introductory lectures on string theory

8/12/2019 Introductory Lectures on String Theory

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Introductory Lectures on String Theory 1

A.A. Tseytlin 2

Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.

We give an elementary introduction to classical and quantum bosonic string theory.

1 Introduction

particles strings (1-d objects)1. effective strings: 1-d vortices, solitons, Abrikosov vortex in superconductors,

cosmic strings in gauge theory; these are built out of matter, have nitewidth, massive excitations (including longitudinal ones).

2. fundamentalstrings: no internal structure (zero thickness), admit consis-tent quantum mechanical and relativistic description.

Fig.1: Open string Fig.2: Closed string

Tension = masslength = T - main parameter of fundamental strings.

String vibration modes quantum particlesdiscrete spectrum of excitations ( number)

m 2 = p2 + T N, N = 0 , 1, 2, . . . , p 2 = p20 + p2i (1.1)Example: straight open relativistic string rotating about c.o.m.

1 These lectures were prepared in collaboration with Alexei S. Matveev.2 Also at Department of Theoretical Physics, Lebedev Instititute, Moscow, Russia
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Fig.3: Rotating string

length Lmass M = T Langular momentum J P LP

Mc (relativistic)J P L M L T 1M 2M 2 T J property of relativistic string

Angular moment J is quantized in QM

Why strings?

consistent quantum theory of gravity and other interactions

effective description of strongly interacting gauge theoriesWhy not membranes (or p-branes, p 2)?

No consistent QM theory of extended objects with p > 1 is known

Quantum theory of gravitation:

point-like particle (e.g., graviton): interactions are local UV divergences non-renormalizable (need number of counterterms)

Graviton scattering amplitudes

UV divergences no consistent descriptionof gravity at short distance

Fig.4: Scattering

String interaction are effectively non-local


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Fig.5: 2-d surfaces asworld-sheets

loop amplitudes are nite consistent in QMgraviton and other light particles appear as par-ticular string states

UV niteness: string scale (tension) 1/ 2 plays the role of effective cut-off Historical origin in the theory of strong interactions (1968-72)

Fig.6: Scattering of hadrons

s = ( p1 + p2)2

t = ( p3 + p4)2A(s, t ) = hadron amplitude

duality observed: A(s, t ) = A (t, s )

Unusual for eld theories

L= ( )2 + m22 + 3Fig.7: s, t channels are not the

same

Fig.8: Resonances

But possible in a theory with number of reso-nances (intermediate states)m 2 n = 1 , 2, . . .


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Fig.9: Regge trajectories

Linear relations between energy-squared E 2 of res-onances and their spins J characterstic to string

theory spectrum were indeed observed in experi-ments.

2 Classical String Theory

Point-like particle

Fig.10: Non-relativisticmassive particle

L=D 1

n =1

m x2n2

(2.1)

Action for a non-relativistic massive particle (assume no higher derivatives) 3 :

S = mc2

dt 1 x2nc2 dt mc

2

+

m x2n2 + . . . , x c (2.2)

Momentum: pn =

Lx n

= mxn

1 x2c2(2.3)

Invariance of the action: Lorentz transformationsManifestly relativistic-invariant form (in what follows c = 1)

S = m d x x , = (0 , n ) = (0 , 1, . . . , D 1) (2.4) =diag(-1,1,...,1), D is the dimension of Minkowski space-time {x

}, is a world-line parameter.Equivalently,

S = m ds, (2.5)3 In what follows we will omit summation sign so that n a

2n a

2n a

2 .


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whereds 2 = dx dx , dx =

dx

d d

S = m d (x0

)2

(xn

)2

(2.6)Symmetries:

1. Space-time: Poincare Group

x = x + a , SO (1, D 1) (2.7) = (x, y ) = xy = ( x , y ) (2.8)

ds 2 = dx dx = invariant

2. World-line: reparametrization invariance

= f ( ), x

( ) = x

( ) = + ( ), x = x + x x = x

Proper-time or static gauge (special coordinate system in 1- d)x0( ) = t one function is xedx 0(f ( )) = x0( ) = xes f ( )(cf. A0 = 0 gauge in Maxwell theory)

(x0)2 (xn )2 1 (xn )2 = usual relativistic actionAction in static gaugeS =

m

dt

1

x2n (2.9)

Equivalently, changing x0( )S = m dt ( dx

0

d )2 (

dx n

d )2 = m dx0 1 ( dx

n

dx 0)2 (2.10)

2.1 String action

Non-relativistic string

Fig.11: Non-relativistic string

small

oscillations (i = 1 , . . . ,D

2)

L 12

T (x2i + x2i )

T tension = masslength


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world surface coordinates xi (, ) (here x0 = t)transverse string coordinates x i = x it , xi =

x i

S 12T d d (x

2i + x

2i ) (2.11)

Center of mass: xi(, ) = x i( ) + . . .

S 12

T L d x2i + . . . = m2 d x2i + . . . (2.12)Relativistic string

Basic principles:

1. Relativistic invariance (Poincare group)

2. No internal structure, i.e. no longitudinal oscillations 2-d reparametrizationinvarianceRelativistic generalization: xi x

The Nambu-Goto action 4 is (we assume no higher than rst derivatives are presentin the action)

S = T d d (xx )2 x2x 2 (2.13)Motion in space-time: {x(, )}, (, ) world-surface coordinatesxx

xx , x2

xx (2.14)

Symmetries:

1. space-time (global) Poincare: x = x + a

2. 2-d world-volume (local) reparametrizations of world-surface

= f (, ), = g(, )with x(, ) transforming as scalars in 2- d:

x ( , ) = x(, ) (2.15)

a = ( , ), a = a + a ( )x ( + ) = x( ) x = a a x , a = ( 0, 1) = ( , )

Meaning of the action: area of world surface4 Y. Nambu, Lectures at the Copenhagen Symposium, 1970; T. Goto, Relativistic quantum

mechanics of one-dimensional mechanical continuum and subsidiary condition of dual resonancemodel, Prog. Theor. Phys. 46 , 1560 (1971).


