The classical bosonic string
Paweł Laskoś-Grabowski, Rafał A
astowiecki
Institute for Theoretical Physics, University of Wrocław
December 17, 2007
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 1 / 50
Introduction Preliminaries
Outline
1
Introduction
Preliminaries
Particles and beyond
2
Basic results in strings
Action functional
Equations of motion and constraints
3
Further results
Fourier expansion of the solution
Momenta, mass and more
The Virasoro algebra
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 2 / 50
Introduction Preliminaries
Conventions
Units choice: = c = 1.
Any metric we will use has the pseudo-Euclidean signature  it has
one negative and d - 1 positive eigenvalues, corresponding to
time-like and space-like coordinates, respectively.
The metrics are usually dependent on the space-time point (curved
space-time), but we will omit (x) in m��(x).
For any tensor T��, its core symbol (without indices) stands for the
absolute value of its determinant:
T a" |det T��|
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 3 / 50
Introduction Preliminaries
A few shorthands
A � B a" A�B� a" m��A�B�, where m�� is the relevant metric,
2
X a" X � X ,
dX
� a" , where � is time-like,
d�
�m a" �0m  one-index Kronecker delta.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 4 / 50
Introduction Particles and beyond
Outline
1
Introduction
Preliminaries
Particles and beyond
2
Basic results in strings
Action functional
Equations of motion and constraints
3
Further results
Fourier expansion of the solution
Momenta, mass and more
The Virasoro algebra
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 5 / 50
Introduction Particles and beyond
Relativistic particle
Recollection
Relativistic action of a particle:
S = -m ds2, where ds2 = -g��dx�dx�
Transforming to integral over d� :
dx� dx�
S = -m d� -�2, where �2 = g��
d� d�
Obviously invariant under � � reparametrisations.
�
It does not apply to mass-less particles.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 6 / 50
Introduction Particles and beyond
Relativistic particle
Mass-less form
Introducing the auxiliary coordinate,  einbein :
1
S = e-1�2 - em2 d�
2
Einbein equation of motion (typical Euler-Lagrange procedure):
�2 + e2m2 = 0
As easy" recovering the previous form of S. Just solve the equation
as
-�2
to e = , substitute in S above and simplify.
m
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 7 / 50
Introduction Particles and beyond
Relativistic particle
Gauge choice, momentum
Invariance wrt infinitesimal transformations:
d(�e)
�x = ��, �e =
d�
We use gauge symmetry to ensure e = m-2. We obtain the following
momentum, Hamiltonian and equations of motion:
p� = m2��
1
H = (1 + m-2p2)
2
W� = 0, �� = m-2p�
Non-trivial fact: e EOM is still a constraint  mass-shell condition of
the form:
p2 = -m2
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 8 / 50
Introduction Particles and beyond
Relativistic particle
Generalisation
0-dimensional particles are just a matter of our custom!
We can consider objects such as strings, membranes, or their
n-dimensional equivalents.
n + 1-dimensional world-manifold is embedded in D-dimensional
space-time, thus D > n.
Generalised action has the following (Polyakov) form:
"
T
� �
S = dn+1� hhą�(�)g��(X )"ąX "�X
2
hą� is a metric describing the world-manifold, g��  the space-time.
�0 a" � is time-like.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 9 / 50
Basic results in strings Action functional
Outline
1
Introduction
Preliminaries
Particles and beyond
2
Basic results in strings
Action functional
Equations of motion and constraints
3
Further results
Fourier expansion of the solution
Momenta, mass and more
The Virasoro algebra
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 10 / 50
Basic results in strings Action functional
"
T � �
S = dn+1� hhą�(�)g��(X )"ąX "�X
2
Generalised string action
"
Invariance: hdn+1� is manifestly invariant, the rest is a scalar.
1
Reparametrisations: out of (n + 1)(n + 2) independent components
2
1
we can gauge n + 1 away through reparametrisations. n(n + 1)
2
components still remain.
Weyl scaling allows one more component to be fixed:
hą� �(�)hą�
" "
1
hhą� �2 (n-1) hhą�
Thus for n = 1 (strings) all the hą� variability can be gauged away.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 11 / 50
Basic results in strings Action functional
Why only strings?
By power counting, the action is not renormalisable for n > 1.
Thus only the strings are usually considered.
