Quantcast
  • 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.

News

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

Attributions

(propose a free ad)

Site Statistics

180 submissions , 140 unreviewed
4,559 questions , 1,832 unanswered
5,175 answers , 22,070 comments
1,470 users with positive rep
722 active unimported users
More ...

  The value of Gravitational Chern Simons theory integration on some three manifolds

+ 2 like - 0 dislike
146 views

Consider the 3d gravitational Chern Simons theory 
$$S= \frac{k}{192 \pi} \int_{M_3} Tr(\omega d \omega + \frac{2}{3}\omega^3)$$
where $\omega$ is the spin-connection on $M_3$. For the theory to be well defined, $k$ has to be an integer. I am interested to know what is the precise value of this integral for certain $M_3$. For instance, when $M_3= \mathbb{R}^3$, the integration clearly vanishes. 

(1): What about M_3= T^3 (the three torus with length R along each direction), S^3 (the three sphere with radius R) and RP^3= S^3/Z_2 (S^3 of radius R with anti-pode identified)? 

(2): Are there compact closed manifolds that can distinguish all the k in Z class (i.e., for different k, e^{iS} yields different phases. )?

p.s.: We know that on a M_3 with bdry, the brdy can have chiral central charge c= k/2 hence probes the value of k.  

Any results or references will be helpful.  

asked Apr 18 in Theoretical Physics by user34104 (10 points) [ no revision ]
recategorized Apr 23 by Dilaton

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:
p$\hbar$ysicsOver$\varnothing$low
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).
To avoid this verification in future, please log in or register.




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

Your rights
...