Field Theory Dynamics
from Branes in String Theory
D i s s e r t a t i o n
zur Erlangung des akademischen Grades
d o c t o r r e r u m n a t u r a l i u m
Dr rer nat
im Fach Physik
eingereicht an der
Mathematisch Naturwissenschaftlichen Fakult at I
der Humboldt Universit at zu Berlin
von
Andreas Karch M A
geb oren am in Bamberg
Pr asident der Humboldt Universit at zu Berlin
Prof Dr Dr h c H Meyer
Dekan der Mathematisch Naturwissenschaftlichen Fakult at I
Prof Dr J P Rab e
Gutachter
Tag der m undlichen Pr ufung
Contents
Motivation
D branes and non p erturbative e ects in string theory
The breakdown of p erturbation theory
A string theoretic description of D branes
D branes and gauge theory
Gauge theory on the worldvolume
Compacti cations and D branes
Engineering Gauge theories
Geometric Engineering
Branes as Prob es
Hanany Witten setups
D branes and dualities
String Dualities and M theory
Worldvolume theory of the NS brane
Exploiting the SYM D brane corresp ondence
Classical Hanany Witten setups
Branegineering
There s so much more one can do
Solving the quantum theory
Bending and quantum e ects
Lifting to M theory
Applications of the brane construction
Dualities in the brane picture
Exact S duality
Seib erg duality and Mirror symmetry
S dual N pairs revisited
Non trivial RG xed p oints i
App earance and applications
Physics at non trivial FP from branes
d Hanany Witten setups
Extracting information ab out the FP theory
Global Symmetries
Deformations and Phase Transitions
Correlators from branes
A chiral non chiral transition
The equivalence of the various approaches
The duality b etween orbifold and NS brane
Hanany Witten versus branes at orbifolds
The orbifold construction
The classical branches
Including D branes and orientifolds
Applying T duality
Adiabatically expanding to d N
Branes and geometry
obtaining the Seib erg Witten curve
The approaches
A T duality for b ent branes and
branes ending on branes
Unifying the di erent approaches
Op en problems and directions for further research
Brane Boxes
Maldacena Conjecture
Summary
Bibliography
Zusammenfassung
Danksagung
Selbst andigkeitserkl arung
Leb enslauf
Publikationsliste ii
Motivation and Introduction
Motivation
It is the main goal of physics to explain all ratios b etween measurable quantities The hop e
is that in the end a very simple b eautiful and unique mathematical structure emerges
Mo dern physics is based on two ma jor building blo cks general relativity and quantum
physics The former expresses all phenomena of the macroscopic world and esp ecially
gravity in terms of simple geometric concepts The latter is p owerful in explaining the
microscopic world by replacing the notion of particles which o ccupy a certain p osition in
space and carry a certain amount of momentum by the more abstract formalism of states in
a Hilb ert space and observables as op erators acting on them Since the distinction b etween
microscopic and macroscopic world seems to b e rather arbitrary these two building blo cks
should b e uni ed in an underlying theory More than that as they stand the two concepts
are even inconsistent Straight forward quantization of general relativity leads to in nities
in physical pro cesses that can not b e tolerated A larger theory unifying gravity and
quantum theory is hence not only desired from an aestethic p oint of view but indeed
required for consistency
After many years of extensive search for this unifying theory a single candidate has
emerged string theory String theory replaces the fundamental p oint like ob jects of
particle physics by d strings thereby removing the in nities encountered in quantizing
general relativity General relativity reemerges as a low energy limit at large distances
where it was tested exp erimentally However at small distances stringy physics takes
over and even our concepts of space and time break down Similarly ordinary particle
physics as describ ed by the standard mo del can reemerge in the limit where gravitational
interactions b etween the particles can b e neglected The energy scale at which b oth
gravity and quantum e ects b ecome imp ortant is set by the Planck scale and is roughly
GeV Since this scale is so huge it is imp ossible to just create the fundamental
degrees of quantum gravity in an accelerator and then lo ok what they are
Even though string theory has all the ingredients required by mo dern physics it is
di cult to make contact with physics as we know it The ma jor obstacle is that string
theory allows for a variety of di erent vacua each of which leads to di erent physics
No precise predictions ab out the low energy physics like answers to the why questions
left by the standard mo del can b e made without nding a pro cess that determines the
right vacuum However the most imp ortant concepts app earing in string theory namely
gauge theory gravity and sup ersymmetry are indeed known or b elieved to dominate the
real world While the former two are the bases for all physics describ ed by the standard
mo del and general relativity the latter is b elieved to b e of similar imp ortance for next
generation collider physics Exp erimental veri cation of sup ersymmetry in the real world
would b e even more supp ort that string theory is not just the only known consistent
quantum theory of gravity but indeed the fundamental theory realized in our world
Of sp ecial interest are also ob jects which prob e the regime of quantum gravity that is
they are small enough for quantum e ects to b e imp ortant and heavy enough to require
gravity Examples of such ob jects are black holes close to their singularity or our universe
in very early times close to big bang Treatment of these imp ortant issues as well as the
vacuum selection problem requires non p erturbative information ab out string theory So
far the p erturbative expansion of string theory in terms of worldsurfaces was the only
de nition we had of string theory Only very recently to ols have emerged that allow us
control certain asp ects of the non p erturbative physics b ehind string theory raising the
hop e that these fundamental issues can nally b e addressed
Introduction
During the so called second string revolution it has b ecome p ossible to gain control over
asp ects of string theory that were not contained in its p erturbation expansion
in terms of worldsurfaces The ma jor achievements were the discovery of D branes
as one of the non p erturbative ob jects in string theory and the realization of the role
of duality symmetries in string theory relating two seemingly di erent theoretical
descriptions to one and the same physical situation
Dualities have b een of similar imp ortance in gauge theories Using dualities it has b een
p ossible to solve the IR b ehaviour of certain quantum eld theories exactly and get
a lot of non p erturbative information even in situations with less restrictive symmetries
However in all those setups sup ersymmetry is a vital ingredient and it is not clear yet
how these metho ds can b e generalized to non sup ersymmetric situations
Non p erturbative string theory and Sup er Yang Mills SYM gauge theories are indeed
deeply related The dynamics of the D branes which are so imp ortant for our understand
ing of non p erturbative string theory is basically governed by SYM This connection can
b e used in a twofold way dualities and other obscure asp ects of eld theory only dis
covered recently like non trivial xed p oints of the renormalization group with mutual
non lo cal ob jects b ecoming massless nd their natural place in string theory where they
can b e easily visualized
On the other hand some problems which seem intractable in string theory can b e
mapp ed to questions in gauge theory which are under much b etter control Indeed it has
b een prop osed that all non p erturbative asp ects of string theory can b e enco ded in large
N SYM theory
It is the purp ose of this work to highlight some of the insights gained with the help of
this deep connection b etween gauge theory and string theory
In Chapter I will review the construction of D branes and explain how gauge theory
determines their dynamics I will also comment on the imp ortant role D branes played
in understanding string theory dualities since these dualities lead to the discovery of an
dimensional theory called M theory that basically summarizes all the non p erturbative
insights we gained ab out string theory and is the natural arena for visualizing asp ects of
SYM which are hard to understand from the eld theory p oint of view
In Chapter I will introduce a setup rst used by Hanany and Witten to study d
gauge theories embedded in string theory In Chapter I will use this setup to study
certain asp ects of dimensional and dimensional physics The interplay will allow us
to understand certain non trivial xed p oints and transitions which are obscure from the
eld theory p oint of view to say the least But we can also learn ab out string theory
from the corresp ondence Among many other things it will allow us to show that in string
theory chiral vacua can b e smo othly deformed into non chiral vacua p erhaps taking a
small step towards a more detailed understanding of the vacuum selection problem
In Chapter I will show how the other asp ects of the gauge theory string theory
connection are related to the Hanany Witten setups by a series of string theory dualities
We will see that the di erent techniques used to explore the corresp ondence might b e more
or less p owerful in various situations but that in the end we are guaranteed to obtain
the same results no matter how we chose to embed our gauge theory under consideration
into string theory
In Chapter I will discuss what op en problems remain and where we can go from
here
Chapter
D branes and non p erturbative
e ects in string theory
The breakdown of p erturbation theory
String theory as we used to know it was only de ned via its p erturbation series That
is a given scattering pro cess receives contributions from worldsheets of various top ology
Higher genus surfaces are weighted with higher p owers of g the string coupling which
s
is the exp ectation value of a dynamical eld the dilaton g e s
+g2 +g4 + ...
Figure Propagation of a string from its p erturbative de nition
To calculate the contribution from a single diagram in the p erturbation series depicted
in Fig we have to solve a conformal eld theory on the worldvolume of the given
top ology and than integrate over all p ossible deformations mo duli This is often p ossible
d conformal eld theories are very constraint due to the high amount of symmetry and
many calculational to ols are available At weak coupling only diagrams of low genus
contribute and we can actually calculate the amplitudes
However this p erturbative de nition clearly fails when we are at strong coupling
Here we really have to calculate an in nite number of diagrams since higher top ologies
are no longer suppressed Worse than that a generic scattering pro cess may also receive
contributions that are not even visible at all in the p erturbation series even if we would
g
s
b e able to sum up the in nite diagrams Such contributions are suppressed as e or
2
g
s
These contributions have a series expansion around strong coupling g as e
s
g
s
e
g g g
s
s s
None of these terms has the right p ower to match any of those app earing in the ordinary
p erturbation series which only contains p ositive p owers of g These are purely non
s
p erturbative e ects
From eld theory it is well known that there are indeed phenomena giving rise to such
non p erturbative contributions The most famous example are instantons Instantons are
stable solutions to the Yang Mills equations of motion that are centered in space and time
Their existence is due to the fact that Yang Mills can have top ologically distinct vacua
Instanton solutions interpolate b etween di erent vacua and their stab elness is therefore
guaranteed by top ology Arguing on the bases of the cluster decomp osition principle
one can show that in order to de ne a consistent quantum eld theory we indeed
have to sum over all p ossible instanton backgrounds when p erforming the path integral
so these con gurations do contribute to scattering pro cesses Calculating the classical
action corresp onding to an instanton con guration one nds that it is indeed suppressed as
2
g
YM
This example also gives us an intuitive feeling why such things will never app ear e
in the p erturbative expansion while p erturbation theory expands around a given vacuum
non p erturbative contributions arise from tunnelling pro cesses and interpolation b etween
di erent vacua But this is also why it is so crucial to understand non p erturbative states
in string theory solving the vacuum problem that is what is the right string theory
ground state and how did nature pick it requires detailed understanding of precisely
these pro cesses
Similar e ects are due to solitonic ob jects like monop oles or domain walls They are
again stable solutions to the equations of motion centered in space This enables us to
interpret them as particles or higher dimensional ob jects in our theory They have
masses which go like g At weak coupling they are very heavy and can b e neglected
YM
However at strong coupling they should b e included Virtual monop oles running in lo ops
2
action g
YM
due to the e factor in the path integral signalling will b e suppressed by e
a non p erturbative contribution
In the same spirit we can try to identify solitonic ob jects with mass g in string
theory in order to identify non p erturbative string states By studying sup ergravity
SUGRA the low energy eld theory limit of string theory one indeed nds a whole
zo o of such ob jects generically called p branes Among more exotic ob jects there exists
the magnetic dual of the string the NS brane with tension g and the so called Dirich
s
let D branes whose tension at weak coupling only grows as g Understanding those
s
ob jects should enable us to learn ab out non p erturbative e ects in string theory They
will b e the topic of the rest of this work
A string theoretic description of D branes
To understand the contribution to the amplitude by a given brane con guration one can
study p erturbative string theory in the background of the branes at weak string coupling
For the D branes this is straight forward once one realizes that D branes are space time
defects at which op en strings can end
open strings can end on
D-branes
(Polchinski)
Figure String in the background of a D brane
Figure shows this basic concept of D brane physics It was known since the early
days of string theory that in op en string theory one can as well imp ose Dirichlet b oundary
conditions the end of the string is at a xed p osition as the usual Neumann b oundary
conditions the end is free to move no momentum is allowed to ow of the end One
usually neglected this p ossibility since it introduced hyperplanes the planes on which
the endp oints are forced to stick which break Lorentz invariance It was the achievement
of Polchinski to show via an explicit lo op op en string calculation that these space
time defects carry charge under the RR gauge elds of string theory and calculating their
tension to b e
T
D p
p
l g
s
s
where l and g denote the string coupling and length resp ectively These prop erties allow
s s
us to identify them with the stringy version of the solitonic solutions of SUGRA which I
already called D branes b efore
Now it is straight forward to do everything we are used to from p erturbative string
theory in the background of the D branes Quantizing the oscillator mo des of the string
theory in the presence of the mo di ed b oundary conditions one nds that the massless
sp ectrum of the op en strings ending on the D brane are given by a SYM multiplet living
on the brane worldvolume That is for the D brane we nd the usual N SYM
multiplet consisting of the vector gauge eld and the gauginos in d All other branes
yield dimensional reductions of those to the appropriate worldvolume dimension
Demanding conformal invariance on the string worldsheet yields equations of motion
for the space time elds by setting the function of the d conformal theory on the
worldsheet to zero order by order in the string tension which plays the role of the
coupling constant in the d theory just like in the well known case of Neumann b oundary
conditions Writing down an action that yields these equations one obtains as an e ective
action for the D brane theory a sup ersymmetric Dirac Born Infeld action with Wess
Zumino couplings to the bulk elds
Z Z
q
p F p
det g F d C e S S S d e
ij ij DBI WZ
P
r
C where F F B C is a formal sum over all the form elds present in
r
the I IA B sup ergravity and the integral always picks out the right form to go with the
right p ower of F from the exp onential The elds should b e understo o d as pullbacks from
sup erspace to the worldvolume
The low energy approximation of this action that is the expansion to lowest order in
l the string length yields SYM on the worldvolume This is in accordance with the
s
analysis of the massless sp ectrum The gauge coupling can b e read o from to b e
p
g l g
s
YM s
As in the case of fundamental string theory the scalars on the worldvolume de ne the
p osition of the brane in the transverse space Via the DBI action these are coupled to
the worldvolume gauge elds This is an imp ortant prop erty of the D brane action which
we will explain in more detail in the following Basically a brane can absorb the ux of a
charged particle by b ending in transverse space balancing the force from the gauge elds
with its tension that is with the worldvolume scalars A at D brane breaks half of the
sup ersymmetries since the op en string sp ectrum only has half of the sup ersymmetries of
the closed sp ectrum The sup ersymmetries preserved have to satisfy
L p R
L R L R
where the preserved sup ercharge is Q Q and Q Q are the sup ercharges gen
L R
erated by left and right moving degrees of freedom in the surrounding type II string
theory Cho osing a non trivial embedding generically breaks all the sup ersymmetries If
1
which can e g b e seen by analyzing the Killing spinor equations in the background of the D brane
soliton solution
the embedding geometry allows for some Killing spinors lower fractions of sup ersymmetry
may b e preserved
If we try to rep eat this analysis for NS branes we run into trouble The NS brane
metric lo oks like a tub e the dilaton blows up if we move towards the core and any
conformal eld theory description breaks down Only asymptotically the NS brane can
b e describ ed by a well known conformal eld theory a WZW mo del The NS brane
worldvolume theory is not accessible by purely p erturbative string techniques However
we will see later that we can deduce its prop erties by string dualities
D branes and gauge theory
By now we have gained some insights in the dynamics governing D branes We have
learned that there is a very deep connection b etween D branes and gauge theories We
will analyse how some of the most interesting asp ects of D branes are captured by simple
eld theoretic phenomena This discussion will pave the way for the discussion in the
following chapters where I will exploit the D brane gauge theory corresp ondence to
learn ab out string theory as well as ab out gauge theory The general philosophy is that
we consider certain limits of string theory in which the gravity and heavy string mo des
the bulk mo des decouple leaving us just with the op en string sector describ ed by SYM
The basic quantities that control this limit are the Planck scale M and the string scale
pl
M the inverse of the string length They satisfy
s
M g M
s
pl s
which just shows the relation b etween string frame and Einstein frame Sending M to
pl
in nity is the same as sending Newton s constant to zero so gravity is decoupled Taking
M to in nity sends all excited string states to in nite mass e ectively decoupling them
s
to o This can b e done at nite string coupling keeping an interacting SYM theory
Gauge theory on the worldvolume
As we have seen the e ective theory on the worldvolume is given by a DBI action We
want to analyse this world volume theory in the limit where the bulk physics decouples
that is we get rid of gravity and other closed string mo des We only keep the degrees
of freedom on the brane Expanding the DBI action in l which explicitely shows up
s
together with every F it is easy to see that in the l limit the theory on the
s
worldvolume of the Dp brane reduces to U SYM in p dimensions The amount of
sup ersymmetry preserved by a given brane is determined by its embedding in space time
as discussed ab ove A at brane always preserves half of the sup ercharges of type II
theory leading to maximally sup ersymmetric Yang Mills on the worldvolume Now let
us consider what happ ens if N D branes coincide This situation was analysed by Witten
A single D brane supp orts on its worldvolume a single U multiplet These massless
states arise from a string starting and ending on the same brane The mass of a state
is given by the length of the string times the string tension The massless vector hence
arises from a zero length string starting and ending at the same p oint Each of the ends
of the strings carries a Chan Paton lab el of the gauge group that is an index in the
fundamental representation so the vector multiplet is correctly left with a fundamental
and an antifundamental index an adjoint eld Clearly nothing happ ens to these states
if many D branes coincide However whenever two branes are close there are new states
that b ecome imp ortant Strings stretching from one brane to the other yield states whose
mass is determined by the distance b etween the branes They carry a fundamental Chan
Paton index of the U of the brane they start and end on resp ectively It is natural
to identify those as W b osons of a broken U gauge group The distance b etween the
branes determines the Higgs exp ectation value When N branes coincide all the W b osons
b ecome massless and the full U N gauge symmetry b ecomes visible
In order to obtain di erent gauge groups one can consider D branes coinciding on
top of space time singularities An example of such a singularity which is under control
from p erturbative string theory is an orientifold plane the xed plane of a Z orbifold
action that combines worldsheet parity with a space time re ection in r co ordinates The
resulting p r space time xed plane is called an Op orientifold plane A similar
calculation like that of Polchinski s determination of the D brane charge shows that the
orientifold is also charged under the same RR eld as the Dp where the relative value of
the charge is given by
p
q q
O D
The sign is determined by a discrete choice of the precise way one p erforms the pro jection
When N D branes coincide on top of the orientifold and hence also coincide with their
N mirrors only oriented strings stretching b etween the branes will yield new massless
gauge b osons leading to an SO N U S p N gauge theory on their worldvolume for
an orientifold of negative p ositive charge The b est known example is the type I string
If we mo d out I IB just by world sheet parity we basically pro duce an O Since this is
a space lling brane we have to cancel the RR charge forcing us to use the negatively
charged orientifold with D branes on top of it yielding an SO gauge theory as
exp ected
2
Here and in what follows I will always consider the D brane and its Z mirror as di erent ob jects
2
each carrying charge q If one wants to consider only physical D branes one should assign them charge
D
q in these conventions
D
Compacti cations and D branes
There is a seemingly di erent way that D branes can b e describ ed by gauge theories If
we consider compacti cations of string theory we will have non p erturbative states in the
resulting lower dimensional theory from D branes wrapping cycles of the compacti cation
manifold The mass of these states is just given by the tension of the brane times the
volume of the cycle and therefore has the g dep endence signalling a non p erturbative
s
state At certain p oints in the mo duli space of compacti cations some of these cycles
may shrink to zero size leading to new massless states in the low energy theory Some of
these states are usually massless vectors giving rise to non p erturbative gauge groups
cycle
shrinks
Figure Non p erturbative states from D branes on shrinking cycles
Engineering Gauge theories
With the two mechanisms at hand we can try to engineer gauge theories that is we make
up a string theory geometry with branes that realize a certain gauge theory we want to
study Combining the two basic mechanisms discussed ab ove in various ways there are
several p ossibilities to do so Basically all these di erent approaches describ ed in the
literature can b e separated in three classes As I will discuss in the last chapter they are
actually equivalent There I will also give a more technical discussion for the sp eci c case
of N theories in d
Geometric Engineering
A geometric engineer tries to co ok up a string background that captures all asp ects of
the gauge theory she wants to study in the geometry of the compacti cation manifold
In order to fo cus on the gauge theory mo des one has to decouple all stringy mo des and
all bulk mo des That is one has to send M and M to in nity Let me for simplicity
s pl
of notation discuss the case of a K compacti cation see and references therein
engineering an N or N sup ersymmetric gauge theory in d for type I IA
or the heterotic string resp ectively It should b e clear that these principles work the same
way in other examples The relevant scale is the d Planck scale given according to a KK
ansatz by
V M M
K
P l P l
Decoupling gravity therefore e ectively amounts to decompactifying the K Since the
d gauge coupling of the p erturbative gauge groups already present in d are also given
via the KK ansatz as
V g g
K
Y M Y M
they decouple in the same decompacti cation limit The only gauge groups that survive
are the non p erturbative gauge groups that arise via wrapping branes around vanishing
cycles in the manifold All information ab out the gauge theory is therefore enco ded in
the lo cal singularity structure of the K
The basic example is I IA on an ALE space that is a non compact version of K
An ALE is a blow up of an R orbifold where is a discrete subgroup of SU
Since spinors transform as a under the SO SU SU spacetime