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Fig.12: Embedding of ( ,)-plane into space-time

Induced metric on world surface ( a, b = 0 , 1)

ds 2 = dx dx = h ab( )d ad b (2.16)

hab = a x bx (2.17)

hab = x2 xx

xx x2 , det h ab = x2x 2 (xx )2 (2.18)

Thus Nambu-Goto action reads

S = T d2 h, h = det h ab (2.19)Euclidean signature ( )

i E , ds 2 = d 2 + d2 d 2E + d2 (2.20)S E = iS M = T d E d x

2x 2 (xx )2 (2.21)Element of area:

d(Area) = |x||x |sin dd E cos =

xx

|x||x | , sin = 1 cos2

Fig.13: Element of area


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S = T d dl 1 V 2 = T d d x 2 + ( xx )2 x2x 2 (2.29)This action (in x0 = gauge) is invariant under

f ().

This coincides with the Nambu-Goto action in the incomplete static gauge x0 = :

det hab = (x x)(x x ) + ( xx )2 = ( x n )2 (x n )2(xm )2 + ( xn x n )2The Nambu-Goto action thus describes a collection of particles coupled by theconstraint that they should move transversely to the prole of the string (i.e. thereshould be no longitudinal motions).

2.3 p-brane action

p = 0: particle

p = 1: string p = 2: membrane (2-brane), etc.

p-brane ( p + 1)-dim world surface p+1 embedded in Minkowski space M DEmbedding: x( a ), = 1 , . . . , D 1, a = 0 , . . . , p

Principles:

1. Poincare invariance (global)

2. world-volume reparametrization invariance (local)3. no higher derivative (acceleration) terms

L 12

T p a x i bx i ab + . . .

where ab = ( 1, +1 , . . . , +1)Static gauge:x0 = 0, . . . , x p = p; xi( ) = x i( 0, . . . , p) (2.30)

x i ( ) dynamical (transverse) coordinates

Action that satises reparametrization invariance condition has geometric inter-pretation: volume of induced metric on p+1

ds 2 = dx dx = h ab(x( ))d a d b, hab = a x bx (2.31)

S p = T p d p+1 h, h = det h ab (2.32)


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Poincare and reparametrization invariance are obvious.Static gauge: use rst ( p + 1) coordinates x0, . . . , x p to parametrise the surface

xa = a , x i

(x1, . . . , xD 1 p)

(x p+1 , . . . , x D 1)

h ab = ab + a x i bx i

det hab = (1 + ab a x i bx i + . . .)S p = T p d p+1 1 + a x i a x i + . . . = T p d p+1 (1 +

12

a x i a x i + . . .) (2.33)

For p = 0 , 1 one can nd a gauge in which the equations of motion become linear;but this is not possible for p 2, i.e. the equations are always nonlinear.Variational principle: x x + x

S p = T p d p+1

det hab(x( )) (2.34)

h ab = a x b(x ) + ( a b), h = 12 hh abh ab (2.35)

Used that det A = eT r ln A . Then

S p = T p d p+1 x b( hh ab a x) T p d p+1 b( hh ab a xx ) (2.36)Assume boundary conditions that the boundary term here vanishes; e.g., for p = 1(0 L):

closed string: x() = x( + L), with the boundary condition x( in ) =x ( f in ) = 0 open string: free ends Neumann boundary condition x = 0 at = 0 , L

Equation of motion:

b( hh ab a x) = 0 (2.37)Highly non-linear; simplies in a special gauge to a linear one for p = 0 , 1 only.


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2.4 Action with auxiliary metric on the world surface

In addition to x( ), let us introduce new auxiliary eld gab( ) metric tensor on p+1 .Classically equivalent action for p-brane is then:

I p(x, g ) = 12

T p d p+1 g(gab a x bx c) , c = p 1 (2.38)Action is now quadratic in x but it contains independent gab( ) eld. We shall callthis auxiliary metric action .5 Eliminating gab through its equations of motion yieldsthe same equations for x as following from (2.32).

Interpretation: action for scalar elds {x} in d = p + 1 dimensional space dwith metric gab( ).Variation of ( 2.38) with respect to x yields

a ( ggab bx) = 0 , (2.39)which is of the same form as for a set of scalar elds x of zero mass in curved spacewith metric gab .

Recall that variations of the inverse of the metric tensor and of the determinantof the metric are

gab = gac gbdgcd , g = 12 ggabgab . (2.40)

With the help of ( 2.40) the variation of ( 2.38) with respect to gab reads

d p+1 g gab

12gab(gcd cx dx c) gac gbd cx dx = 0 . (2.41)

Using the notation h ab = a x bx and lowering the indices with gab we obtain

12

gab(gcd cx dx c) = hab , (2.42)5 In the string ( p = 1, c = 0) case this action originally appeared in the papers [S. Deser and

B. Zumino, A complete action for the spinning string, Phys. Lett. B 65 , 369 (1976)] and [L. Brink,P. Di Vecchia and P. S. Howe, A locally supersymmetric and reparametrization invariant actionfor the spinning string, Phys. Lett. B 65 , 471 (1976)] which generalized a similar constructionfor the (super) particle case ( p = 0, c = 1) in [L. Brink, S. Deser, B. Zumino, P. Di Vecchiaand P. S. Howe, Local supersymmetry for spinning particles, Phys. Lett. B 64 , 435 (1976)]. Anequivalent action with independent Lagrange multiplier elds that led to the correct constraints(but did not have an immediate geoemtrical interpretation) appeared earlier in [P.A. Collins andR.W. Tucker, An action principle and canonical formalism for the Neveu-Schwarz-Ramond string,Phys. Lett. B 64 , 207 (1976)] (see footnote in the Brink et al paper and the discussion at the end of the Deser and Zumino paper). This auxiliary metric form of the classical string action was widelyused after Polyakov have chosen it as a starting point for path integral quantization of the string[A. M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 , 207 (1981)]


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i.e.gab = h ab , 1 =

12

(gabhab c). (2.43)On the other hand,

gabh ab = 1gabgab = 1( p + 1) , (2.44)i.e.

c = 1( p 1). (2.45)Thus, if p = 1 and c = p 1 we get = 1, or gab = hab . This means that on theequations of motion the independent metric gab coincides with the induced metric(for p = 1 it is proportional to it up to an arbitrary factor).

