SL(2,C)Chern–Simons theory, a non-planar graph operator, and 4D quantum gravity with a cosmological constant: Semiclassical geometry

Abstract

We study the expectation value of a nonplanar Wilson graph operator in SL(2,C) Chern-Simons theory on S 3 . 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 F 2 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.

3D holography: from discretum to continuum

“…loop quantum gravity has been hindered by mainly two issues: firstly the construction of models incorporating a negative cosmological constant is much more involved than models with vanishing or positive cosmological constants, but recent progress can be found in [12][13][14][15][16]. These complications can be avoided by turning to a generalization of the AdS/CFT duality to asymptotically flat gravity, which has been considered in [17][18][19][20].…”

Holographic description of boundary gravitons in (3+1) dimensions

“…Another generalization would be to add a cosmological constant. This can also be considered within Regge calculus, if one uses homogeneously curved building blocks [41][42][43][44].…”

“…8 There is, however, even 7 This is true if one uses flat simplices. There is also a Regge action for homogeneously curved simplices [40][41][42][43][44], which is particularly appropriate in the presence of a cosmological constant. The vertex translation symmetries are then present for solutions describing a homogeneously curved spacetime.…”

“…This holds in the original version[25] of Regge calculus. Note that one can also work with homogeneously curved building blocks[40][41][42][43][44], which allows generalization of this argument to (Regge) gravity with a cosmological constant.…”

Quantum gravity kinematics from extended TQFTs

“…The TV model describes Euclidean quantum gravity with a positive cosmological constant. There exists a tight relationship between the quantum deformation parameter and the cosmological constant, and a similar role of the quantum deformation is believed to hold also in + ( ) 3 1 dimensions [46][47][48][49][50][51][52][53]. Another important feature of the quantum deformation at root of unity is that one can expect the Hilbert spaces associated to a fixed graph (which here will be replaced by Hilbert spaces on manifolds with fixed punctures) to be finitedimensional.…”