rotations embedding the orbifold in just one of the SU factors leaves half of the spinors
invariant and hence also half of the sup ersymmetries unbroken Since K can b e written
as an orbifold of T these orbifold singularities can arise lo cally in the geometry of K
The statement that should b e a subgroup of SU is equivalent to demanding that the
holonomies of K only ll up SU and not the full SO of a generic d manifold In
order to obtain gauge dynamics the lo cal description in terms of the ALE is all we need
We exp ect new gauge dynamics when we move to the singular p oint the orbifold itself
The ALE space has top ological non trivial cycles
S 1
Figure Non trivial cycle on R Z
Figure illustrates non trivial S s arising from an R Z orbifold Similarly we get
spheres on the ALE These spheres shrink to zero size at the orbifold p oint New
massless states arise from D branes wrapping these cycles The intersection pattern
of the cycles will determine the gauge group Luckily all discrete subgroups of SU
can b e classi ed by an ADE pattern where the corresp onding Dynkin diagram gives us
precisely the information ab out the intersection numbers of the vanishing spheres The
resulting gauge theory has a non ab elian ADE gauge group
Branes as Prob es
Branes as Prob es is the most natural way if we want to learn something ab out string
theory from Yang Mills theory The idea is that in order to study what happ ens to a given
string background once one takes into account all the quantum e ects one prob es the
background with a D brane On the worldvolume of the D brane we will as usual nd
a gauge theory The background geometry will b e enco ded in this gauge theory via the
matter content the amount of unbroken sup ersymmetry and the interaction p otentials
Solving the quantum gauge theory will teach us ab out the quantum b ehaviour of the
background This technique has b een very successfully used for probing Dp branes and
Op planes with Dp branes as well as probing orbifold singularities
with Dp branes
In b oth cases the matter content and classical sup erp otential of the gauge theory can
b e analyzed by p erturbative string theory For the higher p branes we will nd new states
on the Dp worldvolume corresp onding to the zero mo des of strings stretching b etween Dp
and Dp branes in addition to the gauge multiplet already present from the Dp Dp
strings
In the case of the orbifold we rst include all the twisted sectors required in string
theory for consistency by including all the mirror D branes and strings stretching in
b etween them Then we pro ject onto states invariant under the orbifold group and this
way obtain the corresp onding sp ectrum
Hanany Witten setups
Hanany and Witten HW introduced a setup of intersecting branes realizing d
N gauge theories The gauge theory again lives on the worldvolume of D branes
The other branes make the gauge theory interesting by breaking SUSY and introducing
new matter Since we are now only dealing with at branes in at space many things
b ecome very intuitive Mo duli and parameters just corresp ond to moving the branes
around and are very easy to visualize As advertised ab ove I will show in the end that
3
In this language one could view string theory as we used to know it as probing space time with a
fundamental string
all the approaches are actually equivalent so by studying the intuitive HW setups we
can get non trivial results ab out quantum string backgrounds by considering the dual
branes as prob es setup The next chapter is devoted to an extensive review of the HW
idea so I won t go into any details at this p oint
D branes and dualities
String Dualities and M theory
Probably the most imp ortant application of D branes so far is the idea of string dualities
the statement that one and the same physical system has two dual descriptions The
concept of duality was already discussed long ago in the context of eld theories as I will
explain in more detail in the Chapter In string theory duality was rst detected in
the form of T duality Studying the sp ectrum of b osonic string theory on a circle of
radius R
p p oscillators H
R L
R l
s
p
m n p
R
R l
s
R l
s
p
p n m
L
R l
s
One sees that due to the presence of winding mo des characterized by the integer m as
well as momentum mo des n around the circle the states are invariant under an exchange
of the two if one simultaneously takes R into l R This invariance under R R
s
exchange can b e shown to b e a symmetry of amplitudes to all orders in p erturbation
theory and is b elieved to b e valid even non p erturbatively The two compacti cations
are T dual to each other For the sup erstring this T duality works almost the same For
example type I IA on R is dual to I IB on l R In this case the p form elds from
s
the RR sector T dualize into p and p form elds dep ending on whether we take the
comp onents along or transverse to the compact direction Since the Dp branes couple
to these elds T duality transverse to the worldvolume pro duces a D p brane while
T duality along a worldvolume direction leaves us with a D p brane
More interesting are dualities relating one string theory at weak coupling to another
string theory at strong coupling Many dualities of this type have b een discovered over
the recent years However non of them can b e proven by a direct calculation Since
by de nition we compare a strongly coupled with a weakly coupled theory only one
side is accessible to calculations Duality then amounts to a prediction for the strong
coupling b ehaviour of the other theory The reason why most string theorists nevertheless
b elieve in the validity of these dualities is that they can b e checked in several ways The
most imp ortant check is the matching of ob jects which are BPS They preserve some
fraction of the sup ersymmetry and are therefore protected by the sup eralgebra from any
renormalization We have already encountered some of these ob jects D branes This way
certain prop erties of these non p erturbative states which dominate the strong coupling
theory can b e calculated and they can b e matched onto the p erturbative states at weak
coupling
One of the examples I am going to consider several times in this work is the selfduality
of type I IB string theory Type I IB with coupling g is dual to type I IB with g
s s
a i
Including the axion a we can build a complex coupling Combining the
g
s
g to g duality with the invariance of the axion under shifts of a whole SL Z
s s
of dual theories can b e constructed The NS form eld combines with the RR form
into an SL Z doublet The ob jects coupling to them the fundamental F and the
D string are exchanged under the strong weak coupling duality More general SL Z
transformations take the F into a p q b ound state of p fundamental and q D strings
Similarly their magnetic duals the NS and the D brane form an SL Z doublet
Since there is only one form eld it has to b e a singlet under SL Z and hence the
D brane stays invariant under all duality transformations Since the low energy e ective
actions of the dual theories are supp osed to agree the Planck scale has to remain invariant
therefore using the dual string scale has to b e M g
s
s
Basically all string dualities can b e summarized as the existence of an conjectural d
theory called M theory which contains all the string theories as well as d SUGRA as
p erturbative expansions in certain limits
IIB 11d SUGRA
het SO(32) IIA M
het E8 x E8
type I
Figure All known string theories as well as d SUGRA are just di erent
p erturbative expansions of an overarching d M theory
Since I am going to use this M theory picture in what follows let me brie y present
as a de ning duality of M theory the duality b etween d SUGRA and type I IA which
originally led to the discovery of M theory According to this prop osal M theory
on a circle is type I IA string theory with the I IA coupling and string length given in terms
of the d Planck length and the radius of the th dimension R as
l
R
pl
l g
s s
l R
pl
These relations can b e obtained by comparing the low energy e ective actions The
relation really constitutes a strong weak coupling duality at very large R I IA b ecomes
strongly coupled and we lose all control However in the d picture as R b ecomes bigger
the curvature b ecomes smaller and SUGRA b ecomes a go o d approximation Similar at
very small R the curvatures are Planckian in d so SUGRA fails to capture the physics
however p erturbative string theory is a go o d description To describ e couplings of order
we need the yet unknown full edged M theory
The app earance of the th dimension can b e seen from studying D branes D branes
are non p erturbative states whose mass go es to zero in the strong coupling limit N D
branes are b elieved to form a unique threshold b ound state that is with zero binding
N
energy They therefore led to a tower of states with mass It is natural to
g l
s s
M identify these as momentum mo des around the th dimension of radius R
g l
s s
theory also provides us with a nice organization principle for all the other branes From
d SUGRA we learn that M theory has two extended ob jects the M and the M brane
Together with three more complicated solutions that only arise up on compacti cation of
at least one more direction the wave momentum mo de around the circle with mass R
its magnetic dual the KK monop ole brane with tension R l and an M brane with
pl
they give rise to all brane solutions in the p erturbative limits of M theory tension R l
pl
Matrix Theory
Having said the ab ove it would clearly b e desirable to nd a microscopic de nition of M
theory The only candidate that has emerged so far is matrix theory The idea b ehind
this approach is to quantize the theory in a sp ecial frame called the in nite momentum
frame where only a very limited amount of the original degrees of freedom are visible
What we do is b o ost ourselves as observers in nitely along a compact direction so that
of all mo des with momentum N R only those with p ositive N survive This has to b e
considered as N and R b oth go to in nity with N R also going to in nity
Since we want to work at nite N to do any realistic computation one would like to
study a reference frame that is describ ed by nite N matrix theory and reduces to the
IMF in the N limit Such a frame exists the discrete light cone frame Therefore we
want to study discrete lightcone quantization DLCQ of M theory This was conjectured
to b e describ ed by the nite N matrix mo del in DLCQ formally can b e thought of
as quantizing the theory on a compact lightlike circle This notion seems to b e rather
counterintuitive Indeed it was shown in that the b est way to think ab out the DLCQ
is to consider it as a limit of a compacti cation on an almost lightlike circle that is
q
2
2
R
R
R
s
p p
R
x x x
s
R
R
R
p
p
t t t
where R R For R this reduces to a lightlike compacti cation with radius
s s
R Now this compacti cation is just a Lorentz b o ost transform of an ordinary spacelike
compacti cation on R with b o ost parameter
s
R R
s
q
R
R R
s
Therefore Seib erg s nal result can b e stated as follows DLCQ of any system is Lorentz
equivalent to the R limit of a spacelike compacti cation on R But we know what
s s
M theory on a vanishing circle is it is just weakly coupled I IA Since this is a rather
familiar theory it is very easy to identify the relevant degrees of freedom
So matrix theory is just the weak coupling limit of type I IA string theory The
relevant degrees of freedom surviving are the carriers of p ositive momentum around the
vanishing circle in the th dimension But we already identi ed them as D branes
Studying matrix theory of compacti ed M theory we nd that the limit on the I IA side
shrinks all the radii to zero forcing us to p erform a T duality This way the D branes
turn into di erent branes But after all one nds that matrix theory is de ned via the
worldvolume theory of a certain brane Some of these are rather exotic E g matrix
theory on the T is describ ed by a D brane at very strong coupling so that it turns
into an M brane Similar the T compacti cation is describ ed by D branes at
strong coupling and hence I IB NS branes at weak coupling The T is describ ed
by D branes at strong coupling However in this case one automatically keeps
some of the bulk mo des so that the interacting world volume theory is not decoupled
from gravity Higher dimensional compacti cations are plagued by similar problems
Even though we can still de ne matrix theory as the worldvolume theory of some brane
this description is not any more useful than saying M theory is a consistent d theory
of gravity since the corresp onding worldvolume theories do not decouple from the bulk
gravitons in the matrix limit
4
Existence of limits decoupling the bulk while keeping an interacting theory on the brane are required
for the brane pro of of the existence of certain xed p oints in higher dimensions I will discuss the
existence and non existence of these limits in the following chapters
5
Some doubts have b een voiced whether it is legitimate to neglect all the e ects of the mo des with zero
momentum around the compact circle which b ecame in nitely heavy and were integrated out by keeping
The matrix conjecture this way elevates the SYM non p erturbative string theory
corresp ondence to a principle not only do es the worldvolume SYM capture imp ortant
asp ects of non p erturbative string theory it is used as a de nition of all of M theory
So learning something ab out the worldvolume theory of branes will always automatically
bring us a step further towards the goal of understanding the fundamental theory of
everything
By analyzing the size of D b ound states some evidence can b e found that matrix
theory is holographic that is information in a given space time volume grows like the
area surrounding the volume not like the volume itself This is supp osed to b e
a genuine prop erty of quantum gravity The idea is that the b est you can do is to ll
up your volume with a big black hole and the entropy of the black hole grows with its
horizon area Until recently it has b een totally unclear how such a principle could b e
implemented in string theory Maldacena s prop osal led to a b eautiful realization of
holography in spaces with negative cosmological constant So far matrix theory is our
only candidate for a holographic description of Minkowski space
Worldvolume theory of the NS brane
As we have seen it is easy to understand the worldvolume theory of the D branes from
analyzing the mo des of strings with Dirichlet b oundary conditions The NS and M
branes are a little bit more elusive but using dualities we can say something ab out them
to o First consider the I IB NS brane Type I IB is selfdual that is type I IB with coupling
g and string scale M is dual to type I IB with coupling g and string scale M g
s s s s
s
holding M M g xed The dual theory is the theory of I IB D strings which play
s
pl s
the role of fundamental strings at strong coupling This duality takes NS into D brane
From this we learn that as the D the NS will b e governed by d SYM with gauge
coupling
g
NS
M
s
The sup ersymmetries preserved by an at NS brane living in space will satisfy
L L R R
where the dep ends on whether we consider I IA or I IB
The M brane can b e b est understo o d in the limit where M theory is well describ ed
by d SUGRA Here N coinciding M branes can just b e thought of as a soliton given
by the following SUGRA solution with F b eing the eld strength of the form vector
only the p ositive momentum mo des Doing eld theory these zero mo des carry all the information ab out
the non trivial vacuum structure In DLCQ the vacuum is trivial The description we presented so far
might have to b e mo di ed due to the e ects of the zero mo des
p otential
dr r d ds f dx f
jj
H F
1 4 5 1 4
N l
p
f
r
Analyzing the zero mo des of this soliton one can calculate that the worldvolume supp orts
a d N sup ersymmetric tensor multiplet From d SUGRA I IA duality we
immediately learn that the I IA NS brane hence also supp orts a tensor multiplet
However this time one of the scalars in the tensor multiplet lives on a circle the one
parametrizing the p osition of the brane in the th dimension Using a normalization
in which the scalar has mass dimension the radius of this circle is M
s
6
This is the natural normalization since the scalars sit in the same multiplet as the form B which
must have mass dimension so that the B Wilson line is dimensionless
Chapter
Exploiting the SYM D brane
corresp ondence
Classical Hanany Witten setups
After we have learned in principle how to use the SYM D brane corresp ondence in order
to engineer certain gauge theories in a stringy setup we now want to exploit this cor
resp ondence and study p ossible applications One of the most prominent and intuitive
setups used to learn ab out gauge theories from string theory is the Hanany Witten HW
setup Let me brie y review the basic ideas A very exhaustive review of these setups
and their applications can b e found in
Branegineering
The idea b ehind Hanany Witten setups is to study branes in at space and get interesting
gauge theories by having many at branes intersecting each other The matter content
can b e determined by simple intuitive rules Similarly deformations and mo duli b ecome
easily visible In the last chapter of this work I will establish a dictionary mapping
HW setups to the other approaches where the interesting dynamics is hidden in the
background geometry This way HW setups can b e used to enco de complicated lo oking
information ab out deformations and phase transition in harmless lo oking brane moves
In several cases asp ects of string theory in the background of the intersecting branes can
b e solved leading to interesting results ab out the quantum gauge theory on the brane
We start with the maximally sup ersymmetric sup ercharges SYM on the world
volume of a Dp brane For de niteness let me discuss the case p The idea will b e
the same in the other dimensions
In order to go to more interesting physics with lower sup ersymmetry we have to pro ject
1
As compared to p in the original work of
out some of the degrees of freedom This can b e achieved by letting our D branes which
I henceforth will call color branes N of those will give rise to SUSY QCD with N colors
c c
end on another brane The b oundary conditions will do the job of pro jecting out certain
states
The concept of a brane ending on branes is a straight forward generalization of the
de ning prop erty of D branes as an ob ject on which strings can end Indeed all of the
brane ends on brane con gurations used in this work can b e obtained from this de ning
setup via duality
S-dual F1 on D5 D1 on NS5 T-dual D2 on NS5 D4 on D6 T-dual T-dual S-dual D3 on NS5 D3 on D5 T-dual T-dual D2 on D4
D6 on NS5
Figure Con gurations dual to a fundamental string ending on a D brane
Figure illustrates this chain of dualities Another way to understand the who ends
on whom rules is to study the worldvolume theories The end of a brane is a charged
ob ject In order for a given brane to b e allowed to end on another brane there should
b etter b e a eld on the worldvolume that can carry away that charge In the case of the
fundamental string ending on the D brane the end of the string is charged electrically
under the worldvolume gauge eld Similarly we can explain the other setups of gure
For example the D brane ending on a I IA NS brane is the string like ob ject charged
under the index tensor gauge eld on the worldvolume the D brane ending on the NS
or D is the magnetic monop ole on the worldvolume which is a brane in dimensions
In addition the non vanishing eld strength induced on the brane couples via the DBI
action to the scalars which describ e the embedding of the brane and the brane has
to b end in order to comp ensate the force exerted by the gauge eld Indeed it was shown
in that the DBI action allows for stable soliton solutions which can b e interpreted as
a string ending on a D brane Much can b e understo o d ab out b ending just on the base of
symmetry arguments As we will see in the following b ending is really a quantum e ect in
the eld theory So for now I will neglect b ending and only discuss the classical setup
We choose to let our D branes end on two NS branes This way the D brane
stretches only over a nite interval b ounded by the NS branes Usually one chooses
this nite interval to b e in the direction This will do several things for us for one
certain degrees of freedom will b e pro jected out by the b oundary conditions as desired
In addition since our gauge theory now lives on a compact interval the low energy theory
will b e governed by the KK reduction on the interval Thus e ectively we will b e dealing
with a p dimensional theory Last but not least the addition of the NS brane will break
some more sup ersymmetry We will remain with sup ercharges that is in the p
case with N as can b e checked explicitly using and The following table
displays the worldvolume directions of these branes The additional D brane will b e
explained so on
x x x x x x x x x x
NS x x x x x x o o o o
D x x x x o o x o o o
D x x x x o o o x x x
The presence of these branes breaks the Lorentz symmetry group SO down to
SO SO SO While the rst part is our obvious Lorentz group in d the rest
should b etter have an interpretation as the R symmetry of the corresp onding sup eralgebra
It has to b e an R symmetry due to the fact that we interpreted the worldvolume scalars
as p ositions of the branes in this internal part of spacetime therefore the scalars and
fermions on the worldvolume naturally transform like vectors and spinors under this
internal Lorentz group A symmetry group acting di erently on the fermions and scalars
in a sup ermultiplet is an R symmetry Indeed the R symmetry of the N SUSY
algebra is U SU U SO SO
To determine from p erturbative string theory which degrees of freedom are pro jected
out would require an analysis in the background of NS branes Luckily there is an easier
way to gure out what is going on by remembering once again that the scalars describ e
the p osition of the brane Without the NS branes we had scalars in the d N
vectormultiplet describing the p osition of the D brane in space In addition we
will get a th scalar after KK reduction from the comp onent of the vector itself Under
the d N SUSY of these scalars constitute the b osonic part of a hypermultiplet
HM while the vector together with the other scalars forms the b osonic part of the
vectormultiplet VM Now requiring the D brane to end on the NS brane xes its
p osition in space The only scalars surviving are the motion From sup ersymmetry
it than follows that the surviving multiplet is the VM and the HM is pro jected out The
same will b e true for any D brane ending on NS branes So indeed in order to branegineer
gauge theories a D brane ending on NS branes will b e the starting building blo ck The
gauge coupling of the gauge theory is also enco ded in this simple setup Due to the KK
reduction the inverse gauge coupling is prop ortional to the distance L b etween the NS
branes Using the precise value for the gauge coupling of the p dimensional gauge
theory from a Dp brane susp ended b etween two NS branes is
p
l g
s
s
g
YM
L
Moving the brane along the direction turns on vevs for these scalars This corresp onds
to moving on the Coulomb branch of the gauge theory At generic p oints of the Coulomb
branch all the D branes will b e at a di erent p osition and we are left with an unbroken
rank
U gauge group justifying the name
As already indicated in the brane table ab ove using and there is one more
brane one can add without sp oiling any further SUSY a D brane Analyzing the p ossible
brane motions one nds that a D b etween two D branes supp orts a hypermultiplet
while the vector is this time pro jected out A D susp ended from NS to D is stuck and
hence supp orts no scalars and since we have N this means no degrees of freedom
at all So in order to have a gauge theory we will keep having the D end on the two
NS branes Is there any new multiplet that arise from the presence of N avor D
f
branes The D D system only contains D branes so it is easy to deal with In addition
to the strings ending only on the D brane we will have strings ending on the D and on
the D Analyzing their massless sector one nds N hypermultiplets in the fundamental
f
representation of SU N Naively one can understand this from the fact that such strings
c
will have an N Chan Paton factor on their one end and an N Chan Paton factor on
c f
their other end
The dynamics of this system in the IR will only b e determined by the lightest branes
In the case of HW setups these will b e the lowest dimensional ob jects around that is the
D branes The SU N symmetry is really a global symmetry the gauge b osons from
f
D D strings decouple This general philosophy that only the smallest brane contributes
to the dynamics carries over to HW setups in other dimensions where we susp end Dp
color branes b etween NS branes with D p branes taking over the role of avor branes
Motions of light branes will corresp ond to mo duli while motions of heavy branes are
parameters of the theory
Let us consider the limits involved in more detail Let me discuss the case of D branes
b etween NS branes and avor D branes as in the original work of Since this is in
type I IB the worldvolume of the NS brane supp orts just SYM and is easier to discuss
In order to decouple gravity and higher string mo des we send M and M to in nity
pl s
We hold g g L xed This sets the scale for all the gauge theory mo des In order
s
YM
to decouple the Kaluza Klein mo des from the interval we want this to b e much less than
L that is we have to consider weak string coupling Indeed in this limit the gauge
coupling on the NS and D branes which is M and g M go es to zero justifying
s
s s
in a quantitative manner our assertion that at low energies the only surviving mo des are
those of the lowest dimensional brane leading to a d gauge theory
To summarize we have shown that in the decoupling limit
l l L g xed
s pl
YM
the HW setup with N Dp color and N Dp avor branes describ es an interacting p
c f
dimensional theory with SU N gauge group and N avors
c f
There s so much more one can do
So far we allowed ourselves to freely jump b etween dimensions in order to relate the
classical setups This is indeed p ossible by applying a simple T duality to the original d
setup As long as we act inside the worldvolume of the NS brane it will stay an NS
