Transcript

Silviu Pufu: Precision Studies and Supersymmetry of 2D Adjoint QCD and Its Generalizations

From Simons Foundation · Episode page · Watch on YouTube

· 57 min · 9,147 words · 83 timestamped sections

This is the full, timestamped transcript of "Silviu Pufu: Precision Studies and Supersymmetry of 2D Adjoint QCD and Its Generalizations", an episode of Simons Foundation published January 22, 2026 (57 min, about 9,147 words). It mentions 12 people, works and ideas, including QCD, Super Symmetry, Quantum Field Theory, Tensor Networks, Eigor Benjamin Soggard.

What this episode is about

The discussion focuses on recent advancements in the study of two-dimensional adjoint quantum chromodynamics (QCD) and its generalizations, highlighting collaborative work that explores nonabelian gauge theories as a framework for understanding more complex four-dimensional models. Key points include updates on numerical studies and the exploration of novel quantum field theory concepts, emphasizing the relevance of these two-dimensional theories for various applications such as quantum simulation and algorithm development.

Key takeaways

  • Two-dimensional gauge theories are the sandbox for testing big ideas—like quantum simulations without the heavy machinery.
  • Adjunk QCD reveals mass peculiarities: massive fermions can exist without members of the mass club—how's that for exclusivity?
  • Bizarre feature alert! In 2D, massless theories can still own a mass gap—because physics loves to break the rules.
  • Flux tubes in 2D mimic a dance-off: confinement criteria hinge on whether your area law is in vogue or not.
  • Symmetry isn't just for aesthetics; in gauge theory, it shapes reality—like a well-tailored suit for physics equations.

Who speaks in this episode?

What is discussed, and when

  1. – 5:00: Quantum Field Theory, Super Symmetry, Eigor Benjamin Soggard, Edward Whitten
  2. – 10:00: Super Symmetry, Lattice Gauge Theory, QCD, Quantum Field Theory
  3. – 15:00: Tensor Networks
  4. – 20:00: Tensor Networks
  5. – 30:00: Stonybrook Conference
  6. – 35:00: Super Symmetry
  7. – 45:00: Super Symmetry
  8. – 50:00: Super Symmetry
  9. – 55:00: Super Symmetry, Tensor Networks
  10. – 56:43: Super Symmetry, Tensor Networks

Notable quotes from "Silviu Pufu: Precision Studies and Supersymmetry of 2D Adjoint QCD and Its Generalizations"

  • It's this one in collaboration with Eigor Benjamin Soggard and Edward Whitten and this other one with Ross Demsy, Anamaria Gluke and Benjamin Soggard also mentioned some earlier.
    about Eigor Benjamin Soggard · at
  • Algorithms for instance in litecoin quantization lattice gauge de DMRG to QFT ideas such as non-invertible symmetries anomalies and others and maybe even to quantum simulators…
    about Quantum Field Theory · at
  • Two values of m equals z and m over gang mills is equal to<unk> of n / 2 pi that I'll mention the second half of the talk will be about the super symmetry occurs for this second value.
    about QCD · at
  • And u also brief review of the general features of things that Igor mentioned that so the surprising features are at m equals z and m equals this other value that I mentioned…
    about Super Symmetry · at
  • And the best method so far for this for small values of n I think is this Hamiltonian latis gauge theory combined with infinite tensor networks. and this is what I want to.
    about Lattice Gauge Theory · at
  • So the question is how do we how do we do that and as I said we made some progress using tensor networks. So I want to explain at a high level what the main ideas are. people here are not it's not most people are not working on.
    about Tensor Networks · at
  • That we've studied. This is still work in progress but hopefully it'll appear soon. So this is what Benjamin talked about in the Stonybrook conference zero when mass is zero when you don't have these non-invertible.
    about Stonybrook Conference · at

Who and what is mentioned in this episode?

People

  • Eigor Benjamin Soggard — Person · Referenced collaborator / audience co-author · 2 mentions · first at
    Named as a co-author on one of the speaker’s recent papers and also referenced as having already introduced key background (2D gauge theories).
  • Edward Whitten — Person · Referenced collaborator (high-profile co-author) · 2 mentions · first at
    Named as a co-author on one of the speaker’s recent papers; his presence signals the work’s connection to high-level theoretical physics.
    It's this one in collaboration with Eigor Benjamin Soggard and Edward Whitten and this other one with Ross Demsy, Anamaria Gluke and Benjamin Soggard also mentioned some earlier.
  • Benjamin Soggard — Person · Referenced collaborator / audience co-author · 1 mention · first at
    Listed as a co-author on a second paper and referenced as presenting related material at a conference mentioned in the talk.
    With Eigor Benjamin Soggard and Edward Whitten and this other one with Ross Demsy, Anamaria Gluke and Benjamin Soggard also mentioned some earlier work. I think all the authors are in the audience. so you can also ask them questions.
  • Anamaria Gluke — Person · Referenced collaborator / audience co-author · 1 mention · first at
    Named as a co-author on the second paper; part of the collaboration around numerical/tensor-network studies mentioned in the talk.
    With Eigor Benjamin Soggard and Edward Whitten and this other one with Ross Demsy, Anamaria Gluke and Benjamin Soggard also mentioned some earlier work. I think all the authors are in the audience. so you can also ask them questions.
  • Ross Demsy — Person · Referenced collaborator / audience co-author · 1 mention · first at
    Named as a co-author on the second paper discussed; included among authors likely present in the audience.
    It's this one in collaboration with Eigor Benjamin Soggard and Edward Whitten and this other one with Ross Demsy, Anamaria Gluke and Benjamin Soggard also mentioned some…