String tension T has dimensions of L-2 = M2. It is related to the
Regge slope as follows:
T = (2Ąą )-1
In Minkowski space, the action reduces to:
"
T
�
S = d2� hhą�(�)"ąX "�X�
2
M
For convenience, 0 d" � d" Ą.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 12 / 50
Basic results in strings Action functional
Additional terms in the action
 Cosmological constant term:
"
S1 =  d2� h
"
Not considered  not invariant under Weyl scaling ( hhą� is
"
invariant, but sole h is not). Thus either hą� or  is 0.
Curvature term:
"
1
S2 = d2� hR(2)(h)
2Ą
R(2)(h) is the scalar curvature of the world sheet, significant in string
interactions.
"
The term hR(2) is a total derivative  doesn t contribute to the
results.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 13 / 50
Basic results in strings Action functional
Action symmetries
Reparametrisation generalised:
� �
�X = �ą"ąX
�hą� = �ł"łhą� - "ł�ąhł� - "ł��hął
" "
�( h) = "ą(�ą h)
Weyl scaling:
�hą� = �hą�
Poincar� (full relativistic) invariance (a�� = -a��):
� �
�X = a� X + b�
�
�hą� = 0
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 14 / 50
Basic results in strings Action functional
Energy momentum tensor
We can obtain the e m tensor by varying S wrt the metric:
2 1 �S
Tą� = - "
T �hą�
h
The result is as follows:
� 1 �
Tą� = "ąX "�X� - hą�hą � "ą
X "� X�
2
Under Weyl scaling hą� �hą� the trace transforms:
hą�Tą� �hą�Tą�
ą
This implies T = 0.
ą
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 15 / 50
Basic results in strings Action functional
Energy momentum tensor (cont.)
Field equation 0 = �S/�hą� <" Tą�.
�
With Gą� = "ąX "�X�, the above gives:
1
Gą� = hą�hą � Gą
�
2
1
G = h(hą�Gą�)2
4
" "
1
S = d2� hhą�Gą� = d2� G
2
Ł Ł
The latter is simply the area of the world-tube.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 16 / 50
Basic results in strings Equations of motion and constraints
Outline
1
Introduction
Preliminaries
Particles and beyond
2
Basic results in strings
Action functional
Equations of motion and constraints
3
Further results
Fourier expansion of the solution
Momenta, mass and more
The Virasoro algebra
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 17 / 50
Basic results in strings Equations of motion and constraints
Equations of motion
Gauge choice
From now on, we assume that the gauge and Weyl scaling is chosen so that
the world-manifold metric is locally flat, thus equal to Minkowski metric
-1 0
hą� = �ą� =
0 1
Action in simplified form:
T
S = d2� �ą�"ąX � "�X
2
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 18 / 50
Basic results in strings Equations of motion and constraints
Equations of motion (cont.)
"f
Typical Euler-Lagrange procedure (where f a" ):
"�
"L
= 0
�
"X
"L
= -T "ąX�
�
"("ąX )
� � � �
X a" X - Ś = "ą"ąX = 0
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 19 / 50
Basic results in strings Equations of motion and constraints
Boundary conditions
� � �
Invariance under general variation: X X + �X .
Action varies as follows (omission of terms super-linear in �):
�
S S - T d2� �ą�"ąX "��X�
After integration by parts we obtain (after omission of terms, which
vanish after integration over d� ):
�=Ą
�
-T d� X��X = 0
�=0
Vanishing requirement gives boundary conditions:
� �
X (�, �) = X (�, � + Ą) (periodicity) for closed strings,
�
X = 0 at � = 0, Ą (von Neumann condition) for open strings.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 20 / 50
Basic results in strings Equations of motion and constraints
D-branes
Digression
We could choose Dirichlet boundary conditions, i.e.:
�
�X = 0 at � = 0, Ą
This assumption violates Poincar� (translational) invariance  unless
the endpoints of a string are attached to a moving object, called in
general a D-brane.
Energy flow through the endpoint of a string is nonzero.
Some physical entity (D-brane) has to be a source of such flow
(energy conservation law).
