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I had asked a question on math.stackexchange but did not receive any answers. I hope that this question is appropriate for this website as it is about an advanced subject. Hence I am posting it below.

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and Homological Algebra), self study of Geometry (Manifolds, Differential Geometry, Riemannian Metrics, Curvature and Connections), elementary representation theory of finite dimensional semisimple Lie Algebras and complex analysis. Before I study TQFTs I will try to read Algebraic Topology, Characteristic Classes and K-Theory. On the physics side I am comfortable with scalar field theory and renormalization.
My first question is :

1.) Are there any other subjects that I should learn before embarking on a study of TQFTs from a Mathematics perspective ? In particular should I study Algebraic Geometry (e.g. Hartshorne or Huybretchs) before or concurrently with TQFTs ?

Also there are good reviews of this subject from the Physicists' perspective such as the famous review here and lots others listed here, however I wish to approach the subject from the Mathematics side. One way is to perhaps start with the original papers of Witten but as I mentioned I am more keen on learning the Mathematics side and preferably from a book or review article.

2.) Is there a good book or detailed review article useful for beginners interested in learning TQFTs from a mathematical perspective ?

TQFT is now a very vast subject and I would like a book or article dealing with an overview of the subject rather than a specialized sub area. I am aware of the book by Turaev but am not sure whether it is good for beginners or satisfies the above criteria.

I am more keen on learning the subject for its applications in manifolds rather than knot theory (for which a helpful answer is already available in this answer.)

It would be ideal if someone could suggest a roadmap to learning this subject. Thanks !

Note : I realized that there is a related question but the answers there mostly contain references to original papers rather than books or comprehensive reviews which is what I am looking for.

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1 Answer 1

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A good place to start is in low dimensions and there a good introduction can be found in

J. Kock, 2003, Frobenius Algebras and 2-D Topological Quantum Field Theories, number 59 in London Mathematical Society Student Texts, Cambridge U.P., Cambridge.

There are also old notes of Quinn that take a very neat homotopy theoretic approach to some of the problems:

F. Quinn, 1995, Lectures on axiomatic topological quantum field theory, in D. Freed and K. Uhlenbeck, eds., Geometry and Quantum Field Theory, volume 1 of IAS/Park City Math- ematics Series, AMS/IAS,.

These do not really get near the physics but you seem to indicate that that is not the direction you want to go in.

I like the links with higher category theory. I realise that this is not everyone's `cup of tea' but it does have some useful insights. I wrote a set of notes for a workshop in Lisbon in 2011 which contain a lot that might be useful (or might not!). They are available at

http://ncatlab.org/nlab/files/HQFT-XMenagerie.pdf

My advice would be to raid the net getting this sort of resource (storing it on your hard disc rather than printing it all out!), then as you start working your way through some of the stuff you have found there will be explanations available ready at hand. Start with the main ideas and `back fill', i.e. don't try to learn everything you might need before you start. If when reading some source material an idea that you are not happy with comes up, search it out then, just enough to make progress beyond that point easy. (Of course this is how one progresses through lots of areas of maths so ....)

(Those notes of mine exist in several different forms and lengths, so in a longer version some idea may be more developed.... so ask!)

.... and don't forget the summaries in the n-Lab can be a very useful place to start a search.

(I had at the back of my mind just now a reference to a seminar that I had some notes on .... but no idea of the source. A little search found me:

http://ncatlab.org/nlab/show/UC+Riverside+Seminar+on+Cobordism+and+Topological+Field+Theories

... One other thing, there are lectures by Lurie on this stuff on YouTube:

https://www.youtube.com/watch?v=Bo8GNfN-Xn4 that are well worth watching.)

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Thanks a lot for this very helpful answer ! I will try to follow your advice of 'back filling' but was still wondering if basic Algebraic Geometry (first 3 chapters of Hartshorne) and Complex Geometry (Kahler Manifolds etc Huybretchs) may be a prerequisite or can be skipped. Thanks! –  user90041 Aug 19 at 15:43
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I find that as I read something like TQFTs that I feel the need for some additional material on .... (whatever) but rarely in the order that basic books present it however good they are and I know that Hartshorne is very good (you may need sheaves at some stage, I do not know the other book) but why not start on TQFTs, and as a side dish skim through those other sources, skimming so as to see what is there ... in case you need it. That way you get to know where to look and that is a very good piece of info to have! –  Tim Porter Aug 19 at 20:21

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