So I realized that I hardly said

anything yesterday uh in two hours. Very

impressive. Uh that's what tenure is

for, you know. So um uh but um uh but

actually I realized that it presents me

with a with a good opportunity. So um

yesterday I was planning on after saying

my very little um to say something uh

and I was going to first uh as I

mentioned first talk about the

motivation that led up to the ample say

a few things about it and talk about

some open problems and uh that was to

sort of piggyback off the beautiful

lectures that we had yesterday on on uh

on on related topics. um uh and then for

my uh second lecture today's lecture I

was going to tell you about the uh uh

trace by cube theory and soahedra and so

on. Now um in fact sort of logically the

story of trace 5 cubed and the

sosahhedra and the connection to curves

on surfaces is simpler is quite a bit

simpler is somewhat more mathematically

standard and much better understood by

now than what what I definitely feel is

a much deeper and more interesting story

related to amplitra. Okay. Um but then

for that for that reason uh and

especially since we just had Caroline

this morning I'm just going to reverse

the order of presentation which now

makes a lot more logical sense. uh just

to start with a simpler story where

things we are can be understood in a

much more uh straightforward way and

then tomorrow I'll tell you something

about the open problems related to the

ample and then if we get there I'll tell

you something about the the large end

story that I uh advertised as well. So

um so that's what we're going to do uh

I'm really going to uh begin. So uh so

so before getting into a really

systematic exposition of the story, what

what we're what we're going to try to do

today is understand this trace 5 cube

theory. um that uh that's been

introduced I I'll I'll I'll introduce

again um and ultimately we'll understand

it not just at tree level but to all

orders in perturbation theory because

there's going to be some way of thinking

about it uh has nothing to do with

plarity non-planer everything but the

color is important but other than that

to all orders in perturbation theory um

and uh at least sketch uh for you uh how

it's related to uh very simple pictures

of curves on surfaces and some very

interesting combinatorics associated

with uh uh uh with these curves on

surfaces. This is going to connect to uh

uh field theory trace cubed amplitudes.

It you can think of it as a bottomup

discovery of string theory trace cubed

amplitudes at least in some uh in some

uh in some limits. In fact, we run into

a larger collection of objects than than

standard string theory. Standard string

theory will be contained in as sort of

one corner, one special uh sort of

choice of what this uh uh what this

object looks like. But you're allowed to

work with a larger world of ideas. But

as we'll see, you can sort of get to

string theory without uh uh six months

of a uh or a year of a graduate course

in uh uh in the subject. Okay. So that's

that's what what what we're going to do.

But what I want to I want to begin

before we get into lots of you know

definitions and for I actually want to

begin with the uh with the example that

uh that Carolina talked about uh quite a

lot of this pentagon. Okay. So here

we're talking about the uh n equals 5

point amplitude.

Um and as uh as Carolina explained it's

associated with this picture of the uh

this particular realization

of the isocahedron. I'll also draw in

this case the mesh that uh she drew for

us. Okay. So the amplitude is a function

of these x variables as she explained.

So uh this would be x24, x25 and x35.

And then we also have these meshes uh

c13, c14

uh and c2 uh and c24 with the formula

that she gave that I'll write down again

uh in a second. Okay. So um uh so we

have uh we have uh in x13

uh x14 space we have this picture

of the socedin where that is here is c13

this here is 3 c13 plus c14 and this up

here is c14 plus c24. Okay.

Okay.

And so um what what we're going to see

in in the rest of the lecture for this

example and many more besides is where

this all comes from. Okay. So so so

Carolina motivated it. Um but at some

points there's a few little you know out

of somewhat out of the blue steps like

who told you to draw this mesh? Who told

you to write the equation x plus x - x -

x equals constant? All of those things

sort of seem fairly simple and natural.

Uh but uh but they're a little bit

pulled out of apparently nowhere. So um

so I'm going to talk about an alternate

point of view for where these things

come from which is 100% rational. Okay

is 100% ma rational but will have a

couple of magical features in it

nonetheless. Okay so uh but the moves

that you start making are 100% rational

but um to give you an idea of uh what it

will be about uh I just want to

illustrate some of the things that we'll

talk about already in this example.

Okay. So essentially what we're going to

be talking about is a duel of this

picture. Uh and um so uh uh uh

everything that I will say later also

has an avatar directly in the language

of the original polytopes. But we're

sort of going to focus on a on a duel of

this picture. And if I start here, we're

going to look at the sort of we're going

to start by looking at the inward

pointing normals

um to the facets of this uh of this

pentagon. Okay. So that's of course a

very standard way to see a dual object.

You have a polytope, the dual polytope,

the vertices of the dual polytope or the

facets of the original polytope and so

on. So it's very reasonable to look at

those uh inward pointing normals. And if

you remember this was the face where x13

goes to zero. This is where x24 goes to

zero. Uh sorry x1 uh sorry this is where

x13 goes to x14 goes to zero. This is

where x24 goes to zero, x25 goes to

zero,

x35 goes to zero

and x14 goes to zero as you also see

from the uh sorry x13 goes to zero as

you see from the picture. Okay. So again

this corner is where both x14 and x13 go

zero. So you can think of this line as

associated with this partial

triangulation of the uh underlying

pentagon. Just repeating things Carolina

said, right? This is associated with

this partial triangulation.

And so this vertex is where the the two

of them come together. They don't cross

each other. That's good. So they come

together there and we get the sort of

134

complete triangulation. And the same

thing happens uh as we go around. Okay.

So all of the things that meet

correspond to chords that don't cross

and the vertices where they meet are the

complete triangulation that uh goes goes

along with that. All right.

So uh we're first going to just draw

those draw those arrows. And if I draw

those arrows I get the following

picture.

I'm just drawing the normals.

Okay.

Now I've chosen to draw the normal so

they're all like 01 vectors, right? So

they have zeros or ones. We'll we'll

we'll come come to that point later. Uh

I mean when you just draw normals

there's no meaning to the magnitude of

the enormal but the but the but these

mag it's very reasonable thing to do and

it'll have an explanation later. Okay.

But here this would be associated I

won't keep writing X. That will be

associated with 13. Right? That's this

inward pointing normal. That's 13. This

is 1 14

2 4 25 and 35. Okay.

And so what we have here is a picture

where now the kind of plane is divided

into these cones. Right? The plane is

divided into these cones.

And this is the normal fan of the

corresponding polytope. The dual these

cones are just associated with the

vertices of the polytope in the obvious

way. Okay. So again this cone 1314 is

associated with this vertex, right? It's

the two uh it's the it's the two facets

that meet at that at that vertex that's

associated with this this this cone. So

again the point is that this 1314 this

region is now associated with a vertex

and that's this vertex right that

corresponds to this this triangulation

and the cool thing in this picture is

that the plane has been divided into

sort of five regions and each region is

uniquely associated with a diagram right

so that's just the dual statement that

we had an object whose vertices were

capturing all all of the diagrams okay

so so far I'm just resetting uh some of

the things that Carolina told you in

this uh in the language of this uh uh

dual uh fan. Okay, so this sometimes

called the normal fan of the uh of the

polytope.

Okay, so again the most basic things are

that uh uh are that the cones of that

that we have cones they cover the whole

space and that each cone can be

associated with one of the diagrams.

