717 research outputs found
Discrete and continuum third quantization of Gravity
We give a brief introduction to matrix models and the group field theory
(GFT) formalism as realizations of the idea of a third quantization of gravity,
and present in some more detail the idea and basic features of a continuum
third quantization formalism in terms of a field theory on the space of
connections, building up on the results of loop quantum gravity that allow to
make the idea slightly more concrete. We explore to what extent one can
rigorously define such a field theory. Concrete examples are given for the
simple case of Riemannian GR in 3 spacetime dimensions. We discuss the relation
between GFT and this formal continuum third quantized gravity, and what it can
teach us about the continuum limit of GFTs.Comment: 21 pages, 5 eps figures; submitted as a contribution to the
proceedings of the conference "Quantum Field Theory and Gravity Conference
Regensburg 2010" (28 September - 1 October 2010, Regensburg/Bavaria); v2:
preprint number include
On the Expansions in Spin Foam Cosmology
We discuss the expansions used in spin foam cosmology. We point out that
already at the one vertex level arbitrarily complicated amplitudes contribute,
and discuss the geometric asymptotics of the five simplest ones. We discuss
what type of consistency conditions would be required to control the expansion.
We show that the factorisation of the amplitude originally considered is best
interpreted in topological terms. We then consider the next higher term in the
graph expansion. We demonstrate the tension between the truncation to small
graphs and going to the homogeneous sector, and conclude that it is necessary
to truncate the dynamics as well.Comment: 17 pages, 4 figures, published versio
Torus graphs and simplicial posets
For several important classes of manifolds acted on by the torus, the
information about the action can be encoded combinatorially by a regular
n-valent graph with vector labels on its edges, which we refer to as the torus
graph. By analogy with the GKM-graphs, we introduce the notion of equivariant
cohomology of a torus graph, and show that it is isomorphic to the face ring of
the associated simplicial poset. This extends a series of previous results on
the equivariant cohomology of torus manifolds. As a primary combinatorial
application, we show that a simplicial poset is Cohen-Macaulay if its face ring
is Cohen-Macaulay. This completes the algebraic characterisation of
Cohen-Macaulay posets initiated by Stanley. We also study blow-ups of torus
graphs and manifolds from both the algebraic and the topological points of
view.Comment: 26 pages, LaTeX2e; examples added, some proofs expande
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