On the Classification of Topological Field Theories (0905.0465v1)

Abstract: This paper provides an informal sketch of a proof of the Baez-Dolan cobordism hypothesis, which provides a classification for extended topological quantum field theories.

• The paper introduces a rigorous inductive proof of the cobordism hypothesis using higher categorical frameworks.
• It reformulates Atiyah's definition of TFTs by extending symmetric monoidal structures to ∞-categories with duals and adjoints.
• The work uncovers connections between low-dimensional topology and algebraic invariants, offering new insights for theoretical physics.

Summary of "On the Classification of Topological Field Theories" by Jacob Lurie

Jacob Lurie's draft paper "On the Classification of Topological Field Theories" aims to provide an expository account of contemporary work surrounding the classification of topological field theories (TFTs). The central focus lies on proving a version of the cobordism hypothesis, initially conjectured by Baez and Dolan, and characterizing extended topological field theories.

Key Concepts and Formulations

Lurie begins by revisiting Atiyah's classical definition of TFTs, which are symmetric monoidal functors from a cobordism category to the category of vector spaces. This lays the groundwork for a deeper exploration into higher category theory, suggesting that extended TFTs are naturally framed within this context. In particular, this involves the extension of notions like symmetric monoidal structure and duals to $\infty$-categories, which capture more intricate morphism compositions such as those between bordisms.

Lurie extensively discusses the formulation of the cobordism hypothesis. The hypothesis posits a correspondence between $n$-dimensional framed TFTs and fully dualizable objects in a symmetric monoidal $\infty$-category. This formulation uses the language of $\infty, n$-categories and revolves heavily around the tool of complete Segal spaces as an approach to model such categories.

Inductive Proof Framework

The proof strategy for the cobordism hypothesis is highly inductive, leveraging prior classifications for $n-1$ dimensions to understand the $n$-dimensional case. It relies on the interpretation of TFTs as generators and relations derived from Morse-theoretic perspectives on manifold decompositions into handles.

The inductive approach uses two primary tools:

1. The construction of Bordism categories: These are constructed using n-fold Segal spaces where the interplay between $k$-morphisms representing bordisms of varying complexities is systematically unfolded.
2. Adjoints and Duals: The cobordism hypothesis requires a rich notion of duals, expressed in terms of adjoint functors within these higher categorical settings.

Moreover, Lurie considers both the stable and unstable facets of these theories, treating the broad scope of field theories through structured frameworks like the Baez-Dolan tangle hypothesis, which captures embedded bordisms.

Calculations and Applications

Lurie provides a detailed account of specific dimensions where TFTs have broader implications, such as in dimension 2, where TFTs correspond to Frobenius algebras. The paper also conjectures extensions beyond the cobordism hypothesis, including TFTs for manifolds with singularities and potential connections to string topology. A segment of the paper ties these concepts into calculable results in low-dimensional topology, revealing how algebraic structures in monoidal categories can lead to potent invariants for geometric objects.

Implications and Future Work

The ramifications of this work are substantial in both mathematical physics and higher algebra, driving forward the theory of TFTs and their role as critical tools for classifying quantum field theories and related structures. Furthermore, the rigorous framework laid down offers a bedrock for future developments in higher category theory, specifically in how algebraic and topological data can be synthesized to provide universal properties across categorical landscapes.

In conclusion, Lurie's draft presents a rigorous investigation into TFTs through an intricate blend of cobordism, higher categories, and dualizability, setting a foundational role for future theoretical explorations. By tackling the classification problem within the categorical framework, this paper both synthesizes past insights and propels new conjectures forward in the field of modern mathematical physics.