Then equation for x in (2.37) becomes

a ( hh ab bx) = 0 , (2.46)which is the same as following from the original S p[x] action (2.32).

Thus, I p[x, g ] and S p[x] give the same equations of motion and also are equal if we eliminate gab using the equation of motion gab = hab

I p[x, g ]|g ab = h ab = 12

T p d p+1 h(hab a x bx c)=

12

T p d p+1 h(habhab p + 1)=

12

T p d p+1 h = S p[x].2.5 Special cases

The tensor gab is symmetric, so it has ( p+1) ( p+1) components which are functionsof ( p + 1) arguments a . Reparametrization invariance (gauge freedom) is describedwith ( p + 1) functions a = f a ( ). Thus the number of non-trivial components of gabequals to p( p+1)2 .

2.5.1 p = 0 : particle

In this case a, b = 0 and we have only one metric component gab = g00 = e2. Usingthe following notation a = , T 0 = m, e = e( ) we can rewrite (2.38) in the formI 0[e, x ] =

12

T 0 d e (e2x x + 1) = 12m d (e1x2 e) (2.47)Variation with respect to e gives

e2x2 + 1 = 0 , i.e. e = x2 .


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ThenI 0|e= x2 = m d x2 = S 0[x]. (2.48)

Rescaling e by m , i.e. e( ) = m ( ), leads to an action that admits a regular masslesslimit m 0:

I 0 = 12 d (1x2 m 2) . (2.49)

The limit m = 0 gives an action for the massless relativistic particle

I (0)0 [, x ] = 12 d 1x2. (2.50)

Variation of ( 2.49) with respect to x and gives the equations of motion

d

d 1

x

= 0 (2.51)x x = m 22. (2.52)

Here the number of gauge parameters equals to 1 and is the same as number of components of gab . Thus g00 = m 22 can be completely gauged away. Choosing thespecial reparametrization gauge

( ) = 1

givesx = 0 x = x0 + p, (2.53)

andx x = m 2 p p = m 2 . (2.54)

These are the usual relativistic particle relations.

2.5.2 p = 1 : string

This is also a special case because ( p 1)1 = c = 0 is satised identically for any . Thus, solution is gab = h ab , where = ( ) is an arbitrary function of .The equations for x are still the same for the actions S p and I p since

ggab =

hh ab . (2.55)

This is due to an extra symmetry that appears in this case. Indeed, in 2 dimensions(d = p + 1 = 2) the action ( p = 1 , c = 0 , T = T 1)

I 1[x, g ] = 12

T d2 ggab a x bx (2.56)


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is invariant under the Weyl (or conformal) transformations

gab = f ( )gab . (2.57)This symmetry is present only for massless scalars in d = 2 = p + 1. For any p

ggab = f

d 22 ggab . (2.58)

This is another local (gauge) symmetry in addition to the reparametrization invari-ance. gab has 3 components same as the number of gauge parameters of the Weyl(one) and the reparametrization invariance (two) transformations. This allows oneto gauge away gab completely, i.e. to set gab ab , and then obtain a free action forx .

For general p the equation for gab is the vanishing of the total energy-momentumtensor

T ab =

2

gI p

gab = T p( a x bx

1

2gabgcd cx dx)

c

2T pgab . (2.59)

For c = 0 and gab = h ab we have T ab(x) = T ( a x bx 12 habh cd cx dx) = 0.To summarize, in the p = 1 case we have the following local gauge transformations

1. Reparametrization invariance: a = f a ( ) x ( ) = x ( ), gab( ) =

c

a d b gcd( )

2. Weyl invariance:x ( ) = x( ), gab( ) = ( )gab

Total number of gauge functions equals to the number of components of gab so thatit can be gauged away completely.

Namely, one can choose the conformal (or orthogonal) gauge gab = ab . Itis sufficient even to choose the reparametrization gauge only on the Weyl-invariantcombination: ggab = ab .

Then the equation for x becomes linear. Indeed, consider the reparametrizationgauge gab = f ( )ab , where f is arbitrary. Then, g00 = g11 and g01 = 0 and fromthe equation of motion

a ( ggab bx ) = 0 (2.60)we have

a a x = 0 , i.e. x x = 0 . (2.61)Also we know that hab gab ab , where hab = a x bx . Then, h00 = h11 andh01 = 0. In other words, we have in addition the following differential constraints(containing only 1st time derivatives and generalizing x2 = 0 for a massless particle)

x2 + x2 = 0 , xx = 0 . (2.62)


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2.6 Meaning of the constraints and rst-order actions

2.6.1 Particle case

Let us introduce an independent momentum function p( ). Then the action for

{x( ), p( ), ( )} which is equivalent to ( 2.49) isI 0(x,p, ) = d x p 12( p2 + m2) , (2.63)

where is a Lagrange multiplier imposing the constraint p2 + m2 = 0.The variation of ( 2.63) gives

x : p = 0

p : x = p

: p2 + m2 = 0This action is invariant under the reparametrization with d ( ) = d ( ). Fixingthe gauge as = 1 we get the usual equations

p = x , x = 0 . (2.64)

Eliminating p from the action ( 2.63) yields (2.49), i.e.

