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%%                              TABLEAUX.TEX
%%      This  macro file is for producing a ``Young Tableau'' which is
%%      an array of little squares sometimes used in mathematical physics.
%%      For instance, the command $\tableau{6 3 2}$ will produce a tableau
%%      with 6 squares in the top row, 3 in the next, and 2 in the last.
%%                                  OOOOOO
%%      This tableau will look like OOO    but made of squares instead of O's.
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%%
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%%                                                      -- Doug Eardley 9/19/8%%
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\def\({\left(}
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%%ENGLISH MACROS
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\def\ie{{\it i.e.}}
\def\cf{{\it c.f.}}
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\hyphenation{Di-men-sion-al}


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                                %
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                                % SHOW THIS IFF \eqtrue HAS BEEN DECLARED
                                % \fi



%% SKIP TO HERE


\title{8.821 F2008 Lecture 02: String theory summary continued}
\author{Lecturer: McGreevy~ Scribe: McGreevy}
\begin{document}
\maketitle

Today: \\
1. problems with bosonic string\\
2. superstrings\\
3. D-branes\\
%4. a bold assertion.



Recall:\\
Last time I was showing you how primitive is our description of string theory.
Specifically, we are reduced to doing a path sum over worldsheet embeddings.
To decouple timelike oscillators, we gauged the worldsheet conformal group.
I told you that this leads to a theory of quantum gravity, which reduces to GR plus stuff
at low energies.  

The spacetime effective action which summarizes the scattering amplitudes
computed by the worldsheet path integrals is 
$$ S_{st}[ g, B, \Phi ...] 
\sim 
\int d^D x \sqrt g e^{ - 2 \Phi} \left[ \CR^{(D)} + (\del \Phi)^2 + H^2 + ...\right] \( 1 + \CO(\alpha') \)
\( 1 + \CO( g_s^2 ) \) 
$$
Here $ H = dB$, meaning $ H_{\mu\nu\rho} = \del_{[\mu} B_{\nu\rho]}$,
and 
$$ H\wedge \star H = H_{\mu\nu\rho} H^{\mu\nu\rho} $$

To study other backgrounds (which are only weakly curved), 
we can try to study a worldsheet action in background fields (a nonlinear sigma model):
\be\label{Sws} S_{ws}[X] = {1\over 4 \pi \alpha'} \int d^2 \sigma \sqrt{g} \left[
g_{\mu\nu}(X) \del_\alpha X^\mu \del_\beta X^\nu \gamma^{\alpha\beta} 
+ B_{\mu\nu}(X) \del_\alpha X^\mu \del_\beta X^\nu \epsilon^{\alpha\beta} 
+ \Phi(X) \CR^{(2)}_\gamma+ ...  \right]\ee
%$$ S_{ws} = ... $$
A few observations about this which we made in the first lecture:
-- conformal invariance constrains form of BG fields to solve spacetime EOM\\
-- the dilaton term implies that the contribution to the path sum of a worldsheets
with more handles are weighted by more factors of 
%string coupling expansion in topologies, weighted by 
$g_s = \vev{ e^\Phi}$\\
-- F-strings charged under B-field, and cannot end at some random point
in spacetime.
The specific objection to the ending of the string is the consistency of the 
Gauss' law for the B-field.  This arises from its equation of motion,
which in the presence of a fundamental string looks like:
$$ 0 = {\delta S_{st} \over \delta B} =  - d \star H  +  \delta^{D-2}(string) ;$$
which arises by varying the action 
$$ S[B] = \int_{everywhere} %d^{D}x ~
H \wedge \star H + \int_{string} B $$
with respect to $B$.
Integrating this (D-2)-form over a (D-2)-ball containing the string,
we get the integrated gauss' law:
$$ \int_{S^{D-3}}\star H = Q_1,$$
the number of strings inside the (D-3)-sphere ($4 \pi $ is equal to one in this discussion).
If the string ended suddenly, the answer here 
would depend on which choice we made for the interior of the $S^{D-3}$.


problems:\\
1. bosonic string has a tachyon, a boson field with mass${}^2 <0$.  this is a 
real instability\\
2. conformal anomaly: under a scale transformation on a curved worldsheet,
$$ \delta L_{ws} \propto T^\alpha_\alpha \propto {D-26 \over 2 \pi} \CR^{(2)} $$
3. perturbation expansion does not converge. worse than QFT.  $e^{ - 1/g_s}$.

