Calabi-Yau Manifold

Theory of Superstring, and M Theory :

Extra Dimensions and Supersymmetry, Compactification, Types of Superstring Theory and M Theory,
Modulus, Flux, Brane, and Vacuum Energy

The followings provide a passing glance on the parts of Superstring theory leading to the adoption of the Calabi-Yau space.


• Extra Dimensions and Supersymmetry - In the process of developing the Superstring theory, a number of problems arose such that the theory makes sense only if there are extra-dimensions beyond the usual 3 spatial and 1 temporal dimensions or by including supersymmetry into the theory. The problems can be summarized as :
  
  
  1. Ghost - It usually means negative probability. This is associated with the failure to construct physical states from the time

  component in the mode expansion (of the position vector Xµ) and the subsequent quantization. Such problem always comes up with the bosonic string. The solution is to impose the condition that the dimension D = 26. Then 16 of these 26 extra-dimensions are compactified into small circles leaving 6 more dimensions to be compactified further from the remaining 10 dimensions. Figure 11 shows that an 11-D space can be drawn in a piece of paper or computer screen, if one axis represents many dimensions. The d11 dimension is the

  Figure 11 11-D Space

  additional dimension demanded by the M Theory. The blue line is the 3 brane in which all of us live in.

  2. Anomaly - An anomaly is the failure of a classical symmetry (such as Lorentz invariance) to survive the process of quantization or regularization (a mathematical device to bypass undesirable expression such as infinity). Conservation laws are violated as the consequence. For the bosonic string the solution is to impose the same restriction for the dimension D = 26, while in the superstring theory with both bosonic and fermionic degrees of freedom, the condition becomes D = 10 (after compactifying 16 bosonic dimensions in the Heterotic theories) .
  3. Inconsistency - This is the appearance of infinities in the theory. The same solution of taking D = 26 can be applied to the bosonic string, and D = 10 to the superstring.
  4. Tachyon - In the bosonic string theory, tachyon appears in its formulation. Tachyon is the kind of particle always moving faster than the speed of light. It has never been observed, and it creates instability in the theory - decaying to lower energy state by unknown mechanism. This troublesome feature is removed by incorporating supersymmetry into the theory.
  5. Supersymmetry - In addition, introduction of supersymmetry into the string theory fixes the problems of missing fermions, offers a way to produce realistic mass for the particles, stabilize the vacuum, drops the dimensional requirement from 26 to 10, and generally helps to reduce the labor of calculations.
  
  
• Compactification - Since all of us experience only 3 spatial and 1 temporal dimensions, the 10 and 26 extra-dimensions have to be hidden under some schemes. One of the two alternatives is to roll them up into very small size not observable even under a very powerful microscope. The other one is to consider our existence on a 3 brane floating in the bulk of ten spatial dimensions. The first alternative is called compactification. It is more complicated than merely shrinking the size (of the dimensions). Even in the very simple case of a (4+1) toy model, compactification to a small circle of radius R produces particle in the 3-D space with mass = n/R, where n is an integer. It manifests itself as a scalar particle (spin 0) obeying the Klein-Gordon equation. Compactification of the 16 extra-dimensions for the bosonic string, produces the gluon and electroweak gauge fields. Compactification of the remaining 6 extra-dimensions breaks the

Heterotic string symmetry down to the point where the hadrons and leptons of more conventional theories are recovered. Viewed from a distance, the symmetry-broken Heterotic strings look just like familiar point particles - but without the infinities and anomalies of the particle approach. In order to maintain conformal invariance (i.e., the world sheet should remain unchanged by relabeling), these 6 extra-dimensions have to curl up in a particular way - a more promising one is the Calabi-Yau manifold (see more in "Compactification") as shown in Figure 12, where each point stands for a 3-D space.

