• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,064 questions , 2,215 unanswered
5,347 answers , 22,734 comments
1,470 users with positive rep
818 active unimported users
More ...

  A Question about Atiyah-Patodi-Singer Index Theorem

+ 3 like - 0 dislike

In the paper "Quantum Field Theory and Jones Polynomial", (equation 2.16, page 359), as well as in "Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups" (equation 4.2 in page 33), the authors used a formula which is derived from the APS index theorem.

For example, in Witten's paper, equatin 2.16 says that 


(the eta invariant in Witten's paper is half of the usual eta invariant) where $G$ is a compact simple gauge group, $c_{2}(G)$ is its quadratic Casimir element, $\eta(A)$ is the APS eta-invariant of a Dirac type operator twisted by the gauge field $A$, $\eta(0)$ is the APS eta-invariant of trivial gauge $A=0$, and $I[A]$ is the Chern-Simons action on a three dimensional manifold $Y$. i.e. 

$$I[A]=\frac{1}{4\pi}\int_{Y}\mathrm{Tr}(A\wedge dA+\frac{2}{3}A\wedge A\wedge A)$$

However, the APS index theorem is saying that 

$$\frac{-1}{8\pi^{2}}\int_{M}\mathrm{Tr}\left(F\wedge F\right)+\frac{\dim E}{192\pi^{2}}\int_{M}\mathrm{Tr}\left(R\wedge R\right)-\frac{1}{2}\eta(A)$$

is a topological invariant, where $M$ is a compact four dimensional manifold such that $\partial M=Y$, $F=dA+A\wedge A$ is the curvature $2$-form on the $G$-bundle $E$ over $M$, and $R$ is the Riemann tensor on $M$.

I want to prove Witten's formula (2.16). Let me assume a simple case when $M$ is flat so that $R=0$. By using the Stoke's theorem, the APS index theorem implies that 


is a topological invariant. Therefore, I expect that 


Question: Where does the second order Casimir element come from?

asked Nov 22, 2018 in Theoretical Physics by Libertarian Feudalist Bot (270 points) [ revision history ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights