start.
I don't know if it slows down. Oh, no, it slow weeks.
Perfect. Okay.
So, it's our project to have Marius coming from…
Upstate. I think it…
Maris, gonna go jiza PhD student at NYU.
Ken is your advisor, right? Yeah, okay, good. And I guess you're gonna go to Santa Barbara? That's right, yeah. Natania?
Uh, yes, I'm from the Angola.
Oh yeah, I guess you're gonna have better weather. So, yeah. So, if you want to talk about the diamond loops and Avila and Eastern Time, so I'm so dumb that I thought, like, Avila and Eastern Tom didn't. I'm gonna tell you why they do exist.
Uh, good. Yeah, so indeed, uh, the title of the talk is Diane Loops in a Reline Instant Tons.
And it's about this connection between dions, so particles of electric and magnetic charge, and a notion of Avelian instantones.
Uh, and the talk will mostly be based on this paper that came out last year, uh, with Isabel Garcia-Garcia and then Ken Ventilerk. I also have some work in progress with a student at UCSB, coincidentally, Dan Zayek, that I'll touch upon very lightly near the end of the talk.
Good. Okay, so the outline of the talk is the following. I'm gonna start by just
Motivating why you should be interested in a connection between dions and instantons in a Belie and gauge theories, and why those could possibly be relevant
to physics that we might care about.
Part 2 will then be showing that in a U1 gauge theory, these dions, these charged particles with magnetic and electric charge,
Uh, our type of abelian instanton. They carry abelian instanton number, they're some kind of topological configuration.
Uh, and then part three will be going to a UB completion of the story in here in Part 2. So, you might know that
Monopoles and more about the dions in the Belie and gauge theories are singular objects, right? That's the…
famous direct monopole, for example, and to really get a sense out of this U1 diode configurations, you want to go to some
a gut theory or something like that, and so that'll be what the final part of the talk is about. Okay, so that's that plan.
So, uh, let's just dive into the motivation.
So, in the story today, we have a clear, and we have a stage, and the player is some monopoles and ions.
So, monopoles are pretty ubiquitous, and the completions of the standard model loops.
Uh, theorise, and for example, Grand Unified Theories, uh, pretty generically, in external theories, you can have what are called KK monopoles.
And we understand today that they are almost a necessary ingredient in theories of quantum gravity, and this is captured in what we call the completeness hypothesis, and more recently, the magnetic weak gravity conjecture. We have deep reasons to think
that Monopulse must exist at some energy scale in nature.
So, monopoles are quite the familiar object, I think, to a lot of particle physicists.
What are perhaps less familiar are dions,
So, uh, theories that have magnetic monopoles, so this could be, like, some Grand Unified theory, for example, some non-avelion gauge theory.
They often come with other states as well, these theories. They don't just have single monopoles of some magnetic charge, so this would be this guy right here, some particle of magnetic charge, one and electric charge, zero.
It, in fact, comes with a whole tower of states, uh, where you get other particles as well that have
Magnetic Charge 1 still, uh, but they might have electric charge plus 1, plus 2, minus 1, and so on. So you get this whole tower of ionic states.
Particles of both magnetic and electric charge. Again, this happens in guts, for example.
Uh, and we can see that it's exactly loops of these types of dions that correspond to our type of abelian instant element.
a topological.
So, that's the player. We're going to talk about dions. What is the stage? What is the actual physics where these objects are interesting, in particular in these loop configurations that I'm going to talk about?
Uh, that's… stage is when you have some global U1 symmetry in your theory that is broken by an ABJ anomaly.
So, that just means I have some no other current in my theory, J. It can be another current associated with a bunch of different U1s, I'll give you a couple of examples in just a bit.
Uh, but sometimes you can get a violation of those symmetries, such that the northern current is sourced by FFDUAL.
And it's exactly when you have U1 symmetries of this kind that are sourced by FFDUL, that you care about a notion of a Belanian sentons.
So, examples of theories where this happens could be a theory of massless fermions, or a theory of axions coupled to photons.
So, for example, in massless QED, you just have massless direct permions,
Uh, you have this axial U1 symmetry, it's broken by an AVJ anomaly, it's sourced by this FFDUAL term on the right-hand side, right, so I can plug this J up here, and this will be the right equation for it.
in mass security.
Uh, where this is just, you know, your direct Fermion operator.
Uh, or you can have, for example, a theory of axions coupled to photons.
So, uh, Axions famously have this shift symmetry,
that in a free axion theory, I'm allowed to shift the axion by any constant, and I get back the same Lagrangian.
Uh, well, if you couple axions to photons, so you add a theta FF dual term to your Lagrangian,
Uh, the shift symmetry is also broken exactly by an abelian J anomaly in this type of sense.
So, this symmetry casts the current, uh…
FD mu A, where A is the axion field operator, F is the axion decay constant, and again, as soon as I coupled axions to photons, this shift symmetry current will be violated in this way.
Okay? These are the types of symmetries we'll care about today. Why?
Because these types of symmetries are heavily, heavily affected by the statement
And the phenomenon that there are no abelian instantons. So this is sort of common lore that I think
Many of us have heard uttered, uh, many, many times.
So, it's common lower than in a U1 gate theory in a flat spacetime, there are no topological solutions that carry charge under FF dual.
There are no abelian instantons.
Why is that important?
That's important to theories that have symmetries broken by these FF dual currents.
Uh, because…
There's effectively no object to realize the right-hand side of this equation. I'm saying my symmetry is broken by some abelian ABJ anomaly, the current is sourced by FF dual, but I have nothing that's actually charged under that FF dual. And this has real physical consequences.
So, going back to massless QED, for example, and massless QED, it means that, in spite of the fact that I'm breaking some U1 axial symmetry associated with
the fermions of my theory.
I still have Helicity Conservation. If I scatter particles together,
Uh, so here you're seeing, for example, electrons, massless electrons being scattered off each other, you know, the spin here, all these points away from the direction of momentum. That's saying the helicity, right, has a minus sign. If I look at what comes out of that scattering process,
The helicity will be conserved. That's in spite of the fact that the current, or the symmetry, that we would naively associate with holistic conservation is in fact broken.
I think an even more striking example is the other one I told you about earlier, which is the axion.
So, for the Axion shift symmetry, if you couple the Axion to photons, the axionetry also gets broken by some FF dual term on the right here.
Uh, and you might be familiar with the fact that, for example, if you couple axions to gluons, if this were, like, some GG dual thing,
then I would have instant hunts, I gave the axion some cosine potential, there's this famous plot that people sometimes draw of that, where there's sort of a nice cosine shape here, but if I just couple the axion to photons,
Even though the symmetry looks broken in exactly the same way.
The axion potential remains flat. In a sense, the axion shift symmetry stays intact, right?
The axion shift symmetry tells me I can shift the Axion by any constant, and I don't climb up or down some potential. The potential turn remains constant, and so here are the action shift symmetry, even though it's broken by this equation,
is still intact. And the reason that you still have some notion of some surviving symmetry in spite of this breaking here is exactly because
Uh, you have no abelian instant times.
So the takeaway is really that in a flat spacetimes,
Symmetry is broken by abelian ABJ anomalies are still partially intact.
Due to the non-existence of abelian instant tons.
Okay.
That's a statement that has been known for quite some time, and I haven't yet told you about how monopoles and dions fit into the story.
That'll be the next part. And this is where a modern perspective on these symmetries will become really, really helpful.
So, in particular, we will go to what are called generalized global symmetries. I don't know if you guys are too familiar with those here, so I'll give you just a quick rundown.
Uh, the sort of modern perspective on symmetries is, for example, if an axion shift symmetry, or if there's some U1 axial symmetry.
I can implement the symmetries not just by some transformation rules in a Lagrangian or an action, or in a path integral if you want to be fully general.
Instead, I can implement symmetries by some topological operators, here I'm calling them curly D.