Concepts & topics

  • QCD — Concept · Core topic · 2 mentions · first at
    Quantum Chromodynamics is a fundamental theory in particle physics discussed as part of the study of non-abelian gauge theories.
  • Super Symmetry — Concept · Key argument · 18 mentions · first at
    Discussed as a surprising feature of the theory at a specific mass value, revealing non-linear realization and quantum anomaly.
  • Quantum Field Theory — Concept · Framework · 3 mentions · first at
    Underlying theoretical framework for the discussed models and concepts, including QCD and Yang-Mills theory.
  • Tensor Networks — Concept · Strategy · 7 mentions · first at
    Used as a numerical method to study the spectrum and ground state of adjoint QCD, especially for small values of n.
  • Yang-Mills theory — Concept · Framework · 2 mentions · first at
    Presented as the simplest non-abelian gauge theory in two dimensions, foundational for understanding more complex theories.
    Next more complicated one which is the topic of this talk will be adjunk qcd. So yan's theory coupled to myana fermian in the adren representation. The adren representation is real. So we can take the fermon to be a myonopermian.
  • Lattice Gauge Theory — Concept · Strategy · 2 mentions · first at
    Discussed as a method for numerical studies of gauge theories, particularly in the context of Hamiltonian lattice gauge theory.

Places & events

  • Stonybrook Conference — Event · Referenced event · 1 mention · first at
    Mentioned as the venue where earlier work related to the discussed topics was presented.

Full transcript with timestamps

Um I will talk mostly about um two papers from this uh past summer. Um it's uh this one uh in collaboration with Eigor Benjamin Soggard and Edward Whitten and uh this other one with Ross Demsy, Anamaria Gluke and Benjamin Soggard also mentioned some um earlier work. I think all the um authors are in the audience. Uh so you um can also ask them questions. Um so u Igor already uh mentioned um uh uh 2D gauge theories just to put things in perspective again and remind people. So to study nonobilian gauge theories in four dimensions which is our goal uh eventual goal it is useful to get inspiration from from more tractable models uh in two dimensions and in two dimensions we have um the simplest uh gauge theory nonabilian gauge theory would be 2D yang mills theory um so too simple has no particles the next more complicated would be the tof model which is yangmill's theory plus the fundamental dac firmian which

was solved by to largeen And then the next more complicated one uh which is the topic of this talk will be adjunk qcd. So yan's theory um coupled to myana fermian in the adren representation. The adren representation is real. So we can take the fermon to be a myonopermian. This is the simplest theory that has dynamical degrees of freedom in the adren representation of the gauge group. Um so um uh what what are these to what are we hoping to learn? What what are these 2D theories uh good for? where they're good playground for various things uh ranging from numerical algorithms um for instance in litecoin quantization lattice gauge de uh DMRG um to QFT ideas such as non-invertible symmetries anomalies uh and others and maybe even to um uh quantum simulators um either analog quantum simulators or um putting them on quantum computers So in this talk I want to do two things.

Uh one of them is the first one is to give some updates on numerical studies of adin QCD. Uh and the second one uh is to talk about more in the spirit of new QFT ideas um to talk about u the super symmetry of this theory the specific value of the fermian mass um which is um a little peculiar in that it's not manifest in the classical arrange um so I'll explain how that I'll explain how that works um so adjunk QCD uh just to review Igor already mentioned Um so as you engage the one adana fermion the lrgonian density u is has the gauge kinetic term the kinetic term for the fermian um and also the mass term um the there's only one dimensionless parameter uh which is m over gang ms and two dimensions gang ms has units of mass and so we have to study the theory for various values of this parameter uh but we don't need to do all values because uh we can use an axial transformation where s goes gamma 5 s gamma 5 I denote

by uh this is the kirality matrix um and um uh this can be this kind of transformation leaves the first two terms invariant and changes the sign of the last term so we can use this transformation to set m to be u non- negative uh and the interesting points uh are um two values of m equals z and m over gang mills is equal to<unk> of n / 2 pi uh that I'll mention um the second half of the talk will be about the super symmetry occurs for this second um second value. Um so eager already mentioned this that uh so sum has um because the myanop fermion is not charged under the center of s uh then the zn center u there's a symmetry there's a one form global zn1 form global symmetry and the stage transform under it um and there are n flux tube sectors um that are labeled by the anality of the electric field uh the anality I'll denote it IP. So to create the P flux tube sector, we can just put at infinity um P uh fundamental quarks

and P antifundal quarks to create the flux tube where the electric field has this analogy. Um the flux to vacuum in two dimensions unlike in higher dimensions in two dimensions the flux to vacuum is Lauren invariant. So we should study all these uh we should study all these um flux tubes as vacuum of the theory. In terms of flux tubes, uh one can define precise confinement um criterion which is theory confines if the area if the fundamental Wilson loop obeys an area law um and in terms of the of this flux tube flux tubes what this means is that the string tension which is defined as the energy difference in the energy density in the P equal 1 flux tube sector and the P equals 0 flux tube sector uh this is non zero. So this is string tension line zero theory confines um if the string theor if the string tension um vanishes then theory doesn't confine um and u also brief review of the general features of things that Igor

