Thank you. I want to first and foremost thank the organizers for putting together this conference
and giving me the chance to speak. I am going to talk about Vertex Algebra, so everybody
who was here very recently heard Clay give you some of the lowdown on these, although
then he said other interesting things I don't understand. For the most part, what I'm saying
is going to be related to a paper that just appeared the other day with my collaborator
Leonardo Rustelli, and it was mentioned before that a lot of the sort of foundational work
on this stuff was done by myself with Leonardo and with Bald von Reiss and with Wolf Gore
Pilaris and Madeleina Leymos and Pedro Leando, some of whom are here. So I'm going to start
out with some very general comments. I am, I guess, obviously going to be discussing
algebraic structures that are associated with n equals 2 superconformal theories in four
dimensions, and although my talk today isn't going to focus on this, I just feel contractually
obligated to mention this because of what I do with the rest of my time when I'm not
posing as a mathematician, that these algebras that we're talking about are really some sort
of weak shadows of some really interesting and complicated full CFT operator algebra
that very little is known about. And so a lot of the ideas that I'm going to be using
and sort of the motivations come from trying to more generally sort of develop algebraic
tools for attacking these theories in general that might someday give us access to sort
of the full person of these theories instead of just these shadows. And sort of the relevant
information for thinking about these things as algebras, both the big thing and the little
things, is the so-called CFT data, which is some list of local operators in these theories along
with OPE coefficients that tell you how you multiply them together in this schematic way
that's illustrated. But it's specific subsets of this CFT data that make themselves known through
these sort of more tractable algebras that I'm going to be talking about. And they're sometimes
recycled sort of through various algebras, and that's going to be one of the main points is that
I want to sort of understand how these things get recycled. So to go into marginally more depth than
claded about the actual structure of these operators in general, n equals 2 superconformal
field theories have an SU 2 comma 2 slash 2 superconformal symmetry algebra. And the operators,
the local operators in them are organized into certain nice representations of them. And usually
when we talk about them, we further organize them in terms of this sub algebra that I've written
where SU 2 1 and SU 2 2 are the left and right Lorentz groups or rotation groups in four dimensions.
And the others are r symmetries and r sub e is the real line energy, the algebra generator.
So we label local operators by their charges under the carton sub algebra of this thing that I wrote
down. So they have energy, they have two spins, they have an SU 2 r charge and a U 1 r charge.
And importantly, in addition to these things that I wrote down that are in the ordinary conformal
algebra plus some extra r symmetries, there are these super symmetries, there are 16 no potent
fermionic symmetries. In any one of these superconformal models, eight of them are so called
Poincare supercharges that are labeled by Q's and the other eight are so called superconformal
or special conformal supercharges that are labeled by S's. And in terms of these supercharges, we can
construct simpler algebras out of this full fledged mysterious OPE algebra of the theory,
usually by passing to some sort of cohomology with respect to the supercharges or something
like a cohomology. And so I'm going to run through these real quick, sort of what I consider the main
ones. The first is the Coulomb-Carrow ring that isn't going to make a big appearance in this talk.
It's built precisely out of the operators whose energy is equal to their U 1 r charge.
And as an algebra, this thing is a commutative associative C algebra. As David Benz-V said,
there's some more structure, but you sort of have to go outside of the operator algebra to see it.
And in general, in examples, this is a freely generated algebra. And so there isn't that much
to do with it if you are only studying it at the level of an algebra of local operators and not
going to these higher things you can do with line operators and whatnot. And these are specified by
sort of being the things that are still around after you do the Donaldson-Witten twist and
pass to the cohomology of this DW supercharge. Okay, closer to what I like spending my time on
is the Higgs-Carrow ring. So these are similar to the Coulomb-Carrow ring in that they're specified
by their energies being roughly equal to their r charges, this time the SU 2 r charge. And
generally speaking, this thing is a commutative associative Poisson algebra.
And in fact, it is the coordinate ring of a symplectic singularity, which is the Higgs branch
of vacua. In the theory, this thing is really a hypercalor singularity. But when you are
focusing on sort of a particular cohomological construction in terms of cues, you really see it
as a holomorphic symplectic space. And this thing is defined in one of these strange ways that I
mentioned. It's not really in the cohomology of one supercharge. It's defined via something that
I think all physicists are happy with and many mathematicians get confused by when it gets
raised, or at least at the one workshop where I tried to talk about this before.
It's something that most of us would call the simultaneous cohomology of four supercharges.
And because I'm not going to do much with it, except use the fact that it exists,
I won't talk about it more. But I think it's a vaguely interesting question whether there's
something to this and the fact that it sounds surprising to certain subsets of mathematicians,
or whether it's just a thing that nobody thought to define before, because it's
not usually that interesting. There's a slight upgrade of the Higgs-Carrow ring called the Hall
Littlewood-Carrow ring. This is defined by a slightly more general energy relation. If
little r is equal to zero, then this is an ordinary Higgs-Carrow ring operator. This
thing is also a commutative associative C algebra. It's also actually holomorphic Poisson,
although I'm not sure that it's been noticed that it's holomorphic Poisson before recently.
