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

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,862 answers , 20,637 comments
1,470 users with positive rep
502 active unimported users
More ...

Generalization of Witten's computation of the volume of moduli space

+ 8 like - 0 dislike
2615 views

Let $\Sigma$ be a Riemann surface, and let $X=\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\\!/\operatorname{SU}(2)$ be the $\operatorname{SU}(2)$ character variety of $\pi_1(\Sigma)$. There is a natural symplectic form on $X$ (by Goldman) coming from the intersection pairing on $H^1(X)$, and its top wedge gives $X$ a natural volume form. Witten (in this paper) calculated the volume of $X$ by decomposing $\Sigma$ into a bunch of copies of $\mathbb S^2-\{3\text{ pts}\}$. The final answer is given by an infinite sum over all irreducible representations of $\operatorname{SU}(2)$.

Now the volume that Witten computes can clearly be written as follows: $$\int_X1\cdot\omega^{\wedge\text{top}}$$ Of course, there are a lot of other interesting functions we could try to integrate! The usual way of representing functions on $X$ is by a spin diagram on $\Sigma$. Alternatively, we could think of the function $\rho\mapsto\prod_i\operatorname{tr}_{V_i}\rho(\alpha_i)$ for some $\alpha_i\in\pi_1(\Sigma)$ and $V_i$ representations of $\operatorname{SU}(2)$.

Given a spin diagram on $\Sigma$, is there any known calculation of the integral: $$\int_Xf\cdot\omega^{\wedge\text{top}}$$ where $f:X\to\mathbb R$ is the function associated to the spin diagram?

I believe this should be calculable using Witten's technique. I'd like to know if anyone has seen the answer in the literature, or at least knows what the answer should be.


This post imported from StackExchange MathOverflow at 2014-09-02 20:36 (UCT), posted by SE-user John Pardon

asked Jan 18, 2012 in Theoretical Physics by John Pardon (75 points) [ revision history ]
edited Sep 2, 2014 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$\hbar$y$\varnothing$icsOverflow
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
...