So that, for example, and I'm just going to use a specific example of asymmetry, for an axion with some shift symmetry, one has some associated set of operators, call them dshift,
So that, uh…
in my path integral, or inside of correlation functions, I can put these
operators that are defined as some
fields on a 3 surface.
And I'm purposely being, sort of, schematic here, I'm not giving you the details of how these operators are defined, because I just want you to get the gist before we move on to the actual substance of
If we're talking about diode loops. But if you put these operators down that you can define into a correlation function like this,
Uh, these operators implement the symmetry topologically.
So for an axion shift symmetry, for example, how would these symmetry defect operators implement the axion shift symmetry? Well,
You would say, in my correlation functions, where I say I have some axion field insertion A,
I can put down the shift symmetry operator for free, I'll just put down the sphere, it's some operator that's defined in a particular way that I'm shying away from. It'll be more schematic.
Uh, the way that the shift is implemented is I pull this operator through the field insertion A, right? So this picture here on the right-hand side,
I imagine that I've now gone to some new symmetry defect operator called D'.
that I've created by just taking this red beer that you're seeing, and I'm pulling that it's through the axion field insertion. So I'm sort of pulling the red sphere through the orange point, so now they're completely separated from each other, and the price I pay for doing that is I implement the symmetry transformation. So the axion shift symmetry transformation
will shift the axion field by some constant alpha F,
And that's exactly what happens when I sort of pull these two surfaces through each other.
And once I've pulled the symmetry defect operator through the Axion field insertion,
I can then just imagine shrinking it away to zero, and so in the language of correlation functions, what I'll get is I'll get
Axion field insertion is equal to a shifted axion field insertion. You could have said this not using these pictures at all with ordinary symmetries, you would just have said, in any correlation function, I should be allowed to shift the axion.
This is just sort of the operator version of doing that. And this is what the general isymmetries program is all about. It's saying,
Symmetries, which I would narrowly just think about as some local transformation of a field, I can implement by these operator
And the shift gets implemented by pulling them through each other.
So, alright, so that's kind of a way to just view an ordinary symmetry in this new language of generalized symmetries. Why do we care about that?
And why does that give some insight, that thinking about ordinary symmetries could not do?
We care about it because
People have realized in recent years that if I add a billion AVJ anomaly to a theory, so just had some current sourced by FFDUL, the way that I showed you before,
then you don't actually really break these associated U1 symmetries.
Uh, instead, you get what's called a non-invertible symmetry. There's really a notion of there's some kind of symmetry that survives this explicit breaking that it looks like is going on in the current,
And that's… that surviving thing is what we'd call a non-invertible symmetry. So, for example, in Axion-Maxwell theory, with this coupling here, so I'm coupling an axion theta, theta is, you should think of the axion field here.
Uh, to photons.
I can define some new, new operator, and again, I'm being schematic, I don't want to get too much into the details.
But I can take, basically, the operator D that I showed you before, I can put some new auxiliary fields on that surface,
That's what I'm doing here, and I can define some new operator that implements the axion shift non-invertibly. There's some notion of a
conserved non-invertible axion shift symmetry, even in the presence of these terms that otherwise give me an ABJ.
And this was really first explicitly constructed by these folks back in 2022.
Good. So, in pictures, why do I care about this non-invertible symmetry defect operator?
Well, I care about it because this non-vertible thing that I now call DEP over M, these are just some integers, uh,
It's not too relevant for today's story. This operator still acts invertibly on the axion itself.
So, if this symmetry defect operator that's non-inputable that I constructed, if I just look at how it acts on axion field insertions, so it would act on axions and correlation functions,
It still just shifts to Axion by some constant.
Uh, nothing funky is going on here. This is exactly like the picture I showed you a couple of slides back.
The axion shift symmetry, in some sense, is still intact, even though this is now what's called a non-invertible symmetry you picked operator. And this tells you, indeed, that the axion can't have a potential
Because there's a notion, really, of an intact axion shift symmetry.
The subtlety here…
is that this operator, which, sure, it acts invertibly on Axion field insertions, I'm not really saying anything new, if I look at how
This thing acts on Tuft loops.
The story is very different. So what's a TOFE loop?
You can just think of it as a world line of an infinitely heavy magnetic monopole in a theory, so even if I just have a theory of axions coupled to photons,
Uh, this is still a totally valid operator to write down.
Uh, even though I might not have any dynamical monopoles, I can still sort of put in these probe line operators that make my theory look like there's a local monopole moving in some loop.
Uh, and if I put down one of these top loops in my theory, that's again just a theory of photons coupled to axions,
And I take this surface, this kind of represented as a sheet and not a sphere, and I pull it through the loop. I pull them through each other.
Uh, the price I pay is the TOF Loop now comes with some other surface attachment.
This is some flex, right? This is a UN field strength integrated over some sheet that goes through the middle of that toked loop.
Uh, and so I see that even though the symmetry defect operator, when I've coupled axions to photons, still shifts the axion invertibly,
It has this weird operation on TOF loops.
And why is that helpful to thinking about
Uh, a BLAN Instant Hongs and ABJ anomalies was… it's… it's… it's important because of the following.
If I add dynamical monopoles to my theory,
Now, this tough loops, right, or tough lines, that before I told you you can think about as swirled lines of infinitely heavy magnetic monopoles,
Well, now, if I add to my theory, effectively monopole creation operators, so maybe I have some grand unifi theory that has some monopole in it, if I add some
Monopoly creation operator, now these two plans can stream out of that insertion. So this is, like, I'm creating a monopole here, and it's sort of moving to the right.
Uh, and if I go to the previous slide here…
Doesn't this depend on the relative chart?
It does depend on the relative charge, so everything here you can just think of as being charge 1, but indeed, if the monopoly creation operator was charged 2, then it could only screen some of the tuft lines, so indeed, it's much more subtle that I'm running on here.
Uh, I'm just getting to a main point about why you should think that dions are kind of an abelian instant type.
Uh, good. And so if I… if I have these monopolic creation operators, and I go back to the previous picture…
Which was this non-invergible symmetry effect operator did this weird thing to the TUF clients, it added this sheet,
Well, now, if I imagine that I can create these lines out of
This point operator insertions. You can imagine that I could effectively start breaking open these loops, right? I can add some extra operators to my correlation functions, out of which these lines stream.
And what happens when you do that is even this non-invertible symmetry
that was still there, that you found still told you that the axion couldn't have a potential, because there was an intact axion shift symmetry,
Even the non-invertible symmetry is now broken.
And so what this really tells you…
Uh, to some extent, that… this really tells you
That monopulse must somehow play a role analogous to U1 instant Hans.
And they will generate a potential for the axion, because now I add monopoles to my theory, there's not even a non-invertible symmetry anymore that can protect the axion shift symmetry.
Hence, I expect the axion shift temperature to really be explicitly broken. I really expect the Axion to have some potential. It's no longer shift-symmetric.
And so, in that sense, you really expect monopoles, and you'll see more broadly dions, to play a role at UN incidence.
Good. Okay. Um, good.
Uh, this was sort of one of the hints we took, that there was some kind of connection here. That wasn't actually our original hint,
Our original intent was actually from an earlier paper by Matt DeRissant and collaborators.
But from 2021, where they just took a theory of a monopole, a monopole world line theory in particular, so they weren't thinking about, like, monopole self-interactions or anything like that, they just had a theory of a free monopole,
They tried closing that worldline,
With the monofil being coupled to some axion,
And what they found is that
Uh, in that theory, they could show that some axion potential was generated.
So again, they just took out the theory of a monopolar-lined couple to an axion, they did some clever integrals, and they showed
that gives you an axiom potential.
So this is what that potential looks like.
It'll be some exponential suppression in the monopole mass, and the cost and energy it takes to excite a monopole into a dion.
then there'll be some coefficients, CL, L will be some integer you sum over, and then there'll be some cosine, depending on the axion field here, A over F.
