Transcript

Studying 3D CFTs with the Fuzzy Sphere: Chern-Simons-Matter Theories and Fermionic CFTs - Zheng Zhou

From Institute for Advanced Study · Episode page · Watch on YouTube

· 1 h 29 min · 12,007 words · 236 timestamped sections

This is the full, timestamped transcript of "Studying 3D CFTs with the Fuzzy Sphere: Chern-Simons-Matter Theories and Fermionic CFTs - Zheng Zhou", an episode of Institute for Advanced Study published January 12, 2026 (1 h 29 min, about 12,007 words). It mentions 12 people, works and ideas, including fuzzy sphere, 3D conformal field theories, CFT, U1 level two, transimons matter theories.

What this episode is about

This talk surveys progress in studying three-dimensional conformal field theories on the fuzzy sphere, focusing on (i) Chern–Simons–matter theories and (ii) fermionic CFTs, including how gauging, critical behavior, and conformal windows arise in these systems and the role of topological terms in shaping their infrared dynamics.

Key takeaways

  • Fuzzy sphere framing lets 3D conformal theories reveal IR fixed points with tiny system sizes, enabling precision checks beyond usual lattice limits.
  • Translational matter theories unlock dualities: distinct Lagrangians flow to identical IR fixed points, like a symmetry-rich matchmaking service.
  • Conformal window questions persist in 3D: flavor counts dictating conformality resist bootstrap and standard monolithic analysis.
  • Dualities hinge on symmetry matching and operator content; conjectures survive on global symmetry and counterterm consistency, not proof alone.
  • Monopole-backed quantum Hall-like settings on S^2 expose conformal behavior and phase transitions inaccessible in flat-space approaches.

Who speaks in this episode?

What is discussed, and when

  1. – 10:00: 3D conformal field theories, fuzzy sphere, SU2 QCD, U1 gauge field
  2. – 20:00: fuzzy sphere, SO5
  3. – 30:00: U1 level two
  4. – 40:00: fuzzy sphere, CFT, U1 level two
  5. – 50:00: SO5, fuzzy sphere, U1 level two
  6. – 1:00:00: fuzzy sphere, U1 level two
  7. – 1:10:00: fuzzy sphere, super easing theory, U1 level two
  8. – 1:20:00: fuzzy sphere
  9. – 1:28:32: fuzzy sphere

Notable quotes from "Studying 3D CFTs with the Fuzzy Sphere: Chern-Simons-Matter Theories and Fermionic CFTs - Zheng Zhou"

  • So good afternoon everyone. It's an honor to give the first similar talk this year at IA is and today I'm going to talk about our recent progress studying the 3D conformal…
    about 3D conformal field theories · at
  • So to make things even more interesting, we can add a transimons term to the dynamical gate fields. This gives us the transimus matter theories. These transimus matter theories…
    about transimons matter theories · at
  • On for these two dualities, one side of the duality is known. On the other hand, the U1 level two with a scalar, it is conjectured to flow to a SO to a CTF in the IR with an…
    about U1 level two · at
  • And on the fuzzy sphere we have obtained the first answer to this question. So briefly speaking the fuzzy sphere involves studying interacting electrons moving on a sphere with…
    about fuzzy sphere · at
  • And for example if you want a SU2QCD with an SPN symmetry then you take two end flavors to then to form the S spn fundamental router.
    about SU2 QCD · at
  • Yes. And the local firmas again can be they can be realized as the bzon firmian linear. So the modular frame it has two it is a two component spinner and the two components are realized as BFF and FB and we can indeed see that the.
    about Madana firmian · at
  • Flavor the firm fionic flavor in Madrana and the Z2 even flavor in EC and there is and the CFT ground state has substantial overlap with this reference state fully filled and…
    about CFT · at
  • Here is just the hopping between zero and one flavor and the kak is just the bzonic density. In this way we have realized the super easing theory and we can see that the…
    about super easing theory · at

Who and what is mentioned in this episode?

Concepts & topics

  • fuzzy sphere — Concept · Core topic · 15 mentions · first at
    Central to the discussion on studying 3D conformal field theories, providing a method to analyze quantum systems on a sphere with a magnetic monopole.
  • 3D conformal field theories — Concept · Core topic · 2 mentions · first at
    Main subject of the seminar, discussing their study and properties using the fuzzy sphere method.
  • CFT — Concept · Core topic · 2 mentions · first at
    Abbreviation for conformal field theories, central to the discussion on quantum systems and their behaviors.
  • U1 level two — Concept · Key concept · 6 mentions · first at
    Discussed in the context of transitions between topological orders and its role in describing various quantum phases.
  • transimons matter theories — Concept · Key concept · 4 mentions
    Discussed as theories describing transitions between or out of topological orders, with relevance to materials.
  • SU2 QCD — Concept · Key concept · 2 mentions · first at
    Mentioned as an example of a theory describing exotic critical phases or phase transitions in the infrared.
  • U1 gauge field — Concept · Key concept · 2 mentions · first at
    Part of the discussion on constructing interacting conformal field theories and their transitions.
    So let us look at a simple example. We take a toplet scaler the 54 theory and couple it to a dynamical U1 gauge field A and then add a transmus level two to this to this U1…
  • super easing theory — Concept · Key concept · 2 mentions · first at
    Discussed as a theory with emergent superconformal symmetry, relevant to the study of quantum systems.
  • SO5 — Concept · Key concept · 4 mentions · first at
    Mentioned in the context of deconfined criticality and symmetry breaking in quantum systems.
    Phase phases or critical phase transitions in the infrared. For example, the SU2 QCD with two flavors of fiance, it describes the SO5 deconfined criticality and the direct spin liquid.
  • Madana firmian — Concept · Key concept · 4 mentions
    Mentioned in the context of free fermionic CFTs and their realization on the fuzzy sphere.
  • QED — Concept · Key concept · 1 mention · first at
    Discussed as a theory describing the direct spin liquid in the context of quantum systems.
    Example, the SU2 QCD with two flavors of fiance, it describes the SO5 deconfined criticality and the direct spin liquid.
  • fractional quantum core systems — Concept · Key concept · 1 mention · first at
    Mentioned as experimental systems relevant to the study of transimons matter theories.
    Are based on quantum so-called quantum hall fairman or lowest lamb level at integer feelings. If we go to lowest L level at fractional feeling, we can actually realize a wealth of topological orders. If you we if we look at the experiment.

Full transcript with timestamps

Welcome everyone to the first seminar of spring semester and today we're very happy to have Sheno perimeter institute who will be telling us about how to study 3D conform theories on the passive sphere and we plan to uh take Shen to dinner tonight and we can meet here in the lobby at 6:30 or at the restaurant at 6:45 And uh if you're interested in coming, please uh send Lisa an email by 4 p.m. so we can make

Yes. Was there a question? Mediterrane. So good afternoon everyone. It's an honor to give the first similar talk this year at IA is and today I'm going to talk about our recent progress studying the 3D conformal field theories on the fuzzy sphere. So my talk today will be mainly based on these two papers of ours and I will first give an introduction to crystal gage theories and positive and then I will focus on two different aspects. The first is the transilence matter theories and the second is about interacting and free permanent CFS.

