Okay, thank you very much.
First of all, thank you for giving me the opportunity
to speak here and really for organizing
this fantastic conference in a place like Hamburg,
which is exciting to be back here, actually.
So Ron just pointed out to me that this may be
the first string theory talk at this string math conference.
That's not obvious from the title,
but Ron heard this talk already last week
at Sheldon's birthday.
Anyway, so I'll talk about Enigos for Super Yang Mills
with varying coupling.
And you may all know what Enigos for Super Yang Mills is,
but you might not have heard what it means
to have it with varying coupling.
So I'll explain this and I hope at the beginning
I'm a little sort of very, very slowly pedagogical
and at the end I'll sort of take off
and do some physics with it.
This is work in collaboration in various combinations
with Benjamin Arsell, Ed Cern, Chris Cousins, Craig Laurie,
Dari Matali, Jenny Wong, and Timo Weigand.
Craig is somewhere here in the audience,
I don't know where, so he's a postdoc student
and Dario is a King's team writer.
So as I said, we'll start with something
that we should all somehow be familiar with.
So Enigos for Super Yang Mills has had a lot of activity
for various reasons.
It's a super conformal field theory.
It's maximum supersymmetric.
It's sort of an extremely beautiful theory.
You can study it from various points of view.
Today we're interested in its SL to Z duality.
So let's take this theory with a gauge group G
and then it has sort of an SL to Z monotone olive duality
which generalizes electromagnetic duality
which acts on this complexified combination
of the theta angle and the gauge coupling G.
So tau is this object here and the SL to Z acts
as A tau plus B over C tau plus D.
Under S duality in particular,
G actually maps to its Langland-Stuhl group
which has sort of generated a lot of interest
in this theory and math.
Now I will need a little bit more information
about this theory than just, you know,
this very sort of general statements.
So I wanna look at the theory for G equals to U1.
So it's an abelian theory and just sort of say a few things.
What are the ingredients in that theory
which will become useful later on.
So the action is we have this tau one, tau two
which are the real and imaginary parts
of this complex tau parameter.
They appear in the action in front of the star F
or HF term and the theta term in terms
in front of the F or HF.
F is the field strength here in this case it's abelian
and then there's scalar fields and fermions
and the abelian theory that's basically all there is.
So what's important is going to be
what are the symmetries of this theory.
Apart from the gauge symmetry,
it has the Lorentz symmetry in four dimensions.
So think of it as SU two times SU two
like in the last talk for example
and it has this very large, the maximal SU four R symmetry.
So this will act on the super charges of that theory
and in particular I can now write the fields
and the representation of this group,
the Lorentz group and the R symmetry.
The gauge field is a vector and trivial
under the R symmetry, the scalars are trivial
under the Lorentz and in a six in the SU four
and the lambda, lambda, tilde are fermions
of R spinners and transform in the four of the SU four.
Likewise, this theory actually is of course,
as I said at the beginning, it's super symmetric
and the super charges transform
and very similar representation
to the fermions given here.
So what we like to do today is look at this theory
and think of compactifications of that,
particularly on curves
where we allow this parameter tau to vary.
So normally you would actually look at tau
as a constant object and then you look at
sort of the dual theory, you know, again with constant tau.
I think that's the power, is it?
Anyway, let's leave it.
If we run out of power, we'll be fine.
Okay, so normally tau is constant
and today we'll ask what happens when tau actually varies.
So the first thing, it'll be very important
to make sure that it's consistent with the SL2Z,
this duality symmetry here.
For example, if we have tau goes to minus one over tau,
say we do some loop in space time,
then it undergoes a monodomy like this,
then in fact, you know, this needs to sort of
be compatible with the SL2Z.
And to actually make this sort of happen,
it'll be very useful to think of tau
as actually a geometric object,
namely as the complex structure of an elliptic curve.
And for you who have heard talks about F theory,
that'll be very familiar.
So we think of tau as sort of really a geometric quantity.
There's an elliptic curve that's not fibered
over the world, over the n equals four space time
and that's geometric.
