We are delighted to have Jeff Harvey from. University of Chicago to tell us about Anion C, Bleep.
The level of duality and holomorphic CFT. Take it away, Jeff.
Hey, thank you very much. Um, thanks for the invitation to talk, and my apologies for not being there in person.
But even if I tried to be there in person, given the weather, I probably wouldn't have made it, so, uh…
This is the best I can do.
Um, I'm going to be talking about, yes, any uncondensation.
apparently doesn't like that word, although at times he's told me he does like the word, and I think he's actually used it in a paper of his, so anyway.
I'm talking about connections between the three things in the title, and it's based on joint…
What?
Guilty as charged.
Okay, and it's based on joint work with Clay Cordova and his ex-student, Diego Garcia Sepulveda,
Um, hopefully to appear soon, uh, now that the quarter is almost over.
So, um…
Maybe a quick…
Now, why? Okay, quick word about motivation.
From a scientific point of view, there are a couple of things that we, um, hope to accomplish.
One is to put the general idea of level rank duality, which goes back to the, I think, late 1980s,
within a broader context,
A lot of it's already been done, but we're broadening it even more.
Um, and the second goal is to find concrete examples of non-invertible inion condensation, and I'll explain.
roughly what that is.
Um, it's something that's been studied
at a very abstract mathematical level,
And there are a handful of examples in the literature that are worked out, but really a very small number.
It seems like an interesting tool that hasn't been sufficiently explored.
From a personal point of view, um…
As I'm sure everyone's aware, there are all sorts of new ideas about symmetry, generalized symmetry, topological order,
categorical points of view that are now being used and are finding applications and…
formal theory, particle theory, string theory, condensed matter, quantum computation, and mathematics.
But being a poor physicist, I have a very hard time understanding things, uh, unless I can actually do computations.
So, one of my goals here was to learn some of this material,
Um, in a context where I could actually learn how to do computations,
And luckily, I had two collaborators who taught me a lot of what they know.
So, the main idea and results…
Our first of all, um…
There is a classification of C equals 24 holomorphic conformal field theories.
with currents, that is, with dimension 1 operators,
Um, that's now been proved, and…
That classification can be used to uncover generalizations of
level rank, or rank, level duality.
The dualities to establish them at the level of topological quantum field theories, that is, in three dimensions rather than
relations in two-dimensional conformal field theory,
require… well, let's…
Some people would call any uncondensation,
And this is often invertible, meaning you're just really gauging a finite group.
But it's sometimes non-invertible, in which case one is doing something more general and less well understood.
These lead to new dualities, uh, between three-dimensional topological quantum field theories,
This includes two new infinite families of dualities, which we…
figured out, starting with particular examples and then generalizing.
And, uh, a few of the examples make it clear that it doesn't look at all like level rank duality.
These are dualities between topological quantum field theories, so…
Which involves either no anion condensation at all, because it's trivial, like E7 at level 2,
Uh, is equivalent as a topological quantum filter to spin 11 at minus 1 times F4 at minus 1.
up to a gravitational turn. Simons term.
Um…
Those of you in the know, know that these gravitational Trin Simons terms
are there to soak up the difference in central charges?
Um, and play a role in the full equivalence of
the topological quantum field theory.
But, uh, at the level of…
fusion algebra, the topological lines, etc. They don't really play any role.
And so I will probably often just drop them or ignore them.
Then there are examples where you gauge a non-invertible symmetry. So, for example, SU18 at level 1, if you mod up by Z3,
You get a topological quantum field theory, which is equivalent to E7 at level minus 1.
And they're not invertible examples, so E7 at level 3, mod…
An Algebra A, which is a…
goes by various names that I'll mention later.
is equivalent to a Chern Simons theory with group SU6 with k equals minus 1.
So here's an overview of what I'm going to say. So, I will start with a very brief reminder of the classical level rank duality.
Uh, with the prototypic example being SUN Level K and SUK level N.
then I'll say a little bit about this, uh, classification.
of Shellican theories at C equals 24.
I'll explain why holomorphic CFTs of the form
that the Shellican series give you.
lead to topological interfaces.
And how those topological interfaces are associated
To top a lot equivalences of topological quantum field theories,
of the form G, level K mod AG is equivalent to H…
level minus k tilde mod AH.
Where AG and AH are…
called condensable algebras, or ETAL algebras, or I think…
connected Frobeneous algebras.
But whenever they are, they…
are a non-trivial generalization of the idea of gauging a finite group.
So, I will give examples of each type of condensable algebra.
And just a comment, I guess mainly for Greg,
That, compared to the abstract and, uh…
I don't know, edible, it's a joke. Um, quiche and sandwich formalism of, uh, that he developed with Freed and Telliban,
Our emphasis on, kind of, concrete examples and computations, rather than on…
the abstract and powerful mathematical formalism.
Um, because as I said, I need to be able to compute to understand things.
And I would encourage you to interrupt with questions, um, if you have any.
Because otherwise, I have no idea whether I'm going at the right pace.
Or explaining things at the right level, or whatever, so…
Um, do feel free to interrupt me with questions.
So, the, um, classical level rank duality
is an equivalence between SUN at level K and SUK at level N.
And what does the, kind of,
SIM equal to mean in this equation.
Well, the answer to that has evolved over time.
In the original work, it was purely a statement at the level of two-dimensional conformal field theory.
And involved a relationship between the modular data and the braiding of conformal blocks for certain integral representations.
of SUN at level K and SUK at level N.
that were related by transposing the Young diagrams that labeled the representations.
So, I've drawn two young diagrams, lambda and lambda tilde.
They're related by transposition,
And of course, for integral representations, there are certain conditions on…
length or width of these young diagrams, but assuming that these are both legitimate and
These would be… correspond to particular integral representations of these two groups.
