Transcript

Mithat Unsal: Self-Dual Monopole Loops, Instantons and Confinement

From Simons Foundation · Episode page · Watch on YouTube

· 57 min · 7,285 words · 57 timestamped sections

This is the full, timestamped transcript of "Mithat Unsal: Self-Dual Monopole Loops, Instantons and Confinement", an episode of Simons Foundation published January 22, 2026 (57 min, about 7,285 words). It mentions 26 people, works and ideas, including Gauge Theory, Instantons, Confinement, Yang-Mills theory, Semiclassics.

What is discussed, and when

  1. – 5:00: Instantons, Self-Dual Theory, R4, Gauge Theory
  2. – 10:00: S1, Semiclassics, R3, Gauge Theory
  3. – 15:00: Gauge Theory, Semiclassics, Instantons, R4
  4. – 20:00: Instantons, Confinement, Mass Gap, R4
  5. – 25:00: Michael Peskin, Gauge Theory
  6. – 30:00: Gauge Theory, Yang-Mills theory, U1
  7. – 35:00: Semiclassics, R3, S1
  8. – 40:00: Gauge Theory, Self-Dual Theory
  9. – 45:00: Confinement, Mass Gap, Lance Dixon
  10. – 50:00: Topological Term, Chalmer, Dr. Mark Seagull, Necraso
  11. – 55:00: Topological Term, Yang-Mills theory, Chalmer, Dr. Mark Seagull
  12. – 56:39: Confinement, Mass Gap, R3, S1

Notable quotes from "Mithat Unsal: Self-Dual Monopole Loops, Instantons and Confinement"

  • I will talk about self dual monopol loops. I will explain what it is precisely in a gauge theories and perhaps also in nonabelian gauge theories and instantones and their…
    about Gauge Theory · at
  • This is this was my first attempt to do so. you know I restudied some aspects of U1 latis gauge theory and to my surprise understood something new in that context. the fact that there is also a pro previously not quite so much.
    about U1 · at
  • Okay now let's start with the classical data on instant tones on R4 of course this is well known instant are essentially solution of this cell tool equation or anti-ell tool…
    about Instantons · at
  • Was a retrograde it was a step back and that was the reason I wasn't working on it but Yuya when we were discussing showed me that there are some new phenomena new perspective…
    about Yuya · at
  • They didn't build a semiclassical theory of mass gap or confinement based on them. But they knew fractionalization very clearly, very cleanly. And some similar observations were made by numerical studies by Gonzalezo.
    about Mass Gap · at
  • Mentions it in passing. You see that he understands everything but it is like half of a paragraph really and there is this extended nice paper by Michael Peskin and there is this gauge action.
    about Michael Peskin · at
  • Scaling variant theory however there is something very funny perturb you know topological term does not enter the perturbation theory but in this theory remember that we broke the parity between positive and negative felicity.
    about Topological Term · at
  • Opinion carry some glimpse of hope towards addressing the mascap problem and confinement problem in young males theory. I think they present at least something new. So this is all I wanted to say and I can take questions.
    about Confinement · at

Who and what is mentioned in this episode?

