Next: Spectrum of the Bosonic
Up: STRINGS, BRANES AND DUALITY^{1}
Previous: Introduction
In this section we overview some basic aspects of bosonic and fermionic
strings. We focus mainly in the description of the spectrum of the theory
in the lightcone gauge, the effective action, the description of spectra of
the five consistent superstring theories and the perturbative CalabiYau
compactifications (for details, precisions and further
developments see for instance [3,4,5,6,7,8]).
First of all consider, as usual, the action of a relativistic point
particle. It is given by
, where are functions
representing the coordinates of the dimensional Minkowski spacetime
(the target space),
and
can be identified with the mass of
the point
particle. This action is
proportional
to the length of the worldline of the relativistic particle.
In analogy with the relativistic point particle, the action describing the dynamics of a
string (onedimensional object) moving in a dimensional Minkowski spacetime (the
target space) is proportional to the area of the worldsheet . We know from the
theory of surfaces that such an area is given by
, where
is
the induced metric (with signature ) on the worldsheet . The background metric will be
denoted by and
with are the local
coordinates on the worldsheet. and are related by
with
.
Thus
the classical action of a relativistic string is given by the NambuGoto action

(1) 
where
is the string tension, are embedding
functions of the worldsheet into
the target space . Now introduce a metric describing the intrinsic worldsheet geometry,
we get a classically equivalent action to the NambuGoto action. This is the Polyakov action
originally proposed by Brink, di Vecchia, Howe and Zumino

(2) 
where the 's are scalar fields on the worldsheet. Such a fields can be interpreted
as the coordinates of spacetime (target space), = det and
.
Polyakov action has the following symmetries: Poincaré invariance,
Worldsheet diffeomorphism invariance, and Weyl invariance
(rescaling invariance). The energymomentum tensor of the twodimensional
theory is given by

(3) 
Invariance under worldsheet diffeomorphisms implies that it should be conserved
i.e.
, while the Weyl invariance gives the traceless condition,
. The equation of motion associated with Polyakov action is given by

(4) 
Whose solutions should satisfy the boundary conditions for the open string:
(Neumann) and for the closed
string:
(Dirichlet). Here is
the
characteristic
length of the open string. The variation of with respect to leads
to the constraint equations: .
From now on we will work in the conformal gauge. In this gauge:
the equations of motion (4) reduce to the Laplace equation in the flat worldsheet
whose solutions can be written as linear superposition of plane waves.
truecm
The Closed String
For the closed string the boundary condition
, leads to the general solution of Eq. (4) in the conformal gauge

(5) 
where and are the position and momentum of the
centerofmass of the string and and
satisfy the conditions
(leftmovers) and
(rightmovers).
truecm
The Open String
For the open string the corresponding boundary condition is
(this is the only boundary condition which is Lorentz invariant) and the solution
is given by

(6) 
with the matching condition
truecm
Quantization
The quantization of the closed bosonic string can be carried over, as usual, by using the Dirac
prescription to the centerofmass and oscillator variables in the form

(7) 
One can identify
with the annihilation
operators
and the corresponding operators
with
the
creation ones. In order to specify the physical states we first denote the center of mass
state given by . The vacuum state is defined by
with and
and similar for the right movings (here
). For the zero modes these states have negative
norm (ghosts). However one can choice a suitable gauge where ghosts decouple from the Hilbert
space when .
truecm
Lightcone Quantization
Now we turn out to work in the so called lightcone gauge. In this
gauge it is possible to solve explicitly the Virasoro constraints: . This is done
by removing the lightcone coordinates
leaving only the transverse coordinates representing the physical
degrees of freedom (with
). In this gauge the Virasoro constraints
are explicitly
solved. Thus the independent variables are
. Operators and
can be written in terms of and
respectively as
follows:
and
). For the
open string we get
. Here stands for the normal ordering and is its associated constant.
In this gauge the Hamiltonian is given by

(8) 
where is the operator number,
, and
The
massshell condition is given by
(open string) and
(closed string). For the open string,
Lorentz invariance implies that the first excited state is massless and therefore . In the
lightcone
gauge takes the form
.
From the fact
where
is the Riemann's zeta function (which converges for and has a unique analytic continuation at
, where it takes the value ) then
and therefore .
truecm
Subsections
Next: Spectrum of the Bosonic
Up: STRINGS, BRANES AND DUALITY^{1}
Previous: Introduction
root
20010115