And one of the equivalences that came out of the earlier work
was that the modular S matrix, which tells you how
The characters of, uh, say, the representation associated to lambda mixed with the vacuum character, divided by S00,
for SUN level K, it was equal to the same ratio for SUK level N,
for the representation labeled by the transposed young diagram.
And the level and the, um…
Well, this is at the level of CFT. Eventually, when we talk about topological quantum field theory, this is an equivalence
between quantum dimensions, which can also be thought of as the value of a unnot,
And, uh, turn siren's theory.
And I've listed some, but probably not all of the names of people that developed this at the level
of CFT, and also, to some degree, at the level of churn silence theory.
So, Jeff, I do have a question. So… modular data only gives us a central charge 124.
So, the central charges are actually equal, right? Or not.
The central charges here are not equal.
Uh… no, SQN-level KNS.
They're not… okay, see. These are not isomorphic as conformal field theories. There's no.
No. No, they're not. I mean, I'll get to that in just a minute.
No sense. Okay.
I mean, there's a… well, there's a conformal embedding, which is part of the explanation of this classical equivalence.
But SUN level K and SUK level N have different central charges.
But this… I guess the difference in central charges would show up in the T matrix, but I think at the level of the modular S matrix, they…
This equality is true.
Wait, I was assuming you were… The modular tensor categories were the same. It's not… Even that's not true.
I don't think that's true.
Oh.
You have to… you have to modify things in a small, small way. Ask me again in a slide or two.
Okay, so…
Okay.
There are also versions of these level-right dualities involving SO and simplectic groups.
And they can all be understood.
these classic ones.
Via conformal embeddings,
via the connection between 2D WZW models and…
three-dimensional Chern Simons theories,
And by the idea of quotienting by a common center.
So, for example, there's a conformal embedding, where you take SUN level K times SUK level N,
and embed it in SUNK Level 1.
So, that is a conformal embedding, meaning the central charges on both sides are equal,
And the stress-energy tensor on the right is the sum of the stress-energy tensor
of the two components on the left.
Or another way of saying this is that you can make a GKO coset, where you take SUNK level 1,
And take the GKO coset by SUK level n to obtain SUN level K.
Now, if you view these, uh, WZW models as living at the boundary of a 3D bulk,
And the bulk has turned Simon's theories,
Then this leads you to a duality between
Topological quantum field theories,
Where, when you…
Well, you have to use the interpretation of the GKO coset in terms of Turn Simons terms, the term you're quoting by.
or doing the COSET by becomes a term Simon's theory at negative level.
Um, and you have to quotient by a common ZK.
Which is common to the center of SUNK and SUK.
And that quotient by ZK
Uh, is necessary to ensure that the boundary CFT has a unique vacuum.
And this quotienting by a common center also works in the other classical
level rank dualities.
And I believe, um, this observation that you should quote by the Common Center goes back to
Greg's paper with Nadi on taming the conformal zoo.
Now, there are also what are known as maverick cosets in the literature. They're mentioned in a…
One of the last sections in the big yellow book by DeFrancesco et al.
And in these Maverick cosets,
They're kind of a random collection of cosets, and this procedure fails.
There's no common center that you can quotient by.
But you still end up with an equivalence of modular tensor categories.
And a paper by Clay and Diego showed that these cosets
also lead to topological quantum field theory dualities.
But the most general case requires not a quotient by the center, which you can view as a particular kind of abelian inion condensation.
But a more general gauging procedure known as non-abelian or non-invertible anion condensation.
So, what, uh, the three of us have done is, in a way, to extend their results to other
Topological quantum field theory dualities.
But ones that arise from these theories of Shellicans.
So, I'm going to discuss very briefly the Shellicans theories,
And then explain why they allow you to find new kinds of
generalizations of the idea of level-right dualities.
So, at C equals 24, there is a famous conformable field theory, holomorphic conformal field theory, or vertex operator algebra.
whose automorphism group is the monster, and which has no dimension 1 operators.
As far as I know, it is still not known whether that theory is unique.
That is, there could be other C equals 24 theories with no…
Dimension 1 operators. It's an
It's… it's not… it's just an open question in the math literature.
But, if you assume that there are Dimension 1 operators, then long ago, Shellicans conjectured that there were 70 more holomorphic C equals 24
good formal field theories.
So, these are theories whose partition function…
Uh, if it's just a left-moving part, it would just be a purely holomorphic function.
So, equal to the J function up to a constant.
Or if you combine it with its anti-holomorphic part, the partition function would be the absolute square of a modular
function. And by a very clever analysis, he gave you a list of the F and Lee algebras that could appear.
And there are all sorts of weird things, some examples that are relevant in the later analysis involve things like E6 at level 3,
times G2 at level 1 cubed, or…
A8 at level 2 times F4 at level 2. Anyway, there's a whole list of 70 of these.
And those theories have now been classified, and it's been proved that his classification was correct.
And, um, you should think of…
The classification is involving something that have been called generalized deep poles in the leech lattice.
And the leech lattice conformable field theory, or lattice VOA, is sort of the progenitor.
And the Shellicum's theories are its avatars, and apparently
Plus the monster. So, it seems like…
The Leach Lattice VOA has 71 avatars that…
show different faces involving either the monster or various kinds of WZW
models.
So, I'm not going to explain all of that.
Yeah.
I have a question about a previous slide. So, um, so this is a list of WZW models, right?
Yes, uh, each…
So, um, so the group here is Simply Connected Cover.
Right? Because, you know, the SO2 and the S… S03, sorry, SU2, and the SO3WZW model are not the same.
Yes, I'm aware of that, and I'm not specifying that data right now, but when we get around to actually using these theories, it's going to be absolutely crucial that we do that.
And in general, um, one has to take
Okay.
Quotients by various combinations of elements of the center.
Ah, I see, otherwise you don't get a good… Good homework.
Yeah, I mean, for example, if you take any of the root lattices of these theories, you will not get a self-dual
I don't know, take the example. A171E71.
I… Right.
