Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases

TL;DR

This paper introduces a new dual spin network basis for (3+1)d lattice gauge theories, enabling a better understanding of electric and magnetic excitations and their algebraic structures, with applications to topological phases and boundary conditions.

Contribution

It proposes a novel dual spin network basis derived from a recent encoding strategy, extending the traditional spin network framework to include magnetic excitations and boundary conditions.

Findings

Reconstruction of the spin network basis for (3+1)d theories.

Introduction of a dual basis describing magnetic excitations.

Application to boundary conditions and algebraic structures of excitations.

Abstract

Using a recent strategy to encode the space of flat connections on a three-manifold with string-like defects into the space of flat connections on a so-called 2d Heegaard surface, we propose a novel way to define gauge invariant bases for (3+1)d lattice gauge theories and gauge models of topological phases. In particular, this method reconstructs the spin network basis and yields a novel dual spin network basis. While the spin network basis allows to interpret states in terms of electric excitations, on top of a vacuum sharply peaked on a vanishing electric field, the dual spin network basis describes magnetic (or curvature) excitations, on top of a vacuum sharply peaked on a vanishing magnetic field (or flat connection). This technique is also applicable for manifolds with boundaries. We distinguish in particular a dual pair of boundary conditions, namely of electric type and of magnetic type. This can be used to consider a generalization of Ocneanu's tube algebra in order to reveal the algebraic structure of the excitations associated with certain 3d manifolds.