Dual computations of non-Abelian Yang-Mills theories on the lattice

Abstract

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.

A dual non-Abelian Yang–Mills amplitude in four dimensions

“…3 A similar phenomenon can arise in the three-dimensional case [24], although there it is sufficient to use two closely related network evaluations; alternatively, a translation invariant splitting can be used [20]. to a spin network in the sense of [22,23]; although the two are equal in magnitude, in general there is a sign factor that depends on the spin arguments.…”

“…To do this we follow the procedure described in Appendix A of [20], which relates the original spin network (with oriented edges) defining the 48j symbol 1 Of course, for a given labelling of plaquettes, the sum over intertwiner labels of the product of edge and vertex amplitudes over the lattice is independent of the choice of splitting.…”

“…It was demonstrated in [20] that numerical computations could be carried out for the dual form of LGT with G = SU (2) on a cubic three-dimensional (D = 3) lattice. One major hurdle to extending the algorithm of [20] to D = 4 is the problem of finding a closed-form formula for the vertex amplitude that can be evaluated efficiently on a computer and gives reasonable scaling with increasing spin labels.…”

“…One major hurdle to extending the algorithm of [20] to D = 4 is the problem of finding a closed-form formula for the vertex amplitude that can be evaluated efficiently on a computer and gives reasonable scaling with increasing spin labels.…”

“…a3 U j4 (g (1) is a projection map from the space of all linear maps j 6 ⊗ j 5 ⊗ j 4 → j 1 ⊗ j 2 ⊗ j 3 onto the space of intertwiners (maps commuting with the action of G), and can thus be expanded using an orthogonal basis of intertwiners I i : Carrying out the contractions results in a spin foam model (see, for example, [19,20,21] for details), with the vertex amplitude defined as a spin network evaluation of the network shown in Figure 1. In drawing this figure, we have placed the vertex labelled −t "at infinity".…”