brane while the color and avor branes lo ose or gain a dimension whether we T dualize
a worldvolume or a transverse direction The maximal dimension we can achieve this
way is from D branes b etween NS branes These will b e the main fo cus of this work
Gauge theories in more than dimensions have a minimal of sup ercharges This is a
classical constraint just following from the size of the spinor representation of SO
Our branes know all this In any dimension the sup ercharge HW setup lo oks as follows
x x x x x x x x x x
NS x x x x x x o o o o
p
Dp x up to x x o o o
p
Dp x up to x o x x x
In order to go down to sup ercharges we need yet another kind of brane One thing
we can do is to rotate one of the players already present Checking the unbroken
sup ersymmetries according to and we see that for example an NS brane living
in space will do the job It will break of the sup ersymmetries presented in the
setup so far In addition the SO SU part of the R symmetry corresp onding to
rotations in the plane is broken to SO U as required Analyzing the massless
mo des on a D brane susp ended b etween NS and NS we now nd that all the scalars are
lo cked Only the N vector multiplet survives the pro jection The same amount of
SUSY will b e preserved if we choose to rotate the second NS brane by any nonzero angle
in the plane The adjoint chiral multiplet from the decomp osition of the N
vector multiplet under N will receive a mass m tan For one recovers the
N theory for the adjoint decouples all together In they also found
k
that when we use k coinciding NS branes a sup erp otential W X is created rather
than the mass term X which we obtained for k One can argue for a term like this
by studying which at directions such a term lifts in the classical eld theory and than
comparing with the p ossible brane motions The second p ossibility is rotating the avor
branes This will leave the matter content untouched Instead the sup erp otential
W XQQ required for an N theory will b e turned o continously with
All these rotations are only p ossible for HW setups in and lower dimensions again
re ecting the fact that is the maximal dimension for SYM with sup ercharges Intro
ducing even more branes with other rotations we can as well engineer gauge theories with
and sup ercharges in and dimensions Of course it is no problem to also write down
con gurations that preserve no sup ersymmetry at all A generic setup will do so
Since rotating branes as we have seen just corresp onds to p erturbing N theories
only a very restricted class of N theories may b e constructed this way For example
except for one exotic exception I will introduce later no chiral gauge theories can b e
constructed this way In order to do so it is necessary to generalize the idea of susp ending
a Dp brane on an interval in order to have a p dimensional gauge theory to having D p
branes on a rectangle a brane b ox This way generic mo dels can b e constructed
at least on the classic level Recently it has b een shown that they are also go o d as
a quantum theory as long as we choose an anomaly free matter content This pro of is
done in the equivalent brane as prob e picture Since the relation b etween the various
approaches will b e one of my main sub jects I will p ostp one a discussion of brane b oxes
to later chapters
Let me at this p oint introduce another p ossible construction used to introduce avors
It will also b e the basic building blo ck for pro duct gauge groups The basic question
to analyze is the following let us consider three NS branes at di erent p ositions in
the direction We will put N D branes on the rst and N D branes on the second
c
c
interval What is the low energy eld theory corresp onding to this We will de nitely get
an SU N SU N gauge theory Additional matter will b e pro duced from the D D
c
c
branes They have one Chan Paton factor in each strings stretching from the N to the N
c
c
gauge group So we would exp ect to obtain bifundamental matter that is a hypermultiplet
fundamental under b oth gauge groups This can again b e veri ed by checking that the
allowed brane motions corresp ond to the classic mo duli space of the gauge theory Taking
the third NS brane to in nity the second gauge group decouples and b ecomes a global
symmetry This way one can introduce avors via semi in nite D branes to the left or
right
Indeed the two ways of including avors are related if we take into account the HW
e ect whenever a avor D crosses and NS a D is created in b etween them This
2
An equivalent way of viewing this is to view NS branes at an orbifold as it was suggested in
For us it seems more natural to either have the advantages of branes as prob es or the HW setup However
the hybrid construction yields the same answers
pro cess holds in all the T dual con gurations as well That brane creation must o ccur
can b e seen by lo oking at the so called linking number of the NS and D brane which
is a top ological invariant In order to have it unchanged in brane crossing prcesses a D
brane has to b e created when D and NS pass So if we move all the avor branes to the
far left or right we will create semi in nite branes The number of avors didn t change
just their realization There are many equivalent pictures yielding the same gauge theory
There is one more classical parameter we can introduce the relative p osition of the
NS branes in the space This corresp onds to turning on an FI term that is adding
Z
d xd D
to the Lagrangian where D is the auxiliary eld in the vector multiplet Once more one
can prove this by analyzing the p ossible brane motions in the presence of this term and
compare with classical gauge theory
We have now all the to ols in order to branegineer quite arbitrary classical gauge groups
The space of all p ossible brane motions corresp onds to the classical mo duli space of the
gauge theory We already identi ed the motions as the Coulomb branch The Higgs
branch is seen if we split the color branes along the avor branes As shown earlier such
a brane segment b etween two color branes supp orts the scalars in a hypermultiplet
corresp onding to moving the segment o leaving a broken gauge group A generic p oint
on the Higgs branch then would lo ok as follows
2 2 2 2 2 1 2 0 2 2 2 0 2 2 1
0 2 2
0 2 1 2 2 2 1
2
Figure A maximally broken situation on the Higgs branch in a theory with
sup ercharges The numbers denote the complex scalar degrees of freedom
asso ciated with motions of the given brane piece The s rule has b een taken
into account
In the picture the fat line is the NS brane the fat broken line is the NS brane the
horizontal lines denote color and the broken vertical lines denote avor branes For a
theory with sup ercharges the NS has to b e replaced by an NS branes and b oth sides of
the picture lo ok the same The picture do es not dep end on the dimension it has to b e
or less since we are only dealing with sup ercharges The numbers denote the complex
degrees of freedom asso ciated to moving around the brane pieces In order to repro duce
the results from classic gauge theory we have to take into account the so called s rule
given a single pair of NS and avor brane only a single color brane is allowed to end on
them Summing up all the degrees of freedom we nd that the complex dimension of the
Higgs branch is for N N
f c
d N N N N N N
H ig g s f c c f c
c
in p erfect agreement with the eld theory counting where we just count the scalar elds
that are not eaten by the Higgs mechanism
d number of scalars number of vectors N N N
H ig g s f c
c
The Higgs branch is not visible when we use semi in nite branes to realize the avors
Turning on FI terms that is separating the NS branes in space for sup ercharges or
just in space for sup ercharges kills the Coulomb branch of the N theories as is
exp ected Since the color branes are only allowed to extend in the direction in order to
preserve SUSY they have to split on the avor branes that is we have to b e on the Higgs
branch
Solving the quantum theory
Bending and quantum e ects
Ab ove I argued in a qualitative manner that a brane has to b end if another brane ends
on it in order to balance the force exerted by the ux it has to supp ort since the end of
the other brane represents a charge on its worldvolume Let me discuss this p oint in a
bit more of a quantitative manner following Before doing so I d like to show that
b ending indeed is a quantum e ect from the p oint of view of the SYM
Let me discuss the N setup in d since this was the main example I discussed
on the classical level so far The classical value of the Yang Mills coupling is given by
g l
s s
g where L again denotes the length of the interval In order to fo cus on the
YM
L
SYM mo des in this setup we want to take the decoupling limit The tension of the
D brane and the NS brane are and resp ectively Since g is prop ortional
s
2
g g
s
s
we see that for zero coupling the NS brane is in nitely heavy and hence is not to g
YM
pulled or b ent by the D brane Turning on the lo op corrections in the gauge theory is
re ected in the b ending of the branes
As p ointed out earlier the DBI action couples the scalars and the vector multiplet
so a non trivial ux on the brane as it is induced by the end of another brane can b e
supp orted by simultaneously turning on vevs for the scalars that is b ending the brane
since the scalars describ e the p osition of the brane We are only interested in the l
s
limit so the SYM approximation of the DBI is su cient We are lo oking for a stable setup
with non vanishing ux One way to construct this is to lo ok at BPS setups One can
construct them by demanding some preserved sup ersymmetry The no force condition
is then guaranteed This discussion was carried out in The same results for the
b ending as found by Callan and Maldacena can b e obtained in a more qualitative fashion
following the discussion of Witten and noting that
the b ending should b e prop ortional to the net charge that is
branes ending on the left brane ending on the right
the scalar terms in the SYM action demands a minimal area embedding of the brane
therefore the b ending should b e a solution of a Laplace equation
the b ending can only dep end on the worldvolume directions transverse to the end of
the other brane and for symmetry reasons should only dep end on the radial distance
From this discussion it follows that in order to solve for the b ending of the NS branes
in a p dimensional HW setup we should solve the Laplace equation in the p NS
brane worldvolume directions transverse to the end of the Dp brane leading to r log
r b ending in dimensions where r denotes the radial co ordinate along the
NS brane worldvolume away from the end of the Dp brane The co e cient in front of
the r dep endence will b e prop ortional to the net number of branes In d the transversal
space is zero dimensional No b ending can o ccur no eld strength can b e supp orted The
net number of branes has to b e zero
In theories with sup ercharges the only p erturbative corrections to the function
come from lo op Since the gauge coupling is enco ded in the length of the interval the
r r log r b ending in dimensions re ects precisely the known lo op
running of the gauge coupling in these dimensions
There is one p eculiar e ect due to the b ending As I stated ab ove only motions of the
light branes are mo duli of the setup Taking into account the b ending we should make this
slightly more precise only such motions leaving the asymptotic form of the heavy branes
untouched will corresp ond to mo duli Consider the simplest setup of N color branes
c
on the interval We argued ab ove that the classic theory describ es U N gauge group
c
N
c
A generic p oint on the Coulomb branch will have an unbroken U corresp onding to
moving the N branes indep endently along the NS branes If we have a r or faster fall
c
of in the b ending quantum mechanically the color branes only created a little dimple on
the NS and the asymptotic of the NS branes will stay unchanged However if we have
logarithmic or linear b ending changing the center of mass would corresp ond to changing
the asymptotic b ehaviour Therefore on the quantum level the center of mass U part
is frozen out and we are only left with an SU N gauge theory This re ects the eld
c
theory statement that only in and lower dimensions a U gauge theory can lead to
interacting IR physics
Once we have frozen out the U we have to nd a new interpretation for the
p osition of the branes An FI term is only p ossible for ab elian gauge groups the auxiliary
comp onent of the vector multiplet transforms as an adjoint and is not gauge invariant
unless we are dealing with U On the other hand SU theories have a new branch
the baryonic branch Interpreting the p osition as a deformation that forces us on
this baryonic branch yields the right mo duli spaces when compared with gauge theory
calculations
In addition to the e ects from lo ops there are also corrections due to instantons These
instantons can b e seen directly in the brane picture It is known that a Dp brane
within a Dp brane satis es the dimensional YM instanton equations So D branes
are instantons within D branes To interpret these D branes as instantons in the
spacetime we should consider Euclidean D branes whose world line stretches along the
direction so that they are contained within the D branes b etween the NS branes
This way the eld theory ob jects and quantum e ects have b een mapp ed to
D branes and their prop erties The problem is now to solve the theory after including all
these e ects
Lifting to M theory
So far we have seen how the quantum corrections manifest themselves in the stringy
embedding Now we are going to actually solve them This analysis was p erformed in
the remarkable work of Witten To implement this solution we use the I IA M theory
duality and view our setup as an d setup For large values of the radius R of the
eleventh dimension d SUGRA will b e a go o d approximation Since b oth the NS and
the D lift to an M theory M brane our whole setup will b e describ ed by just a single
M brane in d For this brane to b e a solution of the d equations of motion it will
b e determined by the requirement that it lives on a minimal area cycle this way Laplace
equation sneaks in again Therefore the shap e of our branes will b e solely determined by
solving the problem of soap bubbles we x the asymptotics of the branes and they will
arrange themselves to live on a minimal area cycle
Before I move on and show that indeed all quantum e ects p erturbative as well as
non p erturbative are indeed incorp orated in the shap e of the M brane in the d SUGRA
limit let me rst discuss the validity of this limit Let me discuss once more the precise
3
Going from SU N to U N basically corresp onds to gauging baryon number
c c
decoupling limit from the I IA p oint of view Here we want to decouple gravity heavy
string mo des and KK mo des from the nite interval that is send M and M to in nity
s pl
g
s
and L to zero while keeping the gauge coupling g xed Translating into d
YM
M L
s
R
is supp osed to b e xed as L go es to zero But this requires us to also units g
YM
L
take R to zero This is the opp osite limit of the one where we are able to solve The
only reason why I nevertheless will go on and do some calculations in what follows is
that holomorphic quantities will b e protected and we can calculate them at any value
of R even though they will only corresp ond to gauge theory quantities in the small R
regime Also qualitative asp ects like e g whether the theory con nes are supp osed to
agree But it has b een shown in that in general unprotected terms do not agree
This is a pity since the holomorphic information is enco ded in the SW curve and was
known already from pure eld theory considerations The branes only give us a nice
organizing principle for analyzing the holomorphic quantities in complicated situations
where eld theory guess and check metho ds like they are usually employed in order to
obtain SW curves do not work anymore like in situations with many pro duct groups
or matter content that leads to non hyperelliptic curves e g two index symmetric tensors
In order to get new information we have to solve the full string theory in the back
ground of NS and D branes a task that seems to o hard with to day s to ols This is a
problem that is common to most approaches of trying to get new gauge information out
of string theory including the most recent one the Maldacena large N conjecture ab out
which I will make some more comments in what follows There is always one regime
where we can easily compute and another regime where we want the answer However for
Maldacena s case to get the full answer we have to do string theory on AdS S with
some RR ux turned on This hasn t b een solved so far but due to the large symmetries
of the background at least there is hop e
Ab ove I have argued that in M theory we can solve for the exact shap e of the M
brane by solving the minimal area condition given a set of b oundary conditions which
incorp orate the classical input Requiring SUSY of the low energy e ective action amounts
to restricting to sup ersymmetric cycles a sp ecial sub class of minimal area cycles
A cycle is sup ersymmetric if a brane wrapping it preserves some amount of sup ersymmetry
For cycles this condition directly translates into holomorphicity A cycle obtained as
the zero lo cus of a holomorphic function of complex co ordinates will preserve as
the zero lo cus of equations in variables of the original sup ersymmetries
Now let me go ahead and show that indeed all the quantum e ects are incorp orated in
the shap e of the brane After this I will write down the solution as obtained in We
already found that the b ending of the brane incorp orates the p erturbative corrections
In the type I IA setup the non p erturbative corrections have to b e put in by hand They
are represented by Euclidean D branes stretching along the direction In order to
incorp orate all instanton e ects we would have to sum over all setups b ound with a given
number of D s What b ecomes of the D branes once we lift to d Remember that the
d origin of a D brane is momentum around the compact x circle that is a b ound
state with D branes corresp onds to adding momentum The p osition of the D
brane would b e a function of time
x x x t
Our eld theory instantons corresp ond to Euclidean D branes that is their worldvolume
stretches along the nite interval in the direction instead of stretching in time According
to the same philosophy they should corresp ond to twist in the d setup that is
x x x x
All quantum e ects lead to b ending and twisting in d and are hence incorp orated
in the shap e of the brane
Now let me move on and present the solution Let me discuss the general setup of
n NS branes leading to a pro duct of n SU N gauge groups We choose as complex
6 10
s x ix R
co ordinates t e e and v x ix so that a D brane is at v const
and an NS brane at t const This is the only complex structure in which our two
ingredients can b e written as holomorphic functions Since our cycle asymptotically
will lo ok like NS or D we have to choose this structure Taking the exp onential in
the de nition of t b eautifully takes into account the compactness of x Using x as
parameter one can introduce a complex coupling constant
i s s i
g
where the s denote the p ositions of the various NS branes At the one lo op level the
lo op running of the gauge coupling from the logarithmic b ending is simply generalized
to
X X
s log v a log v b
i j
i j
where a and b denote the p osition of the D branes to the left and right of the NS brane
under consideration
In order to incorp orate all the non p erturbative e ects we have to solve for a surface
in the four manifold R S which is parametrized by the co ordinates s and v As
stated ab ove N space time sup ersymmetry requires that s varies holomorphically
with v such that is a Riemann surface in R S Using t exp s is de ned by
the complex equation
F t v
At a given value of v the ro ots of F in t are the p ositions of the branes i e F is a
p olynomial of degree n in t On the other hand for xed t the ro ots of F in v are the
p ositions of the I IA branes Recall brie y the situation of a mo del with two branes
i e n This mo del is describ ed by the curve
F t v A v t B v t C v
where A B and C are p olynomials in v of degree k More sp eci cally the zero es of
A v C v corresp ond to the p ositions of the semi in nite branes ending from the left
right on the branes On the other hand the p olynomial B v b elongs to the k branes
susp ended b etween the two branes After suitable rescaling and shifting of v and t one
obtains for pure SU k gauge theory simply A C B v is then a p olynomial of
the following form
k k
B v v u v u
k
where the u are the order parameters of the theory In order to include for example N
i f
avors of hypermultiplets from the right C v takes the form
N
f
Y
v m C v f
j
j
It is imp ortant to note that these curves precisely coincide with the Seib erg Witten Rie
mann surfaces The entire holomorphic N prep otential is enco ded in the curve
F t v
Chapter
Applications of the brane
construction
As we have seen in the previous chapter there is a deep connection b etween gauge theories
and string dynamics In the following I will show how this connection can b e used to
understand certain asp ects of gauge theory and the phase structure of string theory
esp ecially the transitions b etween top ologically distinct vacua
Dualities in the brane picture
Exact S duality
General Idea
In the previous chapter I discussed that there are certain symmetries in string theory
relating one theory at strong coupling to another theory at weak coupling More generally
sp eaking in theories with a free parameter e g the coupling we identify theories with
di erent values of this parameter The easiest example is the self duality of type I IB string
theory where this free coupling is the string coupling and the duality symmetry identi es
the theory at g with the theory at g If we also include the axion g is enhanced to a
s s s
complex coupling parameter and we have a whole SL Z symmetry acting on it This
strong weak coupling duality exchanges fundamental string and D string Taking into
account the full SL Z one nds a whole zo o of p q strings in type I IB This kind of
duality is usually called S duality
The idea that such a duality may also exist in eld theory is very old It is easy to
convince oneself that classical Maxwell theory is invariant under the exchange of
E B j j
el mag n
Of course this duality can only b e valid if we introduce magnetic charges as well as
electric charges Since in quantum mechanics consistency requires that electric charge e
and magnetic charge g satisfy the Dirac quantisation condition
eg n
we see that any implementation of this electric magnetic duality symmetry in a quantum
theory would automatically provide an S duality since the charges also play the role of
coupling parameters and hence strong electric coupling corresp onds to weak magnetic
coupling This idea of realizing a quantum version of electric magnetic duality was rst
voiced in and is hence referred to as Montonen Olive duality The problem is that
in a standard eld theory like QED the coupling constant is not a go o d op erator but
runs according to the renormalization group Therefor it is exp ected that S duality is
only realized in nite theories One way to guarantee niteness is to consider maximally
sup ersymmetric Yang Mills where sup ersymmetry guarantees cancellation of all divergen
cies
Indeed by now it is b elieved that N SYM realizes Montonen Olive duality Like
in I IB string theory the g g duality is enhanced to a full SL Z duality once we
include the theta angle and its invariance under shifts Since at least one of the two
theories we want to identify with each other is at strong coupling it is again imp ossible
to directly prove the duality But several consistency checks have b een p erformed the
most convincing b eing Sen s calculation establishing the existence of all the p q
dyon states SL Z dual to the electron multiplet
Another way to establish this Montonen Olive duality is to exploit the gauge theory
string theory corresp ondence The idea is that by embedding the gauge theories into
string theory string dualities directly translate down into dualities of the eld theory Of
course this is not a pro of of the duality but one reduces the number of indep endent
assumptions to just string dualities
Indeed in the case of S duality in d N SYM it is straight forward to realize
this idea We just study at D branes in uncurved space According to the gauge
coupling on the D brane is just g g p q dyons in the eld theory are p q
s
YM
strings ending on the brane S duality of I IB leaves the D brane invariant Field theory
duality is just what is left of the string theory duality after we decoupled the bulk mo des
S duality in N
There are some examples of nite theories with less than sup ersymmetries In N
the function is solely determined at lo op N N So just by choosing the
c f
right matter content the theory is nite It is b elieved that all these theories do p osses
an S duality acting on their coupling constant This was rst established for the SU
case with avors in
In order to nd the S duality in the brane picture one rst p erforms the lift to M
theory Finiteness translates into a no b ending requirement The duality group is then the
homotopy group of the resulting Riemann surface Since this analysis only dep ends
on having the curve and not whether we obtain it from brane physics or just from eld
theory this statement do es not shed much new light on N S duality The story
b ecomes clearer when going to the branes as prob es picture Finiteness corresp onds to
cancellation of tadp oles in the orbifold background as I will show when discussing the
duality b etween these two approaches The gauge theory can b e realized as D branes on
top of an ADE singularity As in the N case S duality of the embedding I IB string
theory directly translates down to S duality of the gauge theory In general the duality
group will b e bigger than just SL Z since it will also include transformations that just
relab el the gauge group factors These are obvious symmetries Since they in general do
not commute with S duality a large discrete duality group is generated
The orbifold construction carries over straight forwardly to N Now we have
to consider D branes on top of an C singularity with SU Here we have to