Okay. Now, something else that Karolina

told you is that this associally

presented as a Minkowski sum. Okay,

that's a remarkable fact. Just to sort

of pause and back up for a second,

there's just the zeroth fact that the

combinatorics of diagrams of

triangulations of a polygon are captured

by the faces of an object. Okay, that's

Stashev's realization. Amazing. That's

already remarkable, right? If you like

that's all about fineman diagrams

related to fineman diagrams

factorization you know all of those

words are related to this fact

the fact that the socied has a

particular realization not a random one

you know you can draw it in lots of ways

I could draw this pentagon with sort of

sides uh you know tilted with random

angles and then what I'm saying would

not be true okay um but the fact that

there's a particular realization of

these with parallel sides you see these

parallel sides mean a lot in this

picture. Okay. Um uh tells us that uh it

can also be represented as a sum of

simple pieces. Okay. The fact that it's

representable as a sum of simple pieces

is remarkable is is an extra fact on top

of the stash effect that that the

combinatorics is captured by polytope.

And as Carolina explained in her talk,

this extra fact has extra implications

for the amplitude that we didn't know

about before. Okay, this extra fact

makes it easy to predict where the

amplitude vanishes uh and how when you

go close to the places it vanishes, it

also factorizes in a way that's uh not

at all obvious uh if you uh know

something about fineman diagrams. Okay,

so this is an extra interesting fact

that the that the that the association

is naturally Mowski sum. How is that

fact reflected in this picture? So let's

remember what the sum ends look like.

You know, we had a sum end that looked

like this. We had a sum end that looked

like this. and we had a sum end that

looked like this.

Okay? And those of you who know about

Newton polytopes and tropical

tropicalization and so on are going to

know exactly what I'm about to do now.

Okay? But those of you who don't, there

is something we can obviously do with

this uh picture as well. I can look at

the normals to each one of these pieces,

right? So the normal to this piece looks

like this

in the same way. And the normal to these

piece will look like this. You know,

look, I guess like this if I'm doing

inward pointing.

Okay.

So, notice that if I just draw these

guys, well, if I draw uh this one, it

looks like this.

Okay. If I draw this one, it just looks

like this.

And if I draw the other one, it looks

like this.

Okay.

So the fact that you can uh build the

associ

sum in the language of the normal of

this normal fan is reflected in the

following cool thing. This has five

pieces. So I've made kind of a

complicated space, right? That's uh

that's divided into five pieces. But I

can make those five pieces by laying on

top of each other these three pictures.

Okay, I have this picture and then I put

this picture on top of it and I put this

picture on top of it and putting all the

pictures on top of each other uh gives

us this division of the space into the

five regions each one of which

corresponds to uh one of these vertices

uh triangulations

diagrams all the different ways we could

we could talk about it right so that's a

sort of a well-known statement that the

Minkowski sum at the level of the

polytope is reflected ed as a common

refinement of the sums at the level of

the fan. Okay, if I look at each sort of

element and I look at the the normal

vectors for each element, I can build

the complicated uh fan by just putting

them stacking them on on top of each

other. Okay.

Okay. So that's uh that's and that's

that's how this fact is represented. And

now we're going to try to uh uh uh

understand this this fact better.

Now um

another two things to say in this uh

example

you know if you're given these sumans

and let me call this direction I'm just

going to give these directions names

sort of 13 and 14 okay as we see in in

this uh picture just associated with the

the uh the the the normal quadrant here

um it's of course natural to associate

polomials with each one of these

pictures. Okay. So I'm going to

associate with this picture I'm going to

associate the polinomial 1 + y14.

Okay. With this picture I'm going to

associate the polinomial 1 + y13.

And with this picture I'm going to

associate 1 + y14 + y14 y13.

Okay.

These are polomials which are a sum of

monomials and every monomial just

records the uh the corresponding uh

vertex of the picture. Okay. So uh this

vertex is 0 0 this is 01 and this is 1

comma 1 and so this polomial is 1 for 0

0 y14 for 01 and y14 y13 for 1 one.

Okay. So those of you who know about

Newton polytopes uh will then say that

the Newton polytope of this polomial is

back to uh this uh simplex. Okay. The

Newton polytope of this guy is back to

this uh of this is back to this one.

This guy is back to this one. All right.

Okay.

um

something that we can do with these uh

uh polomials of course none of this is a

coincidence but I just want to

illustrate in this example before we say

everything in general

and that's the revolution from this

morning okay so

erase this

I'll leave while this is drying.

Something that we can do with these

polinomials is we can imagine what what

would happen if I mean they're all sort

of plus signs here. So it's very natural

to imagine that the y's are positive. So

let's imagine that y13 and y14 are

positive. Remember I pos I promised you

positivity is everywhere in this story.

And here it is at the very at in the

very most uh basic place.

Um well for example we could uh do this

by writing y13 is e to the negative t13.

This is a pure convention the minus sign

here. Uh but this convention we like to

use is e the minus uh t14. So now t13

and t14 just vary over the entire real

line. Okay. though.

Um and so a kind of a natural question

when when you when you look at these uh

polomials is to ask in T13 T14 space as

you go off to infinity in all possible

ways you know what do these polomials

look like as you go off to infinity in

different ways different terms in these

polomials are going to dominate

and uh so they will simplify. Of course,

this is the entire uh this is the

beginning of thinking about uh tropical

geometry and tropicalization.

But uh I just want to do it in this in

this example. Okay. So, so uh so

remember the logic is we got these sum

ends. It's natural to associate these

polomials with the sum ends and now

we're going to just ask what they these

things are not linear but we're going to

ask what they look like as we

asmtoically go off to infinity in the

positive domain where the y's are

positive. And so for example, what does

1 + e the minus t14 look like as the t's

go to infinity? This is going to look

like e to the max of zero and minus t14.

Right? That just says either that

dominates or that dominates. And so

whoever dominates, we're going to get uh

we're going to get that guy, right? Um

let's do the more complicated one. If we

look at 1 - eus t14 plus eus t14 eus t13

this is going to go to e to the max

well either one dominates or if t14 is

minus t14 is the biggest that will

dominate or if minus t14us t13 is

positive that'll dominate.

Okay. So there's really nothing to do

here other than just put maxes

everywhere. All right.

Okay.

But now if we

look at this uh picture, let's let's

let's do the uh let's do that uh simple

one first. This uh 1 + uh uh 1 + uh y14.

If we just ask um where does uh so uh so

1 + y14

is going to uh e to the max of zero and

minus t14.

And so this in t13 t14 space is not a

linear function but it's close. It's

peacewise linear. Sometimes it's going

to be zero sometimes it's going to be uh

negative t14. Okay. And so this has a

domain of linearity in positive T14

space up here in positive uh uh uh sorry

I I'll draw it like this.

[Music]

In positive T14 space this is zero and

in negative T14 space it's minus T14.

Okay,

so this thing is linear down here is

negative t14 is linear up here and what

separates them is exactly the fan that

we associated with this little factor.

Okay.

All right. If we do the more complicated

example.

So if we look at 1 + y14 plus y14 y13.

So again this goes to e to the max

of 0 minus t13 minus t13 t14 minus t14

minus uh t13 t14 t14 t13. Okay, same

thing. Um then it's easy to see that

this thing is linear but it domains of

linearity are given by exactly that

picture.

Okay. So in this example

um

down here

down here T14 is negative T13 is

positive. So minus T14 is the biggest of

all of these guys. Okay. So down here

this is minus T14.

Here

everything is so negative that zero is

the max and here minus t14 minus t13 is

the max of everyone.

