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Dbranes are, despite of the dual fundamental degrees of freedom in string theory,
extremely interesting and useful tools to study nonperturbative properties of
string and field theories (for some reviews see [10,11]). Nonperturbative
properties of supersymmetric gauge theories can be better understanding as the
worldvolume effective theory of some configurations of intersecting Dbranes
(for a review see [14]). Dbranes also are very important to connect gauge
theories with gravity. This is the starting point of the AdS/CFT correspondence
or Maldacena's conjecture. We don't review this interesting subject in this paper,
however the reader can consult the excellent review [15].
Roughly speaking Dbranes are static solutions of string equations which satisfy
Dirichlet boundary conditions. That means that open strings can end on them.
To explain these objects we follow the traditional way, by using Tduality on
open strings we will see that Neumann conditions are turned out into the Dirichlet ones.
To motivate the subject we first consider Tduality in closed bosonic string theory.
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Tduality in Closed Strings
The general solution of Eq. (4) in the conformal gauge can be written
as
, where
. Now, take one coordinate, say and compactify it on a
circle of radius . Thus
we have that can be identified with
where is called the
winding
number. The general solution for with the above compactification condition is

(38) 
where

(39) 
Here and are integers representing the discrete momentum and the winding
number, respectively. The latter has not analogous in field theory.
While the canonical momentum is given by
. Now, by the mass shell condition, the mass of the perturbative states
is given by
, with

(40) 
We can see that for all states with , as the mass
become infinity, while implies that the states take all values for and form a
continuum. At the case when , for states with , mass
become infinity. However in the limit for states with all
values produce a continuum in the spectrum. So, in this limit the compactified
dimension disappears. For this reason, we can say that the mass spectrum of the theories at radius
and
are identical when we interchange
. This
symmetry is known as Tduality.
The importance of Tduality lies in the fact that the Tduality transformation is a parity
transformation acting on the left and right moving
degrees of freedom. It leaves invariant the left movers and changes the sign of the right movers
(see Eq. (39))

(41) 
The action of Tduality transformation must leave invariant the whole theory
(at all order in perturbation theory). Thus, all kind of interacting states in certain theory
should correspond to those states belonging to the dual theory. In this context,
also the vertex operators are invariant. For instance the tachyonic vertex operators are

(42) 
Under Tduality,
and
; and from the general solution Eq. (38),
,
. Thus, Tduality
interchanges
(KaluzaKlein modes
winding
number) and
in closed string theory.
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Tduality in Open Strings
Now, consider open strings with Neumann boundary conditions. Take again the
coordinate and compactify it on a circle of radius , but keeping Neumann
conditions. As in the case of closed string, center of mass momentum takes only discrete
values
. While there is not analogous for the winding
number. So, when all states with nonzero momentum go to infinity
mass, and do not form a continuum. This behavior is similar as in field theory, but
now there is something new. The general solutions are

