Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group

Classical and Quantum Gravity (Impact Factor: 3.17). 02/2002; 19(19):4953-5015. DOI: 10.1088/0264-9381/19/19/313
Source: arXiv


We analyse the Hamiltonian quantization of Chern-Simons theory associated with the real group SL(2, Sigma with punctures. This algebra, the so-called moduli algebra, is constructed along the lines of Fock-Rosly, Alekseev-Grosse-Schomerus, Buffenoir-Roche using only finite-dimensional representations of Uq(sl(2, Bbb C)Bbb R). It is shown that this algebra admits a unitary representation acting on a Hilbert space which consists of wave packets of spin networks associated with principal unitary representations of Uq(sl(2, Bbb C)Bbb R). The representation of the moduli algebra is constructed using only Clebsch-Gordan decomposition of a tensor product of a finite-dimensional representation with a principal unitary representation of Uq(sl(2, Bbb C)Bbb R). The proof of unitarity of this representation is nontrivial and is a consequence of the properties of Uq(sl(2, Bbb C)Bbb R) intertwiners which are studied in depth. We analyse the relationship between the insertion of a puncture coloured with a principal representation and the presence of a worldline of a massive spinning particle in de Sitter space.

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    • "In the 3d case, the resulting q-deformation of the corresponding Lorentz group is very well understood from the canonical point of view in the combinatorial quantization framework for Chern-Simons theory [8] [9] [10] and from the path integral perspective both from the Chern-Simons quantization [11] and the Turaev-Viro spinfoam model [12] with the asymptotics of the q-deformed {6j}-symbols [13]. From the canonical perspective of loop quantum gravity, which is subtlety different from the Chern-Simons theory, this issue is not entirely settled despite some recent interesting efforts [14] [15] and [16] [17] [18] [19]. "
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    ABSTRACT: We investigate the generalization of loop gravity's twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant SU(2) gauge theory. Its classical states are graphs provided with algebraic data. In particular closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in flat space R^3. One then glues them allowing for both curvature and torsion. It was recently conjectured that q-deforming the gauge group SU(2) would allow to account for a non-vanishing cosmological constant Lambda, and in particular that deforming the loop gravity phase space with real parameter q>0 would lead to a generalization of twisted geometries to a hyperbolic curvature. Following this insight, we look for generalization of the closure constraints to the hyperbolic case. In particular, we introduce two new closure constraints for hyperbolic tetrahedra. One is compact and expressed in terms of normal rotations (group elements in SU(2) associated to the triangles) and the second is non-compact and expressed in terms of triangular matrices (group elements in SB(2,C)). We show that these closure constraints both define a unique dual tetrahedron (up to global translations on the three-dimensional one-sheet hyperboloid) and are thus ultimately equivalent.
    Preview · Article · Jan 2015 · Classical and Quantum Gravity
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    • "For Chern-Simons theory with non-compact gauge group SL(2, C) and couplings k and s the situation is more complicated. In [47] it is shown that for the Lorentzian unitary case (i), with s ∈ R, and under the restriction k = 0, the quantum SL(2, C) Chern-Simons theory is related to the unitary irreps of the quantum group SL q (2, C) with real deformation q = exp 2π s . Moreover, a quantum group deformation of SL(2, C) is only known for such a real deformation parameter q [45] [46]. "
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    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.
    Full-text · Article · Dec 2014 · Nuclear Physics B
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    • "It has been studied intensively since the discoveries by Witten that first three dimensional gravity can be reformulated in terms of a Chern-Simons theory [15], and then that the quantum amplitudes of the Chern-Simons theory are closely related to topological three-manifolds invariants and knots invariants [16]. The so-called combinatorial quantization is certainly one of the most powerful Hamiltonian quantization scheme of the Chern-Simons theory because it can be applied not only to a large class of compact [17– 19] and non-compact gauge groups [20] [21] [22] [23] [24] but also to any punctured Riemann surfaces Σ (Note that Loop Quantum Gravity techniques are very successful when there is no cosmological constant [25] [26], and the case of non-vanishing cosmological is still under construction [27] [28]). When the gauge group is G = SU (2) and the surface Σ is a punctured two-sphere, the physical Hilbert space H(j ), where (j 1 , · · · , j n ) color the punctures, has a very simple structure. "
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    ABSTRACT: We define the analytic continuation of the number of black hole microstates in Loop Quantum Gravity to complex values of the Barbero-Immirzi parameter γ. This construction deeply relies on the link between black holes and Chern-Simons theory. Technically, the key point consists in writing the number of microstates as an integral in the complex plane of a holomorphic function, and to make use of complex analysis techniques to perform the analytic continuation. Then, we study the thermodynamical properties of the corresponding system (the black hole is viewed as a gas of indistinguishable punctures) in the framework of the grand canonical ensemble where the energy is defined à la Frodden-Gosh-Perez from the point of view of an observer located close to the horizon. The semi-classical limit occurs at the Unruh temperature T U associated to this local observer. When γ = ±i, the entropy reproduces at the semi-classical limit the area law with quantum corrections. Furthermore, the quantum corrections are logarithmic provided that the chemical potential is fixed to the simple value μ = 2T U.
    Full-text · Article · Jun 2014 · Journal of High Energy Physics
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