distinguish two kinds of tadp oles tadp oles from twist elements that leave a d plane
xed so that they lo ok like N cancel only in a nite theory all other tadp oles
have to cancel in order for the theory to b e free of anomalies This can b e understo o d as
follows a non vanishing tadp ole corresp onds to a net charge In a compact space this has
to vanish If our orbifold has an r dimensional xed plane we are dealing with a charge
in r dimensions If r is bigger than there are no problems with this For r as we have
to deal with in the N case and for the sp ecial tadp oles in N the charge will lead
to a logarithmic divergence I will prove that this divergence is nothing but the running
of the gauge coupling in Chapter So cancellation of tadp oles of the rst kind r is
a necessary requirement tadp oles of the second kind r vanish only in theories with
no running Again one can see the niteness requirement in the dual brane b ox picture as
a no b ending requirement S duality is again established trivially by the embedding
via D brane in type I IB
There are certainly other N S dual pairs The list of nite theories which can
b e constructed using the metho ds of or exhibits many examples which can not
b e realized in any brany way so far and probably most of them exhibit some kind of
S duality One way one might hop e to generate such S dual pairs is to scan through all
nite theories whose Seib erg dual is known Before embarking on this discussion let me
explain Seib erg duality and its brane realization
Seib erg duality and Mirror symmetry
Universality
Another interesting asp ect of eld theory that can b e addressed quite systematically in a
brany language is universality The phenomenon of universality is due to the e ect that
the renormalization group RG ow is irreversible Evolving a theory towards the IR
certain information ab out the theory is lost Several p ossible deformations of the theory
are irrelevant and do not lead to new IR physics Many di erent theories hence ow to the
same xed p oint The detailed information enco ded in the UV physics do es not matter
and all physical IR prop erties are just enco ded in the structure of the xed p oint theory
Since many di erent theories ow to the same xed p oint their IR physics is describ ed by
the universal prop erties of the xed p oint All theories owing to the same xed p oint
are usually referred to as a universality class This is the closest we can get to duality in
the context of theories with a running coupling The dual theories which are usually still
referred to as electric and magnetic no longer are identically but nevertheless describ e
the same physics in the IR
In eld theory it is often very di cult to establish whether two di erent theories
b elong to one and the same universality class Usually one identi es certain p ossible
deformations of the theory as irrelevant This is done analysing their quantum dimensions
Once we know the precise dimension that is after quantum corrections have b een taken
into account of a given op erator we can read o from the dimension if the corresp onding
dimensionful coupling increases or decreases once we multiply all length scales involved
with a certain scaling factor The corresp onding op erators are called relevant or irrelevant
The former lead to a di erent IR description latter leave the IR description unchanged
Of sp ecial interest are dimensionless op erators They are usually referred to as marginal
Existence of an exactly marginal op erator leads to a series of FPs which can b e smo othly
deformed into each other by tuning the marginal coupling that is a xed line
This way one can establish that theories which have the same Lagrangian up to some
irrelevant op erators b elong to the same universality class A much more sp ectacular
example of such a matching was found in where it was established that even two
theories with a completely di erent gauge group can b elong to the same universality
class Usually this phenomenon is called Seib erg duality This nomenclature is due to the
fact that one can view the two di erent UV descriptions as the dynamics of electric and
magnetic variables resp ectively The name duality might b e a little bit misleading since
the two systems do describ e di erent physics away from the xed p oint It is however
quite impressive that they do describ e the same physics in the IR
Most statements ab out dualities of this kind are still conjectural even though a huge
amount of evidence has b een accumulated It would b e desirable to gain a b etter under
standing of this phenomenon of duality from the branes SYM corresp ondence The basic
strategy is the following we should identify certain brane moves as irrelevant This can
for example b e done by matching the resulting gauge theories which are known to b elong
to the same universality class Of course an intrinsic stringy explanation of why such an
brane move should leave the IR physics unchanged would b e desirable It would basi
cally complete the pro of of Seib erg duality But once we have identi ed a certain brane
move as irrelevant we can pro duce a vast variety of eld theories b elonging to common
universality classes by applying the irrelevant brane move to various con gurations
One example of such an irrelevant brane move is changing the p osition of the various
branes involved Since the direction is the one along which the color branes stretch it
seems reasonable to assume that the low energy physics shouldn t dep end on the p osi
tions involved E ectively we p erformed a Kaluza Klein reduction along the direction
to obtain the p dimensional physics from the p dimensional worldvolume of the color
branes In a Kaluza Klein reduction we throw away everything but the zero mo de that
is the constant mo de along the compact direction We do not exp ect that our low energy
physics is sensitive to any structure that is to the brane p ositions in the KK reduced
direction Let me show in the following how one can use this assumption to obtain Seib erg
duality and its d cousin mirror symmetry from this brane move After that I will brie y
comment on the validity of the assumption that this brane move is indeed irrelevant
Seib erg duality
Probably the most famous example of an IR duality of this type is the equivalence of
SUSY QCD with N avors and N colors with another SUSY QCD also with N avors
f c f
but with N N colors and additional singlet elds called mesons coupling via a sup er
f c
p otential M q q with the quark elds Let me explain the phase diagram of SUSY QCD as
it was discussed in For other gauge groups the phase structure will lo ok very similar
with r N N replaced by the ratio where denotes the quadratic index
f c matter adj
phases of SUSY SU(N) QCD
1+1/N 1 3/2 3
2 flavors/colors
QSM
ADS SUPO free free
NACP no vacuum magn. electric flavors/colors 2 3/2 3 s-confine magentic phase diagram self-dual
point
Figure Phase diagram for SUSY QCD
For small r holomorphy and symmetries allow for a unique sup erp otential whose min
imum is at in nity in all the mo duli the ADS sup erp otential An instanton calculation
at N N shows that it is indeed generated for all r For r the quantum
f c
mo duli space is describ ed by mesons and baryons with a unique quantum constraint again
xed by symmetries and holomorphy For N N again baryons and mesons are the
f c
right degrees of freedom The classical mo duli space stays uncorrected This b ehaviour
is referred to as s con ning in the literature It is a sp ecial case of Seib erg duality
with a trivial magnetic gauge group
For r a little bit b elow one can establish the existence of a non trivial IR xed
p oint following the arguments of Banks and Zaks The relation b etween R charge
and conformal dimension contained in the sup erconformal algebra tells us that this xed
p oint b ehaviour has to break down at r Assuming that the non trivial xed p oint
theory usually referred to as Non ab elian Coulomb phase really holds in the whole regime
r a b eautiful and consistent picture emerges It is in this NACP that the dual
description comes into play Using the same b orders in the dual group one can see that
it is also in its NACP The duality then states that this is indeed the same xed p oint
There is plenty of evidence for this conjecture
the t Ho oft anomaly matchings are satis ed
the mo duli spaces match
under p erturbations via sup erp otential terms we ow to new consistent dual pairs
the ring of chiral op erators matches
For r the electric theory loses asymptotic freedom and the theory is free in the
IR For r the electric theory is strongly coupled and intractable however extending
the duality conjecture to this regime one nds a free magnetic phase For sp ecial values
of r one encounters self dual pairs In the case of SUSY QCD this happ ens for r As
shown in in more general gauge theories self duality might show up at r n
for some integer n in this notation SUSY QCD realizes selfduality at n
It was shown by that the two dual theories can indeed b e connected via a
motion of branes in the direction One imp ortant thing one has to remember from our
discussion in Chapter is the Hanany Witten e ect of brane creation when an NS
brane passes an D brane Taking the NS brane all the way through the avor branes
and then around the NS brane one obtains a brane theory that is describ ed by the
dual SYM
Mirror symmetry
Another example of universality is mirror symmetry in dimensions It was rst discov
ered by Intriligator and Seib erg in their study of dimensional N theories that is
sup ercharges In dimensions the vector is dual to a scalar so that b oth the VM and
N
f
1 2
To b e more precise N N ! 1 g ! with g N xed and
f c YM c
YM
N
c
the HM just contain scalars as b osonic degrees of freedom Sup ersymmetry requires the
Coulomb and the Higgs branch b oth to b e hyper Kahler manifolds There might hence
b e a symmetry mapping the Higgs branch of one onto the Coulomb branch of another
theory Since the scalar one obtains from dualizing the vector lives on a circle of radius
g while all the other scalars are in general non compact one should exp ect that such a
YM
symmetry can only exist at in nite coupling that is in the far IR since the coupling has
mass dimension
Taking into account quantum corrections the Higgs branch remains untouched while
the Coulomb branch is corrected at lo op Intriligator and Seib erg constructed theories
which are mirror to each other in the sense that their quantum mo duli spaces agree in
the far IR with the role of Coulomb and Higgs branch swapped
The implementation of mirror symmetry in string theory uses SL Z duality of type
I IB and again the irrelevance of the motion in the direction This analyzes was p erformed
in the original HW pap er and was one of the main motivations for introducing these
brane setups Acting with S duality on a d HW setup changes NS branes and D
branes After rearranging branes in the direction one winds up with a system that has
again a SYM interpretation If we start with a theory with a single gauge group and a lot
of matter we will end up with a pro duct of many gauge groups and just two fundamental
matter multiplets from the two original NS branes turned into D branes I will present
examples later on
k
E g the mirror of U N with bifundamentals and a single fundamental in one of the
gauge groups is U k with an adjoint and N avors In the HW realization we put the
theory on a circle The electric theory is obtained via k NS branes and one D brane in
one of the gauge groups The mirror has just one NS brane which yields U k with an
adjoint The N D branes add the N matter multiplets
Counting quaternionic dimensions of the branches is very simple The dimension of
the Coulomb branch is just the rank of the gauge group so it is N k in the original and k
in the mirror The Higgs dimension is obtained by counting the number of HMs not eaten
by the Higgs mechanism so it is the number of HMs minus the number of VMs that is
N k k N k k for the original and N k k k N k So we nd the exp ected
agreement Doing a lo op calculation one can check that not only the dimension but
really the full hyper Kahler metric agrees
Generalization to U S p groups needs some song and dance since we now have to
deal with the S dual of the O This analysis was p erformed in We can get mirror
symmetry in this case also via combined S and T duality in the brane setup consider
the setup with a single NS and N D s T dualizing this yields k D branes probing a
2 6
Below I will give a more detailed discussion of HW setups with a compact x direction but it is
straightforward to see that a single NS on the circle gives us a single gauge group with an adjoint from
the bifundamental starting and ending in the same group
background of N D branes In the S dual picture we have just a single D and N NS s
T dualizing now yields D branes probing an A singularity with one extra matter
N
multiplet coming from the D We will see later that these give indeed rise to the gauge
groups presented ab ove I think this is a very b eautiful realization of mirror symmetry
since it relates directly the two most prominent backgrounds which were studied using
the brane prob e technique In this picture generalization to U S p gauge groups is straight
forward and yields a corresp ondence b etween D branes probing a D singularity and D
N
branes probing an O N D system
Combining b oth gauge theory with N in d
Mirror symmetry can b e generalized to N in d The mirror can b e
thought of as a theory of vortices In addition one can still p erform the EGK brane move
that yielded Seib erg duality in dimensions so one might exp ect that these theories do
exhibit two di erent kinds of dualities mirror symmetry and Seib erg duality Indeed it
was argued in and from the eld theory p oint of view that the Seib erg duals
suggested by the brane picture still hold in the d setup This way one can pro duce not
just two but really very many gauge theories that live in the same universality class
Irrelevance of the p osition
I have presented several examples of theories in the same universality class by assuming
that the p osition in HW setups is irrelevant This assumption was based on the fact that
in order to identify the low energy eld theory we e ectively KK reduced on the interval
thereby throwing away all mo des that could prob e any structure along this direction
This argument could b e sp oiled if we encounter phase transitions when moving the
branes around Esp ecially dangerous are p oints where we move branes past each other
In the case of a NS brane crossing a D brane we are saved by the brane creation
mechanism and the s rule However there are situations when the IR physics indeed do es
change with the relative p osition of branes This happ ens when NS branes meet parallel
avor branes like it is p ossible in HW setups with sup ercharges the NS is parallel
to the D in the conventions used so far As examples I will discuss the enhancement of
the chiral global symmetry and the phenomenon of avor doubling b oth related to D
branes crossing NS branes Another example of a similar phase transition was discussed
in where it was shown that passing avor branes which are rotated with resp ect to
each other past each other changes the sup erp otential of the corresp onding gauge theory
As a rst example let me discuss the app earance of enhanced chiral symmetry as
suggested in Let us consider a d gauge theory The global symmetry of avor
rotations is visible as the decoupled SU N gauge theory on the D worldvolume With
f
sup ercharges that is all we exp ect since the rotations of fundamental and antifundamental
N chiral multiplets are linked via the N sup erp otential QX Q where X is the
adjoint scalar from the VM For the same reason SO and U S p gauge groups will have
U S p and SO global symmetry resp ectively these are the subgroups of the full SU N
f
avor rotations that leave the sup erp otential invariant However in the situation with
sup ercharges no such sup erp otential is present In the SU case we exp ect the full chiral
SU N SU N symmetry while the orthogonal and symplectic gauge groups will
F L F R
have a full SU N rotation group Generically the branes only see the N remnant
f
If we move all D branes on top of the parallel NS brane they can split this is just a
d HW setup The half D branes will realize the full global symmetry For this to b e
p ossible we had to choose a very particular p osition of the D branes
This is just a sp ecial example of the more general phenomenon of avor doubling
whenever a D brane meets an NS Consider the following cross con guration
7
6
D6
D4
Figure A cross con guration An intersection of a D brane a D brane
and an NS brane
Let me identify the matter content corresp onding to this con guration First let us
recall that when a D brane meets a D brane there is a massless hypermultiplet Q
which transforms under the U U symmetry groups which sits on the D and D
branes The number of sup ersymmetries for such a con guration is and so there is a
sup erp otential which is restricted by the sup ersymmetry to b e m x QQ where m is
the p osition of the D brane x is the p osition of the D brane
We can slowly tune the p osition of an NS brane to touch the intersection of the D
and D branes Lo cally the number of sup ersymmetries is now At this p oint b oth the
D and the D branes can break and the gauge symmetry is enhanced to U U
u r
U U The u d indices corresp ond to the two parts of the D branes and the l r
d l
indices corresp ond to the two parts of the D branes
As usual for the transitions which lead to breaking of the D branes we should lo ok
for an interpretation as a Higgs mechanism Since the number of vector multiplets is
increased by two we need to lo ok for two more massless chiral elds By applying the
same logic as in the case of the enhanced chiral symmetry we see that we have four
copies of chiral multiplets Q Q R R They carry charges
resp ectively under the gauge groups
In addition there are bi fundamental elds for the intersection of the two new D
branes F F with charges and resp ectively Two more bi fundamental
elds come from the two new D branes For the moment we will ignore the bi
fundamentals for the D branes since they have six dimensional kinetic terms and so
are not dynamical for the four dimensional system
The system has sup ercharges and we cannot exclude the p ossibility of a sup erp oten
tial Studying p ossible deformations and comparing with p ossible branches of eld theory
the sup erp otential can b e uniquely xed to b e
Q FR R F Q
What happ ens if we rejoin the D brane and move it away from the NS Dep ending
on whether we move it to the left or right either QQ or RR will b ecome massive since
the strings stretching b etween the one D piece and the D have to stretch a non zero
distance If we embed this cross con guration in an HW setup realizing pro duct gauge
groups we nd that any D brane basically contributes a fundamental hypermultiplet to
all pro duct factors All but one of them will have a nite mass At the sp ecial p oints when
a D touches an NS and is allowed to split the fundamental hypermultiplets in the two
neighbouring groups will b ecome massless simultaneously We have created a situation in
which the matter content do es dep end on the p ositions of the branes involved
S dual N pairs revisited
After this long discussion ab out Seib erg duality and all its cousins we can readress the
question what is the Seib erg dual of a nite theory Many of the examples of Seib erg
duality include theories that can b e made nite up on adding an appropriate sup erp otential
term Is in these cases the Seib erg duality an S duality
Just checking through several examples one nds that in general the Seib erg dual of
a nite theory is not nite so that a g g S duality cannot b e true
YM YM
However in all examples the dual do es have a very sp ecial prop erty it contains at least
one marginal op erator The existence of an exactly marginal op erator corresp onds
to having an arbitrary coupling in the xed p oint theory parametrizing a whole xed
line On the xed line of marginal couplings the theory is sup erconformal i e all
functions are vanishing Following the work of Leigh and Strassler a simple criterion
for the existence of exactly marginal op erators is given by analyzing the exact Shifman
Vainshtein formula for the function The gauge and Yukawa functions get a
lo op contribution and all higher lo op and non p erturbative corrections enter as linear
functions of the anomalous dimensions of the matter elds In general setting the r
functions to zero yields r conditions on the r couplings leaving at most a xed p oint
If some of them are linearly dep endent we get a line of solutions and hence a marginal
op erator
Now we can make the comparison nite mo dels are in general S dual to a sup ercon
formal theory parametrized by a free coupling constant multiplying a marginal op erator
In the sp ecial case that the dual is also nite like in the well known N example
this dual free coupling is just the gauge coupling whereas in general it is a combination
of gauge and Yukawa couplings Several examples along these lines have b een presented
in One of the examples found in actually seems to give a nite dual
of a nite theory The electric mo del is based on the gauge group SO with matter
elds V in N vector and N elds Q in the spinor representations The mo del
f q
with this sup er eld content has vanishing one lo op gauge function Under addition of
a sup erp otential
X
Q Q V W h
i i i
i
the theory b ecomes nite The conjectured S dual is based on an SU U S p
gauge group with a symmetric tensor in the SU factor and several bifundamentals and
fundamentals Again the lo op functions vanish and the sup erp otential is such that
according to the whole theory is actually nite For more technical details see
Non trivial RG xed p oints
App earance and applications
Interacting xed p oint theories
An interesting phenomenon very familiar in and lower dimensions is the app earance of
non trivial xed p oints of the renormalization group Since the function by de nition
vanishes at the FP the theory at the FP is invariant under scale transformations In the
simplest cases the theory at the FP is free that is the coupling vanishes This is for
example the case for the UV xed p oint of asymptotically free eld theories like QCD
Scale invariance of the free theory is somewhat trivial There are just no interactions
present that could set any scale Sometimes however one encounters an interacting xed
p oint In this case at least in the sup ersymmetric version the theory is b elieved to
not only exhibit scale invariance but the full conformal invariance which includes scale
invariance Due to the lack of any scale and hence any mass in this sup er conformal
theory one has to deal with a continuum of states making any particle interpretation
imp ossible All information ab out the theory is contained in its correlation functions
Such conformal theories are needed to describ e critical systems and are of substantial
interest in b oth particle and statistical physics
One of the big surprises coming out of the SYM string theory corresp ondence was
that such xed p oints can also exist in and dimensions This was rst noted in for
maximally sup ersymmetric N form gauge theory in dimensions arising
from compactifying I IB string theory on a K Similar xed p oints where shown to also
exist in d SYM theories From the eld theory p oint of view such xed p oints were
b elieved to b e imp ossible Since the dimension of a gauge eld is in any spacetime
dimension simple dimensional analysis tells us that the gauge coupling must b e dimen
sionful in dimensions other than While in lower dimensions the coupling b ecomes
stronger at low energies in and higher dimensions it b ecomes weaker Therefore all
gauge theories in and and higher dimensions were b elieved to b e infrared free at the
same time b ecoming ill de ned in the UV since for the same reasons the coupling blows
up and the theory is non renormalizable just from naive p ower counting
The caveat in this argument is that we tacitly assumed that at least at some energy
scale gauge theory is a valid description Then it follows automatically that at all lower
scales gauge theory is also a go o d description since the gauge coupling b ecomes even
weaker and we hit the free xed p oint in the IR The only way to avoid this is to have
a theory which is intrinsically strongly coupled so that gauge theory is never a go o d
description A heuristic way to say this is that even though the F gauge kinetic op erator
is an irrelevant op erator turning it on do esn t move us away from the free theory in the
IR we might b e able to reach an interacting theory by taking the coupling parameter of
this irrelevant op erator the gauge coupling formally to in nity leaving us with a strongly
coupled gauge theory From the eld theory this only teaches us that gauge theory is not
the right arena to discuss the app earance of non trivial xed p oints in and dimensions
All we can do is try to write down consistency conditions that have to b e satis ed by the
theory in order to have a chance to have a well de ned strong coupling limit where we
exp ect the xed p oint to b e These conditions where analyzed by Seib erg In
dimensions one has to assure that the theory is anomaly free while in dimensions the
relevant criterion is that once we make a small p erturbation from the xed p oint that
is we turn on small g the resulting infrared free theory b e free of UV divergencies
Surprisingly the branes then teach us that ALL theories satisfying these criteria in fact do
give rise to non trivial strong coupling xed p oints as I will partly show in the following
for the dimensional theories
Another criterion is that there should exist a sup erconformal algebra in the corre
sp onding dimension with the right amount of sup ersymmetry These sup erconformal al
gebras where classi ed by Nahm Again one nds that the branes realize all p ossible
sup erconformal algebras
Applications
Just establishing the existence of these higher dimensional xed p oint theories is certainly
interesting on its own since they were not exp ected to exist But they also have some
interesting applications For one we can basically learn ab out d eld theories from
compactifying d FP theories on a torus In a certain limit the physics of the compacti ed
theory reduces to pure Yang Mills theory Let me brie y discuss the FP compacti ed