Problem 2 is `solved' by string compactification: just study backgrounds where
the dimensions you don't like are too small to see.  This is an interesting thing
that we're not going to talk about.

To fix problem 1: superstrings.\\
The `RNS' description of superstrings is
obtained from bosonic strings by changing the worldsheet gauge group to superconformal group

-- no tachyon, still graviton, B, dilaton AND spacetime fermions (which exist)\\
-- sometimes spacetime susy\\
-- $D_c=10$. optimal for susy (will explain later)\\
-- The supersymmetric ones are the most interesting because we know their
stable vacua.
There are five flavors of 10d supersymmetric superstrings: IIB, IIA, type I, het $E_8$, het $SO$.\\
There duality relations between all of them: 
they are descriptions of different probes of different states of the same theory.\\
We will focus on the type II theories.\\
They are very closely related to each other.
At low energies, they reduce to type II supergravity,
with 32 supercharges.

A few words about the low-energy spectrum of type II. 
In addition to 
the fields which they share in common with the bosonic string,
and the spacetime fermions,
the type II strings (also type I has these) contain various antisymmetric tensor fields,
called Ramond-Ramond (RR) fields.
They are like Maxwell fields and like the B-field.  
They differ from the B-field in that 
strings are neutral under the associated gauge symmetry.\\
Type IIA has RR potentials of every odd degree (and even-degree field strengths $G= dC$):
$C_1, C_3, C_5, C_7, C_9$.\\
They are related in pairs by EM duality (like $ F^E = \star_4 F^B$ of E\&M in 4d): 
\eg\ the 1-form potential is the magnetic dual of the 7-form potential:
$$ d C_1 = \star_{10} d C_7 .$$
Type IIB has even-degree potentials, again related in pairs by EM duality.
The middle-degree 5-form field strength is self-dual.\\
(actually all of this is determined by supersymmetry.)

The simplest duality relation, namely T-duality, is 
worth mentioning here if only to convince you that the two type II theories
are not rival candidate theories of everything or something silly like that.

[T-duality interlude:\\
compactification of
string theory on a circle is described 
on the worldsheet by making some of the fields periodic.\\
Duality of 2d field theory relates strings on big circles
to strings on small circles (in units of $\alpha'$).
The spectrum of strings on a circle is made up 
of a KK tower of momentum modes,
and another tower of winding 
The basic idea is just that as the original circle shrinks,
the winding modes of the strings start to look more and more
like the 

In the type II superstring, this is accompanied by an interchange of 
the two flavors IIA and IIB.
To relate the RR potentials:
A RR-potential with an index along the circle being dualized
loses that index; one without an index along the circle gains an extra index.
]

%kinetic terms for q-forms.



{\bf{(D-branes)}}\\
re problem 3: the few insights beyond perturbation come from
the following set of observations. 

The effective action (derived by expanding around 
particular solutions like flat space) can be used to find other solutions.
If these solutions have small curvatures and field gradients (in string units)
and if the dilaton is uniformly small, we can trust the leading-order effective action
to tell us that it will really be a solution of string theory.
In general, we don't know a worldsheet description of most such solutions;
the ones where the RR fluxes are nonzero are particularly intractable from the worldsheet
point of view. 

One particularly interesting family of solutions 
are analogs of the Reissner-Nordstrom black hole solution of 4d Einstein-Maxwell theory, 
but which carry RR charge.

An object which couples minimally to a $p+1$-form potential
(\ie\ which carries the associated monopole moment) 
has a $p+1$-dimensional worldvolume $\Sigma_{p+1}$, so that we can have a term in the action of the form
$$ S_{st} \ni \int_{\Sigma_{p+1}} C_{p+1} 
= \int d\sigma^{\alpha_1}...d\sigma^{\alpha_{p+1}} \epsilon^{\alpha_1..\alpha_{p+1}}
\del_{\alpha_1} X^{\mu_1} ...\del_{\alpha_{p+1}} X^{\mu_{p+1}} C_{\mu_1...\mu_{p+1}} .$$

In flat space, the stable configurations of such objects are just flat infinite $p$-dimensional
planes in spacetime.
These objects have a finite tension, and hence these are not finite-energy excitations
above the vacuum; they are different superselection sectors.
Nevertheless, they exist (they can preserve half the supersymmetry of the vacuum) and are interesting 
as we'll see.  If the space contains compact factors, we can imagine wrapping such objects 
on compact submanifolds and getting particle-like excitations.