Figure 12 Calabi-Yau Space

• Types of Superstring Theory, and M Theory - Depending on the characteristics assembled in its framework, there are 5 different theories of superstring as shown in Figure 13. The most important characteristics include :
  
  
  1. String Type - Those with no end-points are closed strings (like a loop), while the open strings look like a 1-D curve. Strings with

  different vibrational modes correspond to different particles. Calculations show that the massless closed (bosonic) strings can be either scalar (spin 0) particles, or gravitons. The massless open (bosonic) strings can assume the role of scalar particles or vector particles (such as photons or gluons). Since the next level of massive strings have mass in the order of 1019 Gev, all particles in the Standard Model with mass < 10 Tev can be considered as massless. String theory with

  Figure 13 Superstring Theories [view large image]

  supersymmetry enables the introduction of ferminonic strings (with spin 1/2) to accompany the bosonic ones (in both the closed and open varieties). Not all superstring theories contain open

  strings, but every theory must contain closed strings, as interactions between open strings always end up in closed string, and closed string is required for the theory to include gravitation.
  2. Number of SUSY (supersymmetry) generators - Supersymmetry is characterized by pairing every particle (boson or fermion) with a partner differing in spin by half a unit, but with otherwise identical properties (Figure 14, the different mass size is produced by broken symmetry). The idea of supersymmetry can be expressed in simple mathematics :
     Q_i|F_i> = |B_i> and Q_i|B_i> = |F_i>
     where i indicates the component of the spinor acted upon (there are 2^D/2 components in dimension D), |F_i> and |B_i>

     are fermionic and bosonic states respectively. The operator Qi is called supersymmetry (SUSY) generator (also known as supercharge), which transforms these states into each other. The SUSY generators number N characterizes the effect of supersymmetry on the standard model or other theories such as the theory of string. The altered theory is then adjusted to remain invariant under the SUSY transformation - resulting in new fields and particles.

     Figure 14 Superpartners [view large image]

     For example, N = 1 generates the supermultiplets as shown in Table 02 below :

     Supermultiplet	Particle	h	Helicity	CPT-Conjugate Helicity	Degeneracy
     Chiral	Higgs, Squark, Slepton	1/2	0	0	1
     Chiral	Quark, Lepton, Higgsino	1/2	1/2	-1/2	1
     Vector	Gaugino	1	1/2	-1/2	1
     Vector	Gauge Boson	1	1	-1	1
     Gravitino		3/2	1	-1	1
     Gravitino		3/2	3/2	-3/2	1
     Gravity	Gravitino	2	3/2	-3/2	1
     Gravity	Graviton	2	2	-2	1

     Table 02 N = 1 Supermultiplet

     The h = 1/2 supermultiplet consists of two helicity states 0 and 1/2. The requirement of CPT invariance introduces a second h = 0 state as well as an h = -1/2 state. Together they form a complex scalar field for the Higgs and a left-handed (left-chiral) particle and its antiparticle. In the Vector supermultiplet, SUSY requires the generation of h = 1 state from the h = 1/2. CPT invariance further demands two more states with h = -1 and h = -1/2. The states h = 1 correspond to the two helicity states of massless vector boson - photon, gluon, and the weak interaction gauge bosons (considered to be massless here). There is no known particle corresponding to the Gravitino supermultiplet. The Gravity supermultiplet contains the all important graviton for including gravity into the theory. The supersymmetry generator number N typically occurs in power of 2, i.e., N = 2ⁿ, where n = 0, 1, 2, 3. Theory with N > 8 generates massless fields with spin greater than 2, which may not be associated with consistent quantum field theory. Table 03 for N = 2 supermultiplet below illustrates further the pattern formed by different value of N.
     
     

     Supermultiplet	h	Helicity	CPT-Conjugate Helicity	Degeneracy
     Hyper	1/2	-1/2		1
     Hyper	1/2	0		2
     Hyper	1/2	1/2		1
     Vector	1	0	0	1
     Vector	1	1/2	-1/2	2
     Vector	1	1	-1	1
     Supergravity	2	1	-1	1
     Supergravity	2	3/2	-3/2	2
     Supergravity	2	2	-2	1

     Table 03 N = 2 Supermultiplet

     The Type I, HO, and HE theories has N = 1, therefore their chiral property is guaranteed by the Chiral supermultiplet. On the other hand the Type IIA theory is non-chiral because it selects opposite helicity (-1/2 and +1/2) for the left and right movers (in the closed string). Although the Type IIB theory is chiral by selecting same helicity for the 2 movers, these two kinds of Type II theories ultimately become non-chiral as both of the Vector supermultiplets transformed the same way (as -1/2 and +1/2) under a gauge symmetry. So for N 2, the 2 chiral states (-1/2 and +1/2) must be treated on an equal footing by any gauge force. But this conflicts with parity violation in the weak interaction, which admits only left handed particle or right handed antiparticle; hence only N = 1 SUSY is relevant to the real world. The Gravitino and Supergravity supermultiplets are local supersymmetry (meaning the supersymmetry generator acts differently depending on location, i.e., it turns into a gauge field). It it an extension of the gravity in general relativity to the quantum scale.
     
