# The quaternionic Seiberg-Witten equations

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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?"

recategorized Aug 10, 2018

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 ?

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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 (80 points)
edited Jun 27, 2018

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(?).

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