9. Large N Expansion as a String Theory, Part II

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November 15, 2020
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November 15, 2020

9. Large N Expansion as a String Theory, Part II

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visit MIT OpenCourseWare at ocw.mit.edu. HONG LIU: OK, let’s start. So first let me just

remind you what we did at the end of last lecture. So we see that the large N

expansion of gauge theory have essentially exactly the

same mathematical structure with, say, the mathematics of

the [? N string ?] scattering. And so here the observable

is a correlation function of gauging

[? invariant ?] operators. And then these have a large

N expansion as follows. And on this side you have

just an N string scattering amplitude. Just imagine you have some

kind of scattering of strings, with total number of N strings. And then this also

have expansion in terms of the string

counting in this form. So now, if we identify–

so if we can identify the g string as 1/N. So if we

identify g string with 1/N, then these two are essentially

the same kind of expansion, OK? And you also can identify

these external strings, string states,

within the large N theory which we called

the glueball states for single-trace operators. And then each case is

corresponding to [? sum ?] over the topology. It’s an expansion [? in ?]

terms of the topology. So here is the topology

of the worldsheet string. And here is the topology

of Feynman diagrams. Here is the topology of

the Feynman diagrams. So still at this

stage, it’s just like a mathematical

correspondence. We’re looking at two

completely different things. But probably there’s no–

yeah, no obvious connection between these two objects

we are discussing. Yeah, we just have a precise

mathematical structure. But one can actually argue that,

actually, they also describe the same physical structure

once you realize that when you sum over all possible

Feynman diagrams. So once you realize that

each Feynman diagram, say, of genus-h can be

considered as a partition, or in other words,

triangulization over genus-h surfaces,

[? 2D ?] surfaces. OK. So if you write more

explicitly this fh, so if we write explicitly this

fh, then this fh, this fnh, then will be

corresponding to your sum of all Feynman

diagrams of genus-h. Suppose G is the expression

for each Feynman diagram. Say for each diagram. And then I can

just rewrite this. In some sense, I [? accept ?]

all possible triangulation of [? a genus-g ?] surface. Say there will be some

weight G. And summing over all possible

triangulations of a surface is essentially– so

this is essentially the same as this sum over

all possible surfaces. So this is a discrete version. So sum all possible

triangulations of some genus-g

surfaces, or translations of genus-g surfaces. Then they can be considered

as a discrete version of sum over all possible surfaces, OK? AUDIENCE: So you’re saying

it’s like a sum over [? syntheses, ?] like

a simple [? x? ?] HONG LIU: Exactly. Exactly. Yeah, because, say, imagine

when you sum over surfaces, so you sum over all

possible metric. You can put [INAUDIBLE]. And that’s the

same way as you sum over different discretizations

of that surface once you have defined the

unit for that discretization. So if we can identify–

so for now record this Fh. So this Fh, this Fnh

is the path integral over all genus-h surfaces

with some string action, weighted by some string action. So if we can, say,

identify this G with some string action– the

exponential of some string action. Then we would

have– then one can conclude that large N gauge

theory is just a string theory, OK? That large N gauge theory

is just a string theory, if you can do that. In particular, the

large N limits– so large N limit here,

as we discussed before, can considered as a classical

theory of glueballs. Or a classical theory of

the single-trace operators. So this would be matched to

the classical string theory. So as we mentioned last time,

so I was mentioning before, this expression– so just

as in the as we discussed [INAUDIBLE], the

[? expansion ?] in g string the same as

expansion in the topology. And the expansion

in the topology can also be considered

as the expansion of the groups of a string. Because whenever you add

a hole to the genus– when you add the genus, and you

actually add the string hole, you add the string loop diagram. So in this sense, you can

[? integrate ?] all these higher order corrections,

as the quantum correction to this

classical string behavior. So this is just a tree-level

amplitude for string. And this [? goes ?]

one into the loops. Whenever you add this

thing, you add the loop. OK. Is this clear? Now, remember what we

discussed for the torus. If you’ve got a torus,

then correspondingly you have a string split

and joined together. And this split and join

process you can also consider as a string loop,

a single string going around a loop, [? just like ?]

in the particle case, OK? In the standard

field theory case. And so the large N limit, which

is the leading order term here, would map to a leading order

in the string scattering. And the leading order in the

string scattering– they only consider tree-level

[? skin ?] scatterings, and then corresponding to

classical string theory. And also the

single-trace operator here can be mapped

to the string states. Yeah, can be mapped

to the string states. But this is only– this

is a very nice picture. But for many years,

this was just a dream. And because this guy looks

very different from this guy, but this is

difficult. So this has some [? identification is ?]

difficult for the following reasons. So first, so this G just–

say your Feynman diagrams, amplitude for particular

Feynman diagram. So G is typically

expressed as product of field theory propagators. So imagine how you evaluate

the Feynman diagram. The Feynman diagram,

essentially, is just a product of

the [? propagators. ?] And then you integrate

it [INAUDIBLE] integrated over spacetime. So they just take the

Yang-Mills theory. And if you look at the

expression for this diagram, of course, it looks nothing. So they look nothing like– OK. So let me make a few

comments about this thing. Because if you want to match,

say if I gave you a Yang-Mills theory, so I gave you a QCD,

then you can write down– then you can go to large N. You

can write down expressions for the common diagrams. But if you say, I

want to write it as a string theory, the

first thing you have to say, what string theory do

you want to compare? So first you have

to ask yourself what string action do

you want to compare. So the string action, as

we discussed last time, this describes the

embedding of the worldsheet into some spacetime. OK, so this is worldsheet

into a spacetime. So this is also sometimes

called the target space. So this is a spacetime. This string moves. And the mathematical

of this is just the– this is encoded in this

mapping X mu sigma tau. OK, X mu is the

coordinate for M. And then sigma tau

is the coordinate as you parameterize

your worldsheet. So in order to write

down action, of course, you have to choice

of space manifold. You have to choose

your spacetime. And also you have to–

when you fix the spacetime, you don’t have a choice. And sometimes the way to write

down such kind of embedding is not unique. The action for such

[? finding ?] is unique, so you only need to choose

what action you include. And also often, in

addition to this embedding, sometimes you can have

additional internal degrees freedom. living on worldsheet. For example, you can

have some fermions. Say if you have a

superstring, then you can have some

additional fermions are living on the worldsheet,

in addition to this embedding. So in other words, the choice

of this guy in some sense is infinite. And without any

clue– so you need some clue to know what to

compare the gauge theory to. And otherwise,

even if this works, you’re searching for some

needle in the big ocean. And then there’s another

very important reason why this is difficult, is

that this string theory is formulated in a continuum. It’s formulated in a continuum. And these Feynman

diagrams, even if they’re corresponding to some

kind of string theory, they correspond into a

discrete version of that. So at best, it’s a

discrete version. So we expect such a

geometric picture for G, for these Feynman

diagrams, to emerge only at strong couplings. OK? Emerge only at strong couplings

for the following reason. So if you look at

the Feynman diagram– so the simplest Feynman

diagram we draw before, say for example

just this diagram. And if you draw

it on the sphere, it separated the sphere

into three parts, OK? So this [? discretizes ?]

a sphere into three parts. And essentially, just as

the sphere just becomes three points, because

each particle is wanting to– when you’re trying

to [INAUDIBLE] each part, you approximate it by one point. So essentially, in

this diagram, you approximate the whole sphere

essentially by three points. OK. And of course, it’s hard to

see your [? magic ?] picture from here. And your [? magic picture ?]

you expect to emerge, but your Feynman diagrams

become very complicated. For example, if you have

this kind of diagram, because of the

four-point vertex. In principle, you can

have all these diagrams. And then this

[INAUDIBLE] [? wanting ?] to discretize– yes, I

suppose this is on the torus. Suppose you have

a– for example, this could be a Feynman

diagram on the torus, OK? For the vacuum [? energy. ?] And now this is next some

kind of proper discretization. And this will go to

a continuum limit, say when the number of

these box go to infinity. When the number of

box go to infinity, then you need a

number of propagators, and the number of vertices

goes to infinity, OK? So in order for

continuum, a picture to emerge, so you

want those complicated diagrams– it’s not your number

of vertices or large number of propagators that dominate. And for those things

that dominate, then you need the

strong coupling. Because with this coupling, this

is the leading order diagram. And there’s no

geometry from here, OK? So in order to

have the geometry, you want the diagram are

very, very complicated, so that they

really– [INAUDIBLE] a triangulation of a surface. A weak coupled diagram with

small number of lines will cause [? one ?] [? and two ?]

are very close triangulization of a surface. So we expect this only appears

in strong couplings, OK? Yeah. AUDIENCE: By the cases like

we have to sum over all the [INAUDIBLE]. HONG LIU: Yeah, sum over

the [INAUDIBLE] diagram. AUDIENCE: Including

those simple ones. HONG LIU: Including

those simple ones. So that’s why you

want to– so if you’re in a weak coupling,

then the simple ones– so we sum all those diagrams. And each diagram you can

associate with a coupling power. So at weak coupling, then

the lowest order term would just dominate. And the lowest order term

have a very simple diagrams. And then that’s because

[? one ?] and [? two ?] are very crude triangulization

over the surface. But if you have a strong

coupling– in particular, if you have an infinite

coupling– the diagrams, the infinite number of

vertices will dominate. And then that’s because

[? one ?] and [? two ?] have very fine triangulization

over the surface. And then that can go


interaction a coupling constant has been [? dragging ?]

out from– HONG LIU: No. That’s just N dragged out. AUDIENCE: Oh, I see. HONG LIU: No, there’s what

we call this [INAUDIBLE] still remaining. By coupling, it’s only [? N. ?] AUDIENCE: [INAUDIBLE] HONG LIU: No, no, this isn’t

to [? hold ?] coupling. In coupling we mean

that [INAUDIBLE]. So example we talk about,

[? because one ?] [? and two ?] [INAUDIBLE]. Yeah, and then we

make more precise. So in the [? toy ?] example

we talked about before. So previously we talked

about this example, N divided by lambda, trace,

say 1/2 partial phi squared, plus 1/4 phi to the power 4. And strong coupling

means the lambda large. Because of the N I’ve

already factored out, so you’re coupling just lambda. AUDIENCE: Oh, I see. HONG LIU: Yes. AUDIENCE: So in

these [INAUDIBLE] the propagator in that

version would become the spacetime integration? HONG LIU: Hm? AUDIENCE: I was

just wondering how the propagator can

[? agree, ?] can match to the spacetime [INAUDIBLE]. HONG LIU: Yeah, yeah. So the propogator–

yeah, propagator you do in the standard way. You just write down

your propagator, and then you try

to repackage that. As the question, you

said, whatever your rule, Feynman rule is we just

do that Feynman rule. And you write down

this expression. It’s something very complicated. And then you say, can I find

some geometric interpretation of that? Yeah, what I’m saying is that

doing from this perspective is very hard because you don’t

know what thing to compare. And further, in the

second, you expect that your [INAUDIBLE]

would emerge only in those very

complicated diagrams. And those complicated diagrams

we don’t know how to deal with. Because they only emerge in

the strong coupling limit, but in the strong

coupling limit, we don’t know how

to deal with that. And so that’s why

it’s also difficult. But [? nevertheless, ?] for

some very simple theories, say, if you don’t consider

the Yang-Mills theory, you don’t consider

the gauge theory. But suppose you do consider

some matrix integrals. Say, for very simple systems,

like a matrix integral. So this structure emphasizes–

this structure only have to do with you

have a matrices, OK? And then you can have

matrix-valued fields [? or ?] this structure will emerge. Or you only have

a matrix integral. So there no field at all,

just have a matrix integral. That same structure

will also emerge. For example. I can consider theory–

have a theory like this. Something like this. And have a theory like this, OK? And M is just some

[INAUDIBLE] matrices. So this is just integral. And the same structure

will emerge, also, in this series when we

do large N expansion. So that structure have nothing

to do– yeah, you can do it. So matrix integral is much

simpler than [INAUDIBLE] field theory because you have

much less degrees freedom. So for simple systems like, say,

your matrix integral or matrix quantum mechanics,

actually, you can guess the corresponding

string theory. Because also the string

theory in that case is also very simple. You can guess where is

simple string theory. But it’s not possible

for field theory. It’s not possible

for field theory. Yes. AUDIENCE: So what do you mean

by matrix quantum mechanics? Like that, OK. HONG LIU: So this is

a matrix integral. And I can make it a little

bit more complicated. So I make this M to

depend on t, and then this become a matrix

quantum mechanics. Say trace M dot squared

plus M squared plus M4. Then this become a

matrix quantum mechanics, because it only have time. And then I can make

it more complicated. I can make M be t, x. Then this becomes one plus

one dimension of field theory. AUDIENCE: So in what context is

this matrix quantum mechanics [? conflicted? ?] HONG LIU: Just at

some [? toy ?] model. I just say, and this is a

very difficult question. You said, I don’t know how

to deal with field theories. Then this [? part of it’s ?]

a simple system. And then just try to

use this philosophy, can see whether it can

do it for simple system. And then you can show that

this philosophy actually works if you do a matrix

integral or matrix quantum mechanics. Simple enough, matrix integral

and matrix quantum mechanics. OK. And if you want

references, I can give you references regarding these. There’s a huge, huge

amount of works, thousands of papers, written on this

subject in the late ’80s and early ’90s. So those [? toy ?] examples

just to show actually this philosophy works. I just showed this

philosophy works, OK? But it’s not possible if we

want to go to higher dimensions. Actually, there’s one paper–

let me just write it here. So this one paper

explains the philosophy. So here I did not

gave you many details, say, how you write this G

down, how you in principle can match with this thing. With [? another ?]

