
1.  Introduction 
At  present  the  Standard  Model  (SM)  of  particle  physics  is  the  best  and  most  sensible  theory 
which summarizes our understanding of the basic components of matter and their interactions 
in  an  unified  scheme.       However  there  is  the  known  aesthetic  problem  arising  in  the  SM  of 
particles, that include the hierarchy problem, the abundance of free parameters and the apparent 
arbitrariness of the flavor and gauge groups, etc.  The SM is for this reason commonly regarded 
as the low energy e?ective description of a more fundamental theory, which solves these problems 
(for a nice review on this subject, see [1]).  On the other hand, it is also widely recognized that 
Quantum  Mechanics  and  General  Relativity  (GR)  cannot  be  reconciliated  in  the  context  of  a 
perturbative quantum field theory of point particles.                 Hence the nonrenormalizability of GR is 
also  regarded  as  a  evidence  that  it  is  just  an  e?ective  field  theory  and  new  physics  associated 
to  some  fast  degrees  of  freedom  should  exist  at  higher  energies  (for  a  review,  see  for  instance 
[2]).  String  theory  proposes  that  these  fast  degrees  of  freedom  are  precisely the  strings  at  the 
perturbative level and at the non-perturbative level the relevant degrees of freedom are higher- 
dimensional extended objects called D-branes (dual degrees of freedom). 
    At   the  perturbative      level  String    Theory    has   intriguing    generic    predictions    such    as:  (i) 
Spacetime  supersymmetry,  (ii)  General  Relativity  and  (iii)  Yang-Mills  fields.                   These  subjects 
interesting by themselves are deeply interconnected in a rich way in string theory. 
    Though the study of theories involving D-branes has produced great number of results, it is 
still at an exploratory stage of the whole structure of the string theory.  Therefore the theory is far 
of being completed and it is necessary to explore the structure of the theory before we can give 
concrete  physical  predictions  to  make  contact  with  collider  experiments  and/or  astrophysical observations.  However many aspects of theoretic character, necessary in order to make of string 
theory a physical theory, are quickly in progress.  The purpose of these lectures are to overview 
the  basic  ideas  in  order  to  understand  these  progresses  following  strongly  the  set  of  lectures 
given in Ref.  [3]. 

2.  Basic  Facts  on  Field  Theory 
First  we  overview  the  basic  structure  of  GR  and  Yang-Mills  (YM)  theory  in  four  dimensions. 
They  are  very  di?erent  theories.      GR  for  instance,  is  the  dynamical  theory  of  the  spacetime 
metric g?   while quantum YM theories and in general, Quantum Field Theory (QFT) describes 
the dynamical building blocks of matter in a fixed spacetime background.  Here we survey basics 
aspects of GR and YM theory following closely Ref.  [4]. 

2.1.  General Relativity 
The pure gravitational field is described by a pseudo-Riemannian metric g?  with , ? = 0, 1, 2,3 
(on a four-dimensional space-time manifold M) satisfying the vacuum Einstein equations with 
a cosmological constant ?, 
                                                  R?  = ?g? .                                               (1) 

Einstein equations can be derived from the Einstein-Hilbert action 
                                                1         4  v 
                                   SGR  =                 d  x  -g(R - 2?),                                   (2) 
                                            16?GN      M 

where   GN    is  the  Newtons  constant.      This  constant  together  with        and  c,  determines  the 
Planck  scale  where  gravitational  e?ects  in  the  quantum  theory  are  relevant.          The  mass  scale 
termed  Planck  mass  is  MP l      =   c    =  1.2  x 10-5grams  or  equivalently  the  Planck  length 

                                          GN 
LP l  = M   c ? 10-33    centimeters. 

          P l 

2.2.  Gauge Theories 
Classical Gauge Theories 
    If  one  wants  to  formulate  the  gauge  theory  on  a  fixed  pseudo-Riemannian  manifold  spite 
of  the  metric  g?  we  require  from  an  additional  structure  on  the  spacetime  i.e.       a  connection 
                                                                    ? 
A ? ?(T *E ? G on a G-principal bundle on M :  G > E > M , where G is the SM gauge group, 

G = SUC (3)xSUL (2)xUY  (1).  As usual, the gauge field A (x) given by the connection one-form 
has  associated  the  field  strength  F      = ?   Aa   - ?  Aa  + ifa Aa Ab    with  fa   being  the  structure 
                                          ?        ?      ?        bc     ?         bc 
constants of G.  Given any representation R of G one can construct the  associated vector bundle 
VR . 
    The Yang-Mills action is given by 
                                 SY M  = -     1       g g?? T rRF? F                                 (3) 

                                            4g2      M                        ?  , 
                                               Y M 

where T rR    denotes the trace in the adjoint representation of G. 
    Now we want to introduce fermions.           The chiral fermions are sections of the the chiral spin 
                                                                                  ? 
                                                                                                           
bundles S     over spacetime manifold with Spin structure M , i.e.  S > M , where S = S+  ? S-. 
The fibers are the Cli?ord modules constructed with the Dirac matrices ? .                   Dirac operator is 

         
                            
D ? ?   D   : ?(S) > ?(S) with D  being the spacetime covariant derivative.  In even dimensions 
                                                                                   
Dirac operator decomposes as:  D        =D +? D  -  where D   : ?(S) > ?(S?) with 



