How can one represent a C∞ sheaf/supermanifold from supersymmetry, as relations of simpler gauge groups?

[FlawedBeing]

As seen in the attached image, the paper "Supersymmetric Nonlinear Sigma Models, (June 4, 2000)",  on page 9 describes a manifold/projective space in terms of gauge group relations, including  SU(n) , U(n) etc.

How can one represent a \[C ^\infty (M) \] sheaf/supermanifold from supersymmetry (See Wikipedia/Supermanifold), as relations of simpler gauge groups?

The gauge groups are such an elegant way to understand physics, and this could help me to better understand physics.The world reasonably needs more physicists, and maybe less soccer players.

Edited May 29, 2020 by FlawedBeing
Latex is not rendering properly

[Mordred]

Clarify this is the correct paper.

https://arxiv.org/abs/hep-th/0006025

You forgot to link the paper.

I am also assuming you are looking at [math]CP^{n-1}[/math]

 

[joigus]

2 hours ago, FlawedBeing said:

simpler gauge groups

Sorry, gauge groups simpler than what group?

[FlawedBeing]

3 hours ago, joigus said:

Sorry, gauge groups simpler than what group?

So, I thought that supermanifold structures are very complex, consisting of Poincare space, i.e. both normal space time and anti-commuting coordinates.

I garnered that normal space time could constitute some simpler structures, including the use of SU(n), U(n), etc, looking for example at sources like this paper: http://uw.physics.wisc.edu/~himpsel/449group.pdf

Thereafter, I was thinking that some sheaf/supermanifold from supersymmetry, since a combination of both normal space time, could be expressed in terms of the constituent structures from normal space times, i.e. the simpler gauge groups, since I thought the sheaf/supermanifold concerned a supergroup.

 

4 hours ago, Mordred said:

Clarify this is the correct paper.

https://arxiv.org/abs/hep-th/0006025

You forgot to link the paper.

I am also assuming you are looking at CPn−1

 

1. Yes, that is the paper. I didn't see a link icon, so I thought it was some forum constraint on new members forbidding links. 
2. All the items on the left hand side in the picture are complicated wholes, formed by the right hand side items, i.e. where rhs=combinations of gauge groups. I wanted to find out whether \[C^∞(M)\] was also expressible in terms of these gauge groups, and what the right hand side for \[C^∞(M)\]  would look like.

Notably, \[CP^{n−1}\]  doesn't seem to concern supermanifolds directly?

Edited May 29, 2020 by FlawedBeing

[joigus]

3 minutes ago, FlawedBeing said:

So, I thought that supermanifold structures are very complex, consisting of Poincare space, i.e. both normal space time and anti-commuting coordinates.

Yes. Supermanifolds are the direct product of space-time and an even number of Grassmann (anti-commuting) variables (alt. any number of \vartheta \vartheta^{*} complex Grassmann variables.) Plus a series of prescriptions for the differential calculus of the Grassmann part of superspace. I'm not sure I can help you, but I will try to gather info and make as much sense of it as I can. I'm considerably out of touch. I'm sure Mordred can help you much better.

Poincaré group is a non-compact group. Gauge groups are different. They're all compact. Theorems guarantee compact groups irreducible representations are all unitary. Not so for non-compact. That's why "representing" Poincaré (or the manifold?) as reps. of gauge groups sounded weird to me.

SS for the likes of me is better understood on a back-to-basics kind of way. Its original motivation was both to solve some problems in scalar-field vacuum and to overcome Coleman-Mandula theorem, which says that any symmetry group that combines Poincaré group and gauge groups must do so with a trivial Cartesian product PxG, with P Poincaré group and G the gauge group, the 1st acting on space-time, and the 2nd on particle states. But one of the premises is that the background is commuting space (if I remember correctly.) If BG has anticommuting variables, you can combine M and the fibers \varpsi (fields) in non-trivial way by a fiber bundle with G-group acting on the fibers. So that you can only locally expand the fibre bundle as PxG.