And the geometry of this ring is not something that has really been well understood. It's
something like, spec of this thing is something like an odd fermionic coherent chief on the Higgs
branch. And this is, again, defined by this simultaneous cohomology construction, but with
three supercharges. Finally, there's a, well, not finally, semi-finally, there's a thing that
doesn't really have a name. So just for today, I'm calling it the Schur algebra. It's the algebra
built out of these Schur operators that Clay was talking about. So Schur operators are ones whose
energy is defined to be equal to two r plus their spin. And structurally, this thing is some
commutative vertex algebra, which is not all that interesting a thing usually. It's just some sort
of commutative associative algebra with derivation. But it's actually a little more, it's a vertex
Poisson algebra, which gives it some extra structure. And the geometry of this thing,
I think, is almost completely unstudied, perhaps, outside of people who have talked to Kevin
Costello plus Kevin Costello. And I think that it is, I think that this thing is something like
the coordinate ring of the derived symplectic quotient of the infinite jet scheme of the
representation space that defines the matter content of a Gates theory. And you maybe by the
end of this talk will understand why I think it would be interesting to sort of know what these
geometries are, because we're going to try to work our way back from something else. And this thing
is defined as an honest cohomology. These operators are the things that are in the cohomology
of this so-called Schur supercharge here, which is the thing that gets picked out if you do the
holomorphic topological twist, which goes back to work of Andre Johansson and Anton studied it later.
And I think the person doing the most with it in recent years has been Kevin. So those constructions
I only wrote Q in all the slides. So those are all things that you can do in any n equals to
quantum field theory. You can do more when you have these extra supercharges called S's
that I mentioned before. And in particular, there's this vertex algebra that Clay announced.
So this is not a commutative vertex algebra. It's like a good vertex algebra.
And in fact, it turns out to be a quantization of that commutative vertex algebra. So
this is like the better version of it. And again, you get it by an honest cohomology,
but this time it's a cohomology with respect to a supercharge, which to certain people looks
slightly unfamiliar because it's a combination of an ordinary Poincare supercharge in one of
these things, S's that you only have in a superconformal theory. And I'll mention, as many people
have noticed, that it's certainly the case that there's some sense in which you can realize this
thing not by saying Q plus S, but instead by saying the words Omega background.
If that's true, it's not the usual Omega background in the sense that everybody
knows at least one Omega background for four dimensional n equals two theories.
And I'm under the impression that there's some amount of progress, although I can't speak to
the amount on really understanding this as an Omega background by Dylan, Kevin, and Davide.
Okay. So this is the vertex algebra that I want to start with. This is the main
object of my interest. So let me just take a couple slides to review in the sketchiest
possible way what these things are and some of their terminology. So for my purposes,
as somebody who was trained as a physicist, a vertex algebra is just what I recognize as
an OPE algebra in two dimensions, which is meromorphic. So it looks like what I wrote,
where the dependence on positions appearing in the OPE of two operators only depends holomorphically
or meromorphically. A standard thing that one then does is then you expand these vertex operators,
these O of z's, as some sort of Laurent series, where the individual terms appearing in the Laurent
series act on the space of local operators that you could insert at a point, which without loss
of generality you take to be the origin. And in terms of this, you sort of have a simple
realization of this more general notion of a state operator map. A given operator at the origin
associates a state that you get by acting with this particular mode in the Laurent expansion
on the vacuum state, which is the thing called Omega. And similarly, you can get the derivatives
of O by acting with these lower modes. And on the off chance that there's any really hardcore
mathematical vertex operator algebraists here, I apologize, I'm using the physics moding, so
there's an H here instead of a 1. So just so we all know the words, there are some operations
that you can do in a vertex operator algebra other than just doing the OPE. There's some sort of
smaller things you can do. One is to take just the normally ordered product, which can be defined in
various ways, but it amounts to taking two states and defining the state that you get by acting with
this particular combination of endomorphisms on the space of states. This thing is generally
a non-commutative and non-associative algebra. So on its own, one doesn't do many things with it
usually, except just use it to construct other operators. There's also a secondary bracket.
There's actually infinitely many brackets, but there's a secondary bracket that's particularly
important, which is just defined by adding a plus 1 here. So this is the same thing as taking the
OPE of two operators and just picking out the simple pole. And you can pick out higher poles,
and that gives you other brackets as well, but this one is good enough.
And to keep going with terminology, there's a notion of a strong generator in a vertex operator
algebra, and this is some operator that you can't write as the normally ordered product of anything
else and that is not a derivative of something. And these are useful because then you can build
all of the operators in your theory or all of the states in your space of states by taking
normally ordered products of these strong generators, possibly with relations.
So I'll just mention, because in the rest of the talk I will implicitly assume it,
that we are under the impression, based on examples, and surely things want to be nice,
that in all of the vertex operator algebras that come out of four-dimensional n equals
two superconformal field theories, you have a strongly finitely generated vertex algebra.
So you only have to list a finite number of these strong generators, and this is useful
if there's anyone who likes to bootstrap W algebras in the audience. This is useful because this means
that if you want to fix what the algebra is, if you have an opinion about the list of strong
generators, it's a finite algebra problem. You can put it on a computer, you write down the
most general singular OPEs that you can write down, and you solve some Jacobi identities.
Okay, so now I'm going to say a little bit more about sort of the construction of this
vertex algebra in detail, because that's going to have some bearing further on.
So the vector space, as people have probably concluded, is the space of sure operators in
the superconformal theory. That gives us some infinite dimensional vector space,
which is the state space for this vertex algebra. And then to get a operator that deserves to be a
part of this vertex operator algebra at some general point in space, you have to do it via
a so-called twisted translation operation, which looks like this. So in particular, I didn't say
it out loud before, but notice that this is a translation in Z and Z bar and not other directions.