It's the complex structure of this elliptic curve.
So why is this sort of something interesting to look at?
So in string theory,
we actually think of 40 n equals four
coming from these three brains.
So the low energy effective theory of these three brains
and type 2B, 10 dimensional string theory
is exactly this theory, 40 n equals four super young mills.
And actually, the type 2B string theory
has a very similar duality symmetry.
So there's an SL2Z, which acts on this tau 2B,
which is a complex combination of the axion, C zero,
and essentially E to the minus five,
which is related to the string coupling.
And the type 2B string theory has a self duality
under this SL2Z operation on this tau 2B.
Now this SL2Z descends on the D3 brains
exactly to the monotone olive duality of n equals four.
So actually realizing now in type 2B,
these D3 brains and generalizing type 2B
to actually a varying tau type 2B version,
meaning F theory, the D3 brains that we consider
actually also will have to have a description
in terms of a theory that n equals four
where we now allow the tau to vary.
So two setups that are useful to sort of keep in mind,
and I'll discuss only one of them in this talk here.
The first one is where we have a four dimensional space time
where now tau varies over the full four dimensional
space time.
And then there's a second setup where we say,
well, let's put this n equals four theory
on a complex curve, and tau only varies over the curve,
and there's a transverse, one comma one dimensional space,
and we're looking at a two dimensional theory on it space.
And we're trying to characterize now,
depending on what the variation of tau is,
what the curve is, how it's embedded,
what the two dimensional theory on R11 is.
And we'll see it's a super conformal field theory,
has zero comma four supersymmetry,
so there are lots of things you can do with it.
Okay, so this is motivation for string theorists.
Why would you actually look at these three brains?
So usually in F theory, we would consider
some compactification on Calabi-L,
say Calabi-3-fold, or four-fold, or even five-folds now.
And we have some base, a section of this elliptic vibration.
And the curve C, which appears here,
is in the base of this elliptic vibration.
So above that, above this curve C,
we have a restriction of this elliptic vibration to C,
and this will induce the varying of the tau parameter
of the n equals four.
So in six dimensional F theory vacuum,
these are one comma zero super conformal theories
that have been classified recently,
this beautiful work by Heckman, Morrison, Baffa.
These types of wrapped, these three brains on curves
inside the base of Calabi-L three-folds
actually serve the smoking guns for saying
these are super conformal theories,
so they're extremely important objects there.
They also feature in two-dimensional F theory vacua,
so you can actually put F theory on a Calabi-L five-fold,
elliptically fibered, and then you get a two-dimensional
zero comma two model, and there you actually are forced
to have these kinds of D3 brains for temporal cancellation.
So from the physics, these are good motivation
to look at them.
Just formally asking the question,
does this SL to Z duality, can I now have this,
can I use that to actually have a varying tau
over the world volume of a D3 brain,
is a perfectly fine, totally F theory detached question
to ask, so this is essentially just to back this up
from F theory.
So you should keep in mind for the rest of the talk,
this is sort of the setup,
there's elliptically fibered Calabi-L,
the elliptic fiber restricts to the curve C,
the D3 brain wraps on C times R11,
and the elliptic vibration will have a section,
so it has some biostrust form,
and F and G here will be determined
in terms of the anticononical bundle
to the four and six respectively,
and the elliptic vibration restricted to C
will tell us how the complex coupling varies
over this curve C.
So the question that I'd like to ask is,
what's the theory of these wrapped D3 brains
in this two dimensional R11,
meaning what kind of two dimensional CFTs are these,
and can we calculate interesting physical quantities
for them and do the usual things that physicists do
with control field theories, namely do some comparisons,
for example, what I'll show you,
and also interesting new ADS dual solutions, yup.
I will show you exactly, it's not Waffa-Witten,
so it's something that's a modification
of the standard Bichatsky et al. twist,
so I will actually now get to the twist.
So you can't just do the normal topological twist
because we'll see the super charges
will transform under this SL2Z,
so there's something that's called the duality twist,
and I'll get to that.