mentioned um uh that so the surprising features are at m equals z and m equals this other value that I mentioned so at m equals z um even though there are no um uh dynamic all the dynamical degrees of freedom are massless in this case. Uh there's a mass gap um and numeric show that the lowest fermian mass is this value uh in um uh in the tuft units um and the lowest bzon mass has has this other value. So this was obtained starting in the 90s uh in uh using using lyon quantization mark of eigore and and others. Um one way to see that the theory is gapped is uh to describe the deep IR of this theory. So by bzonizing the the fermians one can get this s S^ squus one level one model and then gauging um um so this is a gauge wel so you have to divide it by sum level n and we can compute the the central char the IR central charge is the central charge of the free fermas minus the central charge of this current

algebra in the denominator and One can check that this is zero. So the theories uh this theories gap. Um the other surprising feature at m equals z is that the theory doesn't confine. Uh so the Wilson loop the fundamental Wilson loop has a perimeter law uh suggesting in the 90s um can also be understood from this co-et which has 2 to the n minus one degenerate vacua that are relatable related by non-invertible symmetry. Um and this implies uh so this is work of several people in the audience. Um and uh this implies that um the energy density um the energy densities are all equal in all the flux tube sectors. So in particular it implies that the string tension vanishes.

Um so this is the m equals0 point. The surprising things about it then m equals this particular value g square of n over 2 pi. Um the surprising thing is that this theory has super symmetry and I'll I'll talk more about it later. Other values of the mass um well there's still a mass gap and there's confinement uh these are not surprising things but to study the precise uh spectrum or the string tension that requires um numeric. So this is a a very brief uh review. So what will I do in this talk? talk about some numerical progress on using uh tensor networks um for uh finding the u spectrum of at small values of n and also um in the second part some I'll give you some better understanding of the super symmetry at this particular value of the of the mass.

So that's the that's the plan. So I'll start with the uh with the first one. So numerical this theory has been studied numerically starting in the 90s uh using lyon quantization um and so initially it was a large n um um uh small n was only more recently now the main issue with lyon quantization so initially it was discretized lyon quantization but then in this paper sorry I should have just said like quantization here um there's other other truncation methods so the But the main issue with Lyon quantization is that the vacuum is uh trivial. Um and in particular um there are no flux tube sectors at least in the way that Lyon quantization is implemented right now. Um so um you know Lyon quantization we get some spectrum and then the qu the question is if the theory has multiple vacua then uh what spectrum does that spectrum correspond to? Which vacuum does it correspond to?

And uh it was confusing for a while but now I think I know the answer to this question. So if you want to know I can tell you. Um so in any case we in order to study all the vacuum I would say we need we need other methods. Um and the best method so far for this for small values of n uh I think uh is this Hamiltonian latis gauge theory combined with uh um infinite tensor networks. Um and this is what I want to um uh I want to briefly mention. Um so Hamiltonian latis gauge theory um eager already introduced it. Uh so discretize only space not time. So then we get a quantum mechanics problem with some Hamiltonian.

So for for our theory what are the ingredients of this of this uh lattice model? Well we have the fermians. So we're going to have latis myana fermians on sites. Uh so those are kai n a n is the site index. A is the color index. They anti-commute uh unless they're on the same site and of the same color in which case they they square to one. Uh and then we have uh gauge variables on links like in any gauge theory. So we have on every link we have an sum valued um operator un and then we have the canonical conjugates because it's a nonabilian gag we have like the left acting and the right acting electric fields um and then one can write down uh lattice Hamiltonian so this is just a discretization of the of the lattice of the of the continuum Hamiltonian uh the first term here is a discretization of the um electrostatic energy. Um uh this term here is a discretization of the fermy kinetic term

and this thing here is a discretization of the um mass term. Like in any respectable gauge theory u this hamiltonian should be supplemented by G law which is that the discretized derivative of the electric field is equal to the charge density which is now the charge on on each site. Um it's a very very interesting lattice model. Um um um it has uh we studied a lot. It has the one interesting thing about it is that it has uh analoges of anomalies from the continuum theory but this is something that I'm not going to um going to going to talk about.

Um so that's the uh that's a that's a lattice model. Um and now the question is how do we um so in principle all we have to do is to implement this on a computer and then diagonalize the Hamiltonian. So the question is how do we how do we do that and as I said we made some progress uh using tensor networks. So I want to explain at a high level what the main ideas are. people here are not uh it's not most people are not working on on this topic so I don't want to go into details but I just want to explain roughly what the what the what what the ideas are of what what's new in what we did um so in general state in a quantum system so if we have some degrees of freedom on some sites that are labeled by these s's uh then the g general state is described by a wave function um a Um so that's a very in general it's a very high dimensional hilbert space.

Um but um in certain cases for example if we have a gap ground states uh it's useful to make an onsat um that takes the form of a matrix product. So we have instead of having a general uh general wave function here we write the wave function as a product of matrices. So for each S we have a matrix and then we take the product of the matrices um except for the first one and the last one where we just have vectors. Um so this is the this is the uh matrix box state ons uh and then the idea is to find uh these tensors you know using variational principle finds these tensors the the tensors that minimize the expectation value of the Hamiltonian. Usually this will be a very good approximation.

Um how do we compute the expectation value of the Hamiltonian? Well, it's useful to write down the Hamiltonian also as a product of um uh uh tensor uh operators uh like this. So h1 * h2* h3 and so on um with their own um um virtual indices. So these are virtual indices and these are physical indices with their own with their own virtual indices contracted. So the size of the matrix correspondent to the the matrices corresponding to Hamiltonian is not related to the size of the matrices corresponding to the these tensors that describe the state.

