Thanks a lot. First of all, thanks to the organizers for this wonderful opportunity.
I will be telling you about the bootstrap program for four-dimensional series with n
equals two and n equals three supersymmetry. And this is mostly based on these works here.
So you probably heard many talks about the conformal bootstrap in the recent years, but
I will just start by a brief summary to set the stage and remind you of what are our goals
with the conformal bootstrap and what we hope to achieve, and then move on to the super
conformal bootstrap where I will also remind you of how we can carve out in any four-dimensional
series with n equals two or more supersymmetry a protected subsector that is a morphic to
two-dimensional chiral algebra or vertex operator algebra. And then I will show how this can
constrain n equals two and n equals three SCFTs. So let me start with the super conformal bootstrap
program. The idea of the bootstrap is that we hope that the symmetries alone and general
consistency requirements are powerful enough that with very little input, a few basic spectral
assumptions, maybe we can solve super conformal field series and I will define what I mean by
solve in a second. And the idea is that we hope to do this only by leveraging symmetries to a
maximum and without having to rely on any Lagrangian description which we might not have for the
series I am interested in or we do not have. So the first question I should answer is what
are our goals, what we hope the bootstrap will tell us, and the first thing we would really like to
have is we would like to have a catalog of all conformal field series or in this case super
conformal field series. If you start with, so I am restricting four dimensions, and if you look at
the series with maximum supersymmetry we believe we have the complete catalog, it is n equals four
supering meals. And okay you could ask and the bootstrap showed that there are no exotic series
but we are kind of satisfied there. But the minute you go to half maximum supersymmetry like
n equals two series we have this vast zoo of known models and many of them do not have a
Lagrangian description. Although we see, and although we see some patterns we really don't
have, we are very far from a complete classification. Until recently we were jumping between these two
cases but half ago we were given examples of n equals three series which had been ignored
so far. And if you really hope that the bootstrap can help you chart out the space of consistent
super conformal field series then n equals three series seems like a nice place to start as they
are more constrained by symmetry than n equals two. And of course this in itself is a very
ambitious goal but after you have done this you would also like to actually start solving series
and I keep using the word solve without defining it, I will do so in a minute. But this would be our
second goal, can we solve specific series which might be particularly helpful if the
series is simply described, if you can describe the series to me in a sentence. So I've been saying
solve, let me say what I mean by this. So we take this abstract viewpoint that the conformal field
series is defined by the set of local operators and all the correlation functions. By doing so I
am ignoring any extended operators such as defects which are interested in their own right, but the
set of local operators forms a closed subalgebra and this is what we choose to focus on. And so we
really want to use all our symmetry so we want to organize our operators in representations of the
conformal algebra or super conformal algebra. So we take our operators, we want to make modules of
the conformal algebra, we start, so these are highest weight representations that are killed by
the special conformal transformations and then you build the whole module by acting with momenta.
So to refer to a representation of the conformal algebra I just denote the quantum numbers of this
conformal primary which is the scaling dimension and spins. And then the nice thing about conformal
field series is that we have an operator part of the expansion so I can replace the product of two
local operators by an infinite sum over all the other operators in the series with some coefficients.
And the nice thing about the conformal field series is that this operator part of the expansion
has a finite radius of convergence. Now of course I don't want to have the sum over all the operators
in the series, again I want to organize my operator in the representations of the super conformal
algebra. So I want the sum to run over the conformal primaries and to get conformal
descendants I act with momenta which is acting with derivatives and so this is something I know
how to do. It's completely fixed by symmetry and it's captured by this c function here.
