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added to references at Higgs bundle.
added to Higgs bundle right after the definition a brief remark on the similarity and difference to the notion of flat vector bundles, and then added a section “Definition – In terms of D-geometry” copied from what I just added at nonabelian Hodge theorem.
added a remark on the terminology by Witten
The link in # 6 is to nonabelian Hodge theory; also added was recent paper that appeared in SIGMA last year.
If somebody can clarify. From the idea section,
Higgs bundles can be considered as a limiting case of a flat connection in the limit in which its exterior differential tends to zero, be obtained by rescaling. So the equation $d u/dz = A(z)u$ where $A(z)$ is a matrix of connection can be rescaled by putting a small parameter in front of $d u/dz$.
First sentence I do not understand even as an English sentence, and the second mathematically. Where is $\Phi$ from the idea section – which limit of what ?
First sentence I do not understand even as an English sentence, and the second mathematically.
Yes, that’s a common problem with the author of revision 3 ;-)
Since I see you are now editing elsewhere, allow me to say:
In such situations you ought to react:
Either invest the work to remind yourself and clarify/fix the passage you wrote – or else delete it.
But do not just leave the entry with its issues alone for others to deal with.
Revision 3 sentence from 2010 was, as you now alerted,
Higgs bundle can be considered as a limiting case of a flat connection in the limit in which its exterior differential tends to zero, what can be obtained by rescaling.
where at least the English does make sense. However, the English in that sentence has been truncated in version 7 by another hasty author :)
I really do not know which reference explains the reasoning in the second sentence, it is definitely taken from some standard reference. WHAT IS…a Higgs Bundle? paper listed does not have this reasoning (and it seems that Hitchin’s paper listed later and also Simpson’s do not provide it either). There is another comparison on Higgs version flat connections in Sec. 3.2 of pdf, along different lines.
Why I need it now: moduli of certain Higgs sheaves have Hall algebra-like structure which is referred in a paper about categorified Hall structures which I review for Zbl. It is a side issue (just one of the background papers) there, but I am trying to get through the references to understand the motivation.
I can say that Hitchin makes me upset with choosing so counterproductive terminology; it is demotivating as one is bound to conflate it with a rather different idea. Witten’s explanation that it may be fit, after twisting, in a supersymmetric model, does not make it fit with physics terminology and being a scalar there. I never understood the subject precisely because conflated terminology is obscuring the motivation for me to understand.
I think I got partial sense of it: if the small parameter is on the right hand side as the sentence says, that is like saying that the connection form is rescaled by a large parameter. Then the Maurer-Cartan equation has a quadratic term which is quadratic in large parameter and the exterior derivative term is linear in large parameter. So in the first order the Maurer-Cartan equation becomes the Higgs equation. (This is in local coordinates and to this extent it is clear; I still do not see however the global definition from this.)
There is a MathOverflow which makes some related comparisons, https://math.stackexchange.com/questions/2498255/difference-between-higgs-bundle-and-vector-bundle-with-connection
Higgs bundles can be considered as a limiting case of a flat connection in the limit in which its exterior differential tends to zero, what can be obtained by rescaling the connection form. So the equation $d u/dz = A(z)u$ where $A(z)$ is a matrix of connection can be rescaled by putting a small parameter in front of $d u/dz$, or equivalently a large parameter in front of $A(z)$. Then the Maurer-Cartan equation has a term quadratic in the large parameter and the exterior derivative term is linear in large parameter and can be neglected, so the flatness reduces to the Higgs equation $\Phi\wedge\Phi = 0$.
P.S. Higgs equation –> Higgs bundle condition
Added
I am now almost happy with the updated formulation
Higgs bundles can be considered as a limiting case of a flat connection in the limit in which its exterior differential $d\Phi$ is small in comparison to the exterior square $\Phi\wedge\Phi$, what can be obtained by rescaling the connection form. So the equation $d u/dz = A(z)u$ where $A(z)$ is a matrix of connection can be rescaled by putting a small parameter in front of $d u/dz$, or equivalently a large parameter in front of $A(z)$. Then the Maurer-Cartan equation has a term quadratic in the large parameter and the exterior derivative term is linear in large parameter and can be neglected, so the flatness reduces to the Higgs bundle condition $\Phi\wedge\Phi = 0$.
Still, while a holomorphic flat connection via rescaling can produce Higgs bundle, it seems to me that it is not true other way around – if we assume $\Phi\wedge \Phi= 0$ this does not say that it must be simultaneously also $d\Phi =0$ or that it came from a limiting procedure. In particular, Higgs bundle condition does not imply usual integrability in the sense of connections, hence it is not clear why one stills calls it integrability (in the entry and elsewhere).
And probably one should just decide to decouple intuition from any meaning in word “Higgs” here and think of a useful degenerate analogue (also rather than the limiting case) of a holomorphic flat connection adapted to introducing strong tools like stability conditions.
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