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

TL;DR

The authors propose a dual framework for (3+1)d lattice gauge theories and topological phases by lifting 2d fusion/basis structures on Heegaard surfaces to 3d via a lifting procedure. This yields a conventional spin network basis alongside a novel dual spin network basis that diagonalizes magnetic (curvature) observables, complementing the electric-flux description; both bases emerge from Drinfel'd double and Ocneanu tube algebra structures, and can accommodate manifolds with boundaries. The work develops two complementary parametrizations of flat connections on 2d surfaces, resolves how to lift them to 3d, and provides explicit examples (including S_3 and torus defects) to illustrate when curvature labels suffice and when extra local observables are needed. Finally, it analyzes cutting and gluing of 3-manifolds with defects, connects to extended quantum triple algebras, and discusses implications for entanglement entropy and coarse-graining in 3+1d lattice gauge theories.

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.

Figures (10)

• Figure 1: The left panel represents a closed line embedded into the three-sphere. The middle panel corresponds to a regular neighborhood of such closed line which is nothing else than a solid two-torus $\mathring{\mathbb{T}}_2$. The right panel represents a graph embedded on the boundary of the solid two-torus which captures the two non-contractible cycles.
• Figure 2: Graphical depiction of the $S$-transform which sends the graph-state $|g,h \rangle$ to $|h^{-1},g \rangle$. The matrix elements of this map relates the two fusion bases which can be defined on the two-torus.
• Figure 3: Graphical depiction of Ocneanu's tube algebra. The cylinders are glued by identifying the punctures and the links ending at these punctures are connected. This leads to a new internal two-valent node and a new face (represented in gray on the figure) at which we need to enforce gauge invariance and flatness, respectively. The resulting state is gauge invariant and flat and can therefore be expressed on a minimal graph. The result defines the multiplication rule of the Drinfel'd double.
• Figure 4: A genus-$\mathsf{g}$ two-dimensional hypersurface can be obtained by gluing $\mathsf{g}$ cylinders to a $2\mathsf{g}$-punctured two-sphere. We then define a graph $\Gamma$ on the surface which captures all the non-contractible cycles.
• Figure 5: Graphical representation of the conventions used to label the holonomies and the nodes of the graph.