I 0(x,p, ) p=

1 x

= 1

2 d (1x2

m 2). (2.65)

Thus g00 = m 22 plays the role of a Lagrange multiplier.2.6.2 String case

Similarly, let us introduce the independent momentum eld p(, ) and consider thealternative action 1-st order action for x , p with the conformal-gauge constraintsadded with the Lagrange multipliers (, ) and (, )

I 1(x,p, , ) = T

dd x p

1

2 ( p2 + x2)

p x (2.66)

The variation of ( 2.66) gives

x : p (x ) = 0 p : p = 1(x x )


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2.7 String equations in the orthogonal gauge

Let us introduce the light-cone parametrization for the world-sheet

=

, (2.73)

= + + , = + . (2.74)Then we get

+ x = 0 equation of motion (2.75)

x x = 0 constraints (2.76)

Consequently, we can easily write the general solution of ( 2.75) as a sum of theleft-moving and right-moving waves with arbitrary proles

x = f + ( + ) + f

( ). (2.77)The constraints then have the form

f 2+ = 0 , f 2 = 0 . (2.78)

2.8 String boundary conditions and some simple solutions

In the conformal gauge the string action reads

S = 12

T dd a x a x . (2.79)Its variation has the form

S =

T

dd x (

2x)

T

dd a (x a x), (2.80)

where the variations of x at the starting and the ending values of are equal tozero, x( 1, ) = 0, x( 2, ) = 0.

2.8.1 Open strings

If the string is open, the second term vanishes in the two cases:

Fig.14

Neumann condition : free ends of the string

x(, )|=0 ,L = 0 (2.81)Dirichlet condition : xed ends of the string

x(, )|=0 ,L = y0,L ( ) (2.82)Here y0,L ( ) are some given trajectories. The Dirichlet condition is relevant for theopen-string description of D -branes. This condition breaks the Poincare invariance.


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In general, one can also impose mixed boundary conditions. Let us divide thecomponents of x into the two sets

Fig.15: Boundary conditions

Neumann components (along the brane)

{x}= {x0, x 1, . . . , x p}Dirichlet components (transverse to brane)

{xk}= {x p+1 , x p+2 , . . . , x D 1}The boundary conditions read

x |=0 ,L = 0 , xk(, ) =0 ,L = yk0,L ( ). (2.83)If the space-time contains no Dp -branes, the open strings have free ends in all the

directions (Neumann conditions). Then the full Poincare invariance is unbroken.

2.8.2 Closed strings

For closed strings we impose the periodic condition x(, ) = x(, + L) and maychoose units so that L = 2 and (, ) are dimensionless. Then we get the set of equations

x x = 0 , x2 + x2 = 0 , xx = 0 (2.84)with the condition

x(, ) = x(, + 2 ). (2.85)

The simplest solution is a point-like string

x(, ) = x( ) = x0 + p , p p = 0 . (2.86)

2.9 Conservation laws

It is useful to split x(, ) into the coordinate of the center of mass x(, ) andoscillations around it x(, )

x(, ) = x( ) + x(, ), (2.87)

x( ) = 1L d x

(, ), (2.88)

d x(, ) = 0 . (2.89)Global symmetries lead via the Noether theorem to quantities that are conserved onthe equations of motion.


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On the other hand, the local symmetries (reparametrizations and the Weyl trans-formation) lead to the restrictions on the energy-momentum tensor

a T ab = 0 , T aa = 0 , (2.90)

or to the constraints on x after the gauge xing.If the action is invariant under some transformation x , then using Lagrange

equations of motion Lx a

L a x

= 0 (2.91)

we get a conserved current

j a = L a x

x , a j a = 0 . (2.92)

Let x = AA , where A are constant parameters. Then j a = j aAA and a j aA = 0.Now it is easy to check that the integral

J A( ) = dj 0A(, ) (2.93)gives a conserved charge

dd

J A( ) = 0 . (2.94)

The string action is invariant under the Poincare transformations

x = x + , (2.95)

where = , i.e. so(1, D 1). Then for space-time translations in theorthogonal gauge gab ab we get the conserved current (momentum density) pa = pa

= L a x

= T a x . (2.96)Recalling that x = 0x = 0x and taking ( 2.87) into account, we get

P = d p0 = T d x = T L x , (2.97)d

d P = 0 . (2.98)

Indeed, using the equation of motion and the boundary conditions we get

P = T d x = T L

0d x = T (x )|

L0 = 0 (2.99)


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This conservation law means that there is no momentum ow through boundary.From ( 2.97) we get

x

( ) = x

0 + p

, p

= P

, T L 1 . (2.100)

Thus, the center of mass moves with a constant velocity determined by the totalmomentum of the string.

Remark:Since the string equation of motion is of the second order formally it has a simple

solution linear in both and

x = x0 + p + q . (2.101)

For the open strings with free ends, i.e. with Neumann boundary conditions we

get q

= 0. In the case of the closed strings we need x

(,

) = x

(,

+ 2

). Ageneralized version of this condition can be satised if x is a compact (angular)coordinate, e.g.,

Fig.18: Compacticationon cylinder

x1 = R , + 2

Then ( 2.101) is consistent with the gen-eralized periodicity

x(, + 2 ) = x(, ) + 2 Rw,

where w = 0 , 1, 2, . . . is the windingnumber.

Then x = x( ) + q, where q = Rw is quantized winding momentum (cf. particlein a box).

Solution (2.101) has an additional symmetry ( T -duality) under p q, .Consider next the Lorentz rotations x = x . The conserved current (angular

momentum density) reads

j a = L a x

x = 1

2 j a

, (2.102)

j a (, ) = T (x a x x a x) (2.103)

The corresponding conserved charge is

J = d j 0 = T d (xx x x). (2.104)


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By construction, on the equations of motion one has

dd

J = 0 . (2.105)

Using (2.87), we can rewrite (2.104) as

J = T d( x x x x) + T d( x x x x) = I + S . (2.106)Cross-terms here vanish due to ( 2.88) and (2.89). I is the orbital angular momen-tum and S is the internal one (i.e. the spin). Using ( 2.97) we can rewrite theorbital moment as

I = P x P x . (2.107)Using (2.97) we have

dd

I = dd

(P x P x) = ( P x P x) = 0 .With the help of the equations of motion we get also

dd

S = T d (x x x x) = T d (x x x x )= T d (x x x x) = T (x x x x )|L0 = 0 .