And really the only data I have in unknown data is the spectrum of primaries, the range of the sum
and these numbers here. And now I told you that conformal field series is defined by set of local
operators and all their correlation functions but I can use this operator product expansion recursively
inside the correlation function to reduce any endpoint function to one point functions
and the only non-zero one point function is that of the identity. And so all I need to define for
you when I say I want to solve the CFT I mean I want to give the set of all local operators on all
the three point couplings, all these numbers lambda's here. So the idea behind the bootstrap
is that if you just give me a random set of CFT data in general it will not define a consistent
theory and this data here is strongly constrained by unitarity. I will choose to focus on unitary
theories and by associativity of the operator product algebra here which I can translate into
the following statement. Let's consider a four point function which I'm decomposing into lower
point functions by doing the OP. So I take this four point function and I can do the OP between
this guy number one and number two here which I'm drawing pictorially like so and then three and four
but I could have chosen very well to start the other way around and let's do two and three first
which I draw like this and this better give me the same answer. So these are the constraints
that the bootstrap aimed at solving and there has a lot there's been a lot of progress recently
since the work of Fertati, Riech, Koff, Tony and Vicky where so this is of course an infinite set
of equations because this should hold for all the operators in your theory for an infinity of data
which is all the spectrum of operators and all these three point couplings here
and this is hard to solve in a CFT in more than two dimensions and recently the emphasis went
from let's not try to solve all of these equations but just try to constrain the space of solutions
and there has been really a lot of progress in getting nice results with this. What I will tell
you about today is not about the numerical approach to these equations but what you can do
if you have enough supersymmetry how you can get exact results for a sub-sector of the operators.
So what happens when you add supersymmetry? Well we were organizing operators in the presentations
of the conformal algebra now I have the supercharges that relates various conformal families
so I'm organizing conformal multiples in a big representation of the super conformal algebra
which amounts to a finitary organization of this infinite amount of data so you could
say that it doesn't buy me much but you could ask the question if there is a solvable truncation
of the crossing equations if you have enough supersymmetry and the answer you might know
already is yes in four dimensions if you have n equals two or more supersymmetry you can
there is a solvable truncation and you can also do it in these other cases here
and so what you really want to do is first to solve this sub-sector and then of course there's
information the sub-sector does not capture and you will need to use the full-blown
numerics for the rest so in my talk I will focus on this step one.
Okay so let me review for you this how to get this sub-sector so like I said I want to organize
operators in the representations of the super conformal algebra I'm taking n equals two in four
dimensions and if the theory happens to have extra supersymmetry that's just something that's
extra there so the operators will be labeled by the scaling dimensions and spins and by the SU2R
charge and U1R charge and other flavor symmetries the theory might have
then the prescription to get this solvable sub-sector is the following I'm telling you to
pick a plane just a plane in R4 with nothing special happening there and I'm giving it
coordinates in Z bar and I choose to restrict my operators to this plane I just focus on operator
that live on this plane and only very specific operators whose quantum numbers satisfy this
relation here so let me consider an endpoint function of such operators on this on the plane
and notice that these are operator that transform non-trivially under the SU2R symmetry so they
have an SU2R index here and now I'm going to do something funny which is to uh contract the SU2R
index with some vectors that depend explicitly on the Z bar position of the operators so at the
if you put the operator at the origin C bar equals zero this vector will just pick a direction so I
say I pick the highest weight of the SU2R representation at the origin and then the Z
bar dependence will come with lower weights of the operator and the claim is that after you do this
what you get is a meromorphic function of Z so uh you've probably seen this before so let me just
briefly review why this is true so the idea is that this sub-sector is obtained by passing to
the cohomology of a certain ill-potent supercharge that is a linear combination of a normal supercharge
a Poincare supercharge and a conformal supercharge because of this conformal supercharge of course
it does not commute with translation so the cohomology will look very differently
at the origin and away from the origin let me start by the origin at the origin the cohomology
uh you can just find operators which are in cohomology you find on empty classes
and now as I want to be able to translate the operators away from the origin preserving them
in the cohomology uh it's a supercharge a particular one I omitted the index
indices for simplicity and a conformal supercharge which is the conjugate of Q
