The cobordism hypothesis (1210.5100v2)

Abstract: In this expository paper we introduce extended topological quantum field theories and the cobordism hypothesis.

• The paper establishes that an n-dimensional TQFT is fully determined by its value on a point, underscoring the central role of the cobordism hypothesis.
• It employs symmetric monoidal (∞, n)-categories to articulate the deep connections between algebraic topology and quantum field theory.
• The work makes strong categorical claims that link manifold classification with theoretical physics, offering new insights into both fields.

An Overview of "The Cobordism Hypothesis"

The paper "The Cobordism Hypothesis" by Daniel S. Freed provides a detailed exposition on the theory of extended topological quantum field theories (TQFTs) with a particular focus on elucidating the cobordism hypothesis. This hypothesis, initially conjectured by Baez and Dolan and proved in various dimensions by Hopkins-Lurie, stands as a central theorem in the intersection of quantum field theory, algebraic topology, and manifold theory. The paper uncovers the multifaceted nature of the cobordism hypothesis, examining its foundations, implications, and connections to wider mathematical and physical theories.

Key Concepts

The cobordism hypothesis fundamentally posits that an n-dimensional TQFT is completely determined by its value on a point. In the language of higher category theory, this translates to saying that the symmetric monoidal (\infty, n)-category of n-dimensional bordisms is free on a single object. This simple but profound statement reveals the elegant manner in which manifold topology and algebraic structures intertwine.

The manuscript explores several essential topics to scaffold the main thesis:

• Bordism and Cobordism: These serve as the categorical backbone for topological quantum field theories. The paper outlines the traditional algebraic topology perspective, where cobordism classes of manifolds are used to define and classify the algebraic structures that TQFTs take as input and output.
• Quantum Field Theories: Freed draws parallels between the evolution of states in quantum mechanics and quantum field theory and the morphisms in bordism categories, emphasizing the connection through structures such as symmetric monoidal categories.
• Category Theory and Dualizability: The paper makes extensive use of categorical constructs, particularly (\infty, n)-categories, to establish the framework for TQFTs. Dualizability conditions in categories are crucial as they determine when an object within the category can serve as the target for a TQFT.

Numerical Results and Strong Claims

The paper does not focus on traditional numerical results as it is rooted in categorical and topological underpinnings. However, strong claims are provided concerning the existence and classifiability of TQFTs given a certain level of dualizability and adjointability. These claims solidify the link between purely theoretical mathematical constructions and theoretical physics applications.

Implications

The cobordism hypothesis has profound implications for both mathematics and physics. From a mathematical standpoint, it provides a structuring principle that ties together different areas of topology, paving the way for new methods in manifold classification and algebraic topology. Practically, in physics, it offers a categorical perspective on quantum field theories, which could illuminate new pathways in understanding quantum gravity and condensed matter physics.

Furthermore, the extended nature of the TQFTs accommodates cutting and gluing operations at all dimensions up to n, broadening the horizons for how physical theories can be constructed and understood.

Future Speculations

Future developments in AI can draw parallels from these topological and categorical methodologies to build more complex and "higher-dimensional" network structures that resemble the robustness and flexibility inherent in TQFTs. Additionally, the abstraction and generalization processes inherent in the cobordism hypothesis could inspire novel algorithmic architectures in computational models.

Conclusion

In summary, Freed's paper not only offers an expository on the cobordism hypothesis but also reflects on the interconnectedness of various branches of mathematics and physics through the lens of topological quantum field theories. It is a testament to how conceptual frameworks in mathematics can offer deep insights into the physical universe and beyond.