Combinatorial quantization of the Hamiltonian Chern-Simons theory II

This paper further develops the combinatorial approach to quantization of the Hamiltonian Chern Simons theory advertised in [1]. Using the theory of quantum Wilson lines, we show how the Verlinde algebra appears within the context of quantum group gauge theory. This allows to discuss flatness of quantum connections so that we can give a mathematically rigorous definition of the algebra of observablesA _CS of the Chern Simons model. It is a *-algebra of functions on the quantum moduli space of flat connections and comes equipped with a positive functional (integration). We prove that this data does not depend on the particular choices which have been made in the construction. Following ideas of Fock and Rosly [2], the algebraA _CS provides a deformation quantization of the algebra of functions on the moduli space along the natural Poisson bracket induced by the Chern Simons action. We evaluate a volume of the quantized moduli space and prove that it coincides with the Verlinde number. This answer is also interpreted as a partition partition function of the lattice Yang-Mills theory corresponding to a quantum gauge group.Supported by Swedish Natural Science Research Council (NFR) under the contract F-FU 06821304 and by the Federal Ministry of Science and Research, Austria.Part of project P8916-PHY of the Fonds zur Förderung der wissenschaftlichen Forschung in ÖsterreichSupported in part by DOE Grant No DE-FG02-88ER25065     

Explicit quantization of the Chern-Simons action

We quantize the three-dimensional Chern-Simons action explicitly. We found that the geometric quantization of the action strongly depends on the topology of the (fixed-time) Riemann surface. On the disk the phase space and the symplectic structure are the same as those of the (chiral) Wess-Zumino-Witten model. On the torus the Hilbert space is the vector space of characters of Kac-Moody algebras. The fusion rules of the primary fields are derived from theclassical matching condition of the holonomy. In general case, the wave-functional of the theory is the generating function of the current insertion in Wess-Zumino-Witten model.     

Supermembrane with non-Abelian gauging and Chern-Simons quantization

We present the non-Abelian gaugings of supermembranes for general isometries for compactifications from eleven-dimensions, starting with an Abelian case as a guide. We introduce a super Killing vector in eleven-dimensional superspace for a non-Abelian group G associated with the compact space B for a general compactification, and couple it to a non-Abelian gauge field on the world-volume. As a technical tool, we use teleparallel superspace with no manifest local Lorentz covariance. Interestingly, the coupling constant is quantized for the non-Abelian group G, due to its non-trivial mapping . Received: 15 September 2004, Revised: 12 November 2004, Published online: 14 January 2005 PACS: 11.25.Mj, 11.25.Tq, 04.50. + h, 04.65. + e     

Becchi-Rouet-Stora-Tyutin quantization and Hamiltonian formalism

An introductory review of BRST hamiltonian formalism is presented. The method of quantization of gauge and string theories is discussed. A few simple examples are presented to illustrate the BRST techniques.     

The quantization of the Hamiltonian in curved space

The construction of the quantum-mechanical Hamiltonian by canonical quantization is examined. The results are used to enlighten examples taken from slow nuclear collective motion. Hamiltonians, obtained by a thoroughly quantal method (generator-coordinate method) and by the canonical quantization of the semiclassical Hamiltonian, are compared. The resulting simplicity in the physics of a system constrained to lie in a curved space by the introduction of local Riemannian coordinates is emphasized. In conclusion, a parallel is established between the result for various coordinates and a proposed procedure for quantizing the semiclassical Hamiltonian for a single coordinate.Partially supported by Fundação Calouste Gulbenkian, Lisboa.     

Discrete Hamiltonian symmetries and semiclassical quantization

Most quantum Hamiltonian systems exhibit discrete symmetries. Allowing for these is crucial when properly calculating the fluctuation properties of the quantal spectrum. These properties are then employed to distinguish between classically chaotic or non-chaotic quantum systems. In general, semiclassical quantization procedures do not take into account irreducible representations of the Hamiltonian. A procedure is presented to take these into account in semiclassical quantization schemes and calculate some of the energy eigenvalues belonging to a specific irreducible representation.     

Hamiltonian N=2 superfield quantization

We present a superfield construction of Hamiltonian quantization with N=2 supersymmetry generated by two fermionic charges Q^a. As a byproduct of the analysis we also derive a classically localized path integral from two fermionic objects Σ^a that can be viewed as “square roots” of the classical bosonic action under the product of a functional Poisson bracket.     

Witten's identity for Chern-Simons theory

We present a simple heuristic calculational scheme to relate the expectation value of Wilson loops in Chern-Simons theory to the Jones polynomial. We consider the exponential of the generator of homotopy transformations which produces the finite loop deformations that define the crossing change formulas of knot polynomials. Applying this operator to the expectation value of Wilson loops for an unspecified measure, we find a set of conditions on the measure and the regularization such that the Jones polynomial is obtained.