Okay. This is better, right? Yeah. So this talk will be about some work, which is mainly
motivated by a paper of Nekosov, Rossley and Schadesvili in 2011. It's an attempt to understand
what they did in that paper in some sense. It's based on some longer collaboration with
Andy Netzke and with two of my students, Omar Kiry and Philip Ritter. And maybe the
simplest example to start out with is this one. So suppose we consider a spectral curve
geometry. Yes. So suppose that the spectral curve is given by this simple equation, which
is a ramified covering of degree 2 over just a complex plane. So we have two branch points,
two simple branch points, which are these positions. This geometry is very well to tell
us about the geometry, about the physics of a single n equals 2 hyper multiplet in 4 dimensions.
So that hyper multiplet is encoded in a spectral network. And the spectral network, which is
relevant to describe that hyper multiplet, is drawn here. So think of this white slide as the
plane. And here are the two branch points of the covering. Then the spectral network is defined
given two pieces of data, namely a one form. This is the one form where I didn't mention the
dz on the plane, as well as a phase. And that phase I've taken to be pi over 4 here, but it's
rather arbitrary. Given this data, you can draw a spectral network. And the trajectories of this
spectral network are given by an equation of this form. So here you see the phase theta and the one
form lambda. So how do you generate a spectral network? We'll start at one of these branch
points. And one of these branch points look at the tension factors at that branch point. Then
there will only be three tension factors for which this constraint is satisfied. So if I contract
lambda with a tension factor, that this is actually a real value. And these three tension
factors, they describe where these trajectories start, in which direction they start. Then if I
move over this trajectory, it's given by, again, this constraint. Now I want to associate a little
bit more of discrete data to the spectral network. First of all, I give these trajectories
orientations, and I orient them in such a way that they start at the branch point. Furthermore,
I associate a tuple of numbers 1, 2, or 2, 1 to each of these trajectories. And these tuples are
defined by having this constraint to be larger or smaller than 0. So for instance, I could call a
trajectory of type 1, 2 if lambda contracted with a tension factor smaller than 0, and I could call
it 2, 1 if it's greater than 0. So obviously, this depends on the trivialization of the spectral
curve and of a particular choice of branch cuts. Now, there is one particular one cycle in the
spectral curve geometry, which is the cycle which you can see by, which is drawn here, illustrated as
this red line. It's a one cycle on sigma on the spectral curve, and I'll call that cycle a. And
if I integrate my 1 for lambda around that cycle, I get 2 pi i times m, the mass of this hyper
multi-blad. Now, this spectral network is very special in the sense that this constraint is
satisfied. So that means that the period integral of lambda along this one cycle is real valued up
to this phase theta. So this is a very special feature of this spectral network. Oh, let's see.
This doesn't quite work. Let's go back. So I can see this spectral network by starting out from us. So
this is a very special spectral network. I can start out with a slightly less special spectral
network where I change my phase theta a little bit. And I see that I have the wrong thetas here. So
this spectral network is a spectral network which is drawn if the phase is slightly smaller than pi
over 4. And if I change the phase, and this should be theta pi over 4, then I find that two of these
trajectories start colliding. And I get what is called a saddle trajectory here. If I make my
phase... Sorry about that. Something went wrong. Okay. It's disappearing. If I make this phase a
little bit smaller, then you see that the trajectories are moving off in different directions.
So let me just skip that. Now, if I consider flat collections, so flat as two connections in some
middle E on C on the complex plane with some irregular singularity, then those flat collections
are not just characterized by the monodomy, but also converged on Stokes data. And here I choose
the type of the irregular singularity such that there are four anti-Stokes lines. Well, this is
an interesting... This defines an interesting modelized space of these particular flat
connections. And it has complex dimension two if I fix a trivialization of the bundle at some point
at infinity. So what I've done in this picture, I've taken a real blowup of the point at infinity,
which gives me this blue circle. And then I fix the trivialization of that bundle E at a particular
point at infinity. Then the monodomy space of these particular flat connections is of complex
dimension two. So given any of such flat connections, I can choose a number of flat
sections, which are asymptotically small if I go along the Stokes, these anti-Stokes lines. And I
choose my anti-Stokes lines to be the asymptotic directions of the spectral network. So I get a
section S1 here, S2 here, S3 here, and S4 here. Then I can do something rather interesting. Namely,
I can bring this flat connection into an abelian form. Of course, I can't bring it into an abelian
form globally, but you can do this in any of the patches of the complement of the spectrum
network. So how do I do this? I choose two sections. I choose a basis of sections in each
of these components, such that the connection is diagonal with respect to those sections.