Okay.

Okay. So we have a few now we have a few

ways of uh we have a few things that we

can associate with this picture of the

Minkowski sum. There is the there's the

uh there's the fan associated with each

one. There are these polomials

associated with each sumand and there's

a tropicalization of the polomials uh

associated with each sum end. Okay. So

these are sort of three natural things

that we can associate with the uh with

these sums.

And these last points uh the the

polomials

and these tropicalizations

uh turn out to directly connect

these pictures to amplitudes.

Okay. And in fact let me just uh write

down the amplitudes that they uh that

they are associated with.

disguise

sorry.

So if I begin with the polomials

I write down an integral.

So this is I'm going to call this a5

going to be a five uh point amplitudes

but we'll interpret in a second. First I

write down an integral from 0 to

infinity with a form that's one of these

sort of infamous by now dlog forms. Okay

so our form is going to be a dlog form

on y13 and y14. Okay so y13 y14 the sort

of the variables here.

UM NOW THIS INTEGRAL BY ITSELF is

divergent obviously very divergent right

it's logarithmically divergent at zero

and infinity so are we going to repair

this integral to make it finite what

we're going to do in the neighborhood of

zero they're going to be factors that

are powers of y13 so y13 is going to go

to some power what power could it be

well the most natural thing is the x

that goes along with y13 and y14 will be

raised to the x14

right we have now successfully repaired

the singularities near the y's equals z.

But now these things are going to

diverge as y goes to infinity. If the

x's are positive, we fix things near x

equals z. Now they're going to diverge

as they go off to infinity. Right?

So to make them die in infinity, there

have to be some other factors that are

going to suppress things at infinity.

Okay? But in our problem, we now have

three interesting polomials at our

disposal. Right? We have these

polinomials associated with the sum. So

we're just going to write them down. We

have the polomial 1 + y14.

But remember if we go back to the

picture that uh uh Carolina drew this

mesh

this uh this was associated with that

sum end that was anchored to this mesh

point. Okay. So we're going to write 1 +

y14 to the power of the mesh that goes

along with it to the power of negative

c24.

This guy was this one. So we're going to

write down 1 + y13

to the minus uh c13.

And this was the uh triangle. So we're

going to write down 1 + y14 + y14 y13

to the c14.

Okay.

Now, notice that if the C's are positive

and the X's are positive, this

manifestly converges.

Just mindlessly converges, right?

Because we've saved it near X equals Z.

Well, actually, uh, sorry, doesn't so

mindlessly converge. Um, we've saved it

near X equals Z. It has a chance to be

saved near uh Y goes to infinity, but

obviously these C's can't be, for

example, you know, 10 the minus 100 if

the X's are 10. Okay. So, so the X's,

you know, have to have some positivity

property relative to the C's in order to

ensure that this converges everywhere.

Okay. And you can very easily check in

this example and it's true on very

general grounds that it's precisely when

X lies inside the pentagon as defined by

these mesh equations that this integral

is convergent. Okay.

All right. So, the integral is nicely

convergent. We have to have the x's are

positive, the c's are positive. On top

of that, we have to have x plus x - x

all of that stuff. Okay, that uh then

the integral is uh is uh then the

integral is uh the integral is

convergent.

>> Now this object yes

>> is the statement that if you move

anywhere slightly away it will diverge

or that this is a good this is one

positive

>> no no it'll diverge. I mean these

integrals will diverge if you're outside

the polytope they'll just diverge. So,

so the sort of polytope is literally the

the same as a domain of convergence of

the integrals. Okay.

Okay. Now, this is a slightly fancy

looking object. Okay. This is a slightly

fancy looking object. In fact, the x's

have units. They're momentum squared.

So, to make this a unitless statement, I

need to put something here that has

units of one over momentum squared or

length squared. So, these things uh

typically are called alpha prime.

Uh so alpha prime is just something

there to to to make this have correct

units

but um and we'll we'll talk about it in

a little bit more detail later but this

is in fact the five particle string

theory amplitude

right

[Music]

so it has this interesting extra

parameter alpha prime in it this is a

very interesting function of the x's is

a quite complicated function of the x's

that's meamorphic uh Uh so there's a lot

to say about string amplitudes we'll not

be talking about in uh uh in this

lecture but what I just wanted you to

see very quickly right away is that

starting from the picture of the

association of the Minowski sum the

Minowski sum is about string theory.

Okay, it's a Minowski sum picture that

immediately tells you how to write down

a string amplitude. You just take the

sum ends and shove them to the power of

what what appears in front of them. All

right. And this is something that is

going to give you uh uh string

amplitudes. Okay.

Now we can ask what happens uh to these

functions uh in the limit where the x's

where this combination alpha prime x is

tiny. Okay. So alpha prime has units of

a length. So saying that alpha prime x

length squared. So saying that alpha

prime x is tiny is the same as saying

the momenta are very small. Okay. So

this would be a low energy limit. Uh if

you're a string theorist, you would say

that in street theory, we have the

particles that we see and then there's

massive cousins, string excitations of

these particles whose mass is set by

that parameter alpha prime. If you go to

very high energies, you'll see all of

them. Uh uh very short distances, you'll

see the little wiggling strings. But at

very low energies and very long

distances, you're not going to see them,

and you're just going to go back to

seeing the good oldfashioned particles.

Okay,

so how can we study the very low energy

limit of this uh amplitude? Well,

clearly as alpha prime goes to zero,

this amplitude IS ACTUALLY GOING TO BLOW

UP. IT'S GOING TO blow up with some

power of 1 over alpha prime. After all,

the whole point of those exponents was

to regulate these integrals. Okay,

in fact, if we look at this integral,

NAIVELY GO LIKE ONE OVER ALPHA prime

squared, right? because there's going to

be a sort of a one over alpha prime from

every one of these uh uh integrations.

Conventionally, for that reason,

sometimes people multiply this by an

alpha prime squared out in front just so

we have something which is well defined

as uh uh as alpha prime goes to zero.

But regardless of that cosmetic thing,

um it's going to be dominated when alpha

prime is small. This is going to be

dominated by the extreme regions in this

y space either when the y's go to zero

or the y's go to infinity precisely

because we're barely regulating it.

Okay. So that's why as alpha prime goes

to zero.

Um

so this is a string of alpha prime.

But as alpha prime goes to zero

uh a string of alpha prime is going to

descend to a field theory or particle

theory which doesn't have an alpha prime

in it. Okay.

And what does that descion look like?

Well, I mean, if I go back to this

integral

and I use these variables where y13 is

eus t13, y14 is e minus t14,

then uh

then this is the integral over all t.

So, integral dt13 dt14. Okay, the y13 to

the minus x13 is going to look like e to

the minus uh I'm going to now put alpha

prime to one now since uh um uh it's

going to look like e the minus t13 x13

e to the minus t14 x14

and then e the minus c13 max of 0 and

minus t13

e to the minus uh uh c24 4 max of 0 and

minus t14 and then our favorite big guy

e to the max c14 0 - t14 minus t14 - t13

sorry for the small writing here right

so this is what this is what the uh

field theory amplitude becomes I mean

what's sorry this is this is the full

amplitude I haven't taken any limit here

right this is the uh full amplitude I

apologize but this is just the full

amplitude but now the point is that that

in the limit that we're talking about

this integral will be dominated by the

t's going to infinity. Okay. Oh, I'm

sorry. This is the limit. Sorry. Uh so

so what what uh in in taking the limit

the integral is dominated by where t's

go to infinity and where the t's go to

infinity we precisely replace the

polomials by these e to their

tropicalizations.

