### 3.10 Coupling to SU(2)-gauge fields

Berg and collaborators [42, 43, 44, 27, 28, 41] coupled the pure-curvature action geometrically to the
Wilson action for -gauge fields via dimensionless weight factors ,
where denotes the -holonomy around and is proportional to the inverse square
coupling constant, . One motivation was to understand whether in the well-defined pure-gravity
region, one can choose the elementary particle masses to be as , as one might expect for
a realistic gravity+matter system. This seems a rather distant hope, since in the simulations performed so
far, the ratio is of order unity.
Initial computations were performed on a -lattice with the scale-invariant measure, and at
[42, 44], and extended to larger -values in [43]. For , one finds some evidence for
a (first-order?) transition; the region of where the transition occurs does not change much with
. Beirl et al. [27, 28] extended this analysis by measuring the static potential of a
quark–antiquark pair on lattices of size and . With and without gravity, one finds
both a confined and a deconfined phase; in the presence of gravity, the transition occurs at a
smaller -value. More recently, Berg et al. [41] have gathered further data on the location and
stability of the well-defined phase in the -plane, and extracted a string tension for various
-values.