     
  3. Gauge Groups - In theoretical physics invariance under a certain transformation signifies some kind of special property. For example, if the Newtonian equation of motion is invariant under the translational transformation x'=x+c, then it implies the

  		conservation of momentum (of the particle). The gauge transformation is more abstract, it is the rotation in an "internal space". The invariance of the Dirac equation under this transformation implies the existence of force (called gauge force) associated with the spin 1/2 particle described by this equation (Figure 15). In the Standard Model, there are three kinds of gauge forces associated with three different kinds of "internal rotation" represented by mathematical object called "group". The U(1) group has only one phase angle corresponding to one boson - the photon for the electromagnetic interaction. The SU(2) group has three phase angles
  Figure 15 Gauge Invariance [view large image]	Figure 16 Gauge Bosons	corresponding to three bosons - the W, and Z for the weak interaction. The SU(3) group has eight phase angles corresponding to eight bosons - the eight gluons for the strong interaction (Figure 16).

  According to superstring theory, the gauge fields exist in all ten dimensions. The gauge fields in the 4-D large dimensions are generated by the internal structure of the 6-D compactified dimensions. In addition, the gauge group helps to cancel out anomalies in the theory. The Type IIA, and B theories possess the U(1) gauge symmetry (the theory is invariant under the U(1)

  	transformation), while the Type I and HO theories has the SO(32), and the E8 in the HE theory is a 248-dimensional (phase angles) symmetry group. It is suggested that these huge number of components can be reduced to the required 12 in the SU(3)xSU(2)xU(1) symmetry groups. The process is called
  Figure 17 Broken Symmetry	symmetry breaking, which is much like a sphere restricted to rotate about the North-South magnetic axis instead of spinning in any of the axes going through the center (Figure 17) - the infinite number of

  symmetrical operations are reduced to one by picking a particular axis.
  
  
  4. D-Branes - In Figure 13 if we start in either the Heterotic-E or Type IIA regions and turn the value of the respective string coupling constants up, what appeared to be one-dimensional strings stretch into two-dimensional membranes. In the Type IIA case the eleventh dimension is a tube, whereas in the HE case it is a cylinder (Figure 18). Moreover, it can be shown that through a series

     		of manoeuvres we can smoothly and continuously move from one string theory to any other. Thus, all the 5 string theories involve 2-D membranes, which become apparent in the strong coupling limit and show up in the 11th dimension. Thus the five superstring theories are nothing but different solutions of a single theory, called "M-theory". In this revised picture, the various string theories merely provide different windows to this M-theory. It is suggested that the "true home" of the
     Figure 18 11th Dimension [view large image]	Figure 19 World Path Dimensions	theory may actually be in the 11th dimension, where we find new, exotic objects, such as the branes.

     
     
