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Title: Stable homology as an indicator of manifoldlikeness in causal set theory
Authors: Major, Seth A.
Rideout, David
Surya, Sumati
Keywords: Lattice and discrete methods
Logic and set theory
Spacetime topology, causal structure, spinor structure
Issue Date: Sep-2009
Publisher: IOP Publishing Ltd.
Citation: Classical and Quantum Gravity, 2009, Vol.26, p175008
Abstract: We present a computational tool that can be used to obtain the ‘spatial’ homology groups of a causal set. Localization in the causal set is seeded by an inextendible antichain, which is the analogue of a spacelike hypersurface, and a one-parameter family of nerve simplicial complexes is constructed by ‘thickening’ this antichain. The associated homology groups can then be calculated using existing homology software, and their behaviour studied as a function of the thickening parameter. Earlier analyticalwork showed that for an inextendible antichain in a causal set which can be approximated by a globally hyperbolic spacetime region, there is a one-parameter sub-family of these simplicial complexes which are homological to the continuum, provided the antichain satisfies certain conditions. Using causal sets that are approximated by a set of 2D spacetimes, our numerical analysis suggests that these conditions are generically satisfied by inextendible antichains. In both 2D and 3D simulations, as the thickening parameter is increased, the continuum homology groups tend to appear as the first region in which the homology is constant, or ‘stable’, above the discreteness scale. Below this scale, the homology groups fluctuate rapidly as a function of the thickening parameter. This provides a necessary though not sufficient criterion to test for manifoldlikeness of a causal set.
Description: Restricted Access. An open-access version is available at (one of the alternative locations)
ISSN: 0264-9381
1361-6382 (Online)
Alternative Location: 10.1088/0264-9381/26/17/175008
Copyright: 2009 IOP Publishing Ltd.
Appears in Collections:Research Papers (TP)

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