So as an introduction tapping critical scalers or critical firmians to to a gau dynamical gauge field is among one of the few known ways to construct interacting conformal field theories in three space dimensions. In matters, they also describe various exotic critical phase phases or critical phase phase transitions in the infrared. For example, the SU2 QCD with two flavors of fiance, it describes the SO5 deconfined criticality and the direct spin liquid. It is described by the QED with four flavors of direct fiance. Despite its importance, view is known about its crit critical data. As an example, there is a famous conformal window problem. That is to say, under what numbers of flavors does these languages flow to a conformal fix point? This is largely unknown in three space-time dimensions.

So to make things even more interesting, we can add a transimons term to the dynamical gate fields. This gives us the transimus matter theories. These transimus matter theories they describe transitions between or out of topological orders. Theoretically they host dualities between different interacting theories and experimentally they are also very relevant to for example the fractional quantum core systems or the materials. So let us look at a simple example. We take a toplet scaler the 54 theory and couple it to a dynamical U1 gauge field A and then add a transmus level two to this uh to to this U1 gauge field and this describes a transition this transition that is controlled by the master M square. So when M square is positive the Bzans are which leaves us with a pure U level two transance theory. This gives us a sim topological order and depending on different context you can describe the kerospin liquid or

the new half bonic ling state. On the other hand when m² is negative the condensation of the bzons hits the hits the gate field a completely which gives us a trivity depth face. So this transition appears in various contexts. For example, in spin system it can describe a transition between a kyospin liquid and a paramagnet and upon doping at this brittle point it can induce the so so-called annual superconductivity and in in code atoms um this transition can be induced by the fa resonance. Now an important feature of these transance metric theory is the duality. The duality describes that different labians they can flow to the same fixed point in the IR.

A famous example would be the partial vortex duality which says that the scalar QED with weight one flavor of complex scaler it is dual to a critical complex scaler. itself which is the ultruism fish. This duality has been has been numerically verified by Dasupta and Hucker. On the other hand, more dualities are stays at the stage of conjecture. These conjectures are made by matching the global symmetry, matching the face diagram which also includes matching the counter terms and or the response and matching matching the operative content of both sides. For example, the UI level one transignments coupled to a complex scaler is conjectured to be dual to one one free direct framing.

On for these two dualities, one side one side of the duality is known. On the other hand, the U1 level two with a scalar, it is conjectured to flow to a SO to a CTF in the IR with an SO3 symmetry. This theory is conjectured to to have three other dual languages namely U1 level minus three half coupled to a direct FMI SU2 level one coupled to a coupled to a scaler and SU2 level minus half coupled to a firmian. So for these SU2 gauge theories, the SO3 symmetry is manifest as the as the flavor symmetry and and on the other hand for the U1 for the U1 gauge descriptions, the manifest symmetry is U1 topological of the U1 gauge field and it it is subjected to be enhanced to a an SO3 symmetry in the infrared.

However, this uh this transition you want level two with a with a scaler it is prop first proposed 30 years ago by when and in the three decades people are people people are people keep being interested in it. However that what is the nature and it its property? This is an unsolved question. People cannot use quantum mono to study this transition due to the intrinsic design problem. Perform bootstrap also does not apply so well to this to the gauge theories and on the fuzzy sphere we have obtained the first answer to this question. So briefly speaking the fuzzy sphere involves studying interacting electrons moving on a sphere with a magnetic monopole at its center. It has been first applied to the 3D easing CFP where where it where people are able to obtain a wide range of spectrum at very high accuracy even on very small system size and people have obtained the first ver numerical verification of the conformal

symmetry of the 3D easing easing transition. So to better explain the idea of multisphere we split it into two key words. first sphere. So we want to study a quantum Hamiltonian that is defined on a D sphere. This is because combined with the imaginary time direction, the whole space-time manifold, the cylinder is conformally equivalent to a flat space time through a through a wire transformation where each time slide of time slice on the on the cylinder is mapped onto a concentric circle. This allows us to make use of some very nice property of the CTFs on the flat space time. A very important one is the state operator correspondence which says that each CFT operator if it has a onetoone correspondence to an aent state of the quantum Hamiltonian that describes the CFTR here. the state and the operator they have corresponding angular momentum corresponding quantum numbers and more importantly since the

Hamiltonian which sends one time size to the other is mapped to the dilotation which sends one one sphere to a larger one their aen values are corresponding so that so that the excited energy of the aen state on the sphere is proportional to the scaling dimension of the corresponding CTF local operator. That is to say to extract the select scaling dimensions of the of the CFT which is a very important piece of CTF data. We no longer need to do very complicated binary size scalings. We only need to read it off from the energy spectrum.

Now in 1 S1D it is very easy to put things on a sphere. we only need to consider a lice chain with periodic boundary condition. However, in 2 + 1d things get a little tricky. This is because there is an there is no lice that preserves the full SO3 symmetry due to the curvature of the sphere. So we take an alternative approach. We we fuzzify the sphere. That is to say we use the spherical London levels instead of the latis as as the single particle states. This is an idea that first come from the quantum and quantum hall physics more specifically coming from Duncan. Others may call it the the sphere but the Hoden himself calls it a fuzzy sphere. Thus comes the name. The word fuzzy essentially means non-commatitivity.

And here the non-commativity comes from the magnetic field. So we consider electrons moving on a sphere with a magnetic molecule at its center. The magnetic field it modifies the single particle Hamiltonian and further the single particle states. Now the single particle states they form highly degenerate spherical l. The lowest n level for example has a degeneracy of 2q + one where q is the charge of the molecule at at the center. So if we set the single particle gap to be much larger than the scale of the interaction and then partially feel the lowest standard level. We can we can constrain the fluctuate the quantum fluctuation within the lowest standard level. So we can project the system onto the lowest L level and just throw away everything that is above. This gives us a finite honian which we can study numerically. So we write down the electron creation operator as a linear combination combination of the creation

operators of the orbitals on the lowest general level. So now we have the these these single particle steps to construct a interacting honian. The mo the simplest building block is the density operator which is local firmian biline. So for example to con to construct the CFT with a which spontaneously breachs a Z2 symmetry we only need to consider two favors of firmians on the lowest L level which carries respectively uh Z2 old and Z2 even and add a add a rep add a attract density density attraction that is that that mimics the easing type interaction.

and then add a polarizing field. This transition by tuning the parameter H can realize the easing CFP and if it is in complete analogy with the transverse field easing model

and X is the density of spin the X direction. Yes. So, so we have two two flavors of firmians.

Okay. And we insert a sigma x. Can I ask a vibration because why are you what is how do you um I don't know.