maybe [INAUDIBLE] you can make this discussion

a little bit more explicit, but I don’t have time. But if you want, you can

take a look at this paper. So this paper discusses the

story for the matrix quantum mechanics. But in the section 2

of this paper– so this is a paper by Klebanov. So in the section

2 of this paper, it explains this mapping

of Feynman diagrams to the string action. And this discretization

picture give you a nice summary of that

philosophy with more details than I have given to you. So you can take a look at that. And this paper also has

some other references if you want to

take a look at it. OK. Any questions? Yes. AUDIENCE: Sorry, but who

was the first to realize this connection

between the surfaces in topology of Feynman diagrams? HONG LIU: Sorry? AUDIENCE: Who first

realized this relation between topology and– HONG LIU: So of

course, already when ‘t Hooft invented this

large N expansion, he already noticed that this

is similar to string theory. So he already commented on that. And he already

commented on that. And for many years people

did not make progress. For many years, people

did not make progress. But in the late ’80s–

in the mid to late ’80s, people started thinking

about the question from this perspective,

not from that perspective. So they started to

think about the order from this perspective. Because just typical string

theory are hard to solve, et cetera. So people think,

maybe we can actually understand or generalize our

understanding of string theory by discretize the worldsheets. And then they just

integrate over all possible

triangulization, et cetera. And then they realized

that that thing actually is like something

over Feynman diagrams. And then for the very

simple situations, say like if you have only

a matrix integral, actually you can make the

connection explicit. So that was in the late ’80s. So people like [? McDowell ?]

or [? Kazakov ?] et cetera that were trying to explore that. Other questions? AUDIENCE: I’m having

trouble seeing how the sum over all

triangulations [INAUDIBLE] each surfaces. How does that correspond to

the discrete version of summing over all [INAUDIBLE]? HONG LIU: Right. AUDIENCE: That’s the discrete

sum over all possible [? genus-h, ?] right? HONG LIU: Yeah. I think this is the example. Yeah, let’s consider torus. So a torus is a box with

this identified with this, and this identified with that. OK. And let me first just draw

the simplest partition here. Just draw like that. Yeah. Let me just look at

these two things. So suppose I give each box–

so if I specify each box, say, give a unit area. OK? And I do this one,

I do that one, or I do some other

ways to triangulize it. Then because [? one and two ?]

give a different symmetric to the surface. And then because

[? one and two ?] integrate over all possible

metric on this surface. And they integrate over

all possible metric on this surface, you can

integrate [INAUDIBLE] all possible surfaces. AUDIENCE: In the case of

the strings for example, [? we put some ?]

over the torus here and the torus and

the torus there. HONG LIU: No, no. You only sum over

a single torus. Now, what do you mean by summing

over torus here, torus there? AUDIENCE: I thought like

in the path integral, in the case of the

string theory– HONG LIU: No, you’re only

summing over a single torus. You’re only summing

over a single surface, but all possible ways to

write– all possible ways to draw that surface. So what you said about

summing torus here, summing torus there, because

[INAUDIBLE] what we call the disconnected amplitudes. And then you don’t need to

consider them in physically disconnected amplitude. You can just

[? exponentiate ?] what we call by connected amplitude. And you don’t need to

do that separately. So once you know how

to do a single one, and the disconnected one

just automatically obtained by [? exponentiation. ?] AUDIENCE: [INAUDIBLE] HONG LIU: Sorry? No, no. Here the metric matters,

the geometry matters. It’s not just the topology. AUDIENCE: [INAUDIBLE]

Feynman diagram [INAUDIBLE]? HONG LIU: Yeah. Yeah, just the key is that

the propagator of the Feynman diagram essentially

[? encodes ?] the geometries. And in encoding a

very indirect way. Yeah. Just read this part. This section only

have a few pages, but contain a little bit more

details on what I have here. It requires maybe one more hour

to explain this in more detail. Yeah, this is just that. I just want to explain

this philosophy. I don’t want to go through the

details of how you do this. OK, good. So now let me just mention

a couple of generalizations. So the first thing

you already asked before, I think maybe

both you have asked. Let me just mention

them quickly. And if you are interested,

I can certainly give you a reference for

you to read about them, or I can put it in

[? your P ?] sets. And so, so far, it’s all

matrix-valued fields, OK? But if you can see the

theory– or in other words, in the mathematical

language, say, it’s an adjoint representation. It’s an adjoint

representation of the– because our symmetries are

UN, it’s a UN gauge group. OK? UN gauge group. But you can also, for example,

in QCD, you also have quarks. So you also have field in the

fundamental representations. So it can also include field in

the fundamental representation. So rather than matrix-valued,

they’re N vector. OK, they’re N [? vectors. ?] So for quarks, of course, for

the standard QCD N will be 3, so you have three quarks. You have three different

colored quarks. And so then your

Feynman diagrams, in addition to have

those matrix [? lines, ?] which you have a double line. And now here you only

have a single index, OK? And then you only

have a single line. So the propagator

of those quarks will just have a single line. And then also in

your Feynman diagram you can have loops over

the quarks, et cetera. So you can again work this out. And then you find it is a

very nice large N expansion. And then you find the

diagrams, the Feynman diagrams. Now you find in this

case the Feynman diagrams can be classified by

2D surfaces with boundaries. So essentially, you have– and

let me just say, for example, this is the vacuum diagrams,

for all the vacuum process. Then you can [INAUDIBLE]

or the vacuum diagrams. And then they can all

be [? collectified. ?] So previously, we have

a matrix-valued field. Then all your vacuum

diagrams, they are corresponding closed

surfaces– so sphere, torus, et cetera. But now if you

include the quarks, then those surfaces

can have boundaries. And then [INAUDIBLE] into

the quark groups, et cetera. And then they [? cannot ?]

be classified. And so these also

have a counterpart if you try to map to

the string theory. So this [INAUDIBLE]

[? one and ?] [? two, ?] string theory. There’s string theory with

both closed and open strings. And so essentially

those boundaries give rise to the open strings. So here, it’s all

closed strings. It’s all closed surface. Well, now you can, by

adding the open strings, and then you can, again,

have the correspondence between the two. OK. So all the discussion

is very similar to what we discussed before. We just apply all this the

same philosophy to the quarks. Yes. AUDIENCE: [INAUDIBLE] do the

same trick on string theory and find some sort

of expression which then will map to some higher

order surfaces, [INAUDIBLE]? HONG LIU: Sorry, say that again? AUDIENCE: [INAUDIBLE]

Feynman diagrams we move to string

theory for surfaces. Is there some [INAUDIBLE]

from surfaces just they go one more [? step up? ?] HONG LIU: You mean higher

dimensions, not strings. Yeah, that will

become– of course, that’s a [? lateral ?] idea. So that will [INAUDIBLE] you can

consider [? rather ?] strings, you can consider

two-dimensional surface, a two-dimensional surface

moving in spacetime. And then [INAUDIBLE] into

[? so-called ?] the membrane theory. But let’s say where it

turns out to be– turns out string is a nice balance. It’s not too complicated

or not too simple. And it give you

lots of structure. But when you go to

membrane, then the story become too complicated,

and nobody knows how to quantize that theory. So the second remark is

that here we consider UN. So here our symmetry

group is UN. Because our phi– phi there is

[? commission. ?] So when you have a [? commission ?] matrix,

then there’s a difference between the two indices, so

we put one up and one down. So they are

propagators that lead to– so it leads to

the lines with arrows, because we need to distinguish

upper and lower indices. OK? Between the two indices. But you can also

consider, for example, phi is a symmetric matrix. Say it’s a real

symmetric matrix. It’s a real symmetric,

or real anti-symmetric. In those cases, then

there’s no difference between the two indices. And then when you draw a

propagator– so in this case the symmetry group

would be, say, SON, say, or SPN, et cetera. And then the propagators,

they will no longer have orientations. OK? They will no longer

have orientations. Because you can

no longer– yeah. So this will give rise–

so let me write it closer. So this will give rise

to unorientable surfaces. Say, for example, to

classify the diagrams, you can no longer just use

the orientable surfaces. You also have to include

the non-orientable surfaces to classify the diagrams. And the [INAUDIBLE] this also

have a precise counterpart into unorientable strings. No, non-orientable strings. Yeah, I think non-orientable,

non-orientable surfaces. Also non-orientable strings. Good. So I’m emphasizing

how difficult it is if, say, we want

to start with QCD and then try to find the

string theory description. But this still, [? none of ?]