Okay, the root lattice of E71 times A171 is a 24-dimensional even lattice.
But it's not self-dual.
So you have to… you can either think of it as adding some weights to that lattice,
I see.
or as a topological quantum field theory, you're taking a Z2 quotient to kill certain Wilson lines.
whichever way you…
I see, so… Yeah, the WDW model, I take the Simply Connected cover, I divide by some suitable subgroup of the center.
Exactly, exactly. And in the examples I treat, I will be more careful about that. Here, I'm just listing
The Lee algebras without worrying about the group.
Certainly.
Okay, so just to give you a very…
vague idea. A deep hole in the leech lattice is a point at maximal distance from all lattice points.
Now, that's impossible to visualize because, at least I can't visualize 24-dimensional lattices.
I can visualize a two-dimensional lattice, like the hexagonal lattice,
And I've… the little red dot is a deep hole.
But unfortunately,
all deep holes here are equivalent modulo, the symmetries, automorphisms of the hexagonal lattice.
I can get from one deep pole to another by either rotating or shifting by lattice vectors.
So what you can't visualize, but is true…
is that in the leech lattice, there are 23 deep poles.
Which are not equivalent up to an automorphism of the leech lattice.
And there's a marvelous fact…
that if you…
take one of these deep holes, and you translate by that deep hole, alright, you get to a point that's not a point of the leech lattice, because it's maximal distance from all lattice points.
But if you keep shifting by that,
deep hole vector, you will eventually get to a lattice point after H…
shifts. Where H is the coxator number of one of the…
23 Niemeyer lattices.
And if you are befooled by that Z sub H,
Given by that translation.
You will construct from the leech lattice
all of the Niemeyer lattices.
So, to do the orbital fold, you would project onto states that were invariant under that
Or, you know, that…
At eigenvalue 1 under that translation, and then you would add the shift… a shifted lattice,
And project onto invariant states, and that reproduces a lattice VOA, but one which has the Niemeyer lattice.
Not the leech lattice.
Now, there are other automorphisms, of course. There's a huge symmetry group of the leech lattice involving the Conway group, which involves not translations, but
Rotations, and there are more general automorphisms, which are called generalized deep poles,
They have to obey some non-trivial conditions in order that the orbifold
is… or the group you're Orible folding by,
Uh, is not anomalous and produces a new holomorphic CFT.
But due to the work of a number of mathematicians, some of whom are listed
Here, uh, it now been proven that all the Shellicans theories can be obtained
by Orbifolds, starting with the leech lattice VOA.
So, Shellikin's classification was complete, and they all come from the leech lattice.
In this generalized sense.
So, to combine these two elements,
We use a connection between holomorphic CFT and topological interfaces in…
TQFT.
And invoke a theorem, um, by these four authors.
Which I will have to confess, I…
don't fully understand, but I…
Uh, am willing to believe a certain interpretation of it, which I will tell you.
And then we'll go from there.
So, if you have a conformal embedding with an equality of stress tensors on both sides, you can build a topological interface
between two, uh…
True and Simon's topological quantum field theories.
The blue line here is the topological interface,
And on the left and right here, I've put
two things that were involved in that previous conformal embedding.
of SUN level K, SUK level N, and SUNK level 1 on the right-hand side.
If you tensor both sides by a U1 level NKTQFT,
And you gauge on the left,
by a common center, ZK.
And on the right, you gauge by a common Z sub K times N.
You will produce a new picture,
Where, on the left, you have SUN level K times UK level N, and on the right, you have UNK level 1.
But UNK Level 1 is really defined as this quotient by a ZNK,
And that quotient kills all the non-trivial
lines. So, on the right-hand side, you have a theory
Which has trivial anion structure. It just has a single anion, it just has the vacuum.
And so, if you now take this picture, and you…
fold to get a new topological interface.
You get an interface between SUN level K and UK minus N.
Up to a churn Simons term.
Sorry, I didn't understand how you got the trivial any on.
So,
Evion structure. By the way. the small K and the large K are the same.
Oh, this… yeah, sorry about that. There's my inability to type correctly.
Little K should be big K.
Oh, that's fine. So… so… But you said something I didn't understand about the UNK Level 1 having the trivial anyon structure.
Right, so SUNK Level 1 has NK anions.
Labeled by the NCality.
Right. Right.
We tense are on left and right by a U1
at level NK.
And then on the right, we quotient by Z sub Kn.
Got it. I see, okay.
Act ZKN kills everything except the thing that has trivial Knality.
And produces just the vacuum edion.
Correct. How is that the U1… UNK Level 1 theory?
Well, I think that's the definition of the UNK Level 1 theory, or at least it's my definition of the UNK Level 1 Theory.
Bill? That's not the… I mean, for every cutback group, there's a WZW model. I could take the UN.
I could take my WWWZW model. Uh, with the group UNK.
Well, alright, I think we're… I mean…
Well, alright, let me just…
And…
The thing on the right is what I've told you. S-U-N-K level 1 times U1…
NK modZKN.
So if I'm… if you're willing to just accept that, and then maybe I've used a notation which is not…
Okay, I'm happy.
Totally standard.
But…
Right, I don't think I would call that… the UNK Level 1 model, but fine, okay.
Okay, anyway, we agree on what it is.
It has trivial NEON data.
And so then by…
Folding, you get a topological interface in this form.
But the idea here that we want to generalize
is that if you have a topological interface between a pair of WZ, or a pair of Chern Simons theories,
to a theory that has trivial anion structure, you can use that to get a duality, which is a generalization of level-rank duality,
between two churn silence theories.
Okay, sorry, this is important. So, the right-hand side, is that a, um, terayo Vero theory?
Is that a what?
I believe that's correct.
Get to Raya Vero theory?
Ah, so it's Merita equivalent to the trivial theory.
On the right, up the first… yeah, yeah.
Yeah, so basically this thing you're calling UNK Level 1.