Yeah, this is just a model to but but how can you get rid of I mean like you said before that you learn something about icing like but in this context of putting a magnetic monopoly inside. Yeah.

How can you get it rid of it? So that you still learn something by it. Yeah. So essentially the monopole it does it is just a setup in the UV. It is m micro microscopic setup. It does not enter the CTF in the infrared. That is one explanation. And the other one this if you consider the thermodynamic limit. So we put therm thermodynamic limit we take thermodynamic limit as Q goes to infinity and correspondingly this the side the radius of the sphere goes to infinity. This is nothing more than just a just a flat space time and acting a acting a magnetic field on it. Can you remind us what m and v are?

Sorry we have an mab. remove and mab is just a just a matrix and that acts on the flavor index. So for example for the easy model we have two flavors of firmians. So a b takes up spin up and spin down for example and the n takes the can take the poly matrices. So it is a mic. So for example for nx we insert sigma x

and for unz we insert sigma. U is u is a function of u u is an density a potential function of the density density interaction. And since to reach the ECCFT you only need to take you only need to tune one parameter. So H can be taken rather arbitrarily. For example, we what we usually do is we take the U as a linear combination of local and superlo interactions. We will see some examples. Uh this is just a head waving expression that shows shows you how the model looks like. We'll see some more concrete examples in the forum.

Can you say again what determines the number of play you know for species you have to use

and usually usually you design the model to have the correct symmetry and phase diagram. So, so for for example, for an easing transition, you need you need two flavors that in the Z2 even and the Z2 odd representation of the ENZ2 symmetry. And for for example if you want a SU2QCD with an SPN symmetry then you take two end flavors to then to form the S spn fundamental router. So it is based on the the CTF that you want to realize. Okay. So this is the simplest example using the fuzzy sphere. We have studied many CFTs that people have no idea how to study before. For example, we have studied the S SO5 deconfined criticality. We have determined that it its nature is pseudocritical and it has approximate conformal symmetry. We have further extended this model

pseudoritical essentially means that uh maybe it's clearer in the context of SU22CD. So when the when the number of flavor is large, you have two two fixed points that describes the QCD and the QCD gross number. When when when the when you decrease the number of flavor, these two fixed points become closer and closer and they collide into complex plans. They become complex CFDs. And if the top CFS they are very close to the real axis then your the physics that you observe in this area which we call cenocritical area it is still controlled by these complex CF what you make

is seven close that's a that's that's a good question I think it is compared with the system size of your the linear scale of of your your system. When your when your system becomes large enough, you will not see this you will not see this error. But but it is only observable when the distance to the real axis is comparable with your system size.

In what units? distance to the real axis in units of some of some of the dimension of some operator.

Yes, you also need to know the velocity that depends if you are sitting at this fix point operator. Yes,

it takes us to the real line. Yes, this is how you measure distances. You say distance you mean the units given by the dimensions. Yes.

How do you compare that? What do you compare that with? Is conformal is not right. So I'm sitting at a fixed point. It's conformal. Yes.

I'm deforming it with some relevant operator. Whatever that means to approach the the the real axis. I want to know when I'm sitting in the at the real axis. I have a theory which is not conformed. In what sense am I close to the sort of critical point?

Since I'm not not talking about this today, maybe we can maybe we can talk talk about this later.

Thank you. Okay. So we we also use the fere to study new CFTs that people does not know of before. And beyond the bulk theories, we can also study various conformal line defects and conformal boundaries. However, all these constructions they are based on quantum so-called quantum hall fairman or lowest lamb level at integer feelings. If we go to lowest L level at fractional feeling, we we can actually realize a wealth of topological orders. If you we if we look at the experiment of the fractional quantum P effect each plateau in the conductivity corresponds to a different topological order. So for example at feeling one over K the so-called laughing state realizes the topological orders U1 U1 level K and beyond that is just the simplest example and there are various uh billion and non billion topological orders beyond that.

So if we construct the transitions between them those are actually the transisence metric theories that we desire. So let us look at as a first example the confinement transition of U level two transments. So our setup involves two flavors of firmians and one flavor of Bzon. We let the two f flavors of firmians they carry the fundamental rep of the of an SU2 global symmetry and we set we we let the BZA to feel the magnetic field as well. We set its electric charge to be twice of twice of the firm. So that the lowest L level of the bzons, it has twice as many orbitals as the as the firm. We allow a conversion between one ba and two two firmians such that the electric charge is conserved.

So on the one side of the transition we set the firmians to be completely filled and the bzons to be completely empty. This will give us a feeling two firmionic integer quantum pole state which is a an invertible T2FT. And on the other side we move all the particles from the fionic flavor to the bonic flavor. In this process the number of orbitals double but the number of particles become half so that the Bzon filling is one quarter of the firmia feeling which is 1/2. So so on the other side we can realize the new equals 1/2 bonic bzonic locking state described by a U level two level two transments. So for for the for the Hamiltonian we only include the simplest terms which are the charge density interaction which appears as the background interaction in many fisphere models and this and this term also favors the bonic lightning state on the right right hand side of the transition.

We include a conversion a hopping between one bzon and two two firmies and we also include a relative chemical potential which we use at as the tuning parameter. So when mu is largely large and negative firmians are favored. So we can realize the firmic integer quantum whole state. when the when mu is large and positive we realize the bonic ling laugh lingling state. So the so this transition we confirm it can be described by by each of the four rand the chemical potential. Yes,

this is what's the U1 symmetry that couples to Yes, the U1 symmetry that U1 symmetry that couples to the

is it just the this is the same you want that is the charge or the fac the charge that couples to the magnetic field. uh the charge that couples to the magnetic field it is conserved and in the infrared the U1 charge is decoupled.

Okay. So so so the so actual conserve charge is NF plus 20. Yes. Yes. That's right. Is that the chemical potential you're

Yes. And it is not this one. it and the chemical potential what what it does is it adds an extra penalty to the firm or to the bots.

Oh, I see I see. So you're calling it chemical potential but it's it's not really coupled to conserve charge. It's just

no it is. Okay. Yeah. What is it coupled to? Final number. So when when you were in charge of the fian

that's not a symmetry. It's not a simple but you can add it as a term in the lranch. It's just confusing to call it chemical potential but

yeah yeah maybe I shouldn't call it. Yeah. So we confirm that this transition can be described by either of the four languages. So we do this theoretically by matching the face diagram. That is to say we can confirm that um that either of the four languages they have the correct topological order on both sides and it has the correct uh response on or the counter terms which include the transance term of the probe field that is the whole conductance and the gravitational transance which is the thermal conductance. Sorry. How do you see all these trans on the sphere?

So I think it is given given by the conductance. So for example one flavor of firmian it has a core conductance one which translates to an ADA joint.

Is dynamical or classical? passing. So what are the dynamical pH fields with temperature and silence?

Um we we we think they emerge and what we can say is that on the when when the theory is it has the it has the same phase diagram as the uh as the transition that is that we are studying. put this system on on your fast sphere.