Okay, the plan of the talk is we'll start
exactly discussing how to actually define that
and equals four theory.
There's going to be a useful six dimensional point of view
coming from M5 brains where you really take literally
in the geometry this tau as an extra direction
in your space time that'll help us understand
certain effects due to the singular fibers here,
and then finally I'll discuss sort of duals
in terms of ADS CFT of these theories.
Okay, so let's start with actually defining
what these theories are.
So these three brains I said we'll be looking
at this theory, it's n equals four super young mills
with this coupling, these are the symmetries,
this is sort of the content.
And we'd like to pack this theory on a curve
times R11 and preserve supersymmetry.
So the classic way of doing this
has appeared many, many years ago in this paper
Brzezinski, Johansson-Sadoff, Waffa,
and also then later in Kapustin-Witten.
What we have to look at is the Lorentz group gets reduced.
We have an SO11 acting on the R11 transfers,
but also there's this U1C factor
which is the holonomy of the curve.
So the spinner, so we're looking at the supercharged Q, say,
the spinner two one will decompose
into these two guys here.
And we see that under the U1C,
they have non-trivial transformation properties.
So as we walk around the curve,
these supersymmetries are not really a constant,
so actually we need to do what the physicist
called topological twist.
So we also decompose the R symmetry,
which is an internal symmetry,
into an SO4 transfers times the U1R.
This is the four bar decomposes in this way here.
And now we can redefine what we mean by the U1C
by sort of twisting it with the U1R.
So define a generator, T-twist,
as the linear combination of the U1 generator TC and TR.
And under this now twisted symmetry algebra,
which is this index here in the back,
we now have four chiral left moving
and four chiral right moving supersymmetries,
which are invariant under this redefined twisted U1C.
And so we would say this resulting theory
is a two dimensional four comma four supersymmetric CFT.
Where it preserves two comma two supersymmetry,
then you still need to check what kind of theory it is.
And what's well known is the BPS equations
of the theory along the curve
are exactly the well-known Hitchen equations.
So FZZ bar minus the commutator of phi Z phi Z bar is zero.
The important fact here is that the scalars
become the one forms along the curve after this twisting.
So we repeat sort of the same analysis for the scalars,
which were transforming in the one, one comma six.
And then they actually become sections
of the canonical bundle on the curve.
Okay, so the resulting four comma four
supersymmetric sigma model
is intersitu moduli space M8.
And this is sort of what came out of this analysis here.
So this is not half a bit and this is
a slightly different thing, right?
It's more like a pushing.
So what actually changes when we now have this varying tau.
So the supersymmetries actually transform under SL to Z.
Say we have some element gamma and SL to Z.
And the supersymmetries Q and Q tilde
pick up a phase rotation.
So this phase here is given in terms of the phase
of C tau plus D.
And so here these would have charge plus minus a half.
And also the fields except for the scalars,
fermions and also the gauge field combination given here
transform non-trivially under this SL to Z.
So this means if we actually have tau varying
over our space time, we actually need to
account for this as well.
So in fact not only do we have to do this topological
twist of U1C with U1R but also this phase rotation
which I'll denote by U1D with the U1R.
And this is what we call topological duality twist.
Essentially what this means is that all the fields
are not just now, the scalars are not just sections of Kc
but actually sections of this restriction
of the elliptic vibration.
So this Kb inverse to the curve C.
So the sections of a line bundle
that sort of characterizes the varying of the tau.
So the connection is precisely given in terms of
D tau one over tau two.
So this is zero if it's just a standard constant tau
and it goes for superimals but in general
this now actually is non-trivial.
Okay so let's just quickly repeat how the analysis
that we did earlier, sort of this recap from the 90s,
how that changes.
So in addition to the Lorentz and the R symmetry
we now have this U1 duality.
The superchargers are charged plus one and minus one
in a certain normalization.
So if we have a curve C say in the base
of an elliptic Calabi R3 then we have the standard
decomposition that I showed you earlier
of the Lorentz and the R symmetry but now we have to
not only twist the U1C with U1R but also we have to
twist with the U1 duality with the U1R symmetry.