on a torus and how it reduces to pure non sup ersymmetric QCD Compactifying the
A theory the theory on N coinciding M branes on a circle of radius R
N
one obtains maximally sup ersymmetric SU N Yang Mills theory in d with coupling
g R the theory on the resulting D worldvolume In order to obtain dimensional
YM
non sup ersymmetric QCD we compactify on yet another circle of radius R and include a
non trivial twist by the R symmetry basically choosing anti perio dic b oundary conditions
for the fermions The resulting N SU N gauge theory has gauge coupling
R
g
QC D
R
according to the standard KK ansatz Besides the gauge b osons of QCD this theory
certainly contains many other states There are the KK mo des with masses R and
R the fermions with masses of order R and the scalars which get masses at one
lo op from the fermions and hence have a mass of order
s
q
g R
QC D
R R R
R R
The question is whether we can nd a limit in which all these states b ecome very massive
while keeping the QCD scale xed
This is indeed p ossible Let us rst read o the QCD scale from the information
we have so far This is we have to take into account the running of the gauge coupling
with the energy scale
g log
QC D
QC D
gives us the gauge coupling at the compacti cation scale hence
R
g R log
QC D
R R
QC D
From this we read o that
R
1
R
2
e R
QC D
Now consider the limit R R with R R All KK states the fermions
and the scalars b ecome very massive in this limit the lightest ones b eing the scalars with
q
R
2
mass R However due to the exp onential suppression in for su ciently small
R
1
R R the QCD scale will b e much smaller than any of the other masses in the problem
leaving us with pure QCD as advertised Note that this limit corresp onds to very weak
coupling in This is not surprising since as discussed ab ove determines the
coupling at the energy scale where all the other elds b ecome imp ortant We want this
to happ en far ab ove the QCD scale that is at very weak coupling due to asymptotic
freedom of QCD
A second very imp ortant application is to study the deformations of the xed p oints
As we will see in what follows this can b e done very easily using branes A given brane
setup represents a certain phase of string theory This will b ecome more transparent once
we have shown that brane setups are actually equivalent to the language of geometric
compacti cations By tuning parameters of this compacti cation that is by moving
around the branes we encounter critical p oints as certain branes collide the non trivial
FP Often at the FP we see new deformations that allow us to p erform a phase transition
into a top ologically distinct vacuum of string theory
Last but not least these FP theories play an imp ortant role in the recent matrix
conjecture As explained in Chapter this conjecture elevates the corresp ondence
b etween gauge theory and non p erturbative string theory to a principle de ning all of M
theory in terms of the world volume theory of certain branes For Matrix compacti cations
on a T the relevant brane theory is the strong coupling limit of the worldvolume of N
coinciding D branes that is the worldvolume theory of N M branes the FP theory
Physics at non trivial FP from branes
General idea
From what we have learned so far it should b e clear that branes are the essential to ol to
prove the existence of strongly coupled xed p oints in and dimensions The princi
pal idea is as follows one considers string theory in the background of certain branes
Since this is a well de ned theory we can try to take the limit in which gravity and the
other bulk mo des decouple In some cases this leaves us with an interacting theory on
the brane The strong coupling limit then corresp onds to having some branes coincide
usually exhibiting an in nite tower of states b ecoming massless as we would exp ect from
a conformal eld theory So proving the existence of an interacting xed p oint amounts
to analyzing whether string theory allows for a decoupling limit that leaves an interacting
theory on the branes To demonstrate this pro cedure let us brie y consider the brane
realization of the xed p oint that was found by using a geometric picture
I will also discuss the maximally sup ersymmetric cases in and higher dimensions
as well as in d with sup ersymmetry In these cases the analysis of tells us
that there are no sup erconformal algebras Indeed we will nd in the brane picture that
in these cases decoupling of the bulk mo des leads automatically to a free theory on the
brane
The theory from the M brane
Consider a system of N parallel M branes In the same spirit as in we want to
decouple bulk gravity by taking the limit M in nity We want this to do in such a
pl
fashion that the theory on the M branes stays interacting Recall that the theory on
the M is that of N N tensor multiplets This theory do es not have a coupling
constant which we could keep xed However we do know that these tensor multiplets
D
couple to the strings that describ e the ends of M branes The tension of these strings is
3
l
pl
where D is the characteristic distance b etween two M branes These tensions corresp ond
to the vevs of the scalars in the tensor multiplet since they are given by the M p ositions
and have mass dimension in the natural normalization Therefor the decoupling limit
D
will b e D l holding u xed At the origin of the mo duli space that
3
pl
l
pl
is if all the u go to zero we exp ect a sup erconformal xed p oint The strings b ecome
tensionless and provide the continuum of massless states of the conformal theory
One might worry that this xed p oint could b e a free theory One way to see this
cannot b e the case is to consider the theory on a large circle R The resulting theory will
b e d SYM with gauge coupling g R We see that this is an interacting theory
YM
and the large R limit even corresp onds to very strong coupling
d=6 N=(2,0)
5 brane
2 brane tensionless string string on 5 brane continuum of massless
modes
Figure Brane realisation of the theory
3
One way to see this relation is to consider the compacti cation in terms of branes The M turns
2
into a D with g l g R
s s
YM
The other maximally sup ersymmetric theories
One can try to do a similar construction for SYM theories in and higher dimensions
which are also maximally sup ersymmetric where this time in d we have non chiral
sup ersymmetry These can b e realized as the world volume theories of D and higher
branes We want to send M and M to in nity in order to decouple all bulk mo des If
pl s
we would again predict non trivial FPs we would b e in trouble since the analysis of
shows that for these cases there are no sup erconformal algebras We exp ect a non trivial
theory once we put several branes on top of each other N colliding branes give rise to
U N gauge theory In order to obtain an interacting theory on the branes we need to
keep g on the branes nite According to
YM
d
g M g
s
YM s
for the d dimensional gauge theory on the worldvolume of a Dd brane Since
g M M
s
pl s
We see that for the D brane g M and hence g go es to zero in the decoupling
YM
YM pl
limit M M This still is true in higher dimensions We are left with a free theory
pl s
For the D g M g we see we can keep g nite if we simultaneously with
YM
YM pl s
M take g to in nity According to strongly coupled I IA string theory is b etter
pl s
thought of as d SUGRA The duality tells us that the d Planck scale M is given
pl
by
M M g g
pl pl s YM
In order to decouple the d bulk we again have to stick to a free theory on the worldvol
ume For the D and D branes a similar story is true In b oth cases g can b e kept
YM
nite only in the strong string coupling limit S dualizing the D brane turns into an
NS brane at weak coupling with gauge coupling g M which is again free in the
NS
s
decoupling limit One can slightly mo dify the decoupling limit by relaxing the condition
that M is supp osed to go to in nity In this case still all the bulk mo des decouple
s
However we are no longer left with just SYM on the brane But whatever it is we are left
with string theory tells us that it exists This way Seib erg proved the existence of a
d string theory It has a string scale M and hence do es not corresp ond to a conformal
s
theory
For the D brane the strong string coupling limit once more decompacti es the th
dimension The D brane b ecomes an M Hence the d SYM at strong coupling grows
an extra dimension We do obtain a non trivial strong coupling xed p oint but it is again
the dimensional theory which we encountered b efore
4
The former is required to decouple gravity while the latter decouples the higher order terms from the
DBI action
d Hanany Witten setups
Motivation
In order to study less sup ersymmetric theories we need more involved brane setups In
what follows I will present a Hanany Witten setup describing xed p oints in d with
N This setup was presented in and further analyzed in The same
xed p oints where analyzed from the branes as prob es p oint of view by and
were geometrically engineered from F theory in The motivation for analyzing these
theories instead of their more sup ersymmetric cousins is threefold
As describ ed ab ove we can learn ab out d gauge theory by putting these xed
p oints on a torus Since the dynamics of maximally sup ersymmetric SYM in d
is rather constrained it is de nitely interesting to do this with less sup ersymmetric
theories If we again want to study non sup ersymmetric QCD by imp osing anti
p erio dic b oundary conditions on the fermions we have this time the choice to put
di erent b oundary conditions on fermions coming from vector and hypermultiplets
op ening up the p ossibility to realize QCD with matter
The xed p oint has only one p ossible deformation which corresp onds to mov
ing the M branes apart To really study phase transitions we need to study the
more elab orate xed p oints with only sup ercharges which in general do allow
several di erent p erturbations leading to a transition b etween distinct phases
When used as matrix mo dels these xed p oints should b e used to describ e DLCQ
string theories with sup ercharges like the heterotic string on a T or type II
string on a K These theories have a much richer dynamics and are hence more
interesting than their more sup ersymmetric cousins
The consistency requirement Anomaly cancellation
As discussed ab ove even though eld theory is not the right arena to discuss the app ear
ance of non trivial FPs it nevertheless imp oses several constraints on the existence of
consistent strong coupling limits In dimensions this criterion is anomaly cancellation
Let me brie y review the relevant mechanism from the eld theory p oint of view In what
follows we will see that the branes actually know ab out this mechanism and only yield
anomaly free theories
The anomaly in six dimensions can b e characterized by an anomaly eight form For
Q
G with matter transforming in the repre a theory with a pro duct gauge group G
sentation R the anomaly p olynomial reads
X X X
0 0
F tr F tr TrF n n trF I
0
R R R R R
0 0
R
RR
Here Tr denotes the trace in the adjoint and tr is the trace in Representation R The
R
symbol tr is reserved for the trace in the fundamental representation n denotes the
R
0
is number of matter multiplets transforming in a particular representation R and n
RR
the number of multiplets transforming in the representation R R of a pro duct group
G G For a consistent theory the anomaly should b e cancelled in some way Either the
anomaly p olynomial vanishes or we cancel the anomaly by a Green Schwarz mechanism
So let us lo ok at the anomaly in more detail First we want to rewrite the anomaly
p olynomial using only traces in the fundamental representation Formally the p olynomial
then lo oks like
X X
0
I c a trF trF trF
0
0
For concreteness let me discuss the case of a single gauge group factor The anomaly
reduces to
I atrF c F
For a c the anomaly cancels completely If a the anomaly can t b e cancelled
and the theory is sick However if a we can cancel the anomaly by the Green Schwarz
mechanism by coupling it to a two index tensor eld In dimensions the tensor splits
into an anti selfdual and an selfdual part While the former sits in the gravity multiplet
the latter is part of the tensor multiplet Dep ending on the sign of c we have to use one
or the other If c we can cure the anomaly by coupling to a tensor multiplet for
c the theory is sick without coupling to gravity If realized in terms of branes in the
latter case there can t b e a consistent decoupling of the bulk mo des For a very detailed
discussion of the anomaly p olynomials for pro duct gauge groups see e g
In the case of SU N gauge theory using the group theoretical identity
Tr F N trF trF
SU N SU N SU N
we nd that
I N N trF trF
f
Therefore the deadly F term cancels precisely for N N The prefactor c of the
f
F term is always bigger than zero so that we can cancel the anomaly by coupling to
a single tensor multiplet
Adding the Green Schwarz counterterm the action contains a coupling of the scalar
in the tensor multiplet to the gauge elds so that the gauge kinetic term b ecomes
p
trF c trF
g
We see that one can absorb the bare gauge coupling into the exp ectation value of by
just a shift of the origin We obtain an e ective coupling
p
c
g
ef f
At we exp ect the p ossibility of a strong coupling xed p oint
The brane con guration
In order to study Hanany Witten setups in dimensions we need as fundamental ingre
dients D branes on whose worldvolume we realize the gauge theory In addition one
has the obligatory NS branes b etween which the branes are susp ended As usual this
compact interval gets rid of one of the worldvolume dimensions of the D brane via
KK reduction In addition the b oundary conditions for the D branes ending on the NS
branes pro jects out some of the elds In total we break of the sup ercharges by
the D and another by the NS leaving N in d
As usual there are two ways to introduce avors in the setup One is to include the
additional avor branes which in our case will b e D branes I will explain how to deal
with them later For now I will include avors via semi in nite D branes to the right
and left With this the players in our game are
x x x x x x x x x x
NS x x x x x x o o o o
D x x x x x x x o o o
D x x x x x x o x x x
The presence of these branes breaks the SO Lorentz symmetry of type I IA down
to SO SO corresp onding to rotations in the space and Lorentz transfor
mations in space The SO can b e identi ed with the SU R symmetry of
the d N SUSY algebra To understand the basic issues that are di erent in d
from the usual Hanany Witten setups discussed ab ove let me rst fo cus on the simplest setup
7,8,9,10
6
L N R
Figure The brane con guration under consideration giving rise to a
dimensional eld theory Horizontal lines represent D branes the crosses
represent NS branes
Figure shows a con guration of N D branes susp ended b etween two NS branes
There are L R semi in nite D branes to the left right Let us rst discuss the matter
content corresp onding to this setup As usual we get an SU N vector multiplet from
the color branes and L R fundamental hypermultiplets from the semi in nite avor
branes But these are not all the matter multiplets we get According to our standard
philosophy we keep all the light mo des coming from the lowest dimensional branes In all
other dimensions this meant decoupling the worldvolume elds of the NS branes But
here the NS branes live inside the D brane they also have a d worldvolume as the
nite D brane pieces we are considering Hence we should also include the matter from
the NS branes
This can also b e discussed in a more quantitative manner lo oking at the precise
decoupling limit As discussed in our introduction to Hanany Witten setups we want
to take M and M to in nity in order to decouple the bulk sent the length L of the
pl s
interval to zero in order to decouple the KK mo des and do this all in such a way to
g
s
keep the gauge coupling of the d theory xed According to we get g
3
YM
M L
s
The theory on the NS branes is a theory of tensor multiplets so there is no gauge
coupling To see what is going on let us translate into d units via We see that
3
l
pl 11
But this is precisely the tension of the M branes stretching b etween the g
YM
L
branes When discussing the decoupling of the theory we found that this quantity
basically governs the interaction strength of the theory on the brane Keeping it xed
means that we should indeed keep the interacting mo des on the branes as well as those
on the worldvolume of the nite D brane
So now what is this additional matter The theory on a I IA NS brane is the theory of
a tensor multiplet This multiplet consists of a tensor and scalars and fermions
Because of the presence of the D branes one half of the SUSY is broken and we are
left with a theory The tensor multiplet decomp oses into a tensor which only
contains one scalar and a hypermultiplet which contains scalars The hypermultiplet
is pro jected out from the massless sp ectrum b ecause the p osition of the semi in nite D
branes xes the p osition of the NS branes Moving the NS brane out of the D brane
remember that we have the brane embedded in the D brane corresp onds to turning
on the FI terms and a theta angle The scalar in the tensor multiplet corresp onds to
motions of the branes in the x direction This is going to b e a mo dulus of our theory
We have two NS branes and therefore two tensor multiplets but e ectively we keep
only one of them b ecause one of the scalars can b e taken to describ e the center of mass
motion of the system The vev of the other scalar gives us the distance b etween the NS
branes On the other hand we know that the distance b etween the NS branes is related
to the inverse Yang Mills coupling of the six dimensional gauge theory according to the
usual KK philosophy The vev of the scalar hence plays the role of a gauge coupling as
exp ected from eld theory
Now let us turn to the quantum picture As in the other theories with sup ercharges
the lo op information will b e enco ded in the b ending of the NS branes and there won t
b e any further corrections We in particular exp ect the b ending to provide us with the
information ab out the anomalies since those are a lo op e ect Indeed the analysis of
the b ending is really easy in d as discussed b efore Since the NS branes are embedded
in the D branes there is no transverse worldvolume The NS brane can t absorb any
ux coming from the charged ends of a D brane The only way we can have a consistent
brane setup is if the net charge cancels The net charge is given by the number of D
branes ending from one side minus the number of D branes ending from the other side
Thus we only get a consistent picture if
N L R
where L R denotes the number of D ending from the left right The total number of
avors is
N L R N
f
Together with the tensor multiplet from the NS branes this is indeed the only right
matter content to cancel the anomaly
Realization of the xed p oint
Our theory has a Coulomb branch parametrized by the vev of the scalar in the tensor
multiplet At the origin the e ective gauge coupling b ecomes in nite and we nd the
strong coupling xed p oint Since the tensor multiplet corresp onds to the distance b etween
the NS branes this happ ens precisely when they coincide The strings from D branes
stretching b etween the NS branes b ecome tensionless Building pro duct groups is straight
forward Again we nd FPs whenever the NS branes coincide
An interesting generalization is to put the theory on a circle In this case we will see
k
an SU N gauge group where k denotes the number of NS branes Tuning the mo duli
in such a way that we obtain the xed p oint theory amounts to moving all the branes
on top of each other Since the distance b etween the branes is the e ective coupling
constant of the corresp onding gauge theory we see that there is always one gauge group
who s color branes stretch around the entire circle no matter where we choose to bring
our branes together This means that we will always have one gauge factor with a
nite gauge coupling that b ecomes free in the IR and hence b ecomes a global symmetry
From the eld theory p oint of view this statement can also b e seen to b e true De ning
5
For SU and SU there are also some other p ossibilities since they do not have an indep endent
fourth order Casimir and hence a vanishes automatically Global anomalies restrict us to N
f
for SU and N for SU The avor case can b e achieved by realizing SU as U S p
f
with an orientifold The other SU theories do not have an HW interpretation
the e ective gauge couplings by absorbing the classic gauge couplings in the vev of the
scalars in the tensor multiplets one nds that there is always one subgroup for which g
ef f
b ecomes negative at large so that we should start of with a nite value for this
coupling
R R
6 6
The value of at which this sign change happ ens is given by g
3
3
dec f act
l g l
s
s
pl 11
remember that has mass dimension That is in the decoupling limit where we send
L
xed this critical value go es o to in nity so we do not see any l keeping
3 s pl
l g
s
e ect of the nite radius If however we choose to also send R to zero as well keeping
R
6
xed we can engineer a theory with the following prop erties
3
l
pl 11
it decouples from all bulk mo des hence it is d
R
6
new degrees of freedom b ecome gauge theory breaks down at energy scales
3
l
pl 11
imp ortant
R
6
the theory contains string like excitations with tension
3
l
pl 11
One usually refers to those theories as little string theories The way we constructed them
is the T dual of the original construction of as a generalization of Seib erg s work
on the theory with sup ercharges
Including branes
So far we have seen how to realize xed p oints in branes Now I will move on to more
complicated setups in order to classify all xed p oints realizable by HW setups In all
anomaly free matter contents in d have b een classi ed Strictly sp eaking this list should
b e seen as the list of all theories that could p ossibly have a consistent strong coupling
xed p oint The nice stringy result is that indeed all of them do HW setups will not
b e able to generate the exceptional gauge groups but will b e very e cient at classifying
those FP allowed with classical gauge groups and pro ducts thereof The few missing cases
can b e realized via geometry
In order to build more general xed p oint theories let us now lo ok at the second
p ossibility to include avors via D branes living in as discussed in D
branes introduce additional complication since D branes are not a solution of standard
I IA theory but require massive I IA that is the inclusion of a cosmological constant m
The D branes then act as domain walls b etween regions of space with di erent values of
m m is quantized and hence in appropriate normalization can b e chosen to b e an integer
The presence of m makes itself known to the brane con guration via a coupling
Z
m dx B F
where B is the form NS gauge eld under which the NS brane is charged and F is the
form eld strength for the form gauge eld under which the D is charged From the
Bianchi identity for the form eld strength dual to F in the presence of a D brane
ending on a brane dF d F x mH where H dB one gets
mdH The function comes from the charge of the D brane end From here
we get back the old result that for m D branes always have to end on a given NS
brane in pairs of opp osite charge that is from opp osite sides However for arbitrary m
this relation shows that RR charge conservation requires that we have a di erence of m
b etween the number of branes ending from the left and from the right
n
L N R
m=p m=p+n
Figure Basic Hanany Za aroni Setup
Figure shows the basic brane con guration involving D branes The n D branes
in the middle give rise to n avors for the SU N gauge group In addition they raise
the cosmological constant m from p to p n In the background of these values of m the
mo di ed RR charge conservation tells us
N L p
R N p n
With the R L avors from the semi in nite D branes the total number of avors is
L R n N still in agreement with the gauge anomaly considerations on the brane
for every p ossible value of p So far we have not succeeded in obtaining any new FPs since
SUSY QCD with N N is just what we were already able to realize with semi in nite
f c
D branes However as we move on we will need the D branes in more elab orate setups
6
Throughout this work I will use the following conventions to x the signs by passing through an D
from left to right m increases by unit In a background of a given m the number of D branes ending
on a given NS from the left is by m bigger than the number of D branes ending from the right
Orientifolds
There is one more element we can incorp orate in the HW brane setup the orientifold
Since an orientifold breaks the same sup ersymmetries as the corresp onding D brane we
can introduce O O or b oth Each of them comes with two p ossible signs Let us rst
consider the O We have to distinguish two p ossibilities O planes with negative or
p ositive D brane charge that is or The former are the T dual of the O
pro jecting I IB to type I On the D worldvolume they pro ject the symmetry group to a
global SO group while on the D we get a lo cal U S p group The p ositively charged O
pro jects onto global U S p and lo cal SO If we want to have vanishing total D charge we
should restrict ourselves to either negatively charged O s with D branes or one O
of each type
There is indeed a physics reason that we should restrict ourselves to satisfy this re
quirement What b ecomes imp ortant is that not only the cosmological constant jumps
when we cross a D brane but also the dilaton If the dilaton is constant to the right
of a given D brane e the inverse string coupling will rise linearly on the left For
symmetry reasons this means that the dilaton will run with slop e into an O from
the left and leave with slop e on the other side In order to b e back to slop e again
b efore hitting the next O one has to put precisely physical D branes in b etween
One interesting application of this b ehaviour is that this way the coupling may diverge
at the orientifold planes while it is nite everywhere else and even constant in b etween
the th and the th physical D Usually one refers to this whole setup as type I string
theory Of course it is no problem to include just one O and fewer D branes by just
keeping far away from the others
m=8 m=-8 m=-8 m=8
K K+8 K K-8 Sp SU SO SU
m=8 m=8 m=-8 m=-8 K K SU SU
+ Asym. + Sym.