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 21 / 50
Basic results in strings Equations of motion and constraints
General solution
Usual solution as a sum of arbitrary functions (left- and right-moving
modes):
� �
�
X (�, �) = XR (�-) + XL (�+)
�ą = � ą �
Conjugate derivatives and Minkowski metric tensor in light-cone
coordinates:
1
"ą = ("� ą "�)
2
�+- = �-+ = -1
2
�++ = �-- = 0
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 22 / 50
Basic results in strings Equations of motion and constraints
Constraint equations
Expanding Tą� = 0 and in ordinary and light-cone coordinates:
T10 = T01 = � � X = 0
1 2 2
T00 = T11 = (� + X ) = 0
2
1
T++ = (T00 + T01) = "+X � "+X
2
1
T-- = (T00 - T01) = "-X � "-X
2
T+- = T-+ = 0
The last follows from tracelessness.
Tąą = 0 implies following statements:
2 2
�R = �L = 0
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 23 / 50
Basic results in strings Equations of motion and constraints
Conformal invariance
Energy momentum conservation law in conformally invariant case:
"-T++ = 0
Infinite set of conserved quantities  for any function f of �+,
current fT++ is conserved.
Setting hą� = �ą� still allows gauge change, consider
"ą�� + "��ą = ��ą�
Then in light-cone coordinates, the �ą = �0 ą �1 may be arbitrary
functions of �ą.
This residual symmetry is generated by:
"
ą
V = �ą(�ą)
"ą
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 24 / 50
Further results Fourier expansion of the solution
Outline
1
Introduction
Preliminaries
Particles and beyond
2
Basic results in strings
Action functional
Equations of motion and constraints
3
Further results
Fourier expansion of the solution
Momenta, mass and more
The Virasoro algebra
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 25 / 50
Further results Fourier expansion of the solution
Decomposition into Fourier series
Closed strings
� �
Periodicity conditions: X (�, �) = X (�, � + Ą).
Corresponding decomposition into Fourier series:
il 1
�
1 1 �
XR = x� + l2p�(� - �) + ąne-2in(� -�)
2 2
2 n
n =0
il 1
�
1 1
XL = x� + l2p�(� + �) + ąne-2in(� +�)
��
2 2
2 n
n =0
Fundamental length:
" "
l = 2ą = 1/ ĄT
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 26 / 50
Further results Fourier expansion of the solution
Decomposition into Fourier series
Closed strings (cont.)
x�  centre of mass position, p�  momentum.
�
Terms linear in � cancel out in X  satisfaction of the boundary
condition.
Reality requirement:
�
�
ą-n = (ąn)", ą-n = (ąn)"
�� ��
Derivatives of the solution:
"
� �
�
"-XR = �R = l ąne-2in(� -�)
n=-"
"
� �
"+XL = �L = l ąne-2in(� +�)
��
n=-"
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 27 / 50
Further results Fourier expansion of the solution
Decomposition into Fourier series
Closed strings  Poisson brackets
At fixed � we obtain:
� � � �
[X (�), X (� )]P.B. = [� (�), � (� )]P.B. = 0
� � -1
[� (�), X (� )]P.B. = T �(� - � )���
After expansion insertion we get:
� �
[ąm, ąn]P.B. = [ąm, ąn]P.B. = im�m+n���
�� ��
�
[ąm, ąn]P.B. = 0
��
�
It remains valid with ą0 = ą0 = lp�, and implies
�� 1
2
[p�, x�]P.B. = ���
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 28 / 50
Further results Fourier expansion of the solution
Decomposition into Fourier series
Open strings
�
Boundary condition: X |�=0,Ą = 0.
Standing wave solution:
� �
X = x� + l2p�� + il ąne-in� cos n�
n =0
"
�
� � �
2"ąX = � ą X = l ą�e-in(� ą�), where ą0 a" lp�
n
n=-"
Contrary to the closed case, left- and right-moving modes are
dependent.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 29 / 50
Further results Momenta, mass and more
Outline
1
Introduction
Preliminaries
Particles and beyond
2
Basic results in strings
Action functional
Equations of motion and constraints
3
Further results
Fourier expansion of the solution
Momenta, mass and more
The Virasoro algebra
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 30 / 50
Further results Momenta, mass and more
Noether method
Noether theorem: current Ją associated with transformation
Ć(�) Ć(�) + �Ć(�) is conserved.
