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Somehow I was under the impression that I had written out on the $n$Lab at several places how the traditional physics way to talk about instantons connects to the correct maths discussion. But now that I wanted to point a physicist to this, I realize that in each entry that touches on this, I just gave a quick remark pointing to Cech cohomology, clutching construction, one-point compactification and Chern-Simons 2-gerbes, but not actually giving an exposition.
So I went ahead and wrote such an exposition finally:
SU2-instantons from the correct maths to the traditional physics story
Beware two things:
1) this entry is meant to be included as a subsection into other entries (such as Yang-Mills instanton, BPTS instanton) therefore it is intentionally lacking toc, headlines and other introductory stuff
2) I just wrote this in one go (trying to get back to somebody waiting for me), and now I am out of steam. This hasn’t been proof-read even once yet. So unless you feel energetic about joining in the editing, better wait until a little later when this has stabilized.
Ideally this kind of account would eventually be beautified with some pictures and the like.
I just wrote this in one go
Impressive! I did some proofreading.
Thanks!!
I now went through the first part and polished and expanded.
I also added sub-section names. The pieces that I have polished and expanded now are the first three: “Instantons”, “Vanishing at infinity” and “Instanton number”.
A small fee for being able to understand these ideas!
Your help is much appreciated.
Once we are done with this exposition, then eventually it would be nice to similarly expand on the closely related discussion at baryogenesis.
Namely the remarkable fact is that the standard idea of how baryogenesis took place (why there is a positive net number of fermions (hence matter) in the universe, not annihilated against anti-matter) is via instantons and says that there is one net fermion in the universe per twist of the universal Chern-Simons 2-gerbe. (Not all that far from the vortex atom idea, really, just a bit better, and crucially, a good fit to experimental reality)
The reason one never hears people say this is that they are typically a little careless with the instanton story, never being clear about the global nature of htat 4-form $\langle F_\nabla \wedge F_\nabla\rangle$ – which serves as the baryon source in this mechanism.
I have a quick note about this at baryogenesis, but I assume its point is missed by most of those readers who would appreciate its physical relevance. Eventually I’d like to expand on that, too.
I’m sure I have only a very naive picture of this, but is the idea that after a certain point in the history of the universe, the net fermionic number is fixed, allowing only pair creation and annihilation events? And there’s a desire to understand why this net fermionic number isn’t zero (if indeed it isn’t), resting presumably on some lack of symmetry?
I know they’re looking to detect whether there’s anything different about anti-hydrogen, spectral lines, etc.
Ethan Siegel had three nice posts: I, II, III.
OK, so from the latter there’s something possible different going on with baryon and lepton numbers. Can we see everything in terms of topological twists?
Re #1,
Somehow I was under the impression that I had written out on the n nLab at several places how the traditional physics way to talk about instantons connects to the correct maths discussion
you did have something already in section 3 of instanton in QCD.
is the idea that after a certain point in the history of the universe, the net fermionic number is fixed, allowing only pair creation and annihilation events? And there’s a desire to understand why this net fermionic number isn’t zero (if indeed it isn’t), resting presumably on some lack of symmetry?
Yes. One needs to be careful that the concept of "particle" itself is only well defined on Minkowski spacetime, hence not on cosmological scales. But nevertheless there is a fermion current 3-form $J$, such that its integral
$Q_\Sigma \coloneqq \int_\Sigma J \in \mathbb{R}$has the interpretation of the total net charge carried by the stuff measured by $J$, hence counts the net "fermion number" in all of space" as seen by $\Sigma$.
This is familiar from, say, Maxwell’s equations, where such a $J$ appears on the right hand side as the electromagnetic source form
$d \star F = J \,.$Now given any such current 3-form, then its de Rham differential $d J$ measures how the charges $Q$ changes as time progresses. Because let $\Sigma_1$ and $\Sigma_2$ be two spatial slices of spacetimes, and $X_4$ a piece of spacetime connecting them, i.e. such that
$\partial X_4 = \Sigma_2 - \Sigma_1 \,.$Then Stokes’s theorem gives that $d J$ is the local change of charge, because:
$\begin{aligned} Q_{\Sigma_2} - Q_{\Sigma_1} &= \int_{\Sigma_2} J - \int_{\Sigma_1} J \\ &= \int_{\partial X_4} J \\ & = \int_{X_4} d J \end{aligned}$So if $J$ is closed, then total charge is conserved, while if $J$ is not closed, then $d J$ is the local measure for how charge is being "created" or how it disappears.