Um so this is a general picture of uh how to compute an expectation value of the Hamiltonian. So the the lines here means contractions of of indices. So the vertical bonds correspond to physical indices uh these ones over here and uh horizontal bonds correspond to virtual indices. Um and the matrix product operators or any operators um so the matrix product operator is just the Hamiltonian is just this operator here. It acts only on physical indices. There's a generalization of this construction that's going to be useful which is that one so one can generalize this um to notion that's called symmetric matrix product states where both the physical and the virtual Hilbert spaces for which we have these these indices uh transform in representations of some symmetry group G um and in addition to that all the tensors here so the tensors for the states and the tensors for the operators are invariant tensors of the group G.

That's what symmetric matrix box states means. Uh this will be important uh uh in the next slide. Um and now uh tensor networks can have can be of finite size. So that would correspond to simulating a theory on an interval or they can be of infinite size uh which would be corresponding to simulating a theory that's on the on the on the real line. and they're efficient algorithms for both finite tensors and also infinite tensors. So this is the general setup. Now we want to to to uh uh to uh use this for our gauge theory. So how do we use this for a gauge theory? Uh there's two naive approaches um that we're not going to use. Um so this is these are the first things you might think about. So the first one would be well we have matter degrees of freedom and we have gauge degrees of freedom. we can keep the matter degrees of freedom and the gauge degrees of freedom together. So say some of our uh

physical sites would correspond to matter degrees of freedom. Some of our physical sites would correspond to uh gauge degrees of freedom. Um the uh gauge degrees of freedom. So the the Hilbert space uh is infinite dimensional. So that would need truncation. So that would be a downside of this approach. And then we would have to figure out what the gauging variant states are. And that gets pretty complicated for nonobilian gauge theories. So for this reason, this would not be a very good approach for nonobilian uh gauge theories. And the other approach would be to solve the Gaus law. So if we're on an interval or on the line in 1 + one dimension, then we can completely eliminate the gauge field. we can solve the Gaus law um and then rewrite the gauge kinetic terms in terms of the u charges but that would introduce non-local interactions. Maybe that's not that big of a problem. um it's not that

big of a problem if we have a tensor of finite extent but if we have a tensor of infinite extent that really would be a big problem uh because we u the only thing that we'd have to our advantage in that case which is the translation invariance would be broken uh by by such a procedure. So we're not going to do that either.

Yeah. Why is translation? Well, because we have to decide where to start solving from.

Non manif not manifest. It would it would not translation invariance would not be manifest. Yeah, that that would be maybe that's that it wouldn't be maybe broken is the wrong word. They're not manifest. Um but that that would mean I mean it really means that we can't if these algorithms that we that exist for infinite tensor networks would not be able we would not be able to use them. So we we'd like we'd like a setup where we'd have we'd have a translation of various

I think there's recent paper where they which case in the shrinker data so in in a billion gauge theories these problems can be overcome they're harder to overcome in nonabilian gauge theories yeah for a billion gauge theories that that's these are yeah not not not not that big a problem but non billing HD becomes becomes bad. So anyway the better approach that that um um we found um comes from the realization that these symmetric MPSS where the uh hilbert space is transforming some representations of a of a group which would be the gauge group actually solve the Gaus law automatically.

So the fact that the nodes are invariant tensors this uh really is the local Gaus law condition provided that on the physical hibers spaces we put only the matter degrees of freedom and not the gauge degrees of freedom and what that implies. So on the physical hibers spaces we just have the matter degrees of freedom and that and the invariant tensors um solve uh the local Gaus law. So that implies so these physical degrees of freedom here these tensors solve the local Gaus law. So that means that on the links we have information about the the states of the gauge field.

So then the question is so the virtual hilbert spaces is the last point here virtual hilbert spaces carry information about the gauge field representations. So the question is then how do we extract uh this information? Um usually when we act with operators we act only on the physical um uh physical hyperspaces. We don't act on the virtual hippers spaces. So in order to read off the expectation value of the um of the uh of this e equare term that appears in the in the um Hamiltonian electric field density um uh we need operators that act on the links in order to be able to extract the representations. So in addition to regular operators uh where that act only on the physical hyperspaces uh we have to uh introduce some link operators that act on the links uh that can figure out from which we can extract the representations um under the group G and uh well to our surprise these kinds of operators had not been had not been

considered before. Um so uh for the gauge kinetic term for instance uh we would want these operators that act on the link to be proportional to the quadratic casmir of of the of the gauge group. Um so now uh these operators the regular ones and the ones that act on the link can be combined together. Um so this only works for symmetric NPS that we can have operators that can act on the links. So they can be combined together and for symmetric MPS's the most general operators um are combinations of them. So we can so before from these ones um we could make a matrix product operator by having several of them and contracting the virtual indices in the horizontal direction. And now we can combine these with those and have a network like this where uh we go from an operator that acts on a physical um uh hbert space uh to the one that acts on the link and then a physical hyperspace and then link

and then so on. So we called these operators uh link enhanced matrix product matrix product operators or lampos. Um so um um the uh nice thing about these operators is that they can be uh incorporated into existing uh codes um such as such as um MPS kit uh code which is a code piece of software that's being developed actually at across the street at the at the flat iron institute at uh at CCQ and we are very grateful for that. Um so um we can uh we can uh we can uh modify to to incorporate these operators. So that's that's the idea.