Yeah, yeah.
Yeah.
That's a trivial MTC. Got it, okay, that's the idea, okay.
Okay, so we have a very similar setup in the Shulkin series. You start with the WZW
model of the foreign GK times HK tilde.
Now, okay, sometimes there are actually 3 or maybe even 4 components.
So, HK tilde here really might stand for a product of several components.
But I'll just look at examples where there are two
groups, G and H.
Um, and so this is okay.
This can be generalized, but just to keep the notation simple, I wrote it like this.
And what Shellicans does from this…
At the CFT level,
is he figures out how to combine representations
So that you produce a holomorphic CFT that is
A theory which, as a topological quantum field theory, has trivial NEON data.
Because it corresponds to a purely…
Holomorphic CFT. The chiral algebra is extended to the point where there's just a single character.
So, there's a theorem in this paper by DMNO
That says, if you have two MTCs,
C and D, and they're WIT equivalent, then there exists these etal algebras.
And an equivalence
between abraded equivalents, between C and D when you
Well, I would say when you condense,
A1 and A2.
So, as I said, this paper's a little too heavy on the math for me to fully understand it, but our interpretation of it, which
I guess I believe, because it makes sense and allows one to compute, and my collaborators tell me it makes sense.
is that this means there's a topological boundary between GK and HK tilde, just like I illustrated
For S-U-N-K and UK-level minus Z.
And condensable algebras such that there's a generalization of rank-level duality,
with this equivalence between G-level K and H at level
Uh, that should be minus k tilde there, sorry.
Once you condense these algebras A, G, and AH.
So, the point of the rest of the talk is to start with the pair G and H,
Given to us by Shellicans,
And try to figure out exactly what AG and H
AHR, so that this… so that this proposition holds.
So that you get an equivalence
of modular tensor categories and equivalents of topological quantum field theories.
So, I see. So, um, DMO guarantee the existence?
That's my understanding.
You have to know, you have to know… the WIT equivalent.
Right.
Then there exists these condensable algebras.
By the way, what's the superscript on the C and the D, the zero subscription?
I'm not sure, I'm not sure.
Super script. Um, okay, but you have to show the WIT equivalent, so the WIT equivalence means that you can tensor with Drenfeld centers on either side and get an equivalence.
of MTCs. As I recall.
So, don't you have… So they say these two things are equivalent.
So, I'm not quite… seeing something here.
Oh, I see, so the cheek up, so because it's a holomorphic CFT.
It had…
So I think… I think the argument that I… that I gave for S&K-level internet and UK-level minus N
Um…
Well, okay, I believe it's equivalent to this statement about wood equivalents. That is that there's a topological boundary between GK and HK tilde because
It's possible to extend
this to a holomorphic CFD.
physical interpretation of WIT equivalence is the existence between two MTTCs.
is the equivalence of a topological boundary between the corresponding three-dimensional TQFTs.
Right.
That's what… that's what equivalence means to me. There's also a math definition, but that's what it means to me.
Yeah. Well, so that's…
So I'm trying to understand the logic here, so… The logic is… That because you have.
a holomorphic C equals 24 theory. Therefore… there should be a topological boundary between.
Well, there should be a topological boundary.
E, H, G, K, and H minus K tilde.
Yes. By… because you have…
You have a boundary between GK times H k tilde,
To a trivial theory, and then by flipping, you get an equivalence between GK and H minus K.
Yes.
minus K tilde.
Ah, okay, and then… and then… and then the DMNO guarantees the existence.
Right.
Of algebras, and now you're trying to go and find them.
Exactly. And…
Got it. Okay.
And etal algebras are more general than gauging abelian symmetries. They include
These condensable alternatives, so…
Right.
So, from a calculational point of view, we're trying to find the etile algebras that are guaranteed
to exist by this proposition,
And as far as I understand, their proof is not instructive. That is, it doesn't
tell you how to find them, just tells you they exist.
Yep.
Now, what's the physical interpretation of ATOL? Atal Oliver?
I wish I knew. I don't have a good physical interpretation, really.
I mean, I would love to have it.
I have learned how to…
do certain calculations, and…
have rules that make some sense to me, but I… one of the things I would like to…
have is a better physical interpretation.
Jeff?
Yeah.
For those of us who are not named Greg,
Okay.
Could you explain what it means to mod out a group by one of these algebras?
Well, I'm going to give you a procedure to follow.
Uh, and so I think the best I can do is to say,
on a practical level, it means you implement a set of steps, which I will sketch for you in the upcoming slides.
Okay.
I mean, I really don't know how to…
I'm lacking a good conceptual physical explanation.
But it's urgent.
obvious… it's obviously a generalization of the idea of gauging a finite group.
But you don't have a group structure, so it's a little hard to…
But…
say exactly what you're doing.
Yeah, so, I mean, where do I find the algebras in the group?
Or, I don't, right?
No, no, no, you'll find them. Let me explain, because I'm going to… I now want to ill… this was supposed to be the general…
kind of abstract setup for why these things should exist.
The rest of it is to illustrate this with examples and actual computations.
And I hope that…
I hope that will partially answer your question.
All right, so I'm going to proceed in level of complication from trivial
to invertible to non-invertible.
So here's a trivial example. One of the Shellicans models has E8 level 2 times Spin 17 at level 1.
Well, E8 at Level 2
has 3 lines,
This figure shows the quantum dimensions and their conformal weights.
Spin 17 at level 1 also has 3 lines. Here are their quantum dimensions, which match the quantum dimensions of E8 level 2.
Here are the conformal weights.
what's relevant in a TQFT is the top logical spin, which is e to the 2 pi iH.
But for Spin 17 at level minus 1,
It's… you have to complex conjugate, so it becomes e to the minus 2 pi ih.