and so at the origin you find that operator at the origin are the ones that obey this
special relation I wrote in the previous slide and now as I move away from the origin if I want
to preserve them in cohomology so on the plane I have an SL2 times an SL2 bar it turns out that
because we chose it like this like that SL2 bar sorry SL2 commutes with the supercharge I picked
so I'm just free to translate the operators in Z but the SL2 bar does not and it's also
not Q exact so uh I cannot just translate the operators in Z bar and maintain them in cohomology
what I need to do are these twisted translations that this vector is doing so the other way to say
this is that the diagonal subalgebra of SL2 bar of the Z bar translation and SU2R is Q exact
and so uh the antialomorphic dependence drops out in cohomology
okay so by doing this procedure if you can start from a four-dimensional equals two or
greater SCFT and passing to the cohomology the cohomology class is defined vertex operator so
you end up with the chiral algebra like the left or left moving side of two-dimensional
form of field theory or vertex operator algebra and uh the conformal weight of these operators
will be given by the quantum numbers of the four-dimensional operator like this
these are positive number which can be integer or half integer
and you can see that each n equals two super informal multiplet contributes at most with one
operator to the cohomology which will be one SL2 primary
then in doing this procedure we we don't end up with a unitary chiral algebra
we have a very specific non-unitary vertex operator algebra which
will be constrained by the fact that the theory I started with is unitary
but the chiral algebra will not be in the sense that some operators will acquire negative norms
but the operators which require norms so you know the sign of the norm of the operator
if you know which type of representation it is of the parent for this theory
so if you give me uh four dn equals to SCFT I can just pass to this cohomology do this procedure
and try to obtain the vertex operator algebra and see what characteristics it should have
what is a harder question is uh if you just give me a vertex operator algebra there's no
reason of course where it should arise from some for the SCFT although we don't still so we don't
have a list of conditions that the vertex operator algebra should obey for it to arise from for the
SCFT and in this talk I will show you an example of what can go wrong when you start from a vertex
operator algebra you saw it could correspond to for the SCFT and in the end something goes wrong
when you try to interpret um and this is related to another question of how much information you
can recover from the vertex operator algebra so uh let's start a bit with the first point
if you give me a four dn equals to cft what can I tell you about the chiral algebra
well there's one operator I'm guaranteed to have in any local n equals to SCFT
as a stress tensor operator you can ask is the stress tensor in the cohomology
the answer is no but it does sit in a super conformal multiple it's a super descendant and
there is one operator in the stress tensor multiple that does uh that is in the cohomology
it's the issue to our current and this gives rise to the to the stress tensor so if I do
this procedure what I get is the op between two 2d stress tensors and so although in the
beginning I only had an SL2 symmetry a global SL2 symmetry which I always have on a plane
it actually enhances to full zero zero because of the existence of a stress tensor in four dimensions
and you get this map between the central charges so here c4d is just a two-point function of the
stress tensor in four dimensions and when I do this twist you see the stress tensor requires
a negative norm there's a sign a minus sign there and so I get an on unitary theory
with 2d central charge that is given by this uh
sorry
the values of c in 4d well unitarity tells you it has to be positive but then um you can show
actually using this type of uh arguments similar to the ones I'll show today that the
smallest central charge you can have for interacting series 11 over 30 which is an arduous
Douglas point um until we have full Verozoro although one important thing to point out is that
if you insist on making uh representations of Verozoro you are forced to mix together
different four-dimensional multiples so it's not from the 4d perspective you want to work
actually with SL2 representations if you don't want to mix different super conformal multiples
in 4d um okay another operator you know you have is if you have a series with a flavor symmetry
you will have a flavor current the flavor current will give rise again not the current itself but
the representative from its multiplet will give rise to an affine gets moody current algebra
where the two-point function of the 4d theory is related to the level of the akm current algebra
by this relation here and you could go on and other multiples you like see if they are in
cohomology and what they correspond to but now um let me tell you how this already constrains
the space of n equals to s e of d's so let's say you have a theory with a given flavor symmetry
so all you can really assume about it is that you have a stress tensor and a current so you'll have
these operators in 2d that i just described to you and of course a trivial statement in 2d is that
you can write all these correlation functions as functions of the 2d central charges
but now um so in 2d still you can by studying these correlation functions you obtain information
about the operators that appear in the op between two t's two j's and a t and a j and one way to do
