Phase space structure of Chern–Simons theory with a non-standard puncture

Abstract

We explicitly determine the symplectic structure on the phase space of Chern-Simons theory with gauge group G ⋉ g * on a three-manifold of topology R × S ∞ g,n , where S ∞ g,n is a surface of genus g with n + 1 punctures. At each puncture additional variables are introduced and coupled minimally to the Chern-Simons gauge field. The first n punctures are treated in the usual way and the additional variables lie on coadjoint orbits of G ⋉ g * . The (n + 1)st puncture plays a distinguished role and the associated variables lie in the cotangent bundle of G ⋉ g * . This allows us to impose a curvature singularity for the Chern-Simons gauge field at the distinguished puncture with an arbitrary Lie algebra valued coefficient. The treatment of the distinguished puncture is motivated by the desire to construct a simple model for an open universe in the Chern-Simons formulation of (2 + 1)-dimensional gravity.

Symmetries of quantum spacetime in three dimensions

“…Since each link-1/2 carries a representation vector space V = C 2 , we can derive the form of the R-matrix associated to the generatorsĤ + [N p ] by studying the action of the crossing operators on the tensor product vector space V ⊗ V . We can then interpret the diagrammatic relations (34), (40) above as relations between operators:…”

“…Given an orthonormal basis of V = C 2 formed by the vectors v 1 , v 2 , we want to define the action of the cup and cap operators, respectively and , on such basis which is compatible with the relations (34) and such that the bracket (40) as well as the identity = = (42) are satisfied. This happens for the following actions of the cup and cap operators:…”

“…The action of the R-matrix (50) on V ⊗ V provides a representation of the noncommutative holonomies braiding (34). More precisely, the crossing of two noncommutative holonomies can be represented as the action of the R-matrix on the tensor product vector space of two copies of V , followed by the switch of the two vector spaces.…”

“…Let us summarize: we found that the R-matrix associated to the crossing of two operators corresponding to the generatorsĤ + [N p ] is the R-matrix of the Hopf algebra su q (2) with q = exp(i ̵ h √ Λ 2κ). By the same argument, the R-matrix associated to the crossing of two operators corresponding to the generatorsĤ − [N p ] is the R-matrix of the Hopf algebra su q ′ (2) with q ′ = exp(−i ̵ h √ Λ 2κ), see (34), (35). The fact that the deformation parameters of these two sectors are inverse to each other is remarkable and, as we will see below, crucial for the existence of the non-singular Λ → 0 limit (see [9]).…”

“…This aspect represents a crucial departure from the four dimensional case and it allows for a different approach to the quantization of the system. In order to clarify this point, let us first recall some basic elements of the inclusion of particles in 3D gravity (see, for instance, [14,[30][31][32][33][34][35] and references therein).…”