And e to the minus 2 pi ih for these three values.
matches E to the 2 plus
E to the plus 2 pi h,
For these three values. So, the number of lines,
their topological spins, and you can check the fusion algebra, are…
equivalent. So, these two theories are just…
kind of trivially,
the same without having to do any gauging, condensing of anything.
up to a Chern Simons term, needed to account for the difference in the central charge.
There are other examples like this, where…
You just get on-the-nose an equivalence between
Um, Trune Simon series.
All right, um…
Invertible condensation.
So, if you have an invertible line, this generates a ZN symmetry, and there's a 3-step procedure to gauge it, which I think
also originally goes back to the Taming the conformal Zoo.
paper of Moore Cyberg.
Um, which… and I believe they developed this procedure by a careful study of the
Path Integral formulation,
of Turn-Simon's theory and what it means to gauge a symmetry where you
integrate over only those gauge bundles that are consistent with the symmetry you're gauging.
So, it boils down to that analysis boils down to a set of rules.
That lines that braid non-trivially with condensing lines are confined, and you should remove them.
Uh, unconfined lines that differ by fusion with the condensing lines get identified.
And you split any remaining line that's invariant under fusion,
Uh, with S of the condensing
abelian enions into S distinct lines.
So, here's a concrete example of that, which starts from the Shellicans theory with A17 Level 1 and E7 Level 1.
So, in that theory,
The MTC for A17 at Level 1, which is SU18 at Level 1, has 18 lines.
And the conformal weights…
are these HIs for I equals 0 to 17.
And the fusion rules are just that I times j is equal to I plus J mod 18.
So, everything is, um…
Well, there's a… there's a Z18 symmetry, which is
generated by the fusion.
And in order to condense an inion, you need to condense something that is a boson.
And there are only two of these lines that have
integer H, so they're bosons.
or top logical spin 1.
And condensing those two lines is the same thing as
gauging a Z3 subgroup of the Z18 center of SU18,
Because these two lines generate that Z3 subgroup.
So you identify the possible
Any ions that you want to condense.
They have to be a subset of the… of the…
lines that are bosons have integer H.
And then, when you do the quotient by this, and you follow the three-step procedure,
You remove all the lines except 036912, and 15,
by the condition of the braiding,
And then this Z3 also makes certain identifications that the line 0, 6, and 12 are equivalent.
3, 9, and 15 are equivalent, and you end up with two lines,
with H equals zero and a quarter, and that matches
the quantum dimensions and the topological spins of E7 at level 1.
So this is slightly more involved, but it's a very concrete recipe.
Where you just have to identify the…
bosonic algebras that you can condense.
Everything is invertible, and it just boils down to gauging
a finite group.
Okay, so now we get to things that are more complicated, and…
By me, less well understood.
So, you can gauge things that are more general than an invertible anion. It can be an algebra, known as an etal algebra, Formedius algebra, a condensable algebra, whatever you prefer.
It's a sum of balsonic anions with some multiplicities, so…
This calligraphic A,
is a sum over the various enions that are in your starting
TQFTC with multiplicities NA.
There's a compatibility with Fusion that the NAs have to satisfy.
But the hard part is not really finding A. There's some conditions, but you can find various choices of A.
The hard part is figuring out what it means to gauge aid.
So, um…
Let me say a little bit about that.
So, as I said, it's not really algorithmic. It's like solving a Sudoku puzzle.
After gauging A, the anions in the original theory will split into anions of the quotient.
with some multiplicities.
So, A will split into a bunch of
lines of the quotient, A sub i, with multiplicities NAI.
And the multiplicities, of course, have to be non-negative integers.
However, these AIs are not
all independent, because the condensation identifies some of them, just as with gauging a discrete
group we had identifications.
And not all are in the spectrum, because some are confined, which also happened in the abelian case.
So, let me tell you what the constraints are that you need to solve. I don't think this is actually a complete set of constraints. In principle, there are some identities involving…
F symbols and Pentagon identities, which you would have to solve, but…
Which, as far as I know, nobody actually does, because it's too complicated.
So, I will give you a set of conditions taken from the literature,
Which I believe produces a consistent
Um… MTC from this quotient.
But I think, mathematically, there are probably some gaps.
So, first of all, you can compute the total quantum dimension after condensation,
If you have an MTCC, its dimension is the sum of the quantum dimension squared.
These quantum dimensions are either the value of the on-naught, or this ratio of modular S-matrices,
And if you take the rational conformal field theory point of view,
The dimension of A is the sum of the aquatic dimensions with the multiplicities of the lines, and that Algebra A.
And the dimension of the quotient is the dimension of C over the dimension of A squared.
So, give it an A, this immediately gives you the dimension of the quotient.
But you still need to figure out what the inions are in the quotient.
And that is challenging?
So, first of all, say that something in the quotient has a lift
and see if it appears in the splitting of some anion in C.
Often, you will find
that a given line in the quotient has a lift to more than one anion in C.
And if some of them have different topological spins, then AI is confined and removed from the spectrum.
This is a generalization of the condition for gauging an abelian group, but one that has the same flavor to it.
Enions in A always contain the identity in their splitting. That's why I think the word condo
Sorry, that's the analog of the removal of the non-trivial breeding.
Yeah.
Uh, the other things that braid non-trivially with your abelian Enions are just tossed out.
Right. Right.
Any ions in A always contain the identity, so that's, I think,
One of the reasons why people like
or why the idea of condensation makes a certain amount of sense.
When we, I don't know, condense the Higgs boson, when we condense a Cooper pair,
We're identifying something that has non-trivial quantum numbers with the vacuum.
And things that differ by the thing that we're condensing,
are now equivalent to the vacuum.
So when we gauge, you know, we condense a Cooper pair,
The U1 gauge symmetry is broken down to a Z2, because it's only the Z2 that's not equivalent
modulo the charge of the Cooper pair.
And then there are some conditions that tell you that the, um, quantum dimensions of A have to add up to the quantum dimensions
of the constituents in the quotient, up to multiplicities.
a rule telling you that the, uh, sort of conjugate NEON has to have a decomposition in terms of the
conjugate of the, um, quotient.
