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 , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  Instantons and Fivebranes

+ 2 like - 0 dislike
517 views

What is the general relationship between instantons and fivebranes?

In the paper ``Magnetic Monopoles in String Theory'' by Gauntlett, Harvey and Liu, the authors state the fivebrane ansatz of heterotic supergravity as

$$F_{\mu\nu} = \pm \frac{1}{2}{\epsilon_{\mu\nu}}^{\lambda\sigma}F_{\lambda\sigma}$$ $$H_{\mu\nu\lambda} = \mp {\epsilon_{\mu\nu\lambda}}^{\sigma}\partial_\sigma \phi$$ $$g_{\mu\nu} = e^{2\phi}\delta_{\mu\nu}, \qquad g_{ab} = \eta_{ab}$$

where $\mu, \nu, \ldots$ denote "transverse space", and $a, b, \ldots$ denote "orthogonal space".

They go on to say that the first equation can be solved by an instanton configuration in an SU(2) subgroup of the gauge group.

Instantons are solutions of the self/anti-self duality equations in Euclidean space.

(a) Is it possible to define instanton solutions because $g_{\mu\nu}$ is conformally Euclidean in this case?

(b) The one-instanton solution with unit winding is given by

$$F_{\mu\nu} = \frac{2\bar{\sigma}_{\mu\nu}\rho^2}{(x^2 + \rho^2)^2}$$

Now, in Euclidean space $\sigma_{\mu\nu} = (\sigma_1, \sigma_2, \sigma_3, i\mathbb{I})$ where the $\sigma_i$ denote the three (usual) 2x2 Pauli matrices and $\mathbb{I}$ is the 2x2 identity. Likewise, $\bar{\sigma}_{\mu\nu} = (\sigma_1, \sigma_2, \sigma_3, -i\mathbb{I})$.

I am assuming this is what was used here.

The authors find

$$\nabla_\rho \nabla^\rho \phi = \mp \frac{\alpha'}{120}\epsilon^{\mu\nu\lambda\sigma}Tr(F_{\mu\nu}F_{\lambda\sigma})$$

and apparently solve this for a charge one self-dual instanton to get

$$e^{2\phi} = e^{2\phi_0} + 8\alpha'\frac{x^2 + 2\rho^2}{(x^2+\rho^2)^2}$$

(Note that $\nabla$ is the Laplace-Beltrami operator, or the Laplacian defined with the respect to the metric $g_{\mu\nu}$).

The steps between the last two equations are what I want to fill in for myself, and its a bit unclear how instantons are being introduced.

This post imported from StackExchange Physics at 2015-07-19 18:15 (UTC), posted by SE-user leastaction
asked Jul 18, 2015 in Theoretical Physics by leastaction (425 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$\hbar$ysic$\varnothing$Overflow
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
...