In the past several decades there have been a number of proposals for computing with dual forms of non-abelian Yang-Mills theories on the lattice. Motivated by the gauge-invariant, geometric picture offered by dual models and successful applications of duality in the U (1) case, we revisit the question of whether it is practical to perform numerical computation using non-abelian dual models. Specifically, we consider three-dimensional SU (2) pure Yang-Mills as an accessible yet non-trivial case in which the gauge group is non-abelian. Using methods developed recently in the context of spin foam quantum gravity, we derive an algorithm for efficiently computing the dual amplitude and describe Metropolis moves for sampling the dual ensemble. We relate our algorithms to prior work in non-abelian dual computations of Hari Dass and his collaborators, addressing several problems that have been left open. We report results of spin expectation value computations over a range of lattice sizes and couplings that are in agreement with our conventional lattice computations. We conclude with an outlook on further development of dual methods and their application to problems of current interest.

We describe here some new results concerning the Lorentzian Barrett–Crane model, a well-known spin foam formulation of quantum gravity. Generalizing an existing finiteness result, we provide a concise proof of finiteness of the partition function associated with all non-degenerate triangulations of 4-manifolds and for a class of degenerate triangulations not previously shown. This is accomplished by a suitable re-factoring and re-ordering of integration, through which a large set of variables can be eliminated. The resulting formulation can be interpreted as a ‘dual variables’ model that uses hyperboloid variables associated with spin foam edges in place of representation variables associated with faces. We outline how this method may also be useful for numerical computations, which have so far proven to be very challenging for Lorentzian spin foam models.

No abstract

Abstract. We derive an explicit formula for the vertex amplitude of dual SU (2) Yang-Mills theory in four dimensions on the lattice, and provide an efficient algorithm (of order j 4 ) for its computation. This opens the way for both numerical and analytical development of dual methods, previously limited to the case of three dimensions.

We propose a gauge-independent mechanism for the area-law behavior of Wilson loop expectation values in terms of worldsheets spanning Wilson loops interacting with the spin foams that contribute to the vacuum partition function. The method uses an exact transformation of lattice-regularized Yang-Mills theory that is valid for all couplings. Within this framework, some natural conjectures can be made as to what physical mechanism enforces the confinement property in the continuum (weak coupling) limit. Details for the SU (2) case in three dimensions are provided in a companion paper.