[Paper Review] Smooth one-dimensional topological field theories are vector bundles with connection

Daniel Berwick-Evans, Dmitri Pavlov|arXiv (Cornell University)|Jan 5, 2015

Homotopy and Cohomology in Algebraic Topology23 references4 citations

TL;DR

This paper establishes that smooth 1-dimensional oriented topological field theories over a manifold X are equivalent to vector bundles with connection on X, using a novel smooth bordism category defined via cutting axioms rather than gluing. The key contribution is a precise formulation of smooth field theories through smooth ∞-categories and descent, proving that field theory data reduces to parallel transport, which classifies vector bundles with connection via the 1-dimensional cobordism hypothesis in the smooth setting.

ABSTRACT

We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth version of Rezk's complete Segal spaces. With such a definition in hand, we analyze the category of field theories using a combination of descent, a smooth version of the 1-dimensional cobordism hypothesis, and standard differential-geometric arguments.

Motivation & Objective

• To define a smooth 1-dimensional bordism category that encodes cutting laws rather than gluing laws, enabling compatibility with differential geometry.
• To establish a precise equivalence between smooth 1-dimensional topological field theories and vector bundles with connection on a manifold X.
• To provide a smooth variant of the 1-dimensional cobordism hypothesis using complete Segal spaces and Reedy model structures.
• To generalize the framework to higher dimensions and non-topological field theories, using descent and smooth ∞-categories.
• To formalize the correspondence between parallel transport data and field theory values on bordisms, showing that the latter are determined by path representations.

Proposed method

• Defining smooth ∞-categories as smooth versions of complete Segal spaces, using Reedy model structures on simplicial objects in a model category.
• Constructing the smooth 1-dimensional bordism category via cut functions and Morse theory, where morphisms are 1-manifolds with maps to X and a height function with regular values.
• Using descent to relate field theories on bordisms to representations of the smooth path category, reducing field theory data to parallel transport.
• Applying the 1-dimensional cobordism hypothesis in the smooth setting, showing that field theories factor through the path category and are classified by vector bundles with connection.
• Employing degeneracy and face maps in the simplicial structure to model identity bordisms and handle decompositions (0- and 1-handles).
• Using the Reedy model structure to analyze fibrant and cofibrant objects in the category of simplicial functors, ensuring the correct homotopical behavior for field theories.

Experimental results

Research questions

• RQ1 How can a smooth 1-dimensional bordism category be defined to respect differential-geometric structures and cutting axioms rather than gluing?
• RQ2 What is the precise relationship between smooth 1-dimensional topological field theories and vector bundles with connection?
• RQ3 Can the 1-dimensional cobordism hypothesis be extended to the smooth setting using smooth ∞-categories and descent?
• RQ4 How do parallel transport data on paths reconstruct field theory values on arbitrary 1-dimensional bordisms?
• RQ5 What is the role of Morse theory and cut functions in decomposing bordisms and reducing field theory computations?

Key findings

• The space of smooth 1-dimensional oriented topological field theories over a manifold X is equivalent to the nerve of the groupoid of finite-dimensional vector bundles with connection over X and connection-preserving isomorphisms.
• The equivalence is natural in X and holds via a smooth variant of the 1-dimensional cobordism hypothesis, where field theories factor through the smooth path category of X.
• Field theory values on connected bordisms are determined by parallel transport data along paths, with 0- and 1-handles reduced to path compositions and sitting instants at critical points.
• The smooth bordism category is defined using cut functions and Morse theory, ensuring that every bordism admits a decomposition into elementary pieces (points, 0-handles, 1-handles) with regular values at cut points.
• The category of smooth field theories is equivalent to C∞-functors from the smooth path category of X to the smooth ∞-category of vector spaces, which classify vector bundles with connection.
• The result generalizes to higher dimensions and non-topological field theories, with the method relying on descent and smooth ∞-categories for future extension.