Superstring Theory

Superstring Theory^§

Classical Theory of String
Quantization
Supersymmetry and Superstring
Compactification
Types of Superstring Theory
M-theory
Problems and Future Development

Classical Theory of String

The first task to construct the mathematical equations was to begin with a string of the right length and tension. Since the theory was supposed to account for, among other things, the quantized force of gravity, the strings could not be longer than the length scale where the granular structure of space-time becomes important, i.e., 10^-33 cm - the Planck length. This is the same length scale associated with the gravitational constant when it is expressed in units of

= 1 and c = 1. The tension of the string can be estimated from the strength of the force transmitted by the graviton. It is inversely proportional to the strength of the gravitational constant by the formula (

c⁵/G)^1/2 = 1.22x10¹⁹ Gev (the point particle approximation corresponds to higher tension in the string). However, it is thought that, this colossal energy may be cancelled largely by the vacuum energy. The net result becomes the observed mass of the elementary particles (it is the vibrational pattern that determine the type of particle). The massless graviton can be constructed simply as the lowest string state. But it is not that easy to arrive at the correct mass for massive elementary particles. Moreover, the vacuum energy estimated by the string theory (and quantum field theory for that matter) is too large in comparison with the observed value derived from cosmic acceleration (the cosmological constant). For particle physics, the failure to derive the correct vacuum energy is unsatisfying, but it is fatal for the string theory, which is purported to be a "Theory Of Everything". The superstring theory contains another fault (or virtue depending on point of view) - it gives rise to an infinite number of different configurations (universes) that don't have the same vacuum energy. According to the "anthropic principle" (which simply states: "it is there because we are here"), we are living in just the right one with exceptionally low vacuum energy out of the many others in the multiverse.

The string is an one-dimensional object, which can move in various ways. Its movement sweeps out a two-dimensional sheet, called the world-sheet (see Figure 01a) similar to the world-line generated by the motion of a point. In the diagram, X represents a vector defined in D-dimensional space-time that begins at the origin of a coordinate system and ends at some point along the two-dimensional world-sheet with components X(

), where

= 0, 1, 2, ... D-1, the 0th index denotes the time component; the rest are treated as spatial components,

is a space-like parameter, and

is a time-like parameter. All the real action takes place on the (

) surface - the inner spacetime (worldsheet). The processes are then translated into events occurring in the ordinary spacetime X - the outer spacetime. The string degrees of freedom X(

) trace out a

curve as

varies at fixed

. The curve may be open or closed and

varies within a range between 0 and

as the string is traced out from one end to the other for an open string, or 0 to 2

round the circle for a closed string. The string sweeps out a worldsheet as

varies from one instant to another. However, these two parameters have nothing whatsoever to do with the real space and real time. Since nobody has ever seen or detected this artificial space-time, they have to be hidden from the observables in a theory, and thus the requirement of reparametrization invariance. It turns out that such theory can be formulated only with the one-dimensional string within the framework of artificial space-time.

Figure 01a String, World Sheet [view large image]

Construction of the superstring theory starts with the "Action Principle", which confines the movement of the string such that the world sheet sweeps out a minimum area (similar to the shorest distance for the case of point particle). If we demand the action S to be Lorentz covariant in form with conformal and reparametrization invariant, then it can be written as (with

= 1, c = 1):

The variations associated with the Poincare (generalized Lorentz transformation), reparametrization (change of coordinates), and conformal invariances (change of artificial space-time) are:

where

= (0, 1) = (

), are coordinates on the world-sheet. The requirement of the invariances largely determines the form of the action. Reparametrization invariance is to ensure that physics doesn't change with relabelling of these coordinates.

The Lorentz transformation in special relativity has been generalized to D-dimensions. The requirement of Lorentz covariant is to make sure that the superstring theory behaves the same in all inertial frames. Just as in gauge field theory or general relativity, there are fewer independent dynamical degrees of freedom than appear explicitly in the action. The degrees of freedom in the formulation can be reduced via the requirement of conformal invariance under the conformal transformation (Figure 01b), which rescale the world sheet metric. Such change in topology makes it feasible to evaluate string diagrams. Among other things this makes it possible to compactify the world sheet, closing off the holes corresponding to incoming and outgoing strings. For example, a world sheet with one incoming and one outgoing string (as in (a) of Figure 01b) can be conformally mapped to the plane of (a') with the incoming string appearing at the origin and the outgoing string at infinity (not shown) or to the sphere of (a") with the incoming and outgoing strings appearing at the south and north poles^¶. The external string states in (b) of Figure 01b with four awkward legs are projected to points as indicated in (b'). By a suitable choice of gauge (known as covariant or conformal gauge with in the following case - a gauge is a mathematical device to fix the redundant degrees of freedom), the string equations of

Figure 01b Conformal Trans- formations [view large image]

motion can be derived by varying X to minimize the action S, thus we obtain:

---------- (2)

which is an one-dimensional wave equation for X. The number of such equation is equal to "D" - the dimension of the coordinate system. The degree of freedom X is independent of each others.

Then by varying the world sheet metric to minimize the action, more equations can be derived in the form:

---------- (3)

where the indices 0 and 1 are used to refer to

and

respectively.

may be interpreted as the energy-momentum tensor for a two-dimensional field theory of "D" free scalar field X.

For a closed string, the general solution of Eq.(2) consistent with the boundary conditions X(

) = X(

) is:

---------- (4)

where l = (

T)^-1/2 (sometimes it is taken to be 1), and n can be a positive or negative integer except zero. The solution has been separated into the sum of a "right-mover" part (the first term) and a "left-mover" part (the second term) as they move in opposite directions. Each part consists of two terms corresponding to the motion of the center-of-mass, and a sum of oscillators with associated coefficients designated as

_n and

_n.

The solution for an open string with the boundary condition

is:

---------- (5)

In the open string case, the left- and right-mover oscillator terms are not independent, having been linked by the boundary condition, and a separation into left and right movers is not particularly useful.

The two examples in the followings serve to visualize the geometry of the closed and open strings. Let us first consider a closed string momentarily at rest in the (X¹, X²) plane. According to Eq.(4), the time component and n=1 mode can be expressed as:

		X⁰ = (l²p⁰) X¹ = R cos(2) X² = R sin(2) where p⁰ = iE, and R ~ Rcos(2) near ~ 0 is the radius of the cylinder. The energy-momentum tensor defined in Eq.(3) yields the relation E = 2RT confirming that T is indeed the energy per unit length of the string. Figure 02a shows the world sheet swept out by this closed string for a small interval of in the form of cylindrical surface. In general, the
Figure 02a Closed String [view large image]	Figure 02b Vibrating String [view large image]	closed string will vibrate in the three dimensional space (X¹, X², X³) in various harmonic modes (see Figure 02b).

The solution for open string is given by Eq.(5). The rotating open string in the (X¹, X²) plane is expressed by:

	X⁰ = (l²p⁰) X¹ = (l) cos()cos() X² = (l) cos()sin() The energy-momentum tensor defined in Eq.(3) yields the relation E² = (/l)² for this case. As the total angular momentum is given by the formula: , it follows that J = ²/2 for this case of spinning string. Thus J / E² = 1/2T, which can be
Figure 02c Open String [view large image]	identified as the Regge slope in Figure 03b. Furthermore, since the velocity of the string V can be computed from the formula: V = {\|X¹/X⁰\|² + \|X²/X⁰\|²}^1/2 = \|cos()\| ,

it means that the endpoints (

= 0, and

) move at the speed of light (recalling that c = 1 in the unit using here). Figure 02c shows the rotating open string, and the world sheet it sweeps out in the form of helical surface.

This is the original string theory developed in the late 1960s. It is called the bosonic string theory. The name bosonic indicates that all of the vibrational patterns of the bosonic string have spins that are a whole number - there are no fermionic patterns. This led to two problems. Firstly, the theory is incomplete because it misses the world of fermions. Secondly, it was realized that there was one un-physical pattern of vibration in bosonic string theory whose mass-squared was negative - a so-called tachyon (elementary particle that moves faster than light - on the other side of the speed of light barrier). As it will be discussed later, both problems are resolved by introducing super-symmetry into the string theory.

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Quantization

The Gupta-Bleuler quantization of the string requires the identification of the canonical momentum conjugate to X, i.e., P, which is defined as:

(which is defined as the integrand in Eq.(1)).

The spectrum of states is calculated from the Hamiltonian:

---------- (16)

For a closed string it takes the form:

---------- (17)

For an open string:

---------- (18)

This Hamiltonian is the simplest possible one for an extended object. While the last term is the energy for the center-of-mass, the sum is over an infinite set of independent harmonic oscillators, which is basically uncoupled from each other. This form admits simple products of the Fock spaces of all possible harmonic oscillators as its eigenstates:

---------- (19)

with the vacuum state defined by the annihilation operators: ---------- (20)
Each term in the sum in Eqs.(17) and (18) is similar to the number operator in quantum field theory. Eq.(15) shows that the norm for the time-like component is negative, i.e.,

---------- (21)

which is called ghost. This kind of negative probability is unacceptable in physical theory. Elimination of ghost states is possible by choosing the "light cone gauge", which defines two new coordinates according to the transformations:

X⁺ = ( X⁰ + X^D-1), X^- = ( X⁰ - X^D-1) ---------- (22a)

In this form, X⁺ can be chosen to describe the motion of the center-of-mass only; while X^- can be expressed in terms of the transverse degrees of freedom with

= 1, ..., D-2. There is no more time component and thus no more ghost states. In the light cone gauge the energy-momentum tensor can be defined as T₊₊ = ( T₀₀ + T₀₁) / 2, and T_-- = ( T₀₀ - T₀₁) / 2. The commutative relation for T₊₊ has the form:

---------- (22b)

In order to define the physical state properly, the second term on the righ-hand side of Eq.(22b) arised from an anomaly (an anomaly is the failure of a classical symmetry to survive the process of quantization and regularization) has to be removed. Thus, the dimension of the coordinate system has to be twenty six, i.e., D = 26. The constraint equations in Eq.(3) can now be written as: T₊₊ = T_-- = 0. Classically, these constraints can be implemented without much difficulty. However, quantum mechanically it is not possible to impose the conditions without inconsistency. It may be formulated in terms of the Fourier components of T₊₊ and T_-- as the conditions for the physical state |

>. Since the mass square M² = pp, the constraint in Eq.(3) yields the following formula for the closed string after integrating over

from 0 to

and evaluating at

= 0:

---------- (22c)

where the index i = 1, ..., D-2, and a is a formally infinite constant arising from the commutators. It is now regularized to be 1. For the open string:

---------- (22d)

		It was found in the 1960s that the spin (or angular momentum) of a family of resonances (short-lived elementary particles) is related to the square of mass by a simple line on a graph, which is known as Regge trajectory. Figures 03a and 03b show the theoretical derivation of such relationship for a few low-lying string
Figure 03a Regge Trajectories, closed string [view large image]	Figure 03b Regge Trajectories, open string [view large image]	states of the closed and open strings respectively.

Considering the

_n⁺'s, or a_n⁺'s (

= n^1/2a) as components of vector, then following the rule of tensor calculus, the "outer product" such as a⁺a⁺ is a tensor of rank 2. Conversely, the "inner product" (contraction) of two vectors such as ka⁺ lowers the rank 1 vectors to a scalar with rank 0. All the bosonic spin states can be constructed from this scheme in accordance with Eq.(19). The value of M² corresponding to the various string states are computed from either Eq.(22c) or (22d) with the pairs of

's acting as number operator. Table 01 shows many of the low-lying string states as depicted in
Figures 03a and 03b.

String Type	Spin	M²	Name
Open	0	-2	vacuum with tachyon
Open	0	0	massless scalar
Open	0	2	massive scalar
Open	1	0	massless vector (photon)
Open	1	2	massive vector
Open	2	2	massive spine-2
Closed	0	-8	vacuum with tachyon
Closed	0	0	massless scalar
Closed	2	0	massless spin-2 (graviton)

Table 01 Low-lying String States

Figure 03a displays a massless spin-2 particle in the spectrum of the closed string. This graviton-like particle was a great embarrassement when the string was first being developed as a model of hadrons. Now it is interpreted as a natural incorporation of gravity with the quantum theory of string. When the mathematics of this harmonic mode is worked out, it is found that the equations governing the large-scale behavior of a collection of gravitons are exactly those of general relativity. Thus, the superstring theory can be considered to give a fundamental generalization of general relativity. But the concepts behind this generalization remain largely mysterious.

Eqs. (19) and (22c,d) in the quantized theory of string show that each vibrational mode corresponds to a particle with distinct spin and mass. Unfortunately, as shown in Table 01 only the massless states have counterparts of particles in the real world (observed in the case of photon or not-yet-observed in the case of graviton), other states are either unreal (such as the tachyon) or too heavy in the range of 10¹⁹ Gev and beyond.