And then a rule that basically tells you that fusion in C
is related to fusion in the quotient in a way that is consistent.
how do I know quantum dimensions of the AIs?
Well, you know the quantum dimensions of the AIs,
You don't… you don't know them, but you have a constraint that their quantum dimensions have to add up
to the…
to the quant mention of the aid that it came from.
So, what I'm sure… what I'm not sure about is what's known and what's unknown. Is this… Business to help define the quantum dimensions? We don't know anything about.
I mean, what do we know about the… modular data of C mod A.
So you're sort of defining things about CMOD A, but I'm not quite sure.
I'm defining things about CMOT by listing a number of properties.
that… that have to be obeyed.
Oh. Yeah.
But I don't know a priori
all the data… the data that I'm trying to find. All I can do is try to impose all these consistency conditions
And find a solution, and hope that it's unique.
Which, in small cases, I think one can check.
But in more complicated cases,
Okay.
Uh, this is why I'm saying we really need a better understanding.
Uh, let me give you… let me give you a very simple
example.
Okay.
Yeah, yeah, yeah.
Sorry, could we go back to the last star on your… Uh, what is the B there?
there's A and B… I thought we had one L… one, uh, curly A.
So, here… so I have one A that decomposes like this.
Uh, so remember, A decomposes like this.
Yeah.
If B is a number…
Oh, oh, I see. There are a set of edions.
If B is another NEON in A, then B decomposes in IBBI.
Uh…
Etc. And then, this is a…
This is the, um…
This is the fusion of A with B and C.
Okay, I thought I understood something. I thought… I thought the Curlier was what's called an algebra object in the modular tensor category.
But it seems like there's… Multiple algebra objects, and they're forming.
Well, A can… A is a linear sum of enions in C.
That's right, yeah, that's right. And then I thought that it was an algebra object, so it's… It's Tensor product had a morphism to itself.
Well, I don't know morphism from Smorfism, but…
If I…
That's perfect. Okay.
But if…
If the algebra object A, or the etal algebra A, has both A and the anions A and B in it,
Both A and B will have separate
Yeah.
decompositions into the anions of the quotient.
Okay, so…
AI and BI are the elements of the quotient that you get by decomposing A and B,
Right. But, I mean, the quotient has some set of anions, you're calling a little a sub i.
I think I would have been happier if you had written NB upper J A sub J.
On the second fact.
Okay. The anions in the quotient are A sub i, b sub j, c sub k, on through the letters A, B, C, D, etc.
for the anions in the sea, in this notation.
Okay, okay.
Which may not be a perfect notation, but…
Okay. Alright. All right, okay, got it.
Alright.
All right, so let me give you an example.
There's a Shellicans model with E7 Level 3 and A5 Level 1 as the gauge group.
or just the group.
Now,
We want to find some…
uh… equivalents.
between E73 and A5 at level minus 1.
Or vice versa.
And it's not going to be easy, because, for example, E7 at level 3 has 11 lines,
A51 has 6 lines.
And… well, there's just nothing that's gonna work that's obvious.
if there's a Z6 and A5 Level 1, but the top logical spin that generates that is not 1, it's not a boson, so it's not gaugeable.
E7 III has a simple current generating a Z2, but it's also not gaugeable.
But A has a non-invertible boson, so E7-3, if you look at
this list of…
Ravel, the representations are labeled by highest weights,
There is a guy here which has conformal weight 2, which means it's topological spin is 1.
It corresponds to the highest weight W6 plus W7.
And there is an indensible algebra, which is the sum of the identity, and this
Anyon. It's not invertible because, well, to shorten the notation, if I call two W6 plus W7,
The fusion algebra is that 2 with 2 is the sum of all the even, 0 plus 2 plus 4 plus 6 plus 8, but 10.
So that's definitely not invertible.
So, this…
Algebra A, consisting of these
two lines, the identity and this line corresponding to high suite W6 plus W7.
is not invertible.
And if we're going to find an equivalence, this is basically the only thing
possible that we could try to gauge.
So, we have to find how the different lines split, and how they get identified.
And in the paper we're writing, unfortunately, it's like a whole page of rather complicated
identities and argumentation.
And I don't want to try to go through all of it, because it's just too tedious, but let me give you a very simple illustration of how…
Some of the constraints can work.
So, W6 plus W7 is being condensed. Therefore, it has to have the identity, which I'll just
Labeled by 0, in the quotient.
When it… when you split it into the anions of the quotient.
Whatever else remains, let me just call it W6 plus W72, kind of this is W6 plus W7 sub 1.
By the dimension formula, I have to have dimension D minus
One, where D is the dimension of W6 plus W7.
All right, Jeff, I'm confused. Are you… are you condensing W6 plus W7, or… 0 plus W6 plus W7.
I'm condensing 0 plus W6 plus W7.
Okay.
But everything that you're condensing has to have the identity of the quotient in its decomposition.
So, this has D equals 1.
So, that constrains the dimension, quantum dimension of this
remaining term.
Now, there may be more than one, but in most of these examples, you only need a small number of terms, and so you have to… but anyway, whatever remains.
If there was just this one term, it would have dimension D-1.
Some of the other lines, there's one called 2W6. It will split into 2W61,
plus 2W6 sub 2, etc.
W1 will split blah blah blah.
But now we have some information. We know the fusion
in the starting C, the starting…
fusion algebra, that 2W6
fused with W1 gives W6 plus W7 plus some other terms.
But W6 plus W7 has the identity from up here.
And because all the lines in this… in E7 III are self-conjugate,
That means that 2W61, the first term here, if there's just one term,
Must be identified with W1 sub 1, because their tensor product contains the identity.
