Extensions of lattice groups, gerbes and chiral fermions on a torus

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Mickelsson , J 2017 , ' Extensions of lattice groups, gerbes and chiral fermions on a torus ' , Journal of Geometry and Physics , vol. 121 , pp. 378-385 . https://doi.org/10.1016/j.geomphys.2017.08.003

Title: Extensions of lattice groups, gerbes and chiral fermions on a torus
Author: Mickelsson, Jouko
Contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2017-11
Language: eng
Number of pages: 8
Belongs to series: Journal of Geometry and Physics
ISSN: 0393-0440
URI: http://hdl.handle.net/10138/304624
Abstract: Motivated by the topological classification of hamiltonians in condensed matter physics (topological insulators) we study the relations between chiral Dirac operators coupled to an abelian vector potential on a torus in 3 and 1 space dimensions. We find that a large class of these hamiltonians in three dimensions is equivalent, in K theory, to a family of hamiltonians in just one space dimension but with a different abelian gauge group. The moduli space of U(1) gauge connections over a torus with a fixed Chern class is again a torus up to a homotopy. Gerbes over a n-torus can be realized in terms of extensions of the lattice group acting in a real vector space. The extension comes from the action of the lattice group (thought of as "large" gauge transformations, homomorphisms from the torus to U(1)) in the Fock space of chiral fermions. Interestingly, the K theoretic classification of Dirac operators coupled to vector potentials in this setting in 3 dimensions can be related to families of Dirac operators on a circle with gauge group the 3-torus. (C) 2017 Elsevier B.V. All rights reserved.
Subject: Extensions of current algebras
K theory
Topological insulators
111 Mathematics
114 Physical sciences

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