And the effect on the superchargers is now that
actually the theory ceases to have four comma four
and actually becomes a chirally supersymmetric theory.
So it used to be in the sort of constant tau setup
these guys were also trivially charged under
the twisted U1s but here you see these are charged
under this U1D twist.
So actually we were left with only these ones here
they are plus chirally in two dimensions
so we get zero comma four supersymmetry.
Okay now you can repeat this for different Calabi R's.
The interesting thing is always there are chiral theories
so you can get zero eight, zero four, zero two
you can also get zero six.
In all these cases these involve this duality twisting
and correspond to N equals four supermules
wrapped on curves in the base of these Calabi R
elliptically fiber Calabi R enfolds.
Sorry?
The fields transform under the SL2Z in this way.
That's right.
Now I take this, huh, that's right.
So I actually whenever, so think of this as the curve C
here there's a singularity of tau set as a monodomy
around this point, an SL2Z monodomy.
Then what I'm doing is I'm undoing this.
I'm saying I would like to have scalar super charges
by twisting it with this R symmetry here.
So this is the only way you can actually.
So all it does is it ensures for the abelian theory
you can do it explicitly in the action
or you can modify the theory.
All it does is it says now everything is still well defined.
The fields, the super symmetries
as you drew these monodomies.
Right, so tau will undergo the duality
whatever A tau plus B or C tau plus D.
And then the super charges will remain.
These ones with this twister will remain in Beren.
We can actually, hello, Clay.
So another way of seeing this break
is you could also, and I'll get to that in a second.
This is the same as just a standard topological twist
of the M5 brain theory on an elliptic surface.
Like an MSW string, but on a very specific surface
which is elliptically fibered surface, okay?
Okay, so we've looked at the super charges.
There's still a question, now what is that theory?
So first of all we could say what happened to the scalars
before it said in the standard topological twist
it becomes sections of KC.
Now, if you follow this through
with this additional duality twisting,
the twist charge for this U1C twist
still says this interprets the fields sections of KC,
but actually if they have non-trivial twist charge
on this U1D twisted,
they become sections of this line bundle LD
which is just a restriction
of the anti-canonical bundle to the curve.
Right, so this is the effect of the elliptic vibration
restricted to the curve C.
This is sort of telling us
how much this coupling is varying there.
It's a question.
So for example, a field two, one, one here,
actually this should be one, one,
is now a section of NCB, but a junction.
So we can write down the full spectrum
and organize it in terms of the zero comma four
supersymmetry.
The thing you should take away from this
is each one of these charges here
will tell us what kind of sections
what the multiplicities of these fields are.
As I said here, these are now sections of KC and LD.
So this is for the abelian theory.
We get this sort of multiplets
for the zero comma four hyper twisted hyperfermis
and the vectors.
So anyone who's sort of thought mildly
about elliptic vibration will say
you forgot something really essential,
namely elliptic vibrations are usually
have interesting loci.
Why the fiber becomes singular.
So if we have a large stress model,
so we have a section, then the singular loci
are sort of given in terms of the vanishing
of the discriminant, delta four FQ plus 27 G squared.
In terms of F theory, these are the three seven,
so the intersections of the seven brains
or the wrapped loci of the seven brains, excuse me.
So delta is the locus where the seven brains are wrapping.
So in our setup, there was this curve C,
generically there's an elliptic fiber above it,
that's smooth, but now the locus delta
will intersect the curve in points.
So these are divisors in the curve C,
above which delta is vanishing
and so the fiber becomes singular.
Now, what happens above these loci
is that the fiber is now not just some smooth torus
but some singular elliptic fiber.
So some good aerofiber, for example,
something like this sort of collection of rational curves.
So from the point of view of our n equals four theory
that's wrapped on this curve C,
at these points, there will be some specific monodomies
of tau that are dictated
by what kind of singular fiber there is.
And so as we encircle these points,
we'll undergo some SL to Z monodomy.
And so these loci precisely are what, you know,
you would call a duality defect.