Figure Various p ossibilities to introduce O planes
The gauge groups corresp onding to brane setups in Figure have b een analyzed in
the equivalent setup for d in If the O is in b etween two branes the center gauge
group is pro jected to U S p K or SO K resp ectively All other gauge groups stay SU
however the SU groups to the right are identi ed with those to the left and one e ectively
pro jects out half of the SU groups In addition to this in d we get the sp ecial situation
that the O also changes the cosmological constant For symmetry reasons we have to
choose m on the two sides of the orientifold Therefore we obtain in total
Y
SU k i or U S p k
i
Y
SU k i SO k
i
resp ectively From strings stretching in b etween neighbouring gauge groups we get bifun
damentals We have taken all the D branes that are required to cancel the O
plane charge to b e far away Of course including them we get some more fundamental
matter elds and new contributions to the cosmological constant
The other p ossible situation considered in is that one of the branes is stuck
to the O In this case all the groups stay SU again with the left and the right ones
identi ed leading to e ectively half the number of gauge groups In addition the middle
gauge group has a matter multiplet in the antisymmetric or symmetric representation for
p ositive negative charge O planes In d we again have the e ect of the cosmological
constant and as a result get
Y
SU k i
i
with antisymmetric symmetric tensor matter in the rst gauge group factor in addition
to the bifundamentals
The other realization of SO and U S p groups is to introduce an O along the D branes
In discussing O planes we have to b e careful to take into account an e ect rst discussed
in whenever an O passes through an NS brane it changes its sign Originally this
was argued for O and NS based on consistency of the resulting d physics Here we could
basically do the same without including this e ect on would construct anomalous gauge
theories The sign ip miracously removes all anomalies providing strong arguments
in favor of this assumption There also exist some worldsheet arguments that further
supp ort the existence of the ip
Due to this sign ip the O also contributes to the RR charge cancellation Since its
charge is on one side and on the other the number of D branes ending to the left
and right also has to jump by We therefor are left with pro duct gauge groups forming
an
Y
SO N U S p N
i
chain with bifundamentals Bringing the ingredients D O and O together we can
clearly co ok up very complicated mo dels I will write down some hop efully illustrative
examples later on
All these gauge theories I constructed with the help of orientifolds are anomaly free
up on coupling to the tensor multiplets asso ciated to the indep endent motions
of the branes Note that in the case of the O already a single NS brane gives rise to
a tensor multiplet parametrising the distance to the xed plane There is no decoupled
center of mass dynamics since presence of the orientifolds breaks translation invariance
along the direction
Again it is straight forward to put the theory on a circle I will restrict myself to the
discussion of O s The O case is pretty obvious The only thing one has to take care of
is that only an even number of NS branes is allowed so that the global groups from
semi in nite D branes are either b oth U S p or b oth SO and we are able to close the
circle We should either take negatively charged O s with D branes that is the
physical D branes and their mirrors or a pair of opp ositely charged O s
16 D8 16 D8
m=-8 m=8
K+16 O8
K+16 K+8 K K K+8
m=8 m=-8
O8
Figure Brane con guration yielding a pro duct of two U S p and several
SU gauge groups with bifundamentals and an antisymmetric tensor
Enco ding the p osition of the D branes in a vector w where the entries in w denote
the number of D branes in the gauge group b etween NS brane number and number
P
w In one can write down formulas for the resulting gauge groups Clearly
addition we introduce a quantity D that enco des in a similar way the information ab out
the orientifold the entry corresp onding to the gauge group that includes an orientifold is
since an O carries times the charge of a D All other entries are zero
Let me only discuss the case of negatively charged O s with D branes This
will have a dual description in terms of SO small instantons The case of opp ositely
charged orientifolds describ es a disconnected part of the mo duli space It is dual to string
theory compacti cations with reduced rank of the gauge theory like the CHL string
We have to distinguish three cases
the number of NS branes is o dd so one of them has to b e stuck at one of the O s
and the others live in mirror pairs on the circle
the number of NS branes is even one is stuck at each of the orientifolds and the
others app ear in mirror pairs
the number of NS branes is even and all of them app ear in mirror pairs on the
circle and are free to move
Let me discuss in some detail the case where k the number of NS branes is o dd
The other two cases can b e worked out the same way As in Figure we have
two orientifolds on the circle b oth with negative charge An example also explaining the
notation is illustrated in the following gure
w4 w5=w2 w3
5 4 m=-8+w0+w1+w2
3 6 O8 2K+16-2 W0-w1
2 m=8 m=-8 m=-8+W0 0 w2 1 2K m=-8+W0+w1 w6=w1
w1
W7=W0 W0 O8
w0=W0+W7=2 W0
Figure Notation in the example of NS branes on the circle
One of the branes is stuck on the one orientifold We lab el the branes with
running from to k as in Figure In addition there are D branes Let again w
denote the number of branes in b etween the th and th NS brane Obviously
P
w Also symmetry with resp ect to the orientifold plane requires w w
k
It is useful to slightly mo dify the de nition of the vector counting the D branes by
introducing W in order to take care of the e ect that for every gauge group whose color
branes stretch over the orientifold half of the w branes have to b e lo cated on either side
w while in all other cases W w Therefore in these cases we de ne W
According to the rules of how to introduce branes and orientifolds we nd that
the gauge group is
k
Y
SU V U S p V
with
X
W V K
k k
We have matter multiplets subscripts la w and w
k
b el the gauge group and k tensor multiplets
Using the symmetry prop erty W w w W W w the require
k k
P
k
ment of having a total of D branes w can b e rewritten as
k
X
W
which just says that of the D branes have to b e on either side of the orientifold
Therefore
k
X X
W W
and b ecomes
k
X X
W W V K
It should come as no surprise that all these theories are anomaly free This was
explicitely checked in
7
Note that in this numbering the stuck brane is number k
8
The D branes have two e ects rst they introduce a fundamental hypermultiplet in the gauge
group they are sitting in second they decrease the number of colors in every following gauge group to
their right by one p er brane in b etween since the value of the cosmological constant m is changed
Extracting information ab out the FP theory
Global Symmetries
So far I have discussed how to realize FP theories in a given brane setup in the limit that
all the bulk mo des decouple In the spirit of Seib erg identifying a decoupling
limit in which interacting physics survives on the brane is basically an existence pro of for
these FP theories As I discussed earlier this is a result that could not b e obtained from
pure eld theory reasoning Field theory gave us consistency requirements branes show
us the existence
But now we can move on and ask ourselves whether we can actually learn something
ab out the FP theories from their brane realization and our partial knowledge of the
embedding string theory One of the facts we can learn ab out are enhanced global
symmetries Some information ab out the global symmetries can b e gotten already from
eld theory once we p erturb the FP by turning on the relevant p erturbation g F
as discussed in length ab ove we will ow to the free IR xed p oint Dealing with a free
theory it is straight forward to identify the global symmetries In our case there will b e
for example a global avor symmetry SU N It will b e realized in the branes as the
f
gauge group on the higher dimensional branes e g those branes whose worldvolume elds
decouple together with the bulk mo des In our case the SU N global avor symmetry
f
is realized on the worldvolume of the N D branes
f
Once we go to the xed p oint that is turn o the g p erturbation and go to in nite
coupling these symmetries are still present But sometimes it happ ens that we gain some
new or enhanced global symmetries at the xed p oint This can o ccur if some symmetry
breaking op erators b ecome irrelevant at the xed p oint This phenomenon is well known
from eld theory where it go es under the name accidental symmetries some symmetries
which app ear to b e valid in the IR may b e broken by higher dimensional op erators So
while in the full theory they are no go o d symmetries they b ecome b etter and b etter
approximate symmetries when we go to the IR and at the xed p oint they are exact
Enhanced global symmetries in the brane setup are again realized as worldvolume
gauge symmetries on the decoupled heavy branes Consider an easy example Take an
HW setup realizing pure U gauge theory with N in d That is we just
susp end a single D brane b etween two type I IB NS branes The coupling is given by
the separation b etween the NS branes This theory is known to lead to a nontrivial IR
xed p oint as can b e seen following arguments like those in Going to the IR xed
p oint is equivalent to going to in nite coupling since in d the coupling has dimension
of mass and therefor sets the scale Taking the coupling to in nity amounts to lo oking at
energies way b elow the intrinsic scale that is the far IR of our theory The strong coupling
xed p oint corresp onds to taking the two NS branes to coincide Since the worldvolume
theory of I IB NS branes is just SYM this leads to an enhanced global SU symmetry
at the xed p oint This same mechanism works in almost all other HW like setups
However in d the situation is sp ecial and we won t get away with this trick as shown
ab ove in d the dynamics of the NS branes do es not decouple from the D brane gauge
theory In fact we needed to include the tensors from the NS branes in the dynamics
in order to cancel anomalies So coinciding NS branes do not corresp ond to global
symmetries On the other hand the global SU N symmetries on the D branes are
f
already visible at nite coupling that is in the free eld theory It therefore seems that
in the d setup we do not see enhanced global symmetries There is however a way to
obtain such enhanced global symmetries Roughly sp eaking one can go to the xed p oint
by taking the surrounding I IA string coupling to in nity instead of letting the NS
branes coincide
It is well known that in the strong coupling limit the SO N global symmetries on N
D branes coinciding on top of an O plane with N is enhanced to E
N
The missing gauge b osons are realized by D branes stuck on the D brane whose mass
and hence they b ecome massless in the strong coupling limit This leads go es like
g l
s s
to xed p oints with exceptional enhanced global symmetries The b est known example
is the one with no D branes and just one NS brane and D branes all sitting on top
of the O At strong coupling this b ecomes Horava Witten M theory with one M brane
sitting on the end of the world brane This is the well known realization of the small
E instanton
Deformations and Phase Transitions
It is very easy to see deformations of the xed p oint theories in the brane picture For
example we can move one of the NS branes away from the xed p oint If we do this
along the direction we go back on the Coulomb branch of our theory if we move it o
along we turn on some FI terms For the FI terms it do esn t matter at all whether
we turn them on at the xed p oint or at any p oint on the Coulomb branch
More interesting are deformations that are not p ossible away from the FP that is if
the Coulomb branch and a Higgs branch meet and the FP sits at the common origin
In this case we can actually p erform a phase transition That is we start with branes
realizing the theory on its Coulomb branch as one phase of string theory Tuning the
mo duli we can go to the xed p oint At the xed p oint we see a new deformation that
was not p ossible b efore taking us out to the Higgs branch Tuning the Higgs mo duli away
from the xed p oint we have p erformed a transition to a top ologically distinct phase of
string theory
Let me show one example of such a transition In hindsight of the relation to geometry
which I will exhibit in the next Chapter I will call this a small instanton transition It
was rst discussed in the context of d brane setups in USp(8) USp(8) USp(8) + asym.
O8 O8 O8
Figure The small instanton transition trading tensor for hypers
We start with a sp ecial case of a theory with O s and D branes on a circle
All of the D s are on top of a single O so we will see an SO global symmetry
In addition we will study a single pair of NS branes The cosmological constant on
each side will b e resp ectively so that we have to put at least D branes on this
interval They will all terminate on the NS branes The corresp onding gauge group can
easily b e read o to b e U S p There are going to b e fundamental hypermultiplets
from the D branes and one tensor multiplet describing the p osition of the NS and its
mirror on the circle Tuning the scalar in the tensor multiplet we can reach a xed p oint
by letting the NS collide with its mirror at the far orientifold At this p oint we see a
new deformation one of the NS branes can b e moved of in space According to
our rules this will still b e a U S p gauge theory However one obtains an additional
hypermultiplet in the antisymmetric tensor representation from the far orientifold This
gives us new hypermultiplets In addition one of the NS branes moves freely
in space corresp onding to a th hypermultiplet But now the second NS brane
is stuck at the orientifold so there is no tensor multiplet left All in all this describ es
a phase transition trading tensor for hypers These numbers are required for any
theory that can b e coupled consistently to gravity the contribution of a single tensor
multiplet to the gravitational anomaly in d is the same as that of hypers So if the
gauge theory can b e coupled consistently to gravity in one phase and since our setup is
a limit of string theory we should de nitely b e able to do so then the other phase which
lost one tensor and has the same number of vectors has to have more hypermultiplets
in order to have a consistent coupling to gravity to o
9
The gauge theory has a c and hence do esn t need a Green Schwarz mechanism to cancel the
anomaly
Correlators from branes
So far we have seen that quite a lot can b e learned ab out d xed p oints using the brane
realization I have presented in this work But the things we are really interested in are the
correlation functions de ning the sup erconformal eld theory So far they have escap ed
our control
Recently a to ol has emerged that might nally help us to tackle this goal According to
Maldacena there is a duality that relates the theory on the worldvolume of a given brane
setup to gravity in the background of the branes At large N the gravity calculation
reduces to a classical calculation It will b e of great interest to apply these techniques
to the theories I have b een discussing Some progress in this direction has b een made
in where the dimensions of the chiral op erators have b een found In this way the
brane setups I have presented serve as the natural starting p oint for any more elab orate
analysis of the physics of these dimensional theories
A chiral non chiral transition
By the very nature of Hanany Witten setups there is a close connection b etween physics
in d and d dimensions As I explained if we realize our desired d dimensional gauge
theory on the worldvolume of D d branes on an interval matter multiplets can easily b e
incorp orated by orthogonally intersecting D d branes Now instead of just using
D d branes we can use a full Hanany Witten setup corresp onding to a gauge theory
in two more dimensions with twice the amount of sup ersymmetry Applying this idea let
me use our d brane setups to learn something ab out d physics We will follow what
happ ens to our setup under the same brane motions we have p erformed ab ove again
giving rise to phase transitions this time for N sup ersymmetric dimensional vacua
As a highlight we will present a transition b etween a chiral and a non chiral phase
Let me start with the standard ingredients realizing an N theory in d that is
we will have to deal with NS branes D branes and D branes We will allow NS and
D to b e rotated from into space by an arbitrary angle As usual we will refer to
NS branes if they live along and NS if the angle is degrees that is they live
in space When using more general branes the we will call them A B or C NS
branes and the following conventions will b e used for the angles
10
The brane realization of this chiral theory was also discussed in B NS
8 A NS
C NS A D6 θ1 C D6 ω1 ω2 θ2
4
Figure Notation for the angles in the pro duct group setup
In what follows I will at some p oint replace a D by a full d HW setup that is by
D branes with NS branes embedded in them In our d analysis we already realized
that in order to get interesting phase transitions we should really include an orientifold
This way a single NS brane will b e stuck while bringing it together with another one
we will realize a non trivial xed p oint which has a new branch op ening up
Hence the easiest setups we are going to study have at least group factors and
NS branes the NS stuck at the orientifold and then another NS and its mirror In
order to study the resulting theory it is very helpful to rst discuss the matter arising
without the orientifold That is we rst study the following setup for a pro duct gauge
group analyzed in
C
Nf’ x^4,5 x^6
x^8,9
Nc’
Nf
Nc
B
A
Figure The pro duct gauge group considered in The thick lines are
NS branes at arbitrary angle in direction The D branes are parallel to
the A and C NS branes
Let me choose the B branes to b e still NS branes just xing the overall orientation
Let k k k denote the number of A B and C type NS branes resp ectively The
resulting SYM will have SU N SU N gauge group with bifundamental matter F
c
c
and F and fundamental chiral multiplets Q Q and Q Q from the D branes in the rst
and second group factor resp ectively In addition there will b e adjoints X and X For
k k k they will b e massive and can b e integrated out
According to the k k k case then leads to a sup erp otential
k k
W m X m X X FF X FF QX Q Q X Q
while the k k case leads to
k
W Q Q Q X F F QX
m m
In the following I will identify these sup erp otentials by their triple number The
co e cients in the sup erp otential are determined in terms of the angles as
sin
sin
m tan
m tan
As usual all these statements can b e veri ed by studying allowed brane motions and
comparing with the classical mo duli spaces of the gauge groups in the presence of these
sup erp otential terms
Now we are in a p osition to introduce the orientifold I will put an O on top of
the middle NS brane so that the NS is embedded inside the O and we are basically
dealing with a d HW setup In principle we can also include an O However this won t
b e related to any d theory In order to b e symmetric with resp ect to the orientifold we
have to restrict ourselves to
N N
c
c
N N
f
f
and the A branes have to b e mirrors of the C branes
equal number of A branes and C branes that is we consider only k k k con gu
rations
one can only include an O or an O the B NS brane carrying one unit of NS
charge has to b e self mirror
Since one p ossible deformation will b e to move the NS brane along the orientifold cor
resp onding to a baryonic branch of the gauge theory I will also have to study setups with
k This will again allow me to uniquely x the matter content and the interactions
by comparing to the at directions exp ected from classical eld theory So we want to
consider the following p ossible setups
8 6
4
SU with A,S Sp with S W=ASAS SO with A
W=SS
SP with A SO with S
W=SS
SU with S + conj.
SU with A + conj.
Figure Theories with orientifolds and their deformations
On the left in gure we nd the two p ossible pro jections of the pro duct gauge group
setup the right hand side displays two p ossible setups involving just NS branes and an
orientifold which arise as p ossible deformations of the theories on the left as we will see
in the following Let us brie y summarize the resulting gauge groups and matter contents
A more detailed discussion of this identi cation will b e presented in the following sections
The con guration in the upp er right corner of gure is just the N setup
analyzed in The corresp onding gauge group is SO U S p dep ending on the sign of
the orientifold pro jection Rotating the NS branes breaks N to N by giving a
mass to the adjoint chiral multiplet in the N vector multiplet We are left with an
SO U S p Note that this mass is already in nite at since this time it is given by
the angle b etween the outer NS branes which is twice the angle b etween outer NS and
NS that determined the mass b efore Rotating further I claim that instead of the adjoint
tensor that b ecame in nitely heavy a new tensor with the opp osite symmetry prop erties is
coming down from in nite mass As usual the way to prove this is to compare the p ossible
brane moves with results from classical eld theory Note that if we were dealing with
D branes instead of the NS branes the rotation we p erformed would precisely change
their worldvolume gauge theory from SO to S p It s therefore reasonable to assume that
something similar happ ens to the NS branes to o This way we identify the gauge group
corresp onding to the brane con guration in the lower right corner of gure to b e an
SO U S p gauge theory with a symmetric antisymmetric tensor
In the two pictures on the left the two SU factors are identi ed under the orientifold
pro jection One adjoint eld is present whose mass is given by the angle b etween NS
and NS brane In addition there are degrees of freedom that gave rise to bifundamentals
in the pro duct gauge groups According to the analysis of one nds that in the
orientifolded theories the O will give rise to a full avor of symmetric or antisymmetric
tensors for the O dep ending on the sign of the pro jection The really interesting case
is the one with the O This now is part of a d HW setup The O changes sign when
passing through the NS This means that in order to conserve RR charge it has to have
half D branes embedded in it on one side The corresp onding matter content will b e an
antisymmetric tensor a conjugate symmetric tensor and fundamentals Analyzing the
eld theory one indeed nds two baryonic branches corresp onding to moving the NS to
the SO or U S p side of the O and leaving one of the two theories on the right side of gure
resp ectively Note that this is a chiral theory with anomaly free matter content The
fundamentals are precisely what we need to cancel the contribution from the tensors
Now we have all the ingredients we need in order to p erform the transition We
p erform the small instanton transition very similar to how it was p erformed in the d
case and follow through what happ ens to our d physics
11
that is the tensor and its conjugate A B
7
6 4 D6 4 D6
D4
NS NS
Figure A small instanton transition which lead to chirality change in the
sp ectrum An NS brane comes from in nity in the x direction and combines
with the stuck NS at the origin At this p oint they can b oth leave the origin
in the x direction The resulting four dimensional theory is no longer chiral
Consider the con guration depicted in gure A It realizes the chiral theory I just
describ ed I included two other NS branes far away so that they have no e ect on the
dynamics We would like to bring them in from in nity in order to p erform the transition
We will call them upp er and lower branes resp ectively The original NS brane to which
the four dimensional system is attached will b e called central
The NS branes are not allowed to move in the directions since they can only
move o the orientifold as a mirror pair On the other hand the relative motion of the
NS branes along the x direction corresp onds to a change in the two tensor multiplets
in the d theory There is however an option for a pair of NS branes to move in the
directions We can move two NS branes to touch in the x direction At this p oint
the orientifold planes from ab ove and b elow the pair of NS branes are identical as is
clear from gure From a six dimensional p oint of view such a motion corresp onds to
taking one of the gauge couplings to in nity and thereby obtaining strings with vanishing
tension a non trivial xed p oint The pair of NS branes can then move in the
directions as in gure B The resulting six dimensional gauge group is now completely
broken
Let us go back to reinterpret this transition in our four dimensional system When
the NS branes are far we have our chiral theory When the NS branes move in the
directions the theory is di erent We can read it o from gure B We have
SO N SU N with a symmetric tensor for the SO group and a pair chiral and its
c c
conjugate of bi fundamental elds Note that this is a non chiral theory as advertised
Chapter
The equivalence of the various
approaches
As mentioned in the b eginning there are several ways discussed in the literature to embed
gauge theories in string theory So far I have only discussed in detail the Hanany Witten
approach The advantage of this approach is that the setups and esp ecially their mo duli
and deformations have a very intuitive meaning One just has to move around at branes
In this section I d like to show that indeed all the approaches are equivalent in the sense
that the resulting string theories are dual to each other This way I will show that the
phase transitions I studied in the previous chapter by moving branes together and apart
again in a di erent fashion can indeed b e interpreted on the dual side as a transition from
one geometrical compacti cation to a top ologically distinct vacuum This analysis will
hop efully provide one further step towards an understanding of the dynamical pro cesses
relating the various string vacua nally leading to a determination of the true vacuum of
string theory
The duality b etween orbifold and NS brane
The basic relation for our discussion will b e the observation of that string theory
in the background of an NS brane is T dual to string theory on an A type ALE space
This is already an example of an equivalence b etween branes as prob es and geometrical
engineering in the restricted case of theories with sup ercharges Embedding a given d
eld theory in string theory via probing at I IA B with an NS brane or by engineering it
via I IB A on an ALE space is guaranteed to give the same results since these two setups
are actually equivalent as string backgrounds
Let me discuss this duality in some detail I will mostly follow the reasoning of
In order to apply T duality we should lo ok for a U isometry of our solution
C Z do es have some U isometries but the corresp onding radius grows as we move
k
away from the orbifold p oint so that at in nity it b ecomes in nite The would b e dual
should then b e some geometry with a vanishing circle at in nity and hence would b e hard
to interpret
In order to make some progress we consider a slight mo di cation of the original ge
ometry to something that lo oks asymptotically like an S b er over R and still has the