The tension of these RR solitons is proportional to $g_s^{-1}$.  They are heavy at weak coupling,
(though not as heavy as ordinary GR solitons, which have tension $g_s^{-2}$)
and are not made of a finite number of fundamental string excitations.
This means that the euclidean versions of these objects
can contribute the mysterious effects of order $e^{ - S_{eucl} } \sim  e^{ - 1 / g_s}$.  

At weak coupling, it turns out that these objects have a remarkably simple description.
We said earlier that the coupling to the NSNS B-field meant that string worldsheets
can't end just anywhere.  \\
A Dp-brane is defined to be a $(p+1)$-dimensional locus where strings can end.

The open strings which end there have tension and hence their
light states are localized on the brane.  These provide a string theory
description of {\it worldvolume} degrees of freedom.  
We saw above that open string spectra contain vector fields;
these worldvolume degrees of freedom very generally include
a gauge field propagating in $p+1$ dimensions.
The physics of this vector field is gauge invariant.
[It would be more interesting if it weren't.  
This could be that this is just a result of the fact that we used the gauge
principle in constructing the string theory.]
So the leading term in the derivative expansion of the effective 
action for the gauge field is 
$$ S = \int d^{p+1}x {1\over g_{YM}^2} \tr F\wedge \star F + \CO(\alpha') $$  
Because this action comes from interactions involving worldsheets with one boundary,
it should go like $ g_s^{ 2 - 2 h - b } = g_s^{-1}$ and therefore $ g_{YM}^2 \sim g_s$.  
In $p+1$ dimensions, $g_{YM}^2$ has dimensions of length${}^{p-3}$;
these are made up by powers of $\alpha'$.

Strings have two ends.  Each end can be labelled by which brane it ends on.
This means that the string states are $N\times N$ matrices, where $N$ is the number 
of D-branes.  
The ends of the strings are charged particles (quarks and antiquarks).

We should try to understand how the ending of the string
is consistent with the B-field gauge symmetry.  
To see this, we use the fact that the end of the string is a charged particle
to guess that the coupling 
of the worldsheet to a background worldvolume gauge field should take the minimal form
$ S_A= \int_{\del \Sigma} A $ where $\del \Sigma$ is the boundary of the worldsheet.
The variation of the bulk worldsheet action under the B-field gauge transformation 
$ B \to B + d\Lambda$ is 
$$\delta S_{ws} = {1 \over 4 \pi \alpha' } \int_\Sigma d\Lambda ={1 \over 4 \pi \alpha' } \int _{\del \Sigma} \Lambda. $$
We can cancel this if we allow $A$ to vary by 
$ A \to A - 4 \pi \alpha' \Lambda$.  
This means that only the combination
$$\CF \equiv 4 \pi \alpha' F +  B  ,$$
where $F = dA$
is invariant under the B-field gauge symmetry.
Hence $F$ must enter the effective action in this combination.

To see how this story is modified by the presence of the brane,
we need to think about the dependence of the D-brane action on the $B$-field.
Given our previous statement that the D-brane action contains
$ L \ni F\wedge \star F $, 
it must actually be 
$$ L \ni \CF\wedge \star \CF .$$
This means that in the presence of a fundamental string ending on a D-brane,
the spacetime action for the NSNS B-field is 
$$ S[B] = \int_{everywhere} d^{10}x ~H \wedge \star H + \int_{F1} B +2  \int_{Dp}  \star B \wedge F + \CO(B^2) $$
The equation of motion is then
$$ 0 = - d \star H  +  \delta^8(F1) + \delta^{10-(p+1)}(Dp) \wedge \star F ;$$
Integrating this over an 8-ball $B_8$ that's pierced by the string gives
$$ \int_{S^7} \star H = n_1 $$
where $S^7 = \del B_8$ is the boundary of the 8-ball and $n_1$ is the number of strings.
If we integrate instead over $B_8'$ which 
is not pierced by the string, it must instead go through the D-brane,
as in the figure.  
Integrating the EOM for $B$ over $B_8'$ gives 
$$ 0 = -  \int_{S^7} \star H + \int_{B_8' \cap Dp} \star F.$$ 
The first term here is the same as before, and is $n_1$.
But this gives a consistent answer 
because $ \int_{S^7\cap Dp} \star F $ is equal to the number of strings ending on the brane,
by the gauss' law for the worldvolume gauge field.
Basically, the worldvolume gauge flux carries the string charge away.

