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
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Symmetry breaking of $SO(10)$ or $Spin(10)$

+ 4 like - 0 dislike
1061 views

The Spin(10) grand unification has a symmetry breaking of SO(10), or Spin(10).

In Wikipedia, it says,

"The symmetry breaking of SO(10) is usually done with a combination of (( a $45_H$ OR a $54_H$) AND ((a $16_H$ AND a $\overline{16}_H$) OR (a $126_H$ AND a $\overline{126}_H$)) )."

I suppose that 16 has something to do with the 16 spinor representation of SO(10), and 45 has something to do with ${10 \choose 2} = 45$, while 126 has something to do with $\frac{1}{2}{10 \choose 5} = 126$.

  • What does 54 stands for in the representation theory?

  • So what are so special about these number: 16,45,54, and 126 in these models? And their roles in the representation theory?

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user annie marie heart
asked Jul 30, 2018 in Theoretical Physics by annie marie heart (1,205 points) [ no revision ]
45 is just the number of independent entries in an antisymmetric $10\times10$ matrix and there is nothing special about the numbers. Some people like to use an $\overline{126}$-plet because it allows one to break the $B-L$ symmetry to matter parity, others don't like any of the large representations because it seems hard to impossible to get them out of string theory.

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user user178876
Table 41 in your Slansky. the 45 is (01000) in dynkin index notation, the 54 the (20000) and the 16 and 126 spinors. write down the SO(10) invariants of them and the SU(5) invariance of the SSB vacua. Now look in the next table at 16x16bar and 126x126bar to see what higgs reps you need to saturate with the fermion bilinears.... what do you see?

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user Cosmas Zachos

1 Answer

+ 4 like - 0 dislike

Let us call the defining representation of $SO(10)$ for $V={\bf 10}$. The vector space $V$ is endowed with an invariant metric form: $V\times V\to \mathbb{R}$ (of positive/Euclidean signature). Then we have:

  1. The antisymmetric tensor product $\bigwedge{}^2V\equiv V\wedge V={\bf 45}$.

  2. The totally antisymmetric tensor product $\bigwedge{}^3V={\bf 120}$.

  3. The symmetric tensor product ${\rm Sym}^2V\equiv V\odot V={\bf 1}\oplus{\bf 54}$. The trivial representation ${\bf 1}$ comes from contraction with the metric. The ${\bf 54}$ may be thought of as the traceless part of ${\rm Sym}^2V$.

  4. $\bigwedge{}^5V={\bf 126}^+\oplus{\bf 126}^-$, corresponding to selfdual and anti-selfdual 5-forms. (The Hodge star operator is defined via the metric.)

  5. ${\bf 16}_{L/R}$ are the left/right Weyl spinors in 10D.

See also this related Phys.SE post.

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user Qmechanic
answered Jul 31, 2018 by Qmechanic (3,120 points) [ no revision ]
You may need to explain the cup expression to us... thanks...

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user annie marie heart
Is that ant symmetric tensor wedge product?

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user annie marie heart
"The trivial representation 1 comes from contraction with the metric." which metric?

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user annie marie heart
I also hope to understand this ${\rm Sym}^2V={\bf 1}\oplus{\bf 54}$ and what is the 54 better?

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user annie marie heart
Thanks, can you explain that "The symmetric tensor product Sym$^2V$?" How do you define Sym$^2V$?

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user annie marie heart
I accept your answer first, but please clarfy

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user annie marie heart
I updated the answer.

This post imported from StackExchange Physics at 2020-11-08 17:31 (UTC), posted by SE-user Qmechanic

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
...