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Can you suggest a reading list, or at least a few papers that you think would be useful, for a beginner in topological quantum field theory? I know what the curvature of a connection is, know basic algebraic topology, and have some basic background in quantum field theory.

Perhaps others with different backgrounds will also be interested in a reading list on TQFTs, so feel free to ignore my background and suggest material at a variety of levels.

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The answers generated from this question were very helpful. Itoohave been looking for references in these topics. Thanks a lot! May be MO can start a separate section which compiles together references in topics like this. –  Anirbit Jan 3 '10 at 11:27
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FYI, the question of whether this should be community wiki was raised in this thread: tea.mathoverflow.net/discussion/6/…. I've added the big-list tag. A hammer or at least a change of the question status to wiki may be imminent. –  Jonas Meyer Jan 4 '10 at 4:33
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Wiki hammer WHAM! [big-list] questions should basically always be community wiki. The reasoning is that the reference should be accumulating the reputation, not the user who posts it. –  Anton Geraschenko Jan 5 '10 at 6:26

13 Answers 13

I found Bruce Bartlett's MSc dissertation Categorical Aspects of Topological Quantum Field Theories a very clear and well-written introduction to TQFTs and related matters.

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I think it might be worth pointing out that there are two kinds of topological quantum field theory, (Albert) Schwarz-type theories and Witten-type theories. In Schwarz type theories (like Chern-Simons theory and BF-theory), you have an action which is explicitly independent of the metric and you expect that the correlation functions computed by the path integral will also be independent of the metric. In Witten-type theories (Donaldson theory, Gromov-Witten theory), metric independence is a little bit more subtle. In these theories, you do have to choose a metric to get started. But you have some extra structure that allows you to compute some quantities which are metric independent.

(Slightly) more precisely: In a Witten-type theory, you have some operator Q which squares to zero, which you think of as a differential. (Witten type theories are also called cohomological field theories.) You also have an operator T, taking values in (2,0)-tensors, which a) is Q-exact ( T = [Q,G] for some G), and b) generates changes in the metric g. The latter means that if we compute the expectation value < epsilon(T)A > as a function of g, we find that it's equal to the expectation value of A computed with respect to g + epsilon. Here epsilon is a "small" (0,2) tensor we pair with T to get a scalar. In these theories, you can show that the correlation functions of operators which are Q-exact must vanish, which implies that small deformations of g don't change the correlation functions of Q-closed operators A. If you choose A so that its expectation value behaves like a function on the space of metrics, this tells you it's constant on the space of metrics. If you choose some fancier A so that the correlation functions behave like differential forms on the space of metrics, cohomological complications can arise.

Most of the references here are for Schwarz-type theories. For a physics treatment of Witten-type theories, it's worth looking at Witten's "Introduction to cohomological field theory". There's also a long set of lecture notes by Cordes, Moore, & Ramgoolam. The mathematical treatments of the idea are less complete. Hopkins, Lurie, & Costello's stuff is about the most comprehensive, but it's pretty far removed from actions and path integrals. For a starter, you might enjoy Teleman's classification of 2d semi-simple "families topological field theories".

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Even though the Chern-Simons action is independent of the metric, the computation of the correlation functions does require introducing a metric. In other words, the Chern-Simons action actually computes invariants of framed knots. One then has to "deframe" in order to arrive at the usual knot invariants. There is also a distinction to be made between "topological field theory" and "cohomological field theory", the latter computing invariants once a class of metrics (say, fixing the holonomy) has been chosen. –  José Figueroa-O'Farrill Jan 4 '10 at 5:02

There's a nice overview of many of the "higher categorical issues" in John Baez's paper Higher dimensional algebra and TQFT. It's also very friendly for beginners.

For the details, from a perspective emphasising the 2- and 3-categories that make everything tick, you should read

  • Bakalov-Kirillov
  • Turaev's big blue book

Dan Freed's paper are also very good. For a somewhat different perspective, try Kevin Walker's TQFT notes, on his webpage.