People

  • Chalmer — Person · Referenced author (eponymous theory) · 3 mentions · first at
    Named (with Seagull) as the origin of “Chalmer–Seagull theory,” i.e., self-dual Yang–Mills as introduced in the amplitudes/supersymmetric YM context; central to the talk’s later connection to self-dual sectors and interpolation back to full YM.
    Understand. There is this young meals theory and there is also something called self dual young meals theory al it's also called chalmer seagull theory which is essentially…
  • Dr. Mark Seagull — Person · Referenced author (eponymous theory) · 3 mentions · first at
    Named (with Chalmer) in the eponym “Chalmer–Seagull theory,” referring to self-dual Yang–Mills; used to motivate the talk’s construction and two-parameter interpolation.
    Understand. There is this young meals theory and there is also something called self dual young meals theory al it's also called chalmer seagull theory which is essentially…
  • Yuya — Person · Collaborator / referenced contributor · 1 mention · first at
    Credited with prompting the speaker to study R2×T2 compactifications and highlighting new phenomena; part of the collaboration that explored fractional instantons and semiclassics.
  • Michael Peskin — Person · Referenced expert (prior work / paper) · 2 mentions · first at
    Cited for an extended paper on compact U(1) lattice gauge theory (Villain formulation) and its dualization to an Abelian Higgs-like model with monopole worldlines/loops—used as an analogy for instanton size-moduli issues.
  • Mandel Guan — Person · Former graduate student / coauthor · 1 mention · first at
    Named as the speaker’s former graduate student and collaborator on the work being presented (self-dual monopole loops and related gauge-theory analysis).
    Theories and instantones and their possible relation to confinement. And this is the work I did with my former graduate student Mandel Guan.
  • Yui Hayashi — Person · Collaborator / referenced contributor · 1 mention · first at
    Credited as working extensively with Yuya on the R2×T2 perspective that revealed deeper structure about instantons and fractionalization.
    Reveals something deeper about the instant tones. Hence we started to work and Yuya and Yui Hayashi worked it a lot in this context. and the very important fact in these cases…
  • Aguchi — Person · Referenced theorist (volume independence) · 1 mention · first at
    Invoked in the phrase “Aguchi and Kawawaii” as the originators of large‑N volume independence (Eguchi–Kawai reduction), tied to the talk’s discussion of adiabatic continuity and compactification.
    Earlier part of the conference You know this is intimately related to this idea of larger volume independence of Aguchi and Kawawaii.
  • Kawawaii — Person · Referenced theorist (volume independence) · 1 mention · first at
    Mentioned with “Aguchi” in connection with large‑N volume independence (Eguchi–Kawai), supporting the compactification/continuity theme.
    Earlier part of the conference You know this is intimately related to this idea of larger volume independence of Aguchi and Kawawaii.
  • Piljingi — Person · Referenced expert (prior work) · 1 mention · first at
    Mentioned alongside Vambal Lee Lu for work demonstrating constituent fractionalization of instantons in compactified settings with non-trivial holonomy.
    Even before this endeavor of weakly coupled you know semiclassical domains people like especially Vambal Lee Lu and Piljingi constructed compactifications either in the context of young meals theory or.
  • Vambal Lee Lu — Person · Referenced expert (prior work) · 1 mention · first at
    Cited for early constructions showing instanton fractionalization in non-trivial holonomy backgrounds upon compactification (constituent monopoles), predating later semiclassical confinement frameworks.
    Even before this endeavor of weakly coupled you know semiclassical domains people like especially Vambal Lee Lu and Piljingi constructed compactifications either in the context of young meals theory or.
  • Lance Dixon — Person · Referenced expert (amplitudes) · 1 mention · first at
    Mentioned as a past discussion partner about helicity decomposition and the (non-)reality of definite-helicity configurations, motivating the speaker’s move to self-dual sectors and complexified fields.
    Knew it was true. Then now you will see the connections. You will start to see the connections. Then I recalled some conversations I had with Lance Dixon when I was a postto.…
  • Gonzalezo — Person · Referenced researcher (numerical studies) · 1 mention · first at
    Credited with numerical studies that made related observations about fractionalization/constituent structure in compactified gauge-theory contexts.
    Very clearly, very cleanly. And some similar observations were made by numerical studies by Gonzalezo.
  • Necraso — Person · Referenced researcher (related work) · 1 mention · first at
    The speaker suspects the derived constrained path-integral / BF-type formulation of self-dual YM is related to “Necraso’s work,” but does not provide details; indicates a potential literature connection.
    I suspect this must be related to Necraso's work but I couldn't come with all the details but I wanted to talk about it here also to discuss with friends here.

Concepts & topics

  • Gauge Theory — Concept · Core topic · 8 mentions · first at
    Fundamental framework for discussing instantons, Yang-Mills theory, and confinement.
  • Instantons — Concept · Core topic · 4 mentions · first at
    Central to the discussion on gauge theories and their relation to confinement.
  • Confinement — Concept · Core topic · 5 mentions · first at
    Central theme in understanding the behavior of gauge theories and instantons.
  • Yang-Mills theory — Concept · Core topic · 4 mentions · first at
    Fundamental theory discussed in relation to instantons and self-dual fields.
    Completed this part and published the paper. That's something I started to understand. There is this young meals theory and there is also something called self dual young meals…
  • Semiclassics — Concept · Framework · 6 mentions · first at
    Methodology used to analyze instantons and their contributions to gauge theories.
    Will go to the notion of ideiabotic continuity and calculability and reliable semiclassics.
  • Self-Dual Theory — Concept · Key argument · 3 mentions · first at
    Discussed as a sector of Yang-Mills theory with deep connections to instantons.
    This almost integraable theory which is oops self dual both young males and young males and you know I will talk about something new it's a two parameter generalization of the young males theory okay now let's start with the.
  • Topological Term — Concept · Key argument · 8 mentions · first at
    Discussed in the context of self-dual Yang-Mills theory and instantons.
  • R4 — Concept · Setting · 5 mentions · first at
    Spatial context for discussing instantons and gauge theories.
    But it is instant on exponential of the instant on factor divided by n and the semiclassics is reliable provided this ln less than lambda inverse. so as I said semicclassical…
  • Mass Gap — Concept · Key argument · 4 mentions · first at
    Discussed in relation to the behavior of gauge theories and instantons.
  • U1 — Concept · Key argument · 2 mentions · first at
    Discussed in the context of lattice gauge theory and its phases.
  • R3 — Concept · Setting · 6 mentions · first at
    Spatial context for discussing semiclassical domains and instantons.
    Then I will these are of course taking place in compact fight setups such as R3 time on R3 times to T2. But about two years ago I decided that it is time to graduate R4. you…
  • S1 — Concept · Setting · 6 mentions · first at
    Compact dimension discussed in the context of semiclassical domains.
    You will have they will all fractionate and you can see this very explicitly and they fractionate to either on R3 cross S1 to monopol instant and on R2 * T2 to vortex instant.
  • T2 — Concept · Setting · 1 mention · first at
    Compact dimensions discussed in the context of semiclassical domains.
    Then I will these are of course taking place in compact fight setups such as R3 time on R3 times to T2. But about two years ago I decided that it is time to graduate R4. you…

Full transcript with timestamps

I will talk about uh self dual monopol loops. I will explain what it is precisely in a gauge theories and perhaps also in nonabelian gauge theories and instantones and their possible relation to confinement. And this is the work I did with my uh former graduate student Mandel Guan. Um okay. Uh first I will describe briefly some basics about instant tones in four dimension and uh describe the famous or infamous size moduli problem. Then I will go to the notion of ideiabotic continuity and um calculability and reliable semiclassics.