No, I think I think my understanding is that these are these are four theories. He has a fifth theory which is what he discussed previously.

Yeah. Yeah. Yes. He's arguing that that the transition between that fifth theory is the same as transition between all these theories.

Yes. In these theories you can tune these transics BNA etc. Every one of them is another parameter integer parameter you can change.

Yes. What's the counterpart of that in theory? Um I I I I would dis disagree that you can tune this transance level. So when you change the transistor level it becomes a different field.

So we so so we so we need a different construction. But what how do you know that this construction of yours corresponds exactly to these four?

Yeah that that that's a good question. So we confirm that on the both both sides it it have the same it it has the same topological order on the one on the one side is a it's a tribute topological order the other side it is a U1 level two trans

you have to compute conductance and the center charge from wave functions or how do You don't have to parry

I I I I I think for example when mu is infinite the limit is well known to you just write a model wave function let's say the fixed point wave function then that you can just say you don't have to do any

also look at the spectrum yes that's true okay that's true and you get extra let's say whole conduct and some whole these are agrees with Yes. Okay.

I'm getting more and more confused. What you actually study is this mouse, right?

Yes. What I actually study is this one. What's the role of the other four? Well, sorry. What's

you gave evidence that the four other theories are both go to each other. No, this we we do not have access to. We actually do do not have confirmation that well we do not have evidence for their duality and this is it is believed that these four theories are dual to each other because they have the same the same face diagram that's the same oper but I'm still not clear if I didn't know about these theories all I had was the Hamiltonian that you wrote on the previous slide I say okay I have these two phases

and there's a critical point in between yes and then how would I go from these to any of these four margins

I would learn that there is a fixed point which has the information that takes me to these two phases

yes that's all I will know yes that that that's more or le more or less what what we have evidence for

so this is nothing to do with the full theories of the next There are only a few candidates that we know that describes these this transition between these two different phases. So it's a a natural belief that this transition is described by this theory that we know of that describes this transition. But so far we have no evidence for that. somebody with other theory.

Yes. Is that it's either dual of these four. Yes. Or might be something else. Yes. It might be something else. But that's the more non-trivial case.

It could still be that the duality is wrong. That's true. And the fixed point that you found corresponds to one of these theories but not the others. That's true. Um, but we can go through the par the critical parameters and so far all of this model at

I'm not disputing your claim. I'm quite convinced your claim is correct. I just want to see how much of that you can actually substantiate. What what what we can say is we have have not found another fixed point that describes this transition yet. All we have found is just this one transition. So it's natural to believe that it flows to these four ranges. Okay.

Okay. Yeah. So the first thing we can say is that the transition is continuous. So we we perform a an exact dual analization uh calculation on the fuzere with our package fuzzifi. And the first thing we have found is that all the bar all the observables they change continuously and the gap and the energy gap they closes in the fashion of proportional to 1 / r which is an ev evidence for continuous transition and what what is more well what is more interesting is about

sorry so the roton doesn't go as Yes, we are just confirming that this is a gap degra freedom that does not interfere with the CFD and the charge gap also does not close. This means that the U1 charge is decoupled at the crypto point. So only the synchapd growth freedom become gapless at the crypto point. Yeah. So the more important information is about the conformal symmetry. So it we extract the information of the conformal symmetry through the state of a operator correspondence. An important and a unique feature of the CFT spectrum is the conformal multipates. So we start from a primary operator. By acting partial derivatives on the primary we get its descendants and the descendants are at integer spacing with the primary. For example partial mu partial mu s it has scale dimension delta plus one and partial partial s has stay dimension delta plus two etc. And we we can find that all the

lowest lying states they can be classified into such perform modificates. And for the lowest primaries for example the lowest singlet and the conserved current we can we can find we can find all all of its lowest lying descendants sitting nicely at integer spacing. So, so in the first picture that's a single yes conformal family, right?

Yes. Presuming I should see this diamond S equals one the second state of S L equals Z.

Yes. So different symbols are data for different system sizes. It doesn't mean it seems like there's still some pretty large error.

Yes. uh you you can define a cost function to measure how different it is it is from the we I don't have I I I haven't shown this data here but you can actually find that the cross functions scales to zero

scales as you increase as you increase the system size you're calling five six and so on is the genericity of each of the fian

uh yes yes what we call the system size or the orital of number of this and this can for J you can you you can you can also have structures that involves the epsilon tensor

and yes also so um I remember in icing case or in some other cases that you had mentioned earlier you have to use say one conser current to fix like the velocity

yes do you use something like this this um yes So, so you see there is a uh there there is a constant factor between delta and E right

and we we determine the this constant factor by optimizing the topform symmetry. That is to say we use multiple criteria.

Okay. Both J equals to two, T= to three and integer space. I see. We make the the best at the same time.

I see. Not a single one of them. Not a single minimize all of them. Yes. Yes. We are trying to optimize the procedure when we are studying N CFTs.

Say again what T was the third panel. What is the third panel? The third panel is a uh it carries the S SO3 spin 2

that is SO3 scalar operator. Yes. Yes. Yes. So it is a it is an anosotropy that breaks SO3 to O2 and we can confirm that this operator is irrelevant highly irrelevant. they have the trunk bonus space. No, we don't know that. And since the the number of bzons has a maximal value.

Oh yeah. So for example when n mf equals 9 there are at most nine bzons. So it is a finite pal space.

Shall I continue? Yes. Can I ask an ignorant question about A couple slides ago, you said that for this fuzzy sphere Hamiltonian, when you tune this relative chemical potential,

yes, you can see that on one side you have this U1 level two topological order and on the other side you have trivial.

Yes. How do you check that you have U1 level two if you only have access to the theory on the sphere?

Um, you you can check it by calculating the entanglement spectrum. Ah there is an Lihoen conjecture that the entanglement spectrum has one to one correspondence with uh wi with the CTF pyro CTF on the edge and I I can show some results later. Yeah I have some results later that gives you how do you diagnose these TFTs. Yeah. So we can ask about its operator spectrum. The most important thing is that there is only one relevant operator besides the conserved current and we have extracted its scaling dimension through a finite size through a finite size scaling. Its scaling dimension is 1.52.

This means that this means and the second singlet is well above three. This means that it is an in genuine criticality. not a pseudocritical point, not a pseudorital region, not a multi point. And this uh this observation is also consistent with duality. So the so so the anotropy that breaks SO3 to O2 is irrelevant and things like any super conductivity etc. And the fact that delta S is less than two means this transition is unstable to address disorder. And we can also look at to look at higher spectrum. Here L is the Lauren spin and S is the flavor SO3 spin. And we can actually see that the higher spectrum it is very sparse.

So it this makes with this input it makes it a very very suitable target for the bootstrap. So each each scale dimension here can be set as a gap for the as a gap for the conformable bootstrap. So as a brief summary of this part, we have we have we have studied the transition between a a a U level two transistor theory and the depth face and this this transition can have four dual ranching descriptions and it appears in various contexts on the fuzzy sphere. We realize it as the transition between a new equals two fionic integer state and a new equals halfing state. We have we have the numerical evidence for continuous transition and the conformal symmetry. Yes.