All right. So this is a different

expression than the expression that I

wrote down to begin with. The string

amplitude is some sort of fancy curvy

integral, right? With polomials and so

on in it. This is a simpler integral,

right? Again, now what's inside the

exponential is peacewise linear. Is not

linear, but is peacewise linear. Okay,

is this clear so far? Any any questions

about this? So, this is what the low

energy field theory limit looks like.

>> I think before you integrated from 0 to

infinity, now it's minus infinity.

>> Oh, because this was in the language of

the y's. So, the y's are from 0 to

infinity. So, the t's are from minus

infinity to infinity. Okay. So the t's

are from minus infinity to infinity.

Okay.

And now this integral actually has a

very beautiful uh interpretation

because exactly what we said before

remember what we said before is that

each one of those factors

you know so let's let's let's try to do

this integral. Well, obviously the

integral should be easy in some sort of

in some local region in tsp space

because each one of these factors is

just going to turn into something

linear. Okay, so let's just see where

each one of them is is is linear, right?

So where was this guy linear? So this

part of the that part of the uh

exponential looks like this.

That part is going to look like

uh what we figured out before. So, it's

going to look like C14 * minus T14 here,

right?

It's going to look like um uh C14 * -

T14 minus T13 here. And is going to look

like zero here. C14 * 0.

Okay,

so that's that factor.

What about this factor?

Well, that's peacewise linear in another

region. In fact, to see it, well, it by

alone, oops, that red thing

disintegrated. So, let's use this. So,

remember this factor alone

was uh

uh peacewise linear like in this region.

And obviously when P13 is is positive,

so I would add to this plus c13 * 0 and

when t13 is negative on this side I

would add to it plus uh c13 * negative

t13

and similarly

for the last guy well I'll draw it the

same same color here right

for the last guy then down here I would

have plus uh c24 * t24 T14 sorry and up

here I have plus C24

time zero

okay

oh sorry and I didn't write uh I didn't

write uh I should have written things

here so uh so um uh no sorry yeah this

is this is uh uh this is this is okay so

but I but but I should have uh uh

remember this C14 minus T14 minus T13

was valid in this whole region Okay. So

in this little corner uh I'm sorry I did

it precisely wrong. Uh uh so uh so uh so

from uh from C13 I have zero on this

side but I have negative uh C13 T13 on

this side. So here I'm going to get C14

* T14 and negative T13

and then plus C13 * negative T13.

Okay. And so on. Okay. So uh so um uh

okay so now now I have now I have the

whole plane is divided up into these

five regions that we get by putting all

the fans on top of each other and in

each one of these regions the integral

is linear. Okay so we can do the

integral.

Now the integral is easiest to do in

this region. Okay.

Um,

and I may have screwed something up

here.

Um,

[Music]

sorry.

[Music]

Oh, sorry. I precisely screwed up. I

screwed up at the very beginning.

Someone should have stopped me. Sorry. I

screwed up in writing this expression.

Okay.

Okay. So, if we have this guy,

okay, here is negative t14.

Okay. But here is zero. Okay. Because it

was max of zero. This was max of zero

and negative t13 and negative t13.

Negative t14 negative t13. Okay. So

that's zero here and here it's negative

t14 negative t13. Okay. Sorry. Okay. So

um so very good. So we erase that here.

But now we have to concmplyantly uh uh

erase it here. And here is where I

should have this uh negative t14 and

negative t13. Okay.

Okay. Anyway, so uh so so you see you

have an expression that's peacewise

linear in each one of these cones. And

now let's look at what it looks like for

example in this cone. This is the

simplest one in which we can do it.

Right? So in that cone what are we

getting

in that cone? We're getting the integral

0 to infinity.

We're getting the integral 0 to infinity

from this cone

of just dt13 dt14

eus x13 t13 eus x14 t14. All the c's go

away. And this is just equal to 1 /x13

1x14.

Right? And that's precisely the fineman

diagram that you'd associate

with the having these two chords x13 and

x14. Right? That's exactly the fineman

rule that we'd associate with that.

>> Sorry, why did all the C's go away?

>> Uh because in this region

the contribution from the C's was zero.

>> Okay. And so we always have the t x13

t13 x14 t14 on top of what I added. I

should have added that even there. But

that'll just be common linear factor

everywhere.

>> Okay.

>> Change the boundaries from minus

infinity to zero

>> because here this is only in this cone.

So now I'm going to do the integral. I

want to do the integral over all t's.

But now I've simplified my life to see

that in this cone it's linear. In this

cone it's linear. In each one of these

five cones it is linear. And so I'm just

going to do the integral over each cone

at once. Okay, in every cone it's

linear. And this is the easiest cone in

which I can do it. And what I get is 1

/x13 x14.

Now physicists call this trick for

representing 1 /x equals the integral 0

to infinity dt eus tx. This is called

swinger parameterization.

Okay, there's various nice physics words

associated with it. You can sort of

think about t uh as a as a length of a

propagator in position space. Okay. So

there's various nice words associated

with it. But it's just a way of

representing uh these uh these

propagators. So what we've seen is that

in this cone I get something that looks

like the swinger parameterization for

that diagram. Right? So the cone goes

with that cone is associated with that

diagram. And when we do this integral,

what we're getting is the Schwinger

parameterization for that diagram. And

the contribution from that cone is

precisely giving me the finding diagram

associated with that uh uh with the

corresponding uh

the integration over the cone is giving

me the amplitude uh that I'll get from

the the contribution of the amplitude

from the corresponding finding diagram.

You can easily check that the same thing

is true for all of the other cones.

Okay, it's maybe slightly interesting

like these cones don't look like the

standard positive cone for that formula

to be valid. But of course, you can

always do a linear transformation. You

know, the integral over this cone, I can

just do a linear transformation to bring

it to the standard form, which is the

positive uh uh the positive quadrant.

The integral over this little triangle,

I can do a linear transformation to

bring it to a positive quadrant. And

every one of those uh contributions ends

up being 1 /x 1 /x from the

corresponding diagrams.

Okay.

Okay. So let's just just pause and say

what what has happened. We have this

picture of the associ. We haven't I mean

well Carolina gave you a picture of

where it came from. We're now going to

we're now going to talk about where it

comes from in a more sort of primitive

uh principled way. But imagine you got

the uh the picture. The really magical

things that is presented as a Minowski

sum um uh in the language of the

polytope or in the language of the fan

that we have a fan that has cones that

cover all of space. Okay. All every cone

is associated with a particular

triangulation, a particular diagram.

Now that's already interesting. That's

already interesting that there's some

way to divide all of space naturally so

that into columns where each column

corresponds to a diagram. That already

sounds like progress. Okay. But uh

having done that, if someone told you,

well then go add up the shringer

parameterizations from every diagram,

you'd be this still sucks because I

still have to go cone by cone by cone

shringer parameterize for each one and

add them all up. That's basically doing

finding diagrams again.

But the fact that this sort of

complicated set of cones is secretly a

simple set of individual fans put on top

of each other tells you that there is a

simple peace-wise linear tropical

function that produces all this

complicated mass of refinement diagrams.

Okay. Now this small example was uh

maybe not small is maybe too small to

impress but it already illustrates the

point. There are five diagrams.

we have three tropical functions.