And the way that I choose these sections, the ordering of these tuples, the sections in this
tuple are part of the framing data, the chosen sections at infinity, the Stokes data.
The ordering of these sections depends on what kind of type of walls I have surrounding this
component. So in each of these connected components, I choose a basis of sections.
So why is this interesting? So in fact, you can see that the gauge transformations which relate
these bases of sections in each of these patches, they are of a very special type.
They are unipotent. And the fact that they are unipotent implies...
Well, actually, so let me first say that what I really want to do is I want to do this
on either of these resolutions. If I change the phase a little bit, I make it a little bit less,
and I make it a little bit larger than the critical phase. So that means that I get some
extra connected components where I also choose a basis of these sections.
So having sort of abelianized, having found this sort of diagonal form of the flat connection
is rather interesting because it implies that I can lift this flat connection,
this non-abelian flat connection, to an abelian, to a GL1 flat connection on the spectral cover.
So let's introduce a couple of some cycles on this spectral cover. So first of all,
I can introduce... Well, I had already introduced the cycle that I called A. I can also introduce
a dual cycle, the cycle B here, which is actually non-compact on the spectral cover.
Still, it gives an invariant of this abelianized flat connection.
That is, I can compute now the hololomies of this abelianized flat connection, this GL1 flat
connection on the cover. So I can write down the holonomy along the A cycle as well as the
holonomy along the B cycle. And these holonomies, they are invariants, they are C star invariants,
and they determine coordinates on this monolive space of irregular flat connections that we
started out with. For later purposes, what we will actually want to do is we want to do this
for both resolutions. So if we deform the spectral network, if we change the phase a little bit,
so that it's less than the critical phase and that it's more than the critical phase.
So to each of these spectral networks, we get a set of coordinates on this monolive space of
flat connections. And I denote the X coordinate, which is obtained by holonomy along the A cycle
by X minus and X plus for the other resolution, and Y the holonomy around the B cycle for
either resolutions. So this turns out to be a special set of coordinates on this monolive
space of flat connections. In particular, if I take the logarithm of X and Y, then these are the
book coordinates. Now there is a distinguished set of flat connections on C,
which is locally given by equation of this form. So it's just a simple differential equation,
a harmonic oscillator. And this is called, this is the so-called subspace of obers.
So these as well to obers, they can always be brought locally in this form. So in this geometry,
the obers are just given by a one-dimensional family parameterized by this flat connection.
It might be interesting to note that these obers quantize in some sense the spectral curve
geometry. They're also known as as epsilon connections. So you could wonder what happens
if I restrict my spectral coordinates that I just defined to this subspace of low of obers.
And it's actually simple to compute. And you get the following formulas.
So the X coordinate is just simply related to the mass in this exponential way. And the Y
coordinate is related, can be written in the form of a ratio of two gamma functions.
Right. So what have we done? So we had this two-dimensional complex two-dimensional model as
space of flat connections, which is illustrated here by M. And then we have some one-dimensional
subspace of obers, which are denoted by L. Moreover, we've introduced a system of
double coordinates on this model space, which I've illustrated here by X and Y.
This space of obers is one-dimensional subspace of that space of complex connections. And in
particular, we have found a way of writing Y, the Y coordinate, in terms of X when restricted to this
to this subspace. That means it's a natural thing, a natural thing to do is to write down
a generating function, W, for this space of obers in this way.
Here I've written it out before. So what is called this generating function? It's a
generating function of this subspace of obers. And in this particular example,
we find a very simple formula for this generating function, which is given by this formula. This is
also sometimes denoted by an epsilon. And this is a very familiar function in the context of
four-dimensional gauge theories. Namely, it computes an effective twisted superpotential
of a free hyper multiplet. Here I should make one remark, and that is that this
effector, so it's just given by a one-loop term. I'll explain a little bit more about this later.