Then I will these are of course uh taking place in compact fight setups such as R3 time on R3 times to T2. But uh about two years ago I decided that it is time to graduate R4. you know I did enough on semiclassical domains to learn something about the theory on R4. So this is this was my first attempt to do so. Uh you know I restudied some aspects of U1 latis gauge theory and to my surprise understood something new in that context. the fact that there is also a pro previously not quite so much emphasized monopol loop size problem and I will describe its resolution actually theory resolves it for itself for you then I will describe these uh general self-dual abelium fields uh and the necessity of complexification of the field space um then I will draw some analogies between instant and self tool monopol.

The final part of the talk is something that took me by surprise after I completed this part and published the paper. That's something I started to understand. There is this young meals theory and there is also something called self dual young meals theory al it's also called chalmer seagull theory which is essentially self dual sector of the young meals theory and I started to understand that there are some very deep connections between this construction and the one here and uh I ended up constructing some interpolation between this almost integraable theory which is um oops self dual both young males and young males and uh you know I will talk about something new it's a two parameter generalization of the young males theory okay now let's start with the uh classical data on instant tones on R4 of course this is well known um instant are essentially solution of this cell tool equation or anti-ell tool equation for

other way around but I will um almost always talk about both of them. uh if you know it will be implied it is a famous for very large dimensional modulized space it's forn dimensional you can decompose it as R4 which is the center of the uh instanton and R plus is called size moduli and um this M head MC is centered moduli it is MC is um MC is 4N minus four dimensional and MC hat is the is this coet essentially it's related to global gauge orientations never mind there is a very elegant parameterization of the of the instanton these are in terms of so-called um ADHM parameters these are a set of um n spinners you know these I denoted with w you can think of it as mx2 two matrix and you can write down um the instant on solution this is in singular gauge because computations are easier in singular gauge in this form. You should notice that if we are looking to instanton at large distances, it reduces

to this to this form. And you must actually remember this form from your undergraduate days, maybe even earlier because this is the field vector field associated with a magnetic dipole. And but now the distinction relative to magnetic dipole is that this object here is an M byN um anti-self matrix and um um M byN matrix it is anti-self tool and it is not your usual magnetic dipole but some nonabelian generalization thereof. Okay. Now let me tell you uh about the instant on uh size moduli problem. In the early days people attempted to do semicassics with the instanton like we do in quantum mechanics. You look to the prolif in quantum mechanics. We look to the proliferation of the instant tones and they lead to level splitting. For example, um in the case of double whale potential and with similar idea, people look to you know um you know proliferation of the instant on 4D. But if you go ahead and write down

one instant on contribution to partition function, it does something um at the time looked weird. So the contribution comes about there is this scale factor and there is this factor coming from running and you see that even for n equal 2 this factor the contribution of the instant on to the vacuum energy diverges at long distances in the infrared and this is a big problem because it's an um long distance IR divergence it tells you that there is something you do not understand about this uh about the about the construction. So so this observation renders the naive semiclassical expansion based on instant meaningless.

However, this divergence that I mentioned here, if you go to semiclassically calculable domains such as R3 * S1 or R2 * T2 which are uh you know in theory is adabatic connected to R4 this problem doesn't exist. So in these cases if you have an instanton with size moduli larger than this and you will have they will all fractionate and you can see this very explicitly and they fractionate to either on R3 cross S1 to monopol instant and on R2 * T2 to vortex instant. Let me just in a slide tell this ideabatic continuity idea. Normally if we start with a theory on R4 and compact by one direction for a long time you know at least in the QCD community this was interpreted as a thermal circle and there is a phase transition despite the fact that by asytoic freedom the theory gets you know weakly coupled in the small circle you do not gain much because yeah you learn about phase structure of theory which is wonderful

but for the purpose for my purpose purpose. It is sort of detrimental because by studying small S1 it is hard to learn something about large S1 you know because there is a phase transition and it is understood that you know in the you can construct theories in which there is this continuity between large circle and small circle. Of course this was very well known widely known in the context of super symmetry. All super symmetric theories naturally satisfies this. But also all nons super symmetric theories were famously not satisfying this. But then you know we came up with many nonsi theories with better judiciously chosen boundary conditions uh crucially non-termal where this idea holds and you can also make it work not only our three * s1 but our two * t2 and just in conjunction conjunction with the uh earlier part of the of the conference You know this is intimately related to this idea of larger volume independence

of Aguchi and Kawawaii. Which one? Yeah. Gauge gauge coupling becomes weak by asytoic freedom. But of course long distance is not weakly coupled. It's