\figuur{ending}{8cm}{}

Note that IIA has D-branes of these dimensions:\\
D0, D2, D4, D6, D8\\
The D0-brane is just a particle, charged under the RR vector field;
The D6 is the associated magnetic charge.
These are interchanged under T-duality with the branes of type IIB.
while IIB has \\
D(-1), D1, D3, D5, D7, D9 \\
The D9 brane fills space and is a big perturbation; they only exist stably in type I. 
The D(-1)-brane is pointlike in spacetime, and hence is best considered as a 
contribution to a euclidean path integral.  
It contributes $e^{ - 1/g_s}$ effects to amplitudes.
\bigskip


I forgot to say in lecture that 
these objects defined by new sectors of open strings can be shown
to be charged under the RR tensor fields
(and in fact saturate the Dirac quantization condition on the charge),
and have tension $ 1/g_s$ times the right power of $\alpha'$ to make up the dimensions.
The gravitational back-reaction of a stack of $N$ parallel such D-branes
is controlled by $G_N T \propto g_s N \equiv \lambda$.
When this quantity is small, the description in terms of open strings is good;
when this quantity is large, the back-reaction is important and the better description is in terms of the 
gravitating soliton.  More on this soon.

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\vfill\eject

{\bf Reference:} Horowitz-Polchinski, gr-qc/0602037

{\bf Assertion:} Hidden inside any non-abelian gauge theory
is a quantum theory of gravity.

What can this possibly mean??\\
Three hints from the Elders:

1) At the least it must mean that there is some spin-two graviton particle,
that is somehow a composite object made of gauge theory degrees of freedom.
This statement seems to  run afoul of the Weinberg-Witten no-go theorem,
which says:\\
{\bf Theorem} (Weinberg-Witten){\bf :} A QFT with a Poincar\'e invariant
conserved stress tensor $T^{\mu\nu} $ forbids massless particles 
of spin $j>1$ which carry momentum (\ie\ with $P^\mu = \int d^Dx T^{0\mu} \neq 0 $).

GR gets around this because the total stress tensor (including the gravitational bit) vanishes by the metric EOM 
$ {\delta S \over \delta g_{\mu\nu}} = 0 $.  (Alternatively, the `matter stress tensor' which doesn't vanish
is not general-coordinate invariant.)

Like any good no-go theorem, it is best considered a sign pointing away from wrong directions.
The loophole in this case is blindingly obvious in retrospect: the graviton
needn't live in the same spacetime as the QFT.

2) Hint number two comes from the Holographic Principle 
['t Hooft, Susskind].  This is a 
far-reaching consequence of black hole thermodynamics.
The basic fact is that a black hole must be assigned an entropy
proportional to the {\it area} of its horizon in planck units
(much more later).  
On the other hand, dense matter will collapse into a black hole.
The combination of these two observations leads to the following crazy thing:
suppose you have in a volume $V$ (bounded by an area $A$)
a configuration with entropy $S > S_{BH} = {A \over 4 G_N}$
(where $ S_{BH}$ is the entropy of the biggest blackhole 
fittable in $V$),
but which has {\it less} energy.  
Then by throwing in more stuff 
(as arbitrarily non-adiabatically as necessary, 
\ie\ you can increase the entropy)
you can {\it make} a black hole.
This would violate the second law of thermodynamics,
and you can use it to save the planet from the Humans.  
This probably means you can't do it,
and instead we conclude that the black hole 
is the most entropic configuration of the theory in this volume.
But its entropy goes like the {\it area}!  
This is much smaller than the entropy of a local quantum
field theory, even with some UV cutoff,
which would have a number of states $N_s \sim e^{ V}$ 
( maximum entropy $= \ln N_s $)
Indeed it is smaller (when the linear dimensions are large!) than any system with local degrees of freedom,
such as a bunch of spins on a spacetime lattice.