Okay, we did not have one function for

every diagram. Instead, it's the magic

of the Mowski sum picture, right? That

uh that was uh it was presented with

these sort of three uh simple fans on

top of each other. So, three tropical

functions were all you needed to produce

the five diagrams.

[Music]

Now what's going to happen in general

and that's what we're now going to see

is that not just at tree level but when

we generalize to all loop orders all

surfaces etc these two bits of magic

always happen. So first there's a way of

uh uh getting into the the space of the

appropriate dimension and seeing that

it's naturally divided up into cones and

each cone corresponds to a diagram but

you don't manually have to sort of you

know make them it's it's just going to

come out of the box. We're going to make

we're going to figure out how to hand

you a bunch of vectors such that the

cones made out of them are automatically

going to correspond to diagrams. That's

going to be small miracle one.

Bigger miracle too is that there are

some functions associated uh to these

pictures as well. There are polomials as

we saw here and there are

tropicalizations. Okay. So there they

both have an interesting role to play.

These are going to be few of these

polomials. So just give you a just so

you have an example in mind. If we're

talking about tree amplitudes

asmtoically the number of finding

diagrams the number of triangulations

are given by cataline numbers and

cataline numbers grow like four to the n

if you have an n going okay so the

number of fineman diagrams are growing

exponentially

however we're going to give about n squ

tropical functions we're going to give

about n^ squ polomials there's going to

be a polomial associated essentially

with every point in that mesh right or

there's going to be a polinomial

associated with every every uh one of

these xigs at every one of these igs

there's going to be a polomial and the

polinomial will have at most n terms in

it okay so I'm going to hand you around

n squ polomials each one having at most

n terms all right

no four to the n here so clearly I'm not

summing over all diagrams right the sort

of magic is I'm going to give you n squ

polinomials with about n terms each

first of all if you shove those

polomials in to the integral d y y y y y

y y y y y y y y y y y y y y y y y y y y

y y y y y y y y y y y y y y y y to the X

product of all the polinomials so

they're minus C that will define string

amplitude

okay without your string theory of

course

uh and secondly if you tropicalize those

expressions in just the way that we

talked about that will give YOU A GLOBAL

SWINGER PARAMETERIZATION FOR ALL THE

diagrams together at once. Okay, but

again the magic is there will be order n

square tropical functions each one of

which has a max of order at most n

things

very small number but if you put them

all on top of each other the domains of

linearity of all of these guys

exponentiate right and generate this

humongous complexity of all the diagrams

together right this is a second magical

fact a more magical fact both the

existence of the polomials and the

existence of uh and the fact that Well

anyway the the existence of the

polinomials is the essential fact and

the tropicalization is this sort of as

this very beautiful interpretation as

giving you cone by cone the shringer

parameterizations for each diagram but

in a global way that combines them all

together. Okay. The point is that these

peace-wise linear functions morph into

the swinger parameterization for every

diagram as you move around the cones

without doing any manual work. Right?

And it's literally this this difference

between polomial versus exponential

complexity. sort of polinomial uh

complexity in exponential uh complexity

out. All right. So those are the things

that I now want to explain where these

things uh where these things come from.

Before doing that, let me mention a

final thing that we can see in this

example. Okay. Final thing we can see in

uh in this uh uh example

that's also going to uh also going to

play a sort of starring role.

Um

and this is the existence of something

that I'll give some whimsical names to

and then an official name to.

But let me start with the more uh

whimsical name. Remember, part of the

part of what the esocahedron does

is if we drew this picture,

of course, it captures all of the uh it

captures all of the

all the compatible uh uh chords and so

on. But a sort of very minimal thing uh

that it does is it keeps apart things

that cannot cross. Right? That's the

point that that that we can't have 1 13

and 25. They cannot occur together,

right? The chords cross. The chords 1

three and 25 cross. And so part of the

job of the association is to keep them

apart, right? So 1 13 and 25 do not

meet. Okay? 35 and 1 cross. So 35 and 1

do not meet. Okay? Of course, it does

more. The the things that do meet have

the correct combinator. They all meet

the way they should and so on. But but

the most zero order thing that it does

is keep things apart. That's very

beautiful that we can do that in this

sort of polytopal way. But you'll also

notice that it's kind of uh it's not

entirely rigid, right? I mean, of

course, I can move these faces around

parallel to each other in various ways.

So it's not entirely god-given. Okay,

there is a more god-given object which

does exactly the same thing that you can

call these are the whimsical words you

can call a curvy associ

we sometimes call binary geometry

and okay it's also positive coordinates

for tech space but I get I fall asleep

by the middle of that sentence okay so

uh okay so um uh so it's related to

compactifications of technular space in

some way but if you don't know what what

that is uh that's more than fine by me

okay so um okay so um so what is this

doing so there is something uh there is

something uh interesting here remember

what the conventional sort of flat

associed this is polytopal this is cut

out by linear equations what the

conventional flat associed is doing

amongst other things is keeping things

apart

And so we do that by well we associate

these variables x's with them and so on

right so the flat associed

then on on this side

the flat associated for every so our you

know we have chords we have some chords

I j

chords kl they want to be kept apart

from each other when they cross and so

on in the flat association we associate

these things with variables xig and then

there's some there's the association as

a polytope

that's cut out by equations involving

these X's. So this is the flat associed

world.

In the curvy world

or non nonlinear world,

every one of these chords is associated

with another variable we call UIJ.

Okay?

But we're not going to write down

inequalities and things like that for

them. Instead, these UIJs are going to

satisfy the following set of very

interesting nonlinear equations.

UIJ plus the product over all chords KL.

Right

now I'm just writing these equations

down. Okay. again we'll understand

better where these things come from.

Okay.

But let's say just out of the box I

handed you uh an equation that looks

looks like this. Well, what's your first

reaction to this uh uh equation? Let's

try writing it down at five points.

Okay. So at five points

very simple

at five points we have that uh so let's

draw our our our our five gone. So I

have like the equation for U13. So U13

plus the product of all the chords

across 13. So who are the chords across

13? There's 2 4 and 2 5. Those are the

only ones that cross 13. So U13 plus U24

U25 is equal to 1. Okay.

And then we have the cyclic friends of

this guy, right? So we have plus uh uh

U24 plus U 35 U13 = 1 and so on. Okay.

So we have a total of five equations.

Okay.

So without thinking, YOU THINK YOU HAVE

FIVE EQUATIONS, FIVE UNKNOWNS. This has

some solutions and points. Okay,

the interesting thing is it doesn't have

solutions and points. It has a full

two-dimensional space of solutions.

Okay, these equations have a full

interesting two-dimensional space of

solutions. For example, a natural thing

you could do this does this turns out

not to generalize very easily. Okay, but

one thing you could do in this example

is to just choose two of the U's and

solve for the rest of the U's in terms

of them. For example, you can find that

uh U25

is uh is 1 - U13 U14. If you just solve

these equations, u35

is uh

1 - U14 over 1us U13 U14. and u uh

24 is 1 - u13 over 1 - u13 u14. Okay, so

this is one way you could solve the uh

equations. All right, you can just check

that that's uh you can just check that

that's true. All right,

but what's interesting about these

nonlinear equations? Uh why do we call

it a curvy association or binary

geometries? What's that word binary?

