• 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
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  What precisely, is the string theory landscape in 10 dimensions?

+ 3 like - 0 dislike

I was reading this Physics.SE thread. The OP said, (changed the last word to type HO)

For D = 10, there are 5 vacua. Or maybe it's more correct to say 4, since type I is S-dual to type HO.

This lead me to the 2 closely related questions:

  1. When considering the landscape of string theories in some $D$ dimensions, are the S-dual theories considered as the same theory, as suggested by the quote? What about the T-dual theories? I don't think the T-dual theories should, because in that dimension itself, they are different until and unless one dimension is compactified, when they become approximately equal.

  2. When it is said that the string theory/M-theory landscape in $4$ dimensions in $10^{520}$, are the S-dual theories counted as the same? I don't think they are, because there is no classification of this landscape yet.

asked Jun 13, 2013 in Theoretical Physics by dimension10 (1,985 points) [ revision history ]
edited Apr 25, 2014 by dimension10

1 Answer

+ 3 like - 0 dislike

The landscape isn't supposed to be just a set of isolated elements ("depressions" in the landscape, I mean minima) but also the "scenery" in between them. The number $10^{500}$ refers to the number of minima.

In 10 dimensions, supersymmetry can't be broken so there are vacua that have moduli – especially the dilaton, i.e. the string coupling (but also the RR axion in the type IIB case) – that are continuous and parameterize inequivalent vacua. For example, 10-dimensional Minkowski type IIB vacua form a 2-real-dimensional set while the remaining 3 inequivalent theories (type I = the $SO(32)$ heterotic, $E_8\times E_8$ heterotic, and type IIA) are 1-real-parameter families. One must distinguish the families of different dimensions – they should never be just "added" because it would be adding apples or oranges.

However, it'c clear that the S-dual (or otherwise dual) theories describe the same $n$-dimensional moduli set, an $n$-parameter "element of the landscape", if you wish. We may say that the 10-dimensional vacua come in 4 classes that are 1-, 1-, 1-, 2-dimensional, respectively. We may also say that there are 5 limiting weakly coupled string descriptions of various 10-dimensional vacua.

However, for all the vacua in the set of $10^{500}$ vacua, as they have been estimated, $n=0$. So they're zero-dimensional classes with no moduli left. Semirealistic vacua can't have continuous adjustable moduli because they would produce new long-range (scalar-field-induced) forces which would conflict with the experimentally verified equivalence principle.

So there are no moduli left. We say that they are stabilized vacua. If some of remaining stabilized vacua are equivalent, they are counted as one vacuum, if they're inequivalent, they are counted as whatever the higher number of inequivalent vacua is. Each vacuum may potentially have several descriptions that are related to each others by dualities but as long as the resulting dynamics is equivalent, it's just one vacuum.

On the other hand, the stable values of the moduli may be numerous – there may be several values on the previously existing moduli space where the potential is minimized (several solutions to the $V'(\mu)=0$ equation) – and if that's so, they must be counted as many vacua despite the fact that they result from the same (one) $n$-parameter family before the potential was generated.

The vacua are counted literally as the number of inequivalent worlds. This says everything. A duality is a useful tool or insight that allows us to describe the physics of a universe in many ways (and is particularly useful if other ways start to fail) but the knowledge of dualities is in no way needed to define the number of the vacua.

This post imported from StackExchange Physics at 2014-03-09 09:12 (UCT), posted by SE-user Luboš Motl
answered Jun 13, 2013 by Luboš Motl (10,278 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:
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