Uh, and in particular, when they looked at this modifold worldline theory and saw it gave a potential for the axion the way, like, say, a CO2 or SU3 instant fund would in axions coupled to the Thaner model,
They noted that if they took the monopoles of a SU2 fixed down to a U1, so this would be what's called the Jirty Clasho monopole,
Uh, this exponential suppression factor you get…
becomes exactly minus 8 pi squared over G squared, in what's called the BPS limit. It's a limit where you take a Higgs self-coupling to zero.
And this at pi squared over G squared is one of the more famous quantities in our field.
That's the action of a BPST instant on, and so already back then, they were speculating that there's maybe some connection between these loops of monopoles,
And instant hunt.
And so the takeaway here is really that prior findings had a deep relation between monopoles, or really generally dions,
and instant tant, and that this can really be of some phenomenological relevance, right? Maybe they can generate some potential for an axion,
They seem to be important whenever there are these abelian APJ anomalies.
of course, I should note that there's been previous work, too, on the lattice and elsewhere, where people have found hints that there's some deeper connections between, uh,
Instantans and monopoles.
So, for example, there's some work by Clifford Taubes, Richard Brower, Olivion, and a few others that sort of point in this direction. So this is certainly not a brand new hint, but we sort of have this nice
new perspective on it with these, uh, genetic symmetries.
Okay, so that was the motivation.
So now I'm going to get to part two of the talk, which is really where the bulk of our work starts.
Uh, which is about Dion loops as a kind of Aveelian instantomb.
So, we want to study this connection between dion loops and instantons.
And the way we do that is…
We just take a Maxwell theory and Euclidean R4, right? We also really care about instantons and relations between diamond loops and instantons, so we want to be in Euclidean spacetime, so R4 rather than R3 comma 1.
Uh, and then I just want to look at a free natural theory.
So I just take pre-maxial action,
And we want to add some…
Magnetic and electric current loops to that theory. We want to be UV agnostic, we don't say,
Which UV completion is giving you those magnetic currents.
We just want to be in the U1 theory entirely.
Okay. Uh, so to do that, we're just gonna begin with the direct monopole.
Uh, that looks like the following. In an abelian gift theory.
Even if, before adding any matter,
I'm always free to just write down a potential that looks like this, where this theta and this phi are the spherical coordinate thetas and phi's.
And if I just write down this potential, what I find is it gives me some singularity in my gear field,
That is that of a monopole. It has magnetic flux coming out of it. That's what these blue lines are supposed to indicate.
It has a magnetic charge M, your signature quantized.
Uh, and it has this funny feature that it has what's called a direct string attached.
A direct string is some gauge artifact, it's some singularity in the gauge field only that can be moved around with gauge transformations. It doesn't appear in the field strength, of course.
The field strength only sees the sort of singular magnetic monopole object.
So, what we want to do is we just want to take that direct monopole, and we want to move it in a loop.
So I just take the direct monopole,
This is sort of showing you three dimensions out of what is really a four-dimensional space, so this is squished down.
Uh, but what you're seeing is I take my direct model, I force it to move in a loop,
Just by hand, right? Put in a source term in my equations, that's a magnetic current that moves in a loop like this.
Uh, and what that ends up giving you is the following topology.
Generally, when people work with monopoles, and they want to be careful about the topology, they have with them,
Uh, you want to go to a description where you've sort of excised the monopoly from your space-time. So you're imagining, you're sort of poking a little hole in your spacetime out of which magnetic flux is flowing, and you can impose
boundary conditions on that surface or on that hole. That's what we're doing here. So we're imagining this Dirac monopole world line, which is a circle, has this little excise region around it.
Uh, and the other thing you have is you have some direct sheet.
So, right, a static monopole has a direct string attached,
If I make that monopole move in a loop,
Now that string sort of drags out a world sheet,
And that world sheet is exactly what you're seeing here in the middle. So this is just a picture of a…
abelian modifiable moving another…
squished down from 4 dimensions to 3 dimensions, of course. This is really an S1
cross S2, because it's, like, at each fixed time, I have a surface that goes around my whole monopole, and then that surface you can drag around a circle.
But here, I'm only showing you what looks like an S1 cross an S1, right? A torus, because I can't show all four dimensions.
Uh, good. In this talk, I will occasionally use coordinates R-hat and tau hat, and those you should just think about as being… one is just the prepper time along the monopoly world line,
And the other one is some local, uh, transverse coordinate to two of the world line itself. Just that point out.
Good. Okay, so that's a magnetic current loop.
Uh, now things get a little more subtle.
Because now we also want to add an electric current.
And we want to be a little more careful about this, and this care comes from Billy.
Knowing where these monopoles come from, or these dions more generally, come from in the UV.
Which is, I want to add some electric current.
That flows along the proper time direction of the die, what will now become a dion, it's both magnetic and electric charge.
Uh, so it'll go in a circle, like that. Uh, but this time, I want to make row E not quite a direct delta function, I want to make it some smooth electric charge.
And in particular, I want to make it so that the time component of my gauge field right on the loop, or if you think about the excise surface right on this excise surface,
is an integer times d tau, where tau is, again, sort of the effective proper time, it's an angular coordinate that goes around the loop.
Uh, with N being an integer,
This ends up giving me a feature in the loop configuration that if I integrate this
that gauge field that followed in this electric current.
I get 2 pi times n as I go around the whole loop, so this is a bit like a Wilson line holonomy, for those that are familiar with that language.
Sorry, I don't understand. So the boundary condition on A, or are you solving the equations of motion? We're not solving the equations of motion, we're putting in some magnetic and electric currents that satisfy these relations. And you will see… I don't know any way, I understand that.
But what are you… now you're saying something about the gauge field.
Then I'm gonna solve the equations of motion in the background of that Gauge field.
I'm sorry, in the background of those currents. So, I mean, is this… is this a constraint on the gauge fields in your method?
This is effectively a constraint on migration. For now, I'm not doing path integration, I'm just thinking about a single field configuration, but indeed, ultimately, you would want to sum over all these configurations.
There's a third, and then you said something about a gauge field. How are those two related?
I'm solving Maxwell's equations.
So, yes, you are saying that if you solve Nansel's equation with that current, you're going to have this property of the solution. Yes. Okay, that's right.
That's… that's right. Yeah, thanks for clarifying.
Uh, good. Yeah, so all I'm doing is I'm sort of by hand putting in the currents, and then I'm solving the equations. That's what I'm doing.
Okay, so together, we have something that is a dion in the following sense. It's some delta function, magnetic monopole, it comes with a smooth electric dressing that in particular imposes this particular boundary condition on the gauge field as it
approach us at the core of filter functions.
And we'll see shortly that this is all actually very sensible from Yogi completion. You might be like, what the heck, like, this is a delta function, this thing is smooth, that seems a bit strange.
Uh, but really, if you just look at any non-abelian monopole, for example,
Uh, which we'll look at shortly. Uh, and you project out the VLAN degrees of freedom, you find that, really, that Boolean projection of those guys will be some monopole delta function with some smooth electric dressing.
Okay.
So now, we've done, sort of, the basic groundwork. We've said, I have this ion that's this electric and magnetic current,
Uh, and now we want to see if that configuration carries some U1 instantown number.
Uh, so here, the instantane number, right, it's just the integral of FF dual over all of spacetime.
Here, I've written it as FHF, just to emphasize that it's really a topological quantity.
Uh, in the limit where you take this XI surface to zero, this turns out to actually be a perfectly integral quantity, in spite of the fact that you have this draft monopole singularity.
Uh, and there are many ways to evaluate this entry.
So, Route 1 is, uh, I'll show you two routes, because I think they sort of emphasize different
nice aspects of this calculation. It gives you different intuition.
Route 1 would just be to write FHF,
Also known as FFDuel. In terms of
E.B, the electric and magnetic fields that we know from
You engage, Sherry.
