# How to learn QFT from mathematical perspective?

I want to learn QFT, because I have heard of its applications in mathematics, I am not interested in scattering cross sections and such. Where can I start to learn? Only books I found are either way above my level or too physics oriented. I know undergraduate math level and have had courses in QM and SR and such. I found Folland's book abit too high level, there must be some easier arxiv papers or something.

• We have a body of math that was created for the sole purpose of understanding topic X, and which is still practiced for almost the sole reason of applying it to X. It seems odd to me to be so insistent on refusing to learn about the application to X. This is not an example like group theory, where the original motivation was much more specific (permutation groups) but now constitutes only an insignificant portion of the whole landscape. If you learn an entire field of math without understanding a single application, I would question whether you understand the field. – Ben Crowell May 14 '15 at 22:33
• Crossposted: math.stackexchange.com/questions/1282617/… – Qiaochu Yuan May 14 '15 at 23:53
• Note that we don't actually understand QFT mathematically! (In general). There are bits and pieces we know, but it's like knowing calculus in the 18th century: hacks, tricks and a few good examples, but no overarching rigorous mathematical framework. – David Roberts May 15 '15 at 3:53
• I personally love this book, although the point of view may not be popular among physicists: amazon.com/Local-Quantum-Physics-Theoretical-Mathematical/dp/… – Jon Bannon May 15 '15 at 21:27
• @BenCrowell, as the OP mentioned, QFT has lots of applications in geometry which have nothing to do with high-energy or condensed matter physics (which I assume is your "topic X"). In these applications, the parts of QFT most heavily used in HEP and condensed matter are not necessarily relevant. I wouldn't question a particle physicist's understanding of group theory just because they know everything about the smooth Lie groups they use daily, but nothing about the finite groups they never encounter; similarly, I wouldn't chastise a geometer for having never computed a scattering amplitude. – Vectornaut May 15 '15 at 21:28

When I was first learning QFT, I found it very helpful to start in the parts that rest on the most solid mathematical foundations, like topological quantum field theories and 2d conformal field theories. This might be an especially good place to start if you're aiming for the geometric applications of QFT, because I think many of those applications currently revolve around TQFTs and CFTs.

There are lots of places you can learn about TQFTs.

• My favorite introduction is Joachim Kock's Frobenius algebras and 2D topological quantum field theories, which starts with the definition of a TQFT and builds up to a detailed proof of the first big theorem in the subject: the classification of 2d TQFTs.
• For a quicker introduction, you might try these seminar talks by John Baez, Julie Bergner, Chris Carlson, and John Huerta. PDF notes are linked from these pages: 1 | 2 | 3 | 4 | 5.
• For a taste of how TQFTs can be used to do geometry, try using using a topological gauge theory with finite gauge group $G$ to count principal $G$-bundles over a compact surface. Ulrich Pennig has some lovely notes on this. There's also a relevant MO question.
• For a taste of how TQFTs can be used to do representation theory, I recommend Qiaochu Yuan's excellent notes on a field-theoretic proof of Mednykh's formula.
• When you're ready to leave the 1-categorical nest and start thinking about extended TQFTs, try David Ben-Zvi's Northwestern lectures, as recorded by Orit Davidovich and and Alex Hoffnung. PDF notes are linked from these pages: 1 | 2 | 3.
• For a really serious application, you can try Turaev's Quantum Invariants of Knots and 3-Manifolds, which discusses the TQFT that Witten discovered for the Jones polynomial. It's tough reading, though.
• I love John Baez's "Quantum Quandaries" for the physics inspiration it provides, though maybe that's not your thing.

Here are a handful of references for CFTs that I like.

Finally, I should make it clear that there's lots of cool geometry related to QFTs more complicated and mysterious than TQFTs and CFTs.

There's also lots of cool math related to QFTs more like the one's you'd encounter in an introductory physics course, although much of it is more analytic than geometric. Some references:

Good luck!

• The field-theoretic proof of Mednykh's formula is surely not due to me; I don't know who it's due to but Dijkgraaf-Witten is not a terrible guess. – Qiaochu Yuan May 15 '15 at 21:44
• @QiaochuYuan: Whoops, I'll edit! Thanks for pointing that out. – Vectornaut May 18 '15 at 20:13

Quantum Field Theory for Mathematicians: The approach to quantum field theory in this book is part way between building a mathematical model of the subject and presenting the mathematics that physicists actually use.

For a collection of resources, see Peter Woit's lecture notes.

• This should be upvoted way more than it currently is. We should also add the 1994 IAS lecture notes "Geometry and QFT". – Chris Gerig May 18 '15 at 23:52