You just have some operators uh that can act on the link on the links uh from from which we can extract uh information about the gauge field. Um so now there are efficient algorithms um uh using um um for for um uh infinite tensors as I said um so for infinite tensors uh the idea is to have a state where I have some tensor A uh that's being repeated an infinite number of times and then optimize over this tensor A. uh this is what the code does um to find minimize the energy density such that we optimize over a to find to minimize the energy density and some algorithms for this. So this is what we did. Um and uh this is to find the ground state. But you know we can find the ground state in the P equals0 flux tube sector or we can do find the ground state in any other flux tube sector. Um and uh from the taking the difference of the of energy densities um we can then find the string tension and this is how

this plot was produced that uh Eigor also also showed in in his talk. So this is the string tension uh for SU2 and SU3 adjunk QCD um normalized uh this is the string tension divided by the value that it would take in um pureang mills theory. Uh so that's why this uh when the mass of the as a function of mass of the fermian when the mass of the ferion goes to infinity this goes to one. Um so this is sort of like this this kind of approach is sort of tailor made for this kind of problem because uh when we optimize over this tensor a we really are minimizing the energy density which is exactly the quantity that we need in order to compute the string tension.

Yes. Why does it get harder when you what it gets harder because the helper space is bigger

but it's all said the gauge group is because number of representations grow dimensions

the number of representations so dimensions of your side number of your legs is related to dimensionality of representations that that contribute

um I think we'd probably need there are several things that happen like we we need we we need more repres representations and also the representations get bigger and um one of the I didn't describe the details of the of the uh lattice model um but um in order to compute the expectation values we'd need for example uh 6J symbols of the gauge group and that those become harder and harder to compute um so we have some results for SU4 as well but uh it's very hard to go beyond SU4 um um with the brute force technique like this from SU2 to SU3.

Yeah, that's kind of surprising. Uh uh so so this is the string tension norm. Well, actually it's it's a little different but not very different. Um and um so here it's in terms of the mass rescaled by g of n. So if we didn't rescale by g of n then of course the the two curves would look different. Ah um what is blue? Uh blue is uh an analytical approximation at large mass

large mass large mass. So the leading order is just one and then one can compute corrections to that.

Yeah. So it matches very well. already. Yeah, it's not the only quantity that doesn't h have strong dependence on n. Yeah, also the bound state the masses of the lowest fian and the lowest bzon also don't depend in in similar units. they don't depend much on num yeah I should as an aside I should say that um if the theory has uh multiple vacua so for example if we're in the trivial flux tube sector for larger n and we have several vacua there uh then this procedure of optimizing over a actually finds several local minima corresponding to all those vacua so one can actually study all those vacu that are related by non-invertible symmetry um separately.

Um so we've seen this in this example for issue 4 and also uh there are several other gauge theories that we that we've studied. This is still work in progress but hopefully it'll appear uh soon. So this is what Benjamin uh talked about in the Stonybrook conference

zero when mass is zero when you don't have these non-invertible symmetries on the latice.

Um well so this is uh this is extrapolated to the continuum limit. Oh that's extrapolated.

Yeah. Yeah. Yeah. So I'm not plotting it at finite lice spacing. I'm just extra.

Yeah. Good. So we can do this finite lice spacing and then we have to extrapolate to to a goes to zero.

But at finite lice spacing even with zero mass you finite lice spacing even at zero mass there would be a non-zero energy density. That's correct.

Okay. And the same would be true for different vacua in the same flux tube sector. uh but if given though that's true the this minimization procedure can find all these vacuum which I find really really uh neat um so um so this is the just the ground state to find the spectrum um one can form uh it's called the quasiparticle ansat so we use the same tensor that we uh got from finding the vacuum and then we change it at one point here and then we sum over where we changed it. So we call this B uh with some weight and then we create a state of some momentum and then we optimize over B.

Um so u uh this way we can find the low-lying spectrum. So um I want to show you some plots of that for n equal 2 and n equal 3. So this is the first one. So this is n equal 2 in the trivial flux tube sector. Um so uh this is the spectrum as a function of the mass. Um the fermians are in red. The bzons are in blue and then the continuum the two particle continuum multiparticle continuum is uh uh shaded. So you can see that the m equals zero we start with like the lowest one is a ferion and the next is a bzon. These are the values that I also wrote on his in in his talk.

Um and um and then as we change the mass these masses change um and um at the at some point which is the super symmetric point uh they cross all of them uh and then they um they they differ again um so this spectrum uh so was obtained using the method that I mentioned it's also also agrees with a spectrum that was obtained in lyon quantization But as I said in Lyon quantization one gets only one spectrum and this is the spectrum that one gets. U now this theory the SU2 theory has another flux tube sector which is the P1 flux tube sector. So one may wonder what's the spectrum in the P1 flux tube sector. Um so we did that too. Uh this is what it looks like. It starts at the same values at uh at m equals z and then but now the masses go down and the gap closes at precisely the point where we had degeneracy between bzons and firmians in uh in uh uh in the p equals 0 sector and then uh becomes massive again. So why

does this happen? Well, because at the super symmetric mass uh uh I showed you that the P equal one flux tube sector has non-zero vacuum energy density. That's the reason why the string tension it was the string tension that was non vanishing there. So this means that there's spontaneous super symmetry breaking. So that means that in the peak one sector there's a massless golden particle and this is the lowest ferion here. This one the red one this is the golden particle. Um it's close to this point. is the myron of fermon whose mass goes to zero precisely at this point.

So I can wonder okay how does this look for SU3 uh this is of SU3 we have a little less precision harder um again we start with the values uh at M equals0 and then so this is a P equals 0 flux tube sector and then the states cross at the super symmetric point and then they they differ again uh again so this P equals 0 this is also seen in in Lyon quantization you might wonder okay what does this in this case we have P equal 1 and P equal two flux sectors by charge conjugation and they have the same spectrum. Um uh so we can find the spectrum in those sectors as well and it looks like this. So again uh we can see the uh with a little less precision we see we see the gap the gap closing um precisely at this uh at this point. Yes.

um continuum. Um well, we see uh if we go higher in the spectrum, we'd see a bunch of states over here. These lines that we drew are based on adding the masses that we the low the lower masses that we found.

