124 research outputs found
Emergence of Riemannian geometry and the massive graviton
We overview a new mechanism whereby classical Riemannian geometry emerges out
of the differential structure on quantum spacetime, as extension data for the
classical algebra of differential forms. Outcomes for physics include a new
formula for the standard Levi-Civita connection, a new point of view of the
cosmological constant as a very small mass for the graviton of around
ev, and a weakening of metric-compatibility in the presence of
torsion. The same mechanism also provides a new construction for quantum
bimodule connections on quantum spacetimes and a new approach to the quantum
Ricci tensor.Comment: 14 pages, Amslatex, conference proceedings introduction; only fixed a
few typos compared to previous versio
Riemannian geometry of quantum groups and finite groups with nonuniversal differentials
We construct noncommutative `Riemannian manifold' structures on dual
quasitriangular Hopf algebras such as with its standard bicovariant
differential calculus, using the quantum frame bundle formalism introduced
previously. The metric is provided by the braided-Killing form on the
braided-Lie algebra on the tangent space and the -bein by the Maurer-Cartan
form. We also apply the theory to finite sets and in particular to finite group
function algebras with differential calculi and Killing forms determined
by a conjugacy class. The case of the permutation group is worked out
in full detail and a unique torsion free and cotorsion free or `Levi-Civita'
connection is obtained with noncommutative Ricci curvature essentially
proportional to the metric (an Einstein space). We also construct Dirac
operators in the metric background, including on finite groups such as .
In the process we clarify the construction of connections from gauge fields
with nonuniversal calculi on quantum principal bundles of tensor product form.Comment: 43 pages, 1 figure. Revised August 2001 to cut page length (eg
deleted appendix) for publication in CMP. Also fleshed out ex. of q-Killing
metric for q-SU_2 previously mentioned (no significant additions
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