So in order to get a local operator that exists well in the cohomology of this appropriate
supercharge, you need to only move it in a single complex plane and leave it at sort of zero in the
transverse plane. And what this twisted translation is doing for you, if you work it out, so this R
minus here is the lowering operator of SU2R, and it's a fact that sure operators are always in the
highest weight state of their respective SU2R multiples. So if you write a basis with indices,
alpha, 2R of them for a spin R representation of SU2R, then the actual operator that you get at
a given point is some goofy linear combination of the highest weight plus Z bar times the next
highest weight plus Z bar squared times the next highest weight and so on. And this is what you
get for dealing with the cohomology of one of these Q plus S supercharges instead of a nice
Q supercharge. And as a vertex algebra, as a vertex operator, the conformal weight of this,
so sort of the thing that controls the powers of Z that appear in the denominators of these OPEs
for this thing, is given by E minus R, which is R plus J1 plus J2, which is the thing that appeared
in the exponent of Q in this expression that Clay wrote for the Schur index.
And then the fact that doing this gives you something which behaves meromorphically at
the level of cohomology classes is something that just follows from sort of conventional
manipulations of the superconformal algebra. Oh, sorry, the L's are the holomorphic and
anti-holomorphic derivatives in the Z bar planes. So if I didn't have the R minuses,
this is just the way that I would move an operator around.
They're called L's because they live in the SL2 times SL2 bar subalgebra of the global conformal
algebra. Okay, so now I have a construction and I want to say something about sort of the
general structural properties that these algebras have. If you know I'm going to give you one of
these vertex algebras that did come from a four-dimensional theory, what do you expect? What sorts
of nice properties will it have? Some things follow nearly immediately. First of all, from the
expression for conformal dimension, we know that it's a half Z at least, sometimes integer, but at
least half integer graded algebra, but the half integer grading is uncorrelated with parity, so
you can have half integer graded bosons and integer graded fermions. If your four-dimensional
theory is local, meaning that it has a stress tensor, that's what I'll take as the definition
of locality for this purpose, then that means my vertex algebra V will have a vertex subalgebra,
which is the Virasoro algebra at central charge minus 12 times the C conformal anomaly of the
four-dimensional theory, and as Clay mentioned, that means that the Virasoro central charge is
always negative. If I have a continuous global symmetry in my theory, so if I have some conserved
currents in the four-dimensional theory, then I necessarily get a vertex operator subalgebra,
which is the affine current algebra at level minus k4d over 2, where k is some number associated to
this flavor symmetry in the 4D theory, which again is positive, and therefore these are
affine current algebras at negative level. And then some facts about operators that I mentioned
in the previous algebras, so I have these Hall-Littlwood-Kauro ring operators that I mentioned
before, those are lying around in my list of operators in my theory, and Hall-Littlwood-Kauro
ring operators are in particular Shur operators, so they have some life within the vertex algebra,
and in particular you can show that anything which is a generator of this commutative ring,
the Hall-Littlwood-Kauro ring, will be a strong generator of the vertex algebra, and furthermore
anything which is a Hall-Littlwood-Kauro ring operator at all will give you something which is
primary with respect to the Virasoro subalgebra. And then there's this formula that we saw before,
which tells us that we actually know quite a bit about these algebras, or at least about the
spectrum of these algebras in many theories, because there's been a lot of work done on this
Shur index, this sub-SCH, means I'm talking about the limit of the Shur index that Clay was talking
about. And finally, a very pedantic point that nevertheless for some certain people might be
might be meaningful, is that these vertex algebras have no null states, so that's something that a
physicist would always assume, but not everybody would always assume, meaning that you have this
vector space V, and there's actually a non-degenerate bilinear form on it that comes from taking two
point functions with appropriate sort of partner operators, and that means that none of the
operators can have regular OPE with every other operator, so there's no nulls. Okay, so those
were all things that in practice are not that hard to see, and they're not that hard to check
in examples. Now I'm going to talk about some structure that's harder to check in examples.
So in particular, we know from where the operators come from, that this vector space V
of Shur operators is triply graded, sort of meaningfully triply graded, by the conformal
dimension, which we'll use this H convention for the conformal dimension, by their SU2R charge and
by their U1R charge. But the OPE violates the SU2R charge, because the way I built these twisted
translated operators was I took the highest weight and added the lower weight and then added the
lower weight. So my OPE doesn't give me a graded algebra on this thing, it gives me a filtered
algebra because the violation is always in one direction. So if I define filtered components
this way, so filtered components of fixed conformal dimension in U1R charge that have
SU2R charge up to some fixed value, then my OPE preserves that filtration.
And also I can ask how these sort of weaker operations act with respect to this filtration,
so the normal ordered product is filtered in the naive sense, so it just adds filtered weights.
And this secondary bracket is also compatible with the filtration as an operator of degree
minus one. So something that starts out in filtered component R1 and R2 goes to R1 plus R2 minus one,
and it actually does the same thing on the conformal grading.
