A group of fuzzy spacetime with SU(3) isometry is studied at the two-loop level in a IIB matrix model. It consists of spacetime from four to six dimensions, namely, from CP{sup 2} to SU(3)/U(1)xU(1). The effective action scales in a universal manner in the large N limit as N and N{sup 4/3} on four- and six-dimensional manifolds, respectively. The four-dimensional spacetime CP{sup 2} possesses the smallest effective action in this class.

Fuzzy spacetime with SU(3) isometry in the IIB matrix model

In this work we focus on the quantum Einstein-Yang-Mills sector quantized by the methods of loop quantum gravity. We point out the improved UV behavior of the coupled system as compared to pure quantum Yang-Mills theory on a fixed, classical background spacetime as was considered in a seminal work by Kogut and Susskind. Furthermore, we develop a calculational scheme by which the fundamental spectrum of the quantum Yang-Mills Hamiltonian can be computed in principle and by which one can make contact with the Wilsonian renormalization group, possibly purely within the Hamiltonian framework. Finally, we comment on the relationship of the fundamental spectrum to that of pure Yang-Mills theory on a (flat) classical spacetime.

Towards the fundamental spectrum of the quantum Yang-Mills theory

The 6-j symbols of a group are independent of the subgroup chain chosen to define the basis states. The authors present an improved algorithm for calculating the primitive 6-j symbols for a compact group with a faithful irrep by recursive building up using only the Kronecker product rules and the general properties of 6-j. Previously one has sometimes needed to search for the useful equations by systematically trying all equations which involve unknown 6-j. The authors show that the primitive 6-j may all be easily solved in terms of a subclass, the core 6-j. They discuss how the core 6-j can usually be solved, proving that the method is complete for SO3. The authors conjecture that the algorithm is complete for all groups.

https://iopscience.iop.org/article/10.1088/0305-4470/21/14/006/pdf

Calculation of primitive 6-j symbols

The symmetries of the SU(3) 3‐j symbols, which are defined as symmetrized matrix elements of the canonical SU(3) tensor operators are investigated. The symmetries considered are those which in SU(2) correspond to the interchange of columns of the 3‐j symbol, as well as the symmetry under conjugation. It is found that for each tensor operator in a multiplicity set the matrix elements (for a fixed operator pattern) carry a one‐dimensional representation of the symmetric group S3.

Symmetry properties of matrix elements of canonical SU(3) tensor operators

Generalizing the SU(3) canonical tensor operator concept (Biedenharn and Louck) to the quantum algebra uq(3), the Wigner–Clebsch–Gordan coefficients of uq(3) with repeating irreducible representations are considered. Extremal projectors of the quantum algebra uq(3) in terms of the ordered generator polynomials are used for evaluation of the bilinear combinations of the uq(3) canonical isoscalar factors. Explicit expressions of the uq(3) isofactors, corresponding to the maximal null space case of the uq(3) unit canonical tensor operators, and their normalization factors (denominator functions) are presented. The transposition and conjugation phase factors for the SU(3) and uq(3) canonical isofactors are correlated with phases and zeros of boundary isofactors. Invariance of the canonical isofactors (or absence of such invariance) under interchange of the tensor operator and the initial or final state parameters is correlated with the existence and invariance (or numerical degeneracy) of the usual splitting ...

Towards the canonical tensor operators of uq(3). I. The maximal null space case

Starting from the s-classified SU(3) Clebsch-Gordan coefficients introduced previously (Pluhar et al. 1985) the s-classified SU(3) 3j-, 6j- and 9j-symbols are constructed and examined. They satisfy simple symmetry relations similar to those valid for the SU(2) case.

P Holan

Papers

Symmetry properties of s-classified SU(3) 3j-, 6j- and 9j-symbols