Allowing variability of  Ć(�) Ć(�) + (�)�Ć(�) leads to
non-invariant action (proportionally to derivative):
�S = d2� Ją"ą
EOM can be satisfied only if �S = 0 for all , which is true when
"ąJą = 0 (obtained through integration by parts)
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 31 / 50
Further results Momenta, mass and more
Linear and angular momentum
Currents associated with Poincar� transformation:
�
Translation �X = b�  density of linear momentum:
� � �
�S = T d2� "ąX "ąb� �! Pą = T "ąX
� �
Rotation �X = a��X  density of angular momentum
� �
�S = -T d2� "ąa��(X "ąX ) �!
�� � � � �
Ją = T (X "ąX - X "ąX )
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 32 / 50
Further results Momenta, mass and more
Linear and angular momentum (cont.)
Conservation laws:
"ąPą� = "ąJą�� = 0
Amount of momentum flowing across a line segment:
� �
dP� = P� d� + P� d�
No momentum flows through the endpoints of an open string:
dP�
� �
= P� |�=0,Ą = T "�X |�=0,Ą
d�
�=0,Ą
which vanishes under von Neumann conditions.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 33 / 50
Further results Momenta, mass and more
Linear and angular momentum (cont.)
Total linear momentum  Fourier terms with n = 0 vanish after

integration:
Ą �
dX (�)
�
P� = T d� = ĄT (lą0 + lą0 = p�
��)
d�
0
� =0
Total angular momentum:
Ą � �
dX dX
� �
J�� = T d� X - X
d� d�
0
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 34 / 50
Further results Momenta, mass and more
Angular momentum (cont.)
After inserting the expansion we obtain:
J�� = l�� + E�� for open strings
J�� = l�� + E�� + ź�� for closed strings
where we used:
"
1
�
�� � � �
l�� a" x�p� - x�p� E a" -i (ą-nąn - ą-nąn)
n
n=1
and an equivalent expression for ź�� in terms of ąn.
��
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 35 / 50
Further results Momenta, mass and more
Examples
Rotating loop
Let s consider a closed string on the xy plane. At � = t = 0
x = R cos 2�, y = R sin 2�
It is easy to see that EOM and constraints can be satisfied if
t = 2R� near � = t = 0
We don t know  maybe the authors used some inconsistent (or
really advanced!) approximation in t 0.
Anyway, if we believe them, clearly
Ą
E = P0 = T d� 2R = 2ĄRT
0
so the T is really the string energy per length  that is, tension.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 36 / 50
Further results Momenta, mass and more
Regge trajectory
Digression
Tullio Regge studied amplitudes of scattering in quantum mechanics.
He assumed that quantum number l may be complex in general, not
only integer.
Thus the amplitudes had to be analytically continued over the
complex plane.
The function had several poles, which  travelled when scattering
energy was varied  this  travel is described by the Regge trajectory.
When this trajectory is (assumed to be) linear, it is described with the
slope and intercept, ą(s) = ą s + ą(0).
One of the discovered properties:
J
m2 <"
ą
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 37 / 50
Further results Momenta, mass and more
Examples
Rotating straight string
Proof that ą is the Regge slope  maximum angular momentum per
2
E . Let us examine the following case:
x = A cos � cos � y = A sin � cos � t = A�
Clearly satisfies EOM and constraints. From previous results we
obtain E = P0 = ĄAT , Jz = J12 = ĄA2T /2, and
J � E-2 = (2ĄT )-1
Observation: endpoints of an open string move at c (consequence of
1 2 2
boundary condition and (� + X ) = 0 constraint).