Why do you call it confining transition? What is confined here? Uh the g field is confining. So the compounding transition essentially what I mean is it is a transition between one level two cons to nothing. I wanted it to call it a h transition of your level two but then as you have said we're not completely sure that this transition is described by

I don't see what's confined here. The sound for example the annions are I say the are confined you want to say it's like a spontaneous symmetry breaking one form symmetry is that

I'm not completely sure you want level two yes maybe the word proliferation Yeah, but maybe that that will it's definitely not confined are no strings in the system, right? Okay.

Okay. So, let's call it the H transition of U2. Okay. So this is the first study that reports the conformal symmetry and the critical data for the smetric theories. What can we do next? The first thing we want to think about is the conformal defects of these stage theories. So to realize a line defect on the physic sphere through the uh through the radioization we only need to put some put some impurities like the north pole and south pole on it. Now these gauge theory descriptions give give us an ample playground for studying the the defects. For example, we we can ask what what for example each of these descriptions has a uh has a Wilson line and the Wilson line can be regarded as some kind of some kind of uh defect and we want to we want to ask how does these wheels and lines behave and how do we construct these wheels and lines. A hint for the construction is the the defect endpoint operator. So, so

for example, for the SU2 Wilson Wilson line, the end point of the defect will carry a SU SU2 fundamentum. So, what we need to put is we need only need to couple the north pole and south pole to an external uh SU2 SU2 spin. This this behaves just like a Bzon Bzantum effect.

So, we can also ask can we go back? Yes. Yes, these boss lines, aren't they screened by the dynamic of files?

I don't think they can be screened and argument is that the end points they carry the they carry the projective rep of the SO3 normal signature. So are the files. The end point is five, right?

Yes. So it's five and there's a line coming out. Yes. And it goes very far. Can I create the two pairs of fives?

Yes. in the middle and just screamed to the line in British. It's a dynamic question.

Yeah. No, you make the stream long enough any finite mass of pi be energetically favorable. But I think he claims that this this this defect with the fi in the fundamental of SO2 protective representation of SO3. So it's

no but the same is true for the dynamical files there's the same five that sits in.

Yes. Yes. The legal operate legal local operators of I are things which are SO3 representations.

Yes. Right. But I'm not I'm just taking process that I create two files also the vacuum.

Yeah. I'm not disputing your process. I'm I think I think what he's saying is that there is this defect where there is a single file at one location to

touch to line but he wants to discuss not the file but the line that comes out of it. wants to discuss a local operator which is at one point and extends in time

right but if it extends in time I can pair create think of it in uklidian space it's the same question it's this is actually identical to the question I asked earlier about confinement there are no strings in this point and these lines are are not protected by any symmetry it's the same thing is that what's lines in QCD can be screened by ques But you can still ask yeah you can still study there but

it's not protected by one for C doesn't mean that it has to be screen. Yeah it's the same thing exactly the same thing and in particular if you try to match them across descriptions I don't see why it should be the same.

Are you saying that in because he's I think mainly interested in the critical points. Are you saying that any line operator in a 3D CTF which can end is trivial or is screened in some sense?

You that's not at the very least you would need to specify more information what you mean by the line.

I think it's argument that it is non-trivial because it's a non-trivial representation projected representation of SO3

but so are the dynamical fields. No the band of all all the safety fields we can observe is in the in the linear rep of S.

I understand that but the dynamical fields in the problem of five in five is not gauging variant but it is in a projective representation of the SO3 so it's going to screen the other line. Why would you be quoting that?

Yeah, I think he's claiming that there's some defect in the CTF whose hersace of endpoint operators transforms projectively under SO3 and that you can interpret as a defect to anomaly. And that implies at least that it's not literally the identity defect, but it it could be either conformal or it could be something like a decoupled cubit or something like this. And it's not clear if I which of those two possibilities it is, but at least it's not a completely trivial defect. I guess you're further going to claim that in fact they're they're going to be conformal. Is that right?

Yes, we we have indeed observed some signature of symmetry. Yeah. Thank you. Yeah. The other thing we can ask is about other transitions out of U1 level two. For example, um we can we can we can ask about the transition between U1 level two to U1 symmetry breaking or S so SO3 symmetry breaking. These are different or even SO5 symmetry breaking. These are a different trans different transmisome matric theories and this composes a a zoo of CFTs and on sphere the construction is very simple. We only need to replace two flavors of firmian with four flavors of fiance and then the the one side is SO5 symmetry breaking and we can add anotropy to break into two different symmetry breaking patterns.

So we can know we can also study a general framework for QVD with or without transments. These are transitions that be transitions between U1 level M and U1 level U1 level N. And we can construct on the sphere the transition between a one / n mapping state paired with some uh some some firm interquing states. So, so examples in this in this framework are the U1 level two hits transition and the direct and the direct spin liquid. So, we can further generalize it to transitions between Jane states and these are relevant to transitions in the mo materials.

So, despite that we have answered many questions about this transition. There are some questions that we do not yet answer. These are questions in the QFT level beyond the deep IR. For example, given a language, what what is its fate in the in the infrared? Given two Langjets, are they dual to each other? Dual to each other. So far the photosphere can only realize theories in the in the infrared and and it will be very very rewarding to realize the reomalizable runes and the first sphere is actually a very good platform to re regularize due because the non-committity sets a natural cut off scale which may which which which cures the UV divergence and on the fuzzy sphere all the local operators or or their coronators they are non- divergent. So what we want to do we want what we want to ask translate the lang to the f funny sphere model. So the f first step has already been made the three theories three scalar has been

realized on this theory and and and following following this we can ask how to realize free free firmian gauge fields or their interactions. So now let us focus on the second step the the firmionic theories the free firmians there are actually more reason to study firmic CTFs some firmionic CTF can have some firmionic theory can have enhanced super symmetry for example the gross nua with one mana firmian it has enhanced n=1 super symmetry where the fields uh sigma pi and epsilon they are connected by the then equals one's susie susie generator and their sting dimension are locked together and experimentally these fermented CFTs also also have uh have cleaner cleaner signature.