A singularity at the origin These are precisely the prop erties of a k centered
k
KK monop ole in the limit that all the centers coincide Now we just T dualize the U
isometry we introduced by hand One nds that the dual metric is the NS brane metric
as advertised A short cut argument is that T duality exchanges the two U elds in d
generated by the comp onent of the metric and the B eld around the circle resp ectively
Since the KK solution is the monop ole of the former while the NS brane is the monop ole
of the latter the two solutions get interchanged under T duality to o
Hanany Witten versus branes at orbifolds
The orbifold construction
Now we want to move on to the more interesting case of sup ercharges Let me rst
demonstrate that Hanany Witten setups are equivalent to branes at orbifolds a particular
realization of the branes as prob es idea Here I will follow the discussion in We
consider D branes moving on top of a C orbifold space
At this p oint it is necessary to actually work out the eld theory of K Dp branes
moving on the orbifold So let me brie y review the relevant analysis due to Douglas and
Mo ore
Even though the orbifolded space is singular string theory physics on the orbifolded
space is smo oth String theory automatically provides additional mo des in the twisted
sectors that resolve the singularity In the case of an orbifold these twisted sectors are
taken care of by including also the k Z mirror images of each original Dp brane At
k
the orbifold p oint the original Dp coincides with all its mirrors so that we start o with
maximally sup ersymmetric SYM in p dimensions and U k K gauge group In
this theory we imp ose the pro jection conditions by throwing away everything that is not
invariant under Z Since we chose our Z action in such a way that it is a subgroup
k k
of SU remember that this is what we mean by the orbifold limit of an ALE space we
are guaranteed that the remaining sp ectrum will b e that of a theory with sup ercharges
1
The decoupled physics in d won t dep end on the radius of the circle as one could see by recalling
once more the decoupling limit It is only enco ded in the lo cal singularity and not in the global structure
2
Of course we have to restrict ourselves to p so that the transverse space is big enough to carry an
d ALE space This restriction again just re ects the classical fact that theories in ab ove dimensions
have at least sup ercharges
so it is enough to study only the b osons and the fermions just follow by SUSY The Z
k
acts
on the SO subgroup of the R symmetry acting on the scalars and fermions given
in the obvious way by embedding of the geometric action in internal spacetime the
scalars are the spacetime Xs
on the U k K Chan Paton indices according to some representation of Z
k
If we want to describ e D branes which are free to move away from the orbifold xed p oint
we should restrict ourselves to embed the orbifold group into the Chan Paton factors via
the regular representation R of that is the j j dimensional representation that accounts
for every mirror once Note that this representation is reducible and decomp oses in terms
of the irreducible representations r as R dim r r Doing so we take into account
i i i i
the D branes and all its mirrors as is required if we want to have branes that are free
to move away from the xed p oint Other representations can b e considered as well at
least at a classical level and lead to fractional branes We will have
more to say ab out these in what follows For the moment let s restrict ourselves to the
regular representation
We get the following actions on the elds i j lab elling the columns of length K of
vector indices transforming under the same irreducible representation r of and a
i
a vector index of the SO R symmetry
Vectors
i i
A r r A
i j
j j
Q
U dim r N gauge group since r r contains the identity only for i j leaving a
i i j
i
For our case of Z all the r are one dimensional since Z is ab elian Together
k i k
with the gauginos and eventually the scalars corresp onding to motions transverse to the
ALE space and transverse to the brane for p they will combine into full VMs
Scalars
ia ia
a r r
k j
j ik j
where a denotes the Clebsch Gordan co e cient in the decomp osition of r a r
i k k
ik ik
where where denotes the dimensional representation of the scalars under the R sym
metry We hence obtain a scalars transforming as bifundamentals under the ith and k th
ik
gauge group For i k this is interpreted as an adjoint For Z r r and we
k
obtain bifundamentals in neighbouring gauge groups Together with the fermions these
scalars will form HMs
3
e g a D brane in with the ALE space in will have a complex scalar corresp onding to
motions that do es not transform under the spacetime action of the orbifold and is hence pro jected as the
vectors
The classical branches
Having constructed the eld theory corresp onding to D branes on top of the singularity
we can move on and study its classical branches The Higgs branch corresp onds to moving
the branes away from the singularity its metric will b e the metric of the ALE space the
resolved orbifold itself Non trivial xed p oints will o ccur when the branes actually sit
on top of the singularity For p we will also have a Coulomb branch from the scalars
in the VM This will have to b e interpreted in terms of fractional branes For p
that is in d there is no such Coulomb branch asso ciated to the VM However we already
found b efore that in d we need to couple the gauge theory to tensor multiplets in order to
cancel the anomalies These bring in new scalars which parametrize a branch that is now
referred to as the Coulomb branch even though it has quite a di erent interpretation
from in the other cases In the HW setup these tensors were automatically provided
by the NS brane motions In the orbifold construction they corresp ond to degrees of
freedom coming from reducing the d two form on the vanishing two cycles Of course
these degrees of freedom are also there in fewer than dimensions But as in the HW
case taking the limit in which the bulk elds decouple we nd in d that these extra
matter multiplets decouple as well whereas in d we have to keep them since their
interaction strength is of the same magnitude as the d SYM coupling
Including D branes and orientifolds
Let me for a moment just discuss the classical theory so that we do not have to worry
ab out anomalies and charge cancellation We are hence free to study K D branes in
the background of N D branes and an ADE singularity Without the D branes the
Higgs branch still describ es the ALE space itself Without the ADE singularity the Higgs
branch corresp onds to the mo duli space of K U N instantons since a Dp brane is just the
instanton within the Dp brane This can b e put together naturally to the statement
that the theory with ALE space and the D branes together yield a theory whose Higgs
branch is the mo duli space of K U N instantons on the ALE Kronheimer and
Nakajima have introduced these theories as a hyper Kahler quotient construction
previously exactly with the goal of describing these mo duli spaces It is b eautiful to see
that they naturally app ear in the classical analysis of brane theory Of course the same
Higgs branch arises in any other Dp Dp system with the Dp brane wrapping the
ALE space
Besides the obvious data N and K we need one more piece of information the Wilson
lines at in nity Since at in nity is the discrete ADE group such non trivial Wilson
lines can exist They are characterized by a matrix representing in the gauge group
That is is given by w R where R are the irreducible representations of
and w some integers so that really is a representation of In addition for
to b e an element of the U N gauge group it b etter b e an N N matrix so we have
P
to demand that n w N where the n are the dimensions of the corresp onding
representation To keep the notation readable I will from now on fo cus on A type that is
ab elian Z orbifold groups only so that all n The formulas for the general case
k
can b e found in In addition one could also include non trivial rst Chern classes
They would corresp ond to turning on B uxes in the type I IB background For simplicity
I will also neglect those From here on it is a tedious but straightforward analysis to nd
Q
k
the resulting gauge group It is U V where
k
X
C w V K V K
j i
ij
j
As matter we have bifundamentals under neighboring gauge groups and w fundamentals
in the th gauge group This indeed coincides with the hyper Kahler construction Now
let us discuss the quantum theory According to anomaly freedom of the ab ove gauge
theory demands N This is no big surprise we are not allowed to put any D branes
in type I IB
Similarly one can discuss K SO N instantons In the brane picture this is done by
introducing an additional O plane Here anomaly freedom will demand N since
we are now dealing with type I However the classical construction can still b e carried out
for any N yielding again a description of instanton mo duli spaces in terms of the classical
Higgs branch of a given gauge theory The resulting gauge group is mo di ed slightly
For k even there is an additional discrete choice for due to the fact that the
space time gauge group is really S pin Z and not SO The Z is generated
by the element w in the center of S pin which acts as on the vector on
the spinor of negative chirality and on the spinor of p ositive chirality Because
only representations with w are in the S pin Z string theory the identity
element e can b e mapp ed to either the element or w in S pin
G
The orientifold pro jection will of course change the gauge group This is again a
straight forward exercise in applying pro jection op erators and one nds for example
for k o dd
k
Y
SU V S p V
w The matter content consists of hypermultiplets transforming according to
k k
w and
k
1
4 1
C denotes the inverse Cartan metric of that is for Z C i k j k
k +1
i j
the V s pick up an additional contribution from a vector D basically re ecting the
orientifold charge so that we obtain
k
X
C w D V K V K
j j i
ij
j
Cho osing D one repro duces the right classical theory whose Higgs branch
is the mo duli space of instantons while D yields an
k k
anomaly free gauge theory once we couple to the additional tensor multiplets from the
form uxes Cho osing the anomaly free theory one nds that one misses some degrees
of freedom that are necessary to describ e the Higgs branch All in all hypermultiplet
degrees of freedom a missing for every tensor present on the Coulomb branch We again
interpret this as a small instanton transition The Higgs branch describ es the D dissolved
as an instanton inside the D If the instanton shrinks down to a p oint we reach a non
trivial xed p oint From there the Coulomb branch op ens up describ ed by the anomaly
free gauge theory living on the worldvolume of the D The new branch is parametrized
by the vev of the scalars in the tensor multiplets which as I mentioned arise in this setup
from reducing the d two form on the vanishing cycles
More intuitive is the same pro cess for the E small instanton Here the Higgs branch
still corresp onds to an M dissolved inside an M theory end of the world brane carrying an
E gauge theory However the Coulomb branch this time has a real geometric
interpretation moving of the brane into the bulk As we have seen b oth these cases
are also b eautifully realized in the HW setup Here the SO as well as the E E
small instanton s Coulomb branch are visible as motions of the brane
Applying T duality
Of course it is no coincidence that we nd the same small instanton transition in b oth
realizations In fact the two approaches are related by a simple T duality Considering
an HW setup on a circle and T dualizing this direction the NS branes turn into the
orbifold singularity as ab ove the D b ecomes the the D the small SO instanton
The O s and D s turn into O s and D s We repro duce precisely the same low energy
theories if we identify the w and D I de ned in the HW setup as the number of D
branes in the th gauge group and the contribution of the orientifold charge to the th
gauge group with the w and D de ned ab ove basically carrying the information ab out
the non trivial Wilson lines at in nity The Z choice for even k number of NS
branes in the HW setup corresp onds to the choice of having all NS branes free to move
or one stuck at each of the orientifold planes These identi cations re ect the well known
fact that under T duality the Wilson lines translate into brane p ositions
Note that this way a small SO instanton transforms into a D brane in the HW
setup with the NS branes playing the role of the singularity On the other hand taking
the strong coupling limit of the HW the O turns into an M theory end of the world
brane So the strong coupling limit of HW p erfectly captures the small E E small
instanton as well but this time the NS branes are the instantons after all they b ecome
M branes in d while the D branes in d turn into the singularity This way one can
repro duce the theories of describing small E instantons colliding on top of an A or
D type singularity by HW setups
Adiabatically expanding to d N
As I explained in the last chapter a HW setup in d dimensions is closely related to a
HW setup in d dimension with twice the amount of sup ersymmetry by replacing the
avor system in d dimensions by a whole HW setup in d This was how we realized the
chiral non chiral transition in d by p erforming the d small instanton transition This
corresp ondence of course has an equivalent in the geometric language having analyzed
string theory on ALE spaces one can next consider theories on an ALE bration over
a P These will also lead to theories with half the amount of sup ersymmetry in less
dimensions In the limit of large base size one can apply the d results brewise This is
usually referred to as the adiabatic argument
Branes and geometry
obtaining the Seib erg Witten curve
The approaches
As we have seen the branes as prob es and the HW approach to eld theories are actually
equivalent In b oth cases we co ok up a certain string background whose low energy eld
theory realizes the gauge setup we want But then it turns out that these two string
backgrounds are dual to each other This duality allowed us to analyze certain asp ects in
one or the other frame e g deformations were obvious in the HW setup the identi cation
of the Higgs branch as the mo duli space of SO instantons on an ALE space can
easily b e motivated from the orbifold p oint of view Let me nally show in the example
of Seib erg Witten theory that this kind of relation also extends to the third approach
geometrical engineering This will complete our dictionary from phase transitions enforced
by brane motions to the purely geometrical language of string compacti cations
The three approaches to engineer gauge theories have already b een mentioned in gen
eral in the introduction Let me show in more detail how they work in the sp ecial case
of N theories in d I will esp ecially fo cus on the quantum asp ects that is I will
identify the quantum contributions and show how they can sometimes b e solved for
5
A similar connection has b een established in d in
Geometrical Engineering
In the introduction I already discussed how geometric engineering works in principle and
how to engineer d theories with sup ercharges Now we want to go down to N in
d So we have to compactify two more dimensions and break half of the sup ersymmetry
This is done via the adiabatic expansion That is we lo ok at non compact Calabi Yau
manifolds which lo cally lo ok like an ALE bration over a d base The simplest base is
just a P but we will also need bases built out of several P s with non trivial intersection
patterns
Let me rst discuss the case of a P base If we choose the ALE b er to b e of A type
k
we will start with an SU k gauge theory in dimensions The massless states arise
from D branes wrapping the vanishing cycles of the ALE in the orbifold limit Taking
into account the bration structure only the mono dromy invariant states survive It turns
out that if one deals with a genus g base with ADE singularity one exp ects to have ADE
gauge symmetry with g adjoint hyper multiplets That is for our case of
genus zero we only keep a pure N SYM as desired For g one obtains the nite
N sp ectrum and for higher g we would lo ose asymptotic freedom
Decoupling gravity is again achieved by e ectively decompactifying the Calabi Yau
manifold and fo cusing on the lo cal singularity structure However this time there are two
scales involved the size of the base P and the characteristic size of the b ers so that
some care is required
As an illustrative example consider the geometric engineering of a pure SU
gauge symmetry which is related to an A singularity in K Lo cally one needs a vanishing
sphere P around which the D branes b eing the W b osons are wrapp ed This P
f
has to b e b ered over the base P in order to have N sup ersymmetry in four
b
dimensions The di erent ways to p erform this bration are enco ded by an integer n
and the corresp onding b er bundles are the Hirzebruch surfaces F The mass in string
n
units of the W b osons corresp onds to the area of the b er P whereas the area of the
f
base P is prop ortional to g g is the four dimensional gauge coupling at the string
b
scale Now let us p erform the eld theory limit which means that we send the string
scale to in nity Asymptotic freedom implies that g should go to zero in this limit thus
the K ahler class of the base P must go to zero t Second in the eld theory
b
b
limit the gauge b oson masses should go to zero i e t In fact these two limits are
f
related by the running coupling constant
M
W
log
g
and the lo cal geometry is derived from the following double scaling limit
t log t
b f
Clearly this picture can b e easily generalized to engineer higher rank ADE gauge
groups In this case there is not only one shrinking P in the b er but several such that the
f
b er acquires a lo cal ADE singularity For the close comparison with the Hanany Witten
set up it is also interesting how pro duct gauge groups can b e geometrically engineered
For concreteness consider the group SU n SU m with a hypermultiplet in
the bifundamental representation To realize this we have an A singularity over P
n
b
and an A singularity over another P The two P s intersect at a p oint where the
m
b b
singularity jumps to A This can b e seen in dimensions as symmetry breaking of
m n
SU n m to SU n SU m U by the vevs of some scalars in the Cartan subalgebra
of SU n m It is straightforward to generalize this pro cedure to an arbitrary pro duct
of SU groups with matter in bi fundamentals One asso ciates to each gauge group a base
P over which there is the corresp onding SU singularity to each pair of gauge groups
b
connected by a bi fundamental representation one asso ciates an intersection of the base
P s where over the intersection p oint the singularity is enhanced to SU n m
b
Now it is interesting to see how the quantum e ects are incorp orated Basically we
to ok care of the lo op e ects by the double scaling limit the non p erturbative e ects are
due to worldsheet instantons that is non trivial embeddings of the string worldsheet in
the target space Actually the way this app ears in string theory is slightly more involved
From the eld theory p oint of view all the information we want is enco ded in the so called
prep otential
l
X
i A
A A log A c F A
l
i A
l
is the classical gauge coupling the logarithmic term describ es the one lo op Here
2
g
correction due to the running of the gauge coupling and the last term collects all non
p erturbative contributions from the instantons with instanton numbers c This is the
l
function that is solved for by the SW solution
The way this prep otential app ears in I IA string theory is by a classical computation
that is no g corrections We only get corrections from worldsheet instantons This
s
is basically due to the fact that the dilaton which controls the g corrections sits in a
s
hypermultiplet and therefore do esn t talk to the prep otential which controls the vector
multiplets For a Calabi Yau it has the following structure
X X
II
F C t t t n Li exp i d t
AB C A B C d d A A
1
h
A
d d
1
h
where we work inside the K ahler cone Re t The p olynomial part of the type I IA
A
prep otential is given in terms of the classical intersection numbers C and the Euler
AB C
number whereas the co e cients n of the exp onential terms denote the rational
d d
1
h
instanton numbers of genus and multi degree d For our non compact version of the
A
CY there are just two mo duli namely t and t and the corresp onding instanton numbers
b f
n denote instantons wrapping the base d times and the b er d times
d d b f
b f
Now we want to see that in the double scaling limit the string prep otential gives
us back the eld theory prep otential with the right logarithmic one lo op b ehaviour First
note that in this limit the n yield precisely the logarithmic terms from the lo op
d
f
corrections well after all that s how we chose our limit The worldsheet instantons
wrapping non trivially around the base will yield the non p erturbative contributions
But how can we sum them up What comes to our rescue is mirror symmetry Type
I IA on a given Calabi Yau is dual to I IB on the mirror Calabi Yau The two ho dge
numbers counting shap e and size h and h get interchanged It is interesting to
see where the quantum corrections come from As I already mentioned the g corrections
s
will only a ect the HM mo duli space In addition worldsheet instantons will correct
mo duli asso ciated with h since they are asso ciated with the non trivial two cycles But
this means that the vector multiplet mo duli space on the I IB side is exact
g I IB g I IA
s s
1 1 2 1
VM x VM
h h
1 1 2 1
x HM x x HM
h h
All we have to do now is nd the mirror of our non compact Calabi Yau There is a
well de ned description of how to p erform this so called lo cal mirror map This
way we can solve for the SW curve by a classical computation in string theory To give
the SW curve a geometrical interpretation we once more p erform a T duality taking the
singularity of the I IB b er into an NS brane This way the b eautiful nal answer
is obtained The SW curve app ears as the physical ob ject a I IA NS brane wraps on
IIA on non-compact CY IIB Worldsheet Instantons on non-compact CY
Log: limit of large base Solve
NP: finite base size using mirror symmetry
Figure Solving in Geometric Engineering via Lo cal Mirror Symmetry
Fractional Branes
Ab ove I have shown how to construct the eld theory of K Dp branes on top of an
orbifold singularity Their Higgs Branch was identi ed as moving the branes away from
k
the singularity breaking the U K gauge group to its diagonal U N subgroup But
as I mentioned for p we will also see a Coulomb branch corresp onding to the scalars
K k
in the VM along which the gauge group is generically broken to U How do we
see this Coulomb branch in the brane at orbifolds description I will explain this in the
case of D branes relevant for the discussion of N in d Generalizations to other Dp
branes should b e obvious
The answer was given in see also on the Coulomb branch a single D
m
D 3
These fractional branes can b e brane splits into j j fractional branes of equal mass
j j
interpreted as D branes wrapping the vanishing cycles of the orbifold Therefore they
are stuck to the singularity in the orbifolded part of spacetime but are free to move in the
transverse space giving rise to the Coulomb branch The example considered in was
D branes moving on an ALE space In the orbifold limit these new degrees of freedom
supp orted on the Coulomb branch are precisely those that are necessary for the Matrix
description of enhanced gauge symmetry for M theory on the ALE space
Lo oking just at a single fractional brane is achieved by embedding the orbifold group
into the Chan Paton factors via a representation other than the regular one This is
consistent with the fact that only by choosing the regular representation can we describ e
ob jects that are free to move away from the orbifold This identi cation can b e proven
by an explicit world sheet calculation showing that the fractional branes carry charge
under the RR elds coupling to D branes This charge vanishes if and only if one chooses
the regular representation So instead of just obtaining fractional branes by splitting D
branes to move on to the Coulomb branch one can just add these fractional branes by
hand This is the approach chosen in to explain the wrapp ed membranes in
Matrix theory We will give a more detailed analysis of the branches parameters and the
quantum b ehaviour of these theories as we pro ceed
Hanany Witten
The d Hanany Witten setup and its solution via the M theory lift has already b een
discussed in great detail in Chapter Let me summarize Witten s solution I reviewed
there in the form of a diagram like in the other case so that we are able to compare
IIA D4 branes suspended M-theory between NS5 branes M5 brane wrapping (HW setup) Σ Log: bending Solve
NP: D0 branes by lifting to M-theory
Figure Solving HW setup via M theory
A T duality for b ent branes and
branes ending on branes
From our discussion of the d case we are by now used to the idea that branes as prob es
are dual to HW setups The T duality as in the d setup will work the same for D
branes susp ended b etween NS branes and D branes at ADE singularities as long as in
the HW setup the brane connects all the way around the circle In d we were forced to
only consider those setups from anomaly reasons Here they only corresp ond to the very
sp ecial case of nite theories at the origin of the Coulomb branch To understand the
more general cases we will need a generalization of the notion of T duality to the case of
branes ending on branes and b ent branes This will b e the purp ose of this section To
summarize what we will nd the dual of a brane living on a interval b ounded by two
other branes will turn out to b e the fractional brane discussed in the context of branes at
orbifolds Or again phrased in our diagrammatic language
IIB D5 brane wrapping vanishing sphere(s)
Log: divergent potential NP: D1 branes on spheres T duality
IIA D4 branes suspended between NS5 branes (HW setup)
Log: bending
NP: D0 branes
Figure Duality relation b etween HW setup and fractional branes
Consider again K D branes at a transverse A singularity that is we take the D
k
branes to live in the space and put these on top of an A singularity Since the D
k
branes can move away from the singularity we should choose to embed the orbifold group
in the Chan Paton factors via the regular representation that is include the D brane
k
and all its k images The corresp onding gauge group is U K Up on T duality in the
direction this can b e mapp ed into an elliptic Hanany Witten setup that is k NS
branes living in the directions with K D branes living in space susp ended
b etween every pair of consecutive NS branes on the circle Let us lab el the NS branes
with i k This is just the T duality b etween branes at orbifolds and HW setups
I discussed ab ove in detail for the d case
But now consider a setup where we have just K D branes stretching b etween the ith
and the i th NS brane and no other D branes By just comparing the resulting
gauge theory a single U K it is natural to assume that the same T duality in the
direction as we used ab ove now translates this into N fractional branes in the sense of
that is N D branes wrapping the vanishing homology two cycle of the A orbifold
i k
in the dual picture Or again in the language of we consider the same orbifold
group but this time choose to embed the orbifold group into the Chan Paton factors via