this tries– I just try. OK, so let’s just consider,

just take large N generalization of QCD. So this, again, will be

some UN gauge theory, UN Yang-Mills theory, say, in

3 plus 1 dimensional Minkowski spacetime. And can we say anything about

its string theory description? So [INAUDIBLE]. So maybe it’s difficult,

but let’s try to guess it. OK. So in physics, in

many situations, a seemingly difficult problem,

if you know how to guess it, actually you can get the answer. On, for example,

quantum hole effects, fractional quantum hole

effects, you can just guess the wave function. So of course, the

simplest guess– so this is some gauge theory in

3 plus 1 dimensional Minkowski spacetime. So now we say this

is a string theory. So natural guess

is that this maybe is a string theory,

again, in the 3 plus 1 dimensional Minkowski spacetime. OK? So we just take what–

so these will, of course, run into a string, propagating

in this spacetime, OK? As I said, when you write

down the string theory, you first have to specify

your target space, which, as the string moves, the

larger question would be just, should it be the gauge

theory’s Minkowski spacetime. Maybe this string

theory should be. OK? And then this. Then you can just try

to– then you can just write down the simplest action. So maybe say Nambu-Goto action,

which we wrote last time, OK? Or the [? old ?]

Polyakov action. So this Nambu-Goto action will

result [INAUDIBLE] Polyakov. And let me not worry about that. For example, you can

just guess, say, maybe this is a string theory also

in the Minkowski spacetime. Say, consider the

simplest action. Or the equivalent of this, OK? Then at least what

you could try– now you actually have an action. Now you think that

you have this object. Now you think you can compare. OK, now you can

essentially compare. Say, in QCD you calculated

your Feynman diagrams, and now just compare. But of course, you still

have the difficulty. Of course, you have to

go to strong coupling to see the geometric

limit, et cetera. But in principle, it’s

something you can do. But this actually does not work. OK? This does not work, for the

following simple reason. Firstly, that such

a string theory– so a string theory, actually

the remarkable thing about the string is that

if you have a particle, you can put the particle

in any spacetime. But strings are very picky. You cannot put them

in any spacetime. And they can only

propagate consistently, quantum mechanically

consistently, in some spacetime but not in others. So for example, if you

want to put the string to propagate in this 3 plus

1 dimensional Minkowski spacetime, then you actually

find that the theory is mathematically inconsistent. So such a string

theory is inconsistent. It’s mathematically

inconsistent. Except for the D

equal to 26 or 10. OK? So 26 if you just purely

have the theory, and 10 if you also add some fermion. So such a string theory does

not exist mathematically. So you say, oh, OK. You say, I’m a smart fellow. I can go around this. Because we want the

Minkowski spacetime. Because those gauge theory

propagating the Minkowski spacetime, so this Minkowski

[INAUDIBLE] must be somewhere. They cannot go away, because

all these glueballs [INAUDIBLE] in this 3 plus 1 dimensional

Minkowski spacetime. And if we want to identify the

strings with those glueballs, those strings must at

least [? know ?] some of this Minkowski spacetime. And then you say, oh,

suppose you tell me that this string theory is only

consistent in 10 dimension. But then let me take a string

theory in 10 dimensions, which itself consistent. But I take this 10-dimensional

spacetime to have the form of a 3

plus 1 dimensional Minkowski spacetime. And the [? time, ?] some

compact manifold, OK? Some compact manifold. And in such case– so if

this is a compact manifold, then the symmetry

of this spacetime, so the spacetime

symmetry still only have the 3 plus 1 dimensional,

[? say, ?] Poincare symmetry. Because if you want to describe

the QCD in 3 plus 1 dimension, QCD has the Poincare symmetry. You can do Lorentz

transformation, and then you can do rotation. Or you can do translation. The string theory should not

have more symmetries or less symmetries than QCD. They should have

the same symmetries because they are supposed

to be the same description. But if you take the

10-dimensional Minkowski space, of course, it’s not right. Because the

10-dimensional Minkowski space have 10-dimensional

translation and 10-dimensional

Lorentz symmetry. But what you can do is that you

take this 10-dimensional space to be a form of the 3 plus

1 dimensional Minkowski spacetime and times some

additional compact manifold, and then you have solved

the symmetry problem. But except this

still does not work because the string theory, as

we know, always contain gravity. And if you put a string theory

on such a compact space N, [? there would be ?]

always leads to a massless spin-2 particle in

this 3 plus 1 dimensional part. But from Weinberg-Witten theorem

we talked in the first lecture, in the QCD you are

not supposed to have a 3 plus 1 dimensional

massless spin-2 particle, OK? And so this won’t work. So this won’t work. Because this contains– In 3 plus 1 dimensional

[? Minkowski space, ?] which does not have– OK? Or in the large N [INAUDIBLE]. So this does not work. So what to do? Yes? AUDIENCE: So does this just

mean that it’s mathematically inconsistent? HONG LIU: No, no. This does not mean it is

mathematically inconsistent. It just means this string

theory cannot not correspond to the string theory

[? describe ?] QCD. The string theory description–

the equivalent string theory for QCD cannot

have this feature. Yeah, just say this cannot

be the right answer for that string theory. This string theory

is consistent. Yes. AUDIENCE: So is

that you were saying if there is a massless spin-2

particle in that string theory, there has to be a

[? counterpart in the ?] QCD. HONG LIU: That’s right. AUDIENCE: If there is not a

[INAUDIBLE], that won’t work HONG LIU: Yeah. This cannot be a

description of that. From Weinberg-Witten theorem,

we know in QCD there’s no massless spin-2 particle. Yes. AUDIENCE: I thought we have

talked about maybe we can do strings to [? find ?] QCD in

a different dimension [? in ?] space. HONG LIU: We will go into that. But now they are in

the same dimension, because this

Minkowski 4, this will have– because this is a compact

[? part, ?] it doesn’t matter. So in this part, [? there are ?]

massless spin-2 particles. This does not

[? apply ?] in QCD. So what can you do? So most people just give up. Most people give up. So other than give up,

the option is say maybe this action is too simple. Maybe you have to look

at more exotic action. OK. So this is one possibility. And the second possibility

is that maybe you need to look for some

other target space. OK. But now, what if you

go away from here? Once you go away from

here, everything else is now becoming such

little in the ocean, because then you don’t

have much clue what to do. We just say, your basic

guess just could not work. So for many years, even though

this is a very intriguing idea, people could not make progress. But now we have hindsight. But now we have hindsight. So we know that even this

maybe cannot be described by a four-dimensional– so

even though this cannot have a– so this cannot have a

massless spin-2 particle in this 3 plus 1 dimension

of Minkowski spacetime. Maybe you can still have

some kind of graviton in some kind of a

five-dimensional spacetime. You have some five dimensions,

in a different dimension. So there were some rough hints. Maybe you can consider there’s a

five-dimensional string theory. So let me emphasize when

we say five or four, I always mention the

non-compact part. So the compact part, it doesn’t

count because compact part just goes for the ride. What determines the

properties, say, of a massless

particle, et cetera, is the uncompact

of the spacetime. Yeah, because this is a

10-dimensional spacetime. This is already not [INAUDIBLE]. So maybe we [? change ?]