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Supersymmetry and Superstring

The superstring theory is based on the introduction of a world sheet super-symmetry that relates the space-time coordinates
X(

) to a fermionic partners

(

), which are two-component world sheet spinors. The action S consists of three parts:

S = S₀ + S₁ + S₂ ---------- (23)
In this equation:

The first term in Eq.(24) is just the action for the bosonic string in Eq.(1), the second term is new for the fermionic partner. S₀ possesses world sheet reparametrization invariance and global world sheet supersymmetry under the supersymmetry transformation. However, if the transformation is local (i.e., it is a function of

and

), it introduces an extra term, which is cancelled by S₁ -

where the two-dimensional supergravity "gravitino" is related to the local supersymmetry transformation by the formula -

This cancellation scheme in turn gives another extra term, which is finally cancelled by S₂ -

The action S in the form of Eq.(23) is the locally world sheet supersymmetric action for the superstring. In covariant gauge, the equations of motion are:

where J is the supercurrent. Eqs.(29) and (30) amount to the constraint equations, which reduce the degrees of freedom in the theory. The solutions for the bosonic string are similar to those Eqs.(4) and (5). The fermionic degrees of freedom can also be separated into right and left-movers, such as:

where

is the Dirac spinor with 2^D/2 components (in a single-column matrix). It can be decomposed into two Weyl spinors

_R and

_L. If the space-time dimension is taken to be D = 10, then there are 32 components for the Dirac spinor, and 16 each for the Weyl spinors. In the Type IIA and IIB theory, these Weyl spinors have opposite and same chiralities respectively. The Heterotic theories have only one copy of the Weyl spinors. The Type I theory is derived from the Type IIB theory by taking one copy from its even parity sector (and adding open strings and Chan-Paton factors to cancel anomalies).

The surface terms arising from the variation of the action vanish with the boundary conditions:

for closed strings;

for open strings.

The periodic boundary conditions are referred to as Ramond boundary conditions denoted by R, and the anti-periodic boundary conditions - Neveu-Schwarz boundary conditions, denoted by NS. For the right movers, the mode expansion is:

A similar mode expansion can be written down for the left movers with oscillators

and

. For the open superstring, the mode expansions of the right and left movers are not independent of each other, i.e.,

, and the "2n or 2r" in Eqs.(31) and (32) is replaced by "n or r" in the exponent. There is also an additional factor of 1/2^1/2.

Quantization of the fermionic degrees of freedom is achieved by imposing the canonical anti-commutation relations (with the momenta (i/2

)

_{R or L}):

In terms of the mode expansion oscillators, the anti-commutators for Ramond boundary conditions are:

with left- and right-mover oscillators anti-commuting. (Notice that the R or NS boundary conditions may be chosen independently for the right and left movers.)

Additional ghosts from the time-like fermionic oscillator appear in superstring theory. All ghosts can be removed via the light cone gauge in the form of Eq.(22a) and

⁺ = (

⁰ +

^D-1)/(2)^1/2,

^- = (

⁰ -

^D-1)/(2)^1/2 ---------- (37)

It can be shown that a local world sheet supersymmetry transformation may be used to choose

⁺ = 0; while

^- can be expressed in terms of the transverse degrees of freedom with

= 1, ..., D-2. Once again, there is no more time component and thus no more ghost states. However, in this case the removal of anomaly demands the dimension of the coordinate system to be ten, i.e., D = 10 (instead of D = 26 for the bosonic string). Tachyon also occurs in the super-string theory. The GSO projection is used to impose extra restriction for the removal of some unwanted states including the tachyon. Following the same procedure as in the case of bosonic string, the mass sqaure for the superstring is:

In the NS sector the correcponding result is:

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Compactification

Since it is via compactification, which yields local non-Abelian gauge symmetries, and other symmetry groups (to describe all the known particles and forces, plus some unknow ones), the mathematical treatment will be presented in the followings for a better understanding of the subject. Compactification of one dimension will be described in more details. The result is then generalized to compactify the extra 16 dimensions and finally to curl up six of the remaining ten dimensions.

		Considering the simple example of a bosonic string with one dimension (for both right and left movers) compactified on a circle, say X²⁵ in a form similar to Eq.(4) with = 25. Therefore, the value of a point in X²⁵ must satisfy: x²⁵ = x²⁵ + 2nR ---------- (40)
Figure 04a One Dimension Compactification [view large image]	Figure 04b Modes of Motion for Strings [view large image]	where R is the radius of the circle, and n is any integer known as winding number for the string configuration (Figure 04a). The momentum p²⁵ is then constrained by the requirement that exp(ip²⁵x²⁵) should be single valued (see also 5-D Space-time).

Consequently,

p²⁵ = m / R ---------- (41)

where m is any integer. Thus, while the rest of the degrees of freedom take on the form of Eq.(4) with

= 0, 1, ..., 24; the 25th one can be expressed as:

Figure 04b shows serveral kinds of motion within the compactified space - a cylinder in this case. Diagram (a) illustrates some point particles moving on this cylinder. A loop of string can undergo similar motion, with one difference being that it oscillates as it moves around on the surface as shown in Diagram (b). The oscillations of the string imbue it with characteristics such as mass and charge. The string can also wrap around the cylinder as shown in Diagram (c). The string will continue to slide around and oscillate, but it will do so in this extended configuration. In fact, the string can wrap around any number of times. This kind of motion is known as motion in winding mode. A wrapped string has a minimum mass, determined by the size of the circular dimension and the number of times it wraps around as shown in Eqs.(45) and (46). The string's oscillatory motion adds more mass in excess of this minimum. In the light cone gauge: X⁺ = ( X⁰ + X²⁴), X^- = ( X⁰ - X²⁴), the 25-dimensional mass-squared operator including all the contributions from different kind of motion has the expression:

where the number operator for the right- and left-movers is given by:

Two massless vectors with N = 1, = 1, and m = n = 0. They are constructed by taking one index to be associated with the 25-dimensional space-time and the other with the compactified dimension:

This is similar to the Kaluza-Klein compactification when a dimension was compactified on a circle with components of the 26-dimensional metric tensor. The one index associated with the compactified dimension becomes the components of a U(1) gauge field for 25 dimensions. There are two vectors in this case corresponding to the left and right mover respectively. They can be considered as the components of the U(1) X U(1) gauge group.