And this is what's known as a good ascending filtration on a vertex operator algebra,
which is a condition that I think was defined by Haisheng Li. And in particular, that implies
something which is not that hard to just verify by hand, which is that if you take the associated
graded of this filtration, then you will recover a vertex Poisson algebra. So again, that's something
with a commutative OPE but has, I didn't define it before, but has some secondary bracket, which is
the remnant of this secondary bracket that I used here, which gives you some sort of a Poisson
structure. And indeed, you can just check from the construction that this associated graded,
that is a vertex Poisson algebra, as a vertex Poisson algebra, is this thing that I called the
Schur algebra before. So that's the sense in which this non-commutative vertex algebra is the
quantization of the commutative vertex algebra. And once you have this commutative thing,
you can further restrict. So it's trivial to get back to the other things that I mentioned,
starting from this commutative thing. So you can just restrict on the nose to subspaces where H
is equal to R or H is equal to R plus R, where you can see on the slide what I mean by upper or lower
case. And that thing will, as commutative Poisson algebras, give me this Higgs branch or this
Holodalwood cover ring. So this is great in some sense. If I have this vertex algebra associated
to a 4D theory plus this filtration on it, I can get a lot of physics back. I can get these other
things that I spent the last 10 years of my life caring about. And I should say here, additionally,
I mentioned before that all of this is secretly part of a plot to attack sort of more the
full body of the SCFT. And if you want to use the information from these vertex algebras
to attack the full OPE algebra via the conformal bootstrap, this filtration is super important
for reconstructing sort of the detailed conformal data associated to these operators.
But there's a problem, at least at the moment, which is that I don't know how to construct this
filtration in an intrinsic way. So if you give me a theory and you tell me everything about it
already, then I can construct the filtration. But if you tell me the theory you care about has an
associated vertex algebra, which is this crazy W algebra, I don't know how to use W algebra data
to construct the filtration. And there can be, in fact, in these cases are multiple good ascending
filtration. So it's not like the unique one. So maybe there's no intrinsic way to get this,
although I think that's too pessimistic. I think eventually we'll understand it. But for the moment,
I want to consider an easier problem. Instead of constructing the full filtration, I just want to
jump to the end. The last thing I was able to do with the filtration if I had it was get myself
back to the Higgs branch, which is sort of a nice, very geometric space that I associate with my n
equals 2 theories already. So to do that, I don't need the whole filtration. I need less. So let me
say what I need. I need to understand some bigger subspaces. So this V h bar is the subspace of the
space of shear operators, which are not Higgs branch operators. It's an easy enough thing to
define. And you can define it in terms of the filtration this way. So it's the set of filtered
components whose R charge starts below their conformal dimension. And then I can define the
Higgs subspace. This is just the set of Higgs branch operators in my theory. And that's
canonical isomorphic, obviously, to the full set of shear operators, modulo the not Higgs branch
operators. Those are things that so I just said I don't know how to build the filtration. But these
are things that require the filtration. But nevertheless, I can define them. There's another
thing that I can define, which is actually intrinsic to the VOA. And it's a classic construction,
I think, due to although it was probably maybe used earlier than that, called the called the
C two subspace. And this is the subspace of states in the vertex operator algebra that can be built
by acting with a derivative of one field on another field in the normal ordered product.
Remember that derivatives give me these minus these extra minuses.
And it turns out it's not hard to verify that this C two subspace and also this also it's even
easier to see that this sort of non Higgs subspace are two sided ideals for a normal ordered product.
And so we have this kind of descending chain of ideals. And there's even more you can you can
check by direct calculation. It's not that hard that that the the commutator and the associator
with respect to the normally ordered product are also in this C two space. And therefore inside of
the not Higgs space. And also the symmetrizer and the Jacobiator of the secondary bracket
are in these subspaces. And therefore, when you pass to the quotient with respect to C two,
or with respect to V H bar, you get some algebra, which is a commutative associative Poisson algebra.
That's just that's just built into it all. And the one that you get on V mod C two is intrinsic
to the vertex algebra. And the one that you get on V mod H bar is not obviously intrinsic because
to define V H bar I needed the filtration and I don't always have the filtration.
So this one that's intrinsic has a name actually has several names people call it Jews C two
algebra or Jews commutative algebra or the C two algebra. I'll probably use them interchangeably.
And on the up. Right, so we so we get that. We get that intrinsically. This is something you can get
if you can do if you're good at doing computations and vertex algebras. And we know from the four
dimensional construction that if we've identified this other subspace and take the quotient, then
we're going to get back this Higgs-Cauer ring that I already have opinions about. And the question
is, what's the difference? Or if I'm planning on trying to get back to the Higgs-Cauer ring,
the question is, what's the difference? Well, just simple, simple like vector spaceology
tells me that there's some ideal inside of inside of C two.
What did I write? Right, there's some there's some ideal left in the C two algebra,
which I can quotient by that will get me the Higgs branch-Cauer ring. It's just defined as
the quotient of the non Higgs operators by the C two operators. And I want to know what that is.
So I've a lot of this wasn't actually that deep. But I've sort of turned the whole problem of
saying, how do I get from intrinsic to not intrinsic to the question of what is a single
ideal in this particular algebra? There's some good ideal. And I don't have a theorem that tells
me what the ideal is. But there's an observation you can make, which is that the Higgs-Cauer ring
is expected to always be a reduced commutative C algebra, because it's just the spectrum of some
affine variety. So in particular, it shouldn't have no potent elements. On the other hand,
the C two algebra generally has no potent elements. So it has to at least be the case
that the no radical, the ideal of all no potent things is somewhere in this ideal I care about.
And so because I like making guesses, you will just guess that the ideal must just be
the no radical that gets rid of the thing that I know is wrong. And so surely that will leave
me with what's right. And okay, this turns out to be a conjecture that then sounds supernatural,
because you find out that there's this amazing guy, Tomoyuki Arakawa, who's been doing like
parallel work for the last six years, which is very much related to our own work,
without using any of the physics words, which is the statement that this this Higgs branch,
which is the spectrum of which is the spectrum of the of the Higgs-Cauer ring,
should then be the spectrum of the reduction of the C two algebra, which is what Tomoyuki Arakawa
has named the associated variety of the vertex algebra.