So at intersection C with delta,
at these points are additional localized degrees of freedom
which are so-called three-seven strings.
Okay, so before I go and discuss more
what these three-seven strings are
and actually how we describe these theories,
and there's sort of one question
that we should get out of the way,
which is what kind of, what replaces
what used to be this sigma model
into the hydrogen-modulized base?
So if we just had constant tau
and we compactify with this twist I explained earlier,
the n equals four on the curve times R11,
this becomes a sigma model into this hydrogen-modulized base.
So one question is now what replaces this?
So the BPS equation actually along the curve C,
they become now something that looks almost like
the HN equations, except that this curly F
is not this combination here.
So A and A bar are the gauge field components
along the curve, and these covariant derivatives
are with respect to this connection,
this duality connection, this U and D connection.
So both the scalars, these phi's and A
are in our sections of, if you ignore this for a second,
Kc and Ld.
And in addition, these loci here now
will also say that there are additional contributions
from these discriminative loci.
So all in all, what replaces the statement
about the hydrogen-modulized base is then here,
the four dn equals four with compactification
and topological twist, which is this duality twist
becomes a sigma model into some sort of
duality-twisted hydrogen-modulized base.
So this is as much as we know.
So since last week, this page already has improved
by talking to Ron and Sheldon,
but there's a question, what actually are the solutions,
what really characterizes this modulized base here?
This is sort of a very vague, but maybe sort of, you know,
slightly motivated thing that I'd like to say about this.
So the S duality really acts on the hydrogen-modulized base
as T duality on the hydrogen fibers.
So we could also say that maybe these maps from C
to this hydrogen-modulized base, they're now,
or the hydrogen-modulized over this curve C,
they really should perhaps be thought of as,
we move on in this curve here, there's this defect,
there's some walls that separate various regions
of different sort of branch cuts for this tau.
It's possible that we could think of this as,
now we have, as we go around the point like delta,
we actually do a T duality in this hydrogen fiber.
I would love to understand this better.
This is as much as I can say about it,
but this certainly would be an interesting thing
to look into how the sigma model actually has
a nice description using this information
that the S duality really is just the mirror symmetry
on this hydrogen-modulized base.
Okay, so as a physicist,
what can I say about these theories?
First of all, in this picture here, you saw these walls.
So I would like to explain what they really are.
So this is where tau sort of undergoes this monodomy.
So if we cut this curve up into these regions, C-I,
and think of tau as, say, it has a single locus
at this delta, then usually we would have, say,
some behavior like log delta is equal to zero.
So, say, if this is equal to zero,
then actually the behavior would be
log Z of this tau parameter.
So there are branch cuts now.
So the action of the N equals four,
I can write it in terms of F, which F dual,
where F dual is this combination here.
And then along these walls, where the branch cuts of tau are,
we can write an action on the wall,
which is essentially just making sure
that the N equals four theory on this four dimensional space,
C times R11, is well defined
when I have a branch cut along W.
So this is something like a mixture in Simon's theory,
and there's not really a lot of interesting physics
on these walls.
What is interesting are these defects.
So this is where exactly we have additional modes
coming from three seven strings.
They are the low size C dot delta,
so naively we would say they're actually,
you know, delta is in the class 12 C1B,
so they're 12 C dot C1B, many modes at these locations.
If we have a smooth barstows model,
we'll find out this is actually not true.
So there's some subtlety in how exactly
we should be counting these modes.
This is sort of the naive thing,
assuming that all the components of discriminant
are essentially mutually local seven brains.
This is actually not true, so actually the right way
to look at this is to go to a six dimensional point of view
where we now really think of the N equals four super angles
with the varying tau as some full six dimensional theory
on an elliptic surface times R11
and study it from that point of view.
So this is essentially saying that we're looking
at an M5 brain on elliptic surface,
and then these three seven defect modes
will actually come from a dimensional reduction
on this elliptic vibration.
Okay, so that's what I'll discuss next.
And this also will answer hopefully Greg's question
about whether this topological twist makes sense
fully directly.