So, you get an identification, you get a dimension,
And you have to keep, you know, imposing all the constraints you have, using the original fusion algebra, consistency with the
With the fusion algebra,
And the conditions that I gave you, in a fine… and after some complicated analysis,
You find a minimal and consistent
set of lines in the quotient.
And the lines that you get are the ones that I've listed here. The subscripts
tell you which term and the decomposition of the original guy you have.
The fact that many of them don't appear means that they were either confined, or they got identified with another line.
And when you find this consistent set,
You compute their topological spins and their fusion, and they match those of A5 at level 1.
Minus 1.
So, this is not very physical, it's not very conceptual, I wouldn't know how to put it into a computer.
But part of the point of this project was to find more non-trivial examples of non…
invertible, anyon condensation, and this is a non-trivial example.
And hopefully with enough examples, we'll be able to learn
Some more general rules, and eventually come to a better understanding.
But anyway, this just gives you, like, a hint of the kind of calculations that you need to do.
In order to determine what the…
unconfined lines are of the quotient.
Uh, Jeff, can I ask a quick question?
Yeah, sure.
If you go back to the previous slide, when you were describing the quotient…
Yeah,
the CMOD… even the… even before this.
Yeah, here. So, you said the… it's… the AIs don't have a…
Clear definition.
But, uh, when the mathematicians are talking about Italian algebras and quotienting, have they discussed how this quotient can be constructed in terms of generators or something, or no?
No such prescription exists.
I see.
As far as I know, no such prescription exists. I'll confess that, uh,
What are the difficulties for me,
Or maybe people like me in learning this subject, is that
There are two main sources in the literature. One
is written by mathematicians.
And it tends to be extremely formal and abstract.
And then there are some…
reviews and papers by Condensed matter theorists,
They tend to be very example-oriented and kind of hand-wavy.
Uh, I mean, I think they're right.
But for someone like me, who kind of likes to live in the middle of those two worlds, where I understand a little bit more
of the mathematical structure and why it works, but I don't want the full…
abstract math paper level, uh, is very hard to find resources.
So… but anyway, my understanding on the math literature is limited, but as far as I know,
There isn't anything of the form you described in the math literature.
They're just theorems that guarantee that certain things exist, but don't tell you how to construct them.
So, we're constructing some.
Uh, okay. So…
There are…
Sorry?
So is everything on the Shelligans list going to give an example, or… I guess some of the… is everything on the Shelicans list going to give you an example, or somewhere just abelian and neons?
Some are trivial, some are abelian, and some are not abelian.
condensation.
I don't… I have no idea.
Most of them… most of the…
How many are not available? Okay. There's 70, and so you just haven't had time to go through 70, right?
No, no, no, it's worse than that.
I chose an example. This example, if you wanted to
To condense something in E7-3,
The only choice is this one
boson.
If you look at the conformal rates, there's
Uh-huh.
But if you pick a random silicons model that you would… where you can show you need non-abelian condensation,
There might be 20 bosons.
Some of them involve rather high levels.
So there's a lot of primaries, and then you're taking a tensor product of two or three of them.
I see.
Yeah.
If there are 20 bosons, I would not be able to even start doing the kind of analysis of identifying and using the constraints.
it just becomes unwieldy.
I would say as soon as you have more than about 3 or 4 bosons, it becomes…
Well, uh…
I mean, bosons in this formula for A is a sum of.
Well, bosons in the Algebra A that you want to condense.
little n sub little a little.
That's what I meant, isn't that what I said?
Yeah, no, yeah, yeah, if there's… yeah, if too many bosons…
You wrote a formula before. Currently, A equals N sub A.
Yeah, yeah, yeah, yeah. Right, if there are too many bosons in that…
Okay. So, that's an object.
Yeah.
And the modular tensor category. So, A, it's currently A tensor, curly A is an object in the modular tensor category.
Uh, you could ask if there's a morphism of that to A.
Which satisfies some kind of associativity. constraint. That's what an algebra object would be. In terms of representation theory.
It just means that there's an intertwiner. Between NA tensor A, and A.
So, I think that part is probably doable, although we haven't done it.
But the part where you have to figure out how to actually
work out the splitting and the identification of the lines of A to the lines of the quotient.
is extremely non-trivial.
And not algorithmic.
Yeah, yeah, yeah, yeah. Right, I see, okay.
So, you might be able to show that an A exists with some desired properties, and the quotient should exist.
But if you want to figure out the actual properties of that quotient,
I don't think there's a way known to do that easily.
And it becomes very, very unwieldy when there are many terms.
Okay, so… So, so on the Shelicans list, there might be many more examples, but you don't know.
I'm… I'm going to mention 3 more examples, but I think there's probably many more.
Just… Okay.
Okay.
Uh… okay. So, here's another example that requires non-invertible any uncondensation. I'm not going to go…
go through the details at all.
But you start with, uh…
D9 at level 2, so SO18 at level 2.
And there is a non-invertible algebra, which I guess I've labeled by the highest weight of the representation.
W6 plus the vacuum.
And the quotient…
gives you the MTC of A7…
minus 1.
There's another known duality between Spin 2 Level 2 and A7-1.
And… if you kind of stare at these two things long enough,
Spin 2, Level 2 SO18 Level 2,
Or spin 18 level 2, you might guess…
spin 2N squared at level 2 might have a quotient that gives you A7-1 for any odd N.
And in fact, we identified an A which satisfies all the constraints,
And in this case,
The structure is simple enough that you could actually work out the condensation.
And check that this is true.
So, this is one example of an infinite set of
very generalized level rank dualities.
Where you have, um…
condensable, non-invertible algebras.
that are needed to establish the…
Um, the duality.
All right, I'm almost done.
Um, there's some very pretty relations between glucodes that are needed to construct DMAR lattices and endion condensation. Here's one little example.
If you start… so one of the Sheldikin's theories, also one of the Niemeyer lattices,
starts from SU-25 at Level 1.
Now, the root lattice of SU-25 is an even, right, 24 lattice, but it's not self-dual.