So if I write the integral of F wedge f instead of… Again, we have the same question. So, is the F here going to be a solution of the equations of motion in…
the, uh, in the current background that you specify? Yes, it will be. Specify electric and magnetic currents, that's fine. Yes. And then you could consider the gauge fields.
Yes, that satisfy Maxwell's equation with those curves.
And that will define a topological sector, and then you could even consider the gauge fields in that topological sector. That's right. Evaluate I for that sector. That's right.
Uh, but in fact, of course, since this is a topological quantity, what you ultimately find is
This quantity doesn't depend on, sort of, smooth wiggles, say, of the gauge field, far away from the monopole or dion singularity. But it does depend on the singular forces, so those are encoding the right topology you need to have instant time numbering, that's what you'll find.
This is the Euclidean equations of motion, yeah. So this E-field is, like, not quite what you'd normally call it an E field. It would be, like, I times the normal E-field. But so this is all indeed solving.
the Euclidean equations of motion and Euclidean space. Because ultimately, we want to relate this back to standard instant time.
Okay. So now I have E.B,
Uh, I hope you'll trust me if I tell you you can raise the E-field like this, it will be integral over this component of the field strength, and I can write the B field as this.
And if you go over this component over the field strength,
And what you find, when you just do this integral, is it factorizes really nicely into
You get a 2 pi m, where M is the integer magnetic charge from this factor.
And you get minus 2 pi n from what is effectively the electric charge integrated around this dion.
Uh, and so all together, your Instanton number ends up being minus M times M. So these are in the appropriate integer quantized unit.
the… effectively, the… it's a winding along the skage field, which you can think about as some kind of electric charge, and then it's the magnetic charge.
multiply it together. So you get something that's really an integer at the end.
And the reason I really like this approach to computing the instant time number is it sort of very intuitively explains why Diane Loop should have some abelian instanton number, why this quantity could be non-zero,
For a die-on loop? Well, it's, of course just because what is a dion? A dion is an object that has
parallel EMV fields coming out of it. And so, of course, if I have some abelian loop of dions,
I will get some instant on number, that's non-zero, uh, right? That's… that's sort of the intuitive explanation for why DionLips carry this atomic rate.
Okay, there are other ways to do it, uh, in particular, I want to show you one more way to evaluate the instantown number that…
Uh, just shows the nice topological nature of this object.
So, I can convert my integral of F poach up using Stokes theorem over effectively all my space.
Time… probably spare time, to…
an integral over all the boundary surfaces.
of this field configuration. So in this case, for example, right, if you're familiar with how to evaluate non-avilion instantown numbers,
One of these surfaces would be the infinite spacetime boundary.
In this case, we have more boundaries, because we have the singular defect, that is,
The direct monopole, and we also have this, uh, direct sheet that was sort of the world sheet of the
drag string that's being, uh, being created as you sort of have the monopole go in a circle, or the diagon go in a circle.
Uh, and so when you just do the U1 violent instantime number integral this way, what you get is, you get 1 over 8 pi squared, integral f y jap,
can, after some simplifications that I'm… that I'm…
Skipping over here, because it's as technical of details.
You get, uh, this nice factorization.
Where you get an integral of the gauge field right on this singular surface, or in the exhibition, depending on how you think about it, there are two equivalent descriptions.
of the gauge field, so this will give me, again, this sort of volatomy factor that I had before, 2 pi n.
And I get another integral that's an integral of, at fixed time, so at any fixed point along this loop.
Just the magnetic flux coming out of the monopole, or the dion. And so, that'll give me a two-piece down.
And so, all together, again, right, this gives me minus n times m. If I just multiply that with the pre-vector. And I like this way of evaluating it, too, because it sort of nicely shows you how
Really, the instantane number of these configurations just comes from the fact that you have this sort of singular surfaces that carry non-trivial.
Okay. And so, ultimately, what we found is that the instantane number is minus N times M, it's itself an integer.
And so it comes from the magnetic charge,
And this worldline polymometer, which, again, you should think about as being proportional to the electric charge.
How does this allowed, right? So somebody asks, I think before I even started the talk, I think it was Nico, how is this allowed, considering that
We know this lore, that there are no avian instantons,
Well, that lore really comes from this story.
So, for those that are familiar with homotopy groups, you might have heard the statement that the third homotropic group pi 3 of the U1s is trivial.
And so no abelian incentones. What this is saying, uh, just this, uh…
Homotopy language is saying is just…
And aphelion Gagefield is topologically so simple,
that if I look at my infinite spacetime boundary, which is normally where the non-trivial topological structure of a non-abelian instanton would come from, there's no way for that abelian gauge field to wind.
Because it's gauge group is U1, and there's just not enough structure in a U1. It's a circle or a group manifold to support
And a billion instant phone number.
And so this is why we don't normally have, uh, abelian instant ons. The U1 group is just too simple to have these
non-trivial topological wrappings of infinity.
Uh, however, of course, the loophole to the construction I just showed you is you're not really in a, uh,
flat spacetime now anymore. You're really in a flat spacetime where you've poked, effectively, a hole or a circle. So I've taken my R4 Euclidean flat spacetime,
And I put down a singular direct monopole that moves in a loop, while that singularity is really poking a hole. And so now my spacetime is not R4, it's R4, subtract the circle, and…
Uh, it's exactly when you have the circle subtraction that you can have the U1 gauge field non-trivially around it,
in such a way that you're allowed to have a billion instantane number. And so this is how you sort of circumvent the standard lore.
It's just saying that a direct module, a singular, pokes a hole in my spacetime, and so now I can have abueling exceptional number.
Okay. Uh, and here's sort of a fun little slide.
Uh, which is just showing you that, uh, one way to think about these, these…
topological quantizations is…
Right. One of… one of the integers I had came from this holonomy factor, the gauge field integrated along the worldline of the dion.
Uh, that you can think about as being a map.
From the world line of the dion, so a circle, onto my U1 group manifold.
Here I'm thinking of my U1 group manifold that's effectively being parameterized by a rainbow, so I imagine if I go from white to red to purple to blue, back to white, as I'm doing that winding, I'm sort of winding along the U1 circle.
And so this is showing a spatial mapping from the region around this diagon loop onto the group manifold, and in particular, right, if I go the long way around, so I go along the loop,
I lined white, blue, red,
back to weight, I'm winding around my whole U1 group manifold,
That landing number is an integer, so that's this integer here.
Uh, and then, uh, for monopoles, too, so that's from the electric part, from this electric colonomy,
Uh, but then… then you're not just talking about an electric particle, you're really talking about a dion,
And the monopole part of that particle also has some non-trivial winding number associated with it. In particular, you can think about this as being sort of a landing around the equator of the monopole. And as I do that, the equatorial winding
that winding number here around the equator of the monopole at fixed time along the loop,
Uh, that gives me another integer, and so altogether, these classes of maps from
space-time to my group manifold are given by a product of two integers. That's what we saw was this n times m. The Holonomy factor times the magnetic charge.
And this just so happens to be the same topology as rainbow bagels, which you guys are, I guess, in New Jersey?
So you're probably… I don't know if you dislike inventions from New York City or not, but this is one of our more atrocious ones, I think, over our history.
Uh, so, yeah, that's the sort of fun feature that, right, you can wind this way around the rainbow bagel. You go around the whole rainbow as I go this way around. I can also go this way around, right, so kind of through the inside of it, and I also wind around all the colors. So this is really the topology of a rainbow bagel, basically sty on Duke instant hunt.
Where do the dielect charges of payments? Yeah, I don't think… look, this is…
I think this one should be till, like, it's… you can sort of wind around the rainbow as you go this… through the interior here, but I actually don't think it sort of winds all the way around, so I think this is actually topologically trivial, but it kind of looks like it isn't.
Uh… yeah.
