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 ...

  The quaternionic Seiberg-Witten equations

+ 2 like - 0 dislike
1068 views

I consider the Spin-H structures (H the field of quaternions) which are like the well-known Spin-C structures, Spin(n).SU(2)/ {1,-1} over space-time of four dimension. So I can define the quaternion Seiberg-Witten equations which are non-abelian theory (like the instantons of Donaldson). My questions are "Are the moduli spaces so defined compact?" and "Are the quaternion SW invariants interesting?"

asked Jun 22, 2018 in Theoretical Physics by Antoine Balan (-80 points) [ no revision ]
recategorized Aug 10, 2018 by Dilaton

EW equations are somehow an approximation of those of Donaldson which showed that these moduli spaces are in general not compact. Or else, are you speaking of vector bundles over the quaternions ?

1 Answer

+ 0 like - 0 dislike

I consider a tensor product between the spinor vector bundle and a SU(2) bundle; a Spin-H structure. I take the Dirac operator D_A with a SU(2)-connection A. And next I consider the quaternion SW equations :

                             D_A (psi)=0

                             F_+ (A)=  q(psi)

psi is a quaternion spinor and F_+(A) is the self-dual part of the curvature of the connection A which is an imaginary quaternion.

The moduli spaces are perhaps not compact (Uhlenbeck's lemma?) but perhaps can also be compactified and may give new invariants of four manifolds.

answered Jun 27, 2018 by Antoine Balan (-80 points) [ revision history ]
edited Jun 27, 2018 by Antoine Balan

perhaps this completes the question but it is not an answer

You can find a beginning of answer at the following link: vixra.org/abs/1804.0003

Very clear background for the question... In general, compacity is 'rare' and authors try to build remarkable cases where it can be found. ie Mod 2 Seiberg-Witten invariants of homology tori

I am trying to show compacity of the moduli spaces, following a book of Thomas Friedrich AMS vol 25. I asked also myself that quaternion numbers may be more deep in physics and that we could rewrite the Hilbert space in Quantum Mechanics with a tensor product with the quaternions. Perhaps that we could make a well-founded Quantum Theory of Hamilton numbers...??? In fact, we could replace the complex numbers by the quaternion numbers(?).

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$\hbar$ysicsO$\varnothing$erflow
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
...