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
5,082 questions , 2,232 unanswered
5,353 answers , 22,786 comments
1,470 users with positive rep
820 active unimported users
More ...

  What is the set of central charges of two dimensional rational conformal field theories?

+ 4 like - 0 dislike
994 views

In this question,  RCFT means a unitary full rational two dimensional conformal field theory. As every unitary two dimensional conformal field theory, a RCFT has a central charge $c$ which is a nonnegative real number. The rationality hypothesis implies that in fact $c$ is a nonnegative rational number: $c \in \mathbb{Q}_{\geq 0}$.   

Let $\mathcal{C}$ be the subset of $\mathbb{Q}_{\geq 0}$ made of rational numbers which are central charge of some RCFT. As it is possible to tensorize RCFTs, $\mathcal{C}$ is an additive subset of $\mathbb{Q}_{\geq 0}$.

For example, the intersection of $\mathcal{C}$ with the interval $[0,1]$ is given by $0$, $1/2$, $7/10$, $4/5$,...., $1$ i.e. by the central charges of the unitary minimal models union $c=1$. In particular, $c=1$ is an accumulation point of $\mathcal{C}$.

My questions are: 

Quantitative: Is the set $\mathcal{C}$ explicitely known ?

Qualitative: Is the set $\mathcal{C}$ closed in $\mathbb{R}$ ? Is it well-ordered ? If yes, what is its ordinal ? (for example, are there accumulation points of accumulation points...)

These questions have two motivations:

1) the claim that the RCFT's are classified: see for example http://ncatlab.org/nlab/show/FRS-theorem+on+rational+2d+CFT  

I did not go through this work but I would like to know if this classification is "abstract" or "concrete". In particular, I would like to know if it gives an answer to the previous questions.

2)Similar questions have been asked and solved for a different set of real numbers: the set of volumes of hyperbolic 3-manifolds. It seems to me that there is a (very vague at this moment) similarity between these two sets of real numbers.

asked Apr 6, 2015 in Theoretical Physics by 40227 (5,140 points) [ revision history ]
recategorized Apr 6, 2015 by Dilaton

I think above 1, the spectrum is continuous, but I need to check the yellow book.

@Ryan Thorngren : as I am restricting myself to rational CFTs, the set of central charges is certainly not continuous. But even considering all the CFTs, I don't think it is true. For example, the existence of some CFTs with irrational central charges was a not so easy question as far as I understand. Are you rather refering to the fact that there exists unitary representations of the Virasoro algebra for any value of the central charge above 1? 

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$\hbar$ysicsOverf$\varnothing$ow
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
...