Another scheme to make massless states is to set N = 1, = 0, m = n = 1, and R = 1 / (2)^1/2. This is purely a stringy phenomenon depending on the existence of winding number. If the state with a particular pair of m and n is denoted by |m,n>, then four extra massless vectors can be constructed according to the different combination of m and n:

Thus, for the particular choice of the radius R of the compactified dimension, there are present in the theory the six massless vector fields required for the adjoint representation of the gauge group SU(2) X SU(2).

In the heterotic string, the left mover is a bosonic string in 26 dimensions; while the right mover is a superstring in 10 dimensions:

---------- (53a)

---------- (53b)

---------- (53c)

---------- (53d)

---------- (53e)

The extra 16 dimensions can be considered as 16 "internal" degrees of freedom. Gauge group can be constructed by compactifying these degrees of freedom on a 16-dimensional torus, which is defined by introducing a lattice with basis vector e_a^I, a = 1, ..., 16, chosen to have length 2^1/2, then the x and p terms in X_L^I can be written as:

where the R_a are radii, the n_a, and m_a are arbitrary integers.

In the light cone gauge: X⁺ = ( X⁰ + X⁹), X^- = ( X⁰ - X⁹), the ten-dimensional mass-squared operator for the physical states is given by:

M² = M_R² + M_L² ---------- (55)
with M_R² = M_L² ---------- (56)
For the superstring right movers,

For the bosonic string left movers with 16 dimensions compactified on a torus,

where a sum over i from 1 to 8, and over I from 1 to 16 is understood.

Similar to the case of one dimension compactification, the massless states (M_R² = M_L² = 0) with vanishing winding numbers generate 16 massless vectors in a U¹⁶(1) gauge group (because only the left mover is involved). The p_L^I in the compactified dimensions produce more massless states. However with the requirements of absence of anomalies and finiteness of loop contributions to the scattering matrix, the radii of the torus is restricted to R_a = 1/2^1/2 and only two forms of lattices in 16 dimensions are compatible. One of them leads to a SO(32) gauge group, and the other can be identified to the gauge fields of an E₈ X E₈ gauge group. The heterotic string contains no tachyons because the only negative mass-squared right-mover state has M_R² = -2 (see Eqs.(57) and (59)); while the only negative mass-squared left-mover state has M_L² = -4 (see Eqs.(60)). These negative mass states are invalid because they fail to satisfy Eq.(56). Thus, unlike the case of the superstring, the absence of tachyons in the Heterotic theory is not enforced by a GSO projection.

Any string theory that is to be a candidate theory of the world we live in will have to possess just four observable space-time dimensions, or, if there are extra spatial dimensions, they will have to be compactified on a sufficiently small scale as to be unobservable with the energies that are currently available to us. To complete the construction of a four-dimensional theory it is necessary next to compactify six of the remaining ten dimensions in some way for both right and left movers. The simplest possibility is to repeat the previous procedure employed for the compactification of the 16 extra dimensions in the left-mover. Unfortunately, it yields massless states not compatible with the world we live in because they are always non-chiral. However, a simple modification of the toroidal compactification to orbifold compactification can overcome this difficulty. An orbifold is a 6-dimensional space obtained by identifying points on the torus that are mapped into one another (by rotating 2

/3) referred to as the point group as shown in Figure 05. The diagram actually shows a two-dimensional orbifold surface. By combining

		three such orbifolds together, it is possible to generate a six-dimensional space with 3 X 3 X 3 = 27 singular points, which can be identified to the 27 fermionic fields in the E₆ group. In this way, one of the E₈ in the 16-dimansional compactification breaks correctly into SU(3) and E₆. The E₆ itself has yet to be broken into an even finer structure. It turns out that the orbifold predicts
Figure 05 Orbifold [view large image]	Figure 06 Calabi-Yau Manifolds [view large image]	36 generations of elementary particles. This is clearly far too many (for the observed 3 generations), but at least the theory is on the right track.

Orbifold compactification is the simplest and it preserves the equation of string in its simple form. However, the formulation is not on a truly manifold because it involves singular points. If one is prepared to pay the price of much more difficult equations of string, the Calabi-Yau manifolds (see Figure 06) may be constructed by cutting off the singular points, patching them up smoothly, and shrinking the patch-ups to zero. A typical Calabi-Yau shape contains holes as shown in Figure 07. There is

a family of lowest-energy string vibrations associated with each hole in the Calabi-Yau portion of space. Because the familiar elementary particles should correspond to the lowest-energy oscillatory patterns, the existence of multiple holes means that the patterns of string vibrations will fall into multiple families. If the curled-up Calabi-Yau has three holes, then we will find three families of elementary particles as observed. Unfortunately, the number of holes in each of the tens of thousands of known Calabi-Yau shapes spans a wide range from 3, 4, 5, 25, ... 480.

Figure 07 Calabi-Yau Holes [view large image]

The problem is that at present no one knows how to deduce from the equations of string which of the Calabi-Yau shapes constitutes the extra spatial dimensions. The properties of the force and matter particles can be extracted from the boundaries of the various multidimensional holes,

which intersect and overlap with one another. The idea is that as strings vibrate through the extra curled-up dimensions, the precise arrangement of the various holes and the way in which the Calabi-Yau shape folds around them has a direct impact on the possible resonant patterns of vibration. It seems then the string theory can provide us with a framework for answering questions - such as why the electron and other particles have the masses they do. Once again, though, carrying through with such calculations requires that we know which Calabi-Yau space to take for the extra dimensions. Since the Calabi-Yau shape can be deformed in many ways (see Figure 07), there are literally an infinite variety of them. They are now referred to as vacua, and each of them could be a different universe with a different set of physical parameters.

See more in "Calabi-Yau Manifold".

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Types of Superstring Theory

	Type I - Type I theory was formulated by Green and Schwarz in 1980. This type contains both open and closed strings. The two spinor fields _R, and _L have same chirality. Gauge invariance can be added into the theory by attaching charges at the end of open strings (the charges are distributed on closed strings). The gauge group must be SO(32) in order to cancel all anomalies. The
Figure 08 Types of Superstring Theory [view large image]	strong coupling limit of the Type I string theory is identical to the weak coupling limit of the Heterotic-O theory. Type I string theory contains D-branes with 1, 5, and 9 spatial dimensions.