for string theory in five-dimensional uncompact. AUDIENCE: Five, so in 4 plus 1? HONG LIU: Yeah. In 4 plus 1 uncompact spacetime. Yes. AUDIENCE: [INAUDIBLE]

compactors. When you say

compact, do you mean the mathematical

definition of compactness? HONG LIU: Yeah, that’s right. Yeah, I just say there

is a finite volume. Just for our purpose

here, we can do it simply. Just let’s imagine–

yeah, compact always has a finite

volume, for example. Yes? AUDIENCE: Why can we just

ignore the compact dimensions? Is there any condition on

how big they’re allowed to be or something, like limit? HONG LIU: Yeah, just

when you have– so if you know a little

bit about this thing called the Kaluza-Klein theory. And you know that the

compact part– the thing is that if you have

a theory [? based ?] on uncompact and

the compact part, and then most of the

physical properties is controlled by the

physics of uncompact parts. And this will

determine some details like the detailed

spectrum, et cetera. But the kind of

thing we worry about, whether you have this massless

spin-2 particle, et cetera, will not be determined

by this kind of thing. AUDIENCE: Is there

any volume limit on the compact

part, like maximum? HONG LIU: No, it’s fine

to have a finite volume. AUDIENCE: Just finite,

but can it be large? HONG LIU: No matter how

large, this have infinite. It’s always much

smaller than this one. Yeah, but now it’s

just always relative. It’s always relative. Yes. AUDIENCE: Tracking

back a little, is there any quick explanation

for 26 and 10 are special, or is it very complicated? HONG LIU: Um. [LAUGHTER] No, it’s not complicated. Actually, we were going

to do it in next lecture. Yeah, next lecture we will

see 26, but maybe not 10. 10 is little bit

more complicated. Most people voted

for my option one, so that means you will

be able to see the 26. Right. AUDIENCE: Who

[? discovered ?] 26 and 10? I mean, they are specific

for this [INAUDIBLE] action rate, so for other action

would be something else. HONG LIU: Specifically for

the Nambu-Goto action is 26. And for the 10, you need to

add some additional fermions and make it into a so-called

superstring, then become 10. And even this 26 one is not

completely self-consistent. And anyway, there’s still

some little, tiny problems with this. Anyway, so normally we use 10. OK so now, then there’s

some tantalizing hints for the– say, maybe you

cannot do it with the 3 plus 1 dimensional uncompact spacetime. Maybe you can do a 4 plus

1 dimensional uncompact. So the first is the

holographic principle, where you have length. Holographic principle we have

learned because there we say, if you want to describe

a theory with gravity, then this gravity should

be able to be described by something on its boundary. And the string theory is

a theory with gravity. So if the string

theory should be equivalent to some kind of

QCD, some kind of gauge theory without gravity, and then

from holographic principle, this field theory maybe should

be one lower dimension, OK? In one lower dimension. Is the logic here clear? AUDIENCE: Wait, can

you say that again? HONG LIU: So here we

want to equate large N QCD with some string theory. But string theory we

know contains gravity. A list of all our

experience contain gravity. But if you believe that

the gravity should satisfy holographic principle, then the

gravity should be equivalent, according to

holographic principle, gravity in, say, D

dimensional spacetime can be described by

something on its boundary, something one dimension lower. AUDIENCE: But I thought

the holographic principle was a statement about entropy. HONG LIU: No, it’s

a state started from a statement about entropy. But then you do a

little bit of leap. So what I call it little

bit of a conceptual leap is that the– or

[? little ?] leap of faith is that you promote that

into the statement that said the number

of degrees freedom you needed to describe

the whole system. Yeah, so the

holographic principle is that for any region, even

the quantum gravity theory, for any region, you should

be able to describe it by the degrees of freedom living

on the boundary of that region. And degrees freedom living on

the boundary of that region, then it’s one dimensional lower. AUDIENCE: Wait, so can I

ask one question about that? If I have some region,

some volume in space, some closed ball or something. And I live in a universe which

is, for example, a closed– like maybe they live

on some hypersphere or something like this. Then how do I know whether

I’m– how do I know that the information is encoded? How do I know whether

I’m inside the sphere or outside of the sphere? For example, we see

that the entropy that has to do with the

sphere basically tells you about how

much information can you contain inside the sphere. But if you live in a universe

which is closed or something, then you don’t

know whether you’re inside or outside the sphere. HONG LIU: Yeah, but that’s

a difficult question. Yeah, if you talk about

closed universe here, we are not talking

about closed universe. AUDIENCE: I see. HONG LIU: Yes. AUDIENCE: I thought the

holographic principle is that the number of degrees

freedom inside the region is actually bounded by the area. HONG LIU: Right,

it’s bounded by– AUDIENCE: Yeah, but

why is it that we use that degree of freedom

living on the boundary? HONG LIU: There are several

formulations of that. First is that the total

number of degrees freedom in this region is

bounded by the area. And then you can go to the

next step, which is maybe the whole region can

be just described by these degrees

of freedom living on the boundary on that region. AUDIENCE: Is that because,

say, the state of density on the boundary [INAUDIBLE]

the state on the boundary is proportional to the

area of the boundary? HONG LIU: Yeah. Exactly. That’s right. AUDIENCE: So here our goal is to

recover the large N theory in 3 plus 1 dimensions

without gravity. So we have no gravity. You can’t 3 plus 1. HONG LIU: Right. Yeah, so if that is supposed

to be equivalent to the gravity theory, and the

gravity [? theory ?] to find the

holographic principle, and then the natural

guess is that this non-gravitational

field theory should live in one dimensional lower. OK? So this is one hint. And the second is actually

from the consistency of string theory itself. So this is a little

bit technical. Again, we will only

be able to explain it a little bit later, when we talk

about more details about string theory. You can [? tell, ?] even though

the string theory in this space is inconsistent. But there’s a simple way. This is– it’s not a simple way. So what’s happening

is the following. So if you consider,

say, a string propagating in this spacetime,

and there are some symmetries on the worldsheet. And only in the 10

and 26 dimension, those symmetries are satisfied

quantum mechanically. And in other dimensions,

those symmetries, somehow, even though

classically it’s there, but quantum

mechanically it’s gone. And those symmetries

become– because they are gone quantum

mechanically, then it leads to inconsistencies. And it turns out that

there’s some other way you can make that consistent,

to make that symmetry still to be valid, is by adding

some new degrees of freedom. OK? It’s just there’s

some new degrees freedom dynamically generated. And then that new

degrees freedom turned out to behave like

an additional dimension. OK. Yeah, this will make

no sense to you. I’m just saying a consistency

of string theory actually sometimes can give you

one additional dimension. AUDIENCE: What is the difference

between these inconsistencies, talking about anomalies and– HONG LIU: It is anomalies. But here it’s called