So, I would… I wish I could bake a better rainbow bagel that really emphasized this point. That's what I had.
had these colors be sort of a little more tilted to the right, so I really had to go all the way. The color doesn't actually wrap all the way around. It doesn't wrap all the way around when I go the short way around.
But this was the best picture I could find, unfortunately. From this angle, it looks like it does. From this angle, it kind of looks like it. Nobody has raised that objection, uh, so I think…
The couple of times I've talked about this episode, I think… I think people buy it. But I'm being… I'm cheating a little bit.
Good. Okay, so that's all great, right? I've put in some electric and magnetic currents by hand, and I've shown that there are some instanton number, but
What does this actually buy me? Uh, that is the question, and…
there's subtleties here, which is, in particular, if I wanted to, for example, start computing a potential for the axion using these diamond insertions,
I want to know what the action associated with that loop is.
That turns out to be a challenge.
Because, yeah. I came in, like, so maybe you address this, but this is independent of the size of the… This is independent of the size of the loop. Particularly, if you just have a straight line, it goes to…
Yes, so you actually… you can show that just for a single ion that's sitting still, it'll have… that configuration will have, effectively, infinite instant ton number.
Because… there'll be some instantane number per unit period.
Uh, and then, yeah.
It'll have infinite instant time under.
So this was actually noted by Jaqiv and Christ all the way back in the 1970s, in particular in
at SU2 UV completion, where you have, like, a TUF-Pulikov monopole, and more generally, what's called Julius E. You're holding fixed sumw winding per unit length along… Here, I'm holding fixed sum winding per unit length, indeed. To get it fitted.
Uh, no. You just take an on-shell Dion,
So it has the right mass that you expect with an energy eigenstate.
And you… it is an infinite world line in R4, it's just sitting still. You compute the instantown number of that thing, and you get infinity.
And it's because, uh,
For a Dion, there's really some…
Instanton number per unit length for an on-shell diamond. But here, I'm fixing the winding number. So this is a really subtle story, uh, that I'm not going to get into here, because we're sort of addressing these questions of quantization in a follow-up.
But if you just take quantum mechanics on a circle with a compact time,
Uh, and you ask…
Uh…
There will be a certain topologically non-trivial configurations, right, that wind,
all the way around the circle in the finite time that you have before, you know, time is a periodic coordinate because you've compactified by time, so you have this winding in one unit of your compact time. Ideally, you want winding number 1, right, per compact time.
But those states that correspond to the integer windings per your full compact time length,
aren't actually quite the same as the energy eigenstates of the theory.
And so there's… yeah, that's that subtlety that's sort of rearing its head here.
But as classical in configuration, this is just something where you've sort of fixed this winding number along the loop to be
an integer, but really in data, right, you sort of scaling it as you're blowing up or shrinking down the dion. And so there's a little bit of a subtlety here about electric charge quantization that I won't talk about too much.
But I'm happy to chat later if you're curious. It's sort of a very rich story.
Okay. So, okay.
So, so good.
Uh, so now, just look at my time. I want to compute the action of this thing, right?
And natively, people have thought about the slots when they thought about, sort of, pear production of…
Electrically charged particles, or monopoles, or even dions in the background of some electric field, or magnetic field, for example. You can very easily just estimate the action of a closed world line of a
of a monofil or a dion in the following way.
It'll just be 2 pi times the radius of that
closed world line, this was just the length of this, right? Times what is the mass of the dion.
So the action of these closed-loop particle configurations will generally just be 2 pi r times M.
And so, if you look at this, uh,
Great! Like, this is the action.
But of course, if I don't have any sort of background fields or anything like that,
Uh, this thing will tend to want to shrink to zero, right? I'm not quite solving the equations of motion when I just sort of plonk down this
The electric and magnetic current loop.
Really, I want to be able to vary those… the placements of those currents as well, right? Ultimately, I want to solve a full set of dynamical equations where I'm not sort of forcing any current loops by hand.
And what this is telling you is that, uh, it looks like these diagon loops
really aren't quite solutions to the equations of motion once I make the currents themselves dynamical.
Uh, they only solve the equations when I take R to 0, at which point the whole action vanishes.
But there's a subtlety here.
Which is, if I just take this Dian loop I've constructed, I try to extremize the action so that it's shrink and shrink and shrink,
Eventually, it'll hit a radius,
that is comparable to the core size of the dion.
So, in reality, right, we know that these direct monopoles, and more broadly dions, they aren't quite
singular objects, right? They really have some UV completion that'll show you that when you go to a small length scale,
you don't just see a point-like direct monopole, what you see is really some, for example, non-abelian degrees of freedom that are localized in this tight core of this object.
And what happens is that you take the loop and you shrink it down and down and down and down, eventually the radius of the loop will be smaller than the core size of the dion. And then you get sort of these inverse R terms that'll end up contributing non-trivally to the action.
Uh, what this ultimately shows you is, right, the abelian-dion loop, you can initialize it at some
finite radius, it really wants to shrink,
In fact, it looks like it really wants to shrink all the way down to zero, but at some point, you'll need the UV theory to tell you what happens.
When the core size of the dion is larger than its radius,
then clearly there's some kind of…
UV physics going out, because you have the cores overlapping in the middle.
So, really, you need a UV theory to estimate the action of this object.
Again, that's just a function of direct modifiers being singular.
Okay, so the takeaway is, again, an abelian diamond effectively carries U1 instantone number with a loop perverse shrink to zero size, and to really understand the action and the effects, physical effects of this configuration,
You need a UVCB. So it brings me to the final part of the talk.
Which is to work in a UVA completion. So now we're going to get rid of all these infinities and smooth charges, and we're gonna look at actual UV-complete modules and diamonds.
So the strategy here is just the following.
I want to look at this action. It's that if the Georgia Dimensional model,
Uh, it's just a SU2GET field.
with some adjurant Higgs that breaks SU2 down to U1.
Uh, with some potential here, find not, will be the Higgs Vev, and lambda will be the Higgs self-coupling.
Okay? So this theory…
past the nice features that it has monofil and dion solutions. In fact, it has sort of the quintessential ones.
Uh, good. So the plan is just to construct an analogous field configuration to this U1 loop that we saw in the U1 gauge theory, but now in the full UE theory.
Then we want to, uh, so the field equations for this theory, unlike just the abelian gauge theory, are nonlinear,
I want to use some numerical relaxation scheme to try to solve these field equations for some… for every value of R, and then I want to study what happens as I let the loop shrink down to zero.
And then finally, I want to relate this to the other type of instanton that exists in the theory, which will be PPST instantons. Spoiler alert, you'll see that they're deformations of each other. Uh, and
Uh, I also want to see how the non-avelion construction projects down into U1, and see whether I can get some notion of this.
a billion dialogue that I showed you before.
Okay, so, uh…
This theory, before we go to a closed loop, if we just look at fixed time, has the…
Uh, classic Tuftopolyakov solution.
It's just some solution to the equations of motion that has some spatial gauge field profiles, so there's some non-trivial AIs.
There'll be some non-trivial HICS profile, and crucially, the zeroth component of the gauge field, at least in this particular gauge I'm working on, will be zero.
I'm specifically not writing down the details of this configuration, because I just want to give you the cartoon, but…
Famously, if you write this solution,
Uh, it is that of a Tuft Polyakov monopole, so it's some Higgs field that points in this hedgehog
configuration. That just means I have some point here in the middle, that'll be my monopole, the Higgs field, it has three isospin indices,
Those are all aligned away from that point, and I'll have some gauge field to match, and that'll be a solution to my equations of motion that I will call the Topst Paul. And indeed, it'll just give me some monopole that sits still
in time, right? So it's just some infinite world line, if I extend it through time, that source of some magnetic phase.
Okay? So that's the classic Tuft Poliakov model. Perhaps a bit less known is that what's called the Julius E.ione, which is a generalization of the solution. So the Tope Polikov solution has this feature that the zero component of the gauge field is zero.