But this effective, this is just a one-loop term in that effective twisted superpotential,
which is not canonically fixed. So this is a particular choice of randomization scheme
to compute this effective twisted superpotential. And this is a choice which appears a lot in the
setting, for instance. It's in some sense a live view gauge for that one-loop term.
Well, what I've just shown you is, in some sense, a small toy example of the proposal by
Necrosov, Ross, Lien, Schade, and Fili. So what they say is that, in general, the effective
twisted superpotential for a theory of class S is equal to a generating function of obers
in a special choice of Darboo coordinates. And what they do in their paper, they define
these Darboo coordinates in a special case, namely, where k is equal to 2,
on a surface with four regular parameters.
So, sorry, let me say a few words about what this theory S means. So theory S means that we
get this theory by a particular compactification of the six-dimensional 2,0 theory on a Riemann
surface C, which might have punctures. And at these punctures, there are some defect data
associated, which I do noted by these d's here. Well, that defines a particular
n equals 2 theory in four dimensions, which is called a theory of type class S.
What's interesting about this way of realizing n equals 2 theories is that microscopic
properties of these theories are encoded in terms of this surface C. For instance, the UV gauge
couplings of the four-dimensional theory are realized as complex structure parameters of this
surface C. Moreover, this spectral curve comes up in the physics setting as describing the
Coulomb-Vecra of this particular theory of this theory of class S. And in general,
that spectral curve is described by giving a number of k-differentials on the curve C. So,
by specifying a number of k-differentials on C, we have specified a Coulomb-Vecra of this theory of
class S. The low-energies theory of the theory T is encoded in a pre-potential, which is defined
in terms of the periods, where lambda is the tautological one form in this spectral geometry,
or in other words, it's the Seibach written form. Now, this pre-potential f may be computed by first
principle from gauge theory in a certain deformation of the theory T. So, what we do is we, instead of
considering the theory T on a four-dimensional in just R4, we turn on what is sometimes called an
omega deformation, which is labeled by two parameters, two complex parameters, epsilon,
which correspond to isometries of these two planes. And if I then do equivary localization
in this geometry, that leads to what is very well known as the necker-soft partition function,
which can be written in terms of a deformed pre-potential, where the pre-potential is deformed
by these two variables epsilon one, epsilon two, where in the limit, if I send epsilon one and
epsilon two to zero, I recover the original Seibach written pre-potential. So, in this talk,
what I'll be interested in is turning off one of the deformation parameters, and instead just focus
on the geometry R2 epsilon times R2. And that theory preserves n equals two comma two,
Poincare symmetry in two dimensions.
So, and necker-soft and shutters really have analyzed this theory in a lot of...
And they have, in particular, they say that the infrared physics of the theory T epsilon
is described in terms of in terms of abelian twisted chiral multiplets, which are coupled
to an effective superpotential. And this particular effective superpotential can be
expressed in terms of the four-dimensional deformed pre-potential in this limit.
So, this effective twisted superpotential for this theory in this omega background
can be computed if T has a Lagrangian description. And in that case, it has the form,
it has a classical contribution, a one-loop contribution, and an instanton contribution.
So, what is interesting about these effective twisted superpotentials
is another conjecture of necker-soft and shutters theory, saying that this
the superpotential is the Yang-Yang function of the quantized Hitchen system. And the Alpers,
in that setting, they appear as the eigenvalues of that quantum integrable system.
So, this is another reason for being interested in that effective twisted superpotential.
Now, if the theory is, of course, as a superconformal, then
the defects are labeled by Yang tableaus. And the building blocks of these theories
are three punctured spheres, where all of these punctures are labeled by Yang tableau.
A Yang diagram, I should say. So, for k is equal to 2, there is a single building block,
which is just a three punctured sphere, which is labeled by three SU2 groups, the flavor groups of
the theory. And it corresponds to a half hyper multiplet in the trifinimental representation
of these three SU2 flavor groups. If you glue two of these building blocks,
using the, for instance, the plumbing fixture method with a parameter Q, then you get a Riemann
surface, a sphere is four printers, where one of the printers is labeled by this parameter Q.
And this describes a superconformal SU2 theory, which is coupled to four hyper multiplets,
where the coupling constant of that SU2 theory is given by this parameter Q.
