From D=3 to D=2 dimensions: a note on topological order

TL;DR

This paper explores the dimensional reduction of a Chern-Simons theory to derive an effective 1+1D theory for superconductors, revealing connections between topological phases and Dirac fields in slab geometries.

Contribution

It introduces a novel dimensional reduction procedure linking topological Chern-Simons theories to effective 1+1D superconductor models with topological distinctions.

Findings

Effective 1+1D superconductor theory derived from Chern-Simons theory.

Topological and ordinary phases distinguished by phase parameters.

Relation established between reduced theory and Dirac field in slab geometry.

Abstract

We construct, by a procedure involving a dimensional reduction from a Chern-Simons theory with borders, an effective theory for a 1+1 dimensional superconductor. 1That system can be either in an ordinary phase or in a topological one, depending on the value of two phases, corresponding to complex order parameters. Finally, we argue that the original theory and its dimensionally reduced one can be related to the effective action for a quantum Dirac field in a slab geometry, coupled to a gauge field.