Topological quantum field theories

Definition

Topological quantum field theories (TQFTs) are mathematical frameworks that connect quantum field theory with topology, focusing on the properties of spaces that are preserved under continuous deformations. They provide a way to assign algebraic invariants to topological spaces, particularly through the use of cobordisms, which relate different manifolds to each other. This connection helps in understanding both physical phenomena in quantum mechanics and deep mathematical structures.

5 Must Know Facts For Your Next Test

1. TQFTs abstractly assign algebraic structures to topological spaces, allowing for computations in both mathematics and physics.
2. The mathematical foundation of TQFTs involves categories and functors, linking spaces through morphisms that respect their topological properties.
3. Cobordism plays a critical role in TQFTs, as it enables the study of how different manifolds are related via their boundaries.
4. TQFTs have applications beyond pure mathematics, including in theoretical physics, particularly in string theory and quantum gravity.
5. An example of a TQFT is the Reshetikhin-Turaev invariant, which associates a topological invariant to 3-manifolds using representations of quantum groups.

Review Questions

• How do topological quantum field theories utilize cobordism to create relationships between different manifolds?
  • Topological quantum field theories leverage cobordism as a key concept to relate different manifolds by considering them as boundaries of higher-dimensional spaces. In TQFTs, a cobordism connects two manifolds by serving as an intermediate 'space' whose boundaries correspond to the original manifolds. This relationship allows for the assignment of algebraic invariants that reflect the topological features shared between these spaces.
• Discuss the significance of functors in topological quantum field theories and how they facilitate computations across different categories.
  • Functors are essential in TQFTs as they map between categories, preserving the relationships and structures within those categories. In TQFTs, functors help relate different topological spaces and their associated algebraic structures by ensuring that morphisms between spaces correspond to algebraic operations. This preservation enables mathematicians and physicists to compute invariants effectively and understand how different topological properties interact within quantum field theories.
• Evaluate the impact of topological quantum field theories on both mathematics and theoretical physics, specifically regarding their applications in modern research.
  • Topological quantum field theories have had a profound impact on both mathematics and theoretical physics by providing new tools and perspectives for understanding complex systems. In mathematics, they facilitate insights into knot theory and low-dimensional topology through their assignment of invariants to 3-manifolds. In theoretical physics, TQFTs contribute significantly to areas like string theory and quantum gravity, where they offer frameworks for quantizing spacetime and understanding the fundamental nature of particles. The interplay between these fields showcases how concepts from topology can yield groundbreaking results in our understanding of the universe.

Related terms

Cobordism: A relation between two manifolds where one can be transformed into the other by adding a boundary, often used to classify manifolds in topology.

Functor: A mapping between categories that preserves the structure of the categories, commonly used in TQFTs to relate spaces and algebraic structures.

Invariants: Properties or quantities that remain unchanged under certain transformations, crucial for understanding the topology of manifolds.