obviously okay. So these works on semiclassical calculity on R3 cross S1 um you know they started some time ago but then there was this uh point where we started to examine our two times T2 to be honest I thought it was a retrograde it was a step back and that was the reason I wasn't working on it but Yuya when we were discussing showed me that there are some new phenomena new perspective ives and it reveals something deeper about the instant tones. Hence we started to work and Yuya and Yui Hayashi worked it a lot in this context. uh and the very important fact in these cases is that the semiclassical expansion parameter in these cases is not the instant on factor but it is instant on exponential of the instant on factor divided by n and the semiclassics is reliable provided this ln less than lambda inverse. Um so as I said semicclassical expansion is an expansion in these uh fractional

instantons or their clusters called u bions and again as I said these regimes are adabatically connected to our four in the sense of gauging varant parameters but it doesn't mean that you can take the effective field theory and extrapolated to large circle effective field theory has a region of validity it is valid provided But this condition holds and beyond that you cannot continue these things to larger circle. You cannot continue your construction. Then you are not talking about you know uh what you want to talk about you know um this is just for the old times memory you know uh just an example of what happens. For example, if you take SU2 gauge theory with adjint matter, okay, you have instant tons for the instanttons which are self dual as well as monopoly instanttons and KK instanttons which are self dual and you have these objects with magnetic charge and half a topological charge and then you can

construct these BPS KK bar type object which is which has magnetic charge two but topological charge zero and even these neutral objects and you can actually when you write down the super potential in n=1 super young ms theory on r3 cross s1 and build the bonic bonic potential out of it they describe these object these composite objects you can show this and hence that means that The semiclassics is based on these objects and you can write down effective lang. So far so good. So what happens with the size moduli problem? With the size moduli problem okay the big issue was this integrant. Okay was growing very large. You know it was lambda row 113 n / 3 minus 5. It's growing like that polomially but at enlarge it grows even faster.

But what happens with these semicological constructions is essentially following. You know if you have a small instant on up to the size of ln still uh this function grows in the same manner but after that the gauge coupling you know there is su gauge structure reduces to something like maximal group something like you want the n minus one and there are no charge degrees of freedom which would make the coupling to run further. So coupling in some sense freezes. This is similar to a generic point in the modulized space of cyber viton theory and then the size mod this integral decays and the integration is finite and the IR divergence does not even exist.

Now, okay, some remark here. Um, now trying to go ahead. Exactly. Yeah. Otherwise, you know, it will continue to grow obviously and the theory will become strongly coupled and you will lose control.

You don't trust it. You don't trust it. Um now this is all nice the previous story it's all nice but unlike uh these weekly coupled calculable domains there is no physical fractionalization for sure you know for the instant on R4 in gauge theories or instantones in sigma models like CPN okay so what is happening with the instant on R4. Okay. Um but uh even before this endeavor of weakly coupled uh you know semiclassical domains people like um especially Vambal Lee Lu and Piljingi constructed compactifications either in the context of young meals theory or um in the context of siting theory and they call for super young meals where they show that you In a non-trivial holonomy background, if your polyanko loop is non-trivial in the compactified direction, there is a fractionalization to these constituents.

They didn't build a semiclassical theory of mass gap or confinement based on them. But they knew fractionalization very clearly, very cleanly. And some similar observations were made by uh numerical studies by Gonzalezo. So what does this mean? If we think about it, if we think about it, it means that whatever is the internal structure of an instanton on R4, when you compactify it, it must be able to accommodate either these n types of monopolant or n types of center vertices. You know these in two dimension these reduces to vertices. Of course, center vertices are point like in two dimension and as I said with UA we established a nice semiclassical description of them and it is a little bit weird they must have some internal structure but it doesn't reveal itself on R4 okay so and I wanted to understand better the instant on R4 and we wanted to understand them better I will give you one more classical data

about the instant ones which is extremely important and um so it is as important as ADHM uh you know description of the moduliz space as a because it's also classical data on the same footing. So if you take two small instantones at a large separation or instant on anti-instant and look at their interactions the instant on instanton interaction is strictly zero because they are bps okay but if you look to instanton or take this formula you know if you look at it you know it is of type 1 /x to the 4 which is long range algebraically falling But it is falling much faster than 1 /x² for sure. And there is this structure and this structure is again something familiar. If you were to write down this thing in three dimensions okay you would get from your electromagnetism book 3 p1 n p2 n minus p1 dot with p2. This is literally dipole dipole interaction but generalized to four dimensions.

Um that is to my mind very curious because if you want to get uh in four dimension and this is also first derived again this paper I mentioned kalan gross and um maybe dashan and and in the general however this formula is also true for su in all in all in all cases of n. So when you think about it you know one one question why the reason is you know we have instantones we thought they were pointike objects in spacetime yes they are different from instantones in other theories because they have a size moduli right but where did these structure came from this structure is something you can affiliate in four dimension with loops with currents. So if you had for example two current loops in four dimension you can write of course in four dimension magnetic dipole moment is a second rank anti-ymmetric tensor. You can write this thing you can calculate the interaction between these loops and they will decay

exactly like this. But for us instant were not loops. To me this is very surprising. This is very weird. Okay. But I must add one more thing. Yes it is. So but we cannot be too naive about this standard current current interaction. For example, if you take two instantones, the interaction must be zero. Whatever you do with dipoles, regular dipoles, you can never make the interaction zero. You know, maybe once in a while in some angle there is some accident. I don't know. Okay. But for instant on instanton, you cannot make it zero. So it cannot be just regular loop. It is for sure. Okay.

