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I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space.

Coming from a background of studying Quantum Field Theory from the books like that of Weinberg, I have myself used Feynman Path Integrals to compute scattering of particles.

Earlier I had done courses in Riemannian Geometry and these days I am also doing courses in Algebraic Topology and hence I think it would be very educative if I can see how exactly the calculation of topological invariants that one does here are related to Feynman's ideas.

It would be helpful if someone can give me references which explain (hopefully starting with simple examples!) how one can use path integrals in geometry.

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You might want to look at the survey article written by Daniel Freed about Chern-Simons theory, even though this is a little closer to topological quantum field theories than just path integrals. You can find it in "Daniel S. Freed. Remarks on Chern-Simons theory. Bull. Amer. Math. Soc. 46 (2009) 221-254.". –  Ulrich Pennig Mar 27 '10 at 7:41
I believe the first significant result of this type was Witten's "proof" of the Atiyah-Singer index theorem using the Dirac operator on the free loop space. Sorry, I don't have the reference to hand. –  Bruce Westbury Mar 27 '10 at 7:45
@Bruce: you are conflating two separate things. Witten's proof of the Atiyah-Singer index theorem uses supersymmetric quantum mechanics and can be made rigorous (Getzler did that). The Dirac operator on the free loop space (the so-called Dirac-Ramond operator) was a separate development and was used for some computations of elliptic genera. –  José Figueroa-O'Farrill Mar 27 '10 at 12:50
Here's a link an answer of José's that is also relevant here: mathoverflow.net/questions/14714/… –  Steve Huntsman Mar 27 '10 at 14:26
Thanks a lot for all the amazing references that have been poured in. I was wondering if one can start off with a simple example. Like say can one re-derive the well-known fundamental group of a circle by doing a path-integral quantization? (Like of a particle confined on a circle?) Like more generally if homotopy group of a space is computable by doing a path-integral on that space? Something along these lines to start off with? –  Anirbit Mar 29 '10 at 13:11
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5 Answers

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Witten, Quantum field theory and the Jones polynomial

Witten, The index of the Dirac operator in loop space

I have found both of these papers quite difficult to understand. I don't know any easier references, and would greatly appreciate it if anybody could suggest some.

Anyway, I guess the basic idea is very simple: Take a manifold, consider some space of "fields" on the manifold (for example a space of sections of a vector bundle), do "integrals" over this space of fields. The results should be invariants of your manifold --- this is not always true, but this is the idea or the hope, anyway.

Edit: I want to also add that (T)QFT has applications not just to geometry/topology but also representation theory. For example check out these nice notes of David Ben-Zvi.

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Have you looked at the Geometry and Physics of knots, by Atiyah? It's a mathematical exposition of the first Witten article you mention. –  Joel Fine Mar 27 '10 at 14:24
This part of the answer... "Anyway, I guess the basic idea is very simple: Take a manifold, consider some space of 'fields' on the manifold, do 'integrals' over this space of fields. The results should be invariants of your manifold." is incorrect. If this were the case, then 3+1 QED would be a topological field theory. It is not. One can verify it is not by the fact that one can see the difference between a coffee cup and a donut. :-) The basic requirement for a TQFT is that the action not depend upon the metric. This is the case in Donaldson-Witten theory, Chern-Simons theory... –  Kelly Davis Apr 6 '10 at 20:23
@Kelly: Yes, I know what I said was incorrect... –  Kevin H. Lin Apr 6 '10 at 20:57
Asking that the action does not depend on the metric is too strong. It is not true in Donaldson-Witten theory, for example. In the Witten type or cohomological type of TQFT, you find a class of observables that are independent of the metric, where this class of observables is actually a set of cohomology classes associated to an odd scalar symmetry of the theory. Even here, however, you can run into problems -- the standard example is the usual $b_2^+$ problems/interesting results occuring with Donaldson theory. –  Aaron Bergman Jul 21 '10 at 21:49
Try the book "The Feynman Integral and Feynman's Operational Calculus", by G.W. Johnson and M.L. Lapidus. Chapter 20 contains some discussion of Witten's know invaraiants, Atiyah-Singer index theorem and some more. –  Anton Fetisov Jan 30 '12 at 21:04
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Witten, Supersymmetry & Morse theory is probably the most accessible reference on "physical methods" in topology & geometry.

Witten, Two Dimensional Gauge Theory Revisited -- contains a path integral construction of the intersection numbers of the moduli space of flat connections

Witten, Topological Quantum Field Theory -- contains a path integral construction of the Donaldson invariants

Witten, Topological sigma models, and Witten, Mirror Manifolds and topological field theories -- use path integrals to compute the intersection numbers of moduli spaces of holomorphic maps.

Anyone seeing a pattern yet?

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I was just waiting for you to post a great answer to this question! –  Kevin H. Lin Mar 27 '10 at 23:57
Thanks. There was a lot of bait on MO today! –  userN Mar 28 '10 at 3:51
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You might find Witten's lectures on the The Dirac index on manifolds and loop spaces from the IAS course on quantum field theory useful.

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Thanks for this reference! –  Anirbit Mar 29 '10 at 12:28
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"Feynman Path Integral can be used to compute geometric invariants of a space."

There several different approaches doing this. Let me try to explain one of them, but remember it is not the only.

The point is that first you should omit the world "Feynman" ! Just integrals are useful to compute geometric invariants - for example Gauss-Bonnet theorem expresses the Euler characteristics as integral over manifold. Word "Feynman" appears when we consider infinite-dimensional manifolds - so we need to "integrate" over infinite-dimensional spaces. However we are NOT really interested in geometry of infinite-dimensional manifolds - we are interested in finite-dimensional manifolds. It appears that in some situations infinite-dimensional manifolds are either contractable to finite-dim ones or there is some heuristics which relates invariants of infinite-dimensional manifolds and finite-dim. For example if you consider loop space of M, manifold itself is embeded into loops(M) as subset of constant loops. If you consider the rotations of loops - then constant loops are fixed-point of this action - so in this case manifold is inf-dim but fixed point set is finite-dim - so we considering equivariant calculations we can get the result on finite-dim results.

So the red-line is the following -

in finite-dim case you integrate closed forms on manifold - and get invariant

in Feynman setup certain integrals reminds closed forms on some inf-dim spaces (loop space or whatever) so integrating it you get invariant.

(In some situations "closed form" menas with respect to BRST differential).

The classical examples are related to Mathai-Quillen formalism and interpretation in terms of QFT.

Let me suggest to look a M. Blau The Mathai-Quillen Formalism and Topological Field Theory http://arxiv.org/abs/hep-th/9203026

And cite the abstract: "These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the Mathai-Quillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the Atiyah-Jeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections."

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@Alexander Chervov, your answer is always wonderful! –  shu Jan 30 '12 at 21:00
@Shu Thanks so much for your kind words ! –  Alexander Chervov Jan 31 '12 at 7:17
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If you read French, Henniart's survey Les inégalités de Morse. Séminaire Bourbaki, 26 (1983--1984), Exposé No. 617, 19 p. might be a good place to start. He explains Witten's analytic proof of the Morse inequalities and calls it natural and elegant.

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If you read Russian - that is also fine :) It was translated into Russian - series of translations of selected Bourbaki seminars published by "Mir". –  Alexander Chervov Jan 30 '12 at 7:12
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