Type IIA - For closed strings, there are two ways to choose the chiralities for _R, and _L. If we choose them to have opposite chirality, then we have the Type IIA string. This is the only theory that is non-chiral (and thus not corresponding correctly to the physics of the real world). In the zero-slope limit (of the Regge trajectory), when only the massless sector of the theory survives, the theory reduces to the point particle N = 2, D = 10 supergravity theory, where N is the number of supersymmetry generators creating 2^N helicity states (N = 8 seems to be the limit beyond which particles with spin greater than two has to be included and the theory becomes inconsistent). Type IIA string theory contains D-branes with 0, 2, 4, 6, and 8 spatial dimensions.

Type IIB - The Type IIB superstring has the same chirality for _R, and _L. In the zero-slope limit, there does not exist any known covariant version of this theory. It seems that the type II string (both A and B) cannot describe the physical SU(3) x SU(2) x U(1) symmetry of the low-energy universe. By compactifying from ten dimensions to four dimensions, the type II string can introduce a wide array of symmetries, but none of them seems to fit the description of this world. Type IIB string theory contains D-branes with -1, 1, 3, 5, 7, and 9 spatial dimensions.

SO(32) Heterotic - This is a theory of closed string. The fields that describe the physical degrees of freedom of the string in its ten-dimensional universe can be divided or decomposed into two independent parts - the left- and right-movers. The right mover is described by Eq.(53a) for the bosonic degrees of freedom, plus Eqs.(53b), (53c) for the fermionic degrees of freedom. It moves clockwise in a 10-dimensional space-time. The left mover is described by the bosonic degrees of freedom in Eqs.(53d), (53e). It moves counter-clockwise in a 26-dimensional space-time. So the heterotic string constructions are a hybrid - a heterosis - in which counter-clockwise vibrational patterns live in 26 dimensions and clockwise patterns live in 10 dimensions. Such combination is possible because the right- and left-movers are independent of each other. Since the extra 16 dimensions on the bosonic side are rigidly curled up (compactified), each of these movers behaves as though it really has 10 dimensions. The extra 16 left mover dimensions provide the gauge group for the resulting 10-dimensional theory. It is found that the possible gauge group consistent with gauge and gravitational anomaly cancellation are SO(32) or E₈ X E₈. It is the latter possibility that has led to phenomenologically promising models. The compactified dimensions carry all the quantum numbers around the loop. The Heterotic theories don't contain D-branes. They do however contain a fivebrane soliton which is not a D-brane. The IIA and IIB theories also contain this fivebrane soliton in addition to the D-branes.

E₈ X E₈ Heterotic - The E₈ group is a large Lie group of rank 8 and dimension 248, much like SU(2), but has no such geometric interpretation as the rotation of vectors. As mentioned earlier, compactification of the extra 16 dimension in term of the Calabi-Yau manifold produces a symmetry breaking to the E₆ group. It is proposed that it further breaks down into SU(3) X SU(2) X U(1). This is exactly the symmetry demanded by the grand unified theory. SU(3) is the symmetry of the quark theory, while SU(2) X U(1) is the symmetry of the electroweak interaction. The mechanism for this symmetry breaking remains to be discovered theoretically. The hope is that when a theory of the compactification process is finally developed, it will indicate the precise steps by which the original heterotic symmetry breaks down. It should also determine the exact symmetry patterns of the elementary particles and their individual masses. As for the other half of the E₈ X E₈ gauge group, some physicists hypothesize that this gauge group corresponds to two universes, each belonging to the smaller symmetry pattern E₈ by itself. Thus, in addition to our own universe, there is a new, hypothetical universe, a shadow world as it were. Other than the gravitational force, each E₈ group describes its own universe, its own pattern of particles and forces. The elementary particles in one group are effectively invisible, or hidden, when viewed from the other group. This hypothesis could provide an explanation for dark matter, which is unseen but is responsible for holding astronomical objects together by gravitational force.

Note that all five versions contain closed strings without which, i.e., without gravity the string theory becomes inconsistent. However, only the E8xE8 Heterotic string provides a faint resemblance to the real world. Anyway, the number of possible universes has now been expanded into 10⁵⁰⁰ (see Manifold, Vacuum Energy, and Multiverse). A new paradigm in the gist of "Anthropic Principle" has now been proposed to turn the absurdity of too many into the advantage of eliminating the fine tuning problem (of the physical parameters). Such peculiar state of affair can be compared to the infinite number of possible planetary orbits and the Newton's Law - there are many solutions but only one law. The differences are that it is impossible to observe the other universes, and the basic law for superstring has yet to be discovered. Nevertheless, the explanation via Anthropic Principle doesn't seem to be as far-fetched as expressed by so many physicists.

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M-theory

The problem with determining a correct Calabi-Yau space is related to the perturbative method used to perform the calculation. In this approximation scheme, each possible Calabi-Yau shape appears to be on equal footing with every other; none is fundamentally singled out by the equations. It is simply not possible to select one Calabi-Yau space from the many others. One of the new approaches to non-perturbative string theory involves M-theory and duality, which in fact force us to reconsider the central role played by strings in supersymmetry. In particular, duality allows us to show that the five previously mentioned superstring theories are nothing but different solutions of a single theory, called "M-theory". In this revised picture, the various string theories are different vacua of the M-theory. While perturbation theory only probes the vicinity of each vacua, duality allows us to make non-perturbative correlations across different vacua. Furthermore, M-theory indicates that the "true home" of the theory may actually be the eleventh dimension, where we find new, exotic objects, such as super membranes and 5-branes. The complete "action" of M-theory is unknown, but is believed to contain membranes (2-branes) and 5-branes. Closed strings in lower dimensions can be viewed as compactifications of these membranes.

T Duality - For the type II string with left- and right-movers in a ten-dimensional spacetime, the T-duality comes about from the curled-up dimension (for example, the 9th dimension). The momentum for the movers are given by the expression similar to Eqs.(45) and (46). It can be shown that with the interchange of (R , 1/2R) and (n , m) the chirality for the left-mover of the 9th dimension spinor flips under this transformation. Since Type IIA string starts out with opposite chirality for the movers, now both movers would have the same chirality, i.e., it changes to a Type IIB string. In other word, by exchanging the radius R and 1/2R (and n, m), one kind of Type II string would change to the other. A more complicated argument shows that similar kind of interchange exists between the SO(32) and E₈ X E₈ superstrings. Since this duality does not involve the coupling constant, which has severe impact on the series expansion, it is applicable even in perturbation theory.

