Topological Quantum Field Theory, definition

[marcus]

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?

 

[marcus]

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

 

[R0man]

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.