Yes. And just to introduce another word, because so so now we can already identify
some nice properties, we know that for the vertex algebra as we care about the Higgs branch is
supposed to be some hypercalor cone. And so in particular, it's supposed to be holomorphic
symplectic. And that's a nice property that an associated variety could have. And so
Tomoyuki gave it a name, which is he calls it quasi least.
Okay. So let's see examples. So that's the guess. Can we can we just check that this works in the
most basic sense and simple examples. So first, the first case is to consider when, when this
C two algebra is actually finite dimensional. So in that case, the whole thing is no potent.
And in that case, if you have if you have a situation where your C two algebra is finite
dimensional, or when your C two algebra is finite dimensional, then the algebra is called C two
co finite. C two co finite, and this is a necessary condition for a vertex algebra to be rational.
So we see a neat thing, which is that if we have any superconformal theories that are supposed to
have a rational vertex operator algebra, of which there are some that it has to be a vertex,
it has to be a theory that has no Higgs branch, which is a special thing. Most,
most superconformal theories that we're interested in have non, non empty Higgs branches.
And so there's an example of these clay mentioned the first of these examples, which is the n equals
to one case, the, the so called a one, a two n, or jeers Douglas theories have been conjectured
originally on the basis of central charges and subsequently on the basis of indices
to be the veresora algebra at the level relevant for the two two n plus three minimal model.
And for those you can compute the C two algebra, it's just generated by the stress tensor,
the only strong generator of the veresora algebra is the stress tensor. But then you get a relation
from the famous null state that makes this an interesting value of the veresora algebra,
which in particular sets t to the two n plus two equal to zero, the Poisson brackets trivial.
And this means that when you mod out by the null radical, you get nothing,
which is what you were supposed to get. So good, it works. And this is the remark I made before.
I don't know if anyone else. It appeals to me that it turns out that the C two
co finite vertex algebras, which were, in some sense, sort of supposed to be like the good ones
to study for a long time in the study of vertex algebras, turn into turn into this obviously,
like very special case of these of these, of these vertex algebras associated to conformal
field theories. So we're getting some, some prediction, which actually was already anticipated,
because I'm not doing this 20 years ago, that there is a there is a much bigger class of vertex
algebras, which generalize the C two co finite ones, they should still have some good properties.
And we'll see more about those good properties. Okay, so I will go quickly through the through
the other examples, I'm not going to read out in words, what I've written on the lines, but it this
this conjecture also immediately passes tests for the other easiest archivorous Douglas theories to
analysis to analyze where there were conjectures about the vertex algebra. So these A one D odd
cases are supposed to be associated to affine vertex algebras associated to the algebra SU two
at these particular levels, which are admissible levels in the sense of Katsumaki modal.
And you can check that the C two algebra is generated by the moment maps or generated by
the things that descend from the currents in the vertex algebra modulo an ideal that looks like
this, where omega is the quadratic Casimir built out of the currents. And this tells you that omega
is actually no potent already. So when you quotient by the no radical, you just get this thing that's
modded out by the Casimir. And that's the same thing as C two mod Z two, which is the right answer,
you can do examples associated to the so called generalized Bresiatsky Polyakov algebras,
and get the canonical singularities as Higgs branches. This is the simplest case where you
get that's supposed to be C two, obviously, by the way, you get C two mod Z three, more
generally, you get C two mod Z N. And I've written all of this out just to make it clear that you're
getting you're doing something and going from the C two algebra to the to the reduction it's there.
You're really sort of missing something when you do what was intrinsic,
and you're getting rid of real stuff when you when you do the reduction and throw away the extra
stuff. Okay, so so that's the conjecture. And we have lots of examples now where it's true,
so it's probably true. So let's consider consequences, assuming that this is always true.
So in particular, this tells us that any strong generator of our vertex algebra that is not
a Higgs-Cauer ring generator has to be no potent in the C two algebra, once we pass to the C two
algebra, because once we mat up by the nopotence, we have to only have Higgs branch stuff remaining.
In particular, this includes the stress tensor. When it's a generator, and by an analogous argument,
even when it's not a generator, and it's constructed as like the Sugawara vector out
of affine currents, it nevertheless has to be the case that the stress tensors image in this C two
algebra is no potent. And it being no potent means that there's a null vector, a non trivial
null vector that I can write down in my vertex algebra that looks like L minus two acting some
number of times on the vacuum, this is k times the stress tensor in the C two algebra, plus something
where that thing is in C two. So this is a null vector that tells me that L minus two to the k,
which doesn't look like it's in C two is actually in C two, once I take the simple question.
And we're, we're accustomed, hopefully from our extensive experience in rational
conformal field theory that when we have these null states, you should try to figure out whether
it has implications for the observables in your theory, in particular, whether it implies some
constraints on partition functions or correlation functions of your theory. And the standard thing
to do with a null vector of this type, which has been studied for a long time going back to the
late 80s, maybe even earlier, if I didn't go far enough back in the in the literature,
is to, is to consider the torus partition function, and to try to impose some non trivial
constraints on the torus partition function from the fact that this vector is no, and in our case,
that's going to be telling us that the vacuum character and so the sure index of the n equals
two theory has some has some interesting constraints. Now a disclaimer, I'm about to sketch a proof,
and the sketch is going to be of sort of a wrong proof. But there's like a better version of the
proof that works, which is due to our Akawa and his collaborator Kawasetsu very recently. But the
version that I sketch is going to actually give the algorithm that basically works in all cases.