S Duality - S-duality can be examined most easily in Type II string theory, because this theory happens to be S-dual to itself. The low energy limit of Type IIB theory is a supergravity field theory, which features a massless scalar field called dilaton. It can be shown that the Type IIB theory is invariant under a global transformation by the group SL(2,R) with the dilaton field transforming as - . Since the gravitational coupling constant G = e, the Type IIB theory thus appears to be unchanged when the strong and weak couplings are interchanged. Again, a more complicated argument shows that similar kind of interchange exists between the SO(32) and Type I superstrings.

U Duality and 11 Dimensions - U-duality is essentially a combination of S-duality and T-duality, e.g., U-duality implies that a theory A which is compactified to small size with weak coupling constant is dual to a theory B which is compactified to large size with strong coupling constant. The T-duality and the S-duality groups together only form a subgroup of this larger U-duality group. For example, if we start in either the Heterotic-E or Type IIA regions (see Figure 08) and turn the value of the respective string coupling constants up, what appeard to be one-dimensional

strings stretch into two-dimensional membranes. In the IIA case the eleventh dimension is a tube, whereas in the HE case it is a cylinder (see Figure 09). Moreover, through a more or less intricate sequence of duality relations involving both the string coupling constants and the detailed form of the curled-up spatial dimensions, we can smoothly and continuously move from one string theory to any other. Thus, all the five string theories involve two-dimensional membranes, which become apparent in the strong coupling limit and show up in the 11th dimension.

Figure 09 U Duality
[view large image]

11D-Supergravity - Although the form of the new theory (M-theory) is not known, it should be approximated by the 11-dimensional supergravity for particle at low energies (low compared to the Planck energy). Supergravity attempts to merge general relativity with quantum field theory via the addition of super-symmetry. The 4-dimensional version ultimately met with failure. The formulation in ten or eleven dimensions turned out to be more promising. In fact, there are four different ten-dimensional supergravity theories that differ in details regarding the precise way in which supersymmetry is incorporated. Three of these become the low-energy point-particle approximations to the Type IIA, IIB, and Heterotic-E string. The fourth is the similar limit to the Type I, and Heterotic-O string. The eleven-dimensional supergravity seems to have been left out in the cold until the M-theory comes along.

p-Branes - M-theory reveals that there are higher dimensional objects in string theory with dimensions from zero (a point) to nine, called p-branes. In terms of branes, what we usually call a membrane would be a two-brane, a string is called a one-brane and a point is called a zero-brane. A p-brane is a spacetime object that is a solution to the Einstein equation in the low energy limit of superstring theory, with energy density of the non-gravitational fields confined to

some p-dimensional subspace of the nine space dimensions in the theory. For example, in a solution with electric charge, if the enrgy density in the electromagnetic field was distributed along a line in spacetime, this one-dimensional line would be considered a p-brane with p=1. Figure 10 shows our 3-brane world (blue line) embedded in a p-brane (green plane, p = d₁₁ + 3), along which the light described by open strings propagates, as well as some transverse dimensions (yellow space), where only gravity described by closed strings can propagate. In most respects p-branes appear to be on an equal

Figure 10 p-brane
[view large image]

footing with strings. It has been shown that a p-brane wrapped around a curled-up region of space acts like a particle; thus drastically increases the number of ways the new vacua can be constructed. A p-brane expanding infinitely far in some spatial directions can act

as black hole in the remaining dimensions, trapping objects that come too close. In this view, it is not necessary to compactify the extra dimensions anymore.

D-branes - A D-brane is a submanifold of space-time with the property that open strings can end or begin on it. Strings can have various kinds of boundary conditions. For example closed strings have periodic boundary conditions (the string comes back onto itself). Open strings can have two different kinds of boundary conditions called Neumann and Dirichlet boundary conditions. With Neumann boundary conditions the endpoint is free to move about but no momentum flows out. With Dirichlet boundary conditions the endpoint is fixed to move only on some manifold. This manifold is called a D-brane or Dp-brane ('p' is an integer which is the number of spatial dimensions of the manifold). For example we see open strings with one or both endpoints fixed on a 2-dimensional D-brane or D2-brane (see Figure 11a). The D9-brane is the upper limit in superstring theory. Notice that in this case the endpoints are fixed on a manifold that fills all of space so it is really free to move anywhere and this is just a Neumann boundary condition. The case p= -1 is when all the space and time coordinates are fixed, this is called an instanton or D-instanton. When p=0 all the spatial coordinates are fixed so the endpoint must live at a single point in space, therefore the D0-brane is also called a D-particle. Likewise the D1-brane is also called a D-string. D-branes are actually dynamical objects which have fluctuations and can move around. For example they interact with gravity. In Figure 11b we see one way in which

		a closed string (graviton) can interact with a D2-brane. Notice how the closed string becomes an open string with endpoints on the D-brane at the intermediate point in the interaction. Compacti-fication of the 11 dimension will generally produce
Figure 11a D-brane [view large image]	Figure 11b D-brane Inter- action [view large image]	even dimensional D-branes for the Type IIA string, and odd dimensional D-branes for the Type IIB string (see Figure 08). M-theory contains only 5-branes, membranes, and gravitons.

The interesting property of D-brane is that the segments of string on the brane behave just like elementary particles. The only thing missing on the D-brane is gravity. That's because the graviton is a closed string - a string with no ends would not be stuck to the brane at all. Instead, they can travel freely through all space. They can interact with other strings by moving in and out of the brane. According to string theorists we are most likely living in a D3-brane with six dimensions tightly rolled up. Such configuration would prevent gravity from spreading out too much.

It has been shown that D-branes and p-branes are actually the same thing. Branes are not merely places; they are also objects that possess finite tension and carry charges. Thus, they can be distorted and can interact with other charged objects and gravitational field. They can move, collide, annihilate, and even form systems of branes orbiting around one another. On the other hand, brane can provide an environment for the strings to play their roles. Since the gauge bosons are also open strings, they would communicate a force that would act on the other brane-bound open strings with charge (at the endpoint). The photon being one of the gauge bosons is also trapped within the D3 brane; thus guarantees that the principle of special relativity (about the constant speed of light in our three dimensional space) is not violated. In short, the D3 brane would contain all the particles and forces in the Standard Model. From the perspective of brane-bound particles, if it weren't for gravity or other bulk particles with which they might interact, the world might as well have only the dimensions of the branes.