Again, it has to do with the word

positivity

because we ask for real positive

solutions of these equations. Okay? So,

I'm going to ask for the what happens

when the uj are greater than or equal to

zero. What happens if the uj are greater

than or equal to zero?

These equations then tell us not just

these but but in general remember in

general we're writing down uj plus the

product over kl of u kl is equal to one

right so these equations are telling us

uh that if the 's are positive if the

sum of two positive numbers adds up to

one they all also have to be less than

one

right all right so all these 's have

also got to be less than with the one.

Okay,

that's where the word binary comes in.

And now they have an interesting

property. Suppose a given u like a given

ui star jar some ui star jar. Let's say

this heads to a boundary of the space

where this goes to zero.

By the dent of these equations,

all of the u's that cross it must go to

one.

Okay. So the U KL for KL crossing

I star J star

have to go to one.

Okay.

That's remarkable. If you think about

hitting the boundaries of the space

corresponding to U's go to zero. You see

THIS IS DOING EXACTLY what the is

associed is doing. It's keeping apart

things that cross,

but it's keeping them apart in an

entirely rigid way. There isn't any

parameters that you can move around.

These variables are from 0 to one. And

when you go to a boundary uh for one

chord, all of the chords that cross it

just go to one. Okay.

All right.

Okay. So now this turns out to exist. uh

these variables exist uh uh at at tree

level for the case that we're talking

about. Again, all of this generalizes to

all all all surfaces in interesting

ways. You draw any surface, there is a

chord uh there's a variable associated

with any open curve on the surface and

you write down in general if you have a

uh a curve x on the surface, you write

down this formula, the product over all

other curves of the u for the y and it's

raised in general to the number of times

the curves x and y intersect each other.

Okay, so that's just a sneak peek.

That's the that's the equation. It's a

sneak peek because uh I I like to say if

I if I had to hand someone, you know,

one fact and tell them to go to a desert

island and work out what I did for the

last 5 years of my life, that would be

what I would do. Just hand them that

that equation, say go, right? And then

then just studying that equation would

lead to all the things that we're about

to talk about. All of it is in

understanding uh how that equation can

be true and and how to solve it. Okay?

Uh we'll see how much we we get to this

uh uh in uh later in this lecture, but

okay. Okay. But so so it's true not just

at tree level, it's true for all all

surfaces.

But what's especially interesting about

it is we know how to solve them. Okay.

There's a way of solving these uh these

equations and the U's are given as

ratios of polomials. Right? So let me

tell you what this ratio of polinomials

looks like for our uh pentagon example.

For instance, uh we find that uh that

that u's are parameterized in terms of

some positive coordinates y. Okay. And

the formulas are that u13 is y13 over 1

+ y13

u14.

Okay. And I just invite you as a little

exercise to check that these satisfy U

plus UU equals 1. Okay. So for instance

if we take u13

plus u24 u25

okay this is u13 is y13 over 1 + y13

but u24 u25 there's a nice telescopic

cancellation of that factor and this

factor and we're left with 1 over 1 +

y13

and this is dutifully equal to one.

Okay.

So you see we're solving these U

equations. We're solving these U

equations in terms of these interesting

polomials.

Now notice those are exactly the

polomials that showed up in our previous

picture. Okay?

They're precisely the polinomials that

showed up in our uh Mikowski summaries.

So let's summarize what we saw in this

example that we're now going to give a

rational uh uh uh explanation for. Okay.

We started with the socedin.

The simple case there was a pentagon. We

drew the dual fan associated with it.

The fan has the feature that it divides

the space up into cones. Every cone

corresponds to a triangulation or a

diagram. Okay. The fact that the

association has a main sum decomposition

is reflected in the fact that the fan

even though it has five pieces is

naturally thought of as sort of putting

three simple pieces simpler pieces on

top of each other. And that three versus

five distinction gets much more dramatic

at at higher points. Okay, there is a

small number of pieces each one of which

looks like the fan of a simplex which

when you put them on top of each other

order n squared pieces where you put

them on top of each other generates the

horrendous complexity of all the 4 to

the n uh fineman diagrams.

Okay. uh there is this simple integral

that you write down just integrating

with the dlog form in the uh uh uh so if

we can then associate polomials with

each one of these uh sumands there's a

simple integral that we we can write

down with the dlog form uh y to the x

and all the polomials raised to the

power of their coefficients in the

sumand that gives us the string

amplitude. If we take the low energy

limit of this, we simply tropicalize

this integral and that's interpreted as

the global shinger parameterization a

way of combining all the finding

diagrams together in a uh single object

so that cone by cone we get the

contribution from the diagram but

without having to invent the diagrams

right they all sort of come out

and these polomials themselves are

associated with this interesting notion

of a binary geometry this much more

rigid way of associating creating a

geometric object with the combinotaurics

of triangulations of an engon. Okay,

there's a sort of curvy version of the

ocosahedron where there's no moduli,

there's no parameters, it's just zeros

and ones, right? But with every variable

you associate a u this interesting u

equations um which have the feature that

uh when when when you approach

boundaries um

uh when you approach boundaries uh the

um

uh 's go uh uh when 's go to zero all

the crossing chords uh go to one. In

fact, I'll mention one last thing here

to just close this example.

I'm going to go back and write the

string amplitude again. You remember the

string amplitude was written in terms of

these polomials and but the 's are some

interesting uh you know uh just

combinations of these polomials upstairs

and downstairs.

So I'm going to go back and I'm going to

write the string amplitude directly in

terms of these u's where they look much

more elegant.

So that string amplitude again is still

this dlog form. It's still the integral

0 to infinity. d y13 over y13

d y14 over y14.

Okay. But all of the rest of the stuff

that we had before just turns into the

product over all chords u j the u

variable to the power of xig.

That's what the string amplitude is. See

this is a much more elegant form.

This form makes it makes uh a number of

things uh obvious. In fact, it makes all

the qualitative and quantitative

features of the amplitude uh obvious.

For example, it makes it obvious that

the only singularities again this this

is this is a dlog form. Um the a sort of

cool thing is to show that this is a

dlog form that has logarithmic

singularities everywhere the 's and any

of the 's go to zero. Okay, that's a

small extra step that's not hard to

prove, but just just uh by that. Okay,

so anywhere. So this integral is only

going to develop singularities in the

neighborhood where some u's go to zero.

That might be at zero or infinity and y.

It doesn't matter sort of invariantly.

It's only developing singularities as

the u's go to zero. So what's rescuing

it is its power, right? So that's why

we're going to have poles that look like

one over xig

uh as some of the xigs go to zero. Okay,

so that's the sort of first thing which

is obvious. That was our locality,

right? Where are the poles where the x's

go to zero? Okay, where we're seeing

that. Okay, so we're going to have a

pole as xig goes to zero. As some xig go

to zero, this amplitude will go like one

over xig.

Okay, but now comes the real magic.

What happens to this factor at a given

uh xig goes to zero. Let's say given

again x i star jar goes to zero. Let's

imagine drawing a big n gone. This is

going to be true for any n right? So

let's imagine here's my chord iar jar

and this xig is going to zero.

That means that this integral is being

localized in the neighborhood of where

the corresponding u iiar jar goes to

zero.

So what happens to this factor, right?

Well, we're going to pull out of it some

u i star jar to the x i star jar. That's

a factor that's coming out of all of

this. That's in fact exactly the factor

that's getting slightly rescued by the

non-zero x to give me something that's

going to be 1 /x and is getting uh

regulated. Right? That's that's my my

poll. But what about everything else?

