5
$\begingroup$

I am interested in learning how to compute $n$-point correlation functions in Chern-Simons theory, thought of as a TQFT (similar to Witten's work linking that theory to knot theory). I am mostly interested in space being $S^3$, with its round Riemannian metric (which is thus in what physicists call Euclidean signature). Could anyone recommend some reference(s) please? I currently do not have access to a library, so I would prefer references that are hopefully free to download online, such as on preprint servers such as arXiv.

I often see these infinite-dimensional integrals called Feynman's path integrals, but I still do not know how to evaluate them.

Here is a simple example. How do you evaluate the integral of $\operatorname{exp}(i\operatorname{cs}(A))$ over the space of all connections on the trivial $SU(2)$ bundle over $S^3$, where by $\operatorname{cs}(A)$ I mean the integral of the (suitably normalized) Chern-Simons form of the connection $A$ over $S^3$.

Edit 1: I am currently reading William O. Straub's notes on the path integral, which can be found at http://www.weylmann.com/pathintegral.pdf. I find his notes helpful to learn how to compute Feynman's path integrals. They do not specifically address what I am interested in, but it is a start.

Edit 2: I am now more familiar with Feynman path integrals (whose idea seems to go back to a "remark" by Dirac). After reading the notes by Straub, I kind of get the current status quo. In general, what physicists do, at least for a scalar field (corresponding to the Klein-Gordon equation), is a manipulation where they write the Feynman path integral as an ordinary multiple integral times a path integral, and they kind of ignore the path integral, which for all I know may be infinite. They kind of justify this by considering a normalized partition function $Z[J]$ depending on a source term, so that the path integrals in the numerator and denominator, which are identical, kind of simplify.

I know I am supposed to cringe, as a mathematician, by such reasonings, but actually, I think it is quite interesting, particularly since QED for instance seems to agree so well with experimental data!

I actually recommend the following column by Tony Phillips based on a lecture by Misha Polyak http://www.ams.org/publicoutreach/feature-column/fcarc-feynman1 to people who would like to start learning about path integrals. Finite-dimensional versions of the Feynman path integrals are studied, and this gives the flavor of what is going on in the case of real Feynman path integrals.

Edit 3: I am now aware of a few references for Feynman path integrals, which are some of the standard references for QFT: Peskin and Schroeder, Michio Kaku, Lewis Ryder, Srednicki and Anthony Zee wrote books on QFT. At a more difficult level, Steven Weinberg wrote also a deep 3-volume book on the topic.

There are also tons of references online (various online lectures, lecture notes, etc.).

So to make this post useful to others, let me ask the following more precise question. What is a reference where people did some calculations using Chern-Simons theory and Feynman path integrals and rederived some knot invariants (such as the Jones polynomial) this way? I have only been recently interested in such matters, so I am not yet familiar with the literature surrounding this beautiful area of mathematical physics.

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ "I often see these infinite-dimensional integrals called Feynman's path integrals, but I still do not know how to evaluate them." Nobody knows in complete generality. For CS there are two relevant constructions: the first is a "perturbative" or formal power series (which all physicists know and is explained in Straub's notes). The second and more advanced is via (supersymmetric) localization, which gives exact information for certain theories (including CS), but for specific observables; you'd have to specify the n-point correlations and see if that's applicable. $\endgroup$
    – AlexArvanitakis
    Dec 24, 2020 at 5:20
  • 1
    $\begingroup$ Your simple example is the partition function, and for that, localisation should (and has, but I don't immediately recall the reference) calculate the integral exactly. Anyway with respect to the perturbative construction a mathematician POV is Getzler and Kapranov (dg-ga/9408003) early in the paper when they talk about Wick's theorem. $\endgroup$
    – AlexArvanitakis
    Dec 24, 2020 at 5:25
  • $\begingroup$ @AlexArvanitakis, thank you so much for your very helpful comments! The perturbative approach sounds easier, so I will start by learning that. I just need to get numerical values to say a few decimal digit, for which the perturbative approach seems well adapted. If you feel like it, maybe turn these 2 comments into an answer? $\endgroup$
    – Malkoun
    Dec 24, 2020 at 14:22
  • $\begingroup$ Well those comments are just generic pointers -- I do not yet understand precisely what you want to calculate or what the motivation is, so I'm happy to leave it to others to reply $\endgroup$
    – AlexArvanitakis
    Dec 24, 2020 at 17:40
  • $\begingroup$ The pointers are helpful to me though. Regarding my motivation, there is a complex-valued function called the Atiyah-Sutcliffe determinant on the configuration space of $n$ distinct points in $\mathbb{R}^3$, and I am trying to find a physical interpretation of it. $\endgroup$
    – Malkoun
    Dec 24, 2020 at 18:12

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.