the d irreducible representation r asso ciated with the ith no de of the extended Dynkin
i
diagram Or more generally if we have K D branes connecting the ith NS brane with
i
the i th NS brane the dual setup will again b e given by a Z orbifold this time
k
with the orbifold group embedded in the Chan Paton indices via a general representation
R given by
R N r
i i i
As a rst check note that even in this more general case the resulting gauge groups
Q
U K gauge group with hyper and matter contents agree b oth constructions yield a
i
i
multiplets transforming as bi fundamentals under neighboring gauge groups Similarly we
can compare the classical parameters and mo duli First consider the case of all K K
i
where the duality is well established This theory has two branches a Coulomb branch
and a Higgs branch As free parameters we can add FI terms for each gauge group
i
factor which are triplets under the SU symmetry and totally lift the Coulomb branch
R
Since we have k gauge group factors we have in addition k gauge coupling constants
g which at least in the classical theory are additional free parameters Together with
i
i
i
the angles they form the complex coupling constants
2
i
g
i
On the orbifold side the Higgs branch just corresp onds to moving the D branes away
from the orbifold p oint in the ALE space Therefore the Higgs branch metric coincides
with the metric of the transverse ALE space Since all the N are equal in the HW setup
i
all the D brane pieces can connect to form N full D branes which can then move away
from the NS branes in the direction As usual the th real dimension of this branch
is given by the A comp onent of the gauge eld on the D branes
When the D branes meet the NS branes in space that is at the origin of the
Higgs branch they can separate into the pieces which are then free to move around
space along the NS branes giving rise to the Coulomb branch In the orbifold picture the
same pro cess is describ ed by the K D branes splitting up into k K fractional branes
These are now lo calized at the singularity in space but are also free to move in the
space Now consider turning on the FI terms For the orbifold this corresp onds to
i
resolving the singularity by blowing up the ith vanishing sphere The mass of the states
i
on the Coulomb branch which are after all D branes wrapping these spheres b ecome
of order The Coulomb branch is removed as a sup ersymmetric vacuum In the dual
i
P
6
where
k +1 i
i
picture the FI terms corresp ond to motion of the NS branes in space This resolves
the singularity since the branes no longer coincide and again the mass of states on the
Coulomb branch which in this case are D branes stretching b etween the displaced NS
branes is of order
i
Last but not least we have to discuss the coupling constants On the HW side they
are simply given in terms of the distances b etween the NS branes in the direction
i
or to b e more precise
A
l g g
s i
s i
A
where g and l denote the string coupling constant and string length of the underlying
s
s
type I IA string theory resp ectively Since we are dealing with a theory with compact
direction we have
X
R
i
i
where R is the radius of the direction The easiest case to consider is if all
i
that is the NS branes are equally spaced around the circle In this case we get g
i
A B
g R g k l Applying T duality to type I IB this means that g g k since
s
i
s s
B A
g g l R in accordance with the orbifold analysis The same con guration
s
s s
can b e also viewed as N fractional branes that is N D branes wrapping the ith vanishing
i i
cycle According to in this case the coupling constant is given in terms of uxes of
two form charge on the vanishing sphere
Z Z
RR NS
B i F B
i
i i
B
At the orbifold the values of the B elds are xed to yield again g g k Next
i
s
consider the case where all the NS branes coincide In d this will corresp ond to an
enhanced gauge symmetry on the NS brane As shown in this only happ ens if the
B ux is zero on the dual side again in accordance with our formulas Under T duality
R
NS
F B Wilson lines translate into the p ositions of the NS branes on the circle these
i
as exp ected
It is easy to see that this discussion generalizes without any problems to arbitrary
values of K In addition we can introduce D branes in the HW setup living in
i
space to give extra matter These simply T dualize into D branes wrapping the ALE
space in the dual picture
One may wonder whether this is consistent as a quantum theory Since as we discussed
ab ove the fractional branes are charged under the RR gauge elds we need enough non
compact space in the transverse dimensions for this to b e consistent Since we are dealing
7
The one gauge factor with the nite gauge coupling is asymptotically non free In the IR it is
just a global symmetry At higher energies new dimensional degrees of freedom come in from the self
intersection of the NS branes it is imp ossible to put b ent branes on a circle without any sel ntersection
with N SUSY there are only two kinds of corrections one lo op and non p erturbative
corrections All higher lo op contributions vanish exactly The one lo op b eta function gives
rise to the usual logarithmic running of the coupling constant The instanton contributions
have b een summed up by the Seib erg Witten solution
On the HW side it is well known how to incorp orate quantum corrections In the
full string theory the D brane ending on the NS brane actually has a back reaction on
the NS brane leading to a logarithmic b ending of the NS brane Since the coupling
constant is given in terms of distance b etween the NS branes this fact just re ects the
lo op correction the running of the gauge coupling
The non p erturbative e ects in this case are given by Euclidean D branes stretching
along the direction using the well known relation that SYM instantons
inside a Dp brane are represented by Dp branes Like the D branes themselves
these Euclidean D branes can of course split on the NS branes and b ecome fractional D
branes According to the same T duality in the direction we applied to the D branes
they should b ecome fractional D branes that is Euclidean D branes wrapping the
vanishing cycles This ts nicely with the picture that this way the instanton corrections
on the orbifold side are again given by Dp branes living inside the Dp brane
But how do es the lo op e ect arise in the orbifold picture Note that the fractional
branes carry charge under the appropriate RR elds as can b e b orn out by a world sheet
computation of the corresp onding tadp oles They represent a charge sitting in the
space transverse to the D brane and transverse to the orbifold In our example this is the
dimensional space In two dimensions the Green s functions are given by logarithms
Therefore we would exp ect a theory of a charged ob ject in dimensions to b e plagued
by divergences To b e more precise we have the following trouble we want to consider
e ectively a brane sitting in d spacetime Since the transverse space is only d the
corresp onding form vector p otential C grows logarithmically
From the D brane p oint of view the brane vector p otential couples to the world
volume theory via
Z
dx F C
After compacti cation on this leads to a term in the brane action of the form
i
Z Z
F dx C
i
R
F as the gauge coupling But recall that we identi ed The growing of C can
2
i g
i
b e absorb ed by introducing an e ective running coupling constant The divergences we
encounter are just due to the lo op running of the gauge coupling Charge neutrality
which is obtained if we choose the regular representation corresp onds to niteness
In this way we can see that the correct dep endence of the b eta function on N and N
c f
is repro duced This can most easily b e seen by observing that the charge clearly dep ends
linearly on the number of fractional branes and is zero when N N corresp onding to
f c
the case where the fractional branes can all combine to form a full brane The relative
contributions originate from the self intersection numbers of the spheres given by the
i
extended Cartan matrix here for A
k
In this way the b ending of the branes directly translates into the fall o of the RR
eld in transverse space in the orbifold picture The corresp ondence also holds in other
dimensions for D branes the transverse space is dimensional and we need neutrality
re ecting anomaly freedom of the underlying gauge theory in dimensions we
get linear logarithmic r r fall o for the eld strength This re ects the
appropriate running of the gauge coupling just as in the HW picture
Unifying the di erent approaches
Now we are in a p osition to put all the three approaches together into an unifying picture
To visualize what we are ab out to do let me rst show a diagram that summarizes the
connection I will then explain the dualities involved step by step IIB D5 brane wrapping vanishing sphere(s)
Log: divergent potential NP: D1 branes on spheres T-duality T +S-duality duality
IIA D4 branes suspended IIA between NS5 branes on non-compact CY (HW setup) Worldsheet Instantons Log: bending Log: limit of large base NP: D0 branes NP: finite base size
Solve using mirror symmetry by lifting to M-theory
IIB on non-compact CY
T-duality
M-theory IIA M5 brane wrapping equivalent description NS5 brane wrapping Σ (transverse radius Σ
irrelevant)
Figure Connection b etween the various approaches The non compact
Calabi Yau spaces are ALE spaces b ered over a sphere or several intersecting
spheres These can b e T dualized via T duality on the b er The last T
duality connecting to the HW picture is the one discussed in this work
In Figure the connections are displayed diagrammatically putting together the
pieces of Figures and For the geometric engineering approach of
the starting p oint is I IA string theory on a non compact Calabi Yau which is an ALE space
b ered over some base a non compact version of a K bration Quantum corrections
can b e solved by using mirror symmetry to a I IB picture A T duality on the ALE b er
can b e used to map this to a I IA NS brane wrapping the Seib erg Witten curve
While in this approach the solution is obtained via mirror symmetry this last T duality
leads to the very nice interpretation of the SW curve it app ears as the physical ob ject
the brane is wrapping
Applying the same T duality which we have just applied to the I IB solution directly
on the ALE b er in the I IA setup which is the starting p oint for the geometric engineering
approach followed by an S duality in the resulting I IB string theory it is straight forward
to connect this to a setup of D branes wrapping the base of the bration
Using the T duality b etween fractional branes and branes ending on branes prop osed
ab ove one can translate this into the very intuitive language of a HW setup which
nally can b e solved for by lifting to M theory This way we not only obtain the
same solution the SW curve but also the same physical interpretation the SW curve
app ears directly as the space an M brane wraps on
In the following we will discuss the steps in the ab ove chain of dualities in more detail
There are two imp ortant p oints one should account for For one thing there are the
quantum corrections Since we are dealing with N there are only two types of these
corrections the lo op contribution which gives rise to the logarithmic running of the
gauge coupling and the instanton corrections It is interesting to see how these quantum
e ects are incorp orated in the various pictures and how they are nally solved for
The other thing one should treat with some care is how the singularity type a ects the
gauge group In all the examples we are considering there are always two pieces of infor
mation which we will refer to as the singularity types determining the gauge group and
the pro duct structure This is easiest to understand in a particular example Consider
k
N D branes at an A singularity This gives rise to an SU N gauge group In this
k
example N determines the gauge group the size and type of the single gauge group
factors and k the pro duct structure we have k SU N factors Since in the various
duality transformations we are using we rep eatedly map branes into singularities and vice
versa it is quite imp ortant to distinguish which ingredient in the picture determines gauge
group and which determines pro duct structure In the cases where the gauge group is
determined by a geometric singularity generalization to D or E type singularities leads
to SO or E gauge groups while in the cases where the pro duct structure is deter
mined by a geometric singularity the generalization to D or E type just leads to a more
involved pro ducts of SU N groups determined by the corresp onding extended Dynkin
diagram For example in the D brane case N D branes at an E singularity lead to an
SU N SU N SU N gauge group
HW fractional branes This is the duality b etween branes ending on branes
and fractional branes I have discussed ab ove A system of k NS branes with N D
i
branes susp ended b etween the ith and the i th NS brane is dual to an A orbifold
k
singularity with N fractional branes asso ciated to the ith shrinking cycle While in the
i
HW setup the gauge group is determined by the D branes and the NS branes determine
the pro duct structure the corresp onding roles are played by D branes and the orbifold
type in the dual picture That is the vanishing cycles are given by spheres intersecting
according to the extended Dynkin diagram The gauge group can b e generalized to D type
8
In the spirit of the di erence b etween I IA and M brane is irrelevant since the radius of the
transverse circle do es not a ect the low energy SYM on the brane
by including orientifold planes on top of the D branes It is not known how to achieve E
type gauge groups this way Using an E type orbifold in the fractional branes only a ects
the pro duct structure
The lo op quantum e ects corresp ond to b ending of the NS branes in HW language
and get mapp ed to the logarithmic Green s functions in the orbifold picture as discussed
in the previous section Non p erturbative e ects are due to Euclidean D branes which
are mapp ed by the same T duality into Euclidean D branes wrapping the vanishing
cycles fractional D instantons
Fractional branes I IA on non compact CY We can now apply a di erent
T duality on the setup describ ed by the fractional branes First we S dualize taking the
N D branes into NS branes and the Euclidean D branes into fundamental strings
i
that is world sheet instantons Performing a T duality in the overall transverse space
that is space the NS branes turn into an A singularity according to the duality of
N
i
which we have already used several times The vanishing cycle which is still the
space built out of the k spheres intersecting according to the extended Dynkin diagram
stays unchanged The fact that the NS branes only wrap parts of this base N NS
i
branes on the ith sphere translates into the fact that the type of the ALE b ers over
i
the base changes from one sphere to the other From the I IA p oint of view this lo oks like
T duality acting on the b ers as describ ed in Since now everything is geometric
generalizations to E type are straight forward the pro duct structure is determined by
the intersection pattern of the k spheres the ADE type of the b er determines the gauge
group This way we can even engineer pro ducts of exceptional gauge groups
The non p erturbative e ects are now due to world sheet instantons as exp ected The
log corrections coming from one lo op are incorp orated in the particular limit one has to
choose to decouple gravity The system can b e solved via lo cal mirror symmetry
Chapter
Op en problems and directions for
further research
Brane Boxes
As I mentioned in the b eginning there is a natural generalization of HW setups the
brane b ox In order to branegineer generic mo dels with sup ercharges one should
consider living on a rectangle b ounded by two kinds of NS branes Only recently it has
b ecome clear that these mo dels are indeed consistent also at the quantum level This
calculation is done using a generalization of the HW branes as prob es T duality I have
presented here Brane b oxes are T dual to D branes at a C orbifold where this time
SU so that we are left with N instead of N Again a brane on a single
b ox is T dual to fractional branes characterized by a given irreducible representation of
the orbifold group just as in the case I was discussing
A tadp ole asso ciated to a generic orbifold element without any xed plane corresp onds
to a charge in a d space So it has to vanish Indeed it was shown that vanishing of
these generic tadp oles is equivalent to anomaly freedom However there are also tadp oles
corresp onding to a twists that leave a d xed plane For the same reasons as I discussed
in the previous chapter one should interpret the resulting logarithmic divergence as the
running of the gauge coupling Vanishing of these tadp oles therefore is not necessary for
consistency but implies niteness This condition once more corresp onds to a no b ending
condition for the brane b oxes
It would de nitely b e desirable to p erform the lift to M theory for the brane b oxes
For one this should explain the anomaly in terms of b ending and hence explore some
yet unknown asp ects of brane physics It is still the wrong limit to actually solve the
theory but again it should b e p ossible to solve for all the holomorphic quantities Some
intrinsic N phenomena such as dynamical sup ersymmetry breaking can b e studied
this way and nd their natural place in string theory It is obvious that such a lift has to
b e p erformed via a SUSY cycle instead of the SUSY cycle The conditions for SUSY
cycles have also b een worked out in These equations are technically much more
involved but hop efully the lift can b e p erformed
Maldacena Conjecture
Recently a really remarkable prop osal has b een made by Maldacena generalizing the
asp ects of the brane SYM corresp ondence I have b een discussing so far It states that
the worldvolume theory of a brane is really dual to string theory in the background of the
brane For macroscopic systems that is large number of coinciding branes the curvature
of the brane solution b ecomes small so that sup ergravity is a go o d approximation of
string theory Taking the limit M reduces the worldvolume theory to SYM while
s
we fo cus in to the near horizon region of the soliton leading to the statement that large
N N SU N SYM is dual to sup ergravity in an AdS S background
This conjecture already led to a variety of b eautiful results Again it basically helps
to understand asp ects string theory as well as of eld theory As I explained in Chapter
it is the most promising approach for actually solving some of asp ects of eld theory
that so far have escap ed our control On the other hand we have learned several new
asp ects ab out quantum gravity As I mentioned it is the rst realization of the concept of
holography which is supp osed to b e a genuine prop erty of quantum gravity So it seems
that there are still many asp ects of brane physics that need to b e explored Hop efully
in the end we will nd a true understanding of the fundamental problems we set out to
solve
Summary
In this thesis I discussed several applications of the connection of non p erturbative string
theory and SYM theory In Chapter I reviewed the physics of D branes as one example of
a non p erturbative e ect in string theory Their dynamics is dominated by gauge theory
This fact can b e used to engineer certain string backgrounds which yield interacting SYM
theories as their low energy description
In Chapter I then introduced one of the approaches in detail the HW setup I gave
a summary of the identi cation of the classical gauge theory showed how quantum e ects
manifest themselves in the brane picture and how to solve them
This way of embedding gauge theories into string theories has several interesting appli
cations These were the topic of Chapter First I discussed dualities in eld theory and
showed how they arise as a natural consequence of string duality As a second application
I used branes to prove the existence of non trivial xed p oint theories in dimensions
and to study their prop erties Some of these xed p oints describ e phase transitions b e
tween two di erent brane con gurations From a d p oint of view these d transitions
can induce a chiral non chiral transition
In Chapter I discussed the relation of the HW setup with the other approaches
of embedding gauge theory into string theory esp ecially the branes as prob es approach
The di erent ways of embedding gauge theories in string theory are shown to b e actually
T dual as string backgrounds For one this allowed us to explore several new asp ects of
T duality like T duality for b ended branes and branes ending on branes In addition this
relation can b e used to show that the transitions found in the brane picture can as well
b e understo o d as transitions b etween top ologically distinct compacti cations of string
theory
Some op en problems and directions for further research were mentioned in Chapter
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D Berenstein R Corrado and J Distler Asp ects of ALE matrix mo dels and
twisted matrix strings hep th
A Lawrence N Nekrasov and C Vafa On conformal eld theories in
four dimensions hep th
P S Aspinwall Enhanced gauge symmetries and K surfaces Phys Lett B
hep th
N Seib erg and S Sethi Comments on Neveu Schwarz ve branes
hep th
S Kachru A Klemm W Lerche P Mayr and C Vafa Nonp erturbative results
on the p oint particle limit of N heterotic string compacti cations Nucl Phys
B hep th
A Hanany and A M Uranga Brane b oxes and branes on singularities
hep th
Zusammenfassung
Nach jahrelanger Suche hat sich bis heute Stringtheorie als einziger Kandidat einer kon
sistenten Quantentheorie der Gravitation herauskristallisiert Um aus der Stringtheorie
pr azise Vorhersagen f ur unsere Niederenergiewelt zu gewinnen ist es notwendig das Vaku
umproblem zu l osen das hei t einen Mechanismus zu nden der aufzeigt in welchem
Stringvakuum wir leb en und warum die Natur dieses ausgew ahlt hat Die Beantwortung
dieser Frage b en otigt nicht perturbative Informationen Diese wurden erst in j ungster
Zeit zug anglich
Eine b esondere Rolle in dieser Entwicklung spielten die sogenannten D branes Sie
stellen m ogliche nicht perturbative Beitr age zu Stringamplituden dar Die Identi zierung
da D branes einfach Ob jekte sind auf denen Strings enden k onnen erm oglicht sie zu
handhab en und zu zeigen da ihre Dynamik im wesentlichen durch Eichtheorien erfa t
wird D branes erlaubten zahlreiche Dualit atssymmetrien zu etablieren deren Hauptaus
sage zu sein scheint da alle Stringtheorien sowie d Sup ergravitation nur verschiedene
p erturbative Limites einer fundamentalen d Theorie sind M Theorie
In dieser Arb eit hab e ich mich mit einigen Anwendungen dieser Ideen b esch aftigt Die
Tatsache da D branes durch Sup er Yang Mills Theorien b eschrieben werden erlaubt uns
einen Stringhintergrund derart zu pr aparieren da wir nahezu jede Eichtheorie als rel
evante Niederenergieb eschreibung erhalten k onnen Eine b esonders verbreitete Variante
dieser Idee sind die sogenannten Hanany Witten setups in denen dieser Stringhinter
grund nur aus achen branes im achen Raum b esteht Mit Hilfe dieser Technik hab e
ich verschiedene Dualit atssymmetrien in Feldtheorien auf Stringdualit aten zur uckgef uhrt
Ferner ist es m oglich mit Hilfe der branes die Existenz nicht trivialer Fixpunkt Theo
rien in sechs Dimensionen zu b eweisen und einige ihrer Eigenschaften zu analysieren
Einige dieser Fixpunkte b eschreiben Phasen uberg ange zwischen verschiedenen brane Hin
tergr unden Unter anderem l a t es sich auf diese Weise zeigen da es in Dimensionen
Ub erg ange zwischen chiralen und nicht chiralen Vacua gibt
Ferner wurde gezeigt da alle anderen Zug ange zu dem Problem Eichtheorien in
Stringtheorie einzub etten im wesentlichen aquivalent zum HW Ansatz sind in dem Sinn
da die entsprechenden Stringhintergr unde dual zueinander sind Dadurch k onnen neue
Asp ekte der String T Dualit at verstanden werden so wie z B T Dualit at f ur brane Seg
mente und geb ogene branes Au erdem erlaubt uns diese Verbindung die Phasen uber
g ange die wir im HW Bild entdeckt hab en tats achlich als Ub erg ange zwischen top ologisch
verschiedenen Stringkompakti zierungen zu verstehen
Danksagung
An dieser Stelle m ochte ich mich b ei Herrn Dr Dorn f ur die Betreuung dieser Arb eit
b edanken und b ei Prof Dr L ust f ur seine tatkr aftige Unterst utzung meiner Forschung
Ein Gro teil der Arb eiten ist in Zusammenarb eit mit Ilka Brunner entstanden ihr gilt
daher mein b esonderer Dank Auch Amihay Hanany Andre Miemiec Douglas Smith und
George Zoupanos danke ich f ur die gute Zusammenarb eit F ur Diskussionen danke ich
au erdem M Aganagic B Andreas G Curio J de Bo er J Distler M Faux S Kachru
A Krause S Mahapatra T Mohaupt H J Otto Y Oz C Preitschopf R Reinbacher
W Sabra I Schnakenburg und E Silverstein
Selbst andigkeitserkl arung
Hiermit versichere ich die vorliegende Arb eit selbst andig angefertigt zu hab en und keine
weiteren als die angegeb enen Hilfsmittel verwendet zu hab en
Andreas Karch
Leb enslauf
Name Andreas Karch
Nationalit at Deutsch
Geburtsort und datum Bamberg
Bildungsweg
seit Doktorand an der Humboldt Universit at Berlin
Mitglied des Graduiertenkollegs
Strukturuntersuchungen Pr azisionstests und Erweiterungen
des Standard Mo dells der Elementarteilchenphysik
Betreuer Dr H Dorn
Graduate Student in Physics
an der University of Texas at Austin USA
Abschlu Master of Arts Januar
Titel der Arb eit
Dualities in N SUSY Yang Mills Theories
Betreuer Prof Dr J Distler UT Austin
Studium der Physik Universit at W urzburg
Dientzenhofer Gymnasium Bamberg
Neb ent atigkeiten im Studium
Mitarb eit in der Fachschaft Physik
Studentischer Vertreter im Fachbereichsrat
w ochiges Ferienpraktikum am MPI f ur Astronomie in Heidelb erg
Softwareentwicklung f ur das ISO Satelitten Pro jekt
HIWI T atigkeit b ei Prof W Kinzel Lehrstuhl Computational Physics
Erstellung von Ubungsaufgab en zur Vorlesung Computational Physics
Publikationsliste
A Karch D L ust und G Zoupanos Sup erconformal N gauge theories b eta
function invariants and their b ehaviour under Seib erg duality hep th
zur Ver o entlichung akzeptiert in Phys Lett B
A Karch D L ust und D Smith Equivalence of Geometric Engineering and
Hanany Witten via Fractional Branes zur Ver o entlichung akzeptiert in Nucl Phys
B hep th
I Brunner A Hanany A Karch und D L ust Brane Dynamics and Chiral non
Chiral Transitions zur Ver o entlichung akzeptiert in Nucl Phys B hep th
I Brunner und A Karch Branes at Orbifolds versus Hanany Witten in Six Di
mensions JHEP hep th
A Karch D Lust and G Zoupanos Dualities in all order nite N gauge theo
ries zur Ver o entlichung akzeptiert in Nucl Phys B hep th
I Brunner und A Karch Compacti cations of Matrix Theory on T Phys Lett
B hep th
I Brunner und A Karch Branes and Six Dimensional Fixed Points Phys Lett
B hep th
A Karch Seib erg duality in three dimensions Phys Lett B hep
th
A Karch More on N selfdualities and exceptional gauge groups Phys Lett
B hep th
A Karch and J Distler N dualities for exceptional gauge groups and quantum
global symmetries Fortschr Phys hep th
Konferenzb eitr age
I Brunner und A Karch Field Theory from Branes wird ver o entlicht in den Pro
ceedings of the st International Symp osium Ahrensho op on the Theory of Elementary
Particles Buckow Germany September
I Brunner und A Karch Matrix Theory and the Six Torus wird ver o entlicht in
den Pro ceedings of the st International Symp osium Ahrensho op on the Theory of Ele
mentary Particles Buckow Germany September