Um this is zero. And how do we reconcile the fact that for example instant on anti- instant on interaction is exactly like this while um instant on instant on interaction is strictly zero. And I said as I said it is very important to keep in mind that both this data as well as classical moduli space are classical data about instantones. They are on the same footing. But you know if you look to generic review on for example instantones in super symmetric givers or something like that you will never see this uh this information.

Maybe in QCD oriented reviews there is this uh formula but for SU2. Okay. Now the question I want to ask really what's an instant really on R4. Okay. It's a very basic question um you know but I think it is I put this question mark intentionally there because we do not quite understand it. what is the internal structure of it which give rise to all these different semiclassical uh you know construction semiclassical configurations. Okay, I will return to this question. This is my target. But first I want to say a few things about a theory of loops a well understood theory of loops and the fact that the size moduli problem may very well be fake it is and maybe a little bit provocative statement but you will see what I mean and maybe more appropriately it is an artifact of what we did maybe in reality it doesn't exist so let me show what I mean? Okay, I will take a one of the best studied models in four

dimension. It is not on R4, it is on Z4. Okay, it is uh U1 um uh latis gauge theory with compact gauge field. I use villain formulation because vain formulation is much more useful for computations. You can do many things. Exactly. Um there are beautiful works about this and they are actually solved quite nicely especially in quite some detail. Pyako mentions it in passing. You see that he understands everything but it is like half of a paragraph really and and there is this extended nice paper by Michael Peskin and there is this gauge action.

So when we do partition function we integrate over these gauge fields real value gauge fields and we sum over all of these integers. These integers um encode the information about monopol. They encode information about discrete magnetic flux mu. Actually we can describe this object and that will be the um the current associated with the word lines of these objects which is very nice. Um big achievements in my opinion of Michael Peskin. This is is that to at least in the strong coupling regime he was able to dualize this theory rigorously and he obtained a latis abelonian hick model where the meadow fields are monopoles and the gate fields are um dual photons. It is like in some sense like a bonic truncation of cyber pit and construction. Basically there is a little bit less matter here than there. Okay. Um this is very nice. So what do we learn from there? So if you work out with these currents and you can

uh do the take the current and make a HD composition of it you know you can actually write this thing um partition function here sorry it doesn't go back it's okay it factorizes to a photon part dual photon part and monopole part and the monopole part is actually a classical gas of loops interacting where loop segments are interacting with each other by this 1 /x² uh interaction as they would do in four dimensions and I decompose these following paschat and g and g not is the self energy of the loop we will which we will associate with the mass of the monopulse. Okay, you can go ahead write down a scalar QED abelionics model. There is dual photon and there is this matter field. You can actually derive this formula from you know from the word line representation of this latis model. you know a scalar monopole coupled to dual gauge fields and incorporating some potential and most things in this model as I said

are very well understood. Okay. Um so you can ask why am I talking about it? I already alluded. The reason is very likely the second most important lesson emanating from this U1 lat theory are not discussed earlier in literature and they may shed light on a problem in pure young meals theory okay so you can study phases of this theory people study it it's strong coupling these monopoles proliferate that's equivalent to condensation of the dual hick field you know this uh dual superc conductivity that people talked often here. But if you go to V coupling, there are still monopols, but they are very dilute. They are very heavy and there is a phase transition at some critical point.

I want to describe for you this phase transition in a very physical and intuitive and beautiful explanation by piles type argument. You can talk you can think about you know the weight factor of loops of length L. Take a loop arbitrary loop of length L. Of course it is Boltzman suppressed is mass times length of the loop. And you would think that it is not quite relevant. But then you have to count how many loops are there. How many nack tracking loops are there? starting at some point going around making L steps and coming back at the same point. If you count this, this gives you the entropy factor associated with these loops. And to my surprise, you can do this this counting quite precisely. It is roughly speaking uh 7 to the^ L which is something growing exponentially with size L and you can put a bound for this C between these two numbers. This seven is really coming from you know 2D minus one. The fact

that we are in um four dimensions and this is something um well known in um let's comm community lot community also graph community. Um so good then the this argument tells you that if you are at strong coupling e square very large this entropy factor dominates and loops proliferate. If you if you are at very weak coupling um boltzman factor energy dominates and you are in the column phase. These are the two phases of lat gauge theory. What is there to learn? Well, okay. I mentioned these are the two phases. You can also construct the same thing in U1 AION Hicks and let me skip this. It's not that important. The important thing is following. Let's take again we are in the context of U1 L gauge theory. Take these two monopol loops and let me call the current K as magnetic current.

These are my currents and this is the standard current current interaction. But let me take them far apart. I already gave the answer but to if you calculate this go ahead calculate this you will find this dipole dipole interaction which is very nice. Now what is the problem in Yan me's theory based on the fact that small instant at large separations exactly same calculation interact by this 1 /x to4 potential it is argued that oh this is much weaker than column potential which is 1 /x² hence one it can be ignored despite it being long range the lesson we learn from semiclassics one should Never ignore long range interactions between topological configurations.