Yes. Is the value of MSI predicted by the continuum? The value of M Suzie is predicted by so it exactly matches.

Yeah. This is what I'll talk about in the rest of the talk. Um, okay. So that's this is these are the updates on the spectrum. Now I want to uh move on to uh talking about the uh super symmetry at this uh at this uh point uh where m is g of n over 2 pi and super symmetry was first noticed in lyon quantization by kasovv in the 90s who constructed two supercharges uh that squared the momentum generators um but as I said there's some issues with lyon quantization so we we would like to have a a better expl explanation.

Um so uh uh two relatively obvious things is that super symmetry must be realized nonlinearly in this model because we only have dynamical firmians and the other one is that it must be something about it must be non-trivial because uh the action is not as I said the action is not invariant under any classical transformation uh rules. Um so um uh for the rest of the talk just to clear what formulas I'm writing I'm going to use I'm not going to work in Lyon quantization but I'm going to use Lyon indices so plus minus indices on vectors just means v mean v 0 plus minus v1 over roo<unk>2 and for spinners I'm going to use also kyal indices um so I'm going to work in a basis where the kality matrix is diagonal sigma 3 and then the top one is going to be s minus and the bottom one is going to be s Um so you know based on Laurens transformation properties or also you know looking at the formulas from that

that Kasav had for uh the charges one can guess um some expression for the supercurren. Um so these are the components of the supercurren. The first the first index here is mu the space time index and the second one is the spinner index. Um so J minus minus would be trace I minus cubed uh which can also be written as trace I minus J minus where J minus is the minus component of the SU current that's gauged which is I minus I minus and then J plus minus would be some number alpha that we might want to determine uh times trace of C plus F. C plus is the component of the FMION and F um is the field strength. So in 1 plus one dimension the field strength is a scalar. Um so we can contract f mu and f with epsilon mu to get f or we can think of f as being just f plus minus.

Um and then we have the other two components j++ and j minus plus which are obtained from the top ones by flipping minuses to pluses and pluses to minuses. And then uh to find alpha we should require conservation. So d mu j mu equals z which means in these ly con coordinates means d + j minus plus d minus j plus for the spinner index minus is equal to zero and a similar equation for the spinner index equal to plus the second equation follows from the first one by parody. So just focus on the first one. Um so we have to compute two quantities two derivatives this derivative of j minus minus and the d minus derivative of j plus minus. uh for that the equations of motion the relevant equations of motion are the derivative derivatives of the fermians der d plus of s minus is proportional to c plus d minus of c plus is proportional to s minus and d minus of f is is j minus there's a little bit of algebra if we

plug in the derivatives um uh into uh the expressions for the currents we find that the derivative d plus of j minus minus is proportional to trace i plus i minus i minus and d minus j plus is proportion. It has two terms now. One proportional to I minus F and the other one proportional to C plus I minus I minus. So you can see that you we want them to add up to zero. They won't be able to add up to zero. We could we could choose alpha such that these terms cancel but this this term the first term is not going to cancel. Um however the subtlety is that product of fermians at coincident points uh is divergent. Uh so if we expand the product of two ferian fields the leading divergence is proportional to the unit operator coefficient 1 /x minus. So a proper definition of trace of s minus cubed would need regularization.

What kind of regularization? Well, we can just take our three points. So we have three s minuses. We can split them and insert small in order to preserve gauging variance. We should insert small Wilson lines between between the three points. And this is the expression that's that's that's written here. And now it's some mechanism like the anomaly mechanism. It is an anomaly mechanism where if we take a derivative with respect to x plus the derivative will receive some contribution from um uh when it acts on the small Wilson line. Now when the derivative acts on a small Wilson line, we're going to get a small contribution because the Wilson line is small. However, the product of the two fermians is large goes like one over epsilon. So then we're going to get a finite contribution.

Uh one way to see this is in equal time quantization if we separate the points in the spatial direction. Um we can see that even though this regularized version of trace uh well even though naively trace I minus cubed would not depend on the on the gauge field actually because of the regularization it does not commute with the electric field. So the commutator between this and the electric field which comes from the commutator between the a and the electric field would give something non zero and it gives something proportional to just a s minus and then that implies that the derivative of trace i minus cubed which we can compute by commutator with a hamiltonian is um also uh receives an extra contribution. So in the end there's one extra term in the derivative d plus derivative of tra i minus cubed which is - n / 4 pi tra i minus f.

So then I wrote down this expression here again. So the derivative of j minus minus has these two terms. This is the new term that we that we got from this regularization. You could also think of the extra term as coming from a fineman diagram. It looks like this where like there's two two ferians get that get turned into an F and it's exactly the same diagram that one would compute if one were to study the axial anomaly. Um it's actually the same. So it's the same diagram as for the axial anomalies just applied to a different operator.

U so now that we have two terms here in the other derivative of D minus J plus minus we also had two terms and we can then make them add up to zero by choosing M and alpha appropriately and uh supercurren is conserved if M is exactly this super symmetric value um and alpha is given by this expression that's that's how it works um we can uh check that the super you know supercurren and super geometric theory should belong to a super multiplet. Uh so we can check that the supercurren belongs to the same multiplet as the as the stress tensor in oneplus 1 dimension the multiplet contains another scalar operator. Um anyway you can we using all this technology we can we can check the various commutation relations. So computation relations between same kality between the supercharge and supercurren give components of the stress tensor and opposite kalities these give derivatives of this scaler Z.

