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

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

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,862 answers , 20,637 comments
1,470 users with positive rep
502 active unimported users
More ...

Why does closed string theory have only one dilaton field instead of $22$?

+ 6 like - 0 dislike
206 views

Looking at $5D$ Kaluza-Klein theory, the Kaluza-Klein metric is given by

$$ g_{mn} = \left( \begin{array}{cc} g_{\mu\nu} & g_{\mu 5} \\ g_{5\nu} & g_{55} \\ \end{array} \right) $$

where $g_{\mu\nu}$ corresponds to the ordinary four dimensional metric and $g_{\mu 5}$ is the ordinary four dimensional Maxwell gauge field, $g_{55}$ is the dilaton field.

As there is one dilaton for one extra dimension, I naively would expect that the zero mass states of closed string theory, which can be written as

$$ \sum\limits_{I.J} R_{I.J} a_1^{I\dagger} \bar{a}_1^{I\dagger} ¦p^{+},\vec{p}_T \rangle $$

and the square matrix $R_{I.J}$ can be separated into a symmetric traceless part corresponding to the graviton field, an antisymmetric part corresponding to a generalized Maxwell gauge field, and the trace which corresponds to the dilaton field.

Why is there only one dilaton field given by the trace of $R_{I.J}$, instead of $22$ dilaton fields corresponding to $22$ extra dimensions of closed string theory which has critical dimension $D = 26$? For example, why are there not $22$ dilaton fields needed to parameterize the shape of the 22 extra dimensions if they are compactified?

This post imported from StackExchange MathOverflow at 2014-06-17 09:17 (UCT), posted by SE-user Dilaton
asked Oct 23, 2013 in Theoretical Physics by Dilaton (4,295 points) [ no revision ]
reshown Jun 17, 2014 by Dilaton

Ok, as I am getting unexplained downvotes here too, it is probably appropriate to hide this ...

But the answer is nice, so I am not sure ... ?

@Dilaton You got exactly 1 silent downvote and 6 upvotes. Why would you hide it?

Ok, that was me, sorry, removed it. I was annoyed, because the question was asking about compactification scalars, not string dilatons, and the answer seems clear--- there's compactification scalars and there's a dilaton! They're two different things. But the IIA superstring dilaton is a compactification scalar in disguise, so there can be legitimate confusion. Sorry again.

2 Answers

+ 4 like - 0 dislike

You would actually not expect 22 dilatons. Let me try to explain.

As you have pointed out, a putative field theory limit of the closed bosonic string would consist of a metric, a 2-form (which is the potential for a 3-form) and a dilaton.

Let us assume that such a theory exists and let us dimensionally reduce to four dimensions à la Kaluza-Klein.

The 26-dimensional dilaton $\Phi$ gives a scalar field in four dimensions. The 26-dimensional metric $g_{MN}$ gives the following four-dimesional fields: a metric $g_{\mu\nu}$, 22 gauge fields $g_{\mu n}$ and $\binom{23}2 = 253$ scalars $g_{mn}$. Finally, the 26-dimensional 2-form $B_{MN}$ gives the following: $\binom{22}{2} = 231$ scalars $B_{mn}$, $22$ gauge fields $B_{\mu n}$ and $\binom{4}{2} = 6$ scalars dual to the 2-forms $B_{\mu\nu}$. (In four dimensions, 2-forms can be dualised to scalars: via $\star dB = d\varphi$.)

So from a 4-dimensional perspective the role of the “dilaton” is now the $22 \times 22$ symmetric matrix $g_{mn}$.

This does not imply that you should expect any number of scalars in the massless sector of the closed string. The massless sector of the string theory corresponds to a field theory in 26 dimensions and in order to interpret it in 4-dimensional terms, requires dimensional reduction.

This post imported from StackExchange MathOverflow at 2014-06-17 09:17 (UCT), posted by SE-user José Figueroa-O'Farrill
answered Jun 16, 2014 by José Figueroa-O'Farrill (2,135 points) [ no revision ]
reshown Jun 17, 2014 by Dilaton
+ 2 like - 0 dislike

Because the bosonic string theory is uncompactified. When you compactify you get new scalars.

answered Jun 17, 2014 by Ron Maimon (7,535 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$\hbar\varnothing$sicsOverflow
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).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...