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

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

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Do the real polarization and Kahler polarization of the character varieties of closed surfaces give rise to equivalent representations of the Mappping Class Group?

+ 3 like - 0 dislike
4891 views

This is a question about the Witten--Reshetikhin--Turaev representations of the mapping class group of a closed surface $\Sigma_g$. For simplicity, we'll stick to the case $G=SU(2)$.

These representations are constructed as follows. To a closed surface $\Sigma_g$, we associate the character variety $X=X(\pi_1(\Sigma_g),SU(2))$. There is a natural line bundle $\mathcal L$ over $X$, whose curvature is a natural symplectic form on $X$. We now let $Z(\Sigma_g,k)$ be a certain vector space consisting of sections of $\mathcal L^{\otimes k}$ over $X$. There are various ways of constructing this vector space.

(1) The standard construction for the WRT invariants is to give a holomorphic polarization of $X$ (i.e. equip it with a Kahler structure), which depends on a choice of complex structure on $\Sigma_g$. The space of holomorphic sections of $\mathcal L^{\otimes k}$ then forms $Z(\Sigma_g,k)$. Given two holomorphic structures on $\Sigma_g$, there is a natural isomorphism between the corresponding spaces of holomorphic sections of $\mathcal L^{\otimes k}$, so we get a canonical vector space $Z(\Sigma_g,k)$, and an action of the mapping class group on it.

(2) Alternatively, one can use a "real polarization" of $X$ as in this paper. In this setting, one constructs a Lagrangian fibration $X\to B$ based on a decomposition of $\Sigma_g$ into pairs of pants, and the space $Z(\Sigma_g,k)$ consists of flat sections of $\mathcal L^{\otimes k}$ over the leaves of this fibration for which the monodromy of $\mathcal L^{\otimes k}$ is trivial. In the reference given above, it is shown that the number of such fibers is the same as the dimension of the space of holomorphic sections from (1), so this $Z(\Sigma_g,k)$ has the "correct" dimension. In this construction, the vector spaces associated to different polarizations can also be related by a natural isomorphism, the BKS pairing. Thus there is a resulting representation of the mapping class group on $Z(\Sigma,k)$ here as well.

Question: are these two constructions equivalent in that there is a natural isomorphism between the two spaces I've called $Z(\Sigma_g,k)$ above? Here "natural" means at least that the isomorphism is equivariant with respect to the action of the mapping class group.

Comment: There are 3-manifold invariants coming out of (2) which do NOT agree with the WRT 3-manifold invariants (see this paper), though this is perhaps due to the construction of the canonical elements associated to handlebodies being different, not the quantization of the moduli space being different.


This post imported from StackExchange MathOverflow at 2014-09-02 20:38 (UCT), posted by SE-user John Pardon

asked Aug 7, 2011 in Theoretical Physics by John Pardon (75 points) [ revision history ]
retagged Nov 9, 2014 by dimension10
To my knowledge, holomorphic polarizations are related by the 'Hitchen connection' (further studied by Anderson and others) which is only projectively trivial, and so allows us to define only a projective action of the mapping class group.

This post imported from StackExchange MathOverflow at 2014-09-02 20:38 (UCT), posted by SE-user Blake

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$\hbar$ys$\varnothing$csOverflow
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
...