Uh, if I add… if I make that non-zero, and I instead make it proportional to, sort of, this
takes, uh, ISO has been direction.
This hedgehog configuration here,
Then, I get something that also sources electric charge. So if I add to my TOF-Polyakov multiple,
Uh, this, uh, A0 component, then it becomes a dion, and this is what's called a Julius dime.
This is sort of the first time people found, and it was kind of a language series.
Good. Uh, okay, so that's a static dion just sitting still. Now things get, uh, subtle.
Sure, you do it all in time.
Uh,
To embed at Julia C. Dion, which has this sort of
Pigs, hedgehog structure.
Right? It's like a… at every point along my worldline,
I have a hedgehog that points away from that point.
In R4, I need to be really careful. I need to be really careful because, uh, if you imagine right, I just have a hedgehog going in a circle,
Uh, in the middle of that circle, the sort of two hedgehogs will point right into each other, and that means there'll be some singularity in the middle. And I want to construct, in particular, I should mention, a field configuration that has no singularities in it. And I know I should be able to, because we're in a fully UV-complete theory, there should be no singular behaviors anywhere. And so, to really embed a dion loop in R4, I need to twist the Higgs field as I go along the whirlwind.
So here I'm imagining I go forward in this proper time coordinate that I showed you before, and the Hague sort of has this interesting feature that it twists. So this is, again, showing the direction the Higgs field is. It twists,
at tau equals zero, it's a normal monopole. At tau equals pi, it ends up looking like an anti-monopole. A hedgehog that points inwards.
Uh, and so, as I go around in time, somehow I drew it clockwise here, I suppose. That's not my normal number.
would go to the positive. Uh, you have this twist feature.
Uh, in doing so will end up creating a diamondidium pair along the loop. Uh, so if I just…
take a slice through the loop, like this. So here, I'm again showing you sort of a loop.
This time, there's actually no direct sheets, because, I'm in a yearly complete theory, it'll just be aloof, but this is just the cartoon I recycled. If I take a splice through the loop here in the middle, what it'll look like to me is…
I find a point here that corresponds to this guy. That's, like, a anti-dion.
This is pigs arrows that are coming into that point in space.
And if I look here, that's corresponding to this point up here, that's like a dion, a positive ion.
It has 6 arrows that point out of it and flow into the bottom guy, right? So again, this picture is showing you this
Two out of my fourth space-time dimensions along this slice.
Uh, in this particular showing you the direction of the Higgs field along every point. So again, this is like a hedgehog, and this is like an anti-hedgehog, so this is like a monopole or dion antidion pier.
Uh, it turns out that having this sort of twisted hedgehog structure that gives you
But it always looks like a Diane, anti-diene pair if you take a slice through the loop.
Uh, it'll actually force…
very interesting, non-trivial topology on the Higgs field at infinity. In particular, it'll force what's called a Hoof map
Uh, on the Hays at infinity. So to have this sort of twisted hedgehog in the middle of my space-time,
Without introducing any singularities whatsoever to my fields.
Uh, out at infinity, I'm required to give the Higgs this
very, very non-trivial structure. This is just showing the three components of the Higgs field in, sort of, Cartesian, uh, coordinates here.
Uh, and these hope maps have these interesting features that… they're these topologically non-trivial maps.
From S3 onto S2.
So S3, right, is my infinite space-time boundary again. That's where I said that Higgs gets this non-trivial structure when I force the middle of my spacetime to have the Higgs twist.
Uh, this map will be from S3 onto S2, where S2 is the Higgs group manifold. The Higgs group manifold is defined by being SU2 mod U1.
the full group, mod, whatever you're breaking down to.
Uh, and these maps, uh, look kind of interesting. This is showing you a
S3, so my infinite space-time boundary, stereographically projected onto R3.
And then squeeze down to a ball.
And so this is really a representation of my infinite S3, and what you're seeing is each of the colors here sort of map onto the corresponding color on the S2, and so this is really some very interesting topologically non-turial map called a Hoof map.
And so, importantly, again, having the sixth twist that really gives me a proper Dion loop in the middle of my spacetime forces the structure on me.
And it also forces something on the gauge field in return, because
For this field configuration to have finite action, I know that the covariant derivative of the Higgs has to vanish at infinity. If it didn't, my action would blow up.
Uh, if this thing was non-zero at infinity,
This is just the Higgs kinetic term, right, or the square root of it, if you want.
Uh, and if this thing goes non-zero, the Higgs… Higgs connection would just blow up.
And so, the fact that this has to be 0 at infinity also forces me to impose some condition on the gauge field, not just on pigs.
Uh…
And you'll see that the condition that I imposed on the gauge field, because of this,
this non-trivial pix structure will be that the gauge field itself lives in the instanton number 1 sector of the theory. So just topological considerations of your… by themselves,
considerations of how can I create a Dion, anti-dion pair, or dion loop?
in R4 forces you to pick a gauge field configuration that has instant turn number 1. Before I've even solved any equation.
Okay. So, uh, you can… you can then look at what these configurations look like, these non-AVLAN configurations, at fixed R, so here I'm sort of forcibly holding R fixed,
I'll eventually take R to 0, uh, and what you get is something that has this, uh, this feature that's… it's a very nice sort of…
monopola, anti-monopole pair, if you look at the B field. So this is, again, just a cut through the loop.
So you get a monopole, and you get an anti-modifole, and similarly, if I look at the E fields, I get something that looks like an electric monopho, electric, anti-monopole.
Although, there is a subtlety here, which is…
The magnetic charges are really nicely localized.
The electric charges are not. And in particular, as I go to really, really large separation,
And I try to solve the equations of motion, enforcing just this sort of twisted Higgs topology. What I'll find is that the B fields still look really nicely like dipoles.
Uh, when I solve the equations of motion, subject to this constraint that I have some loop at finite radius, but if I look at the electric fields, things begin to look kinda strange.
The electric charges don't quite peak right on these points anymore. In fact, they kind of… the field wants to be strongest right in the middle, rather than right around the points themselves.
And what you're seeing here is that there's really some tension between an instantron solution in this theory and thision-loop solutions.
Uh, the Dion loops aren
As I said earlier,
But what you see is that the loop really wants to shrink down to zero. The magnetic charges are forced by topology to be localized right on these two points.
But the electric charge is not quite constrained by topology in the same way, and so it can, in fact start migrating inwards.
Uh, to resemble more of what would be a normal instanton.
And so, again, this is just showing you that
Uh, just fixing topology and solving the equations of motion actually still gets you some kind of object that
While it has the topology of a dial loop, really wants to look like an instant on. It really wants to trink down. And indeed,
As I let the radius of this topologically non-trivial loop vary,
I take the radius to zero, and this is…
looking at a lattice, the following happens.
So, what you're seeing here on the y-axis, you should think about this as being the total action. I want you to look at the blue dashed line here, I think I spoke the cleanest thing to look at. If I just look at the action of this configuration as a function of radius,
But this is radii and inverse Higgs Web units.
Uh, I see the following, that the action is large, at large separation, and then it shrinks and shrinks and shrinks and shrinks as I take R to 0.
And indeed, write at r equals 0, as I take the loop radius all the way down to zero,
this quantity here you're seeing, it's, like, slightly less than 80.
is exactly 8 pi squared over G squared, where I've set G equal to 1, because again, this is done for a specific
like a classical lattice, where we're just solving equation solutions.
Uh, what this is ultimately showing you is that the diagon loop
topology forces it into a
field configuration that carries instant ton number 1, and in fact, when you let the loop shrink all the way down to 0, what you get is a standard BPST instanton.
in a non-emely engaged theory. There's subtleties here about the theories Higgst,
Uh, but I'm not going to get into that, uh, but indeed, what happens is what you get is an honest-to-God instanton at zero radius.