And that is exactly the main example for a Decker-Sulfur-Rasly etchater-Sphealy.
So, what they did is they introduced some Darby coordinates on the modular space of
flat SO2 connections with regular singularities at these punctures. And they then noticed that,
they noticed as well that the braid of operas in this for this geometry is characterized by a simple
Fuchsian differential equation, which has regular singularities. So, it is just a regular,
well, it is just a generic Fuchsian differential equation on the four-punctured sphere.
Where this is also known as the Hundsoper.
So, now, a remark is that these Darby coordinates, they can just as well be found
by abelianizing by the abelianization method that I described in the first example.
And so, abelianization is a way of bringing flat connections into an abelian cage,
for which we need, the input data is just one spectral network. And in this case, to find these
Darby coordinates that Decker-Sulfur-Rasly etchater-Sphealy defined,
we need a spectral network, which is of this kind. So, all of the trajectories here
in this network are saddles. So, these are all trajectories where some trajectories have
collided. And in some sense, this trajectory is a double trajectory in the sense that it goes
both ways around that picture. So, again, this is a very special class of spectral networks
that you can think of. And these spectral networks, they are generated by what is called
strable differentials. And the strable differential, in this case, is characterized
by the property that if I introduce an A-cycle, which is wrapping around a pants curve. So,
suppose I choose a pants decomposition of my surface E. Well, this network defines a pants
decomposition of the of the four-punctured sphere, where there's one distinguished cycle,
namely this A-cycle, the pants curve. And the strable differential is characterized by the
property that if I integrate the corresponding one form along this A-cycle, then it's a real
number up to this phase theta. Here's a picture which shows you what happens if I deform my,
well, if I change my face slightly, namely if I change the face slightly, then I'll get this,
these trajectories which are winding infinitely many times around the puncture.
So, this is one resolution of the spectral network. It's of the of the strable network,
of the financial Nielsen network. There is another one where I change the phase
to the other side, so where I make it slightly larger than the critical phase.
And so, the claim is that these are book coordinates that Nacrosophorous VHSFIDI,
they are obtained as a spectral coordinates where I average over these two, over these two resolutions.
Right. So, our geometric recipe now tells us that suppose we consider the modelized space
of flat connections with regular singularities on a four-pintered sphere, then there is a one
dimensional, well, there is a, sorry, complex Lagrangian subspace of operas in this, in this
modular space, and it can associate a generating function to that subspace, and that generating
function should compute me the effective twisted superpotential. So, how do I verify this?
Well, I could compute the monodomy of a generic flat connection on the four-pintered sphere,
in terms of these double coordinates that I just defined, and this is actually quite a
straightforward, there's quite a straightforward way to do this using the abelization.
Then I can compute the monodomy representation for the Hern differential equation. This is
quite non-trivial, but it can be done in an expansion of Q, where I use perturbation theory
in my Q, and that means that if I compare the two, I extract the generating function of
operas in terms of these double coordinates, which I will then find in an expansion in this
complex structure parameter Q. So, here I've elaborated a little bit about how to compute
this generating function for the, sorry, how to compute the monodomy representation for the Hern
zoper. So, here it's crucial to know that if you send the limit, if you send Q to zero,
then these Hern zopers are just given by hypergeometric operas, which are of this form,
in which, for instance, came up yesterday as well. And for these hypergeometric operas,
it's very well known. An exact formula for the monodomy representation is very well known.
So, by perturbing, by using perturbation theory in Q,
we can get the monodomy representation for the general Hern zoper in an expansion in Q.
And as a result, we'll find this generating function. So, this is just to show you the
result that you get. So, the one-loop factor is in terms, is again written in terms of these
epsilon functions, which are the same ones as appeared in the first example. And,
well, one thing to maybe say is that, so, this is the one-instant contribution
to this effect of this superpotential. And it's not quite the result that we know. It differs
to the result that we know up to an A-independent term, which is just like the spurious factors,
which appear in the AGT correspondence. Right. So, most interesting,
the most interesting thing about this is that this way of computing
this effect of trusted superpotentials can be generalized to any theory of class S.