But in terms of particle you do not have gauge particles right there are no g so so it's sing then

so uh yes so fionic particles would be super partnered their super partners would be bzonic particles

no but there's no bionic particles right there are we have two fmians for example

that's right so so then it's superity between two fmian So uh I mean not just uh you know kind of normally would you would say that it is fion and bzon right but there is no

there is but there are bzonic particles that are just bound states of fermians.

Uhhuh. And those are the super partners of a fermy. I see. And the reason why that that happens is that the super symmetry is realized nonlinearly.

I see. So you can write down the super symmetry transformation rule of S and S would go to U the current J

which is quadratic in S. I see. Yeah. This is very nice but it's not real. So let's say I gave you tomorrow a sexic operator and asked you does it preserve super symmetry. It would take you probably a lot of effort to compute.

It would take you some effort to compute. So it would be nice to find an offshell realization of super symmetry. So we know what are the like

it would be nice to find an optional realization. Yeah, that's right. Um yeah, I guess one one other thing I can say about this uh is that one can also think about it from a path integral approach and uh there one can write down some transformation rules and you see that the action classically is not invariant. I have an extra slide at the end if you'd like to see it. Um and uh the uh non-invariance of the action is canceled by a factor that comes from the change in the measure just like the usual anomaly. Um okay so um so this is the this is the story of uh super symmetry um in uh um our better understanding now of super symmetry in the adjent QCD theory. So for the rest of the talk uh I wanted to um uh talk about the generalization of this construction um unless there are other questions.

Okay. So uh one can generalize this to a larger class of theories that were studied by uh Fedor uh in 2022 um where we start with our adin QCD theory. So gauge field and myana fermian with some mass and to that we add um some massless fermians. So I wrote here the massless fermians as if they're complex. If they're real we can just put a 1/2 in front of the kinetic term. Um so in this case the total gauge current um is has a contribution from the adrenons which is I'll call J. So J is just I bar gamma mu SI and also contribution which I call J J tilda from the massless fermian Q. Um now this J tilda forms a U. So J forms a current algebra SU current algebra at level N and J tilda forms an SU current algebra um at some level K that can be computed from the representation um that these fermians transform in under under sum.

Um so what's the physics that we expect? Well, in general for m equals z m m m m m m m m m m m m m m m m m m m m m m not not equal to zero. So the mass of the adjint fian not being equal to zero which is where we expect some value will have super symmetry. Um uh if the mass if the adro fermion is massive then we can just integrate it out and then we just have uh the gauge field and uh these massless fermians and in the dir uh this is described by a coset cf which is so dimension of r level one divided by the thing that's gauged which is this sum level k um uh current algebra and uh the central charge in the IR can be computed u And in general it's given by you know the formulas for this in general it's non zero. So um in this class of models generically in the infrared we have a non-trivial CTF.

Um so again we can guess the supercurren um so I should say that fedor found the in lyon quantization um found the um supercharges. So we can guess the sorry we can no the other way we can get we can guess again the supercurren. So it's the same expressions as before except that J minus minus and J++ receive an additional contribution um this one proportional to S minus time J tilda minus which is the contribution from the um uh the contribution uh to the current from the massless fermians and this one has u J to the plus instead of J to the minus um and before it was this term the trace I minus cub that requires regulariz required regularization by point splitting. Uh and now it's also this term that will require regular second term that will require regularization by point splitting.

Um yeah so uh the analogous computation I'm not going to go through it uh will gives that this current J mu is conserved um if the mass is now a slightly different value uh very similar formula to the one before the only difference is that instead of having n / 2 pi now we have n + k over 2 pi but before it was The tilt was defined with uh for which way you are away or whatever is here.

Okay, I'm taking a trace trace. Yeah. So I write it in the fundamental representation say and then I multiply by sign and I take a trace

or I just sum the color components and I divide by two whatever and what was K?

K is the level of the current algebra that the that J tilda has. So J tilda is the current of the masses fermians and it forms a current algebra surren algebra at level k that's what's k is it can be determined in terms of the representation r that the fmian transform under um so uh we find that this supercurren is conserved provided that m is given by this value as I said uh and this value agrees with what uh fedor But there's a surprise. The surprise is that if we try to look at the supercurren multiplet and again it has the supercurren a scalar operator and a conserved stress tensor. And now if we were to compute the conserved stress tensor that's related by super symmetry to the supercurren. It actually isn't equal to the stress tensor that one would normally write down for this theory.

Um so this theory actually has two conserved stress tensors. actual stress tensor and the stress tensor that that's in the super symmetric multiplied. Um so the stress tensor in the super symmetric the super symmetric stress tensor um is this. I wrote down the formulas for it. Um it's uh it's a bit hard to to see right away if you haven't uh done the computation but this is exactly the formula that you write down for the full stress tensor except that the contribution from the massless fermian to this stress tensor is replaced by a quartic term in the ferian proportional to trace j tilda squared over here and over here. So the contribution from the contribution to the stress tensor from the Q's is replaced by by these quartic ferion which is usually which is quadratic is replaced by these cortic ferm terms again the cortic femian terms u trace j tilda squared again this should be defined by gauging variant

point splitting um so this is what I said here the QA contribution is replaced by the cordic terms so so then there's two there's two stress tensors there's this one and the usual stress tensor. So then there's a difference. So the the full stress tensor is a super symmetric stress tensor plus some um uh uh difference delta delta t which is also a conserved stress tensor. Um so we found that the this delta t is the contribution to the stress tensor that one would have from the massless fermians uh minus these cordic fian terms.

