Matter Fields in the Loop Representation of the Partition Function

TL;DR

This paper extends the loop representation of the lattice U(1) gauge theory to include matter fields, expressing the partition function as a sum over surfaces representing electric fluxes and matter interactions.

Contribution

It introduces a novel extension of the loop representation to incorporate matter fields and establishes a duality with a topological string description.

Findings

Partition function expressed as sum over surfaces with matter fields.

Duality between electric flux surfaces and Nielsen-Olesen string world sheets.

Open and closed surfaces represent different flux excitations.

Abstract

We present the extension of the Lagrangian $loop$ representation in such a way to introduce matter fields. The partition function of lattice compact U(1) Gauge-Higgs model is expressed as a sum over closed as much as open surfaces. These surfaces correspond to world sheets of loop-like pure electric flux excitations and open electric flux tubes carrying matter fields at their ends. There is a duality transformation between this description in terms of loop world sheets and the $topological$ representation in terms of world sheets of Nielsen-Olesen strings both closed and open joining pairs of monopoles.