And, so, the way to do that, the way to find the generalization of the double coordinates
for any theory of class S, is to consider a higher-rank generalization of these
financial Nielsen networks. So, let me tell you a little bit about what it means to be a higher,
well, to be a higher-rank spectral network. So, to define a spectral network for a theory
where K is larger than 2, we need to fix a tuple of differentials. So, this tuple of
differentials that defined you, for instance, the spectral curve, as is actually written in here.
And we also need to fix a phase. And then the trajectories of the spectral network,
they are now defined as this equation, where lambda i and lambda j are the restrictions
of this tatological 1 form to i and j. So, these networks, for higher-rank, they are much more
complicated than the ones for K is equal to 2. They are labeled by a tuple i and j, which is
fixed by this condition. And, in this case, for K is larger than 2, they can, for instance,
cross each other, new trajectories can be born, etc., etc. But, again, there is a special class
of spectral networks, which are defined by this generalized treble constraint, as I call it.
So, for any spectral curve geometry, you can fix a number of a-cycles, and you can require
these period integrals to be real up to this phase theta.
So, let me give a number of examples of this. So, maybe the first sort of generalization of the
ASHUT 2 geometry is where the ASHUT 3 geometry, where I fixed the... So, this is for K is equal
to 3, where I look at the 4-punctured sphere. But now, in the case K is equal to 3, there are
two types of young diagrams, which correspond to two types of punctures. So, what I could do is,
I could choose a U1 flavor symmetry at two of these punctures, and an ASHUT 3 one at the other
two punctures. This corresponds in four dimensions to considering an ASHUT 3 superconformal gauge
theory, which is coupled to six hypermultiplads. In that case, the constraint that I showed you
to define these financial Nielsen networks gives examples of like this. So, this is an example
of a higher-rank financial Nielsen network on the 4-punctured sphere, where we have chosen two
minimal punctures, which were labeled by the U1, and two maximal punctures, which were labeled by
the ASHUT 3. So, examples of this type have also popped up in work by Longy and by Longy and
collaborators. So, if you want to generalize the computation of the superpotential to the
setting, then we also need to know what is the particular family of operas that we should
consider. And these operas, as you might guess, they are generalized versions of Hund's operas,
although I've never quite seen them being written down in the literature. So, again,
they have the same property that if you take the limit of Q going to zero, you get a generalized
hypergeometric equation. So, how do I find them? So, start out with just a generic function
over of degree 3 on the 4-punctured sphere, and then I can restrict those differential equations
in a certain way at the minimal punctures. So, what I do is I write the differential equation
in this particular form, and then I require that these coefficients, P1 up to P3, are analytic
around the minimal puncture, which is here, denoted by this C*. But what that does is it
enforces that two solutions at the minimal puncture are holomorphic, and therefore,
that the monodomy around that puncture is diagonalizable. So, then I have all the
ingredients necessary to generalize Nacrosar-Frosley and Schottes-Fielly, namely, I can
similarly, as before, compute the generating functions of these generalize Hund-Obers,
and again, it gives the known result. So, what is interesting is these theories give
computations, which are in a perturbation theory in Q, but they are exact in this omega-deformed
parameter epsilon.
Now, of course, not all theories of class S have a simple description like that,
and perhaps the simplest example is where my Riemann surface is a 3-punctured sphere,
but now all these punctures are labeled by issue 3, flavor symmetry groups. And this geometry
corresponds in four dimensions to what is known as the E6 Miele-Hahn-Nimshansky theory,
where the E6 comes in the infrared. So, these three issue 3 symmetry groups in the infrared,
they enhance to this E6 symmetry group, and that's why it's called an E6 theory.
Right. So, this theory is intrinsically coupled, so there's no way to compute, for instance,
this necker-soft partition function.
Let me say a little bit about the spectral curve geometry.
So, in the massless limit, where we turn off all the masses,
the differentials are simply given by phi 2 is equal to 0, and phi 3 is proportional to U.
So, there's one kilometer per meter U in this theory, and in that case, it's given by a
covering over this 3-punctured sphere by Torres, which is not ramified, it's ramified over these
punctures. Well, this is illustrated right here. So, here we have the punctures. So,
the branch points in that massless limit, they have really moved to the punctures,
so two branch points of different types, they move to the punctures. And in this geometry,
there is a sort of canonical way, a canonical basis of one cycles, which is given by gamma 1,
gamma 2, and gamma 3, which are defined. So, for instance, gamma 1 is defined on the spectral
curve by running over the second sheet, then going on to the third sheet, and then coming back.
