Extensions of Lattice Groups, Gerbes and Chiral Fermions on a Torus
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 classication 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.
Keywords
Cite
@article{arxiv.1702.01643,
title = {Extensions of Lattice Groups, Gerbes and Chiral Fermions on a Torus},
author = {Jouko Mickelsson},
journal= {arXiv preprint arXiv:1702.01643},
year = {2018}
}
Comments
Invited talk at the conference "String Geometries and Dualities", IMPA, Rio de Janeiro, December 2016. Details added around eq. (2.2), def. of the Dirac operator on p.8, and the definition of the groupoid in Sect.5