2.10 Rotating string solution

One simple solution is provided by a folded closed string rotating in the ( x1, x 2)-planewith its center of mass at rest

Fig.16: Rotating string

x0 = p0

x1 = r ()cos ( )

x2 = r ()sin ( ),

r () = a sin , ( ) =

This solves x x = 0 with a being an arbitrary constant. From the periodicitycondition it follows that w is integer, i.e. = 0 , 1, 2, . . . . The constraint x2 + x 2 =0 yields p0 = a , and the constraint xx = 0 is satised automatically.The Lagrangian of the string is

L= 12

T a x a x


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so that the energy related the time-translation symmetry x0 x0 + is (we xL = 2 )E = P 0 = T

2

0

d L x

0 = T

2

0

d x0 = p0

=

a

, (2.108)

where = 12T .The spin is related to the rotation symmetry + S = T

2

0d

L

= T 2

0d r 2() = T a2

2

0d sin2 =

a22

. (2.109)

Combining (2.108) and (2.109) we obtain relation between the energy and the spin

E 2 = 2 S, w = 1 , 2, . . . (2.110)

The congurations with the lowest energy for a given spin (or smalest slope on E 2(S )plot) with = 1 belong to the leading Regge trajectory. For them E 2 = 2 S orrecalling that L = 2 we have

E 2 = 2 TLS . (2.111)

We can generalize the above rotating solution by adding motion to its center of mass:

Fig.17: Rotating string withmoving c.o.m.

x0 = p0

x1 = x1(, ), x2 = x2(, )

x i = x i0 + pi, i = 3 , 4, . . .

The constraint x2 + x 2 = 0 then gives p20 p2i = a22. As in the case of the rotatingstring at rest we can nd the expressions for the energy and the spin ( 2.110)E 2 = P 2i +

2

S, P i = pi

. (2.112)

2.11 Orthogonal gauge, conformal reparametrizations andthe light-cone gauge

In the orthogonal gauge the string action has the form

I = 12

T d2 + x x . (2.113)


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The equations of motion and the constraints read

+ x = 0 , x

x = 0 . (2.114)

These are invariant under the residual conformal symmetry which is a subgroup of reparametrizations a = F a ( 1, 2) ( = ) + = F + ( + ), = F ( ). (2.115)

Under such transformations the 2d metric changes by a conformal factor

ds 2 = d + d = dF +d +

dF d

d + d (2.116)

i.e. the orthogonal-gauge condition is preserved by such conformal reparametriza-tions.

One can choose a special parametrization by using this additional invariance, i.e.

on the equations of motion for x one can impose an additional gauge condition the light-cone gauge

x+ (, ) = x+ ( ) = x+0 + p+ , x+ x0 + xD 1 . (2.117)

That means that x+ (, ) = 0, i.e. there is no oscillations in x+ . Indeed, since

+ x+ = 0 x+ = f +1 ( + ) + f 2 ( ),

and one can use conformal reparametrizations to choose these functions so that x+ =12 p( + ) +

12 p( ) = p , where we also set x+0 = 0 by a shift.

This is a physical gauge, in which only the transverse oscillations xi (i =

1, 2, . . . , D 2) are dynamical. Indeed, x is then determined from the orthogo-nal gauge constraints x

x = 0 x+ x + x i x i = 0 , (2.118)i.e.

12 p+ x = x

i xi , (2.119)

which is a linear equation on x determining it in terms of x i .

Fig.19: Light cone

We end up with

x+ = x+ ( )x = x( ) + F [x i (, )]x i = x i ( ) + x iL ( + ) + x iR ( )

x i = dynamical degrees of freedom.


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This gauge where xi are dynamical is analogous to the light-cone gauge in quantumelectrodynamics ( A+ = A0 + A3 = 0) where Ai (i = 1 , 2) are dynamical degrees of freedom.

3 General solution of string equationsin the orthogonal gauge

Let us use uniform notation for open/closed case: = (0 , ), i.e. choose L = Boundary conditions:

x(, ) = x(, + ) closed string x |=0 = x |= = 0 open string

Equations of motion and constraints read + x

= 0 , x x = 0 . (3.1)

We set as beforex(, ) = x0 + p

+ x(, ). (3.2)

Then x is given by Fourier mode expansion.

Open string:Boundary conditions x |=0 , = 0 are satised by

x =

n =0

a n ein cosn (3.3)

Reality condition of x implies: (an ) = a

n . The momentum is

P = T

0d x = T p =

12

p (3.4)

The angular momentum (=orbital+spin) is

J = I + S = x p x p + T

0d (x x x x) (3.5)

Rescaling

an = i 2

n n


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we get

x = i 2 n =0

nn

ein cos n. (3.6)

Closed string:

Here x() = x( + ) and we can represent the oscillating part of the solutionas a sum of the left-moving and right-moving waves x(, ) = xL ( ) + x

R ( + )

( = ), i.e. x(, ) = xR ( ) + xL ( + ) , (3.7)where

xR ( ) = 12

x0 + 12

p +

n =0

an e2in

(3.8)

xL ( + ) =

12

x0 + 12

p + +

n =0

an e2in+

(3.9)

Reality condition implies: ( an ) = a

n , (an ) = a

n .After the rescaling a i

2 nn , a i

2 nn , we get

x = x0 + 2 P + i 2 n =0 n

n e2in ( ) + i 2 n =0

n

n e2in ( + ) (3.10)

The constraints imply the vanishing of the energy-momentum tensor T = ( x)2 =

0. Let us introduce its Fourier components. For closed string we get

Ln = T 2 d e2in T ( ) (3.11)

Ln = T 2 d + e2in + T ++ ( + ) (3.12)

where we integrate over in the range (0 , ) at = 0. For the open string

Ln = T

0d (ein T ++ + ein T

) =

T

4

d e in (x + x)2. (3.13)

The substitution of the solutions ( 3.3) and (3.8) into these integrals yields

Ln = 12

k= nk k ,

Ln = 12

k= nk k (3.14)


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for the closed string and

Ln = 12

k= nk k (3.15)

for the open string. Here we introduced the following denition:

0 = 0 = 2 p closed string (3.16)

0 = 2 p open string (3.17)The angular momentum is found to be

J = I + S ( ) + S ( ), (3.18)

where

I

= x p

x p

,

S ( ) = i

n =1

1n

( n n n n )

The string Hamiltonian is

H = T

0d (xp L) =

T 2

0d (x2 + x2) (3.19)

i.e.