gauge anomalies. It’s gauge anomalies is at

the local symmetry anomalies, which is inconsistent. AUDIENCE: So just–

maybe this is not the time to ask this– but

are the degrees of freedom that you need to save you from

this inconsistency problem. So do they have to be

extra dimensions of space? Or what I’m saying is that if we

need to do string theory in 10 dimensions, is it really

four dimensions plus six degrees of freedom? Or are they actually six

bona fide spatial dimensions? HONG LIU: Oh, this is

a very good question. So if you have–

yeah, this something we would be a little bit more

clear just even in– oh, it’s very late. Even the second

part of this lecture is that here you have

four degrees of freedom, you have six degrees of freedom. But turns out, if you

only consider this guy, then this four degrees freedom

by itself is not consistent. It’s [? its own ?] violation

of the symmetry at the quantum level. And then you need to add

more, and then one more, because of course one and

two have extra dimension. Anyway, we can make it more

explicit in next lecture. Here I just throw a remark here. Anyway, this guy– this

is purely hindsight. Nobody have realized this

point, this first point, nobody have realized it before

this holographic duality was discovered. Nobody really made

this connection. And at this point,

saying there should be a five-dimensional

string theory describing gauge theory,

that was made just before the discovery. I will mention that

a little bit later. Anyway, so now let’s– let

me just maybe finish this, and we have a break. So now let’s consider–

suppose there is a five-dimensional

spacetime, string theory in some five-dimensional

spacetime, say 4 plus 1 dimensional

spacetime that describes QCD. Then what should be

the property of this Y? So this Y denotes

some manifold Y. OK? So as I mentioned, it must have

at least all the symmetries of the QCD, but not more. Should have exactly the

same amount of symmetries. So that means it must have

the translation and Lorentz symmetries of QCD. OK? So that means the only

metric I can write down must be of this form. The only metric

I can write down, the metric must

be have this form. So this az just some function. And z is the extra dimension

to a Minkowski spacetime. And this is some Minkowski

metric for 3 plus 1 dimension. AUDIENCE: You mean it’s like

a prototype to four dimension, we have to get the

Minkowski space. HONG LIU: Yeah. Just say whatever

this space, whatever is the symmetry of this– so

the symmetry of this spacetime must have the

Poincare– must have all the symmetry of the

3 plus 1 dimensional Minkowski spacetime. Then the simplest way, you’re

saying that the only way to do it is just you put the

Minkowski spacetime there as a subspace. And then you have

one additional space, and then you can have

one additional dimension. And then, because

you have to maintain the symmetries and [INAUDIBLE]

to be thinking then you can convince yourself that

the only additional degrees freedom in the

metric [INAUDIBLE] is the overall function. So the function of this

z, and nothing else. OK. AUDIENCE: Can that

be part of kind of a scalar in Minkowski space? HONG LIU: Yeah. Let me just say,

this is most general metric, consistent with

four dimensional, 3 plus 1 dimensional,

Poincare symmetries. AUDIENCE: Why this additional

dimension always in a space part? Can it be in a time-like part? Like a 3 plus 2? HONG LIU: Both arguments

suggest it’s a space part. So because this is just

the boundary of some region there’s a spatial dimension

[? reduction ?], not time. So is this clear to you? Because you won’t have

a Minkowski spacetime, so you must have

a Minkowski here. And then in the prefactor

of the Minkowski, you can multiply by

anything, any function, but this function

cannot depend on the X. It can only depend on

this extra dimension. Because if you have anything

which depend on capital X, then you have violated

the Poincare symmetry. You have violated the

translation [? X. ?] So the only function you can put

before this Minkowski spacetime is a function of this

additional dimension. And then by redefining

this additional dimension, I can always put this

overall factor in the front. Yeah, so this

tells you that this is the most general metric. OK? So if it’s not clear to you,

think about it a little bit afterwards. So these are the

most as you can do. So that’s the end. So you say, you cannot

determine az, et cetera. So this is as most you

can say for the QCD. But if the theory, if the field

theory is scale invariant, say, conformal field theory,

that normally we call CFT, OK? So conformal field theory. Then we can show this metric. So let me call this equation 1. Then 1 must be


symmetry on the boundary as well, [INAUDIBLE]? HONG LIU: Yeah, I’m

going to show that. So if the field theory

is scale invariant, that means that the fields theory

have some additional symmetry, should be satisfied

by this metric. And then I will show that this

additional scaling symmetry will make this to precisely

a so-called anti-de Sitter spacetime. AUDIENCE: Field theory,

and then the 3 plus 1. HONG LIU: Yeah. Right. If the field theory, say the–

QCD does not have a scale. It’s not scaling right, so

I do not say a QCD anymore. Just say, suppose

some other field theory, which have

large N expansion, which is also scale invariant. And then the corresponding

string theory must be in anti-de

Sitter spacetime. AUDIENCE: Are we ever

going to come back to QCD, or is that a– HONG LIU: No, that’s it. Maybe we’ll come back to QCD,

but in a somewhat indirect way. Yeah, not to your

real-life, beloved QCD. AUDIENCE: So no one’s

solved that problem still? HONG LIU: Yeah, no one’s

solved that problem yet. So you still have a chance. So that remains very simple. So let me just say, then

we will have a break. Then we will be done. I think I’m going

very slowly today. So scale invariant theory–

is invariant under the scaling for any constant,

constant lambda. So scale invariant

theory should be invariant under such a scaling. And then now we want to

require this metric also have this scaling. OK? So now, we require 1

also have such scaling. That’s scaling symmetry. OK, so we just do a scaling

X mu go to lambda X mu. And then this term will give

me additional lambda squared. So we see, in order for this

to be the same as before, the z should scale the same, OK? So in order for this to be– so

we need z to scale as the same, in order I can scale

this lambda out. After I scale this

lambda out, I also need that a lambda z should be

equal to 1 over lambda az, OK? So the scaling symmetry

of that equation requires these two conditions. So on the scaling of

z, this a lambda z should satisfy this condition. Then the lambda will cancel. So this condition is

important because we did scale them homogeneously. Otherwise, of course,

lambda will not drop out. And the second condition just

makes sure lambda is canceled. OK, is it clear? So now this condition

just determined that az must be a

simple power, must be written as R divided by z. See, R is some constant. And now we can write

down the full metric. So now I’ve determined

this function up to our overall constant. So the full metric is dS square

equal to R squared divided by z squared dz squared plus

eta mu, mu, dX mu, dX mu. And this is

precisely AdS metric, written in certain coordinates. And then this R, then you adjust

the curvature radius of AdS. So if you don’t know about

anti-de Sitter spacetime, it doesn’t matter. So this is the metric, and

the name of this metric is anti-de Sitter. And later we will

explain the properties of the anti-de Sitter spacetime. So now we find, so now

we reach a conclusion, is that if I have a

large N conformal field theory in Minkowski

D-dimensional space, time. So this can be applied

to any dimensional. It’s not necessary

[? to be ?] 3 plus 1. In D– so this, if it can be

described by a string theory, should be string

theory in AdS d plus 1. And in particular, the 1/N here

is related to the g strings here, the string coupling here. So this is what we concluded. Yes? AUDIENCE: So all we’ve