     Branes have at least 3 key characteristics:
     • Dimensions - The p-branes were discovered in mid-1990s as solutions to the field equations in General Relativity. The p represents the number of spatial dimensions, which can go from 0 to 9. A p-brane can make a massless particle by wrapping around the curled-up dimension. Thus it helps to supplement the kind of particles not produced by the theory of string. The D-branes were invented in 1995 to anchor the end-points of open strings. Later on it was shown that these 2 different kinds of branes are the same thing. The brane with the special case of p = -1 denotes all the space and time coordinates are fixed, this is called an instanton or D-instanton. For p=0 all the spatial coordinates are fixed so the endpoint must stay at a single point in space, therefore the D0-brane is also called a D-particle. Likewise the D1-brane with p = 1 is called a D-string, D2-brane with p = 2 is called a D-membrane, ..., Dp-brane up to p = 9. The M Theory contains only gravitons, 2-D membranes, and 5-branes. Branes are actually dynamical objects which have fluctuations and can move around. It is the generalization of string, and may be more fundamental than string. Figure 19 shows the path traces out by the p = 0, 1, and 2 branes in (2 spatial + 1 temporal) space, and the applicable theory for each case.
     • Electrical Charge - The electrical charge (a point charge) is associated with the massless gauge filed in QED or the Standard Model. In the one dimensional open string the charge is carried at its ends if the theory includes a proper gauge field. For Dp-brane with p > 1 (with proper gauge field), the electrical charge is distributed in the electric field lines. There is no charge on the closed string (graviton doesn't carry charge). Compactification of the 11 dimension will generally produce even dimensional D-branes for the Type IIA string, and odd dimensional D-branes for the Type IIB string. Conservation of charge and energy insures that D-branes with charge (either electric or magnetic) would not decay into open or closed string states with lighter mass.
     • Tension - Tension in the brane determines its stiffness. If a D-brane has a tension of zero, then a minor disturbance would have a major impact. An infinite tension would mean the exact opposite - the brane could not be changed by any interaction. The size of the string is about 10^-33cm, Which implies an enormous tension and thus energy or mass of 10¹⁹Gev
     - about 10¹⁹ times the mass of proton. It has been suggested that such huge mass could be canceled by quantum fluctuation of the vacuum. The cancellation turns out to be exact for the graviton, but beyond the capacity of theoretical calculation for the other particles at present. Within the five superstring theories, the branes (with p > 1) have even higher tension than the string, they have only a small effect on much of physics. However, they become lighter and hence increasingly important in the M theory. It seems that the point particle description and the five superstring theories are only an approximation to the more fundamental M theory, within which the branes possess all the physical properties such as mass, charge, and spin. The problem is to deduce a proper value for this entities in agreement with observations in the real world.
  
  
• Modulus, Flux, Brane, and Vacuum Energy -
  1. Modulus - Modulus is the parameter that determines the appearance of a geometric object or "landscape". For a cylinder it needs

  		only one modulus, which is the radius. A one hole torus requires three moduli to specify the size, shape, and twist as shown in Figure 20. But a typical Calabi-Yau has hundreds, and it becomes more troublesome as the moduli can vary from one point to another in the compactified space as if a force (a massless scalar field) is acting on it. Since the mass and charge of particle is determined by the moduli, thus these fundamental constants are not constant
  Figure 20 Moduli of a Torus [view large image]	Figure 21 Vacuum Energy	anymore contrary to observation in the real world. Fortunately, M theory provides the flux and brane to resolve the problem.

  2. Flux - The fluxes in M Theory is similar to the electric fluxes but have nothing to do with electrons or photons. The presence of fluxes has the effect of holding the manifold's shape in place. The electric fluxes from an enclosed surface is equal to the number of charges within, similarly the fluxes in M Theory also comes with whole numbers of a certain unit (through each hole in the manifold). It drastically increases the complexity of the landscape. Especially when they act on the pointy end of the compactified manifold stretching it into a long, narrow neck. The result is to produce lot of valleys on the landscape with negative vacuum energy (cosmological constant), which is contrary to observation in the real world. Now the brane comes to the rescue.
  3. Brane - Similar to the antiparticle in the point approximation, every brane also has its antibrane. Anitbrane has a tendency of attracting to the pointy end and add energy into the valley to make the vacuum energy positive. Thus, by a mix of a little of everything, a point on the landscape turns out to have a small positive cosmological constant - just like the observation in the real world. It is also found that D-brane can stabilize the size as well as the shape of the compactified manifold (like the steel-belt in radial tire) at least in the Type IIB theory. This function is crucial in the superstring theory, otherwise the 6 hidden dimensions would become unwinded and getting infinitely large. Then we would be living in ten dimensional space instead of the usual three.
  4. Vacuum Energy - Since the Calabi-Yau space has at least 500 donut holes through which the fluxes wind about, suppose that the flux integers are constrained to 10, the number of different configuration would be 10⁵⁰⁰. Moreover, each valley in this gargantuan list has some vacuum energy, and no two of them are likely to have exactly the same value. Figure 21 produces only a few for illustration purpose. It is argued that there must be a huge number for the "window of life" within such gargantuan number. There is no need to wonder why the cosmological constant is so fine-tuned.

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