Cosmic String -Recently in 2003, a peculiar object known as CSL-1 was found by an Italian-Russian group. It consists of two apparently identical elliptical galaxies roughly at a distance of 10 billion light years from Earth and a mere 2 arc-seconds apart. The most intriguing property of CSL-1 is that the object is clearly extended and the isophotes of the two sources show no distortion at all. Both images have a redshift of 0.46, and the two spectra are identical at a 99.96% confidence level (see Figure 11c). There is no intervening galaxy or cluster of galaxies to produce the images by gravitational lensing. Follow-up observation reveals 11 other double images in a field 16 arc-minutes square centred on CSL-1. Researchers are wary of rushing to conclusions. More observations is needed to confirm such an explanation.

		There are at least three theories for producing long thin and heavy cosmic string from the early universe. It is used to explain the identical images such as the CSL-1 (Figure 11d). 1. Force fields pointing to different directions were frozen into long string during the phase transitions as the universe cooled down rapidly after the initial inflation. It is similar to the cracking ice and defects formed in liquid helium and superconductor. 2. Superstring theorists have found that by wrapping the extra dimensions in a special way, the tiny superstring could be inflated into the cosmic string. 3. Another explanation involves a D-brane, which intersect with only one dimension of our universe. As a result, it looks like an
Figure 11c CSL-1 [view large image]	Figure 11d Cosmic String Models [view large image]	one dimensional object. The energy within distorts the space around and bends the light from more distance galaxies to produce the double images.

Scientists in 2006 offer another explanation for the paired images of CSL-1 via the reformulated theory of cosmic strings, which emerged from the cosmic inflation in a spaghetti-like tangle. Although thinner than subatomic particles,

cosmic strings are boundless in length, stretched by cosmic expansion across the universe. They are characterized by huge linear density of 1 million metric mega-tons per centimeter. If a cosmic string runs between the Milky Way and another galaxy, light from that galaxy would go around the string symmetrically, producing 2 identical images near each other in the sky as shown by CSL-1. Figure 11e depicts a simulation of the evolution of cosmic strings from the radiation-dominated universe (left) to the matter

Figure 11e Cosmic Strings
[view large image]

dominated era (right), which shows a much lower density of both long strings and loops, and fewer wiggles in the long strings.

It is shown in 2008 by the Hubble space telescope that the two images in CSL-1 were two different, deceptively similar galaxies. Thus like so many exotic objects concocted by theorists, cosmic string remains just a hypothetical entity.

[Top]

Problems and Future Development

Mass Scale - As mentioned in the very beginning, the tensions of the string is estimated to be an immense 10³⁹ tons, which corresponds to a mass of 10¹⁹ Gev. Thus, only the massless state particle in the string theory is consistent with the real world. In fact, the observed mass of all the elementary particles is negligible in comparison to the Planck mass of 10¹⁹ Gev. It is thought that at the very high temperatures during the creation of the universe, it was totally symmetric with all particles in the lowest string state having zero mass. It is though symmetry breaking that the particles acquire their masses. Working out the fine details of all this is a current problem in phenomenology, which builds models to fit the observational data. At the moment, such details and the precise values of the particle masses cannot be obtained from the superstring theory. What is needed is some new insight, some deeper principle that would explain the mechanisms of symmetry breaking and allows particle masses to be calculated accurately. Attempt has been made to

compute the low energy mass by cancellation with the negative energy from vacuum fluctuation. A noticeable success is the prefect cancellation for graviton. Cancellation to such a high level of precision is generally beyond theoretical capability at present. Figure 12 shows the large gap between the Planck mass and the mass of known particles. There is "nothing" in this enormous region labelled "energy desert". Note

Figure 12 Mass/Energy Scale
[view large image]

that the mass/energy is referred to binding energy for some composite systems such as molecules, atoms, and nuclei.

Fractional Charge - In GUT's, the quarks are required to have 1/3 or 2/3 of the electric charge of the electron by the symmetry of the theory. In strings, the same result is achieved by the way strings wrap around the extra dimensions. Other wrappings would produce particles with 1/5, 1/7, or 1/11 of the electron's charge. Searchs for fractionally charged particles have been unsuccessful so far.

Large Dimensions - Since the Planck length is the natural scale of strings, the most probable universe would be the one in which all of the dimensions are comparable to the Planck length. Why are three spatial dimensions so large in our universe? Why not four, five, six, ...? A novel explanation suggests that if a dimension has a small circumference, strings can actually wrap around the circle as shown in Figure 13. Like rubber bands around a rolled-up newspaper, these wrapped strings tend to keep the universe from expanding in that dimension. When strings collide, though, they can unwrap and the dimension expand rapidly, like the inflationary model. String theorists discovered that these collisions were likely to happen if (at most) three spatial dimensions were involved. The sudden expansion of the three spatial dimensions is interpreted as the moment of the Big Bang. There are other string-base scenarios on the origin of the universe. A better understanding of the structure of string theory must be developed before these issues can be resolved.

Figure 13 Large Dimensions
[view large image]

Calabi-Yau Manifolds - As mentioned earlier there is a lot of difficulties to figure out the Calabi-Yau shape that agrees with the observed physical properties. A sensible start is to focus only on those Calabi-Yau shapes that yield three families. This cuts down the list of viable choices considerably, although many still remain. There are a few entries in the Calabi-Yau catalog that are closely akin to the particles of the standard model. If many of the Calabi-Yau shapes were in rough agreement with experiment, the link between a specific choice and the physics we observe would be less compelling. On the other hand, if none of the Calabi-Yau shapes came even remotely close to yielding observed physical properties, it would seem that string theory could have no relevance for our universe. Then finding a small number of Calabi-Yau shapes appear to be an extremely encouraging outcome. Because of such impasse of selecting an unique CalabiYau manifold that is fine tuned to our world, some string theorists now turn to the last resort - the "Anthropic Principle". It suggests that we just happen to live in one of the 10⁵⁰⁰ manifolds, which permits life to evolve to its present form. They insist that this is a matter of probability; it is not a "cope out" (see Manifold, Vacuum Energy, and Multiverse).

Light Cone Gauge - Superstrings were created to be space-time covariant as manifested by the form of its "action". However, the actual calculations are always made using a particular frame of reference called the light cone gauge, which destroys the covariance. The light cone gauge could be thought of as a frame of reference that moves through space-time at the speed of light. It violates the full relativistic covariance that is inherent in superstring theory and decrees that the details of the theory should not really depend on the particular choice of gauge. By obscuring this underlying symmetry, physicists are really working with the shadow of a much deeper theory. It also turns out that using this single gauge is intimately connected to perturbation theory along with all its limitations. One of the important tasks is to rewrite string theory so that it is free from any particular choice of gauge or frame of reference. In this way, it may be possible to penetrate much deeper into the theory and resolve some of its present difficulties.