2
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 38 / 50
Further results Momenta, mass and more
Squared mass
String squared mass is equal to -p�p�:
"
1
M2 = ą-n � ąn for open strings
ą n=1
"
2
M2 = (ą-n � ąn + ą-n � ąn) for closed strings
� �
ą n=1
These are the mass-shell conditions, analogous to the one derived
earlier for the particle.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 39 / 50
Further results Momenta, mass and more
Two-dimensional Hamiltonian
The Hamiltonian of the 2-D theory:
Ą Ą
T
2 2
H = d� (� � P - L) = d� (� + X )
2
0 0
In our case(s):
"
1
H = ą-n � ąn for open strings
2
n=-"
"
1
H = (ą-n � ąn + ą-n � ąn) for closed strings
� �
2
n=-"
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 40 / 50
Further results Momenta, mass and more
Constraint Fourier series
Closed strings
2 2
Expanding the constraints �R = �L = 0 in Fourier series for closed
strings:
Ą
T
Lm = d� e-2im�T--
2
0
"
Ą
T
2 1
= d� e-2im��R = ąm-n � ąn
2
2
0
n=-"
Ą
T
�
Lm = d� e2im�T++
2
0
"
Ą
T
2 1
= d� e2im��L = ąm-n � ąn
� �
2
2
0
n=-"
�
H = L0 + L0.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 41 / 50
Further results Momenta, mass and more
Constraint Fourier series
Open strings
First we need to assume periodicity in � beyond the interval [0, Ą]:
XR/L(� + Ą) = XR/L(�)
Then the expansion is as follows:
Ą
Lm = T d� (eim�T++ + e-im�T--)
0
"
Ą
T
1
= d� eim�(� + X )2 = ąm-n � ąn
2
4
-Ą
n=-"
H = L0.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 42 / 50
Further results The Virasoro algebra
Outline
1
Introduction
Preliminaries
Particles and beyond
2
Basic results in strings
Action functional
Equations of motion and constraints
3
Further results
Fourier expansion of the solution
Momenta, mass and more
The Virasoro algebra
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 43 / 50
Further results The Virasoro algebra
The Virasoro algebra
Virasoro algebra is spanned by elements Li " C, i " Z and so-called
central charge c, which obey the following:
Ln + L-n " R
[c, Ln]P.B. = 0
ic
[Lm, Ln]P.B. = i(m - n)Lm+n + (m3 - m)�m+n
12
Now we ll prove that Ln satisfy these conditions for c = 0.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 44 / 50
Further results The Virasoro algebra
Virasoro algebra of e m tensor modes
From the definition of Ln:
1
[Lm, Ln]P.B. = [ąm-k � ąk, ąn-l � ąl]P.B.
4
k,l
Applying the P.B./commutator/etc. identity:
[AB, CD] = A[B, C]D + AC[B, D] + [A, C]DB + C[A, D]B
we obtain:
i
[Lm, Ln]P.B. = (kąm-k � ąl�k+n-l + kąm-k � ąn-l�k+l
4
k,l
+ (m - k)ąl � ąk�m-k+n-l + (m - k)ąn-l � ąk�m-k+l)
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 45 / 50
Further results The Virasoro algebra
Virasoro algebra of e m tensor modes (cont.)
After summation over �-indices:
i i
[Lm, Ln]P.B. = kąm-k � ąk+n + (m - k)ąm-k+n � ąk
2 2
k k
Finally, index change k k + n finishes the proof:
[Lm, Ln]P.B. = i(m - n)Lm+n
�
The same can be proved for Ln.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 46 / 50
Further results The Virasoro algebra
Virasoro algebra interpretation
Consider a circle S1 on a plane, parametrised by � " (0, 2Ą].
Generator of an infinitesimal, general transformation � � + a(�):
d
Da = ia(�)
d�
A complete basis of these transformations is given by:
d
Dn = iein�
d�
These clearly obey the Virasoro algebra conditions with c = 0.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 47 / 50
Further results The Virasoro algebra
Analogy to residual symmetries
d "
By replacing ein� �ą and i in generators of S1
d� "�ą
transformations, we recover the generators of residual symmetries.
Variables �ą are not a priori angular.
To fulfil the EOM of the theory, mode expansions of strings contain
exp in�ą only for integer n.
Thus �ą may be considered angular.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 48 / 50
Bibliography
Sources and further reading
Michael B. Green, John H. Schwarz, Edward Witten
Superstring Theory, vol. 1.
Cambridge University Press, 1987.
Lev L. Landau, Evgeny M. Lifshitz
Quantum Mechanics  Non-relativistic Theory
Pergamon Press, 1977.
ż141 of [2] covers Regge poles.
Laskoś-Grabowski, A
astowiecki (IFT UWr) Classical string December 17, 2007 49 / 50
Bibliography
Sources and further reading (cont.)
Wikipedia, the free encyclopedia
Articles on string related topics:
http://en.wikipedia.org/wiki/Stringtheory
http://en.wikipedia.org/wiki/D-branes
. . .
Article on Regge theory
http://ru.wikipedia.org/wiki/
Laskoś-Grabowski, A
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