From 3D topological quantum field theories to 4D models with defects

TL;DR

The paper develops a concrete strategy to lift defect excitations from a well-understood $2D$ TQFT framework to a $3D$ theory with defects by using Heegaard splittings to encode 3D topology on a 2D surface. It constructs a triangulation-based Hilbert space and an operator algebra, including ribbon operators, that generate curvature excitations while preserving flatness constraints on the Heegaard surface; the BF theory with finite groups serves as an explicit, worked example, including a detailed boundary-of-a-4-simplex case. The work outlines a path to generalize to $q$-deformed BF theories and Turaev–Viro models, potentially enabling a triangulation-independent quantum-geometry interpretation in four dimensions and connections to spin-foam gravity. Overall, it provides a principled framework to study and realize $(3+1)$D TQFTs with defects by exploiting the well-developed $(2+1)$D structure and Heegaard-surface reductions, promising new avenues for quantum geometry and quantum gravity models.

Abstract

(2+1) dimensional topological quantum field theories with defect excitations are by now quite well understood, while many questions are still open for (3+1) dimensional TQFTs. Here we propose a strategy to lift states and operators of a (2+1) dimensional TQFT to states and operators of a (3+1) dimensional theory with defects. The main technical tool are Heegard splittings, which allow to encode the topology of a three--dimensional manifold with line defects into a two--dimensional Heegard surface. We apply this idea to the example of BF theory which describes locally flat connections. This shows in particular how the curvature excitation generating surface operators of the (3+1) dimensional theory can be obtained from closed ribbon operators of the (2+1) dimensional BF theory. We hope that this technique allows the construction and study of more general models based on unitary fusion categories.

Figures (5)

• Figure 1: A ribbon operator along a non--contractible cycle on a torus.
• Figure 2: The Heegaard surface induced by the triangulation of the 3--sphere with the boundary of a 4--simplex. We label the vertices of the triangulation and the associated punctured spheres of the Heegaard surface by numbers 1, … 5. The bold line indicates the curve $t \cap \Sigma(\Delta)$ for a triangle $t$ with vertices $v=1,2,4$.
• Figure 3: This figure shows a part of a Heegaard surface consisting of two punctured spheres connected by a tube. It also shows the graph $\Gamma$ on this part of the surface. On the spheres we have links surrounding the punctures as well as links running from a central node to the punctures. There is also a link along each tube that connects the pieces of graphs on different spheres.
• Figure 4: This figure shows the five punctured spheres ${\cal S}_i,\,i=1,\ldots 5$ (in a planar representation) and the links of $\Gamma$ on these spheres. The punctures are shown as grey disks. The spheres are glued to tubes, surrounding the edges of the triangulation, via the punctures. Thus each puncture on the sphere ${\cal S}_i$ can be labeled by the sphere ${\cal S}_j$ to which the glued tubes lead. The punctures are surrounded by links oriented clockwise as shown for the sphere ${\cal S}_4$. The thin lines connecting the punctures show how the triangles cut through the punctured spheres and lead to curves on the Heegaard surface. In the picture for ${\cal S}_3$ we have indicated how to deform the curves induced by the triangles so that these curves run along the links of the graph $\Gamma$. The dashed line represents the path $\alpha$ defining the action of the ribbon operator $\mathcal{R}_{451}[H]$.
• Figure 5: Example of a ribbon operator that requires a refining of the graph. The bold-solid line represents the graph on the boundary of the tube while the dashed line represents the path $\alpha$ followed by the ribbon. The small square represents the target node of $\alpha'$. Each time the ribbon crosses a link $l$, it acts on it. In the situation depicted on the left, the ribbon crosses the links labeled by $h_{ij}$ and $k_{ij}$ which are also part of the shadow path $\alpha'$. To circumvent this difficulty, we refine the graph and introduce auxiliary holonomies so that the path $\alpha'$ associated with the holonomy $(k_{ij}')^{-1}c\,h_{ij}'g_{ij}$ avoids the links crossed by $\alpha$.