So the six dimensional two comma zero theory
is sort of this unique super conformed field theory
in six dimension has some ADE gauge group.
And one important sort of relation
to low dimensional theories, of course,
if we put it on an elliptic curve
with complex structure tau times R13,
it just becomes N equals four super angles
or the coupling is tau.
So our generalization of N equals four with varying tau
translates trajectory into we're looking
at a six dimensional theory now on elliptically fiber space
where now the variation is encoded in elliptic vibration.
So the three brain wrapped on this curve C
now becomes an M5 brain wrapped on this elliptic surface
C hat times R11 and the C hat essentially encodes
already the variation of this coupling.
So if we do a dimensional reduction
on of the M5 brain theory on such elliptic surfaces,
we should be getting the same theory
as in the D3 brain case,
and we should be also able to actually describe
directly what these defect modes are
since here everything is completely geometric.
Okay, so one nice sort of fact
is this slightly strange duality twist
here is now just a topological twist on a careless surface.
So C hat here is elliptic surface,
there's a standard topological twist that we can do
and then if we apply that this actually elliptically fiber
it just reduces to this topological duality twist.
So in fact, this also is the argument
why this would hold for the non-abelian theory as well.
It's less clear actually in N equals four
where we can actually do this duality twisting
only for the abelian case.
Okay, so just to give you some flavor of why that is true,
so the six dimensional theory is, it lives in six teams,
so Euclidean SO6 Lorentz and an awesome entry SP4.
They are in this super conform OSP6 slash four,
the bosonic part of that.
And the abelian, so it's a single M5 brain theory
has this field content which is fermions,
again in four comma four under this group,
scalars and the self-dore form.
So B is two form and DB is self-dore.
And equations of motion are very, very simple.
And now what we can do is really dimension reduce
this theory, again doing a topological twist
where we replace this SO6 now with the specialization
to an SO11, this is the R11,
and a U2 for a Kayla surface.
So doing all this, actually let me skip most of this.
Once identifying what the topological twist is,
we can really see that indeed this is the twist
that in four dimensions would reduce
to the topological duality twist I defined earlier.
So from that point of view, you can also argue now
that this twist makes sense.
But the nice thing is really the six dimensional point
of view is giving you really an angle
how to discuss these two defect modes.
So naively in the type two B point of view
we said C dot delta should count the modes.
As we'll see in a second, this is actually not true.
So the important sort of insight is that we can reduce
this self-dore three form field along one one forms
of these special fibers, these cudaria fibers
in cuda mentioned one in this elliptic vibration.
So we expand this field and post the self duality
and these give us chiral modes,
which are exactly these defect modes.
And doing this counting properly, we can account for these
and the rest of the table is the same
as I showed you earlier, but actually the additional piece
which comes from these additional modes are 8C1 of B dot C.
So this is actually the right contribution of these defects.
Yes, so it's a smooth virus stress model.
You can also include additional singularities
that will change the count.
Well in MC you can always resolve this.
So here, I'm not just drawing this because I like
to draw this in this way, but actually you can really
resolve this and look at that.
Resolve phase, and then in the reduction,
you're going to the N equals to four type 2B version of it.
So in this counting here, this really assumes
it's a smooth virus stress model.
So all you have is I1 fibers, but if you had more,
you would get additional half fermis here.
Okay, so what a physicist would now do
is you have a two dimensional zero four theory,
you compute the central charges.
Actually, earlier I might have forgotten to say
this is a zero comma four theory
with a small super conformed algebra,
and so the central charges here are now given
in terms of intersection numbers of the curve
and C1 of B, let me see.
And this actually is also things well with
early results by Molsonin and Witten and Narsian more,
and I think Simpis, who actually looked at related
signal models now for the M5 brain directly
reduced on a generic yellow surface.
Okay, so the important point about a six dimensional
perspective is that we have an angle,
how to calculate this contribution from the defects,
so this is already accounted for here.
And now we can actually ask, so what can we do
with the central charges?