H = 12

n =

n n = L0 open string (3.20)

H = 12

n = ( n n +

n n ) = L0 + L0 closed string (3.21)

From ( 3.16), (3.20) and mass condition p2 = M 2 it follows that

M 2 =

n =1

n n open string (3.22)

M 2 = 2

n =1

(

n n +

n n ) closed string (3.23)

Let us introduce the Poisson brackets

{x(, ), x ( , )}= {x , x }= 0 , (3.24){x(, ), x ( , )}= T 1 ( ) , (3.25)


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Using the -function representation

() = 1

n =

e2in

one can check that

{ m , n}= { m , n}= im m + n, 0 , (3.26){ m , n}= 0 . (3.27)

For the zero modes we get { p, x }= .One nds then that

{Lm , L n

}= i(m

n )Lm + n , (3.28)

{Lm , Ln}= i(m n )Lm + n , {Lm , Ln}= 0 , (3.29)i.e. the constraints are in involution forming the Virasoro algebra.

From ( 3.14) and (3.15) it follows that

(Ln ) = Ln , (Ln ) = Ln . (3.30)

4 Covariant quantization of free string

4.1 Approaches to quantization

I. Canonical operator approachHere one starts with promoting classical canonical variables to operators and

imposing standard commutation relations.In the Lorentz-covariant approach one uses covariant orthogonal gauge and im-

poses constraints on states. The Weyl symmetry is broken at the quantum levelunless dimension D = Dcrit = 26 (or D = 10 for superstrings). In this approachthe theory is manifestly Lorentz-invariant, but there are no ghosts and the theory isunitary only in D = D crit .

In the light cone (physical) gauge approach the constraints are rst solvedexplicitly at the classical level, and then only the remaining transverse modes arequantized. Here the theory is manifestly unitary but is Lorentz-invariant only inD = D crit .


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II. Path integral approachHere one starts with the path integral dened by the string action

< . . . > =

Dg ab

Dx eiS [x,g ] . . .

The 2d metric is assumed to be non-dynamical (the Weyl symmetry is assumedto be an exact symmetry of the quantum theory, which happens to be true onlyif D = Dcrit ), and the integral over the continuous (local) modes of the metric isremoved by the gauge xing.

The observables here are correlation functions of vertex operators (correspond-ing to string states) dening the scattering amplitudes of particular string modes.

4.2 Covariant operator quantization

We promote canonical center-of-mass variables ( x0 , p) to operators ( x0 , p), and do thesame for the oscillation modes: , .The Poisson brackets {, } are replaced by the quantum commutator i[ , ].The string state ( p, , ) then corresponds to a quantum state vector | > in Hilbertspace.The reality condition n = n becomes n =

n .Then we get [x0 , p ] = i , where x = x

0 + 2 p + . . . and also (below we shall

often omit hats on the operators)

[ m , m ] = m , [ m , n ] = 0, m = n (4.1)

The Fock vacuum |0, p > satises p|0, p > = p|0, p > . (4.2)The vacuum is annihilated by m , m > 0

am |0, p > = 0 (m > 0) , am 1 m

m (4.3)

The general string state vector is

| > = ( an 1 )k1 . . . (an l )kl |0, p > (4.4)Not all states are physical. They must satisfy the constraints and have positive norm.

But [ 0n , 0n ] = 1 , n > 0,implies existence of negative norm states

|| 0n |0 > ||= 1These can be ruled out by imposing the quantum version of the constraints.


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4.2.1 Open string

Here the Virasoro operators are

L0 = p2 +

n =1 n n (4.5)

Lm = Lm = 12

n =0

mn n , m > 0. (4.6)

In the classical theory the constraints are L0 = 0 , Lm = 0. In the quantum theoryL0 and Lm become operators and the problem of ordering of the operators that enterthem arises.

The normal ordering is automatic for m = 0 since m n = n

Lm Lm = 12

n =0

mn n , m = 0 . (4.7)

For m = 0 we have in general (c1 + c2 = 1)

L0 L0 = p2 +

n =1

(c1 n n + c2 n n )

= p2 +

n =1

n n + c2

n =1

[ n , n ]

Then

L0 =: L0 : +a , a = c2 D

n =1n. (4.8)

It turns out that for consistency of the theory one is to choose a = D24 . This value isfound if one uses the -function regularisation and chooses also c1 = c2 = 12 . Considerthe series

I 0() =

n =1

en = 1e 1

0 1

12

+ O()

I 1() = dd

I 0 =

n =1

ne n = e

(e 1)20

12

112

+ O()

I 1() = dI 0() = n =1 1n en = ln(1 e) 0 ln + O()

The -function is dened by

(s) =

n =1

1n s

, s C (4.9)


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and then at s = 0 , 1, 1, . . . by an analytic continuation. This gives the same valuesas nite parts in the above series (0) =

1

2, (

1) =

1

12, (1) =

(4.10)

With this prescriptiona =

12

D (1) = D24

. (4.11)

In fact, the true value of the normal ordering constant a is not D24 but a = D 224 :the additional -2 contribution to D comes from the ghosts of the conformal gauge(in the light-cone gauge, only D 2 transverse modes xi contribute).

The quantum version of the constraints is then

(: L0 : +a)| > = 0 , (4.12)Lm | > = 0 , m > 0 (4.13)where

: L0 := p2 +

n =1

n n , Lm =

n =0

mn n (4.14)

The mass shell condition p2 = M 2 then gives the expression for the mass operator M 2

| > = ( a +

n =1

n n )

| > (4.15)

In particular, the value a thus determines the mass of the ground state

M 2|0, p > = a|0, p > , (4.16)which is tachyonic for D = 26: a = D 224 = 1. In the supersymmetric string casethe ground state turns out to be massless.

