# Extensions of Lattice Groups, Gerbes and Chiral Fermions on a Torus

## TL;DR

This paper explores the topological and K-theoretic relationships between chiral Dirac operators on tori in 3 and 1 dimensions, revealing equivalences and structures involving gerbes, lattice extensions, and gauge connections.

## Contribution

It establishes a novel connection between 3D and 1D Hamiltonians via K theory, and relates gerbes and lattice group extensions to gauge transformations on tori.

## Key findings

- 3D Hamiltonians are K-theoretically equivalent to 1D Hamiltonians with different gauge groups
- Moduli space of U(1) connections over a torus is homotopy equivalent to a torus
- Gerbes over an n-torus can be realized through lattice group extensions

## 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.