Now the remarkable fact is that in the standard model of particle physics, $J$ comes out non-closed. It’s differetial instead comes out proportial to the 4-form which measures instanton number
$d J \propto \langle F_\nabla \wedge F_\nabla \rangle \,.$This is the famous chiral anomaly.
So far this is highlighted in every textbook. But the following further crucial subtlety tends not to be recognized for what it is.
Namely on cosmological spacetimes that carry instantons, then (as I tried to explain a bit in the entry that started this thread here) the 4-form $\langle F_\nabla \wedge F_\nabla\rangle$ is not in fact globally exact. The above formulas hold only locally, on a chart of spacetime. But on intersections two such pieces of data need to be glued by a gauge transformation. If we do make the usual simple assumptions (for simplicity of the discussion), then this gauge transformation is that famous "Chern-Simons winding number" $S^3 \to SU(2)$, which they keep handing the physics students without properly explaining it (such as the one who I am reacting to here).
As a result, in the presence of instantons, then the integral of $\langle F_{\nabla} \wedge F_{\nabla}\rangle$ may be non-zero on a closed cosmological spacetime.
The usual picture (which you see displayed in many popular accounts) is: imagine a 4d cup $D^4$ (a unit disk thought of as a "cup" cobordism from nothing to $S^3$) where the big bang expands spacetime from nothing. Then glue on something like $S^3 \times LongInterval$ and think of the result as being a simple model for the universe. Then assume some boundary condition saying that far in the future from the big bang the gauge fields $\nabla$ that carry those instantons decay away, hence are vanishing at infinity. So then for computing the total fermion charge in this universe, we are effectively dealing with its one-point compactification. In the present simple example this is the 4-sphere $S^4$.
So since the four form vanishes "at infinity", we may just as well assume that it is supported on that cup $D^4$ in our model, the neighbourhood of the big bang.
We learn that that:
The total net fermion charge in the universe is $\int_{S^4} \langle F_\nabla \wedge F_\nabla\rangle \in \mathbb{Z}$;
this fermion number is picked up incrementally increasing from zero (at the origin of our $D^4$-cup, the "big bang singularity) and reaching at "comoving time $t$ \lt 1" the value
(where $D^4_t$ is the disk of radius $t \lt 1$).
Then at $t = 1$ (in this paramneterization) the four form goes to zero (and in this fashion eventually extends to a global 4-form on our one-point compactified cosmological spacetime).
So after the comoving time $t = 1$ there is no net particle creation anymore. Moreover, the net particle number picked up until then is equal to the second Chern class of the cosmological gauge field, hence equal to the number that masure the "knottedness" of the cosmological gauge field (which we here think of as all being concentrated around the big big).
One particle per knottedness of the cosmological gauge field.
That story of baryogenesis remains hypothetical (due to the evident difficulty to test it directly), but it is the best available story that exists and is something like the standard model of baryogenesis, due to Sakharov 67.
Thanks! This must be worth placing somewhere on nLab.
Baryons, of course, aren’t the only fermions, or even the only hadrons. So what else do I add to your fermion current 3-form, $J$, to be able to distinguish baryon, meson, lepton? There is a baryon number current, I know. How does that relate to the fermion current?
I always feel alarm bells should be ringing when physics is turned into ordinary language prose, which you signal by things like
the concept of “particle” itself is only well defined on Minkowski spacetime, hence not on cosmological scales
Then there is the further worrying ’quantum’ business, such as
The theoretical/experimentally observed vacuum of QCD is some superposition of gauge fields in various instanton number sectors.
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