And the point is that there's some obstruction that can happen sometimes
that I basically have never run into in very many examples, but it might occur.
And their theorem shows that even when that obstruction occurs, you can work harder and get
around it. So the point is that I have this null vector. And so if I consider the trace on my
vertex algebra of q to the l naught my c over 24, the usual thing for the torus partition function
and insert the zero mode of that null vector, I have to get zero because that null vector is zero.
So in particular, because my null vector was written as these two pieces, the part that looks
like stress tensors and the part that looks like something in C two, I can set it up as this
equality. And then the trick is that there's ways to evaluate the left hand side and the right hand
side in terms of different operations on the vacuum character to get constraints on the vacuum
character. So on the left hand side, I just have factors of l minus two, and I have zero modes. So
it turns out that this situation of zero modes of the stress tensor in the in the torus partition
function, you will be unsurprised can be turned into differential operators acting on the vacuum
character, because zero modes of l minus twos are things like l zeros, which are the things that
appear in the exponent here, only it's a little bit more complicated. So you don't get really simple
differential operators, you actually get sort of modular covariant differential operators when you
do it right, which have some specified form, depending on how many copies of l minus two you
had. And they look like this in general in terms of various Eisenstein series. The right hand side
is a little bit harder, but still doable. If I have something in C two, I know that I'm supposed
to be able to write it this way with a derivative acting on something else. And then there are
recursion relations, let me replace that derivative with things that are higher moded at the expense
of Eisenstein series, which in particular can be twisted Eisenstein series in the case where the
conformal dimension of this thing is not integers. So in my case, it can be half integers. So these
are particular types of twisted Eisenstein series that can have negative ones here, the form of
these things is known. And what you do then is then you have something over here that has a
positive mode number, and you do a bunch of commutators, and you hope that you either get
something again that looks like it's in C two, or you get something with a stress tensor. If it's
something with a stress tensor, then you can get a derivative. If it's something with a derivative,
then you can do the recursion again. And there are obstructions that can happen. And this is the
thing that never happens. But if it does happen, you can get around it. So at the end of the day,
if you've if you've gone through this analysis and either not run into the obstruction or gotten
around the obstruction, what you find is some sort of modular differential equation, either modular
with respect to the full modular group, if you didn't have any of these half integers, or with
respect to the conjugacy subgroup gamma zero of two, which is generated by T squared and STS.
You'll get modular differential equations with respect to those to those modular groups,
which gives you an extraordinary fact about these sure indices, which as clay said,
have been studied quite a lot. But somehow, nobody quite noticed that. Okay, I shouldn't
do injustice. So slo-mo Raza noticed a number of years ago that there was something interesting
going on with modular properties of the sure index. And for the very simplest cases like
hyper multiplets, he saw exactly what was going on. And then for the more general case,
it was very hard to do the analysis. But there was something going on. And this tells you
exactly what's going on. It tells you that as long as this Higgs branch picture is right,
then the sure index of any n equals two super conformal field theory is the solution of some
finite order linear modular differential equation, whose coefficients are polynomials in the
appropriate kinds of Eisenstein series. Thus, these sure indices of which we have an enormous
collection, give us a big source of vector amount vector valued modular forms of weight zero,
and they can actually be pseudo modular in the sense that they sometimes have logs in them.
But that's okay. They can have log cues in them. We will see that actually they almost always have
logs in not sorry, not sorry, the sure index itself does not have logs in it. But the other
elements of the vector valued modular form. So additionally, as a practical matter, this is very
nice, because the dimension the the dimension of the space of these modular differential operators
at fixed weight is finite. So there's some finite number of parameters. And so if you have a sure
index up to order q to the 50, you can check with with very little mental work and some computer
work, whether it solves a modular differential equation, or you can plausibly check whether it
looks like it solves a modular differential equation of order up to some reasonably high value.
So here's an extremely simple example. So there are these rank one f theory theories. These are
theories that you get by putting a d three brain at a singular fiber of an elliptically fiber k
three, and f theory, and those are labeled by this set of the algebras where there's a formal one
called a zero. And it turns out that for all of these, there's a uniform second order equation
that you can write just in terms of the dual coxitor number. And then once you do this,
a zero has to be dual coxitor number six fifths formally. And if you then this is a fun fact,
if you then go in and analyze the solutions of this differential equation, and ask for them to
have integer coefficients, then you recover the examples you already had, you had to, but you
recover a few extra examples, you recover one that is associated to g two, one that's associated to
f four, and one that is goofy, which in some sense is associated to a thing called e seven and a half.
And these are a famous sequence of the algebras, called the Delene Svetanovich exceptional series
of the algebras. And that's a nice discovery. And there's lots more to say that I don't have
time to say about those. But that's a fun mathematical widget. But in connecting to
four dimensional physics, I should admit that at the moment, for f four, and g two, there aren't
known SCFTs. And there's actually an argument argument by, by Shimitsu, Tajikawa, and Zafir
from recently that they can't exist. Although, and I know this is being recorded, so no offense,
guys, but I'm not totally convinced yet. I think they might exist. E seven and a half definitely
doesn't exist as a 4d theory. Now I'm going to flash some data at you. So because it's easy to,
it's easy to do these analyses of modular differential equations and characters, as I
was saying before, so you can go and take n equals four super young males, where people know the
indices to very high rank. And you can look for these modular differential operators. And you can
find lots of examples, you find modular differential operators of the order given here for the
appropriate modular group. And then you can look at the solutions. Oh, okay, we'll come back to
these conjugate dimensions, you see you, you, you, you get solutions with some list of weights
for the, for the exponents. And the interesting thing is that for a gamma zero to
modular differential operator, if you have weights that are separated by half integers,
or if you have a gamma, gamma, modular differential operator with weights separated by whole integers,
then you're going to get logs, or you're almost certainly going to get logs. And you see that in
all of these cases, you have that. So these logarithms are generic in these theories.
