There is an elegant alternative to the quenched Eguchi–Kawai model, described in the previous chapter, which also preserves the U(1)d symmetry. It was proposed by González-Arroyo and Okawa [GO83a, GO83b] on the basis of a twisting reduction prescription. The corresponding lattice version of the reduced model lives on a hypercube with twisted boundary conditions. The twisted reduced model for a scalar field was constructed by Eguchi and Nakayama [EN83].

The twisted reduced models reveal interesting mathematical structures associated with representations of the Heisenberg commutation relation (in the continuum) or its finite-dimensional approximation by unitary matrices (on the lattice). In contrast to the quenched reduced models which describe only planar graphs, the twisted reduced models make sense order by order in 1/N and even at finite N. In this case they are associated with gauge theories on noncommutative space, whose limit of large noncommutativity is given by planar graphs thereby reproducing a d-dimensional Yang–Mills theory at large N.

We begin this chapter with a description of the twisted reduced models first on the lattice and then in the continuum and show how they describe planar graphs of a d-dimensional theory.

Twisting prescription

We start by working on a lattice to make the results rigorous and then repeat them for the continuum.

The twisting reduction prescription is a version of the general reduction prescription described in Sect. 14.1.