But there are notes that I'm sitting in the use therapy. There are… there are constrained incidents. So, so what… what happens is you, you, say you force your topology to be in the instant on number 1 sector,
You find an instanton solution, but the effective row scale parameter of that thing wants to go to zero. That's… that happens here, too, because we're solving this on some, like, classical lattice.
You're effectively stabilized at some finite radius. But, of course, in a real quantum mechanical treatment,
You do want to have some cutoff in that size row. That's the story of these constrained instantons, where
That scale will correspond to, basically, the Higgs-Vev, uh, there might be some extra…
Gabe's coupling, uh, multiplied on, but it's effectively the Higgsv.
Uh, and so, yeah, there's a subtlety here, but this is indeed the sort of instanton that wants to shrink to zero that you're finding here in this limit as R goes to zero.
but very tiny.
No, the Higgs map is always non-zero, so this radius is… we've set it to 1 numerically, yeah.
Uh, but so everything is just in units of the Higgs valve. But the other couplings are the… well, the quartic, we vary, the gauge cap length language up to 1.
So, I have another question. Yeah.
In your description of the billion things to talk, they were characterized by two integers.
But… Magnetic charge and wiping.
ones are characterized by a single integer.
That's right. So, in this theory…
Georgie last child model.
How many had the troops characterized the… They're still just one integer. Just one. And it'll… it'll ultimately stem from the fact that, indeed, right, the finite radius loops aren't quite solutions on their own.
And, uh, if I take, say, two, uh, two configurations that vary by
say they have instantown number 2, but I've sort of shuffled, which
Which quantum number is 1 and which one is 2, they'll both end up giving me just the standard instanton at the end of the day. Once I embed them in the UE theory. So indeed, there's this very subtle thing about quantization.
And so, importantly, right, as I blow up the loop, it's not a solution to the cushions of motion anymore. What I found is, effectively, the loop is some kind of version of a blown-up non-AV lining instantane.
And so that's takeaway 1 from the conclusion. I have one minute, so let me just finish the last bit here. Dial loops or deformations of Higgs, BPST incentones, and they have higher actions. They're not quite saddle points, you can expand around the way you normally would find something, but…
Uh, there's still kind of interesting in the following way.
Which is, there's some notion of abelian-ness coming from these guys. So, for example, if I work in this Georgia Glasgow model, and I just diagonalize the Higgs,
So I do a singular gauge transformation that takes the Higgs from having this hedgehog structure to now I just make it point in the 3 direction.
And ISO spends space everywhere.
then I can sort of very nicely separate out what I would call my abelian and my non-abelian degrees of freedom. There's a…
get… uh… Gage and Ron, we are way of doing this as well. I'm showing you the gauge-dependent way right now, because it's simpler to think about, I think.
Uh, but I can just take my VLE Engage field and define it as the third component as my full SU2 gauge field, right? It's a component that points along the same direction as the Higgs.
And similarly, I can define my non-abelian degrees of freedom as being packaged in these W fields.
Okay. And so, what I'm doing here is I'm taking this Dian loop configuration at some fixed finite radius,
I'm combing the hedgehog, I'm making a point in the same direction everywhere in space.
So that looks like the following.
Uh, what happens when I do that is exactly that this abelian dion loop returns.
So, what I get is I'll get some singular defect if I just look at A3, what I call my abelian component of my full SE2 gauge field.
That gauge field has a direct monopoly singularity along this loop,
You'll also find that this direct string, and then correspondingly direct sheet singularity comes into play when you do this projection out of the B-Lane increaser.
And so, that is to say, uh, when I… when I sort of separate out the BLAN degrees of freedom, what I get is the U1 dian loop that I showed you before. It also has the smooth electric charge, for example, if you were thinking about that.
Okay. And then this sort of very interesting thing happens, which is I can just compute the instant tau number of this configuration,
In this case, if I wanted to look at the pool of non-avilionelian instantown number, that's my almost last slide.
If I just separate that into an abelian and a non-abelian contribution, this looks…
not quite good at the very end, but again, there's a gauge around way of raising this.
I will find that the non-avelion contribution vanishes, and so as soon as I have these monopold defects,
Uh, even when I look at the full non-abelian instantane number, so the full
G-U-H-G-U-H. This is really the full SU2 field strength. Uh, the entire contribution will come from the Avelian silk.
Once I have the loop defect.
Uh, and so the instant power number is, in a sense, carried in the abelian sector when I have these
background dialogues. Okay, so that's the second takeaway. Projecting that violin degrees of freedom… uh, for sorry, projecting out the BLAN degrees of freedom,
confirms that there's indeed some notion of this object as Dian loop, although it's not quite solving the full anonymlin equation of motion,
it's somehow in a veil and instant time.
And so, my final slide before the conclusion…
What does that mean? Uh, we don't quite know yet. So the question is,
We've established a dye loop, so effectively a kind of U1 instant on somehow a mascot for these non-abelian instantons. They can somehow carry the instanton number in their VLAN component,
Given that, you know, at least a phil the degree completion, they really are related to non-avilion instantons, BPST instantons.
How does this… how is this useful? Can… can we do anything with this? And so there are questions like, can I use…
this abelian dion loops.
to somehow compute, for example, a potential for the axion, knowing just the mass of the monofil and, say, the excitation energy associated with promoting that to a dion. So we can compute things in a UB-agnostic way, even though I know these things are really
mascots of non-abilience.
And so that's… that's an open question that we're thinking about at the moment. So the conclusions are just, uh, engaged series, we have VLAN ABJ anomalies, so hints from non-invertible symmetries and axiom potentials that suggest a deep connection between dions and instantons.
We found that, indeed, dynamics are effectively a type of view on instant hunt,
And in simple UV completions,
Uh, their deformation of standard. Pigsby PST incident, as I showed you.
And this all begs for further explanation. Like, can I actively compute physical instant time effects? For example, for the axion?
Using just these U1 diode loops, knowing no details about the UV completion.
And so that's the talk. Thanks.
In the Georgia Glassgell model, in the Higgs phase, you have an instant pond that has…
Instant Pond number one. Yes.
Then, in your deformation.
So, extrem with the action and pitching.
It's a picture you have. What is the…
electric charge? What's the magnetic charge?
of something that has instant part number one thing. So, indeed. So here, the magnetic charge would be 1.
It would just be the single TUF Polyakov monofil. You might know that in a Georgia Glacier model, there, single monopole solution that has charge 2. It'll be, like, some, like, two monopole configuration, not, like, a single object with a single spherically symmetric core. So it'll be the…
The standard Tofoyakov monopolo of integer charge 1.
And then it'll have this feature that the electric winding number is 1.
But that's, in fact, uh…
not quite corresponding again, and this is going back to this very interesting subtlety about quantization.
Not quite corresponding to a single charged sort of electric eigenstate.
Instead, the electric charge sort of scales of the rate of this winding.
And so what happens is as you shrink the thing down, down, and down, it'll end up going faster and faster and faster, and so localized, it'll look kind of like the electric charge is growing.
As you're taking this operation down. And this is, again, because you're not quite working with an on-shell field configuration. You're working with something slightly different. But it has electric windy one and magnetic one? Yes, and then you can go to electric winding 2, 3, 4, and you'll get the…
Instantile number 2, 3, 4 solutions when you sort of shrink it down. Although we didn't actually confirm that, we just, on the lattice, looked at
the classical, that is, I should say, looked at the single instant time number configuration, but that's what we fully expect to happen.
despite topological consideration.
Yes. So, if you work in the civilian debt,
Yes. And I take…
the different MNs that correspond to one value of the non-billion each other number.
what happens, because you said everything comes from the billion configurations, but they're different.
Uh, so… Right, so somehow it looks like there's maybe… I mean, we haven't… we haven't studied this explicitly,
For the 100-year rate, there's maybe multiple ways of blowing up an instant, right, of taking what is really this
SQ2, Jody Glash out, Instanton, it's stabilized by some finite size. I blow it up into some Dion loop.
