Topological Quantum Field Theory, definition

In summary, the Axioms for TQFT were defined by Atiyah in 1990 and one of the equivalent definitions is in terms of categories. A TQFT is a functor from the category of n-dimensional cobordisms to the category of Hilbert spaces, satisfying certain conditions. This definition is further explained by Baez on his webpage, which states that a TQFT is a functor from cobordisms to the category of Hilbert spaces.
  • #1
marcus
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I believe that the Axioms for TQFT were set out by Atiyah
in 1990 and that one of the equivalent definitions of a TQFT is in
category terms: a TQFT is a functor from the category of n-dimensional cobordisms to the category of Hilbert spaces, satisfying certain conditions.

Is anyone familiar with the category theory definition? If not too convoluted maybe you could run through it?
 
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  • #2
Baez page on TQFT

In case anyone's interested here's a Baez page

http://math.ucr.edu/home/baez/planck/node3.html

it basically says the same thing, a top quant. field theory
is a functor from cobordisms to the category of hilbertspaces
 
  • #3


Yes, I am familiar with the category theory definition of TQFT. To understand it, we first need to understand what a cobordism is. In topology, a cobordism is a manifold that represents the boundary between two other manifolds. For example, a 2-dimensional cobordism would be a cylinder, which represents the boundary between two circles (1-dimensional manifolds). In the context of TQFT, we consider n-dimensional cobordisms, which can be seen as "gluing" together n-dimensional manifolds in a way that preserves their boundaries.

Now, a functor is a mathematical concept that maps objects from one category to another. In this case, the TQFT functor maps n-dimensional cobordisms to Hilbert spaces. A Hilbert space is a mathematical structure that is used to describe quantum states in quantum mechanics. So, we can think of the TQFT functor as assigning a Hilbert space to each n-dimensional cobordism.

The TQFT functor must also satisfy certain conditions, known as axioms. These axioms ensure that the functor is consistent with the physical principles of quantum field theory. For example, one of the axioms states that the TQFT functor should be invariant under diffeomorphisms, which are transformations that preserve the topological structure of a manifold. This is important in TQFT because we want the functor to be independent of the particular way in which we "glue" together the n-dimensional manifolds.

In summary, the category theory definition of TQFT is a mathematical framework that describes a functor from the category of n-dimensional cobordisms to the category of Hilbert spaces, satisfying certain axioms that ensure its consistency with quantum field theory principles. This definition provides a powerful tool for studying topological aspects of quantum field theory and has been used in various areas of mathematics and physics.
 

1. What is a Topological Quantum Field Theory?

A Topological Quantum Field Theory (TQFT) is a quantum field theory that describes the behavior of certain physical systems in a topological space. It focuses on the topological aspects of a space and ignores its geometric and metric properties.

2. How is a TQFT defined?

A TQFT is defined as a functor from a category of topological spaces to a category of vector spaces. This means that it assigns a vector space to each topological space and maps between vector spaces to each continuous map between topological spaces.

3. What are the key properties of a TQFT?

There are three key properties that a TQFT must satisfy: the cobordism hypothesis, the gluing axiom, and the functoriality axiom. The cobordism hypothesis states that a TQFT must assign a vector space to each closed surface and a linear map to each cobordism between surfaces. The gluing axiom states that the TQFT must assign the same vector space to a surface regardless of how it is decomposed into smaller pieces. The functoriality axiom states that the TQFT must preserve compositions of maps between topological spaces.

4. What are the applications of TQFTs?

TQFTs have applications in many areas of physics and mathematics, including topological quantum computing, condensed matter physics, and knot theory. They are also used to study the behavior of topological phases of matter and topological defects in physical systems.

5. Are there different types of TQFTs?

Yes, there are several different types of TQFTs, including differential, unitary, and modular TQFTs. Differential TQFTs use differential geometry and calculus to study topological spaces, while unitary TQFTs have a stronger notion of symmetry. Modular TQFTs are a type of unitary TQFT that are used to study the mathematics of conformal field theory.

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