Of course, these cycles, they add up to a trivial cycle.
Now, financial Nielsen networks for this spectral curve geometry, they are labeled
by an A cycle, and in this case, the A cycle is labeled by two parameters, P and Q,
as I've written down here. So, we'll get a financial Nielsen network for any choice of P and Q,
which is determined by this constraint. So, the simplest network is when
P of P, Q is equal to 1, 0. In that case, the network is just basically, well, it's just a circle,
where each of the strands of the circle is labeled by a different tuple of integers.
And if I deform my masses slightly, then I'll get this picture to the right.
So, then at the punctures, there is some additional geometry that pops up.
Just for curiosity, it's nice to draw these for other tuples, P comma Q, for instance, for
P comma Q is 1 comma 2, I'll find a star-shaped network of this kind. For P comma Q is 1 comma 3,
I'll find a much more complicated shape, and if you go up in P and Q,
then you'll find nicer and nicer geometries. So, you can wonder what happens if we apply
our ablinization method to these spectral networks. So, what kind of double coordinates
will we find? And this is actually a very interesting topic, which I don't have a lot
of time to expand on, but what I want to say is that it's, for instance, very simple to find the
monodomy representation in terms of these spectral coordinates. So, this is for the simplest spectral
network, where X and Y are the corresponding double coordinates. C is just given by this
function, and actually, it's kind of mysterious why there is a square root of 1 minus 14 X plus
X squared appearing here. So, if you have any idea, we would love to know. So, these are the
spectral, these are the monodomy matrices in the limit that we send. The mass is to zero,
but equally, you can find them where you turn on the masses.
Right. So, now we've done one thing. Namely, we've introduced these spectral coordinates
for this particular E6 theory, but we still need to have a look at what the family of
Obers is. Well, that is rather simple in this case, because they are just the most general
fuxian differential equations of degree 3 that you can write down on this 300 sphere.
However, unlike for hypergeometric and generalized hypergeometric differential equations,
there's no way to compute the monodomies analytically in terms of known functions for
these particular differential equations. Well, that seems to be a slight problem, but
fortunately, there is a good solution. Namely, what we can do is maybe we shouldn't try to do
everything analytically in terms of epsilon, but rather, we should try to compute these
generating functions in an expansion in this epsilon parameter. And then, there's a really
nice feature of the spectral coordinates that this can actually be done. Namely, if we compute the
spectral coordinates for an Ober, then it's known that it has an exantotic expansion, so this should
be a fiddle, actually, an exantotic expansion of this particular form, where if you write down
this function, that is an exantotic solution to the Ober differential equation.
So, what I just did is I wrote down an exantotic expansion in epsilon of these spectral coordinates.
If a Borel sum with respect to the given spectral network, then you find exact results for these
spectral coordinates. And so, what's interesting is that these exact results are, of course,
very dependent on the choice of spectral network, but the exantotic results are not quite.
So, that means that, for instance, for this E6 theory, we can compute this effective
tested superpotential in an analytic expansion in epsilon. And now, we need someone to tell us what
to compare to. As a last thing, let me say something about how we motivate this correspondence
from a physics perspective. So, to do that, we need to go back to the geometry in which this
effective tested superpotential appeared. And now, we write it down in a little different way, namely,
we choose a metric on the plane such that this geometry R2 epsilon is actually cigar-like.
So, the metric is given by this formula, where f of R approximates some radius
when little r is going to infinity. In this background, this n equals 2,2 supersymmetry is
still preserved, and in particular, the IR physics is still preserved by the same effective
tested superpotential. Now, there is a really nice argument that this omega deformation could
actually be undone away from the tip of the cigar by a field redefinition. So, I've illustrated the
geometry like this. So, in this geometry, we've undone the omega deformation around infinity.
