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stub for instanton
This seems not to distinguish instantons from multiinstanton solutions ?
I mention instanton number. What should we say instead?
Instanton is a solution with instanton number one. Multiinstanton is a similar solution with instanton number finite n>1. Instanton is an analogue of a soliton “solitary wave”. If you nonlineary superimpose 2 you do not get an instanton but a 2-instanton solution roughly, though not linearly, composed out of 2 instantons. You do not want to call it instanton any more, though instanton solution is OK, if instanton serves as a modifier, meaning that multiinstantons are higher instanton-type solutions.
That’s not my impression of the common use of the term.
As far as I am aware the common use is to say that “an instanton” is a self-dual connection. This may have any instanton number.
But your point is maybe about notions of instanton’s in theories other than Yang-Mills theory? I’ll rename the entry into “Yang-Mills instanton”.
Okay, I have written a more general Idea-section at instanton and split off Yang-Mills instanton.
It is the original distinction from soliton theory, when the instantons are discovered, including I think e.g. the original ADHM paper (should be checked), and it applies both to Yang-Mills and in general. You are probably right, nowdays, the distinction is less often taken care of. Wikipedia gives 484 thousand hits for instanton and 14 thousand for the multiinstanton and 982 thousand for multi-instanton. A bit odd.
If you feel like it, give a discussion of the terminology issue in the entry instanton.
I think one should also keep in mind that in Yang-Mills theory the term is being used in a way rather detached from what may have been its original meaning.
Sorry, I can not afford distract myself in yet another topic. I have only few hours left for math for the entire rest of the week, the rest being already occupied with other things.
added some pointers to instantons in string theory (i.e. branes wrapped on spacelike cycles) to the Idea-section and the References-section of instanton.
Instanton configurations on a G gauge theory are usually classified by a homotopy group (e.g. π3(G) in 4d). How does this classification change in the case of a Higgs phase? Is it some sort of combination of homotopy groups of the reduced gauge group H≤G and of the homogeneous space G/H?
Some old folklore related to this matter is due to:
and
which is commented on at QFT with defects in the section Topological defects from spontaneously broken symmetry.
I see, thanks. In particular, the first reference does speak of the homotopy groups of G/H. As observed in the literature of Lie groupoid gauge theory (e.g. here recently), one can think of Higgs phases in terms of Lie groupoid gauge theory where the relevant Lie groupoid is the G-action groupoid 𝒢 on the homogeneous space G/H. So defects in this phase would be classified by the homotopy groups of 𝒢, involving homotopy groups of G and G/H. Are you aware of a reference that perhaps does not phrase it in this language but speaks of how the homotopy groups of G and G/H combine nontrivially to describe the defects in this Higgs phase?
Am not aware of other references off the top of my head, and haven’t really looked at the one you link to now.
But beware that the action groupoid G\\G/H is equivalent to *⫽H≡BH (see for instance at induced representation the lemma here).
So if you compute homotopy groups using the Borel construction do you just end up computing the homotopy groups of BH?
Yes!
But if not the action Lie groupoid, then what is the Lie groupoid such that if H is a normal Lie subgroup of a Lie group G becomes Morita equivalent to the Lie group G/H presented as a one-object Lie groupoid?
So of course G/H≃G⫽H — but I understand you want a natural homotopy quotient which is B(G/H). I don’t think that exists unless H is abelian.
When H is abelian, then the homotopy fiber sequence
G/Hϕ⟶BH⟶BG,which exhibits BH≃(G/H)⫽G, partially deloops to
B(G/H)−Bϕ−⟶B2Hand that exhibits B(G/H) as the homotopy quotient of the homotopy fiber fib(Bϕ) by the 2-group BH
B(G/H)≃fib(Bϕ)⫽(BH).(Here I am using the general fact that any homotopy fiber sequence of the form F⟶X⟶BG exhibits a homotopy quotient X≃F⫽G, cf. at infinity-action or Prop. 0.2.1 here)
I see, thanks, Urs.
By the way, its entertaining to note that there is a “calculus of homotopy quotients” at work here (not to say “calculus of fractions”):
For ∞-groups 𝒢, ℋ the homotopy fiber of Bℋ⟶B𝒢 is 𝒢⫽ℋ.
With this we have that
fib(Bϕ)≃(*⫽H)⫽(G⫽H).So the homotopy quotient of that by BH is
fib(Bϕ)⫽BH≃((*⫽H)⫽(G⫽H))⫽(*⫽H).Now thinking of “*” as “1” and then cancelling fractions as in arithmetic, we find
(1/H)/(G/H)1/H=1/G1/H=1/(G/H),which is the intended result.
(This is just a side remark. I don’t have another way to justify this “calculus” than by the facts mentioned previously. But it can be used to make quick heuristic checks for the kind of computations we are talking about.)
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