So but about but before all these let us try to realize the simplest instance the free madana fmy. So it is actually difficult to realize if CTF that it contains perionic local local operators on the on the fossphere which the the difficulty is that the microscope firmians they experience strong non-commatitivity and they cannot flow to CFD operators in the infrared. So our so so our solution is to use a bzon firmia mixture. So we include one flavor of firmian and one flavor of bzon. So that their local combination F dagger B and B dagger F they carry no electric charge so they do not experience strong non-competitivity and they can flow to CN this operator carries fionic statistics and if we set the firmian and bzon there's their angular momentum to differ by 1/2 and this operator also have half integer angular momentum thus uh thus satisfying the spin statistics So our model again contains an electric

density interaction, a conversion between two bzons and two firmians and a a a term that favors firm or favors bzon. We set the total feeling to be one. So when when everything is fionic it realizes a fionic integer quantum whole states and when everything is bonic it when the bzon feeling is one it realizes the so-called bonic fian state which is which is described by a s2 level two transmos and numerically we have observed an intermediate phase so the transition between the intermediate phase with the fionic inter in draconium hole or U1 level one and the intermediate phase with the SU2 level two they are both continuous so the so the energy data both closes

what's the phase in the middle that can read tsc it is supposed to be unreadable so that I can introduce in the next slide

yes it's invertible face yes so how do we determine the nature of inter the intermediate case we determine it through calculating the entanglement spectrum. So there is a lio conjecture that says that the the the broad state entanglement spectrum of the of of the TQFT has a correspondence with the spectral counting of the paral CTF on the edge. And from the from the the the special counting we can actually determine that it the tyro mode is three three mana firmates. Thus it is described by described by a SO3 level one transamus theory. It is an invertible T2FT with a power charge three halves.

You actually study like the system on a half sphere hemisphere. Yes. Would you see the same edge states?

So supposedly but we have haven't tried that. What what what we mainly do with the hemisphere is we study the boundary CFTs.

This is but easier than that gap space. I I think the common common belief belief is that calculating the entanglement spectrum is easier than calculating spectrum of hemisphere. Okay. Can question. Yes. Do you at some point make some kind of choice about like boundary conditions of the entanglement cut when you're computing the left?

I I I think people do not yet know how to do that because in principle the TQ could have many different chyro edges.

Yes. But you're finding one of them. So I wonder if at some point you made some implicit choice or something.

Yes. I I think the common belief is we do not yet know how to take the different boundary conditions. Do you have something like orital car to cut? You do or we we can do both.

You can do both. Yes. And the this figure I think is from the real space experience space.

Yes. But the orbital space cut gives you the exactly the same result. Exactly same.

Exactly the same compounding. Yeah. Counting. What is the space cut? So there are or orbitals that are uh loosely speaking localized in real space. So each each

you say m bigger than zero versus yes. Yes. And now we can look at the transitions. So the first transition is between two invertible TQFTs which has has the gravitational transance term one and three halves. So naturally we can expect that this is described by a single free Madran firmy. And indeed we can see that this the spectrum is has exactly the correspondence with the spectrum of mronapmian and all the sailing dimensions scale towards in in scales towards integers in the thermody and

so is there invertible theories right the level one is an invertible theory for classical not It's invertible in a sense that it does not contain any I'm just trying to say it's a

the degeneracy on this on the Taurus. There's only one state. Yes. Yes. This is not the same notation you use. the same location of

native fields. Yes. I I think this might be easier for high energy audience. Yes. And the local firmas again can be they can be realized as the bzon firmian linear. So the modular frame it has two it is a two component spinner and the two components are realized as BFF and FB and we can indeed see that the the correlation function it agrees it scales towards the uh the transformal correlator and what is interesting is that if you adopt if you take that kai equals these two component stuff the fatonian of the on the sphere can be rewritten in a fashion that is exactly corresponding to the QFT Hamiltonian of a single madrren firming.

This hint has us hints at us that there might be some correspondence between the sphere and the and the langian QFDs. Now we can now we can put different theories on the sphere together. For example, the mar free magana and the crit critical real scalar or they have something in common. They they both involve a with so-called reference flavor the firm fionic flavor in Madrana and the Z2 even flavor in in EC and there is and the CFT ground state has substantial overlap with this reference state fully filled and the ho and there is another uh excite it excitation flavor the bosen or the Z2 B flavor so the hopping between them are realizes the elementary fields Kai or sigma and the density of the excitation is corresponding to the mass fields parch or epsilon. So we can stack these two setups together.

We we we come up with three flavors or three flavors one one bonic and two firmionic and we set the hoppings to correspond to sigma and kai respectively and the density to be k by kai and epsilon just yes

obviously bondic it's also a two flavor fian additional two flavor firm you have to realize this basonic layer is that right

um we we put we put literal boz on the lowest. Oh, icing. Yeah, we put bzans on the lowest level.

This is unlike the icing c where it's firm. No. So, so if it is a if it is a a bonic field, we need two two firmians. If it it is a if it is a fionic field, we need one bon and one fl one fian. Yeah. So in this in a in a similar fashion we can translate the super easing like round term by term to a to a term on the to terms on the fuel. For example the coupling involves sigma kai and sigma here is just the hopping between zero and one flavor and the kak is just just the bzonic density. In this way we have realized the super easing theory and we can see that the the operator spectrum has exact correspondence with the super super easing theorem and we can for for the low lowest bing states we can find that they form the super super with half integer spacings.

The super symmetry is something that exists in the continuum limit or is there a remnant in your

No, the super symmetry is not manifest. It is emergent. Yeah. So, uh my talk today is um I I would like to acknowledge my collaborators David and Chwan at parameter. And my talk today is mainly based on these two these two papers with them. And as a final summary, we have we have studied the first transisance matter theory on the fuzzy sphere the U1 level two with the stator. We have we have found this transition continuous and conformal and studied its spectrum. In the future we want to study the def defects of the den theories the other other other various transomics matter theories and we have also we have also studied the first firmionic CTFs the free madrron firmian and the super easing theory on a setup with bzon firmian mixture and we realized the local firmament as bzon firmian binius and we have established the a correspondence between the fere model and Ranji and for the supering

theory we have observed emergent superformal symmetry and agreement with bootstrap. In the future, we want to we want to study more about how to regularize like run QFTs on the fuzzy sphere and we want to make connection with previous attempts for example the non-competitive field theories and we want want to also we want to also explore the ideas of this fuzzy super which gives us an implementation of manifest super symmetry and the key message here is that we connect fuzzy sphere with the fractional quantum p effect. Using the fractional quantum p effect, we can study study many many theories on the f fe fossphere that we do not know how to study before. On the other hand, feathersphere gives us a tool to answer many questions about fractional transitions. And last we noticed that all of the setups today uh all involve one involve the mixture between bzons and firmians. So it is really a bos.

Thank you for listen. Yes. When you regularize a a theorem in latis, you know that you get you expect to get the CFT and then plus some other irrelevant operators that have to

Yes. How it breaks the latis symmetries. Yes. How the lis breaks the continuum symmetry.

Yes. Yes. Is there here some story about what irrelevant operators might be present

due to the fisphere? And the irrelevant operators they must be singlet under global symmetry. singlet under Laurens symmetry.

Yes. But I mean you're breaking some of the Laurens symmetry. So and the Lawrence the Laurens symmetry I think is pretty is fully manifested on the physics sphere. But only the rotation symmetry was symmetry between time and space. It's not not

so on the sphere we are we we have the SO3 rotation symmetry right that's manifest.

