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
5,352 answers , 22,786 comments
1,470 users with positive rep
820 active unimported users
More ...

  How to compute the propogator for Chern-Simons on a torus?

+ 5 like - 0 dislike
1214 views

I'm looking to better understand Chern-Simons theory on a torus. We are given the action $$ S(\phi) = \int_E (\partial \phi)(\overline\partial \phi) + \frac{\lambda}{6}(\partial \phi)^3 $$ which yields (apparently) the propagator $$ P(z) = \frac{1}{4\pi}\wp(z;\tau) - \frac{1}{12}E_2(\tau) $$ (at least for $z \neq 0$)

What I would like to understand is the meaning of the terms in the action, as well as how the propagator is derived.

Classically, the action $$ \int_E (\partial \phi)(\overline\partial\phi) $$ yields the equation of motion $$ \partial\overline\partial\phi = 0 $$ or that $\phi(z, \overline z)$ is of the form $\phi_1(z) + \phi_2(\overline z)$. This then seems suggestive that the interaction term $(\partial \phi)^3$ tells us that we are only interested in holomorphic functions on $E$, which of course are going to be given by some combination of $\wp(z;\tau)$ and its derivatives. This seems reasonable, but other than an intuitive feel, I don't see why this is actually the case.

My questions are:

  1. Is this a reasonable interpretation of the terms in the action?
  2. Since my reasoning was based on classical ideas, how does this carry over to the quantum setting?
  3. Given the above, how does one use this to derive that the propagator is as given above and not, say, $$ P(z) = \wp'(z)\big(\wp(z) + E_4(\tau)\big) $$ or something else similar?

I should probably specify that I am a mathematician, not a physicist.

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Simon Rose
asked Apr 30, 2014 in Theoretical Physics by Simon Rose (70 points) [ no revision ]
To ensure that we are on the same page, which references are you mainly using for this?

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Qmechanic
Well, not really any reference. The papers that I'm looking at are arxiv.org/abs/hep-th/9609022 and arxiv.org/abs/hep-th/9407176, say, but the genesis of my interest is Dijkgraaf's "Mirror Symmetry and Elliptic Curves".

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Simon Rose
I don't think this is related in any obvious way with Chern-Simons theory, could you explain why do you call it CS? Your references don't seem to involve CS at all either.

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user fqq
That's a good question. I'm not really sure---this is just something that I've seen it called. I know that formally the CS action $$\int tr(A \wedge dA + \frac{2}{3}A\wedge A \wedge A)$$ looks similar, but that's about all that I can say.

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Simon Rose

1 Answer

+ 1 like - 0 dislike

Comments to the question (v1):

  1. The theory that Ref. 1-3 are studying is not Chern-Simons theory, but an interacting scalar boson $\phi: S^1\times S^1\to \mathbb{R}$, living on the torus $S^1\times S^1$, i.e. a Riemann surface of genus one.

  2. The propagator $\langle \phi(z)\phi(w)\rangle $ is here meant to be the free propagator. Thus we are considering a free scalar boson $\phi: S^1\times S^1\to \mathbb{R}$. This is a standard exercise, which can be found in many string theory textbooks, see e.g. Ref. 4-5.

  3. In mathematical terms, the propagator $G(z,w)=\langle \phi(z)\phi(w)\rangle$ is just the Greens function for the Laplacian $\partial \bar{\partial}$ on the torus. In other words, we need to solve a double-periodic Dirichlet problem.

  4. The double-periodic Greens function $G(z,w)$ can in principle be uniquely determined from knowledge of the pole structure. A perhaps more constructive type of derivation would involve the formula $$G(z,w)~=~\lim_{\varepsilon\searrow 0^+} \sum_n{}^{\prime} \frac{\psi_n(z)\psi^{\ast}_n(w)}{\lambda_n}e^{-\lambda_n\varepsilon },$$ where $\psi_n(z)$ denotes the eigenfunction for the eigenvalue $\lambda_n\geq 0$ of the Laplace operator $-\partial \bar{\partial}$, cf. e.g. Ref. 6. Here $\varepsilon>0$ is a regularization parameter. (Alternatively, one may use other regularization schemes.) The prime in the sum indicates that zeromodes should be excluded.

  5. It turns out that the second derivative of the propagator $\partial_z\partial_w G(z,w)$ is a meromorphic function, i.e. it is holomorphic away from its poles. [Perhaps confusingly, the second derivative of the propagator $\partial_z\partial_w G(z,w)$ is referred to as 'the propagator' in Ref. 1 eq. (4.44).]

References:

  1. R. Dijkgraaf, Chiral Deformations of Conformal Field Theories, Nucl. Phys. B493 (1997) 588, arXiv:hep-th/9609022.

  2. R. Dijkgraaf, Mirror Symmetry and Elliptic Curves.

  3. M.R. Douglas, Conformal Field Theory Techniques in Large N Yang-Mills Theory, arXiv:hep-th/9311130.

  4. J. Polchinski, String Theory Vol. 1, 1998; Section 7.2.

  5. E. Kiritsis, String Theory in a Nutshell, 2007; Section 4.18.3.

  6. E. Cohen, H. Kluberg-Stern, H. Navelet, and R. Peschanski, Regulated propagator on the flat torus, CERN preprint.

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Qmechanic
answered May 3, 2014 by Qmechanic (3,120 points) [ no revision ]

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$\hbar$ysicsOverflo$\varnothing$
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
...