You can also go to very high order for the so called a one theories of class S, and just see
similar themes, although something you start running and start running into that I would
like to understand better is that, as you go to very high rank theories, you start running into
modular differential operators that have that have solutions to the initial equation that are complex
or rational. And I think that that's a sign that for these high rank theories, you actually get
systems of modular differential equations that annihilate the vacuum character. And we're only
looking at a subset, but that is to be understood better. Later. These are awful punctures. These
are a one. So it's all if you have a good way of producing the indices to high order and q,
it's very simple to do these analyses, regardless, for the particular case of the a one theory,
a one theories, there's an enormously simple expression for the index in general as a
function of G and the number of functions. So that's why I focused on this.
So just some some comments on interpretation, which is going to which connects with with
Clay's discussion a little bit. The derivation of the modular differential equation goes through
if instead of looking at the vacuum character, you're looking at characters of non trivial
modules, if the modules have some nice properties. So if they have finite dimensional L zero weight
spaces, and if they're conformal dimensions are bounded from below. So a natural guess for how
to think about this, the other entries in this in these vector valued modular forms, is that they
should be the they should be the characters of the surface defects. And for some of the simple for
like the classic cases, like these are your Douglas theories that we're giving you minimal,
minimal models and, and su two at admissible levels, this is something you can just check.
But more generally, I think it's what you should expect.
That said, these conditions of finite dimensional weight spaces, and bounded below conformal
dimension are not things that you need to demand for a healthy surface operator. So you're going
to, you're likely to be probing sort of classes of surface operators, as opposed to getting some
super strong statement that like all surface operators correspond to elements of this one
vector valued modular form. In particular, one thing that one thing that definitely happens is
that you can have reasonably nice modules for these vertex algebras that do have infinite
dimensional L zero weight spaces. But if you turn on flavor fugacities, so if you grade by an extra
sort of global symmetry grading, then the combined weight spaces are finite dimensional. So these
things have good characters if you grade for everything, but they can have but they will have
a bad limit where you turn off the second grading. And in those cases, a couple of things can happen,
you can either have finite limits for combinations of characters, this is what happens with the
admissible level, su two, affine current algebras. Or you can have some sort of genuine singular
behavior that somehow gets regularized. And this is something we haven't understood well enough.
But this is actually the this I think is the origin of the logs in the generic in the generic
non vacuum solutions. Okay. So that's great. Let's say even more. If we have now we now we have now we
have very strong modular control over these over these sure indices over these vacuum characters.
And in particular, that means that we under we can understand very clearly, at least up to computing
these s matrices schematically, we understand how they transform under s transformations where tau
goes to minus one over tau. So q being e to the two pi tau goes to q equals e to the minus two pi
high over tau. These are non rational theories. I'm getting this result, not from some general
considerations, but from the existence of the modular differential equation. So this is like
this is like the admissible. I think I think the hallmark example should be the admissible level
f on current algebras. Those are those are generally irrational, but they have some good
set of characters that transform into each other. But but you don't expect the s matrix you don't
expect the verlinde formula to work for fusion. But there's still some modular s matrix that controls
characters. So that's a finite sum. Yeah, yeah, exactly. So so the key was the existence of this
finite order modular differential equation independent of my opinion about the representation
category of these things. I'm going to get a finite sum, which means I can really look at the
expansion of these characters. And let me just get the whole slide on. And I can look at the at the
weight, which is the minimum one which controls the so called effective central charge that controls
the q to one limit. So this is the so called Carty limit. And the nice thing here is that anomaly
analysis due to G Pietro and Komar Gotsky, or really a slight generalization of it, that that
Leonardo worked out with them tells you that the high temperature limit, so the q to one limit
of these sure indices should be controlled by this combination of the four dimensional
anomaly coefficients. And so you're getting relationship between the four dimensional
anomaly coefficients that includes a, which we didn't see earlier in this talk to this quantity
Virasoro central charge minus 24 times the minimal dimension of the characters that appear
in the s transformation. So this tells you the relationship between the a anomaly and these in
the smallest of these module dimensions that appear in the in the s transformation. And if
you combine that with the Hoffman-Maldesena bounds, which are some unitarity bounds,
so if your four dimensional theory is unitary, there's certain bounds on a and c. And this tells
you that this minimal module dimension has to sit in some finite range between c over eight,
which is negative and zero. And I'd like to make an observation, which is that in these tables that
I showed you, and in fact, in almost all examples, and in the counter examples, I don't I don't think
they're real counter examples yet. It appears to be the case that not only h min is less than zero,
but all the h's are less than zero. So it may be that all of the characters that transform
in this in this vector-valued modular form have to live in this finite range,
which would be, I think, a pretty strong property.