Yes. And we are studying 3D CFTs. So the law symmetry there signature is also SO3. So

that's the rotation around the okay one point in the uklidian space. Yeah. Yeah.

But then I can you know translate and try to rotate around some other point then that that would not be my best

that that that's true. So out of the conformal group what is manifest is the dilotation and the rotation. Right. And here the translation and the special conformal transformation is not measured.

One's question is what's the lowest dimension operator? Yeah. That preserves the symmetries that you do have and violates the other symmetries.

That will tell you how fast you approach the critical value. Yeah. I I think that's that that you can only answer case by case. So let us look at some examples. For example, for the for the bonafirian you you actually only have one relevant operator and I think the next operator is a descendant of this kind. So

what's the lowest dimension operator that preserves the symmetry of your problem but violates the full performance symmetry?

That would be the defamation. I I think it will be the box fab pointer for example for the madana it's the so all the all the operators with finite learn spin are forbidden so you cannot you can you can look at only the spin zero sector

spin you mean spin under the so3 that's true okay right that makes sense yeah and the lowest operator is kakai This is our tuning parameter and the higher operator is the leading is the leading environment.

In this case, is that a descendant? Yeah, it's a descendant. Yes, I think there's no more primary in this.

So, so I guess if you want to do componation theory, you would need to add scalar primary leading rather than scale primary.

That's true. which seems that doesn't exist in this case. That's why this model works so well even for

doesn't exist doesn't exist within your computation or the system clearly has these operators. For example, if you if you take the free madrronia, the only scalar lens scalar primary you have is the kabakar and you do not have a higher lens primary I think

for example. Yes. Yes. Let let us look at another case. Yeah. Why don't you have others? You have you have you have the one he said and then you have currents with two

an infinite number of operators spin zero descendants all of them are descendants

are you talking about two fian operators or operating fourian eight fian maybe it is very high I take it back it should be very high

they exist but have very high dimension yes yes so maybe it works so well because the leading operator has such third dimension.

Yes. So, so for example for the SU2 level one hits transition what we have is so so the same dimension we observe is is a function of the system size and here different different lines we tune a another parameter which which is equivalent to tuning an irrelevant operator. We see that these lines go to the same same value at a power and this power essentially determines the the strain dimension of the environment operator which is this operator. So indeed we see that the size finals pointed to is the one that has dimension 3.7.

Yes. Yes. Can I ask a question about line effects? So in one of your motiv motivational slides you said that the main reason to consider a fuzzy sphere is because it manifos to manifest SO3 on the lattice. Now when you insert a line defect I guess you're only preserving the U1 that rotates around the north and south poles. So is there still a compelling reason to treat this with fuzzy sphere as opposed to a lattice regularization or

you and here you still have the state over the correspondence. So you can read you you can read the defect spectrum from the energy spectrum in this setup and you cannot do that by

why can't you do it? It seems like it's analogous to the situation in 2D where on the lattice you can't manifest the full U1 but you can manifest a you know preserves all the symmetry defects as much as

well there's the can be transverse you want and then the not all of it but all symmetry is not

oh but I thought do you do you preserve anything besides the transverse U1 and uh um dilotations

um on the defect conformal group we preserve only the uh you are UI rotation and the datation indeed. So of the defect of the defect on this for the sphere we can still interpret it as the radio quantization of the def of a defect configuration. So that we can still make use of the stereo operate correspondence and you cannot do this or for example defect latice you cannot interpret it as you cannot make use of such nice properties of CTFs

why not maybe that's the part if I put a lice on the sphere that preserves the U1 rotation

well that this this I not sorry not the human but you know a discrete subgroup thereof um

this energies on the sphere with the defects inserted yes I I think you can do yeah you can do that but I don't I I don't know anyone whether anyone has tried that before your examples of divergent super symmetry normally it's hard to uh think about uh super symmetry emerging at at low energies if it's not present uh microscopically uh because there's all kinds of relevant operators that uh that only super symmetry.

That's why it's so hard to put super symmetry on the lattice. There are sort of dumb examples like you can have uh you can arrange to get a massless myana genome uh together with the gauge field and accidentally become super symmetric. But anything with scalers you're doomed, right? that um so how's it working in your examples what what what uh how are you managing to get an immersion super symmetry are these are they are there I didn't see the details of the I didn't the details went by quickly do you have do you have massive scalers in these in these examples

so here we like you have said we have many uh we have many uh relevant operators so we actually have to tune through parameters

fine fine all right yeah it's just fine fine tuning is easier on the posere then on the latice. So what we have to to tune we have to tune away the sigma the the epsilon and the sigma prime we have three tuning parameters

because you're you're maintaining more symmetries. That's true. Okay fine. Sure.

I have a related question. The symmetry that is built in is the rotation symmetry of of the sphere.

This one interacts in the quite complicated way with super symmetry. Yeah. Yes.

So I I don't even see the symmetry group that you're going to tune to. So it's slightly different. It's in the same neighborhood with this rebas.

Yeah. I think he doesn't care though because just a total number of relevant operators is smaller fitting I think than than it would be for a garden.

How much I think the way it's traditionally been thought of you got translation symmetry. Now you want to tune also the super

right. He doesn't have translation. He has translation on the sphere. Let alone that it's also not not commuting. That's not part of the super symmetry algebra. And it's very hard to get super symmetry on the sphere.

So that's why I'm I'm questioning what which symmetry is it that you you have and which symmetry do you feel?

So what I have is only the SO3 sphere rotation symmetry. Right. 70. So anything that is not in this column is forbidden.

Any any primary conformal primary in this column is is allowed. So I I don't have parity either. So what is allowed is this sigma this epsilon and this epsilon this sigma prime. That's three tuning parameters and we indeed tune three parameters on the physics field to reach this value

as opposed to how many if you're just doing something dumb I mean uh just what's the total number of relevant operators if you're just a naive continuum person how many more just to have an idea I mean there are some scalar mass

is meaningful if you say which symmetries you impose and the symmetries he imposes

I'm I'm comparing to what you do on the lattice on conventional lab I just want to have an idea

on the lice you relatively easily get the translation symmetry he has the rotation symmetry the rotation symmetry interacts in in a very complicated way with super symmetry so in order to get invariance yes

say that's a subgroup and I have the subgroup and I'm only interested in those operators that preserve the subgroup but for super symmetry it's it's different now it might be that you're lucky and once you got the loren variance by tuming the appropriate numbers

then you get also the super symmetry yes but that's far from obvious but the point is that the SO3 that you have

yes is not a subgroup of what you're trying to get it is not well if you just put imagine you have a continued sphere

yes so imagine you do the limit and you have a continuous sphere with exact SO3

yes it's not easy to put super symmetry on the on the sphere I I'm not in I'm not implementing super symmetry on the on the microscopic level I let the super symmetry emerge so

once you have the super conformal symmetries the the the rotation has a subgroup of the super super

can we go back to uh just simplest example just the simplest super symmetric model Yeah,

this one right so like super naively if you're going to try to get the series to merge on the lattice you've got to tune of course you got to tune the scalar mass is zero and you have to tune for the relation between the are there relations between couplings here or no

and yes you have you have to have a relation between exactly yeah yes you have to tune the relation between coupling anyway I don't know how many but there's maybe four or five things you have to tune to zero is the difference are you saying that on the fuzzy sphere you only have which one of these uh happen automatically on the fuzzy sphere versus on the lattice. That's I think my question.

