• 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.


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

198 submissions , 156 unreviewed
4,894 questions , 2,072 unanswered
5,309 answers , 22,537 comments
1,470 users with positive rep
801 active unimported users
More ...

  Theories with a symmetry group not isomorphic to a matrix group

+ 1 like - 0 dislike

Suppose we have a Lie group, \(G\), and most importantly, there does not exist any continuous, injective homomorphism of $G$ into any $GL(n; \mathbb C)$. Are there any physical theories for which there is such a Lie group $G$ as a symmetry? Or does any such type of group appear in physics more generally?

One example I have been able to find is that the metaplectic group, that is, the double cover of $Sp_{2n}$, arises in the context of duality groups, and indeed it is not isomorphic to any matrix Lie group. 

asked Jan 1, 2017 in Theoretical Physics by anonymous [ no revision ]

2 Answers

+ 2 like - 0 dislike
Folland's book Harmonic analysis in phase space contains the application of the metaplectic group to geometric quantization. It is also relevant for the study of squeezed states in quantum optics.

Infinite-dimensional Lie groups are also relevant in physics; e.g. the Virasoro group in conformal field theory.
answered Jan 2, 2017 by Arnold Neumaier (15,747 points) [ no revision ]
+ 2 like - 0 dislike

One of the most important examples is a stringy $\sigma$-model on $AdS_5 \times S^5$ background. The global symmetry group of this theory is $PSU(2,2 \vert 4)$ which is non-matrix, and $AdS_5 \times S^5$ itself is equal to $(SO(1,5) \times SO(6))/(SO(1,4) \times SO(5))$ which is a bosonic part of $PSU(2,2 \vert 4)/(SO(1,4) \times SO(5))$ supergroup.

answered Jan 7, 2017 by Andrey Feldman (904 points) [ revision history ]
edited Jan 7, 2017 by Andrey Feldman

PSU(2,2|4) is a supergroup rather than a group. Do you mean that PSU(2,2|4) cannot be realized as a subsupergroup of some GL(m|n)? If so, why?

$AdS_5 \times S^5$ is more precisely the bosonic space underlying the superspace $PSU(2,2|4)/(SO(1,4) \times SO(5))$.

@40227 You are right that $AdS_5 \times S^5$ is a bosonic part of SUSY $\sigma$-model target superspace. I was somewhat imprecise.

$PSU(2,2 \vert 4)$ cannot be realized as a matrix supergroup due to the property of projectivity. It is not a specific feature of supergroups -- the  simplest example is $PSL(2, \mathbb{R})$, which is also cannot be realized in terms of matrices.

@Andrey Feldman  When one projectivizes, one divides by the center so one gets the image of the group under the adjoint representation, so one always gets a matrix group. In fact, $PSL(2,\mathbb{R})$ is the connected component of the identity in $SO(1,2)$, which is obviously a matrix group. Similarly, $PSU(2,2)$ is the connected component of the identity in $SO(2,4)$, which is obviously a matrix group.

@40227 Sorry, of course you are right. The fact that $PSU(2,2 \vert 4)$ is non-matrix is unrelated to projectivity. In order not to write a very long comment, let me refer you to section 3.2 in this review on $AdS/CFT$: https://arxiv.org/pdf/1104.2604v3.pdf, where this group is discussed.

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).
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