Bullivant, A, Calçada, M, Kádár, Z et al. (2 more authors) (2020) Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1)D with higher gauge symmetry. Reviews in Mathematical Physics, 32 (4). 2050011. ISSN 0129-055X
Abstract
Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group.
The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | This article is protected by copyright. This is an author produced version of a journal article published in Reviews in Mathematical Physics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Kitaev model; topological phases in (3 + 1)D; topological quantum computing; topological quantum field theory; higher gauge theory; surface holonomy; crossed module; lattice gauge theory |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Funding Information: | Funder Grant number Leverhulme Trust RPG-2018-029 EPSRC (Engineering and Physical Sciences Research Council) EP/I038683/1 |
Depositing User: | Symplectic Publications |
Date Deposited: | 08 Oct 2019 11:41 |
Last Modified: | 04 Nov 2020 01:38 |
Status: | Published |
Publisher: | World Scientific Publishing |
Identification Number: | 10.1142/S0129055X20500117 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:151859 |