this for example just let's just do the block decomposition of this correlators and extract
these op coefficients these are two d op coefficients there's no reason they should have
of a non-linearity theory so there's no reason it's a number squared but it doesn't have to be
positive or negative but now i want to be able to lift this to 4d and reinterpret them as
four-dimensional quantities and i can do this with some assumptions so these are the
and i was asking what kind of information can you extract from the 2d about the 4d theory
here i had to make some assumptions like the series interacting as a unique stress tensor
to be able to unambiguously go from the 2d op coefficient to the 4d one
but then i do know i have a unitary for this theory so i should impose unitarity in 4d
and this turns out to give non-trivial constraints on so this lambda into the
were of course fixed in terms of c2d and k2d but they should be positive well after reinterpreting
them as 4d quantities and this gives rise to a new unitarity bounds so let me show you
one such example so let's say we have n equals to zero with the sixth flavor symmetry
and so i am i want to think what is the space of n equals to fts with the sixth flavor symmetries
and i will look at the slice of this space that is parameterized by the values of the two central
charges c and k and here these little dots are known theories that happen to have a sixth
flavor symmetry and live here and these unitarity bounds end up ruling out the blue and red regions
and so they give you new constraints on the space of theories and you see that
there are theories that sit in special places in this intersection of bounds so for example this
is d6 theory of minahan and emeshansky
okay so now let me just quickly show you how to do the same thing but not for for the n equals 3
as cfts so if you have more than equals to supersymmetry so when we did this construction
of passing this cohomology i used up all the supersymmetry of the 4d theory but if i started
from a 4d theory with actually more supersymmetry then some would be left over in the chiral algebra
so some of the extra supercharges commute with the supercharger pass the cohomology off
so if you have 4d n equals 4 you end up with the small n equals 4 chiral algebra into d
and if you have 4d n equals 3 you end up with 2d n equals 2 chiral algebra as was shown by these
people and so now the stress tensor which is something i'm guaranteed to have in any local
theory is promoted in the chiral algebra to a super multiplet so this is the stress tensor and
these are the supercharges through supercharges that made it that commute with the supercharger
i passed the cohomology off and so it becomes this super multipot here and so i can again make
the same trivial statement as i did before the correlation function of four stress tensors is
fixed in terms of the 2d central charge and doing the same i can now read off the
OP coefficients of operators being exchanged here and try to interpret them as for the
OP coefficients and i get a new unitarity bound for n equals 3 series where we just assumed again
that the series is interacting and that it has a stress tensor so this is the value we get and
this is not saturated by any known SCFT this is unlike the cases of n equals 4 and n equals 2
were similar bounds were saturated by known series so for n equals 2 it's the 11 over 30 i mentioned
it was shown in this paper here and now i will just briefly mention why there is actually what
is wrong with a theory that would a candidate theory that would saturate this bound
so let's say i have a theory which has exactly the central charges and all i know about it is
that it has a stress tensor so now i can start looking at the TTOP the super stress tensor OP
and try to reconstruct the 4d operators not just the OP coefficients but the actual operators
themselves and what i realize is that although from 4d i would expect uh the operators to give
a specific sign for their norms which i can just map from representation theory if i try to do
this from the operators i see appearing in this OP something is wrong the norms the signs of
norms are just wrong which tells you that actually there is the theory that a candidate
series will have the central charge cannot be consistently interpreted as a 4d theory
and so this bound should actually be seen as a strict inequality
so to summarize i have briefly shown you some new constraints on the space of allowed n equals
2 and n equals 3 and try to argue well mention how you can argue that there is no theory with
a central charge of course if we have a strict inequality we should ask can we actually push
the bound further what is stopping us from getting stronger bounds maybe understanding a
bit better how to uh what conditions a vertex operator also has to have to correspond to a
consistent four-dimensional as cft and of course you could ask if the numerical bootstrap can help
here if we fail to do it analytically and another question you can ask is i showed you this bound
before and there are no series living in this region here and the reason i don't have dots here
is because we couldn't find any series that had this type of central charges so low and so there
seems to suggest that there is a bound that should exclude this region so perhaps a similar reasoning
could help here and finally the e6 theory of minahan and machansky set at a very special point
in the intersection of two bounds that were obtained by this system of mixed
correlators so if you actually want to solve this theory maybe you should do it numerically
with this system of mixed correlators thank you