(3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces

TL;DR

The article constructs a (3+1)D topological quantum field theory with line defects by lifting the (2+1)D Turaev–Viro state space to a Heegard surface framework, imposing 2‑handle constraints to obtain a boundary WRT sector and two dual bases that diagonalize complementary holonomy and flux observables. It reveals a self‑dual quantum geometry with a spin‑network (curvature) basis and a curvature (area) basis, providing an improved Walker–Wang model across arbitrary lattices and topologies. The work connects to Crane–Yetter and HHKR phase spaces, supports coarse‑graining and entanglement constructs, and suggests generalizations to (pre‑)modular categories and dichromatic invariants, opening routes toward background‑independent descriptions of quantum geometry and gravity with a cosmological constant. Key contributions include the explicit 2‑handle constraint implementation, the dual basis transformations via ${\\mathbb S}$ and ${\\mathbb F}$ moves, and the demonstration of basis‑diagonalized Wilson loops corresponding to area and curvature operators.

Abstract

We apply the recently suggested strategy to lift state spaces and operators for (2+1)-dimensional topological quantum field theories to state spaces and operators for a (3+1)-dimensional TQFT with defects. We start from the (2+1)-dimensional Turaev-Viro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects. This work has important applications for quantum gravity as well as the theory of topological phases in (3+1) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably. We in particular show that the fusion bases of the (2+1)-dimensional theory lead to a rich set of bases for the (3+1)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian. Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.

Figures (5)

• Figure 1: A basis for a genus 3 surface. Three of the shown vacuum loops can be contracted to a trivial cycle after using repeatedly the sliding property with the remaining vacuum loops.
• Figure 2: The left panel shows a Heegard diagram for the 3--sphere, determined by a triangulation consisting of two tetrahedra. Three attaching curves, corresponding to three of the triangles are depicted as red dashed lines. The attaching curve associated to the fourth (bottom) triangle can be obtained from combining the other three curves. On the right panel the Heegard surface is deformed into the standard form for a genus 3 surface. Three 2--handle constraints are imposed via vacuum loops. The 2--handle constraint for the fourth triangle (determined by a curve surrounding all three holes) follows from the other three constraints. To see this one has to use the sliding property of the vacuum loops. We have also indicated a basis, which is neither a ${\cal B}_1$ basis nor a ${\cal B}_2$ basis.
• Figure 3: The left panel shows a ${\cal B}_1$ basis for the two--tetrahedra--triangulation of the 3--sphere. Here we did not depict the over--crossing graph copy ${\cal F}_u$ and the vacuum loops associated to the 2--handle constraints. This copy can be transformed, using sliding across the 2--handle vacuum loops, to the same 2--handle vacuum loops. The basis is labelled by six quantum numbers and diagonalizes Wilson loop operators around the (six) edges of the triangulation. The right panel shows a ${\cal B}_2$ basis. The strands and vacuum loop on the backside of the genus 3 surface are depicted in grey. Note that for both bases we have 6 vacuum loops. But always three vacuum loops can be generated from the other three loops, using the projection property (modulo factors of ${\cal D}$) and the sliding property of the loops. The graph ${\cal F}_u$ can be identified with the one--skeleton of the dual complex to the triangulation: two four--valent nodes (expanded into three--valent ones) representing the two tetrahedra and connected with each other by four links, representing the four triangles. This basis diagonalizes Wilson loop operators around the four triangles of the triangulation and in addition two Wilson loops around pairs of triangles.
• Figure 4: The left panel shows the Heegard diagram for the 4--simplex triangulation. The Heegard surface is of genus six. Six of the ten attaching curves for the 2--handles are shown. The other four curves are generated from these six curves. Note that the curve for the triangle $t(124)$ under--crosses the handle representing the edge $e(15)$. The right panel shows a ${\cal B}_1$ basis for the 4--simplex triangulation. For each vertex of the triangulation one can choose a cutting of the associated four--punctured sphere into two three--punctured ones. Again we have not shown the over---crossing graph copy. Using the 2--handle vacuum loops, as well as the vacuum loop around the edge $e(15)$, one can transform the over--crossing graph copy to the 2--handle vacuum loops.
• Figure 5: The left panel shows a Heegard diagram for the 3--torus. The $x,y$ and $z$ directions have to be periodically identified. The Heegard surface is based on a lattice constructed from a cube whose sides have been identified pairwise. This surface is of genus three: there is a punctured sphere corresponding to one vertex of the triangulation and three one--handles corresponding to the three edges of the lattice. There are three two--handles corresponding to the three pairs of identified sides. The attaching curves are shown in blue, green and red. To obtain the Heegard surface for a lattice with more cubes we just need to glue more of the basic building blocks, shown in the panel, to each other. The right panel shows a choice for a ${\cal B}_1$ basis, where for clarity we have omitted vacuum loops and the over--crossing graph ${\cal F}_u$.