And somehow there's not a unique way of doing that. But I'm not fully confident about this one. I agree, it's very interesting. There should be only one way
to go from a fixed solution
in the regular gauge.
to the singular diff.
Only one way of planning to fix. Yes, that's right. Uh, so in… oh, you mean, like, in this case? Yeah, or especially if I think about it, instead of thinking that…
The singular gauge, think about the gauge invariant way. Yeah, yeah. I'm just defining…
gave you invariant functions of the original fields.
Yeah, so I did. So the question is…
Indeed, if I do this VLAN projection, or if I do it virtually, like, point-like instanton, right, I don't really have a good notion of a magnetic charge.
Because it's like, I've taken a magnetic charge loop and I've shrunk it to zero.
Uh, so I don't know if you'll get well-defined
I don't know, that's my… that's my sh…
My short answer. I think that these questions of quantization, we realize, are very interesting, and so it's something we're thinking about at the moment, but I don't have a great answer for you.
Thank you. Yeah. I'm sure you know, there's a lot known about the supersymmetric version… Yeah. Yeah, that's right.
We haven't studied it explicitly. We're sort of phenomenologists, and we're ultimately interested in computing these potentials with these guys, so we just took
the Georgia Goshma model, but indeed, right, there's a BPS styons and supersymmetric theories that are much more controllable. It's understood what the effects have been, you know, at the die on points. Yeah, right, so you're thinking about… yeah, yeah. So you're thinking about a cyber equipment theory, right? Yeah, yeah. So we have thought about that a little bit, but I don't have anything super profound to say, I think.
So, if you have a…
Do you have a tilling vector?
When you look at the, um, institon equations, and you reduce them…
And you use the killing vector to separate one component of the gauge field from the other and call it a Higgs field.
get the monopolic questions.
So that's a rather direct way. The way it's usually done is you look at time-independence solutions with it.
That's a final question. You get the monopolic questions. So you can reduce…
Hans a ton of questions from four dimensions and security. That's right. Yeah, sorry, I think you're referring to mentions, yeah. But really, that you can go with any killing plan.
So I wonder if there's, like, a killing vector that is a zero along the loop.
Yeah, that's… I don't know, that's a good question. Uh, but I think you're talking about what's called the Julius E. correspondence sometimes, which is…
I can sorta…
even in just an SC2 gauge theory, I can treat my zero gauge field component as kind of its own Higgs
Edger and Higgs, and then I can relate my instanton and monopole solutions in that way. Yeah, yeah, right. So this is a pretty standard thing, too.
reducing the acetone equations and getting the monopole equations.
But it really… all it requires is a color lift.
Yeah, so maybe there's some…
Got some funny telling there. Yep, that's possible, yeah.
Yes.
When you say you wanted some time to prepare, I guess, specifically to deal with configurations.
I guess it's gonna be, like, S4 minus the circle? Yes.
Okay, so your statement is…
that is going to have non-trivial V1…
bundles over it, basically? Yes. Okay, and then those are going to be…
the field configurations in each of those sectors are going to be die-hard. Yes. In the U1 picture, that's the same? That's right, that's exactly right. Yeah.
And then you can imagine relevant, right? There's a subtlety of, like,
Right, Dion is really… I hope I got the point across here, but it's really a singular object.
And so, you always have to be very careful about how you regularize the…
configuration, uh, and in particular, as you take the loop radius to zero,
kind of don't really know what happens. You believe I need a full UE theory. But indeed, for some…
finite radius, where the loop radius is larger, the core size of the dion,
And this is really an honest-to-God, sort of, you want them, like,
So, um, I don't know if this is going to be important to you, Richard.
something to be aware of. If you take, um, the U1 gauge theory,
And you want to study its partition function? Yes.
As a function of background electric and magnetic field currents.
As a function of background in electric.
magnetic and electric currents.
Okay. Um, there are some subtleties. It's not a well-defined function.
It becomes a section of a line bundle over the space of background, electric, and magnetic.
And that's… that's related to the fact that there are anomalies.
And, uh, one-form symmetries for electric lights.
So, if you start to look at a partition function, I don't know, maybe you'll just divide out, you know, correlation functions. Yeah, yeah. You don't divide out of bounds.
You know, some operator insertions divided about the partition function, or the subtlety counts what it might.
Yeah. That subtlety is there in principle. Right, yeah, so thankfully, in our case, I could just…
We really got this, uh, shape for the two currents from just looking at a UE complete theory and made them descend down, but indeed,
You have to be very careful. I mean, there's also a question of, right, I told you about AVJ anomalies and, uh, how they're present in chiral gauge theories, right? Uh, and in that case, you'll also have the fact that
Uh, the Kyle Permians will be screening this dion.
And what does that mean? So there's all sorts of subtleties here of the anomaly structure. Of course, that's a zero-form symmetry anomalies.
Uh, you're referring to the one-form symmetry anomalies, and I think that's interesting. For now, what we're trying to do is…
We're just treating as sum over Wilson tough lines, for example. So we've put in some regularized Wilson tough line inserts. But what are you calculating? What I'm saying is, if you try to calculate the partition function, there's going to be extra subtleties.
Yeah. If you're going to calculate the two-point function, divide it by the partition function, you might get lucky and subtleties cancel out. So there would be two… the two-point function of just the field. The two-point function of two gauge invariant. Okay, of two gauge, okay.
Yeah, that's… yeah, we're thinking about the path integral right now. I might get lucky. We might get lucky.
Yeah, so the question is, can you… can you…
do reasonable calculations of this, or are they just kind of deformations of non-abilience at times? That's still… still possible, and really, I mean, if I was a practical person, and I cared about computing potentials for axions, or instanton effects,
I would just say, this is really a SU2 instanton in disguise, and so I just use the standard SU2 instanton solution.
But of course, there would be some power in doing some UV-agnostic calculation, but again, there might be that there are subtleties that can't be overcome very easily. Yeah.
So, good comments.
Okay.
So what other funds can you do this? Like, sorry, like, I'm confused. So, like, for example, can you change the…
prediction of the action mass, now doing like this.
So, phenomenologically, we're thinking a little bit about whether… okay, so if you take… if you take recent companies' calculation at face value, which
We don't think that thing is quite…
quite sensible, but if you take it… sorry, I hope this is not being recorded. I mean, I think there was a great paper in that that sort of sent a bunch of things in this direction, but…
if you're, for example, worried about real-definology of axions, for example, like, you might worry about the quality problem, for example. If you get some potential induced by
Dions from some guts, for example. You can show that in most cases, you're fine, uh, just because these potentials generally still come with these large exponential suppressions.
And, like, in a real physical theory, what would these ions really be coming from? They would probably be coming from some, uh, grand unifi theory, and so you're really suppressed by the scale associated with their
And so, uh, if, for example, you compute axiom potentials from these ion loops, uh, it looks like, and again, the calculation hasn't been done carefully, so I can't say this with utmost confidence, but it looks like you'll at least get this large exponential suppression that'll keep
any physical effects from Grand Unified theory, monopoles at least negligible when it comes to an accident. Then there's a whole set of questions, like I was saying, about the effects of chiral fermions, but there the story also gets really subtle, because
Chiral fermions can kind of screen the dion itself. So, like, super dumb. So what happens if I have, like, this and a electron, and then I do, like, a… Did you scatter them? Yeah, scatter, and then they…
Right.
So this is a Euclidean diamond loop, right? So you're not really scattering anything, but indeed, right, this plays into, ultimately, like, the Kelland-Rubikov effect, then it's, like, fractional fermion number puzzle, the way that dions and fermions talk to each other, and that's… yeah.
There's a rich story, sort of, going on in that area right now, especially sort of new enthusiasm from this general isymmetries program.
Any more questions?
If not, you know, think it's pure game.
And let me know if you want to go to dinner. Are you gonna stay for dinner? First, let me ask you. I can stay for dinner with you guys, so I can…
I'm just… okay.