And this not not the observation, but this original statement is already enough to do
a cute little test, which is you can just say, let me think of vertex algebras that I like,
that I might wonder whether they could be the vertex algebra associated to a 40 theory,
and then just look for their h min's and figure out if it if it obeys this rule. And the only
point of this comment is that that is a non-empty exercise. Many things obey the rule, but there's
also things that don't. So there's an infinite set of virasoro of non-unitary virasoro algebras
that don't obey it. And I believe that clay and Shu-Heng have have used this trick to find even
more impressive results in the context of of affine current algebras.
Okay, so I'm about done. I just wanted to show one complete example. So now I've said,
like, a bunch of things you can do if everything works. And now I just want to, like, go through
what happens when you do this in an example. So let's consider everyone's favorite theory,
which is n equals 4 super Yang-Mills with SC2 gauge group. In this case, the associated
vertex operator algebra is just the small n equals 4 super conformal algebra in the 2D sense
at a particular value of c. So this is that algebra if you don't know what it looks like.
It actually turns out that for this value of the central charge t, the stress tensor
is not a generator. It is the Sugawara stress tensor generated by these currents.
But I'm going to leave it there anyway. These g's are the super currents. They're
fermionic operators of dimension three halves, which transform in the one half of the SC2r symmetry.
So you put this algebra on your computer and start computing norms of states, and you find a bunch
of null states that look like this. And you ask, oh, and one more null state at dimension four,
which is of the type that we were hoping for. This is a null state that says that l minus
2 squared is in the c2 algebra. Look at these things. These are all in c2. And then you say,
what does this all mean in the c2 algebra? That's my game. I'm going to start out with
the full thing, pass to the c2 algebra, throw away no potents. So my c2 algebra, that means,
is the thing generated by the currents, by the stress tensor, which is actually a composite,
and by these fermionic guys, modulo and ideal generated by these relations, which are just the
images of the things that I wrote down before in c2. And you can check what's no potent,
and you can see that after you mod out by what's no potent, you just get c2 mod c2. So all that's
left is the currents. You have a t here that becomes 0 because t squared is 0. So that means
that the currents quadratic chasm is 0, which is, again, this c2 mod c2 space. The Poisson bracket
comes along for the ride as well, so you get c2 mod c2 as a holomorphic symplectic space.
And then you had this modular null that gave you the t squared equals 0 condition,
and you can run recursion on it and derive some modular differential equation, which in this case
looks like this. And you can solve it, or more simply, you can solve the indicial equation,
and you can find what the weights of the characters are. So first of all, the second
solution to this equation has h equals minus one half, so that means that you have logarithmic
solutions. And here's an advertisement for work in progress. So with Wolfgrove, we've analyzed
this particular case when you turn on flavor fugacities. So this thing has flavor,
and so you can turn on additional grading. And indeed, when you do that, it annihilates this
logarithm. So the logarithm is a fake associated with there being a module that doesn't have a
good limit where you turn off the extra flavor grading. And then to take the Q to one limit,
because this is half integers, I need to write down the conjugate equation, so the s transformed
equation where I flip the plus ones and minus ones in these twisted Eisenstein series. And you solve
that, and both modules have h equal to minus three eighths, and you plug that into my equation from
before, and you find out that A4T is equal to three quarters, which is something you already knew,
but it's sure nice to see again. So to conclude, there are some obvious extensions of this just
to get a complete handle on sort of what is going on with all these modular transformations. The
most obvious thing is to include these flavor fugacities. That means that instead of working
with modular forms or working with Jacobi forms, this stuff can get pretty hairy, but hopefully
you'll have a nice story. That should probably be the right set of characters to study their
modular transformation properties. Setting all the flavor fugacities to one seems like a
wrong thing to do. And you can also modify these recursion relations if you want to understand the
case where you're not looking at ordinary modules, but twisted modules. And twisted modules are
actually something that should come up in physics. So even these canonical surface operators,
in class S, can be twisted modules with respect to the vertex operator algebra.
And I will finally conclude with some open questions. So the first, which might not even
be a meaningful question the more I think about it, but it might be interesting, which is what
is the property that controls this number, which is the order of this modular differential equation,
which to my mind, as long as this obstruction I was ignoring before never arises, is roughly
speaking sort of the degree of no-potence of the stress tensor in the C2 algebra. It seems to have
some power over these modular properties, so I don't know if it might be an interesting
structure. However, I should mention that due to recent work by Fredrickson Pei, Wen Bin Yan,
and Ye, it appears that it may be really the case that you should spend more of your time
thinking about the set of characters for the flavored thing. And there there's some
shocking to me evidence that sort of the modular behavior of the characters with the flavors
is associated with structures on the Coulomb branch, which I haven't said anything about
this whole day, because that's not where I usually work. Another question, I extracted the Higgs
branch, I think. I would like to be able to extract the Hall of Littlewood cower ring as well, which
was this sort of intermediate ring. That's going to be trouble because elements of the Hall of
Littlewood cower ring that are not elements of the Higgs cower ring are usually no-potent,
so my no-potent trick is not going to work. The Hall of Littlewood cower ring has important
no-potent elements, so I would really need to do something better, and probably what I need to do
is just find a way to construct this whole good filtration. And there's been some work on this by
J. Wan Song, which seems to work in some very simple cases. I think that it's going to take
qualitatively new ideas to get it in the more general case, but that would let us extract these
rings more easily, but B, it would actually lead to an enormous number of very strong constraints
on these vertex algebras from unitarity. It's very hard to impose the constraints of four-dimensional
unitarity on these vertex algebras precisely because we don't have this filtration that lets
us really nail down which operator corresponds to which operator. So, thank you.