Okay. So this is the same situation with monopol loops. So let us do what we did exactly in the dilute instanton. Yes. Let us ignore the interactions between loops you know and look what happens. So this was my moduliz size integral. But now I want to calculate one monopol contribution to the partition function. So I have to sum over all possible lengths of L with all starting points and all starting points will give me some volume of the latis. But the summation over L if you look at it whenever you are in this strongly coupled regime it's an integral of exponentially growing function it is much worse divergence than in um young meals theory so this is clearly divergent so this is when you do non-interacting yes clearly this is nonsense and when we construct dual Abelionix model it has no such problem. So what is the error? This 1 / x24 is indeed much weaker than kum and and we ignored it despite it being

long range. Um the crucial thing although small loops interact by 1 /x to the 4 the current bits interact by 1 /x² which is interesting in the phase in which loops proliferate. It is certainly unjustified. We are ignoring 1 /x² type interaction. You know it is like ignoring kum interaction in a colum gas. you will not get a D by length period. Okay, this is essentially in my opinion what happened in the first semiclassical attempts with instantones in Yang Mill's theory. However, the difference crucial difference in Yang Mill's theory is that we do not know the counterpart of these underlying 1 /x² type interaction or what give rise to it. This is the thing that we do not know. Okay,

go ahead. Current not no it's current segments they think of yeah current segments

oh it is yeah there is of course I uh apart from very good apart from the the most basic one you know choose a loop you can say I started here end up there but you cannot say hey I started here, end up there. You have to divide at least by L. But that exponential factor is there for these nonback tracking loops. I actually communicated with some mathematicians who were doing these kind of things and I put these upper and lower bound after discussing with them. the spiral argument for

of course of course it's a very very very beautiful argument and it's always extremely useful now okay um I also want to say one more thing about R3 cross S1 you know um R3 cross S1 also tells us something aligned with this you know it cannot prove anything about R4 but it has to be consistent with whatever is emerging on R4 indeed as I said instanton splits up to n monopoly instant and these are monopol operators if you are in the foro model you can forget this part um because it is gapped but in general monopol operator is this there is a complex field in the exponents it's very important Um and Alin Stanton is essentially putting these monopolles together. Okay.

And indeed the classical interaction between monopose and antimonoples are carried with these pi and sigma channels and both of them are 1 /x. But um if you look on the other hand these these were charge operators but if you look to instanton operators you see something interesting again this is on our three cross s1 let's say you take two monopolles at at x1 and x2 let's say they are centered at some x center and the distance between them is d you can just do a tailor expansion on this and you will see that instanton operator is actually on our three cross s1 is some sort of a dipole. It's actually not a very standard dipole. This part is a standard dipole but this part if you want to interpret it really it has to be an imaginary dipole and I will come back to this don't confuse yourself about this if this part sounded confusing but there is clearly a dipole there.

Now I I wanted to put this thing because you know uh this um also inspired a little bit from polyacos. He likes this analogy as well but I will use it slightly differently. You know this cave allegory there is this very weird object instanton we do not understand but we are people trying to understand some of its images some of its reincarnations and on one wall we were understanding monopoles then then came and we started to understand the other wall and they don't look much alike but now we can continuously connect them and whatever this is you know once we understand it we know that it's images its shadows must reproduce those. Okay.

Uh let's call this mother theory. Okay, you know and then therefore my goal is try to understand something directly on um R4 or Z4. Now somebody said that in vacuum of the Maxwell theory there are no instant there are no you know non-trivial self configurations. However, in a gauge theory, I am taking a uklidian setup. It's trivial to construct self dual fields. Turn on some pay attention is ukidian current and current J. So if you take these sources correlated like this you know it is like double loop the field is automatically self tool or anti-self tool and naively uh you know and these these fields are actually nothing but the instant on field instant on field strength okay it's very interesting now we are in a billion okay um now naively if Oh okay one more thing before and if you look to interactions now you did something correct now if you look to interactions um these things are vanishing self tool

self tool are vanishing you can show this and self tool anti-el tool has this form um and it has the same form as instant now we can think of n² minus one components of maxwell field viewing them as components of a matrix. And we can actually reproduce large fields of a you know the field of an instanton at large distances by putting uh these magnetic dipole moments coming from these currents together.

Question. Yeah. So what exactly did you do? So what does monopole loop? Oh, naively

naively it looks like I introduced an magnetic current and electric current. Okay, but I will tell you that this is not a dion. Okay, the I will tell you because this is in Ukidian. I will tell you it's Minkoski interpretation where we do Gauss's law you know live in Minkoski space. Yeah. Um now however these things in young meal theory we have these um dipole moments and you know of course it is self-dual there has to be a multiplicity of three and there are three n² minus one such objects but it is it is something very bizarre because all of these 3 n² minus one things are determined by 4 n minus 4 parameters of the centered moduli space.