So now this expression depends on the gauge field in two ways. It depends on the gauge field in the in the coariant derivatives here in the first terms. And it also depends on the gauge field in a more subtle way through the regularization of the composite operator trace J tilda squared because those have to be regularized by gauging variant point splitting. And it turns out that the two effects actually go away uh cancel each other out. So if we were to compute the com the commutator between the electric field and this delta t we actually get zero. So this means that this this delta t actually does not depend on the u on the gauge field. So we can write down an expression for delta t by replacing the coariant derivative by a regular derivative and replacing this gauge invariant point point splitting by just a normal order.

So the this delta t then is uh as I said the terms that we'd have this is this is the stress sensor that one would have in just a free theory of the Q's and then we subtract uh this quartic fermian terms uh but with a normal normal order prescription. Um now what are we subtracting really is the sugara stress tensor of the of the tilda. So that means that delta t u really is the stress tensor of the s so mod su coex in general. This isn't super symmetric. Um and now delta t susi is the stress sensor that we got in the stress sensor multiplut. Uh and this is super symmetric. So if the first one is the stress sensor of the of the uh CTF in the infrared uh this is the stress sensor of uh whatever else is there uh which is a massive sector. So in general these theories have only partial uh super symmetry. The massive sector is super symmetric but the CTF is not.

Um so then the question is um can one have um uh so in general partial super symmetry but is it possible to have examples where we have um a theory of this type um that's fully super symmetric with with additional with additional massless fermians that's fully super symmetric or in other words can we cancel are there examples where we can cancel the cordic fian terms and uh the answer is Um there's two possibilities. Uh one of them is um if this u if this delta t actually vanishes or in other words if the co-et of the would be cf actually has c equals z. Um so in that case the full theory is super symmetric and an example of that is sun gauge theory with one massive adjint and one massless adjint. So now the Q's are just like one one massless adjint that produces a current algebra at level n.

So then the mass for super symmetry is gang mills times it's usually was n plus k over 2 pi and now it's n plus n over 2 pi. So this theory is fully super symmetric. And another possibility to have full super symmetry is if the CTF itself is super symmetric. Um and that happens if we have sum plus one massive adjint and then the q fermians are two massless adjints and this generates a current algebra level 2n and then this massive adjint for super symmetry would have to have mass gals * n + 2n over 2 pi and the reason why this is super symmetric is because this co-et in this case this so 2n^ 2us 2 level one mod s level 2n has 2 comma 2 super symmetry and this is studied in this paper over here.

Um so that's um sort of what I um wanted to to say about this. Uh let me conclude. So the first part of the talk explains how infinite tensor networks um and lampos allow for precise studies of the ground state and low-lying spectrum of adjunk QCD also generalization we're work generalizations we're working on that um and in the second part uh of the talk I explained another perspective on u the super symmetry of adjin qcd at m equals this value gals time square root of n / 2 pi and in this case a supercurren conservation requires a quantum anomaly. Um I should contrast that with the usual situation. The usual situation is that we have a theory which classically has some symmetry um and then because of quantum anomaly that at the quantum level that symmetry is no longer uh the current is no longer conserved. This is the opposite situation in the classical level the current is not conserved but because of

the anomaly at the quantum level current is conserved. And then there's a large class of theories with partial super symmetry that are obtained from adin QCD by adding massless fermians. U didn't discuss many things like more details about the lattice model anomalies tensor networks for other gauge theories uh vacu related by non-invertible symmetry strong coupling expansions and so on. about the strong coupling expansion. This this I should I want to emphasize that I want to point out the fact that for these um infinite tensor networks the only parameter is the lattice spacing. Um so we don't have to send the number of lice sites to infinity. We only have the lattice spacing. We only have to send the lattice spacing to zero in order to go to the continuum limit. This kind of setup is exactly the setup that's usually used in the lattice strong coupling expansion. We're on an infinite lattice um and we compute as a function

of the lattice spacing. Um so we can compare the results that we get from the infinite tensors directly with a with a ladder strong coupling expansion. Um so um in the future um I think it would be nice to think about scattering um either by setting up like some scattering states um or looking at time dependent processes uh and also uh looking at gauge theories in two plus one dimensions in the Hamiltonian lattice approach. So um thank you very much.

Questions about this episode

Who speaks in "Silviu Pufu: Precision Studies and Supersymmetry of 2D Adjoint QCD and Its Generalizations"?

Silviu Pufu (Host), Eigor Benjamin Soggard (Guest), Edward Whitten (Guest), Ross Demsy (Guest), Anamaria Gluke (Guest).

What is "Silviu Pufu: Precision Studies and Supersymmetry of 2D Adjoint QCD and Its Generalizations" about?

The discussion focuses on recent advancements in the study of two-dimensional adjoint quantum chromodynamics (QCD) and its generalizations, highlighting collaborative work that explores nonabelian gauge theories as a framework for understanding more complex four-dimensional models. Key points include updates on numerical studies and the exploration of novel quantum field theory concepts,…

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12 people, works and ideas are mentioned, including QCD, Super Symmetry, Quantum Field Theory, Tensor Networks, Eigor Benjamin Soggard, Yang-Mills theory. Each one on this page links to its own page with every episode that mentions it.

How long is "Silviu Pufu: Precision Studies and Supersymmetry of 2D Adjoint QCD and Its Generalizations"?

The episode runs 57 min; the transcript is about 9,147 words in 83 timestamped sections.

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