[Paper Review] From 3D TQFTs to 4D models with defects

Clement Delcamp, Bianca Dittrich|arXiv (Cornell University)|Jun 8, 2016

Homotopy and Cohomology in Algebraic Topology5 citations

TL;DR

This paper proposes a method to lift states and operators from 2+1D topological quantum field theories (TQFTs) with defects to 3+1D TQFTs using Heegaard splittings, which encode 3D manifolds with line defects via 2D surfaces. The key result shows how curvature-generating surface operators in 3+1D BF theory emerge from closed ribbon operators in the 2+1D theory, enabling construction of higher-dimensional models from lower-dimensional ones using unitary fusion categories.

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.

Motivation & Objective

• To develop a systematic method for constructing 3+1D TQFTs with defects from known 2+1D TQFTs.
• To address the lack of understanding in 3+1D TQFTs with defect excitations compared to their 2+1D counterparts.
• To establish a bridge between 2+1D BF theory and 3+1D BF theory using topological techniques.
• To enable the construction of more general 3+1D models based on unitary fusion categories through this lifting procedure.
• To clarify the origin of surface operators in 3+1D BF theory as arising from ribbon operators in 2+1D theory.

Proposed method

• Utilizes Heegaard splittings to reduce the topological data of a 3D manifold with line defects to a 2D Heegaard surface.
• Encodes the 3D defect structure—such as Wilson lines and surface operators—into data on the 2D surface, enabling 2+1D TQFT techniques to be applied.
• Applies the lifting procedure to BF theory, a TQFT describing flat connections, to study curvature excitations in 3+1D.
• Maps closed ribbon operators in the 2+1D BF theory to surface operators in the 3+1D theory, showing their emergence from lower-dimensional structures.
• Relies on the algebraic structure of unitary fusion categories to generalize the construction beyond BF theory.
• Uses the duality between Heegaard splittings and 3-manifold decompositions to ensure topological invariance in the lifted theory.

Experimental results

Research questions

• RQ1 How can states and operators from a 2+1D TQFT with defects be systematically lifted to a 3+1D TQFT with defects?
• RQ2 What role do Heegaard splittings play in encoding 3D topological data with defects in terms of 2D structures?
• RQ3 How do surface operators in 3+1D BF theory arise from ribbon operators in 2+1D BF theory?
• RQ4 Can the lifting procedure be generalized to other TQFTs beyond BF theory using unitary fusion categories?
• RQ5 What is the topological and algebraic mechanism that ensures consistency of the lifted theory in 3+1D?

Key findings

• The lifting procedure successfully maps 2+1D states and operators to 3+1D counterparts using Heegaard splittings, preserving topological invariance.
• Surface operators in 3+1D BF theory are shown to originate from closed ribbon operators in the 2+1D BF theory, establishing a direct algebraic and topological correspondence.
• The curvature excitation in 3+1D BF theory—responsible for surface operators—emerges naturally from the 2+1D ribbon operator structure.
• The method provides a constructive framework for building 3+1D TQFTs with defects from 2+1D models, particularly those based on unitary fusion categories.
• Heegaard splittings serve as a powerful tool to reduce 3D topological complexity to 2D data, enabling the application of 2+1D TQFT techniques in higher dimensions.
• The construction demonstrates a viable pathway toward classifying and analyzing 3+1D TQFTs with non-trivial defect excitations, which remain poorly understood.