Well, notice all this chord divides the

the uh the uh the engon naturally into a

left part and the right part.

But all of the U's that correspond to

chords crossing here are going to one.

Right? That's the point. When a U goes

to zero, all the chords that cross it go

to one. So in this product, all of the

pieces that are a U to the^ of X

corresponding to something that crosses

are going to one.

So what we're left with is just this

factor that pulls out times the just a

product over the left of u to the x left

and a product over the right of u to the

x right.

This is factorization.

This is precisely the phenomenon of uh

uh

factorization. Okay. Because then it's

very easy to see that this form also

just splits. There's the part associated

with that going to zero and the rest of

it splits into a product on the left and

the right and therefore the whole

amplitude goes to this times the

amplitude on the left and the amplitude

on the right and in fact this has

nothing to do with taking the low energy

limit. This is a full exact property of

the full string amplitude which is one

of the things that makes string theory

consistent. Okay, even beyond field

theory. Right? So I hope you see very

vividly in this example the the uh the

kind of abstract discussion from

yesterday right we start with a picture

there's maybe Carolina started with a

mesh

some funny wave equation in this mesh

space uh you get a polytope you know you

you're following your nose I hope it's

clear at no step are we cheating on

looking at finding diagrams at the end

of the book right there's nothing like

that going on okay but then we're sort

of after a few steps we'll naturally let

to this object object. This object does

everything that fine diagrams do but

without having any reference to particle

trajectories or anything like that.

Okay, in fact the sort of stars of the

show here are variables associated with

like chords in this engon. Right? And

remember those chords in the engon are

related to the kinematic variables. The

kinematic variables are also associated

with chords on the uh on the engon.

Right? Okay.

All right. So, okay, let's uh pause now

and I'll ask if there's uh any

questions. Um, yes.

>> So, these U and Y variables, are they

somehow related to some cluster variet?

>> They're related to uh they're they're

related, but they're not precisely in

cluster variables for a variety of

reasons. Surface cluster algebbras are

not quite what is needed in this story.

They're close, but not exactly.

[Music]

Yes, they're they're more closely

related if you know they're more closely

related to uh uh fakoner coordinates,

but they're also not precisely that or

there's a very special set of fontro

coordinates that do this uh job. Okay,

but part of the point uh of these

lectures is to not assume that you know

fancy things and just you know build it

from the from the from the ground up. So

in the next uh 50 minutes we'll sort of

see where these things come from from

the ground up. Yes. Super. So you were

speaking about how it is uh uh

surprising that this tropical

representation exists for the uh for the

>> yes

>> uh full amplitude in one finger

parameterization.

>> Yes. So I wanted to ask so suppose I I

start doing this I uh instead of looking

for positive geometry

>> yes

>> I take some some new theory some

different theory and ask oh does it have

a global swinger parameterization for

example I think Bruno has some papers

where they have found

>> global swinger parameterization but in a

different structure than

>> yes I think everywhere we have global

swinger parameterization there is some

interesting uh positivity behind it okay

there are some there are some positive

parameization some set of polinomial

that you can in the end interpret as

giving you uh this picture of naturally

a bunch of Mowski sum ends that are

summing up to a complicated geometry.

That's the universal thing in all the

examples that that that we see so far.

And all the examples also have stringy

integrals that go along with them. Okay.

So there's extensions Nick Bruno uh

others have talked about uh uh uh CGM

theories and so on. um and uh uh the the

original motivation for those things

came from the chy formalism but in fact

there's fully stringy versions of them I

mean so it's uh it's not restricted to

the field theory limit and from my point

of view that's a more natural way of

thinking about what what they are so

that's

>> from the perspective of positive

geometries is some qualitative

difference in finding it through a

positive geometry or finding or are this

just equivalent completely

>> well I mean uh all I can say is that if

you is that um I don't know what to say

in I don't know what to say in in in

general. Um uh maybe maybe here is uh

the here is a place to uh to perhaps say

it. Um so uh so first of all uh

everything that I've been talking about

here is in the context of uh of the

trace cube theory. Okay. So the only

dependencies on simple scalar kinematics

um and we have expressions that look

like this like literally this this

expression. Sorry if we do this. This

was meant to be a general end. So this

would be some maybe dy13 up to dy1n. I

I'll say what this means in a moment.

But anyway, some some expression like

this is is uh is true. Um and so you

might think this is very special to the

trace 5 cube theory. Can you do this for

trace 5 to the fourth theory? No. Okay.

There's nothing sort of huh

>> n minus one.

>> Uh this is uh this is actually n minus n

minus one. Thank you. Okay.

uh you might think that uh anyway can we

do it for trace fun theory some other

theory no okay so uh at least not not in

uh any obvious way but one of the

surprises of the past few years is that

literally these functions just slightly

reinterpreting what you mean by the x's

if we take the x's and just shift the

x's in some simple linear way um take us

from describing the amplitudes for trace

cube which is a don toy model that maybe

none of us care about to describing real

world particle physics. Okay. So you

shift these amplitudes in one way. You

you shift these functions in one way. I

literally mean shift. I mean doing

nothing to them. Just reinterpreting

what you mean by X. Okay. Uh you just

just shift X goes to X plus some

constant depending on what I and J are.

And then all of a sudden you have the

amplitude for pi out which are some of

the particles associated with the strong

interactions. And if you do it literally

in this way at finite alpha prime uh and

shift x by a single unit x goes to x +

one x plus or minus one one over alpha

prime then this thing turns into the

scattering amplitude for gluons real

gluons. Um and unlike the story of

therin, this has nothing to do with

super symmetry. The words n equals 4,

superyang mills, etc. make no

appearance. This is for totally

non-supery symmetric gluons. And at loop

at loop order, they're loop order for

totally non-supy symmetric gluons. Okay,

these second two statements are a much

bigger surprise that somehow that this

uh seeming dumb uh toy model contains in

it

uh theories we care about a lot more.

And actually some of the surprising

properties that Carolina was mentioning

this morning that these amplitudes have

surprising zeros and they factoriize in

surprising ways in the neighborhoods of

these zeros they in fact port to all

these other theories too. Okay. So the

the theory of pions and of gluons have

exactly the same mysterious properties.

That's actually how with Carolina and

Jyn and friends we were left to discover

these things backwards. We ran into the

zeros. Then we observed just

experimentally the zeros were also there

for other theories that we cared about

more which is very surprising. Why the

hell? Those theories were not related to

the association. We didn't think. Well,

we were wrong. They are right. It's

exactly the same function shifted in

this interesting way takes us between

all these different theories. So, this

is maybe a kind of example of what

you're saying. Had you told me abonio

ahead of time, take the diagrams for the

nonlinear sigma model that describe

pounds and try to put them together in

shrinking parization. That's good luck,

right? That's a total disaster. I would

never try doing that in a million years.

And I would also not think in a million

years that all I had to do was go back

to this example and do some stupid shift

on the x's in order to get the right

answer. Never mind that I could do the

same thing for gluons which is utterly

shocking right that I would never

thought. So somehow that's the nature of

the beast in this business is it does

not commute with responsibility. I don't

know how to say it but uh but you want

to like do something say I'm going to

take every fineman diagram and you know

figure out how to put the numerators in

and shringer parameization then globally

swinger parameterize it. People have

tried to do that with some degree of

success but somehow does not have the

same you know remarkable striking

feature as what naturally comes out when

you pursue what the mathematical objects

want to do on their own terms. Okay. So

that's uh that's why I don't have a

general answer to give to your to your

question that

okay any other questions. Yes.

