We study the expectation value of a nonplanar Wilson graph operator in
SL(2,C) Chern-Simons theory on S3. In particular we analyze its asymptotic
behaviour in the double-scaling limit in which both the representation labels
and the Chern-Simons coupling are taken to be large, but with fixed ratio. When
the Wilson graph operator has a specific form, motivated by loop quantum
gravity, the critical point equations obtained in this double-scaling limit
describe a very specific class of flat connection on the graph complement
manifold. We find that flat connections in this class are in correspondence
with the geometries of constant curvature 4-simplices. The result is fully
non-perturbative from the perspective of the reconstructed geometry. We also
show that the asymptotic behavior of the amplitude contains at the leading
order an oscillatory part proportional to the Regge action for the single
4-simplex in the presence of a cosmological constant. In particular, the
cosmological term contains the full-fledged curved volume of the 4-simplex.
Interestingly, the volume term stems from the asymptotics of the Chern-Simons
action. This can be understood as arising from the relation between
Chern-Simons theory on the boundary of a region, and a theory defined by an
F2 action in the bulk. Another peculiarity of our approach is that the sign
of the curvature of the reconstructed geometry, and hence of the cosmological
constant in the Regge action, is not fixed a priori, but rather emerges
semiclassically and dynamically from the solution of the equations of motion.
In other words, this work suggests a relation between 4-dimensional loop
quantum gravity with a cosmological constant and SL(2,C) Chern-Simons theory in
3-dimensions with knotted graph defects.Comment: 54+11 pages, 9 figure