shown is that there is no obvious inconsistency

with that correspondence. HONG LIU: What do you

mean there’s no obvious? AUDIENCE: As in, we didn’t

illustrate any way that they– HONG LIU: Sure, I’m just saying

this is a necessary condition. AUDIENCE: Right, so at

least that is necessary. HONG LIU: Yeah, this is

a necessary condition. So if you can describe a large

N CFT by our string theory– and it should be a

string theory– yeah, this proposal works. This proposal passed

the minimal test. AUDIENCE: I have a question. So when Maldacena presumably

actually did figure this out, you said that this resulted

from the holographic principle, like it was just figured

out right before he did it. Was he aware of

the holographic– HONG LIU: No, here is

what I’m going to talk. So Maldacena, in 1997, Maldacena

found precisely– in 1997, Maldacena found a few examples

of this, precisely realized this. And not using this mass or using

some completely indirect way, which we will explain next. So he found this through

some very indirect way. But in principle, one

could have realized this if one kept those

things in mind. So now let me tell you a

little bit of the history, and then we will have a break. Then we can go home. It depends on whether

you want a break or not. Maybe you don’t want a break. Yeah, let me tell you a

little bit of history. So yeah, just to save time,

let me not write it down, just say it. So in the late ’60s

to early ’70s, so string theory was developed to

understand strong interactions. So understanding strong

interactions was the problem. At the time, people were

developing string theory to try to understand

strong interactions. So in 1971, our friend

Frank, Frank Wilczek, and other people, they discover

the asymptotic freedom. And they established

the Yang-Mills theory as a description of

strong interaction which now have our QCD. And so that’s essentially

eliminated the hope of string theory to describe QCD. Because the QCD seems

to be very different. You [? need ?] the

help of string theory to describe strong

interaction because the QCD [INAUDIBLE] gauge theory, it’s

very different from the string theory. So people soon abandoned

the string theory. So now we go to 1974. So 1974, a big number of

things were discovered in 1974. So 1974 was a golden year. So first is ‘t Hooft realized

his large N expansion and then realized

that this actually looks like string theory. And then completely

independently, Scherk, Schwarz,

and [? Yoneya, ?] they realized that

string theory should considered a theory of

gravity, rather than a theory of strong interaction. So they realized

actually– it’s ironic, people started doing string

theory in the ’60s and ’70s, et cetera. But only in 1974

people realized, ah, string theory

always have a gravity and should be considered

a theory of gravity. Anyway, so in

1974, they realized the string theory should

be considered as a gravity. So that was a very, very

exciting realization, because then you can have

[? quantum ?] gravity. But by that time, people had

given up on string theory. So nobody cared about this

important observation. Nobody cared about this

important observation. So, also in the

same year, in 1974, Hawking discovered

his Hawking radiation. And they established that

black hole mechanics is really a thermodynamics. Then really established

that the black hole is a thermodynamic object, And in 1974 there’s also a

lot of important discovery– which is related

to MIT, so that’s why I’m mentioning it– is that

people first really saw quarks experimentally, is that, again,

our friend, colleague Samuel Ting at Brookhaven,

which they discovered a so-called charmonium, which

is a bounce state of the charm quark and the anti-charm quark. And because the charm

quark is very heavy, so they form a

hydrogen-like structure. So in some sense, the charmonium

is the first– you first directly see the quarks. And actually, even after the

1971, after asymptotic freedom, many people do not believe QCD. They did not believe in quarks. They say, if there’s quarks,

why don’t we see them? And then in 1974, Samuel Ting

discovered this charmonium in October. And so people call it

the October Revolution. [LAUGHTER] Do you know why they laugh? OK. Anyway. Yeah. Yeah, because I

saw your emotions, I think you have

very good composure. Anyway, in the same

year, in 1974, Wilson proposed what we now

call the lattice QCD, so he put the QCD

on the lattice. And then he

invented, and then he developed a very

beautiful technique to show from this putting

QCD on the lattice that, actually, the quark can

be confined through the strings. So the quarks in QCD can be

confined through the strings. And that essentially

revived the idea maybe the QCD can be a string

theory, because the quarks are confined through the strings. And this all happened in 1974. So then I mentioned the

same, in the late ’80s and the early ’90s,

people were looking at these so-called

matrix models, the matrix integrals, et cetera. Then they showed they related

to lower dimensional string theory. But nobody– yeah, they

showed this related to some kind of lower

dimensional string theory. And then in 1993 and

1994, then ‘t Hooft had this crazy idea of

this holographic principle. And he said maybe, things

about the quantum gravity can be described by things

living on the boundary. And again, it’s a crazy idea. Very few people paid

attention to it. But the only person who picked

it up is Leonard Susskind. And then he tried to come up

with some sort of experiments to show that that

idea is not so crazy. Actually, Susskind wrote a

very sexy name for his paper. It’s called “The

World As a Hologram.” And so that paper

received some attention, but still, still, people did

not know what to make of it. And then in 1995, Polchinski

discovers so-called D-branes. And then we go to 1997. So in 1997, first in

June, so as I said, that QCD may be some

kind of string theory. This idea is a

long idea, starting from the ‘t Hooft and

large N expansion, and also from the Wilson’s picture

of confining strings from the lattice QCD, etc. But it’s just a

very hard problem. If from QCD, how can you

come up with a string theory? It’s just very hard. Very few people

are working on it. So in 1997, in June,

Polyakov finally, he said, had a breakthrough. He said that this

consistent [? of ?] string theory give you

one extra dimension, you should consider a

five-dimensional string theory rather than a

four-dimensional string theory. And then he gave up

some arguments, anyway. And he almost always actually

write down this metric And maybe he already wrote down

this metric, I don’t remember. Anyway, he was

very close to that. But then in November,

then Maldacena came up with this idea of CFT. And then he provided

[? explicit ?] examples of certain large

N gauge theories, which is scale invariant

and some string theory in certain

anti-de Sitter spacetime. And as I said, through

the understanding of these D-branes. But even Maldacena’s

paper, he did not– he was still thinking from

the picture of large N gauge theory corresponding

to some string theory. He did not make the connection

to the holographic principle. He did not make a connection

to the holographic principle. But very soon, in February

1998, Witten wrote the paper, and he made the connection. He said, ah, this is precisely

the holographic principle. And this example, he said,

ah, this example is precisely the holographic

principle Susskind and ‘t Hooft was talking about. So that’s a brief history

of how people actually reached this point. So the next stage,

what we are going to do is to try to derive [INAUDIBLE]. So now we can– as I

said, we have two options. We can just start

from here, assuming there is CFT [? that’s ?]

equivalent to some string theory. And then we can see how we

can develop this further. And this is one

option we can take. And our other option

is to really see how this relation actually

arises from string theory. And many people voted

for the second option, which in my [? email ?]