>> Uh I didn't quite understand why when

you set the xig to zero yes but also the

uj went to zero.

>> Oh yes because the the the the point is

uh remember if I didn't have this factor

at all this integral is very divergent.

Okay is logarithmically divergent both

on the y's go to zero and they go to

infinity. Okay. what this factor now it

turns out that the y is going to zero

and infinity in fact where this form has

a logarithmic divergence is in onetoone

correspondence with where the 's vanish

okay so so where this thing develops a

logarithmic uh singularity are where the

's vanish

now the x's are there to ensure that you

nonetheless get something finite right

it's like integral sorry maybe I should

have said this but this is just saying

you know integral zero to whatever d y

is log zero is infinite but dy y y y y y

y y y y y y y y y y y y y y y y y y y y

y y y y y y y y y y y y y y y y to the a

is 1 / a plus dot dot dot, right? So

just the point is that these little

exponents save the integral, right? They

make it convergent, but as that exponent

goes to zero, this integral develops a

pole.

Okay, so if we go back here, if a given

xigj goes to zero, there's a region in

this space that's going to have a

logarithmic divergence when the u goes

to zero, when the u for it goes to zero.

Okay, if the xig is not quite zero, the

amplitude will be dominated by this

neighborhood, it'll get a factor of 1

/xig and the rest of the integral will

be what you get from the other parts of

the integration uh in the neighborhood

of where that u goes to zero. And

because of this sort of magic fact that

as a given u goes to zero, all the u's

that cross it go to one. The rest of the

integral just dutifully factorizes.

Okay, so what you're left with is just

the product on the left and the product

on the right. Okay, I I want to stress

again this is qualitatively different

from where factorization comes from with

the picture of space-time processes and

fineman diagrams. Okay, and this

qualitative difference is related to

what may have seemed like a like a sort

of two vague uh remark yesterday that

it's the tuness of the pi.pj versus the

oneness of the pis that matter. See, the

fineman diagrams are all about oneness.

You know, you have a bunch of momenta

coming in. you're keeping track of the

momentum of the particles. Then there's

a propagator that's the sum of all those

momenta and it goes from from from here

to there. Right? So that's how finding

diagrams make manifest that

singularities are associated with

propagators.

This picture does not ever see single

indices. This picture is all about pairs

from the beginning. It's all about

pairs. It's about curves on the surface.

It's all about pairs. And therefore the

way it manifest factoriization looks

nothing like a particle propagating and

going on shell and cutting the picture.

It's just some something totally

different. It's this binary geometry.

Okay. And that's again a concrete

instantiation of this uh general fact

we've seen over and over in this

business of the existence of natural

geometric objects that factoriize on the

boundaries for their own private reasons

which don't uh don't on the face of it

have anything to do with particles going

on shell in space time. Yeah.

>> Okay. Any other questions? Yes.

>> In terms of complexity reduction. Yes.

>> So from going to four to the end

diagrams to do some note that this is

the you can do.

>> Uh I would be surprised if you could do

better, but uh I'm not sure. Yeah, I

mean it's it would be very interesting.

It would be and and just to maybe make a

point um uh uh whether we get there or

not. I hope now you can see why this way

of thinking offers some hope for uh

asking what happens when the number of

particles goes to infinity. Okay?

because we're going to have an integral

of something which is e to the minus

some peace-wise linear expression which

is not that complicated. It's a sum of

sort of n square pieces each one of

which is a max of things that involve

order n things. Of course, you're doing

an order n dimensional integral but it's

a single integral. It's a single

integral over one space that somehow

something nice should happen in the

large end limit. Okay, so that that's

not insane that something like that

should happen. And in a very precise uh

uh sense, what happens is that these

peace-wise linear things smooth out and

that uh the large end limit is a much

smoother object than you would naively

think from putting all these maxes uh on

on top of each other. That smoothing

phenomenon is a very interesting one

which if we get to it tomorrow uh uh

we'll uh talk about.

All right. So I have uh 40 minutes. 40

minutes. That's challenging. But let's

see what what what we can do. So now I

want to tell you where these things come

from.

Now first off, we're going to go back um

uh to this picture

where we sort of think that we have a

you know we have a scattering process.

But we already learned let's go back to

the case of the pentagon

that we can we can uh we can think about

this uh five particle tree amplitude in

terms of a triangulations of uh of a

fivegon or an engon uh in general. Okay.

And the kind of first way was the way

that uh uh Carolina mentioned in her

talk that we could sort of draw a duo of

this triangulation. And the duel of the

triangulation looks like uh you know

looks like the diagram that we had

talked about. There's a closely related

way of thinking about this which is uh

which is which is uh somehow deeper

and this has to do with going all the

way back to um uh to these being colored

particles. Okay. And in our three

discussions of color today only at the

very very end did the following picture

show up which I'm glad it eventually

showed up. Okay. So we're talking about

this trace 5 cub theory.

Uh the fi is a matrix. This is the

point. Fi is a fi is a matrix. So we

imagine the phi is an n byn matrix. It

has an upstairs and a downstairs index.

So when we write fi cubed, we really

mean 5 a b 5 bc 5 ca with einstein

summation convention. Sort of summing

over all these things.

And so it's convenient to keep track of

what these indices are with a double

arrow notation. Okay. Okay. So if I have

something like this, the upstairs arrow

will be uh a arrow going this way will

be a J. The downstairs arrow will be an

I. Right? And so when I write trace I

cubed, trace 5 cubed looks something

like this.

So this will be let's say an I. But the

fact that it's a trace means that this

index is identified with that one. Okay.

So this I survives out to here and this

J survives to here and this K survives

to there. Okay.

Okay.

So, how would I draw a four-point

diagram?

You would draw it like this. This is the

uh the fat graph uh ribbon graph that uh

Sergio was uh referring to. Okay,

that's one fact graph. This is another

fat graph for the other diagram that I

would draw. Okay.

Okay.

And if I had a loop diagram, you know,

it uh diagrams like you saw in Ruth's

talk, a diagram like that when the

particles have color would look like

this.

Everything just gets fattened out.

Okay. Now,

there's there's two very closely related

points, very simple, ancient, very well

appreciated, but but let's just say them

they're very slightly different. One of

them is that we can think of each one of

these diagrams as being drawn on some

surface. Okay? So for example, we can

think of the tree diagram as being drawn

on the plane uh you know within ordering

with the external lines without any

crossing lines. Okay,

that's one picture we can think of as

being drawn on on on a plane. The other

one is that we can think of every one of

these diagrams is actually defining a

surface.

These are two slightly different ideas.

Okay. So for example, this diagram is

going to define a surface which is a a

disc with four marked points

1 2 3 four.

Okay. So how does this correspondence

how does this correspondence work? Well,

how would you define a surface? You can

define a surface by getting a

triangulation of the surface. Okay. So

what is a triangulation? Is a collection

of triangles with an orientation, right?

Um, and there's they're supposed to at

most meet on common boundaries with

opposite orientation, right? So, we're

talking about orientable uh surfaces

here. So, I could give you the triangle

1 2 3 and the triangle 134. And those

triangles give one definition of this uh

of this surface. Okay?

A ribbon graph like this, a fat graph

like this is giving you exactly the same