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Thanks, the Baez-Dolan paper does seem beginner-friendly. –  user142 Oct 13 '09 at 3:33
    
As a side note, when I asked for the reading list I was expecting to see discussions along the lines of: (1) Here is what a QFT is (quick discussion of path integrals etc.), (2) Now here is an action to quantize, we'll calculate path integrals (partition function, etc.) and derive topological invariants from it. I see that instead, many mathematical discussions focus on the algebraic/categorical structures extracted from this physicists' approach. There is the "topological" stuff with algebraic structures, but the "quantum field theory" part seems to have disappeared. –  user142 Oct 13 '09 at 3:35
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Dan Freed's paper Scott mentioned is here, I think: ma.utexas.edu/users/dafr/OldTQFTLectures.pdf And these seem to be the notes by Kevin Walker: canyon23.net/math/tc.pdf –  user142 Oct 13 '09 at 4:10
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Thanks, AO, for tracking down the references: I was bouncing a baby as I wrote my answer, and decided not to take too long! I'm pretty sure one of Dan Freed's papers starts along the lines you laid out; I'm not sure if it's the one you found. –  Scott Morrison Oct 13 '09 at 6:15
    
Thanks for mentioning them. I've just taken a look at Dan Freed's paper, it does indeed start along those lines. –  user142 Oct 13 '09 at 15:08

There is a very accessible book by Joachim Kock, "Frobenius Algebras and 2D Topological Quantum Field Theories"

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Kock himself admits in the introduction that the book has very little to do with TQFT proper; nevertheless this is a great book for getting a grip on the simplest case. –  Qiaochu Yuan Sep 21 '10 at 17:09

I've found the following articles useful in the past:

Segal's notes: http://www.cgtp.duke.edu/ITP99/segal/

Atiyah's paper "Topological quantum field theories"

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I should add that TQFTs/QFTs is by now a very huge subject, and it's hard to give references unless you are more specific about what you are interested in, just because there are way too many references. But in any case the references I gave above are my favorite basic/introductory references. –  Kevin H. Lin Oct 12 '09 at 21:30
    
Thanks, I had seen the Atiyah paper but didn't know about Segal's notes, they seem useful. These will be sufficient for now. –  user142 Oct 13 '09 at 3:25

If you want something from a more QFT point of view, Witten's 1989 paper on the Jones polynomial is pretty readable.

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104178138

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Thanks, I had taken a look at that paper some time ago, I'm hoping to get back to it soon. Now that I have a place to ask my questions when I get stuck... –  user142 Oct 13 '09 at 15:09

I once tried myself on collecting a reading list, or at least a reference list at

nLab:FQFT

("FQFT" for "Functorial QFT" as opposed to other fomalization approaches likes "algebraic" QFT, which have not been applied that much to the topological case).

This also points to more introductory stuff by John Baez, if that's a selling point, namely to his article "Quanmtum quandaries" with is one of the best exposuitions for why that idea of representations of cobordisms categories is a good one in the first place.

But it also contains links to (some of) the historical articles and then to those describing further development, trying to put it all in a big picture.

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You might also be interested in new "fermionic" theories, with new and almost totally unexplored features. If yes, see e.g. arXiv:0907.3787 and arXiv:0911.1395. To understand these properly, you should read, however, at least something in the beginning of Turaev's book on torsions (but not his papers on quantum invariants) and, of course, some book on Grassmann-Berezin calculus of anticommuting variables (a few relevant pages in Berezin's book on Second Quantization will work).

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Try This Week's Finds in Mathematical Physics first. It has some classic references, e.g. Atiyah's book:

The Geometry and Physics of Knots, by Michael Atiyah, Cambridge U. Press, 1990.

(update: Atiyah has many books and to my knowledge any of them is worth a look)

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Thanks, I've ordered Atiyah's book. –  user142 Oct 13 '09 at 3:30

There is a new book available, Dirichlet Branes and Mirror Symmetry, written by both mathematicians and physicists. It contains a chapter on TQFT written by Moore and Segal which is based on this paper. This book is supposed to be written such that mathematicians are able to understand it, and I think that the authors achieved this goal.

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A recent addition is provided by the recent text Topological Field Theory, Higher Categories, and Their Applications of Anton Kapustin

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This might be a little too simple/review-like for someone with your background, but it is a cute little article and might be a nice introduction: http://arxiv.org/abs/0810.0344

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Some new developments relating TQFTs and Morse theory by Losev, Frenkel and Nekrasov

hep-th/0610149 arXiv:0803.3302

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