I mean if we compare doing it on the lattice it's also it's like I don't know I can't count but maybe five tunings. You're saying there's three tunings. The tunings are ascend masses to zero or or equal and uh equate couplings that are related by super I I think you have to have a relation between this lambda and g.

Yes. Exactly. I I I think this is um this happens automatically. This is not relevant.

Okay. Good. So so so somehow that relation between the couplings emerges automatically from the fuzzy sphere from from the underlying model.

Why? So you have mass of sigma and mass of kai. You have both of them you have to tune.

Yes. And there's a relation between So what what I have to tune is mass of sigma mass of k and a polarizing field which is also allowed. This these three are my tuning properties. This above I let it happen naturally.

So lambda and g which are related in the that's done automatically. I think that's done automatically.

Wow. How how can you explain how can you explain why why it happens that those things are related? I think it's because they are irrelevant of

yeah they are irrelevant there. So the the scaling that the operator relevant. Yeah these are irrelevant. Oh

okay fine. Yeah. Nice. Sorry. Sorry. We're I'm sorry. We're in you. I think you can tell this from for example postruct results.

So in free field theory they're not. That's true. And the symmetry that you this starting point assums that you have one cores.

Yes. Yes. And then you say you have a cheap concentry. Now you ask how many different parameters do I need to tune

to get through the symmetry. That's true. But you don't have you have to tune something in order to get the point cor you mean the super. So what I'm what what I'm saying is that you have to tune the all all these terms on on the second line to be zero and you have to relate lambda to g this this is all that you need to do agree on that in the ways. Yes,

is also emergent. It's emergent because you tune something. You have to make sure that the way to organize it. There are some symmetries exist in the problem symmetry.

Yes. And you look at the at the target and you ask whatever I can use the symmetry that I already have. How many additional ones do I need to

tune in order to get super one correct? Yes. How many of them are relevant? What is the operator that violates the law? Let let me say a different thing. If you put the continu which is super symmetric and you all the parameters are super symmetric in the contin is the spectrum super symmetric the answer is no. The exact continum theory is super symmetric. You put it on the sphere, it's not super symmetric.

Okay? Because the sphere does not preserve the super symmetry algebra. That's true. So you need then you need the super part to emerge. Sorry, I thought you could put it on the sphere in a way that does preserve some modified version of the

that's you need to work in order to do that. But if you just take this branch and it is perfect and take this to the parameters to be super symmetric put it in the on the sphere

it will get only through the metric from the same first term and if G the result does not preserve any super charge

but this modified CI algebra that that does contain the SO3 as a subgroup right

there is a way to put this here but it's not with this it's for custom correction terms.

Sure. I just mean correction terms need to be tuned so that the model is super symmetric.

Right. I think you but originally you were asking about his fuzzy sphere model. Is it correct to just say that he's tuning until the SO3 that he has in the problem becomes enhanced to the modified Suzie algebra on the sphere?

No. But I the way I understood what he did, he had SO3 on the sphere. Yeah. And he t the parameters that you get the full ponis. But he's on the sphere. So there's no punk, right?

Whatever whenever you have the full SO4 should be full and and also super symmetry for the I thought super symmetry should come for free. That that's a surprise here. Take this model tune it to be conformal and you get for free that it is super conformal. This is fascinating.

The question is why did it happen? Normally if you do if you take this this is not true and it's particularly other way because when you put a and standard conqueror varian theory like this one tune it to be super symmetric you put it on the sphere you do not preserve super so in the same model you could also tune to other points where you realize icing

that's true that's true right you tune three parameters so you get to this super icing

with some additional fine tune not just tune it so that it has component symmetry

right but what is it that's what is it that you could be tuned parameters I guess

I guess it's I guess it's conceivable that uh I mean certainly it happens in examples I know in four dimensions that when you just ask for the beta functions to vanish uh uh just happens to force the relation between the couplings that makes it accidentally super symmetric that definitely happens Maybe it's like that.

The question is, is this a property of this field theory regardless of how we or putting it on the fuzzy sphere in general,

right? Well, examples. Yes. Right. Is it the fuzzy sphere is something smartly does?

I think M was saying that the you need different parameters, right? How many fixed points does this theory have? Is it is there a line of fixed points? So if you are asking about the fit points yeah it can flow it can flow to easing free mad free madrren and super easy

but there's no exactly marginal there it's just a some discrete fix points is that what you're saying

there's no exactly marginal there's no exactly marginal but I guess he he's then turning something

that that makes a relevant operator at some point. Mhm. Yeah. So there's some threedimensional frame of space and there's

co dimension probably there's some large code dimension icing and some manifold and there's one isolated part of you just need to get to that.

Yes. Yeah. Yes. Yeah. Yeah, I guess I guess the point is since if if if they're isolated, you know, one of them is super symmetric. So some uh you're going to get there some basic but the super symmetry is kind of accidental. I mean

yes it it is accidental and our knowledge about the spectrum of super easy tells us that there are only three relevant operators. For example, what you see sigma sigma to the power fourth it is slightly irrelevant. It is string dimension 3.7* something like that. So it is irrelevant. You do not need to tune that. Thank you.

That's ranking.

Questions about this episode

Who speaks in "Studying 3D CFTs with the Fuzzy Sphere: Chern-Simons-Matter Theories and Fermionic CFTs - Zheng Zhou"?

Zheng Zhou (Host), Sheno (Guest), Lisa (Guest), Mediterrane, D. (Speaker on collaboration) David and Chwan (Guest).

What is "Studying 3D CFTs with the Fuzzy Sphere: Chern-Simons-Matter Theories and Fermionic CFTs - Zheng Zhou" about?

This talk surveys progress in studying three-dimensional conformal field theories on the fuzzy sphere, focusing on (i) Chern–Simons–matter theories and (ii) fermionic CFTs, including how gauging, critical behavior, and conformal windows arise in these systems and the role of topological terms in shaping their infrared dynamics.

What is mentioned in this Institute for Advanced Study episode?

12 people, works and ideas are mentioned, including fuzzy sphere, 3D conformal field theories, CFT, U1 level two, transimons matter theories, SU2 QCD. Each one on this page links to its own page with every episode that mentions it.

How long is "Studying 3D CFTs with the Fuzzy Sphere: Chern-Simons-Matter Theories and Fermionic CFTs - Zheng Zhou"?

The episode runs 1 h 29 min; the transcript is about 12,007 words in 236 timestamped sections.

More from Institute for Advanced Study

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 October 29, 2026.