It is something extremely orchestrated. You know, I I intentionally use this word because they don't play a game distinct from each other. They are all in harmony. They do something in unison in some sense. Okay, it's interesting. Um, now Tin asked what did I do exactly? And uh I want to bring this thing up is self dual monopolion loop. You know if you look at it naively we introduced a magnetic currents and electric currents but remember this was in uklidian space. And actually you can analytically continuous duality equation to minkski space it becomes this. It tells you that magnetic field is real then electric field must be purely imaginary. So we are working in the space of complexified fields.

Um I will tell that the cell dual version of a monopole in Maxwell theory must be accompanied by a real magnetic charge but purely imaginary electric charge with the same magnitude. Same magnitude it's important. Um this will also guarantee that cell tool cell tool interaction is zero. So let me do my Gauss's law. Gauss's law was this for the electric charge for magnetic charge Gauss's law was this and E time G satisfied direct quantization I think the right setup for this problem is complex by E complex B so now electric field uh at the electric charge has a component which is in units of E which is real the thing that we know and love it also has a purely imaginary part in units of G where G is 1 / E and it's purely imaginary.

Now these are the standard things. If you have a monopole we you know we have some charge G electric charge zero. But if I want to make it self tool man monopole I do get this. However dion is normal dion is not self tool. It is magnetic charge G electric charge E. And if you want to write down a cell tool di you have to have this mixture both real and imaginary electric charges you cannot write a di field which is manifestly real neither in self tool neither in Minkoski space nor uklidian space. I can this was the discussion for the charges.

These are charge minks charge real. There is also similar discussion for currents. You know the role of the currents is sort of flipped because in the current there is a time derivative and uh you can there is such a structure but let's not worry about this. Okay. So now let's go back um instant despite early obsession to mass gap and confinement problem are deemed somewhat irrelevant for this phenomena in the long run. It is commonly accepted that they do not play a role in confinement large they are viewed as completely suppressed in large and mascap.

So I am pushing the opposite idea. However, you know what we understood in the last 15 years is that in all these cases fractionalization of the instant tones lead to all interesting nonperturbative phenomena and they are not inconsistent with large n especially there they they would survive in standard large. Um now let me let me pause here and ask following question. Apparently in the story whatever the story is this existence of long range classical interactions between self tool and anti-eltool configurations play a most prominent role. By the way, uh when I made this table, I thought I I was sort of fooling myself a little bit, but I knew it was true. Then um now you will see the connections. You will start to see the connections. Then I recalled some conversations I had with Lance Dixon when I was a postto. You know we was working on these amplitudes with you know uh positive negative you know they

are decomposing gons to positive negative philistian I recall that you know you cannot really make a positive philisticon real you know sort of if you try to make it real it will be some superposition of self positive and negative but mathematically there is such a solution Okay, then I ask myself a question. This question, let me turn off anti- instant on out of the game. Can I do this? Yes, I can. All I have to do is to insert this direct delta into the pet integral. Insert a direct delta which tells you that I must integrate over cell tool field configurations.

And this took me to a big surprise to some very interesting literature in the context of amplitudes. Lots of wonderful developments. So let's start with theory and for clarity let's set theta angle to zero. Okay. Now I start with pet integral and insert this constraint which tells me that I must integrate only over cell to all fields. Let us see what will happen. It is surprising. Okay, let us impose this by a lrange multiplier and let us also write trace square as f minus square and a topological term. Clear? Now this is my trace f square part. I wrote it a little bit differently and this is my constraint.

Now the very interesting thing is that you can do a field redefinition and absorb these part which include kinetic of the gauge fields to the BF interesting um and I will take B to be anti-self tools okay so it will impose that anti-self tool part of the field strength must be zero hence I must integrate over cell to all parts. The outcome is this. This is the theory. There is a topological term. There is BF term. Okay. You can also do this integration. There is a direct delta which tells you that you are only summing over instantones and there is only this topological term.

I suspect this must be related to Necraso's work but I couldn't come with all the details but I wanted to talk about it here also to um discuss with friends here this is good this action then came to realize that is something well studied it's called self young meals theory. It is introduced in the context of an equal for super young males by chalmer and seagull. But now note that this 1 / g square multiplies only topological term. If you wanted to add theta, you can add theta. But I wanted to reduce the cluster. Okay, no worry about it.

And this is not a problem since we you may think that oh this thing will lead to a divergence but since topological charge is positive now this thing will give a sum over instant this part will be only exponentially decaying some structure that appears also in localization discussions. However, at this stage, I want to draw your attention to the curious fact that G must be interpreted as a topological coupling. It is not a coupling, kinematical coupling. It appears only in the topological term. The perturbative aspects of this theory are very well studied actually. Um, but about the nonperturervative aspects, there is almost no literature. There is a very nice paper by los which is um completely ignored and um costello wrote something mathematically about it. It's also no it's also physical. It's quite nice. Um but I wouldn't call them trying to understand the theory nonperturervatively. they

understand something extremely important about this theory in the instant sector. Actually um I will tell you what they understand. Okay. the perturbative in the perturbative QCD context um I must say one thing people do not even bother to write this term um they just write this part and they do not even bother this term because it's topological it will not enter perturbation theory okay um and remember BF term does not have a coupling constant it's a classically scaling variant theory Um however there is something very funny perturb you know topological term does not enter the perturbation theory but in this theory remember that we broke the parity between positive and negative felicity there is no longer you know a parity symmetry so in principle the topological term is a classically marginal term you have to add to the lranian it may not affect to the the theory it may not affect perturbation theory but it will