So, suppose let's go back to this geometry. So, let's consider the theory T in the background,
where we have just a cylinder times R2. Then, we very well know that if we compactify the theory
T on a circle, it has a description in the infrared, but in three-dimensional sigma model
into the H2 model space on C given by these equations. So, we also very well know that this
H2 model space is hypercalor with respect to some parameter zeta, and we also know that if zeta is
equal to, say, zero, then this model space has a description in terms of Higgs bundles, and it has
a structure of an integrable system, where each point here is a point in the Coulomb branch,
and it's labeled by a tuple of differentials. Now, zeta is not one of these two special points,
then this H2 model space has a description in terms of a model space of flat connections,
which can be constructed out of the H2 data in this particular way, and we know that there is
this distinguished complex Lagrangian of operas in that model space M. So, here we start getting
back to our original setup, because what Nakrasoff had written argued is that if we go to this
background, where we have end-on this omega deformation, then the IR description is actually
in terms of a three-dimensional sigma model, where we have an additional brain wrapping this space of
operas. So, this doesn't completely satisfy yet, because we also have to choose a boundary condition
at the infinity of the cigar, and such a boundary condition is exactly labeled by a phase theta.
It's labeled by a choice of a three-dimensional n equals 2 subalgebra. So, this is where the phase
theta of the spectral network is eventually going to come in. So, our claim is that generic
infrared boundary conditions in this theory are labeled by a choice of differentials,
and a choice of phase theta, together with a choice of polarization of your spectral curve,
so a choice of a cycles and b cycles. Namely, if I've chosen such a boundary condition, such an
infrared boundary condition, that defines me a set of spectral coordinates, which corresponds,
which you can find by abelianizing with respect to the spectral network, which is defined by this data,
and I can use those spectral coordinates to define a boundary condition.
Namely, the boundary condition at infinity, I just defined, corresponds to setting this y-coordinate,
the spectral y-coordinate, equal to 1. Now, I've ended up with a three-dimensional sigma model
with two types of boundary conditions, a boundary condition at the tip given by this brain of
obers, and a boundary condition at infinity by setting a spectral coordinate equal to 1.
If I then reduce to just considering the zero modes in that three-dimensional sigma model,
the zero modes along the C car, then it's known that this gives a two-dimensional sigma model,
which is characterized by a superpotential, which is given by this generating function of obers.
So, you might wonder, so where come, where do these financial Nielsen networks come in?
So, so far, I've just explained to you about IR boundary conditions, but if we compare the
Necrosoft-Shatasvili computation to a sigma model computation, then we should say how
the boundary conditions in the Necrosoft-Shatasvili computation correspond to
UV boundary conditions in the sigma model setup. And so, our claim is that this can be obtained
by a UV-IR duality wall, which is of the type discussed by Cukati, Cordova, and Vafa, as well
as Dimov, Gail, and van Leeuwen. So, but more specifically, what we do here is we define a duality
wall by setting the real part of the central charge of the four-dimensional central charge
to a constant up to a phase theta, and we let the imaginary part run with a parameter t, which can
be identified in a theory which has a Lagrangian description with the log of the, log of the
coupling constant. What this means is that in the IR, the theory is going to be characterized by
light particles which satisfy this constraint, which are really trapped on that UV-IR duality
wall, and that is precisely the condition of the strable differentials. So, our claim is that a
natural set of UV boundary conditions in these theories of class S is given by the strable
differentials. In particular, in a weekly couple description, what is this? It exactly traps the
W bosons in the IR theory. Let's see. So, what I'm saying here is using this UV-IR duality wall,
we construct, so we've constructed a set of UV boundary conditions which are labeled by these
higher financial Nielsen networks, and so that means, let's see, well, so,
let me say, quickly say something, what happens if we don't choose a spectral network which is
of this financial Nielsen type, because that also has a physical interpretation. Namely, we can
think about walking along a path in this space of IR boundary conditions where all of these points
are labeled by spectral networks, so what will happen is that the spectral network will undergo
a number of flips and other transformations if you walk along this path. In terms of the
four-dimensional theory, we can think of walking along this path in terms of coupling the
four-dimensional theory to some 3D boundary, 3D theory on the boundary, and using the 3D 3D
correspondence, you can in particular interpret the flips that the network undergoes as coupling
to a three-dimensional billion theory which corresponds to this tetrahedron. So, our claim
is that the generating function of operas in these other coordinates now corresponds to the
effective two-proportational of the total theory, where we don't just consider the two-dimensional
theory but also the coupling to the boundary. So, I think that's where I want to end my talk.