Before we get to this point, let me point out
one other thing, if we had an N equals four theory
where we now allow the coupling to vary
over a full four dimensional space time,
that would correspond to an M5 brain
wrapped on elliptic three fold.
In that case, these defects are now curves,
and these curves can intersect.
So we have these two dimensional chiral defects
in the base here, and they can intersect now in points.
So you actually, if you now look at what this
self dual field B does along these curves,
essentially there'll be relation between the chiral modes
on one of these defects with the modes on the other one.
And that's essentially dictated as to how
the chiral fibers behave as we go to these higher
chiral dimensions, single or low side.
So that's essentially one of the other feature
that here you have the network of these duality defects
that can have intersections at points,
where from a physics point of view,
I actually get whatever flavor symmetry we had here
from the chiral, single fiber and flavor symmetry here,
enhances at these points,
you can enhance flavor symmetry.
Another way of saying this is you have these chiral CFTs
along these defects, which now have
sort of point like intersections.
Okay.
So, I said we could calculate central charges.
You can do this for Calabia four-folds,
where you get a zero comma two theory
and Calabia five-folds as well.
But the central charge calculation,
if you look at this expression here,
this is for still a single M5 brain
or the N equals four supine meals theory
with gauge group U1.
And we'd really like to know how to do this more generally.
Now, both the 60 theory as well as the duality twisted
N equals four with varying coupling,
we don't know really how to access this.
As a physicist, you then would say,
okay, is there maybe sort of another way
I can get an angle to describe these theories.
And so this will lead us to say,
well, perhaps there's sort of a holographic dual
which characterizes the same system.
So if there's some gravity solution,
some in this case an ADS-3 solution
that characterize the same system
where we can also calculate the central charges
and match these with what we found here.
So that's what I'll talk about in the next 15 minutes.
But let's take stock for what we've done so far.
So we discussed N equals four with G equals to U1.
Then we have a complete description of this duality twist
in the computed essential charges in the spectrum.
And the BPS equations
are these duality twisted Hitchen equations.
So they should be giving rise to some sort of sigma model
into this two-dimensional theories
or some sigma models into these duality twisted Hitchen
modelized spaces.
And so one useful point of view is the 60 point of view
which tells us how to calculate the defect modes
from these three seven intersections.
But indeed, this is sort of limited to the abelian setup
so one question is indeed how do we actually extend this
say to NM5 brains or SUN or UN N equals four.
And for this I'll actually utilize sort of holography.
So I actually write down for you a solution
that characterizes exactly this brain setup
but in the limit when we have sort of
a very large rank gauge group.
And so we have a dual description in terms of
a dual string theory or super gravity solution.
And so this is this work here
in collaboration with these people.
Okay, so just in one slide, what is ADS-CFT?
In fact, I think it was mentioned in the math talk already
but let's just briefly recap.
So if we have a super control field theory in D dimension
then the dual holographic dual setup
is a D plus one dimensional ADS base time
or in fact super gravity solution
which say for N equals four super N mills
happens to be this A is five times it's five
tied to B super string theory or super gravity.
What's the point of this is that we can sort of
access certain things here.
So at the weak coupling and here dual A
we could calculate these strong coupling.
So there's a strong weak coupling duality
but certain protected quantities we can calculate
in both sides and try to match them.
And this is what we'll do here.
So our setup is a two dimensional control field theory.
So we'd like to construct an ADS-3 solution
and the only sort of caveat is that we have this varying tau.
So usually we'd look at N equals four super N mills
in this case it's just to say vanilla
and ADS-5 times it's five tied to B super gravity solution.
Here we have this varying tau
and it'll actually enter into the construction
of the solution.
So what is the solution that we'll find
without writing down the metric and everything.
So it's an ADS-3 solution.
It has an S3 which captures the asymmetry
of the super control field theory.
So it's a zero four small super control field theory
where the asymmetry is exactly the SU-2 acting on the S3.
And then the compact space is an ellipticalabi R3
and that's ellipticalabi R3
where the actual type to B solution is a solution
ADS-3 times the three times the space surface B4.
