What would be the pre-requisites to learn Topological Quantum Field Theory?

[yicong2011]

Personally, I am interested in Topological Quantum Field Theory. And now I am battling against Quantum Field Theory. I am not sure how much Quantum Field Theory is needed to do Topological Quantum Field Theory. And I am not sure what should be the mathematical pre-requisites of Topological Quantum Field Theory.

Your reply would be appreciated. Thanks a lot.

 

[Demystifier]

I don't know much about topological QFT, but it is my impression that most of things one learns in ordinary QFT are pretty much useless in topological QFT. Particle interpretation, perturbative techniques (including Feynman diagrams), methods for calculation of differential cross sections and decay widths, Standard Model, ... all these things seem not to play any role in topological QFT.

Disclaimer: If I am wrong please correct me, because I would also like to better learn what topological QFT is really about.

 

[genneth]

I don't know much about topological QFT, but it is my impression that most of things one learns in ordinary QFT are pretty much useless in topological QFT. Particle interpretation, perturbative techniques (including Feynman diagrams), methods for calculation of differential cross sections and decay widths, Standard Model, ... all these things seem not to play any role in topological QFT.

Disclaimer: If I am wrong please correct me, because I would also like to better learn what topological QFT is really about.

Indeed TQFT's seem to have very little contact with "conventional" QFT observables, and therefore theory. Atiyah proposed some straightforward axioms (http://en.wikipedia.org/wiki/Topological_quantum_field_theory#Atiyah-Segal_axioms) but experts do not entirely agree with them (too restrictive, etc.). I guess the OP is a mathematician?

 

[yicong2011]

Thank you very much.

I am thinking about my future research interest: mathematical relativity, topological QFT, conformal field theory, and maybe statistical physics...

 

[Monocles]

In fact, I'd say that topological quantum field theory (at least at the basic mathematical level) is probably more accessible than 'ordinary' quantum field theory. There is a text called Frobenius Algebras and Two-Dimensional Topological Quantum Field Theories that develops the theory of 2D TQFTs at a level that a motivated undergraduate with no physics background can understand. The physical content of TQFT is barely mentioned (if at all), but it is definitely a surprisingly more mathematically intuitive topic than ordinary QFT.

 

[TFT]

It depends on your point of view.

As a physicist, a solid foundation on QFT is the first thing you need. Then you need to familiar yourself with supersymmetric QFT in various dimensions. From there, you can follow all sorts of topological twists to get to topological field theories.

To really understand what's going on, you will need some more mathematics. For example, to understand 4d topological gauge theory, aka Donaldson-Witten theory, you will need working knowledge of differential geometry, fiber bundles, algebraic topology, moduli spaces and so on.

If you like more formal stuff, you will need category theory and higher category theory. Formally a simplest TFT is typically defined as a functor from the cobordism category to the category of vector spaces, with some additional properties.

 

[Alesak]

From what I've seen on the wiki, this book might come in handy, as it explains things like cobordisms.

 

[QGravity]

to learn Topological Quantum Field theory, you must know things about Quantization of gauge theories using BRST method [for cohomological field theory or witten-type theories], path integral & also its approximation methods like stationary phase approximation [which is used in TQFT approach to knot invariants], on the mathematical side you must know (as our friend TFT said) things about manifolds, fiber bundles & connections on them, algebraic topology and so on..., the books that our friend Alesak said,The Wild World of 4-Manifolds, is a very good book to learn these things from. in my idea the best book that bring together both mathematical and physical ideas is Topological Quantum Field Theory and Four Manifolds. a brief survey of which can be found in Topological Quantum Field Theory and Four Manifolds. also the lecture by Dijkgraaf, Lectures on four-manifolds and Topological gauge theories, is a very good survay! also Loops, Knots, Gauge Theories and Quantum Gravity has introduction to TQFT as applied to Quantum Gravity!

for a pure mathematical treatment, Categorical Aspects of Topological Quantum Field Theories is a very nice introduction to the basic ideas!