d\ge2$, where each unbranched sheet contributes a factor $\exp(-T_\rho \mathcal{A})$, with a representation-dependent tension $T_\rho(a) = a^{(d-6)} \gamma^2(a) C_\rho$, thus mapping the running gauge coupling to edge tensions. To achieve background independence in gravity, the author imposes a symmetry that amplitudes depend only on the geometry of spin foams, leading to a sum over abstract spin foams (equivalence classes under homeomorphisms) and a Barrett–Crane example, while acknowledging potential divergences requiring truncation. The work connects gauge theory, gravity spin foams, and group field theory, offering a pathway to a background-free, non-perturbative quantum gravity framework and motivating future exploration of fermions, large-$N$ limits, and YM–gravity couplings.
Abstract
We review the dual transformation from pure lattice gauge theory to spin foam models with an emphasis on a geometric viewpoint. This allows us to give a simple dual formulation of SU(N) Yang-Mills theory, where spin foam surfaces are weighted with the exponentiated area. In the case of gravity, we introduce a symmetry condition which demands that the amplitude of an individual spin foam depends only on its geometric properties and not on the lattice on which it is defined. For models that have this property, we define a new sum over abstract spin foams that is independent of any choice of lattice or triangulation. We show that a version of the Barrett-Crane model satisfies our symmetry requirement.