For the Virasoro algebra one nds

Classical: {Lm , L n}= ( m n )Lm + n (4.17)Quantum: [ Lm , L n ] = (m n )Lm + n + D12

(m 2 1)m m + n, 0 (4.18)where the quantum algebra turns out to contains a central term. When Lm aremodied by the ghost contributions the coefficient D in the central term is shifted toD 2.


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Note that for n = m we get[Lm , Lm ] = 2mL 0 +

D12

(m 3 m ) (4.19)so that in view of (L0 + a)| > = 0

[Lm , Lm ]| > = 2 m L0 + D24

(m 2 1) | > = 0 , (4.20)which is in contradiction with Lm | > = 0 for m = 0. To avoid this contradictionit is sufficient to impose the constraints on average, i.e. to demand that matrixelements of Lm with m = 0 should vanish:

< |Lm | > = 0 , m = 0 (4.21)The spectrum is then described in terms of the oscillator states:

L0 = L0 + a = p2 +

n =1

nN n + a, (4.22)

N n 1n

n n = an an , [an , a n ] = . (4.23)

N n has the following properties

N n |0 > = 0 , N n a n |0 > = a n |0 > . (4.24)For generic oscillator state

| > = ( an 1 )i 1 . . . (an k )ik |0, p > , p|0, p > = p|0, p > (4.25)where (an 1 )i1 stands for a1 n 1 ...a

i 1 n 1 we can dene the level number as

= i1n 1 + . . . + ikn k (4.26)

Then

n =1

nN n | > = ( i1n 1 + . . . + ikn k)| > = | > (4.27)and thus the mass of this state is determined by ( T = 12 )

M 2| > = 1

n =1

nN n + a | > = 2T ( + a)| > . (4.28)


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In the open string case we may call the physical states those that satisfy thecondition

Lm | > = 0 for m 1. (4.29)Let us dene the true physical states as those that have positive norm

< | > > 0. (4.30)Other physical states that have zero norm are called null states. One can provethe no ghost theorem: iff D = 26 all physical states have non-negative norm.

Fig.20: String states

One can dene spurion states as those obtained byacting by Lm , i.e. | > = Lm | > for m > 0.Such states orthogonal to physical since

Lm

| > = 0

<

|Lm = <

|L

m = 0

< |Lm | > = 0 < | > = 0 .

Null states may be both physical and spurion; true physical states are then non-spurion ones, i.e. physical ones that have positive norm.

4.2.2 Closed string

Here we get two independent sets of creation-annihilation operators m and m . Thenormal ordering ambiguity in going from classical to quantum theory implies

L0 L0 + a, L0 L0 + a.

L0 + L0 = p2

2 +

n =1

( n n +

n n ) + 2 a, (4.31)

Lm =

n = mn n ,

Lm =

n = mn n (4.32)

a closed = 2aopen = 2a = D12

, (4.33)

where D is shifted to D 2 after one accounts for the conformal gauge ghosts or usesthe light-cone gauge.Here one gets two copies of the Virasoro algebra for Ln and Ln .


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The vacuum state is dened by

p|0, p > = p|0, p > , n |0, p > = 0 , n |0, p > = 0 , n > 0 (4.34)and

| > = ( n 1 )i1 . . . ( n k )ik ( m 1 ) j 1 . . . ( m s ) j s |0 > (4.35)form an innite set of states in Fock space. The level numbers are dened by

= i1n 1 + . . . + ikn k , = m 1 j1 + . . . + m s j s . (4.36)

The constraints in classical theory are L0 = 0 , L0 = 0 and Lm = 0 , Lm = 0 for m =0. In quantum theory we need to impose them on average to avoid contradictionwith the centrally extended Virasoro algebra; that means we need to assume that

(L0 + a)

| > = 0 , (L0 + a)

| > = 0 , (4.37)

Lm | > = 0 , Lm | > = 0 for m > 0. (4.38)Equivalently, ( 4.37) reads

(L0 + L0 + 2 a)| > = 0 , (4.39)(L0 L0)| > = 0 . (4.40)

Then the constraints are satised for the expectation values

< |Lm | > = 0 , < |Lm | > = 0 , m = 0 . (4.41)

Equation ( 4.39) determines the mass spectrum and ( 4.40) is the level matchingcondition.From ( 4.39) it follows that

p22

+

n =1

n (N n + N n ) + 2 a | > = 0 , (4.42)

whereN n = an an =

1n

n n , N n = an an = 1n

n n . (4.43)

Then the closed-string states have masses

M 2 = 2

( + + 2a). (4.44)

Eq. (4.40) gives

n =1

n (N n N n )| > = 0 , i.e. = (4.45)


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so that we getM 2 =

4

( + a). (4.46)

The mass of the ground state is thus

M 20 = 4

a . (4.47)

As in the open string case with D = 26 here we get 2a = D 212 = 2, so that theground state is tachyonic.The structure of the physical state space here is similar to the one in the open

string case.

5 Light cone gauge description of the free string spectrum

The equations of motion and the constraints in the orthogonal gauge

x x = 0 , x2 + x2 = 0 , xx = 0 . (5.1)have residual conformal reparametrization symmetry

+ f ( + ), f ( ), = (5.2)As was already mentioned above, it can be xed by imposing an additional lightcone gauge condition

x+

(, ) = x+

( ) = x+0 + 2 p

+

, x = x0

xD

1

, (5.3)implying that there is no oscillator part in the classical solution for x+ .

Then the constraints determine

x(, ) = x0 + 2 p + F ( p+ , x i(, )) . (5.4)

in terms of x i , i.e. D 2 independent transverse degrees of freedom.

5.1 Open string

Using constraint x x = 0 we get0 = xx =

12

(x+ x + xx + ) + x i x i = p+ + x i x i , (5.5)so that

x = 1 p+ d x ix i + h( ) (5.6)


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and thus

n = 1 2 p+

m =0

inm im . (5.7)

From x2 + x2 = 0 it follows that2 p+ x = x i x i + xi x i . (5.8)

Here we quantize as independent ones the transverse oscillators only. The mass shellcondition p2 = M 2 = N = N is then determined by the transverse-modeoscillation number

N =

n =1

in in . (5.9)

In quantum theory

N = : N : + a , a = D 224 (5.10)If D = 26 then a = 1 and one can show that the resulting spectrum is consistentwith the requirement of the Lorentz invariance of the theory.