is option one. So you want to see

how this is actually deduced from string theory. So now we will do that, OK? But I should warn you, there

will be some technicality you have to tolerate. You wanted to see how

this is derived, OK? So we do a lot of

[? 20 ?] minutes today? Without break? Good. OK. Yeah, next time, I

will remember to break. OK. So now we are going

to derive this. So first just as

a preparation, I need to tell you a little

bit more about string theory. In particular, the

spectrum of closed strings, closed and open strings. And so this is

where the gravity– and from a closed string

you will see the gravity, and from the open string, you

will see the gauge theory. OK. We will see gravity

and gauge theory. So these are the first

things we will do. So the second thing we will do–

so the second thing we will do is to understand the

physics of D-branes. So D-brane is some

object in string theory. And it turned out to play

a very, very special role, to connect the gravity

and the string theory. OK. Connect the gravity

and the string theory. Because this is the

connection between the gravity and the string theory. And in string theory,

this [? object will ?] deeply and precisely

play this role, which connects the gravity

and the string theory. So that’s why you can

deduce such a relation. OK. Yeah, so this is the

two things we will do before we can derive this. So this is, say,

the rough plan we will do before we can

derive this gravity. So first let’s tell you a little

bit more about string theory. So at beginning, just say

some more general setup of string theory. So let’s consider a string

moving in a spacetime, which I denote by M, say, with

the metric ds squared equal to g mu mu. And this can depend

on X, dX mu, dX mu. OK? So you can imagine some

general curved spacetime. Say mu and nu will go

from 0 to 1, to D minus 1. So D is the total number of

space dimensions for this M. So the motion of the string, as

we said quite a few times now, is the embedding of the

worldsheet to the spacetime. So this is in the form

of X mu sigma tau. OK, you parameterize the

worldsheet by two coordinates. So I will also write

it as X mu sigma a. And the sigma a is equal

to sigma 0, and the sigma 1 is equal to tau sigma, OK? And we will use this notation. So now imagine a surface

embedded in some spacetime. And this is the

embedding equation. Because if you know

those functions, then you know precisely how

the surface are embedded, OK? And because the original

spacetime have a metric, then this induced metric

on the worldsheet. And this induced metric is

very easy to write down. You just plug in this

function into here. And when you take

the derivative, you only worry

that sigma and tau, because then that means you’re

restricted on the surface, when your only

[? value is ?] sigma and tau. And then you can

plug this into there. So you can get the metric, then

can be written in this form. Here’s sigma a and this sigma b. OK? So remember, sigma a and

sigma b just tau and sigma. And this hab is just equal to

g mu mu, X, partial a, X mu, partial b, X nu. OK? So this is trivial to see. Just plug this into there,

to the variation with sigma and tau, you just get

that, and it’s that. OK? Is it clear? So this Nambu-Goto action

is the tension– tension we always write this 1

over 2 pi alpha prime– dA. So alpha prime is the

[INAUDIBLE] dimensions square. So we often also write

alpha prime as ls square. So alpha prime, just

a parameter, too. Parameterize to [? load ?]

the tension of the string. So this area, of

course, you can just write it as d squared sigma. So again, you use the

notation d squared sigma just d sigma d tau. d squared sigma minus delta h. OK. So this is just the

area, because this is the induced metric

on the worldsheet. Then you take the determinant,

and that give you the area. So this is the standard

geometric formula. So now let me call

this equation 1. So I have a [? lot ?]

equation 1 before, but this is a new chapter. OK. So this is the explicit form

of this Nambu-Goto action. But this action is a

little bit awkward, because involving

the square root. A square root, it’s

considered to be not a good thing in physics. Because when you

write down action, because it’s a non-polynomial. We typically like

polynomial things. Because the only integral we

can do is a Gaussian integral, and the Gaussian is polynomial. So this is inconvenient, so one

can rewrite it a little bit. So you write down the answer. So we can rewrite it

in the polynomial form. And this polynomial

form is corresponding– it’s called the Polyakov

action, so I call it SP, even though Polyakov had

nothing to do with it. And this action can be

written in the following form. And let me write

down the answer. Then I will show the equivalent. AUDIENCE: Wasn’t it invented

by Leonard Susskind? HONG LIU: No, it’s

not Leonard Susskind. [INTERPOSING VOICES] AUDIENCE: Why is it

called Polyakov– HONG LIU: Polyakov–

yeah, actually Polyakov had something to do with it. Polyakov used it mostly

[INAUDIBLE] first. OK, so you can rewrite

it as that, in this form. And the gamma ab is a

new variable introduced. It’s a Lagrangian multiplier. OK. So let me point

out a few things. So this structure is

precisely just this hab. So that’s if you look

at this structure, so this structure is

precisely what I called hab. So now the claim is

adding [INAUDIBLE] to original variable

with just X. Now I introduce a new

variable, gamma. And gamma is like a

Lagrangian multiplier, because there’s no

connected term for gamma. So if I eliminate gamma,

then I will recover this. OK, so this is the claim. So now let me show that. This is very easy to see. Because if you just do

the variation of gamma, do the variation of gamma ab. OK. So whenever I wrote in

this is in [? upstairs, ?] it always means the inverse. OK, this is the standard

notation for the metric. So if you look at the equation

of motion, [INAUDIBLE] by variation of this gamma

ab, then what you’ll find is that the gamma ab– just do

the variation of that action. You find the equation

of motion for gamma ab is given by the following. So hab, just that guy. And the lambda is arbitrary

constant, or lambda is arbitrary function. So this I’m sure you can do. You just do the variation. You find that equation. So now we can just verify

this actually works. When you substitute this

into here, OK, into here. So this gamma ab, when you take

the inverse, then [? cause ?] one into the inverse, hab,

inverse hab contracted with this hab just give you 2. And that 2– did I put

that 2 in the right place? That gave you 2. And that have a 2 on– yeah,

I’m confused about 2 now. Oh, no, no, it’s fine. Anyway, so this

contracted with that, so gamma ab contracted with hab

give you 2 divided by lambda, times 2. OK? Because you just invert this

guy and invert the lambda and 2. And then square root of

minus gamma give me 1/2 lambda, square root minus h. OK? So sometimes I also approximate. I will not write this

determinant explicitly. When I write [? less h, ?] it

means the determinant of h. And the minus gamma,

determinant of gamma. OK? So you multiply these two

together, so these two cancel. And this two, multiply

this 4 pi alpha prime, and then get back that, OK? So they’re equivalent. Clear? So this gives you [? SNG. ?] So now the key– so now

if you look at this form, this really have a

polynomial form for X, OK? So now let me call

this equation 2. So equation 2, if you

look at that expression, just has the form–

so this is just like a two-dimensional

field theory– has the form of a

two-dimensional scalar field theory in the curved spacetime. Of course, the curved spacetime

is just our worldsheet sigma with metric gamma ab, OK? So this is just like–

but the key here, so sometimes 2 is called the

nonlinear sigma model, just traditionally, a

theory of the form that equation 2 is called

the nonlinear sigma model. Nonlinear because typically

this metric can depend on X, and so dependence

on X is nonlinear. So it’s called

nonlinear sigma model. But I would say it’s both

gamma ab and X are dynamical. Are dynamical variables. So that means when you

do the path integral, so in the path

integral quantization, you need to integrate

over all possible gamma ab and all possible X mu. Not only integrate

over all possible X mu, but also integrate all possible

gamma ab with this action. OK. So this is a

two-dimensional [? world ?] with some scalar field. And you integrate over

all possible metric, so over all possible intrinsic

metric in that [? world. ?] So this can also be

considered as 2D gravity, two-dimensional gravity,

coupled to D scalar fields. So now we see that when

you rewrite anything in this polynomial form,

in this Polyakov form, the problem of

quantizing the string become the problem of quantizing

two-dimensional gravity coupled to D scalar fields. OK. So this may look very scary,

but it turns out actually two-dimensional

gravity is very simple. So it’s actually not scary. So in the end, for

many situations, this just reduced to,

say, a quantizing scalar field with a little

bit of subtleties. So yeah, let’s stop here.