Lattice Chern-Simons gravity via the Ponzano-Regge model

2011

Physics Letters B, 2000

We revisit the lattice formulation of the Abelian Chern-Simons model defined on an infinite Euclidean lattice. We point out that any gauge invariant, local and parity odd Abelian quadratic form exhibits, in addition to the zero eigenvalue associated with the gauge invariance and to the physical zero mode at p = 0 due to translational invariance, a set of extra zero eigenvalues inside the Brillouin zone. For the Abelian Chern-Simons theory, which is linear in the derivative, this proliferation of zero modes is reminiscent of the Nielsen-Ninomiya no-go theorem for fermions. A gauge invariant, local and parity even term such as the Maxwell action leads to the elimination of the extra zeros by opening a gap with a mechanism similar to that leading to Wilson fermions on the lattice.

Journal of High Energy Physics, 2003

We study Chern-Simons theory with a complex G C or a real G × G gauge group on a manifold with boundary -this includes Lorentzian and Euclidean (anti-) de Sitter (E/A)dS gravity for G = SU (2) or G = SL(2, R). We show that there is a canonical choice of boundary conditions that leads to an unambiguous, fully covariant and gauge invariant, off-shell derivation of the boundary action -a G C /G or G WZW model, coupled in a gauge invariant way to the boundary value of the gauge field. In particular, for (E/A)dS gravity, the boundary action is a WZW model with target space (E/A)dS 3 , reminiscent of a worldsheet for worldsheet mechanism. We discuss in some detail the properties of the boundary theories that arise and we confront our results with various related constructions in the literature.

Physics Letters B, 1990

The abelian gauge-invariant lattice theory is constructed which in the continuum limit and in the case of the U( 1 ) gauge group tends to the Chern-Simons topological gauge theory.

Journal of Physics A: Mathematical and General, 2003

Chern-Simons type Lagrangians in d = 3 dimensions are analyzed from the point of view of their covariance and globality. We use the transgression formula to find out a new fully covariant and global Lagrangian for Chern-Simons gravity: the price for establishing globality is hidden in a bimetric (or biconnection) structure. Such a formulation allows to calculate from a global and simpler viewpoint the energymomentum complex and the superpotential both for Yang-Mills and gravitational examples.

Nuclear Physics B, 1997

We consider the role of the diffeomorphism constraint in the quantization of lattice formulations of diffeomorphism invariant theories of connections. It has been argued that in working with abstract lattices, one automatically takes care of the diffeomorphism constraint in the quantum theory. We use two systems in order to show that imposing the diffeomorphism constraint is imperative to obtain a physically acceptable quantum theory. First, we consider 2 + 1 gravity where an exact lattice formulation is available. Next, general theories of connections for compact gauge groups are treated, where the quantum theories are known -for both the continuum and the lattice-and can be compared.

We propose an extension of Vasiliev's supertrace operation for the enveloping algebra of Wigner's deformed oscillator algebra to the fractional spin algebra given in arXiv:1312.5700. The resulting three-dimensional Chern-Simons theory unifies the Blencowe-Vasiliev higher spin gravity with fractional spin fields and internal gauge potentials. For integer or half-integer fractional spins, infinite dimensional ideals arise and decouple, leaving finite dimensional gauge algebras gl(2l+1) or gl(l|l+1) and various real forms thereof. We derive the relation between gravitational and internal gauge couplings.

Classical and Quantum Gravity, 2003

In the formulation of (2+1)-dimensional gravity as a Chern-Simons gauge theory, the phase space is the moduli space of flat Poincar\'e group connections. Using the combinatorial approach developed by Fock and Rosly, we give an explicit description of the phase space and its Poisson structure for the general case of a genus g oriented surface with punctures representing particles and a boundary playing the role of spatial infinity. We give a physical interpretation and explain how the degrees of freedom associated with each handle and each particle can be decoupled. The symmetry group of the theory combines an action of the mapping class group with asymptotic Poincar\'e transformations in a non-trivial fashion. We derive the conserved quantities associated to the latter and show that the mapping class group of the surface acts on the phase space via Poisson isomorphisms.

Nuclear Physics B - Proceedings Supplements, 2001

We analyse the pure Chern-Simons theory on an Euclidean infinite lattice. We point out that, as a consequence of its symmetries, the Chern-Simons theory does not have an integrable kernel. Due to the linearity of the action in the derivatives, the situation is very similar to the one arising in the lattice formulation of fermionic theories. Doubling of bosonic degrees of freedom is removed by adding a Maxwell term with a mechanism similar to the one proposed by Wilson for the fermionic theories.