So, at the lattice level, you have to add certain…
Um, weights of SU-25 that are even, such that you construct a self-dual lattice.
In the Lattice literature, that's called a glucode, and
the glue code is Z5.
From the point of view of topological field theory, this means you need to take a Z5 quotient, which kills some of the Wilson lines in the turn-Simon's theory.
Whichever way you view it, you produce one of the Shellicans theories, which is just one of the Niemeyer-Lattice theories.
Now, there's also, uh, spin 25 at level 2, and there's a Z2 invertible quotient of that that gives you SQ25 at level 1.
And there is a non-invertible
algebra, you can quotient spin 25 at level 2 by,
And that's required to get to a different one of the Shellicans theories.
Now, interestingly, you have a Z2 here and a Z5 here, and the algebra that you need to condense here is the algebra of Rep D5, where D5
is the dihedral group, the semi-direct product of Z2 and Z5.
So, this is one of a number of examples where
There's some very beautiful and intricate
structure relating different shellicans theories, how they're constructed from lattices,
And generalizations of level rank duality.
And in general, it seems like, in a sense,
When you look at the topological quantum field theory analogs of the Shellicans theories,
They should all… I mean, this is… I'm not… I think, morally speaking, they should be all connected by some kind of any on condensation.
But I'm not sure I could…
make that very precise at the moment.
But there are enough non-trivial relations that it seems like it's possible.
So, uh, Outlook and Conclusions.
There are many other dualities that, uh, are suggested by Shellican's models that involve non-invertible anion condensation.
I write two exam… two more examples here. We haven't worked out all the details, we just know
Uh, basically that the dimensions match,
When you condense with these two algebras,
But we haven't verified them in detail by working out the splitting of all the lines.
There are other questions, are there exotic MTCs?
For example, there's a class… there's a famous one associated to the Hagarab Fusion category.
can it or other exotic things be obtained by
non-invertible any uncondestation, starting from Chern Simons theories.
This might be viewed as kind of an extension of the…
Old conjecture that you get everything all rational conformal field theories from True and Simons theories.
And the thing that I would most like, and that I think is bothering some of you, is that we don't really have a good physical picture of what's going on with non-invertible, anyung condensation.
or a conceptual description of these very
Generalized rank-level dualities.
And, um…
So, like I said, it's kind of like…
Someone gives you some complicated set of rules, you're not…
Totally sure where they come from, but they look a little bit like the rules of, I don't know, checkers or chess.
But the game's a little bit different. You play the game, and sometimes you win.
But that's not a very satisfying point, you know, place to end as a physicist.
So, we really need a better understanding, and my hope is that by…
Having enough examples and more trivial non-trivial examples,
that… that picture will eventually emerge, and that would be a long-term goal of this
kind of work.
So, thank yous, I think I ran slightly over time, but not too much.
And I'm happy to take any more questions.
Lexi, thanks.
Hey, thanks, Jeff. Excellent talk. So, uh, I asked a lot of questions already, so, um… Maybe some other people would like to ask questions.
Uh, it may have been a bit on the technical side, so it felt like I was, like, giving a talk to Greg, I apologize, but…
Okay. Thank you.
Um, I've done it with lemons occasionally.
Have you encountered something called zesting?
No, I wasn't adjusting?
the context of this project.
Zesting is the kind of generalization of, uh. Um, the civilian anion condensation that is a little bit algorithmic.
And I think is related to this, not a billion.
Anyone condensation, but I can't say much more about it.
But you start with invertible Indian condensation?
No, no, I mean, it's supposed to generalize it.
Okay. Do you have a reference you could send me?
Uh, yeah, I'll send you, um… There's… there's some authors that have been working a lot on testing.
But, I mean, they basically start with a billion condensation.
And… in modular types of categories, and say, let's generalize this.
Yep.
Well, I guess one of the appeals of this subject to me at least, is that, um…
When all this sort of started, I was often skeptical that it seemed like a lot of what was being done was just taking known things and
describing them in more formal language, or maybe, you know, generalizing it slightly.
But I've, you know, over time, I've been convinced that there really are some very non-trivial new results.
that come from this point of view, and…
Um, this non-invertible, any uncondensation definitely
Seems like an example of things that we wouldn't have known how to do before. We don't understand that well yet, but there's definitely…
non-trivial identities and relationships that you can get by thinking about it.
And uh…
Well, I mean, if it turns out that Hagerup really is.
A counterexample. To the idea that, um… Orbifolds and GKOs.
Well, um, applied to WZW gets you all rational conformal field theories. I don't think we know that's.
No, no.
True, for sure, yeah. I think we know for sure that Hagerup.
The Hagrup MTC… Which I think is the Drenfeld center of the Hagelup fusion category.
Yeah, exactly.
I felt center of a fusion category is an MTC, and I think if we take.
Um… the, um, Drenfeld Center of Hagerup, that's an MTC, and I think it's known for sure.
That that is not… Uh, turn silence.
theory for any compact group at any level.
But I… but I don't think we know that it's not what you get by condensing non-invertible anions in a churnsite serum.
Well, that's exactly where I'm going. So, um, so I think HagerUp is a potential.
counterexample to the idea that starting with WZW models and using GKOs and cosets.
Oh, sorry, GKOSES, and um… And Orvifolds by using cosets and orbifolds.
You can produce any old rational conformal field theory. I think that's still… a live conjecture.
Yeah. Of course, you and Naughty…
But it could be… but HackerUp is definitely threatening that conjecture.
So if it turns out that that conjecture is wrong.
Um, but the right revision. is to, you know.
Say, you can caution MTCs by, uh… By non-abilian any oncondition, then I would say it's really good for something that would be a very striking application.
Yeah, yeah.
Yeah, I plan to think about that a bit more.
Yeah, that would be pretty interesting.
Any other questions?
Okay, well, let's thank Jeff again for a great talk.
And, uh