affect the theory even if you do not add it will be generated and this is what lov you know the perturbation theory alters the topological term there is a exact one loop running of G if and only if the theory is considered in the instant on background and the running coupling is 11 to n and this is now one loop exact I we also have an independent derivation of this essentially uh mapping this problem to some nsp type computation maybe arad is here I don't know um in chalmer and seagull also in later literature stated that if we deform the action with this g square trace p square we of course land on back on pure young meals theory which is um very good which is very good But I want to do I do not want to do that. Okay. Um I will do something hopefully far more interesting and I have a good backup for it. Okay. Instead of deforming the theory by exactly by this coupling which will conspire with that to bring it back

to pure young mills I will deform the theory with an arbitrary dimensional coupling epsilon square. epsilon square is my new coupling and again to my surprise viton um I I told you I have a good backup for this in this very interesting paper perturbative g is a string theory in twister space he actually constructed the same idea to you know to you know to to bring back positive fist amplitudes to the game because you know the game was based on negative fist amplitude and you of it to bring them both uh together to to you know to make the symmetry between positive and negative validity back again make make the theory back meals I I constructed the same thing um I introduced this thing actually to generate the disparity between instanton and anti-instant now instanton action is still this but anti-instanton action is this in particular In the limit where epsilon goes to zero, anti-instant action diverges. It

decouples from the theory which is very nice. We are left purely with instant. If you take epsilon / g1 you land on pure young meals theory. So this is a theory two parameter theory two parameter young meals theory interpolating between cell to theory which is almost integraable because the smetrix is classically trivial and quantum mechanically there is only one loop contribution and bardin claimed that this is related to some sort of anomaly it's called integraability anomaly in the context of amplitudes um so it intermolates between this theory You can ask what is the IR behavior of this self to young theory.

In particular, if you take epsilon over G to be much less than one, we can maybe say something. Now I will tell you the other surprise. This is one loop exact. If you continue there are corrections of the form G cube epsilon square but the running of the epsilon is also same. So they all they both run in the same manner but their initial condition differ by a parameter. So this becomes strong this sector becomes strong once and this becomes strong much much later.

G because from the perspective of epsilon G is topological coupling. Isn't it wonderful? Yeah. Um but here there will be by an SVZ type computation. Um so in the epsilon limit you can ask is the theory gapped confined very likely the answer to both are no at least on our three cross one I can show it is no and it's easy. Um despite the existence of monopose and no antimonoples there is no mass gap. uh this is so both in uh self tooling me theory also self tool QCD with adjint fian since collaboration loves a joint fiance I had to do this okay and in fact in the theory with in the cell tool QCD joint with adjint fianions in the interpolating theory it's very easy to construct the mass cap mass cap comes from these these two factors instanton divided by ni instanton divided by n and the crucially Mass cap has this form.

Okay. In the epsilon goes to zero limit even at finite L mass cap vanishes. Confinement vanishes. Okay.

How do you calculate? Uh you can ask if it later you on R4 no on R this is on R3 cross S1. I will make a claim on R4. So I am actually done. I will answer that question. So both constructions I think the one based on the self to loops as well as the interpolating theory between self tool young males and young males in my opinion carry some glimpse of hope towards addressing the mascap problem and confinement problem in young males theory. I think they they present at least something new. So this is all I wanted to say uh and I can take questions.

Questions about this episode

What is discussed in "Mithat Unsal: Self-Dual Monopole Loops, Instantons and Confinement"?

In order: 0:00 Instantons, Self-Dual Theory, R4, Gauge Theory; 5:00 S1, Semiclassics, R3, Gauge Theory; 10:00 Gauge Theory, Semiclassics, Instantons, R4; 15:00 Instantons, Confinement, Mass Gap, R4; 20:00 Michael Peskin, Gauge Theory; 25:00 Gauge Theory, Yang-Mills theory, U1; 30:00 Semiclassics, R3, S1; 35:00 Gauge Theory, Self-Dual Theory; 40:00 Confinement, Mass Gap, Lance Dixon; 45:00 Topological Term, Chalmer, Dr. Mark Seagull, Necraso; 50:00 Topological Term, Yang-Mills theory, Chalmer, Dr. Mark Seagull; 55:00 Confinement, Mass Gap, R3, S1.

What is mentioned in this Simons Foundation episode?

26 people, works and ideas are mentioned, including Gauge Theory, Instantons, Confinement, Yang-Mills theory, Semiclassics, Self-Dual Theory. Each one on this page links to its own page with every episode that mentions it.

How long is "Mithat Unsal: Self-Dual Monopole Loops, Instantons and Confinement"?

The episode runs 57 min; the transcript is about 7,285 words in 57 timestamped sections.

More from Simons Foundation

About this transcript: the episode's spoken words as captioned on YouTube, split into timestamped sections. Reclip.Me locates every person, work and idea in the text, keeps the time it is first named, and quotes the words around it verbatim. Last updated January 27, 2026.