9 research outputs found
Noncommutative Riemannian Geometry of the Alternating Group A_4
We study the noncommutative Riemannian geometry of the alternating group
A_4=(Z_2 \times Z_2)\cross Z_3 using a recent formulation for finite groups.
We find a unique `Levi-Civita' connection for the invariant metric, and find
that it has Ricci-flat but nonzero Riemann curvature. We show that it is the
unique Ricci-flat connection on with the standard framing (we solve the
vacuum Einstein's equation). We also propose a natural Dirac operator for the
associated spin connection and solve the Dirac equation. Some of our results
hold for any finite group equipped with a cyclic conjugacy class of 4 elements.
In this case the exterior algebra has dimensions
with top-form 9-dimensional. We also find the
noncommutative cohomology .Comment: 28 pages Latex no figure
Differential Geometry of Group Lattices
In a series of publications we developed "differential geometry" on discrete
sets based on concepts of noncommutative geometry. In particular, it turned out
that first order differential calculi (over the algebra of functions) on a
discrete set are in bijective correspondence with digraph structures where the
vertices are given by the elements of the set. A particular class of digraphs
are Cayley graphs, also known as group lattices. They are determined by a
discrete group G and a finite subset S. There is a distinguished subclass of
"bicovariant" Cayley graphs with the property that ad(S)S is contained in S.
We explore the properties of differential calculi which arise from Cayley
graphs via the above correspondence. The first order calculi extend to higher
orders and then allow to introduce further differential geometric structures.
Furthermore, we explore the properties of "discrete" vector fields which
describe deterministic flows on group lattices. A Lie derivative with respect
to a discrete vector field and an inner product with forms is defined. The
Lie-Cartan identity then holds on all forms for a certain subclass of discrete
vector fields.
We develop elements of gauge theory and construct an analogue of the lattice
gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear
connections are considered and a simple geometric interpretation of the torsion
is established.
By taking a quotient with respect to some subgroup of the discrete group,
generalized differential calculi associated with so-called Schreier diagrams
are obtained.Comment: 51 pages, 11 figure
Towards Spinfoam Cosmology
We compute the transition amplitude between coherent quantum-states of
geometry peaked on homogeneous isotropic metrics. We use the holomorphic
representations of loop quantum gravity and the
Kaminski-Kisielowski-Lewandowski generalization of the new vertex, and work at
first order in the vertex expansion, second order in the graph (multipole)
expansion, and first order in 1/volume. We show that the resulting amplitude is
in the kernel of a differential operator whose classical limit is the canonical
hamiltonian of a Friedmann-Robertson-Walker cosmology. This result is an
indication that the dynamics of loop quantum gravity defined by the new vertex
yields the Friedmann equation in the appropriate limit.Comment: 8 page