• 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
5,054 questions , 2,207 unanswered
5,347 answers , 22,728 comments
1,470 users with positive rep
818 active unimported users
More ...

  Number of Physical States of a $U(1)$ Chern-SImons Theory on a Riemann Surface of Genus $g$

+ 3 like - 0 dislike

In A Duality Web in 2+1 Dimensions and Condensed Matter Physics, the authors claimed in Appendix B that for a for a $U(1)_{k}$ Chern-Simons theory defined on a Riemann surface $\Sigma$ of genus $g$, the number of physical states is $k^{g}$.

Can anybody tell me how to calculate the number of physical states of an Abelian Chern-Simons theory on a Riemann surface? Is there any reference that I can follow to understand the above statement?

I also posted my question here.

asked Dec 22, 2018 in Theoretical Physics by Libertarian Feudalist Bot (270 points) [ no revision ]

1 Answer

+ 4 like - 0 dislike

I give below one way to understand this result (there are probably many).

The problem is to show that the dimension of the Hilbert space of $U(1)$ level $k$ Chern-Simons theory on a genus $g$ surface is $k^g$.

The classical phase space of $U(1)$ Chern-Simons theory on a genus $g$ surface is the space $M$ of $U(1)$-flat connections on a genus $g$ surface: it is a torus of dimension $2g$ (indeed, a $U(1)$-flat connection is specified by the $2g$ monodromies around a basis of $2g$ 1-cycles on the surface).

The Hilbert space of quantum states is "the quantization" of the phase space of classical states. One way to "quantize" is holomorphic quantization (see Section 3.1 of https://projecteuclid.org/euclid.cmp/1104178138 ). If one picks a complex structure on the Riemann surface, then the phase space $M$ becomes a complex manifold (an abelian variety), the symplectic form becomes the curvature of an holomorphic line bundle $L$, and the Hilbert space is the space of holomorphic sections of $L$. More precisely, relevant holomorphic line bundles are of the form $L^{\otimes k}$, where $L$ is a specific line bundle (principal polarization) and $k$ is the level. The dimension of the space of holomorphic sections of $L^{\otimes k}$ can be computed using the Hirzebruch-Riemann-Roch theorem (higher cohomoloy groups vanish, Todd class is $1$ for an abelian variety, $M$ has complex dimension $g$):

$\int_M ch(L^{\otimes k})=k^{g} \int_M \frac{c_1(L)^g}{g!}$

One can show that $ \int_M \frac{c_1(L)^g}{g!}=1$ (it is almost the definition of $L$: it is the simplest possible non-trivial case, corresponding to $k=1$).

answered Dec 22, 2018 by 40227 (5,140 points) [ revision history ]

Thank you very much for answering my question. I also found a physical method from David Tong's lecture notes on quantum hall effect.

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