Non-perturbation Theory - Interactions in superstring theory are expressed in terms of the splitting and joining of strings (see Figure 14 and 15). The approach used to calculate all the quantities of interest in the theory remains old-fashioned perturbation theory, which assumes a reasonable starting point and then tries to home in on the correct result by adding an infinite series of corrections. In quantum electrodynamics, these perturbation corrections were represented by Feynman diagrams, and, on summing up infinite numbers of terms, results of surprising accuracy were achieved. On the other hand, when it came to gravity, the perturbation series failed totally to represent a curved space-time by adding a finite number of corrections to an initially flat space-time. Perturbation theory is often plagued with infinity, and infinite sum (when the coupling constant is not small). Superstrings are supposed to be a theory about gravity and matter. Yet physicists continue to treat them as if they exist in a flat background space-time, a picture they hope will be corrected by using a perturbation series. Clearly this whole approach is inadequate and obscures the power of the superstrings themselves. M-theory and duality were introduced to overcome the difficulty. Although this approach have yet to give us

		definitive information about four-dimensional vacua, they have already clarified much of the nonperturbative nature of string theory in 10, 8, and even 6 dimensions, giving us a complex web of dualities between different string compactifications.
Figure 14 String Inter- actions [view large image]	Figure 15 Sum of Interactions [view large image]

Superstring Field Theory - Only the first quantization has been applied to the superstring theory up to now. Classical strings are created according to an action principle that is made manifestly covariant in order to produce minimal surfaces in space-time. The various modes of vibration of these minimal surfaces are then quantized to produce string wave function. Following the development of quantum theory, the next step would be to generate, out of the wave functions for individual string vibrations, a super wave function for the string field. It can be shown that the equation of motion for the superstring wave function is:

where the field functional is defined as:

It has been discretize into a series of points along the string. At the limit N , it becomes a continuous function of X . Thus while in the first quantization the basic object is X_i, which represents just one possible configuration of the string; the string functional is simultaneously a fucntion of every point along the string. Now the Hamiltonian H takes the form:

where and P_i= are the canonical momenta conjugate to Xⁱ.
Although superstrings are formulated in a ten-dimensional space-time, in another sense a string field is also a theory about two-dimensional surfaces. A string field theory is therefore about the quantum properties of two-dimensional surfaces. Considerable research is now being directed toward this approach. It has been pointed out that the program can proceed in two directions. On one hand, physicists can study the topological properties of these surfaces and in this way produce powerful insights. But it is also possible to consider the geometry of these surfaces in term of algebra. It turns out that physicists had already for some time been looking at these branches of algebra (the sorts of algebras first discovered in the 19th century). Their idea was to discover ways of probing deeper into the quantum theory and freeing it from its attachment to an underlying space-time. String field theory is currently too difficult to solve with the non-perturbative approach.

Space-time Background Dependence - Superstrings have resolved many of the problems that faced earlier attempts to explain the elementary particles; they are free from infinities and associated with just the right symmetry groups. Yet all these theories are fundamentally flawed because they still regard strings as moving in a fixed, background space-time. Such an approach just has to be wrong; the superstrings in a proper string theory have to interact with the space in which they move and are inseparable from it. We need a theory to describe an universe, which evolves from a more primary state to the fabric of space, time, and by association, dimensions. The graviton in string theory does suggest an idea. It shows that a gravitational field is composed of an enormous number of gravitons all executing some vibrational pattern. Gravitational fields, in turn, are encoded in the warping of the spacetime fabric, and hence we are led to identify the fabric of spacetime itself with a colossal number of strings all undergoing the same, orderly, graviton pattern of vibration. Another approach is to replace ordinary geometry by something known as noncommutative geometry. In this geometrical framework, the conventional notions of space and of distance between points melt away, leaving us in a vastly different conceptual landscape. Nevertheless, as we focus our attention on scales larger than the Planck length, physicists have shown that our conventional notion of space does re-emerge. It seems that string theories work very well above the size of Planck length. The more fundamental theory is buried below this size.

Testing - The Superstring Theory is often criticized for the lack of testable predictions. By 2007 string theorists have come up with at least 4 ways to put their models to test (Figure 16, - evidence for theory, - against theory):

cosmic string

pulsar

3. It is suggested that string theory has already been tested by the RHIC, which produced liquid quark-gluon plasma as predicted by the string theory. According to the holographic conjecture, such plasma in three dimension should correspond exactly to a black hole in the higher dimensional space. It has been shown that the property of the liquid plasma is described by the same equations as a higher dimensional black hole.
4. Inflation generated ripples through space-time with imprint of these primordial gravitational waves in the CMBR (Cosmic Microwave Background Radiation). String theory sets a limit on the strength of such waves. If it is too energetic, the six curled-up dimensions posited by string theory would have unfurled and grown just

Figure 16 Tests for String Theory [view large image]

as large as the three we see around us. This is a test that can falsify the string theory if strong gravitational waves are detected in CMBR. On the other hand, the lack of CMBR gravitational wave can falsify the theory of inflation.

LHC

string theory

theory of supersymmetry

^§The official string theory website is at http://www.superstringtheory.com. It is at the level approachable by general audience.
A more rigorous treatment of the subject can be found in: http://arxiv.org/PS_cache/hep-th/pdf/0007/0007170.pdf

^¶This is an example for the mathematics of conformal transformation on the world sheet. As shown in Figure 01b diagram a, the original world sheet is a single incoming closed string and a single outgoing closed string. In cylindrical coordinates with
-

< z <

, and 0

< 2

the metric is in the form:
ds² = dz² + R²d

²,
where R is the radius of the cylinder. If the coordinate z is changed to:
z = 2R ln(tan

/2) with 0 <

,
then ds² = R²[(sin

)^-2 d

² + d

²].
The conformal invariance allows the rescaling of the metric by a factor:

= sin²

.
The metric is now in the form:
ds² = R²(d

² + sin²

²),
which is the metric for a 2-sphere of radius R. The initial and final string states at z = -

, +

correspond to

= 0,

, i.e. the north and south poles of the 2-sphere, as shown in Figure 01b diagram a".