(43) 
where is a constant.
Thus,
Taking the limit , only the mode survives. Because of this,
the string seems to move in 25 spacetime dimensions. In other words, the strings vibrate in 24
transversal directions. Tduality provides a new Tdual coordinate defined by
. Now, taking
we have
Using the boundary conditions at one has
and
Thus, we started with an open bosonic string theory with Neumann boundary
conditions, and Tduality and a compactification on a circle in the
dimension, give us Dirichlet boundary conditions in such a coordinate. We can
visualize this saying that an open string has its endpoints fixed at a hyperplane
with 24 dimensions.
Strings with lie on a 24 dimensional plane space (D24brane). Strings with
has one endpoint at a hyperplane and the other at a different hyperplane which
is separated from the first one by a factor equal to
, and so on. But
if we compactify of the directions over a torus (). Thus, after
Tdualizing
them we have strings with endpoints fixed at hyperplane with dimensions, the Dbrane.
Summarizing: the system of open strings moving freely in spacetime with compactified
dimensions on is equivalent, under Tduality, to strings whose enpoints are fixed at a
Dbrane i.e. obeying Neumann boundary conditions in the longitudinal
directions () and Dirichlet ones in the transverse coordinates
().
The effect of Tdualizing a coordinate is to change the nature of the boundary conditions, from
Neumann to Dirichlet and vice versa. If one dualize a longitudinal coordinate this coordinate
will
satisfies the Dirichlet condition and a Dbrane becomes a Dbrane. But if the
dualized coordinate is one of the transverse coordinates the Dbrane becomes a
Dbrane.
Tduality also acts conversely. We can think to begin with
a closed string theory, and compactify it on to a circle in
the coordinate, and then by imposing Dirichlet conditions, obtain a
Dbrane. This is precisely what occurs in Type II theory, a theory of closed strings.
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Spectrum and Wilson Lines
Now, we will see how does emerges a gauge field on the Dbrane worldvolume. Again, for
the mass
shell condition for open bosonic strings and because Tduality
. The massless state
(, ) implies that the gauge boson
(
gauge boson) lies on the D24brane worldvolume. On the other hand,
has a vev (vacuum expectation value) which describes
the position
of the Dbrane after Tdualizing. Thus, we can say
in general, there is a gauge theory over the world volume of the Dbrane.
Consider now an orientable open string. The endpoints of the string carry
charge under a nonAbelian gauge group. For Type II theories the gauge group is
. One endpoint transforms under the fundamental
representation of and the other one, under its complex conjugate representation
(the antifundamental one) .
The ground state wave function is specified by the center of mass momentum and by
the charges of the endpoints. Thus implies the existence of a basis
called ChanPaton basis. States
of the ChanPaton basis are those
states which carry charge 1 under the generator and under the
generator. So, we can decompose the wave function for ground state
as
where
are
called ChanPaton factors. From this, we see that it is possible to add degrees of freedom
to endpoints of the string, that are precisely the ChanPaton factors.
This is consistent with the theory, because the ChanPaton factors have a Hamiltonian
which do not posses dynamical structure. So, if one endpoint to the string is
prepared in a certain state, it always will remains the same. It can be deduced from
this, that
with . Thus, the worldsheet
theory is
symmetric under , and this global symmetry is a gauge symmetry in spacetime. So the
vector state at massless level
is a gauge boson.
When we have a gauge configuration with non trivial line integral around a
compactified dimension (i.e a circle), we said there is a Wilson line. In case of
open strings with gauge group , a toroidal compactification of the
dimension on a circle of radius . If we choice a background field given
by
a Wilson line
appears. Moreover, if , and
,
then gauge group is broken:
. It is
possible to deduce that plays the role of a Higgs field.
Because string states with ChanPaton quantum numbers
have charges
under factor (and under factor) and neutral with
all others; canonical momentum is given now by
Returning to the mass shell condition it results,

(44) 
Massless states () are those in where (diagonal terms) or for which
. Now, Tdualizing we have
Taking
,
and
This give us a set of Dbranes whose positions are given by
, and each set is separated from its initial positions
() by a factor equal to
.
Open strings with both endpoints on the same Dbrane gives massless gauge bosons.
The set of Dbranes give us gauge group. An open string with one endpoint
in one Dbrane, and the other endpoint in a different Dbrane, yields a
massive state with
. Mass decreases when two
different Dbranes approximate to each other, and are null when become the same.
When all Dbranes take up the same position, the gauge group is enhanced from to
On the Dbrane worldvolume there are also scalar fields in the adjoint representation
of the gauge group . The scalars parametrize the transverse positions of the Dbrane in the
target space .
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DBrane Action
With the massless spectrum on the Dbrane worldvolume it is possible to construct a low
energy effective action. Open strings massless fields are interacting with the closed
strings massless spectrum from the NSNS sector. Let (with
) be the worldvolume coordinates on . The effective action is the gauge invariant
action well known as the DiracBornInfeld (DBI)action

(45) 
where is the tension of the Dbrane, is the worldvolume induced metric,
is the induced antisymmetric field, is the Abelian field strength on
and is the dilaton field.
For Dbranes the massless fields turns out to be matrices and the
action turns out to be nonAbelian DBIaction (for a nice review about the BornInfeld
action in string theory see [16])

(46) 
where
. The scalar fields representing the transverse
positions become matrices and so, the spacetime become a noncommutative
spacetime. We will come back later to this interesting point.
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RamondRamond Charges
Dbranes are coupled to RamondRamond (RR) fields . The complete
effective action on the Dbrane worldvolume which taking into account this coupling
is

(47) 
where is the RR charge. RR charges can be computed by considering the anomalous
behavior of the action at intersections of Dbranes. Thus RR charge is given
by

(48) 
where
. Here is the ChanPaton bundle over ,
is the genus of the spacetime manifold . This gives an ample evidence
that the RR charges take values not in a cohomology theory, but in fact, in a KTheory.
This result was developed by Witten in Ref. [17] in the context of nonBPS brane
configurations worked out by Sen [18].
Finally, RR charges and RR fields do admit a classification in terms of topological
Ktheory. The inclusion of a field turns out the effective theory noncommutative and
a suitable generalization of the topological Ktheory is needed. The right
generalization seems to be the KHomology and the Ktheory of algebras
[19]. This subject is
right now under intensive investigation.
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