My interest went even further back to basics since LHC seems to have given no signs of any SS partners, so I tend to look at it as an interesting extension to quantum formalism, not in the sense of giving rise to multiplets, but I may well be the only person in the world that feels so. 

And that's about what I remember. Anyway, sorry for the lengthy / useless answer, but I wanted to have an anchor to this post for the follow-ups.

[Mordred]

As it's been awhile since I last looked into MSSM based models The notes you put above serves as useful reminders. It's been awhile since I looked at Grassmann variables.

 So I am currently looking for some useful direction in literature for the OP.  At the OP have you studied Pati Salam via SO(10) MSSM ? Many of the group's used will be applicable particularly SU(N) and Z_2.

 I should have time tonight to do some digging as well as self reminders.

 

Edited May 30, 2020 by Mordred

[FlawedBeing]

2 hours ago, Mordred said:

As it's been awhile since I last looked into MSSM based models The notes you put above serves as useful reminders. It's been awhile since I looked at Grassmann variables.

 So I am currently looking for some useful direction in literature for the OP.  At the OP have you studied Pati Salam via SO(10) MSSM ? Many of the group's used will be applicable particularly SU(N) and Z_2.

 I should have time tonight to do some digging as well as self reminders.

 

I have not studied that GUT. A glance at that Pati Salam GUT, and things already seem exciting, seemingly very relevant to what I was looking for! Thanks Mordred.   

The last type of grand  fundamental physics I looked on, was this recently published call to participate, via Stephen Wolfram (wolfram language creator/genius).

Now, I'm going somewhat off topic about Wolfram  

"Wolfram: Finally We May Have a Path to the Fundamental Theory of Physics… and It’s Beautiful"

The Wolfram Fundamental Physics Project page.

 

6 hours ago, joigus said:

Yes. Supermanifolds are the direct product of space-time and an even number of Grassmann (anti-commuting) variables (alt. any number of \vartheta \vartheta^{*} complex Grassmann variables.) Plus a series of prescriptions for the differential calculus of the Grassmann part of superspace. I'm not sure I can help you, but I will try to gather info and make as much sense of it as I can. I'm considerably out of touch. I'm sure Mordred can help you much better.

Poincaré group is a non-compact group. Gauge groups are different. They're all compact. Theorems guarantee compact groups irreducible representations are all unitary. Not so for non-compact. That's why "representing" Poincaré (or the manifold?) as reps. of gauge groups sounded weird to me.

SS for the likes of me is better understood on a back-to-basics kind of way. Its original motivation was both to solve some problems in scalar-field vacuum and to overcome Coleman-Mandula theorem, which says that any symmetry group that combines Poincaré group and gauge groups must do so with a trivial Cartesian product PxG, with P Poincaré group and G the gauge group, the 1st acting on space-time, and the 2nd on particle states. But one of the premises is that the background is commuting space (if I remember correctly.) If BG has anticommuting variables, you can combine M and the fibers \varpsi (fields) in non-trivial way by a fiber bundle with G-group acting on the fibers. So that you can only locally expand the fibre bundle as PxG.

My interest went even further back to basics since LHC seems to have given no signs of any SS partners, so I tend to look at it as an interesting extension to quantum formalism, not in the sense of giving rise to multiplets, but I may well be the only person in the world that feels so. 

And that's about what I remember. Anyway, sorry for the lengthy / useless answer, but I wanted to have an anchor to this post for the follow-ups.

No, your response seems considerate and useful.😤

It is giving me something to think of/process, similar to Mordred's response.

Thanks joigus.

Edited May 30, 2020 by FlawedBeing

[Mordred]

 On the [math]P(n]:Q(n],[/math][math]\mathcal{C}[/math] and [math]\mathbb{Z}_2[/math] groups.