We conclude from this that a quantum theory of gravity
must have a number of degrees of freedom 
which scales like that of a QFT in a smaller number of dimensions.


3) A beautiful hint as to the possible identity of the extra dimensions is this.
Wilson taught us that a QFT is best thought of as being sliced up by length (or energy) scale,
as a family of trajectories of the renormalization group (RG).  
A remarkable fact about this is that the RG equations for the 
behavior of the coupling constants as a function of RG scale $z$
is {\it local} in scale:
$$ z \del_z g = \beta(g(z) ) $$ 
-- the RHS depends only on physics at scale $z$.
It is determined by the coupling constant evaluated at the scale $z$,
and we don't need to know its behavior in the deep UV or IR.  
This fact is not completely independent of locality in spacetime.
This opens the possibility that we can associate the extra dimensions
raised by the Holographic idea
with energy scale.


Now we make several simplifying 
assumptions, so we can find concrete examples.\\
1) Many colors:  The basic idea here is that 
(as Steve Shenker puts it)
``you can hide a lot in a large-$N$ matrix."  
In our holographic gedanken experiment,
it is only when the linear dimensions of the system
are large compared to the Planck length that 
areas are bigger than volumes in Planck units.  
Large $N$ will help with finding extra dimensions
of macroscopic size.

2) Strong coupling: The assertion certainly isn't obvious
at weak coupling.  
This constrains how we take $N$ large:
we want the theory to remain interacting;
this is not the case for \eg\ large-$N$ vector models.
This is one motivation for the 't Hooft limit:
$N \to \infty$ at fixed $g_{YM}^2 N$.  More soon.

3) Supersymmetry: ick.  This is a (for many, horrifying) technical assumption, 
but has many points in its defense.  \\
a) Without it, strongly-interacting constituents will tend to bind together,
and in general QFTs do not exist beyond a maximal coupling
(familiar examples are QED with its Landau pole, and the Thirring model, whose exact solution just stops at $g^2 = 8 \pi$).\\
b) A supersymmetric theory has many more adiabatic invariants, quantities independent of the coupling.
There are many more things to check.
This is perhaps the main reason why there are so many more papers about 
duality in supersymmetric theories.\\
c) Supersymmetry constraints the form of interactions, and hence there are many fewer candidates
for the dual description.\\
The {\it most} supersymmetric field theory in four dimensions (more on this later)
is the $\CN = 4 $ super-yang-mills theory (often just called the $\CN=4$ theory).
It has the very strange property that the familiar classical dimensionlessness of $g_{YM}$
of 4d YM theory 
has no anomaly: $\beta_{g_{YM}} = 0$.
The theory is exactly scale-invariant for any value of the coupling,
there is a line of fixed points of the RG. So:\\ 
d) supersymmetry allows this line of fixed points, so that the bare coupling 
constant is really a {\it parameter} (not just an initial condition for the RG flow),
and we can ask what happens as we vary it.  Which brings us to:\\
4) Conformal symmetry: 
in particular, we'll have scale invariance: $x^\mu \to \lambda x^\mu, ~~ \mu = 0...3$.
(Known relativistic QFTs which are scale invariant are also invariant under the larger conformal group,
which is $SO(4,2)$ in $3+1$ dims.)

NOW: if $z$ is energy scale, we must have $ z \to z/\lambda$ under the scale transformation.
The most general metric with this symmetry (plus Poincar\'e) is 
$$ ds^2 = {\tilde r^2 \over \tilde L^2} \eta_{\mu\nu } dx^\mu dx^\nu + { d\tilde r^2 \over \tilde r^2 } L^2 $$
where $L, \tilde L$ are some length scales.
By a change of coords $$ \tilde r \equiv {\tilde L \over L } r$$ we can set $\tilde L = L $.
This is $AdS_5$.  

Guess: $\CN = 4 $ SYM is related to a theory of gravity on $AdS_5$.  







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