Since the constraints are solved already, by acting on the Fock vacuum by thecreation operators in = (

in ) we get only the physical states

| > = i1 ...i m ( p) i 1n 1 . . . imn m |0, p > (5.11) in> 0|0, p > = 0 , p|0, p > = p|0, p >, [x0 , p ] = i (5.12)

The angular momentum J = I + S here is

I = x p x p, S = 2i n

( n n n n ). (5.13)

The Lorentz algebra is dened by

[J , J ] = J J J + J , (5.14)where

J = ( J ij , J + i , J i , J + ).

The commutation relation [ J i, J

j] = 0 realized in terms of the quantum stringoscillators turns out to be valid only if D = 26: the light cone gauge preserves

Lorentz symmetry iff D = 26 (in superstring theory one nds this for D = 10).The lowest-mass states in the spectrum are:

N = = 0 : M 2 = a = 1


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N = = 1 : M 2 = 1 1 = 0

| > = i( p) i1|0, p >, i = 1 , . . . , D 2 i( p) is a vector of SO (D 2) having D 2 = 24 physical polarisations; this isconsistent with Lorentz invariance since it is massless.

N = = 2 : M 2 = 2 1 = 1 There are two possibilities

| > = i( p) i2|0, p > (5.15)| > = ij ( p) i1

j

1|0, p > (5.16)The total number of components is

D 2 + (D

2)(D

2 + 1)

2 = D (D

1)

2 1.This is a dimension of SO (D 1) representation (symmetric traceless tensor of SO (D 1)) as it should be for massive state in a Lorentz-invariant theory in Ddimensions. Thus the = 2 state is a spin-2 massive particle.

Higher levels are described by SO (D 1) representations as well. Let us recallthat the irreducible representations of SO (r ) are described by tensors tm 1 ...m k , m i =1, . . . , r which are symmetric traceless, or antisymmetric or have mixed symmetryand can be represented by the Young tableaux.

Higher level states in the light cone gauge spectrum are described by tensors of

SO (D 2) which combine into irreducible representations of SO (D 1), i.e. bymassive particles in D dimensions.The maximal-spin state at level with M 2 = l 1 is

| > = i1 ...i l ( p) i11 . . . i l1|0, p >, (5.17)

where i1 ...i l is a symmetric tensor of SO (D 2). To get the full state of spin onehas to add lower tensors of S O (D 2).For instance, for = 3:

ijk i

1 j

1 k

1|0, p > symmetric 3rd rank tensor

ij i2 j

1|0, p > symmetric + antisymmetric 2nd rank tensor i i3|0, p > vector


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Fig.21: Regge trajectory

Leading Regge trajectory (highest spinstates at given level):

J = = (E ) = (0) + E 2 = 1 + M 2

5.2 Closed string

Here the transverse oscillators are ( in , in ); these are the only ones that remain after

we set n = 0 and n = 0 (light cone gauge) and use the constraints to determine n , n as functions of in , in . Then the relevant oscillator number operators are

N =

n =1

in in , N =

n =1

in in (5.18)

The mass shell conditions are

(L0 1)| > = 0 , (L0 1)| > = 0 . (5.19)The physical states are

| > = i1 ...i n ,j 1 ...j k ( p) i1m 1 . . . inm n j 1s 1 . . . jks k |0, p > , (5.20)For them

M 2 = 2( + ) 4, = 0 , (5.21) = i1m 1 + . . . + in m n , = j 1s1 + . . . j ksk (5.22)

i.e. M 2 = 4( 1). We thus nd: = 0: M 2 = 4 scalar tachyon = 1: M 2 = 0 massless state

This state ij ( p)i

1 j

1|0, p > can be split into irreps of S O (D 2): ij = (ij ) + ij + [ij ] (5.23)

The symmetric traceless tensor (ij ) represents spin-2 graviton which lies on the lead-ing Regge trajectory. The second term is a scalar dilaton. The antisymmetrictensor corresponding to [ij ] is called the Kalb-Ramond eld.


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The graviton has (D 2)( D 2+1)2 1 = D (D 3)2 components, and the Kalb-Ramondeld has (D 2)( D 3)2 components (in D = 4 the graviton has two degrees of freedomand the Kalb-Ramond eld has one, i.e. is equivalent to a scalar).

Higher massive levels are described by SO (D 2) tensors combined in irreps of SO (D 1).The superstring spectrum has similar structure with the ground state being mass-

less (representing D = 10 supergravity states) and with fermions as well as bosonsas physical states.

Many more details and various extensions can be found in the books listed below.

Acknowledgements These lectures were delivered 17-21 August 2006 at theSchool Physics of Fundamental Interactions in Protvino supported by the Dynasty

Foundation.

Some books on string theory

[1] M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory. Vol 1: introduc-tion. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology, Cambridge,UK: Univ. Pr. ( 1987) 596p.

[2] A. M. Polyakov, Gauge elds and strings, Chur, Switzerland: Harwood (1987)301p.

[3] B. M. Barbashov and V. V. Nesterenko, Introduction To The Relativistic StringTheory, Singapore, Singapore: World Scientic (1990) 249p

[4] J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string; Vol.2: Superstring theory and beyond, Cambridge, UK: Univ. Pr. (1998) 402p;531p

[5] B. Zwiebach, A rst course in string theory, Cambridge, UK: Univ. Pr. (2004)558p

[6] K. Becker, M. Becker and J. H. Schwarz, String theory and M-theory: A modernintroduction, Cambridge, UK: Cambridge Univ. Pr. (2007) 739p

[7] E. Kiritsis, String theory in a nutshell, Princeton, USA: Univ. Pr. (2007) 588p

introductory lectures on string theory

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