Here is an article covering the Super lie algebra covering the above groups.

https://www.google.com/url?sa=t&source=web&rct=j&url=https://core.ac.uk/download/pdf/81957395.pdf&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjAAegQIBhAB&usg=AOvVaw3fK1HirnZXUKfwWAxrV938

Here is an arxiv coverage via "Dictionary of Lie Superalgra"

https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-th/9607161&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjABegQIAhAB&usg=AOvVaw0r1Jr_tcDdLlP9PqtUwe2f

You will find the details on the second article incredibly useful in the P(n) ,Q(n) in particular the even odd operators. As well as a wide collection of the superalgrebra groups.

The CP(n-1) of the paper you linked I would think is a tangent from

[math]C^\infty M[/math] but not positive on that. 

 I would recommend studying Clifford and Lie algebra as well. If you haven't already. 

Edit found how CP(n-1) as per direct limit ie the use For the infinity suffix.

https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-th/9607161&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjABegQIAhAB&usg=AOvVaw0r1Jr_tcDdLlP9PqtUwe2f

Which is a complex projective space.

https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-th/9607161&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjABegQIAhAB&usg=AOvVaw0r1Jr_tcDdLlP9PqtUwe2f

Hope that helps.

 

Edited May 30, 2020 by Mordred

[FlawedBeing]

11 hours ago, Mordred said:

 On the P(n]:Q(n], C and Z2 groups.

Here is an article covering the Super lie algebra covering the above groups.

https://www.google.com/url?sa=t&source=web&rct=j&url=https://core.ac.uk/download/pdf/81957395.pdf&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjAAegQIBhAB&usg=AOvVaw3fK1HirnZXUKfwWAxrV938

Here is an arxiv coverage via "Dictionary of Lie Superalgra"

https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-th/9607161&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjABegQIAhAB&usg=AOvVaw0r1Jr_tcDdLlP9PqtUwe2f

You will find the details on the second article incredibly useful in the P(n) ,Q(n) in particular the even odd operators. As well as a wide collection of the superalgrebra groups.

The CP(n-1) of the paper you linked I would think is a tangent from

C∞M but not positive on that. 

 I would recommend studying Clifford and Lie algebra as well. If you haven't already. 

Edit found how CP(n-1) as per direct limit ie the use For the infinity suffix.

https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-th/9607161&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjABegQIAhAB&usg=AOvVaw0r1Jr_tcDdLlP9PqtUwe2f

Which is a complex projective space.

https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-th/9607161&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjABegQIAhAB&usg=AOvVaw0r1Jr_tcDdLlP9PqtUwe2f

Hope that helps.

 

Thanks, and I also find that the Minimal Supersymmetric Pati Salman seems to be my target.

The gauge group namely namely \[ (SU(4) × SU(2)_L × SU(2)_R)/Z_2 \], seems to concern supersymmetry, i.e. it seems to be a good candidate for what the RHS of \[ C^∞(M) \] may look like.

Edited May 30, 2020 by FlawedBeing

[Mordred]

Latex notation that you find handy the symbol commonly used for tensor multiplication. \otimes

[math]\otimes[/math]

You likely already know that but handy for other readers.

 However the group relations you have above is precisely what you looking at for the parity and helicity operations. Example  (Hand fasteness) between left and right handed particles via the right hand rule.

This is an essential part of your complex topology treatments. 

 From the above for hermitean groups which is part of all special unitary groups be sure to understand the Kronecker delta relations. It will also apply to hermitean and orthogonality. (That includes Hamilton's)

Edited May 30, 2020 by Mordred

[FlawedBeing]

9 hours ago, Mordred said:

Latex notation that you find handy the symbol commonly used for tensor multiplication. \otimes

⊗

You likely already know that but handy for other readers.

 However the group relations you have above is precisely what you looking at for the parity and helicity operations. Example  (Hand fasteness) between left and right handed particles via the right hand rule.

This is an essential part of your complex topology treatments. 

 From the above for hermitean groups which is part of all special unitary groups be sure to understand the Kronecker delta relations. It will also apply to hermitean and orthogonality. (That includes Hamilton's)

Intriguing that the cosmos can potentially be described in terms of rich matrix operations/structures. Very intriguing indeed. Hmmm

Thanks Mordred.