Topological quantum field theory

A topological quantum field theory (or topological field theory, TQFT) is a special kind of quantum field theory used in gauge theory and mathematical physics. Unlike other quantum field theories, TQFTs focus on calculating things called topological invariants. These are properties of shapes that stay the same no matter how you twist or stretch those shapes.

Although TQFTs were first created by physicists, they have become very important in mathematics too. They connect to many areas, such as knot theory, the study of four-manifolds in algebraic topology, and the theory of moduli spaces in algebraic geometry. Some of the biggest prizes in math, called Fields Medals, have been awarded to people like Donaldson, Jones, Witten, and Kontsevich for their work related to these ideas.

In the world of condensed matter physics, TQFTs help describe the behavior of special states of matter. These include fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states. These theories give us a way to understand the low-energy properties of such complicated systems.

Overview

In a topological field theory, certain calculations stay the same no matter how the space they are used on is stretched or bent. This makes them useful for studying the shape of spaces.

These theories aren't very useful on simple flat spaces, so they are usually used on more complex curved spaces instead. They are mostly studied in spaces with fewer than five dimensions, though some theories might exist in higher dimensions but aren't well understood yet.

Specific models

There are two main types of topological quantum field theories: Schwarz-type and Witten-type. Schwarz-type theories calculate important values using paths that do not depend on the shape of space. One famous example is Chern–Simons theory, which helps us understand knots.

Witten-type theories began with work in 1988 and include topological Yang–Mills theory. These theories look complex because they include space’s shape, but they end up not depending on it. They follow special rules involving symmetry and observables, making their results focus on the topology of space rather than its exact shape.

Michael Atiyah suggested a set of rules, or axioms, for what a topological quantum field theory should follow. These ideas were inspired by other mathematicians and physicists. The main point is that a topological quantum field theory connects shapes and spaces to mathematical objects called vector spaces.

There are two ways to think about these rules: either for a single space of a fixed size or for all possible spaces of that size. These rules help mathematicians and physicists study complex shapes and their properties in a more structured way.