For those of you who have seen these solutions before
if you put constant tau then in fact this reduces
to the standard ADS-3 times the three times K3 solution.
So here in the type to B super gravity nators
an actual non-trivial profile for the axodilatone.
The flux for the solution is given in terms
of the Kaler class of this base manifold B2 of this Kaler
and this of course if we require this elliptic vibration
to exist in bicalabi L the base B2 is constrained
to be a delpezo by FN or blow up the Roth
when it wreaks surface.
And as I said for constant tau this is just becoming
an ADS-3 times the three times Kaler B2 solution.
Now of course we don't write down an actual metric
for this elliptic vibration so in fact this will give us
some more sort of puzzle in a bit.
So the supersymmetry is realized in terms of
there's a two of the SU2R that acts on this S3
and this is exactly telling us there's an R-symmetry
which is SU2 of a small superconformal algebra.
We can also actually orbifold this S3
and retain a zero for supersymmetry
and that will correspond to having additional
for example for gamma this orbifold group being ZM
additional KK monopoles just as an aside.
Okay so we said we'd like to actually calculate
the central charges of this.
So the central charge is in this holographic setting
is actually computed in leading order in this large
parameter of the rank of the gauge group
by just a classical gravity result.
So the left and the right central charges
are just given in terms of effectively the volume
of the base times six N squared.
So the volume of the base now sits in this elliptic
vibration over here where now even though say
this is a smooth viruses model there are singular fibers
here so there are I1s located above this base.
So the metric on B2 is in fact singular.
So we can't really use the metric to calculate
this volume but we can sort of calculate this volume
by say just thinking of this in terms of a divisor
in this column now it's a section effect
and just sort of pulling back the killer form of the base.
So doing this in fact the leading order central charge
just becomes so this volume of B is then just given
in terms of half C dot C is the leading order
is three N squared C dot C.
Then so if we can continue the sub leading term
can be calculated from the couplings
the transverse couplings of the seven brains
the CL minus CR is then six N1 of B dot C
and if you all put this together then the holographic
result gives us this combination here
so the left and right moving central charges
are given in terms of this expression
and N squared is rank of the gauge group.
So now we can compare that with what we got
for the other side in the field theory
from the spectrum calculation of N equals four.
There we had the result for N equals to one now
and being this parameter here.
If you compare these two things these two leading orders
they agree and it's a leading order
there can be corrections now to the left moving charge
because there's this additional constant piece.
There are also other projects that we can do
like looking at the fiber anomaly polynomial
and comparing that to the anomaly there
which also then gives the same as this result
in the holographic setting.
Huh?
That's right, so there's actually,
okay I can show you the expressions
but there's a one over N thing that's,
okay we can discuss it in here.
So, right so actually one important thing
that Clay's actually pointing out here
is these behave like they scale with N squared
if you're familiar with the central charges
for the A to three times three times K three
these are very different sort of behavior to this.
Okay so that,
actually we've completely watched through this anyway.
So let me just summarize
and we've sort of found these two dimensional CFTs
from the right dimension reductions of N equals four
so bearing those with varying tau
which equals to U one
we can have sort of a complete description of all these
using either description in N equals four
or in six dimensions from the M five brain
by dimension reduction.
To go beyond this the only way at the moment
we know how to do this is to go to a holographic
dual description, calculate the central charges there
and that then conjecturally computes them
in for a UN gauge group at large N.
The nice upshot here is that we have
this new holographic sort of framework
where we have solutions of type to be super gravity
where the tau parameter, this axodilaton
is allowed to be have a non true profile
over the compact part of the space time.
In fact there are solutions also
which preserve zero comma two supersymmetry
of the same sort of type.
One sort of interesting math question
is what actually characterizes now
this duality twisted hitchen modulus space
that I talked about in the first part of the talk.
So we have not only contributions from this duality bundle
but also these modes from the defects.
So this combined system gives now some modification
of this hitchen moduli space
so it would be very interesting to understand
what exactly is that moduli space
is in fact a description using this action of the s duality
in terms of t duality on that target space.
Okay, so thank you very much.