35,017 research outputs found
Stable Flags and the Riemann-Hilbert Problem
We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise
logarithmic modifications of the classical R\"ohrl-Deligne vector bundle. We
show that the solutions of the Riemann-Hilbert problem are in bijection with
some families of local filtrations which are stable under the prescribed
monodromy maps. We introduce the notion of Birkhoff-Grothendieck
trivialisation, and show that its computation corresponds to geodesic paths in
some local affine Bruhat-Tits building. We use this to compute how the type of
a bundle changes under stalk modifications, and give several corresponding
algorithmic procedures.Comment: 39 page
On Semi-Classical States of Quantum Gravity and Noncommutative Geometry
We construct normalizable, semi-classical states for the previously proposed
model of quantum gravity which is formulated as a spectral triple over holonomy
loops. The semi-classical limit of the spectral triple gives the Dirac
Hamiltonian in 3+1 dimensions. Also, time-independent lapse and shift fields
emerge from the semi-classical states. Our analysis shows that the model might
contain fermionic matter degrees of freedom.
The semi-classical analysis presented in this paper does away with most of
the ambiguities found in the initial semi-finite spectral triple construction.
The cubic lattices play the role of a coordinate system and a divergent
sequence of free parameters found in the Dirac type operator is identified as a
certain inverse infinitesimal volume element.Comment: 31 pages, 10 figure
Line arrangements and direct sums of free groups
We show that if the fundamental groups of the complements of two line
arrangements in the complex projective plane are isomorphic to the same direct
sum of free groups, then the complements of the arrangements are homotopy
equivalent. For any such arrangement, we construct another arrangement that is
complexified-real, the intersection lattices of the arrangements are
isomorphic, and the complements of the arrangements are diffeomorphic.Comment: 15 pages, 4 figure
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